text stringlengths 4 2.78M |
|---|
---
abstract: 'This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.'
author:
- 'Beatrice Acciaio[^1]'
- 'Irina Penner[^2]'
date: 'first version: November 13, 2009; this version: '
title: Dynamic risk measures
---
Introduction
============
Risk measures are quantitative tools developed to determine mimimum capital reserves, which are required to be maintained by financial institutions in order to ensure their financial stability. An axiomatic analysis of risk assessment in terms of capital requirements was initiated by Artzner, Delbaen, Eber, and Heath [@adeh97; @adeh99], who introduced coherent risk measures. Föllmer and Schied [@fs2] and Frittelli and Rosazza Gianin [@fr2] replaced positive homogeneity by convexity in the set of axioms and established the more general concept of a convex risk measure. Since then, convex and coherent risk measures and their applications have attracted a growing interest both in mathematical finance research and among practitioners.
One of the most appealing properties of a convex risk measure is its robustness against model uncertainty. Under some regularity condition, it can be represented as a suitably modified worst expected loss over a whole class of probabilistic models. This was initially observed in [@adeh99; @fs2; @fr2] in the static setting, where financial positions are described by random variables on some probability space and a risk measure is a real-valued functional. For a comprehensive presentation of the theory of static coherent and convex risk measures we refer to Delbaen [@d0] and Föllmer and Schied [@fs4 Chapter 4].
A natural extension of a static risk measure is given by a conditional risk measure, which takes into account the information available at the time of risk assessment. As its static counterpart, a conditional convex risk measure can be represented as the worst conditional expected loss over a class of suitably penalized probability measures; see [@rse5; @rie4; @dt5; @bn4; @ks5; @cdk6]. In the dynamical setting described by some filtered probability space, risk assessment is updated over the time in accordance with the new information. This leads to the notion of dynamic risk measure, which is a sequence of conditional risk measures adapted to the underlying filtration.
A crucial question in the dynamical framework is how risk evaluations at different times are interrelated. Several notions of time consistency were introduced and studied in the literature. One of todays most used notions is strong time consistency, which corresponds to the dynamic programming principle; see [@adehk7; @d6; @dt5; @ks5; @cdk6; @bn6; @fp6; @ck6; @dpr10] and references therein. As shown in [@d6; @bn6; @fp6], strong time consistency can be characterized by additivity of the acceptance sets and penalty functions, and also by a supermartingale property of the risk process and the penalty function process. Similar characterizations of the weaker notions of time consistency, so called rejection and acceptance consistency, were given in [@Samuel; @ipen7]. Rejection consistency, also called prudence in [@ipen7], seems to be a particularly suitable property from the point of view of a regulator, since it ensures that one always stays on the safe side when updating risk assessment. The weakest notions of time consistency considered in the literature are weak acceptance and weak rejection consistency, which require that if some position is accepted (or rejected) for any scenario tomorrow, it should be already accepted (or rejected) today; see [@adehk7; @Weber; @tu8; @burg; @ros7].
As pointed out in [@jr8; @er08], risk assessment in the multi-period setting should also account for uncertainty about the time value of money. This requires to consider entire cash flow processes rather than total amounts at terminal dates as risky objects, and it leads to a further extention of the notion of risk measure. Risk measures for processes were studied in [@adehk7; @rie4; @cdk4; @cdk5; @cdk6; @ck6; @fs6; @jr8; @afp9]. The new feature in this framework is that not only the amounts but also the timing of payments matters; cf. [@cdk6; @ck6; @jr8; @afp9]. However, as shown in [@adehk7] in the static and in [@afp9] in the dynamical setting, risk measures for processes can be identified with risk measures for random variables on an appropriate product space. This allows a natural translation of results obtained in the framework of risk measures for random variables to the framework of processes; see [@afp9].
The aim of this paper it to give an overview of the current theory of dynamic convex risk measures for random variables in discrete time setting; the corresponding results for risk measures for processes are given in [@afp9]. The paper is organized as follows. Section \[setup\] recalls the definition of a conditional convex risk measure and its interpretation as the minimal capital requirement from [@dt5]. Section \[sectionrr\] summarizes robust representation results from [@dt5; @fp6; @bn8]. In Section \[sec:tc\] we first give an overview of different time consistency properties based on [@tu8]. Then we focus on the strong notion of time consistency, in Subsection \[subsec:tc\], and we characterize it by supermartingale properties of risk processes and penalty functions. The results of this subsection are mainly based on [@fp6], with the difference that here we give characterizations of time consistency also in terms of absolutely continuous probability measures, similar to [@bn8]. In addition, we relate the martingale property of a risk process with the worst case measure, and we provide the explicit form of the Doob- and the Riesz-decomposition of the penalty function process. Subsection \[subsec:rc\] generalizes [@ipen7 Sections 2.4, 2.5] and characterizes rejection and acceptance consistency in terms of acceptance sets, penalty functions, and, in case of rejection consistency, by a supermartingale property of risk processes and one-step penalty functions. Subsection \[subsec:wc\] recalls characterizations of weak time consistency from [@tu8; @burg], and Subsection \[recur\] characterizes the recursive construction of time consistent risk measures suggested in [@cdk6; @ck6]. Finally, the dynamic entropic risk measure with a non-constant risk aversion parameter is studied in Section \[entropic\].
Setup and notation {#setup}
==================
Let $T\in{\mathbb N}\cup\{\infty\}$ be the time horizon, ${\mathbb{T}}:=\{0,\ldots,T\}$ for $T<\infty$, and ${\mathbb{T}}:={\mathbb N}_0$ for $T=\infty$. We consider a discrete-time setting given by a filtered probability space $(\Omega, {\mathcal{F}}, ({\mathcal{F}}_t){_{t\in{\mathbb{T}}}}, P)$ with ${\mathcal{F}}_0=\{\emptyset, \Omega\}$, ${\mathcal{F}}={\mathcal{F}}_T$ for $T<\infty$, and $\displaystyle{{\mathcal{F}}=\sigma(\cup_{t\ge 0}{\mathcal{F}}_t)}$ for $T=\infty$. For $t\in{\mathbb{T}}$, $L^{\infty}_t:=L^{\infty}(\Omega, {\mathcal{F}}_t,P)$ is the space of all essentially bounded ${\mathcal{F}}_t$-measurable random variables, and $L^\infty:=L^{\infty}(\Omega, {\mathcal{F}}_T,P)$. All equalities and inequalities between random variables and between sets are understood to hold $P$-almost surely, unless stated otherwise. We denote by ${{\mathcal{M}}_1(P)}$ (resp. by ${{\mathcal{M}}^e(P)}$) the set of all probability measures on $(\Omega, {\mathcal{F}})$ which are absolutely continuous with respect to $P$ (resp. equivalent to $P$).
In this work we consider risk measures defined on the set $L^{\infty}$, which is understood as the set of discounted terminal values of financial positions. In the dynamical setting, a conditional risk measure ${\rho_t}$ assigns to each terminal payoff $X$ an ${\mathcal{F}}_t$-measurable random variable ${\rho_t}(X)$, that quantifies the risk of the position $X$ given the information ${\mathcal{F}}_t$. A rigorous definition of a conditional convex risk measure was given in [@dt5 Definition 2].
\[defrm\] A map ${\rho_t}\,:\,{L^{\infty}}\,\rightarrow\,{L^{\infty}_t}$ is called a *conditional convex risk measure* if it satisfies the following properties for all $X,Y\in{L^{\infty}}$:
- Conditional cash invariance: For all $m_t\in{L^{\infty}_t}$ $${\rho_t}(X+m_t)={\rho_t}(X)-m_t;$$
- Monotonicity: $X\le Y\;\,\Rightarrow\;\,{\rho_t}(X)\ge{\rho_t}(Y)$;
- Conditional convexity: for all $\lambda\in{L^{\infty}_t},\,0\le \lambda\le 1$: $${\rho_t}(\lambda X+(1-\lambda)Y)\le\lambda{\rho_t}(X)+(1-\lambda){\rho_t}(Y);$$
- [Normalization]{}: ${\rho_t}(0)=0$.
A conditional convex risk measure is called a *conditional coherent risk measure* if it has in addition the following property:
- [Conditional positive homogeneity]{}: for all $\lambda\in{L^{\infty}_t},\,\lambda\ge0$: $${\rho_t}(\lambda X)=\lambda{\rho_t}(X).$$
In the dynamical framework one can also analyze risk assessment for cumulated cash flow *processes* rather than just for terminal pay-offs, i.e. one can consider a risk measure that accounts not only for the amounts but also for the timing of payments. Such risk measures were studied in [@cdk4; @cdk5; @cdk6; @ck6; @fs6; @jr8; @afp9]. As shown in [@adehk7] in the static and in [@afp9] in the dynamical setting, convex risk measures for processes can be identified with convex risk measures for random variables on an appropriate product space. This allows to extend results obtained in our present setting to the framework of processes; cf. [@afp9].
If ${\rho_t}$ is a conditional convex risk measure, the function $\phi_t:=-\rho_t$ defines a conditional monetary utility function in the sense of [@cdk6; @ck6]. The term “monetary” refers to conditional cash invariance of the utility function, the only property in Definition \[defrm\] that does not come from the classical utility theory. Conditional cash invariance is a natural request in view of the interpretation of ${\rho_t}$ as a conditional capital requirement. In order to formalize this aspect we first recall the notion of the *acceptance set* of a [conditional convex risk measure ]{}${\rho_t}$: $${\mathcal{A}}_t:={\left\{\,}
\newcommand{\rk}{\right\}}X\in{L^{\infty}}{\;\big|\;}{\rho_t}(X)\le0\rk.$$ The following properties of the acceptance set were given in [@dt5 Proposition 3].
\[acceptset\] The acceptance set ${\mathcal{A}}_t$ of a conditional convex risk measure ${\rho_t}$ is
1. conditionally convex, i.e. $\alpha X+(1-\alpha)Y\in{\mathcal{A}}_t$ for all $X,Y\in{\mathcal{A}}_t$ and $\alpha$ ${\mathcal{F}}_t$-measurable such that $0\leq\alpha\leq 1$;
2. solid, i.e. $Y\in{\mathcal{A}}_t$ whenever $Y\geq X$ for some $X\in{\mathcal{A}}_t$;
3. such that $0\in{\mathcal{A}}_t$ and $\operatorname*{ess\,inf}{\left\{\,}
\newcommand{\rk}{\right\}}X\in{L^{\infty}_t}{\;\big|\;}X\in{\mathcal{A}}_t\rk=0$.
Moreover, ${\rho_t}$ is uniquely determined through its acceptance set, since $$\label{defviaset}
{\rho_t}(X)=\operatorname*{ess\,inf}{\left\{\,}
\newcommand{\rk}{\right\}}Y\in{L^{\infty}_t}{\;\big|\;}X+Y\in{\mathcal{A}}_t\rk.$$ Conversely, if some set ${\mathcal{A}}_t\subseteq{L^{\infty}}$ satisfies conditions 1)-3), then the functional ${\rho_t}\,:\;{L^{\infty}}\rightarrow{L^{\infty}_t}$ defined via (\[defviaset\]) is a conditional convex risk measure.
Properties 1)-3) of the acceptance set follow easily from properties (i)-(iii) in Definition \[defrm\]. To prove (\[defviaset\]) note that by cash invariance ${\rho_t}(X)+X\in{\mathcal{A}}_t$ for all $X$, and this implies “$\ge$” in (\[defviaset\]). On the other hand, for all $Z\in {\left\{\,}
\newcommand{\rk}{\right\}}Y\in{L^{\infty}_t}{\;\big|\;}X+Y\in{\mathcal{A}}_t\rk$ we have $$0\ge{\rho_t}(Z+X)={\rho_t}(X)-Z,$$ thus ${\rho_t}(X)\le\operatorname*{ess\,inf}{\left\{\,}
\newcommand{\rk}{\right\}}Y\in{L^{\infty}_t}{\;\big|\;}X+Y\in{\mathcal{A}}_t\rk.$\
For the proof of the last part of the assertion we refer to [@dt5 Proposition 3].
Due to (\[defviaset\]), the value ${\rho_t}(X)$ can be viewed as the minimal conditional capital requirement needed to be added to the position $X$ in order to make it acceptable at time $t$. The following example shows how risk measures can be defined via (\[defviaset\]).
\[ex:entr\] Consider the set of all positions having non-negative conditional expected utility, i.e. $${\mathcal{A}}_t:=\{X\in{L^{\infty}}{\;\big|\;}E[u_t(X)|{\mathcal{F}}_t]\geq 0\},$$ where $u_t$ denotes some non-increasing and concave utility function. It is easy to check that the set ${\mathcal{A}}_t$ has all properties 1)-3) from Proposition \[acceptset\]. A basic choice is the exponential utility function $u_t(x)=1-e^{-{\gamma}_tx}$, where $\gamma_t>0$ $P$-a.s. denotes the risk aversion parameter such that $\gamma_t,\frac{1}{\gamma_t}\in{L^{\infty}_t}$. The corresponding conditional convex risk measure ${\rho_t}$ associated to ${\mathcal{A}}_t$ via takes the form $${\rho_t}(X)=\frac{1}{{\gamma}_t}\log E[e^{-{\gamma}_t X}|{\mathcal{F}}_t],\qquad X\in{L^{\infty}},$$ and is called the *conditional entropic risk measure*. The entropic risk measure was introduced in [@fs4] in the static setting, in the dynamical setting it appeared in [@bek4; @ms5; @dt5; @cdk6; @fp6; @ck6]. We characterize the dynamic entropic risk measure in Section \[entropic\].
Robust representation {#sectionrr}
=====================
As observed in [@adeh99; @fs4; @fr2] in the static setting, the axiomatic properties of a convex risk measure yield, under some regularity condition, a representation of the minimal capital requirement as a suitably modified worst expected loss over a whole class of probabilistic models. In the dynamical setting, such robust representations of conditional coherent risk measures were obtained on a finite probability space in [@rse5] for random variables and in [@rie4] for stochastic processes. On a general probability space, robust representations for conditional coherent and convex risk measures were proved in [@dt5; @bn4; @burg; @ks5; @fp6; @bn8] for random variables and in [@cdk6] for stochastic processes. In this section we mainly summarize the results from [@dt5; @fp6; @bn8].
The alternative probability measures in a robust representation of a risk measure ${\rho_t}$ contribute to the risk evaluation to a different degree. To formalize this aspect we use the notion of the minimal penalty function $\alpha_t^{\min}$, defined for each $Q\in{{\mathcal{M}}_1(P)}$ as $$\label{pf1}
{\alpha_t^{\min}(Q)}=\operatorname*{Q\text{-}ess\,sup}_{X\in{\mathcal{A}}_t}{E_Q[-X\,|\,{\mathcal{F}}_t\,]}.$$
The following property of the minimal penalty function is a standard result, that will be used in the proof of Theorem \[robdar\].
\[erwpf\] For $Q\in{{\mathcal{M}}_1(P)}$ and $0\le s\le t$, $$E_Q[{\alpha_t^{\min}(Q)}|{\mathcal{F}}_s]=\operatorname*{Q\text{-}ess\,sup}_{Y\in{\mathcal{A}}_t}E_Q[-Y|{\mathcal{F}}_s]\quad Q\text{-a.s.}$$ and in particular $$E_Q[{\alpha_t^{\min}(Q)}]=\sup_{Y\in{\mathcal{A}}_t}E_Q[-Y].$$
First we claim that the set $${\left\{\,}
\newcommand{\rk}{\right\}}E_Q[-X|{\mathcal{F}}_t]{\;\big|\;}X\in{\mathcal{A}}_t\rk$$ is directed upward for any $Q\in{{\mathcal{M}}_1(P)}$. Indeed, for $X,Y\in{\mathcal{A}}_t$ we can define $Z:=XI_A+YI_{A^c}$, where $A:=\{E_Q[-X|{\mathcal{F}}_t]\ge E_Q[-Y|{\mathcal{F}}_t]\}\in{\mathcal{F}}_t$. Conditional convexity of $\rho_t$ implies that $Z\in{\mathcal{A}}_t$, and by definition of $Z$ $$E_Q[-Z|{\mathcal{F}}_t]=\max\left(E_Q[-X|{\mathcal{F}}_t],E_Q[-Y|{\mathcal{F}}_t]\right)\quad Q\text{-a.s.}.$$ Hence there exists a sequence $(X^Q_n)_{n\in{\mathbb N}}$ in ${\mathcal{A}}_t$ such that $$\label{folge}
{\alpha_t^{\min}(Q)}=\lim_nE_Q[-X^Q_n|{\mathcal{F}}_t]\qquad Q\text{-a.s.},$$ and by monotone convergence we get $$\begin{aligned}
E_Q[{\alpha_t^{\min}(Q)}|{\mathcal{F}}_s]&=\lim_nE_Q\left[\,E_Q[-X_n^Q|{\mathcal{F}}_t]\,\big|\,{\mathcal{F}}_s\,\right]\\
&\le\operatorname*{Q\text{-}ess\,sup}_{Y\in{\mathcal{A}}_t}E_Q[-Y|{\mathcal{F}}_s]\quad Q\text{-a.s.}.\end{aligned}$$ The converse inequality follows directly from the definition of ${\alpha_t^{\min}(Q)}$.
The following theorem relates robust representations to some continuity properties of conditional convex risk measures. It combines [@dt5 Theorem 1] with [@fp6 Corollary 2.4]; similar results can be found in [@bn4; @ks5; @cdk6].
\[robdar\] For a conditional convex risk measure ${\rho_t}$ the following are equivalent:
1. ${\rho_t}$ has a robust representation $$\label{rd0}
{\rho_t}(X)={\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_t}}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-\alpha_t(Q)),\qquad X\in{L^{\infty}},$$ where $${\mathcal{Q}}_t:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{{\mathcal{M}}_1(P)}{\;\big|\;}Q=P|_{{\mathcal{F}}_t}\rk$$ and $\alpha_t$ is a map from ${\mathcal{Q}}_t$ to the set of ${\mathcal{F}}_t$-measurable random variables with values in ${\mathbb R}\cup\{+\infty\}$, such that ${\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_t}}(-\alpha_t(Q))=0$.
2. ${\rho_t}$ has the robust representation in terms of the minimal penalty function, i.e. $$\label{rd1}
{\rho_t}(X)={\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_t}}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}(Q)}),\qquad X\in{L^{\infty}},$$ where $\alpha_t^{\min}$ is given in (\[pf1\]).
3. ${\rho_t}$ has the robust representation $$\label{rd2}
{\rho_t}(X)=\operatorname*{ess\,sup}_{{\mathcal{Q}}\in{\mathcal{Q}}^f_t}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}(Q)})\quad P\text{-a.s.},\qquad X\in{L^{\infty}},$$ where $${\mathcal{Q}}_t^f:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{{\mathcal{M}}_1(P)}{\;\big|\;}Q=P|_{{\mathcal{F}}_t}\;E_{Q}[{\alpha_t^{\min}(Q)}]<\infty\rk.$$
4. ${\rho_t}$ has the “Fatou-property”: for any bounded sequence $(X_n)_{n\in{\mathbb N}}$ which converges $P$-a.s. to some $X$, $${\rho_t}(X)\le\liminf_{n\to\infty}{\rho_t}(X_n)\quad{P\mbox{-a.s.}}.$$
5. ${\rho_t}$ is continuous from above, i.e. $$X_n\searrow X\;\,P\text{-a.s}\quad\Longrightarrow\quad {\rho_t}(X_n)\nearrow{\rho_t}(X)\;\,P\text{-a.s}$$ for any sequence $(X_n)_n\subseteq{L^{\infty}}$ and $X\in{L^{\infty}}$.
3\) $\;{\Rightarrow}\; $ 1) and 2) $\;{\Rightarrow}\; $ 1) are obvious. 1) $\,{\Rightarrow}\, $ 4): Dominated convergence implies that $E_Q[X_n|{\mathcal{F}}_t]\rightarrow E_Q[X|{\mathcal{F}}_t]$ for each $Q\in{\mathcal Q}_t$, and $\liminf_{n\to\infty}{\rho_t}(X_n)\ge{\rho_t}(X)$ follows by using the robust representation of ${\rho_t}$ as in the unconditional setting, see, e.g., [@fs4 Lemma 4.20].
4\) $\,{\Rightarrow}\, $ 5): Monotonicity implies $\limsup_{n\to\infty}{\rho_t}(X_n)\le{\rho_t}(X)$, and $\liminf_{n\to\infty}{\rho_t}(X_n)\ge{\rho_t}(X)$ follows by 4).
5\) $\,{\Rightarrow}\, $ 2): The inequality $$\label{ungl1}
{\rho_t}(X)\ge\operatorname*{ess\,sup}_{Q\in{\mathcal Q}_t}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}(Q)})$$ follows from the definition of $\alpha_t^{\min}$. In order to prove the equality we will show that $$E_P[\rho_t(X)]\le E_P\left[{\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_t}}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}(Q)})\right].$$ To this end, consider the map $\rho^P\,:\,{L^{\infty}}\,\rightarrow\,{\mathbb R}$ defined by $\rho^P(X):=E_P[{\rho_t}(X)]$. It is easy to check that $\rho^P$ is a convex risk measure which is continuous from above. Hence [@fs4 Theorem 4.31] implies that $\rho^P$ has the robust representation $$\rho^P(X)=\sup_{Q\in{\mathcal{M}}_1(P)}(E_Q[-X]-\alpha(Q))\qquad X\in{L^{\infty}},$$ where the penalty function $\alpha(Q)$ is given by $$\alpha(Q)=\sup_{X\in{L^{\infty}}: \rho^P(X)\le0}E_Q[-X].$$ Next we will prove that $Q\in{\mathcal{Q}}_t$ if $\alpha(Q)<\infty$. Indeed, let $A\in{\mathcal{F}}_t$ and $\lambda>0$. Then $$-\lambda P[A]=E_P[{\rho_t}(\lambda I_A)]=\rho^P(\lambda I_A)\ge E_Q[-\lambda I_A]-\alpha(Q),$$ so $$P[A]\le Q[A]+\frac{1}{\lambda}\alpha(Q)\quad\mbox{for all}\quad \lambda>0,$$ and hence $P[A]\le Q[A]$ if $\alpha(Q)<\infty$. The same reasoning with $\lambda<0$ implies $P[A]\ge Q[A]$, thus $P = Q$ on ${\mathcal{F}}_t$ if $\alpha(Q)<\infty$. By Lemma \[erwpf\], we have for every $Q\in{\mathcal Q}_t$ $$E_P[{\alpha_t^{\min}(Q)}]=\sup_{Y\in{\mathcal{A}}_t}E_P[-Y].$$ Since $\rho^P(Y)\le0$ for all $Y\in{\mathcal{A}}_t$, this implies $$E_P[{\alpha_t^{\min}(Q)}]\le\alpha(Q)$$ for all $Q\in{\mathcal Q}_t$, by definition of the penalty function $\alpha(Q)$.
Finally we obtain $$\begin{aligned}
\label{rdbeweis}
E_P[{\rho_t}(X)]=\rho^P(X)&=\sup_{Q\in{\mathcal{M}}_1(P), \alpha(Q)<\infty}\left(E_Q[-X]-\alpha(Q)\right)\nonumber\\
&\le\sup_{Q\in{\mathcal{Q}}_t, E_P[{\alpha_t^{\min}(Q)}]<\infty}\left(E_Q[-X]-\alpha(Q)\right)\nonumber\\
&\le\sup_{Q\in{\mathcal{Q}}_t, E_P[{\alpha_t^{\min}(Q)}]<\infty}E_P[E_Q[-X|{\mathcal{F}}_t]-{\alpha_t^{\min}(Q)}]\nonumber\\
&\le E_P\left[\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_t, E_P[{\alpha_t^{\min}(Q)}]<\infty}\left(E_Q[-X|{\mathcal{F}}_t]-{\alpha_t^{\min}(Q)}\right)\right]\\
&\le E_P\left[\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_t}E_Q[-X|{\mathcal{F}}_t]-{\alpha_t^{\min}(Q)}\right]\nonumber, \end{aligned}$$ proving equality (\[rd1\]).
5\) $\,{\Rightarrow}\, $ 3) The inequality $${\rho_t}(X)\ge\operatorname*{ess\,sup}_{{\mathcal{Q}}\in{\mathcal{Q}}^f_t}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}(Q)})$$ follows from (\[ungl1\]) since ${\mathcal{Q}}_t^f\subseteq{\mathcal Q}_t$, and (\[rdbeweis\]) proves the equality.
The penalty function ${\alpha_t^{\min}(Q)}$ is minimal in the sense that any other function $\alpha_t$ in a robust representation of ${\rho_t}$ satisfies $${\alpha_t^{\min}(Q)}\le\alpha_t(Q)\;\,{P\mbox{-a.s.}}$$ for all $Q\in{\mathcal Q}_t$. An alternative formula for the minimal penalty function is given by $${\alpha_t^{\min}(Q)}=\operatorname*{ess\,sup}_{X\in{L^{\infty}}}\,\left({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-{\rho_t}(X)\right)\quad\mbox{for all}\;\, Q\in{\mathcal Q}_t.$$ This follows as in the unconditional case; see, e.g., [@fs4 Theorem 4.15, Remark 4.16].
\[abg\] Another characterization of a conditional convex risk measure ${\rho_t}$ that is equivalent to the properties 1)-4) of Theorem \[robdar\] is the following: The acceptance set ${\mathcal{A}}_t$ is weak$^\ast$-closed, i.e., it is closed in ${L^{\infty}}$ with respect to the topology $\sigma({L^{\infty}}, L^1(\Omega,{\mathcal{F}},P))$. This equivalence was shown in [@cdk6] in the context of risk measures for processes and in [@ks5] for risk measures for random variables. Though in [@ks5] a slightly different definition of a conditional risk measure is used, the reasoning given there works just the same in our case; cf. [@ks5 Theorem 3.16].
For the characterization of time consistency in Section \[sec:tc\] we will need a robust representation of a conditional convex risk measure ${\rho_t}$ under any measure $Q\in{{\mathcal{M}}_1(P)}$, where possibly $Q\notin{\mathcal{Q}}_t$. Such representation can be obtained as in Theorem \[robdar\] by considering ${\rho_t}$ as a risk measure under $Q$, as shown in the next corollary. This result is a version of [@bn8 Proposition 1].
\[corrobdar\] A conditional convex risk measure ${\rho_t}$ is continuous from above if and only if it has the robust representations $$\begin{aligned}
{\rho_t}(X)&=\operatorname*{Q\text{-}ess\,sup}_{R\in{\mathcal{Q}}_t(Q)}(E_R{[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}}(R))\label{rd3}\\
&=\operatorname*{Q\text{-}ess\,sup}_{R\in{\mathcal{Q}}^f_t(Q)}(E_R{[-X\,|\,{\mathcal{F}}_t\,]}-{\alpha_t^{\min}}(R))\quad Q\text{-a.s.},\quad \forall X\in{L^{\infty}},\label{rd3f}\end{aligned}$$ for all $Q\in{{\mathcal{M}}_1(P)}$, where $${\mathcal{Q}}_t(Q)={\left\{\,}
\newcommand{\rk}{\right\}}R\in{{\mathcal{M}}_1(P)}{\;\big|\;}R=Q|_{{\mathcal{F}}_t}\rk$$ and $${\mathcal{Q}}_t^f(Q)={\left\{\,}
\newcommand{\rk}{\right\}}R\in{\mathcal{M}}_1(P){\;\big|\;}R=Q|_{{\mathcal{F}}_t},\;E_{R}[{\alpha_t^{\min}}(R)]<\infty\rk.$$
To show that continuity from above implies representation , we can replace $P$ by a probability measure $Q\in{{\mathcal{M}}_1(P)}$ and repeat all the reasoning of the proof of 5)${\Rightarrow}$2) in Theorem \[robdar\]. In this case we consider the static convex risk measure $$\rho^Q(X)=E_Q[{\rho_t}(X)]=\sup_{R\in{\mathcal{M}}_1(P)}(E_R[-X]-\alpha(R)),\qquad X\in{L^{\infty}},$$ instead of $\rho^P$. The proof of follows in the same way from [@fp6 Corollary 2.4]. Conversely, continuity from above follows from Theorem \[robdar\] since representation holds under $P$.
One can easily see that the set ${\mathcal{Q}}_t$ in representations and can be replaced by ${\mathcal{P}}_t:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{{\mathcal{M}}_1(P)}{\;\big|\;}Q\approx P\;\text{on}\;{\mathcal{F}}_t\rk$. Moreover, representation is also equivalent to $${\rho_t}(X)=\operatorname*{ess\,sup}_{Q\in{{\mathcal{M}}_1(P)}}({E_Q[-X\,|\,{\mathcal{F}}_t\,]}-\hat{\alpha}_t(Q)),\qquad X\in{L^{\infty}},$$ where the conditional expectation under $Q\in{{\mathcal{M}}_1(P)}$ is defined under $P$ as $$E_Q[X|{\mathcal{F}}_t]:=\frac{E_P[Z_TX|{\mathcal{F}}_t]}{Z_t}I_{\{Z_t>0\}},$$ and the extended penalty function $\hat{\alpha}_t$ is given by $$\begin{aligned}
\hat{\alpha}_t(Q) = \left\{ \begin{array}{ll} \alpha_t(Q) & \textrm{on}\;\{\frac{dQ}{dP}|_{{\mathcal{F}}_t}>0\}; \\
+\infty & \textrm{otherwise}.
\end{array} \right.\end{aligned}$$
In the *coherent* case the penalty function ${\alpha_t^{\min}(Q)}$ can only take values $0$ or $\infty$ due to positive homogeneity of ${\rho_t}$. Thus representation takes the following form.
\[rdcoherent\] A conditional coherent risk measure ${\rho_t}$ is continuous from above if and only if it is representable in the form $$\label{rdcoh}
{\rho_t}(X)=\operatorname*{ess\,sup}_{{\mathcal{Q}}\in{\mathcal{Q}}^0_t}{E_Q[-X\,|\,{\mathcal{F}}_t\,]},\qquad X\in{L^{\infty}},$$ where $${\mathcal{Q}}_t^0:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{\mathcal{Q}}_t{\;\big|\;}{\alpha_t^{\min}(Q)}=0\; Q\mbox{-a.s.}\rk.$$
\[avar\] A notable example of a conditional coherent risk measure is *conditional Average Value at Risk* defined as $$\begin{aligned}
AV@R_{t,\lambda_t}(X):=\operatorname*{ess\,sup}\{E_Q[-X|{\mathcal{F}}_t]{\;\big|\;}Q\in{\mathcal{Q}}_t, \frac{dQ}{dP}\leq \lambda_t^{-1}\}\end{aligned}$$ with $\lambda_t\in{L^{\infty}_t}$, $0<\lambda_t\leq 1$. Static Average Value at Risk was introduced in [@adeh99] as a valid alternative to the widely used yet criticized Value at Risk. The conditional version of Average Value at Risk appeared in [@adehk7], and was also studied in [@Samuel; @vo6].
As observed, e.g., in [@cdk6 Remark 3.13], the minimal penalty function has the local property. In our context it means that for any $Q^1, Q^2\in{\mathcal{Q}}_t(Q)$ with the corresponding density processes $Z^1$ and $Z^2$ with respect to $P$, and for any $A\in{\mathcal{F}}_t$, the probability measure $R$ defined via $\frac{dR}{dP}:=I_AZ^1_T+I_{A^{\text{c}}}Z^2_T$ has the penalty function value $${\alpha_t^{\min}}(R)= I_A{\alpha_t^{\min}}(Q^1)+I_{A^{\text{c}}}{\alpha_t^{\min}}(Q^2)\qquad Q\text{-a.s.}.$$ In particular $R\in{\mathcal{Q}}_t^f(Q)$ if $Q^1, Q^2\in{\mathcal{Q}}_t^f(Q)$. Standard arguments (cf., e.g., [@dt5 Lemma 1]) imply then that the set $${\left\{\,}
\newcommand{\rk}{\right\}}E_R[\,-X\,|\,{\mathcal{F}}_t]-\alpha_t^{\min}(R){\;\big|\;}R\in{\mathcal{Q}}^f_t(Q)\rk$$ is directed upward, thus $$\label{erwrho}
E_{Q}[{\rho_t}(X)|{\mathcal{F}}_s]=\operatorname*{Q\text{-}ess\,sup}_{R\in{\mathcal{Q}}^f_t(Q)}\left(E_{R}[-X|{\mathcal{F}}_s]-E_{R}[\alpha_t^{\min}(R)|{\mathcal{F}}_s]\right)$$ for all $Q\in{{\mathcal{M}}_1(P)}, X\in{L^{\infty}(\Omega, {\mathcal{F}},P)}$ and $0\le s\leq t$.
Time consistency properties {#sec:tc}
===========================
In the dynamical setting risk assessment of a financial position is updated when new information is released. This leads to the notion of a dynamic risk measure.
\[dcrm\] A a sequence $({\rho_t}){_{t\in{\mathbb{T}}}}$ is called a *dynamic convex risk measure* if ${\rho_t}$ is a conditional convex risk measure for each $t\in{\mathbb{T}}$.
A key question in the dynamical setting is how the conditional risk assessments at different times are interrelated. This question has led to several notions of time consistency discussed in the literature. A unifying view was suggested in [@tu8].
\[sina\] Assume that $({\rho_t}){_{t\in{\mathbb{T}}}}$ is a dynamic convex risk measure and let ${\mathcal{Y}}_t$ be a subset of ${L^{\infty}}$ such that $0\in{\mathcal{Y}}_t$ and ${\mathcal{Y}}_t+{\mathbb R}={\mathcal{Y}}_t$ for each $t\in{\mathbb{T}}$. Then $({\rho_t}){_{t\in{\mathbb{T}}}}$ is called *acceptance (resp. rejection) consistent with respect to $({\mathcal{Y}}_t){_{t\in{\mathbb{T}}}}$*, if for all $t\in{\mathbb{T}}$ such that $t<T$ and for any $X\in{L^{\infty}}$ and $Y\in{\mathcal{Y}}_{t+1}$ the following condition holds: $$\label{definition1}
\rho_{t+1}(X)\le\rho_{t+1}(Y)\;\;(\mbox{resp.}\,\ge)\quad\Longrightarrow\quad\rho_{t}(X)\le\rho_{t}(Y)\;\;(\mbox{resp.}\,\ge).$$
The idea is that the degree of time consistency is determined by a sequence of benchmark sets $({\mathcal{Y}}_t){_{t\in{\mathbb{T}}}}$: if a financial position at some future time is always preferable to some element of the benchmark set, then it should also be preferable today. The bigger the benchmark set, the stronger is the resulting notion of time consistency. In the following we focus on three cases.
\[cons\] We call a dynamic convex risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$
1. *strongly time consistent*, if it is either acceptance consistent or rejection consistent with respect to ${\mathcal{Y}}_t={L^{\infty}}$ for all $t$ in the sense of Definition \[sina\];
2. *middle acceptance (resp. middle rejection) consistent*, if for all $t$ we have ${\mathcal{Y}}_t=L^\infty_t$ in Definition \[sina\];
3. *weakly acceptance (resp. weakly rejection) consistent*, if for all $t$ we have ${\mathcal{Y}}_t={\mathbb R}$ in Definition \[sina\].
Note that there is no difference between rejection consistency and acceptance consistency with respect to ${L^{\infty}}$, since the role of $X$ and $Y$ is symmetric in that case. Obviously strong time consistency implies both middle rejection and middle acceptance consistency, and middle rejection (resp. middle acceptance) consistency implies weak rejection (resp. weak acceptance) consistency. In the rest of the paper we drop the terms “middle” and “strong” in order to simplify the terminology.
Time consistency {#subsec:tc}
----------------
Time consistency has been studied extensively in the recent work on dynamic risk measures, see [@adehk7; @d6; @rie4; @dt5; @cdk6; @ks5; @burg; @bn8; @ipen7; @fp6; @ck6; @dpr10] and the references therein. In the next proposition we recall some equivalent characterizations of time consistency.
\[def2\] A dynamic convex risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ is time consistent if and only if any of the following conditions holds:
1. for all $t\in{\mathbb{T}}$ such that $t<T$ and for all $X,Y\in{L^{\infty}}$: $$\label{tc4}
{\rho_{t+1}}(X)\le{\rho_{t+1}}(Y)\;\,P\text{-a.s}\quad\Longrightarrow\quad{\rho_t}(X)\le{\rho_t}(Y)\;\,P\text{-a.s.};$$
2. for all $t\in{\mathbb{T}}$ such that $t<T$ and for all $X,Y\in{L^{\infty}}$: $$\label{tc2}
{\rho_{t+1}}(X)={\rho_{t+1}}(Y)\;\,P\text{-a.s}\quad\Longrightarrow\quad{\rho_t}(X)={\rho_t}(Y)\;\,P\text{-a.s.};$$
3. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is recursive, i.e. $${\rho_t}={\rho_t}(-{\rho_{t+s}})\quad P\text{-a.s.}$$ for all $t,s\ge 0$ such that $t,t+s\in{\mathbb{T}}$.
It is obvious that time consistency implies condition (\[tc4\]), and that (\[tc4\]) implies (\[tc2\]). By cash invariance we have ${\rho_{t+1}}(-{\rho_{t+1}}(X))={\rho_{t+1}}(X)$ and hence one-step recursiveness follows from (\[tc2\]). We prove that one-step recursiveness implies recursiveness by induction on $s$. For $s=1$ the claim is true for all $t$. Assume that the induction hypothesis holds for each $t$ and all $k\le s$ for some $s\ge 1$. Then we obtain $$\begin{aligned}
{\rho_t}(-\rho_{t+s+1}(X))&={\rho_t}(-{\rho_{t+s}}(-\rho_{t+s+1}(X)))\\
&={\rho_t}(-{\rho_{t+s}}(X))\\
&={\rho_t}(X),\end{aligned}$$ where we have applied the induction hypothesis to the random variable $-\rho_{t+s+1}(X)$. Hence the claim follows. Finally, due to monotonicity, recursiveness implies time consistency.
If we restrict a conditional convex risk measure ${\rho_t}$ to the space $L^\infty_{t+s}$ for some $s\ge0$, the corresponding acceptance set is given by $${\mathcal{A}}_{t,t+s}:={\left\{\,}
\newcommand{\rk}{\right\}}X\in L^\infty_{t+s}{\;\big|\;}{\rho_t}(X)\le0\;\,P\text{-a.s.}\rk,$$ and the minimal penalty function by $$\label{ats}
{\alpha_{t, t+s}^{\min}(Q)}:=\operatorname*{Q\text{-}ess\,sup}_{X\in{\mathcal{A}}_{t,t+s}}\,{E_Q[-X\,|\,{\mathcal{F}}_t\,]}, \qquad Q\in{{\mathcal{M}}_1(P)}.$$
The following lemma recalls equivalent characterizations of recursive inequalities in terms of acceptance sets from [@fp6 Lemma 4.6]; property was shown in [@d6].
\[4.6\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measure. Then the following equivalences hold for all $s,t$ such that $t,t+s\in{\mathbb{T}}$ and all $X\in{L^{\infty}}$: $$\begin{aligned}
X\in{\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}&\iff-\rho_{t+s}(X)\in{\mathcal{A}}_{t,t+s}\label{eqacset1}\\
{\mathcal{A}}_t\subseteq {\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}&\iff{\rho_t}(-\rho_{t+s})\le{\rho_t}\quad{P\mbox{-a.s.}}\label{eqacset2}\\
{\mathcal{A}}_t\supseteq {\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}&\iff{\rho_t}(-\rho_{t+s})\ge{\rho_t}\quad{P\mbox{-a.s.}}.\label{eqacset3}\end{aligned}$$
To prove “${\Rightarrow}$” in (\[eqacset1\]) let $X=X_{t,t+s}+X_{t+s}$ with $X_{t,t+s}\in{\mathcal{A}}_{t,t+s}$ and $X_{t+s}\in{\mathcal{A}}_{t+s}$. Then $$\rho_{t+s}(X)=\rho_{t+s}(X_{t+s})-X_{t,t+s}\le-X_{t,t+s}$$ by cash invariance, and monotonicity implies $${\rho_t}(-\rho_{t+s}(X))\le{\rho_t}(X_{t,t+s})\le0.$$ The converse direction follows immediately from $X=X+\rho_{t+s}(X)-\rho_{t+s}(X)$ and $X+\rho_{t+s}(X)\in{\mathcal{A}}_{t+s}$ for all $X\in{L^{\infty}}$.
In order to show “${\Rightarrow}$” in (\[eqacset2\]), fix $X\in{L^{\infty}}$. Since $X+{\rho_t}(X)\in{\mathcal{A}}_t\subseteq {\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}$, we obtain $$\rho_{t+s}(X)-{\rho_t}(X)=\rho_{t+s}(X+{\rho_t}(X))\in-{\mathcal{A}}_{t,t+s},$$ by (\[eqacset1\]) and cash invariance. Hence $${\rho_t}(-\rho_{t+s}(X))-{\rho_t}(X)={\rho_t}(-(\rho_{t+s}(X)-{\rho_t}(X)))\le0\quad{P\mbox{-a.s.}}.$$ To prove “$\Leftarrow$” let $X\in{\mathcal{A}}_t$. Then $-\rho_{t+s}(X)\in{\mathcal{A}}_{t,t+s}$ by the right hand side of (\[eqacset2\]), and hence $X\in{\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}$ by (\[eqacset1\]).
Now let $X\in{L^{\infty}}$ and assume ${\mathcal{A}}_t\supseteq {\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}$. Then $$\begin{aligned}
{\rho_t}(-\rho_{t+s}(X))+X&={\rho_t}(-\rho_{t+s}(X))-\rho_{t+s}(X)+\rho_{t+s}(X)+X\\&\in{\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}\subseteq{\mathcal{A}}_t.\end{aligned}$$ Hence $${\rho_t}(X)-{\rho_t}(-\rho_{t+s}(X))={\rho_t}(X+{\rho_t}(-\rho_{t+s}(X)))\le 0
$$ by cash invariance, and this proves “${\Rightarrow}$” in (\[eqacset3\]). For the converse direction let $X\in{\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}$. Since $-\rho_{t+s}(X)\in{\mathcal{A}}_{t,t+s}$ by (\[eqacset1\]), we obtain $${\rho_t}(X)\le{\rho_t}(-\rho_{t+s}(X))\le0,$$ hence $X\in{\mathcal{A}}_t$.
We also have the following relation between acceptance sets and penalty functions; cf. [@ipen7 Lemma 2.2.5].
\[setpen\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measures. Then the following implications hold for all $t,s$ such that $t,t+s\in{\mathbb{T}}$ and for all $Q\in{{\mathcal{M}}_1(P)}$: $$\begin{aligned}
{\mathcal{A}}_t\subseteq {\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}&\Rightarrow{\alpha_t^{\min}(Q)}\le{\alpha_{t, t+s}^{\min}(Q)}+E_Q[{\alpha_{t+s}^{\min}(Q)}|{\mathcal{F}}_t]\quad{Q\mbox{-a.s.}}\\
{\mathcal{A}}_t\supseteq {\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}&\Rightarrow{\alpha_t^{\min}(Q)}\ge{\alpha_{t, t+s}^{\min}(Q)}+E_Q[{\alpha_{t+s}^{\min}(Q)}|{\mathcal{F}}_t]\quad{Q\mbox{-a.s.}}.$$
Straightforward from the definition of the minimal penalty function and Lemma \[erwpf\].
The following theorem gives equivalent characterizations of time consistency in terms of acceptance sets, penalty functions, and a supermartingale property of the risk process.
\[eqchar\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then the following conditions are equivalent:
1. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is time consistent.
2. ${\mathcal{A}}_t={\mathcal{A}}_{t,t+s}+{\mathcal{A}}_{t+s}\,$ for all $t,s$ such that $t,t+s\in{\mathbb{T}}$.
3. ${\alpha_t^{\min}(Q)}={\alpha_{t, t+s}^{\min}(Q)}+E_Q[\,{\alpha_{t+s}^{\min}(Q)}\,|\,{\mathcal{F}}_t\,]\;\; Q$-a.s. for all $t,s$ such that $t,t+s\in{\mathbb{T}}$ and all $\,Q\in{{\mathcal{M}}_1(P)}$.
4. For all $X\in{L^{\infty}(\Omega, {\mathcal{F}},P)}$ and all $t,s$ such that $t,t+s\in{\mathbb{T}}$ and all $\,Q\in{{\mathcal{M}}_1(P)}$ we have $$E_Q[\,\rho_{t+s}(X)+{\alpha_{t+s}^{\min}(Q)}\,|\,{\mathcal{F}}_t]\le{\rho_t}(X)+{\alpha_t^{\min}(Q)}\quad{Q\mbox{-a.s.}}.$$
Equivalence of properties 1) and 2) of Theorem \[eqchar\] was proved in [@d6]. Characterizations of time consistency in terms of penalty functions as in 3) of Theorem \[eqchar\] appeared in [@fp6; @bn6; @ck6; @bn8]; similar results for risk measures for processes were given in [@cdk6; @ck6]. The supermartingale property as in 4) of Theorem \[eqchar\] was obtained in [@fp6]; cf. also [@bn8] for the absolutely continuous case.
The proof of 1)${\Rightarrow}$2)${\Rightarrow}$3) follows from Lemma \[4.6\] and Lemma \[setpen\]. To prove 3)${\Rightarrow}$4) fix $Q\in{{\mathcal{M}}_1(P)}$. By we have $$E_{Q}[{\rho_{t+s}}(X)|{\mathcal{F}}_t]=\operatorname*{Q\text{-}ess\,sup}_{R\in{\mathcal{Q}}^f_{t+s}(Q)}\left(E_{R}[-X|{\mathcal{F}}_{t}]-E_{R}[\alpha_{t+s}^{\min}(R)|{\mathcal{F}}_t]\right).$$ On the set ${\left\{\,}
\newcommand{\rk}{\right\}}{\alpha_t^{\min}(Q)}=\infty\rk$ property 4) holds trivially. On the set ${\left\{\,}
\newcommand{\rk}{\right\}}{\alpha_t^{\min}(Q)}<\infty\rk$ property 3) implies $E_Q[\alpha_{t+s}^{\min}(Q)|{\mathcal{F}}_t]<\infty$ and $\alpha_{t,t+s}^{\min}(Q)<\infty$, then for $R\in{\mathcal{Q}}^f_{t+s}(Q)$ $${\alpha_t^{\min}}(R)=\alpha_{t,t+s}^{\min}(Q)+E_R[\alpha_{t+s}^{\min}(R)|{\mathcal{F}}_t]<\infty\quad{Q\mbox{-a.s.}}.$$ Thus $$E_{Q}[{\rho_{t+s}}(X)+\alpha_{t+s}^{\min}(Q)|{\mathcal{F}}_t]=\operatorname*{Q\text{-}ess\,sup}_{R\in{\mathcal{Q}}^f_{t+s}(Q)}\left(E_{R}[-X|{\mathcal{F}}_{t}]-\alpha_{t}^{\min}(R)\right)+{\alpha_t^{\min}(Q)}$$ on ${\left\{\,}
\newcommand{\rk}{\right\}}{\alpha_t^{\min}(Q)}<\infty\rk$. Moreover, since ${\mathcal{Q}}_{t+s}^f(Q)\subseteq {\mathcal{Q}}_{t}(Q)$, implies $$E_{Q}[{\rho_{t+s}}(X)+\alpha_{t+s}^{\min}(Q)|{\mathcal{F}}_t]\le\operatorname*{Q\text{-}ess\,sup}_{R\in {\mathcal{Q}}_{t}(Q)}\left(E_{R}[-X|{\mathcal{F}}_{t}]-\alpha_{t}^{\min}(R)\right)+{\alpha_t^{\min}(Q)}={\rho_t}(X)+{\alpha_t^{\min}(Q)}\quad{Q\mbox{-a.s.}}.$$ It remains to prove 4)${\Rightarrow}$1). To this end fix $Q\in{\mathcal{Q}}_t^f$ and $X,Y\in{L^{\infty}}$ such that $\rho_{t+1}(X)\le \rho_{t+1}(Y)$. Note that $E_Q[\alpha_{t+s}(Q)]<\infty$ due to 4), hence $Q\in{\mathcal{Q}}_{t+s}^f(Q)$. Using 4) and representation for $\rho_{t+s}$ under $Q$, we obtain $$\begin{aligned}
{\rho_t}(Y)+\alpha_t^{\min}(Q)&\geq E_Q[\rho_{t+1}(Y)+\alpha_{t+1}^{\min}(Q)|{\mathcal{F}}_t]\\
&\geq E_Q[\rho_{t+1}(X)+\alpha_{t+1}^{\min}(Q)|{\mathcal{F}}_t]\\
&\geq E_Q[E_Q[-X|{\mathcal{F}}_{t+1}]-\alpha_{t+1}^{\min}(Q)+\alpha_{t+1}^{\min}(Q)|{\mathcal{F}}_t]\\
&=E_Q[-X|{\mathcal{F}}_t].\end{aligned}$$ Hence representation yields $\rho_t(y)\ge \rho_t(X)$, and time consistency follows from Proposition \[def2\].
Properties 3) and 4) of Theorem \[eqchar\] imply in particular supermartingale propeties of penalty function processes and risk processes. This allows to apply martingale theory for characterization the the dynamics of these processes, as we do in Proposition \[worstcase\] and Proposition \[riesz\]; cf. also [@d6; @fp6; @ipen7; @bn8; @dpr10].
\[worstcase\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a time consistent dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then the process $$V_t^Q(X):={\rho_t}(X)+{\alpha_t^{\min}(Q)},\qquad t\in{\mathbb{T}}$$ is a $Q$-supermartingale for all $X\in{L^{\infty}}$ and all $Q\in{\mathcal{Q}}_0$, where $${\mathcal{Q}}_0:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{{\mathcal{M}}_1(P)}{\;\big|\;}{\alpha_0^{\min}(Q)}<\infty\rk.$$ Moreover, $(V_t^Q(X))_{t\in{\mathbb{T}}}$ is a $Q$-martingale if $Q\in{\mathcal{Q}}_0$ is a “worst case” measure for $X$ at time $0$, i.e. if the supremum in the robust representation of $\rho_0(X)$ is attained at $Q$: $$\rho_0(X)=E_Q[-X]-\alpha^{\min}_0(Q)\quad Q\text{-a.s.}.$$ In this case $Q$ is a “worst case” measure for $X$ at any time $t$, i.e. $$\rho_t(X)=E_Q[-X|{\mathcal{F}}_t]-\alpha^{\min}_t(Q)\quad Q\text{-a.s.}\quad\text{for all}\quad t\in{\mathbb{T}}.$$ The converse holds if $T<\infty$ or $\lim_{t\to\infty}{\rho_t}(X)=-X$ $P$-a.s. (what is called asymptotic precision in [@fp6]): If $(V_t^Q(X))_{t\in{\mathbb{T}}}$ is a $Q$-martingale then $Q\in{\mathcal{Q}}_0$ is a “worst case” measure for $X$ at any time $t\in{\mathbb{T}}$.
The supermartingale property of $(V_t^Q(X)){_{t\in{\mathbb{T}}}}$ under each $Q\in{\mathcal{Q}}_0$ follows directly from properties 3) and 4) of Theorem \[eqchar\]. To prove the remaining part of the claim, fix $Q\in{\mathcal{Q}}_0$ and $X\in{L^{\infty}}$. If $Q$ is a “worst case” measure for $X$ at time $0$, the process $$U_t(X):=V_t^Q(X)-E_Q[-X|{\mathcal{F}}_t],\qquad t\in{\mathbb{T}}$$ is a non-negative $Q$-supermartingale beginning at $0$. Indeed, the supermartingale property follows from that of $(V_t^Q(X)){_{t\in{\mathbb{T}}}}$, and non-negativity follows from the representation , since $Q\in{\mathcal{Q}}_t^f(Q)$. Thus $U_t=0$ $Q$-a.s. for all $t$, and this proves the “if” part of the claim. To prove the converse direction, note that if $(V_t^Q(X))_{t\in{\mathbb{T}}}$ is a $Q$-martingale and $\rho_T(X)=-X$ (resp. $\lim_{t\to\infty}{\rho_t}(X)=-X$ $P$-a.s.), the process $U(X)$ is a $Q$-martingale ending at $0$ (resp. converging to $0$ in $L^1(Q)$), and thus $U_t(X)=0$ $Q$-a.s. for all $t\in{\mathbb{T}}$.
The fact that a worst case measure for $X$ at time $0$, if it exists, remains a worst case measure for $X$ at any time $t\in{\mathbb{T}}$ was also shown in [@ck6 Theorem 3.9] for a time consistent dynamic risk measure without using the supermartingale property from Proposition \[worstcase\].
\[remarksm\] In difference to [@fp6 Theorem 4.5], without the additional assumption that the set $$\label{qstar}
{{\mathcal Q}^{\ast}}:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{{\mathcal{M}}^e(P)}{\;\big|\;}{\alpha_0^{\min}(Q)}<\infty\rk$$ is nonempty, the supermartingale property of $(V_t^Q(X)){_{t\in{\mathbb{T}}}}$ for all $X\in{L^{\infty}}$ and all $Q\in{{\mathcal Q}^{\ast}}$ is not sufficient to prove time consistency. In this case we also do not have the robust representation of ${\rho_t}$ in terms of the set ${{\mathcal Q}^{\ast}}$.
The process $({\alpha_t^{\min}(Q)}){_{t\in{\mathbb{T}}}}$ is a $Q$-supermartingale for all $Q\in{\mathcal{Q}}_0$ due to Property 3) of Theorem \[eqchar\]. The next proposition provides the explicit form of its Doob- and its Riesz-decomposition; cf. also [@ipen7 Proposition 2.3.2].
\[riesz\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a time consistent dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then for each $Q\in{\mathcal{Q}}_0$ the process $({\alpha_t^{\min}(Q)}){_{t\in{\mathbb{T}}}}$ is a non-negative $Q$-supermartingale with the Riesz decomposition $${\alpha_t^{\min}(Q)}=Z_t^Q+M_t^Q\quad {Q\mbox{-a.s.}},\qquad t\in{\mathbb{T}},$$ where $$Z_t^Q:= E_Q\left[\,\sum_{k=t}^{T-1}{\alpha_{k,k+1}^{\min}(Q)}\,\big|\,{\mathcal{F}}_t\,\right]\quad {Q\mbox{-a.s.}},\quad t\in{\mathbb{T}}$$ is a $Q$-potential and $$M_t^{Q}:=\left\{
\begin{array}{c@{\quad \quad}l}
0 & \text{if $T<\infty$},\\
\displaystyle\lim_{s\to\infty}E_Q\left[\alpha_s(Q)\,|\,{\mathcal{F}}_t\,\right] & \text{if $T=\infty$}
\end{array}\right.\qquad {Q\mbox{-a.s.}},\quad t\in{\mathbb{T}}$$ is a non-negative $Q$-martingale.
Moreover, the Doob decomposition of $({\alpha_t^{\min}(Q)}){_{t\in{\mathbb{T}}}}$ is given by $${\alpha_t^{\min}(Q)}=E_Q\left[\,\sum_{k=0}^{T-1}{\alpha_{k,k+1}^{\min}(Q)}\,\big|\,{\mathcal{F}}_t\,\right]+M_t^Q-\sum_{k=0}^{t-1}{\alpha_{k,k+1}^{\min}(Q)},\quad t\in{\mathbb{T}}$$ with the $Q$-martingale $$E_Q\left[\,\sum_{k=0}^{T-1}{\alpha_{k,k+1}^{\min}(Q)}\,\big|\,{\mathcal{F}}_t\,\right]+M_t^Q,\quad t\in{\mathbb{T}}$$ and the non-decreasing predictable process $(\sum_{k=0}^{t-1}{\alpha_{k,k+1}^{\min}(Q)}){_{t\in{\mathbb{T}}}}$.
We fix $Q\in{{\mathcal{M}}_1(P)}$ and applying property 3) of Theorem \[eqchar\] step by step we obtain $$\label{lim}
{\alpha_t^{\min}(Q)}=E_Q\left[\,\sum_{k=t}^{t+s-1}{\alpha_{k,k+1}^{\min}(Q)}\,\big|\,{\mathcal{F}}_t\,\right]+E_Q[\,{\alpha_{t+s}^{\min}(Q)}\,|\,{\mathcal{F}}_t\,]\quad {Q\mbox{-a.s.}}$$ for all $t,s$ such that $t,t+s\in{\mathbb{T}}$. If $T<\infty$, the Doob- and Riesz-decompositions follow immediately from , since $\alpha_T(Q)=0\; {Q\mbox{-a.s.}}$. If $T=\infty$, by monotonicity there exists the limit $$Z_t^Q= \lim_{s\to\infty}E_Q\left[\,\sum_{k=t}^{s}{\alpha_{k,k+1}^{\min}(Q)}\,\big|\,{\mathcal{F}}_t\,\right]=E_Q\left[\,\sum_{k=t}^{\infty}{\alpha_{k,k+1}^{\min}(Q)}\,\big|\,{\mathcal{F}}_t\,\right]\quad {Q\mbox{-a.s.}}$$ for all $t\in{\mathbb{T}}$, where we have used the monotone convergence theorem for the second equality. Equality implies then that there exists $$M_t^Q= \lim_{s\to\infty}E_Q\left[\,{\alpha_{t+s}^{\min}(Q)}\,|\,{\mathcal{F}}_t\,\right]\quad {Q\mbox{-a.s.}},\quad t\in{\mathbb{T}}$$ and $${\alpha_t^{\min}(Q)}=Z_t^Q+M_t^Q\quad {Q\mbox{-a.s.}}$$ for all $t\in{\mathbb{T}}$.
The process $(Z_t^Q){_{t\in{\mathbb{T}}}}$ is a non-negative $Q$-supermartingale. Indeed, $$\label{fin}
E_Q[\,Z_{t}^Q\,]\le E_Q\left[\,\sum_{k=0}^{\infty}{\alpha_{k,k+1}^{\min}(Q)}\,\right]\le{\alpha_0^{\min}(Q)}<\infty$$ and $E_Q[\,Z_{t+1}^Q\,|\,{\mathcal{F}}_t\,]\le Z_t^Q$ $Q$-a.s. for all $t\in{\mathbb{T}}$ by definition. Moreover, monotone convergence implies $$\lim_{t\to\infty}E_Q[\,Z_t^Q\,]=E_Q\left[\,\lim_{t\to\infty}\sum_{k=t}^{\infty}{\alpha_{k,k+1}^{\min}(Q)}\,\right]=0\quad{Q\mbox{-a.s.}},$$ since $\sum_{k=0}^{\infty}{\alpha_{k,k+1}^{\min}(Q)}<\infty\;Q$-a.s. by . Hence the process $(Z_t^Q){_{t\in{\mathbb{T}}}}$ is a $Q$-potential.
The process $(M_t^Q){_{t\in{\mathbb{T}}}}$ is a non-negative $Q$-martingale, since $$E_Q[\,M_{t}^Q\,]\le E_Q\left[\,{\alpha_t^{\min}(Q)}\,\right]\le{\alpha_0^{\min}(Q)}<\infty$$ and $$\begin{aligned}
E_Q[M_{t+1}^Q-M_{t}^Q|{\mathcal{F}}_t]&=E_Q[{\alpha_{t+1}^{\min}(Q)}|{\mathcal{F}}_t]-{\alpha_t^{\min}(Q)}-E_Q[Z_{t+1}^Q-Z_t^Q|{\mathcal{F}}_t]\\
&={\alpha_{t, t+1}^{\min}(Q)}-{\alpha_{t, t+1}^{\min}(Q)}=0\qquad{\mathcal{Q}}\mbox{-a.s.}\end{aligned}$$ for all $t\in{\mathbb{T}}$ by property 3) of Theorem \[eqchar\] and the definition of $(Z_t^Q){_{t\in{\mathbb{T}}}}$.
The Doob-decomposition follows straightforward from the Riesz-decomposition.
\[mnull\] It was shown in [@fp6 Theorem 5.4] that the martingale $M^Q$ in the Riesz decomposition of $({\alpha_t^{\min}(Q)})_{t\in{\mathbb{T}}}$ vanishes if and only if $\lim_{t\to\infty}{\rho_t}(X)\ge-X\,P$-a.s., i.e. the dynamic risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ is asymptotically safe. This is not always the case; see [@fp6 Example 5.5].
For a *coherent* risk measure we have $${\mathcal{Q}}_t^f(Q)={\mathcal{Q}}_t^0(Q):={\left\{\,}
\newcommand{\rk}{\right\}}R\in{\mathcal{M}}^1(P){\;\big|\;}R=Q|_{{\mathcal{F}}_t},\;\; {\alpha_t^{\min}}(R)=0\;{Q\mbox{-a.s.}}\rk.$$ In order to give an equivalent characterization of property 3) of Theorem \[eqchar\] in the coherent case, we introduce the sets $${\mathcal{Q}}_{t,t+s}^0(Q)={\left\{\,}
\newcommand{\rk}{\right\}}R\ll P|_{{\mathcal{F}}_{t+s}}{\;\big|\;}R=Q|_{{\mathcal{F}}_t},\;\; \alpha_{t,t+s}^{\min}(R)=0\;{Q\mbox{-a.s.}}\rk\quad \forall\, t,s\ge 0\;\, \textrm{such that}\;\, t,t+s\in{\mathbb{T}}.$$ For $Q^1\in{\mathcal{Q}}_{t, t+s}^0(Q)$ and $Q^2\in{\mathcal{Q}}_{t+s}^0(Q)$ we denote by $Q^1\oplus^{t+s} Q^2$ the pasting of $Q^1$ and $Q^2$ in $t+s$ via $\Omega$, i.e. the measure ${\widetilde{Q}}$ defined via $$\label{pasting}
{\widetilde{Q}}(A)=E_{Q^1}\left[E_{Q^2}[I_A|{\mathcal{F}}_{t+s}]\right],\qquad A\in{\mathcal{F}}.$$ The relation between stability under pasting and time consistency of coherent risk measures that can be represented in terms of equivalent probability measures was studied in [@adehk7; @d6; @ks5; @fp6]. In our present setting, Theorem \[eqchar\] applied to a coherent risk measure takes the following form.
\[coherent\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic coherent risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then the following conditions are equivalent:
1. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is time consistent.
2. For all $Q\in{{\mathcal{M}}_1(P)}$ and all $t,s$ such that $t,t+s\in{\mathbb{T}}$ $${\mathcal{Q}}_t^0(Q)= {\left\{\,}
\newcommand{\rk}{\right\}}Q^1\oplus^{t+s} Q^2{\;\big|\;}Q^1\in{\mathcal{Q}}_{t, t+s}^0(Q), \;Q^2\in{\mathcal{Q}}_{t+s}^0(Q^1)\rk.$$
3. For all $Q\in{{\mathcal{M}}_1(P)}$ such that ${\alpha_t^{\min}(Q)}=0\;Q$-a.s., $$E_Q[{\rho_{t+s}}(X)\,|\,{\mathcal{F}}_t] \le {\rho_t}(X)\quad\text{and}\quad \alpha_{t+s}^{\min}(Q)=0\;\,{Q\mbox{-a.s.}}$$ for all $X\in{L^{\infty}(\Omega, {\mathcal{F}},P)}$ and for all $t,s$ such that $t,t+s\in{\mathbb{T}}$.
$1){\Rightarrow}2)$: Time consistency implies property 3) of Theorem \[eqchar\], and we will show that this implies property 2) of Corollary \[coherent\]. Fix $Q\in{{\mathcal{M}}_1(P)}$. To prove “$\supseteq$” let $Q^1\in{\mathcal{Q}}_t^0(Q)$, $Q^2\in{\mathcal{Q}}_{t+s}^0(Q^1)$, and consider ${\widetilde{Q}}$ defined as in . Note that ${\widetilde{Q}}=Q^1$ on ${\mathcal{F}}_{t+s}$ and $$E_{{\widetilde{Q}}}[X|{\mathcal{F}}_{t+s}]=E_{Q^2}[X|{\mathcal{F}}_{t+s}]\quad Q^1\text{-a.s. for all}\quad X\in{L^{\infty}(\Omega, {\mathcal{F}},P)}.$$ Hence, using 3) of Theorem \[eqchar\] we obtain $$\begin{aligned}
\alpha_t^{\min}({\widetilde{Q}})&= \alpha_{t,t+s}^{\min}({\widetilde{Q}})+E_{{\widetilde{Q}}}[\alpha_{t+s}^{\min}({\widetilde{Q}})|{\mathcal{F}}_{t}]\\
&= \alpha_{t,t+s}^{\min}(Q^1)+E_{Q^1}[\alpha_{t+s}^{\min}(Q^2)|{\mathcal{F}}_{t}]=0\qquad{Q\mbox{-a.s.}},\end{aligned}$$ and thus ${\widetilde{Q}}\in{\mathcal{Q}}_t^0(Q)$. Conversely, for every ${\widetilde{Q}}\in{\mathcal{Q}}_t^0(Q)$ we have $\alpha_{t+s}^{\min}({\widetilde{Q}})=\alpha_{t,t+s}^{\min}({\widetilde{Q}})=0\;{\widetilde{Q}}$-a.s. by 3) of Theorem \[eqchar\], and ${\widetilde{Q}}={\widetilde{Q}}\oplus{\widetilde{Q}}$. This proves “$\subseteq$”.
$2){\Rightarrow}3)$: Let $R\in{{\mathcal{M}}_1(P)}$ with ${\alpha_t^{\min}}(R)=0\;R$-a.s.. Then $R\in{\mathcal{Q}}_t^0(R)$, and thus $R=Q^1\otimes^{t+s} Q^2$ for some $Q^1\in{\mathcal{Q}}_{t,t+s}^0(R)$ and $Q^2\in{\mathcal{Q}}_{t+s}^0(Q^1)$. This implies $R=Q^1$ on ${\mathcal{F}}_{t+s}$ and $$E_R[X|{\mathcal{F}}_{t+s}]=E_{Q^2}[X|{\mathcal{F}}_{t+s}]\quad R\text{-a.s.}.$$ Hence $\alpha_{t, t+s}^{\min}(R)=\alpha_{t, t+s}^{\min}(Q^1)=0\;\,R$-a.s., and $\alpha_{t+s}^{\min}(R)=\alpha_{t+s}^{\min}(Q^2)=0\,R$-a.s.. To prove the inequality 3) note that due to $$\begin{aligned}
E_R[\,{\rho_{t+s}}(X)\,|\,{\mathcal{F}}_t]&=\operatorname*{R\text{-}ess\,sup}_{Q\in{\mathcal{Q}}_{t+s}^0(R)} E_{Q}[-X\,|\,{\mathcal{F}}_t]\\
&\le \operatorname*{R\text{-}ess\,sup}_{Q\in{\mathcal{Q}}_t^0(R)}{E_Q[-X\,|\,{\mathcal{F}}_t\,]}={\rho_t}(X)\quad R\text{-a.s.},\end{aligned}$$ where we have used that the pasting of $R|_{{\mathcal{F}}_{t+s}}$ and $Q$ belongs to ${\mathcal{Q}}_t^0(R)$.
$3){\Rightarrow}1)$: Obviously property 3) of Corollary \[coherent\] implies property 4) of Theorem \[eqchar\] and thus time consistency.
Rejection and acceptance consistency {#subsec:rc}
------------------------------------
Rejection and acceptance consistency were introduced and studied in [@tu8; @Samuel; @ipen7]. These properties can be characterized via recursive inequalities as stated in the next proposition; see [@tu8 Theorem 3.1.5] and [@Samuel Proposition 3.5].
\[rejrecursiveness\] A dynamic convex risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ is rejection (resp. acceptance) consistent if and only if for all $t\in{\mathbb{T}}$ such that $t<T$ $$\label{rcdef}
{\rho_t}(-\rho_{t+1})\le{\rho_t}\quad(\mbox{resp.}\ge)\quad{P\mbox{-a.s.}}.$$
We argue for the case of rejection consistency; the case of acceptance consistency follows in the same manner. Assume first that ${({\rho_t}){_{t\in{\mathbb{T}}}}}$ satisfies (\[rcdef\]) and let $X\in{L^{\infty}}$ and $Y\in L^{\infty}({\mathcal{F}}_{t+1})$ such that $\rho_{t+1}(X)\ge\rho_{t+1}(Y)$. Using cash invariance, (\[rcdef\]), and monotonicity, we obtain $${\rho_t}(X)\ge{\rho_t}(-\rho_{t+1}(X))\ge{\rho_t}(-\rho_{t+1}(Y))={\rho_t}(Y).$$ The converse implication follows due to cash invariance by applying (\[definition1\]) to $Y=-\rho_{t+1}(X)$.
\[weakmiddle\] For a dynamic *coherent* risk measure, weak acceptance consistency and acceptance consistency are equivalent. This was shown in [@Samuel Proposition 3.9].
Another way to characterize rejection consistency was suggested in [@ipen7].
A dynamic convex risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ is rejection consistent if only if any of the following conditions holds:
1. For all $t\in{\mathbb{T}}$ such that $t<T$ and all $X\in{L^{\infty}}$ $$\label{pruddef}
{\rho_t}(X)-\rho_{t+1}(X) \in{\mathcal{A}}_{t,t+1};$$
2. For all $t\in{\mathbb{T}}$ such that $t<T$ and all $X\in{\mathcal{A}}_t$, we have $-{\rho_{t+1}}(X)\in{\mathcal{A}}_t$.
Since $${\rho_t}(-{\rho_{t+1}}(X))={\rho_t}({\rho_t}(X)-{\rho_{t+1}}(X))+{\rho_t}(X)$$ by cash invariance, implies rejection consistency, and obviously rejection consistency implies condition 2). If 2) holds, then for any $X\in{L^{\infty}}$ $${\rho_t}({\rho_t}(X)-{\rho_{t+1}}(X))={\rho_t}\left(-{\rho_{t+1}}(X+{\rho_t}(X))\right)\le0,$$ due to cash invariance and the fact that $X+{\rho_t}(X)\in{\mathcal{A}}_t$.
Property was introduces in [@ipen7] under the name *prudence*. It means that the adjustment ${\rho_{t+1}}(X)-{\rho_t}(X)$ of the minimal capital requirement for $X$ at time $t+1$ is acceptable at time $t$. In other words, one stays on the safe side at each period of time by making capital reserves according to a rejection consistent dynamic risk measure.
Similar to time consistency, rejection and acceptance consistency can be characterized in terms of acceptance sets and penalty functions.
\[eqcharprud\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then the following properties are equivalent:
1. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is rejection consistent (resp. acceptance consistent).
2. The inclusion $${\mathcal{A}}_t\subseteq{\mathcal{A}}_{t,t+1}+{\mathcal{A}}_{t+1}\quad\text{resp.}\quad{\mathcal{A}}_t\supseteq{\mathcal{A}}_{t,t+1}+{\mathcal{A}}_{t+1}$$ holds for all $t\in{\mathbb{T}}$ such that $t<T$.
3. The inequality $${\alpha_t^{\min}(Q)}\le(\text{resp.}\ge){\alpha_{t, t+1}^{\min}(Q)}+E_Q[\,{\alpha_{t+1}^{\min}(Q)}\,|\,{\mathcal{F}}_t\,]\quad{Q\mbox{-a.s.}}$$ holds for all $t\in{\mathbb{T}}$ such that $t<T$ and all $Q\in{{\mathcal{M}}_1(P)}$.
Equivalence of 1) and 2) was proved in Proposition \[rejrecursiveness\] and Lemma \[4.6\], and the proof of $2){\Rightarrow}3) $ is given in Lemma \[setpen\].
Let us show that property 3) implies property 1). We argue for the case of rejection consistency; the case of acceptance consistency follows in the same manner. We fix $t\in{\mathbb{T}}$ such that $t<T$, and consider the risk measure $${\widetilde \rho_t(X)}:={\rho_t}(-{\rho_{t+1}}(X)),\qquad X\in{L^{\infty}}.$$ It is easily seen that ${\widetilde \rho_t}$ is a conditional convex risk measure that is [continuous from above]{}. Moreover, the dynamic risk measure $({\widetilde \rho_t}, {\rho_{t+1}})$ is time consistent by definition, and thus it fulfills properties 2) and 3) of Theorem \[eqchar\]. We denote by ${\widetilde {\mathcal{A}}_t}$ and ${\widetilde {\mathcal{A}}_{t,t+1}}$ the acceptance sets of the risk measure ${\widetilde \rho_t}$, and by ${\widetilde \alpha_{t}^{\min}}$ its penalty function. Since $${\widetilde \rho_t(X)}={\rho_t}(-{\rho_{t+1}}(X))={\rho_t}(X)$$ for all $X\in L_{t+1}$, we have ${\widetilde {\mathcal{A}}_{t,t+1}}={{\mathcal A}_{t,t+1}}$, and thus $${\widetilde {\mathcal{A}}_t}={{\mathcal A}_{t,t+1}}+{{\mathcal A}_{t+1}}$$ by 2) of Theorem \[eqchar\]. Lemma \[setpen\] and property 3) then imply $${\widetilde \alpha_{t}^{\min}}(Q) ={\alpha_{t, t+1}^{\min}(Q)}+E_Q[{\alpha_{t+1}^{\min}(Q)}|{\mathcal{F}}_t]\ge{\alpha_t^{\min}(Q)}$$ for all $Q\in{\mathcal{Q}}_t$. Thus $${\rho_t}(X)\ge{\widetilde \rho_t(X)}={\rho_t}(-{\rho_{t+1}}(X))$$ for all $X\in{L^{\infty}}$, due to representation (\[rd2\]).
\[corm\] Similar to Corollary \[coherent\], condition 3) of Theorem \[eqcharprud\] can be restated for a dynamic *coherent* risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ as follows: $${\mathcal{Q}}_t^0(Q) \supseteq {\left\{\,}
\newcommand{\rk}{\right\}}Q^1\oplus^{t+1} Q^2{\;\big|\;}Q^1\in{\mathcal{Q}}_{t, t+1}^0(Q), \;Q^2\in{\mathcal{Q}}_{t+1}^0(Q^1)\rk\quad (\mbox{resp.} \subseteq)$$ for all $t\in{\mathbb{T}}$ such that $t<T$ and all $Q\in{{\mathcal{M}}_1(P)}$.
The following proposition provides an additional equivalent characterization of rejection consistency, that can be viewed as an analogon of the supermartingale property 4) of Theorem \[eqchar\].
\[mrc\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then $({\rho_t}){_{t\in{\mathbb{T}}}}$ is rejection consistent if and only if the inequality $$\label{superm1}
E_Q\left[\,\rho_{t+1}(X)\,|\,{\mathcal{F}}_t\,\right]\le \rho_t(X)+{\alpha_{t, t+1}^{\min}(Q)}\qquad Q\text{-a.s.}$$ holds for all $Q\in{{\mathcal{M}}_1(P)}$ and all $t\in{\mathbb{T}}$ such that $t<T$. In this case the process $$U_t^Q(X):={\rho_t}(X)-\sum_{k=0}^{t-1}{\alpha_{k,k+1}^{\min}(Q)},\qquad t\in{\mathbb{T}}$$ is a $Q$-supermartingale for all $X\in{L^{\infty}}$ and all $Q\in{\mathcal{Q}}^f$, where $${\mathcal{Q}}^f:={\left\{\,}
\newcommand{\rk}{\right\}}Q\in{{\mathcal{M}}_1(P)}{\;\big|\;}E_Q\left[\sum_{k=0}^{t}{\alpha_{k,k+1}^{\min}(Q)}\right]<\infty\;\,\forall\,t\in{\mathbb{T}}\rk.$$
The proof of Proposition \[mrc\] is a special case of Theorem \[optdec\], which involves the notion of sustainability; cf. [@ipen7].
\[sustainable\] Let ${({\rho_t}){_{t\in{\mathbb{T}}}}}$ be a dynamic convex risk measure. We call a bounded adapted process $X=(X_t){_{t\in{\mathbb{T}}}}$ *sustainable with respect to the risk measure* ${({\rho_t}){_{t\in{\mathbb{T}}}}}$ if $${\rho_t}(X_t-X_{t+1})\le0\qquad\textrm{for all $t\in{\mathbb{T}}$ such that $t<T$}.$$
Consider $X$ to be a cumulative investment process. If it is sustainable, then for all $t\in{\mathbb{T}}$ the adjustment $X_{t+1}-X_t$ is acceptable with respect to ${\rho_t}$.
The next theorem characterizes sustainable processes in terms of a supermartingale inequality; it is a generalization of [@ipen7 Corollary 2.4.10].
\[optdec\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above ]{}and let $(X_t){_{t\in{\mathbb{T}}}}$ be a bounded adapted process. Then the following properties are equivalent:
1. The process $(X_t){_{t\in{\mathbb{T}}}}$ is sustainable with respect to the risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$.
2. For all $Q\in{{\mathcal{M}}_1(P)}$ and all $t\in{\mathbb{T}}, t\ge 1$, we have $$\label{superm2}
E_Q\left[\,X_{t}\,|\,{\mathcal{F}}_{t-1}\,\right]\le X_{t-1}+\alpha_{t-1,t}^{\min}(Q)\qquad Q\text{-a.s.}.$$
The proof of $1){\Rightarrow}2)$ follows directly from the definition of sustainability and the definition of the minimal penalty function.
To prove $2){\Rightarrow}1)$, let $(X_t){_{t\in{\mathbb{T}}}}$ be a bounded adapted process such that holds. In order to prove $$X_t-X_{t-1}=:A_t\in-{\mathcal{A}}_{t-1,t}\quad\mbox{for all}\quad t\in{\mathbb{T}}, t\ge 1,$$ suppose by way of contradiction that $A_t\notin-{\mathcal{A}}_{t-1,t}$. Since the set ${\mathcal{A}}_{t-1,t}$ is convex and weak$^\ast$-closed due to Remark \[abg\], the Hahn-Banach separation theorem (see, e.g., [@fs4 Theorem A.56 ]) ensures the existence of $Z\in L^1({\mathcal{F}}_t,P)$ such that $$\label{e15}
a:=\sup_{X\in{\mathcal{A}}_{t-1,t}}E[\,Z(-X)\,]<E[\,Z\,A_t\,]=:b<\infty.$$ Since $\lambda I_{\{Z<0\}}\in{\mathcal{A}}_{t-1,t}$ for every $\lambda\ge0$, (\[e15\]) implies $Z\ge0\;P$-a.s., and in particular $E[Z]>0$. Define a probability measure $Q\in{{\mathcal{M}}_1(P)}$ via $\frac{dQ}{dP}:=\frac{Z}{E[Z]}$ and note that, due to Lemma \[erwpf\] and , we have $$\label{e172}
E_Q[\alpha_{t-1,t}^{\min}(Q)]=\sup_{X\in{\mathcal{A}}_{t-1,t}}E_Q[\,(-X)\,]=\sup_{X\in{\mathcal{A}}_{t-1,t}}E[\,Z(-X)\,]\frac{1}{E[Z]}=\frac{a}{E[Z]}<\infty.$$ Moreover, (\[e15\]) and imply $$E_Q\left[\left(X_t-X_{t-1}-\alpha_{t-1,t}^{\min}(Q)\right)\right]=E[Z]\left(E[ZA_t]-E_Q\left[\alpha_{t-1,t}^{\min}(Q)\right]\right)= E[Z](b-a)>0,$$ which cannot be true if holds under $Q$.
In particular, property 2) of Theorem \[optdec\] implies that the process $$X_t-\sum_{k=0}^{t-1}{\alpha_{k,k+1}^{\min}(Q)},\qquad t\in{\mathbb{T}}$$ is a $Q$-supermartingale for all $Q\in{\mathcal{Q}}^f$, if $X$ is sustainable with respect to $({\rho_t})$. As shown in [@ipen7 Theorem 2.4.6, Corollary 2.4.8], this supermartingale property is equivalent to sustainability of $X$ under some additional assumptions.
Weak time consistency {#subsec:wc}
---------------------
In this section we characterize the weak notions of time consistency from Definition \[cons\]. Due to cash invariance, they can be restated as follows: A dynamic convex risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ is weakly acceptance (resp. weakly rejection) consistent, if and only if $${\rho_{t+1}}(X)\le0\quad(\mbox{resp.}\;\ge)\quad\Longrightarrow\quad{\rho_t}(X)\le0\quad(\mbox{resp.}\;\ge)$$ for any $X\in{L^{\infty}}$ and for all $t\in{\mathbb{T}}$ such that $t<T$. This means that if some position is accepted (or rejected) for any scenario tomorrow, it should be already accepted (or rejected) today. In this form, weak acceptance consistency was introduced in [@adehk7]. Both weak acceptance and weak rejection consistency appeared in [@Weber; @ros7].
Weak acceptance consistency was characterized in terms of acceptance sets in [@tu8 Corollary 3.6], and in terms of a supermartingale property of penalty functions in [@burg Lemma 3.17]. We summarize these characterizations in our present setting in the next proposition.
\[weaktc\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be a dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then the following properties are equivalent:
1. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is weakly acceptance consistent.
2. ${\mathcal{A}}_{t+1}\subseteq{\mathcal{A}}_t\quad$ for all $t\in{\mathbb{T}}$ such that $t<T$.
3. The inequality $$\label{supermart}
E_Q[\,{\alpha_{t+1}^{\min}(Q)}\,|\,F_t\,]\le{\alpha_t^{\min}(Q)}\quad{Q\mbox{-a.s.}}$$ holds for all $Q\in{{\mathcal{M}}_1(P)}$ and all $t\in{\mathbb{T}}$ such that $t<T$. In particular $({\alpha_t^{\min}(Q)}){_{t\in{\mathbb{T}}}}$ is a $Q$-supermartingale for all $Q\in{\mathcal{Q}}_0$.
The equivalence of 1) and 2) follows directly from the definition of weak acceptance consistency. Property 2) implies 3), since by Lemma \[erwpf\] $$\begin{aligned}
E_Q[\,{\alpha_{t+1}^{\min}(Q)}\,|\,F_t\,]&=\operatorname*{Q\text{-}ess\,sup}_{X_{t+1}\in{{\mathcal A}_{t+1}}}E_Q[-X_{t+1}|{\mathcal{F}}_t]\\
&\le\operatorname*{Q\text{-}ess\,sup}_{X\in{\mathcal{A}}_t}E_Q[-X|{\mathcal{F}}_t]={\alpha_t^{\min}(Q)}\qquad{Q\mbox{-a.s.}}\end{aligned}$$ for all $Q\in{{\mathcal{M}}_1(P)}$.
To prove that 3) implies 2), we fix $X\in{{\mathcal A}_{t+1}}$ and note that $$E_Q[-X|{\mathcal{F}}_{t+1}]\le{\alpha_{t+1}^{\min}(Q)}\quad{Q\mbox{-a.s.}}\quad\mbox{for all}\;\,Q\in{{\mathcal{M}}_1(P)}$$ by the definition of the minimal penalty function. Using (\[supermart\]) we obtain $$E_Q[-X|{\mathcal{F}}_{t}]\le E_Q[\,{\alpha_{t+1}^{\min}(Q)}\,|\,F_t\,]\le{\alpha_t^{\min}(Q)}\quad{Q\mbox{-a.s.}}$$ for all $Q\in{{\mathcal{M}}_1(P)}$, in particular for $Q\in{\mathcal{Q}}_t^f(P)$. Thus ${\rho_t}(X)\le0$ by .
A recursive construction {#recur}
------------------------
In this section we assume that the time horizon $T$ is finite. Then one can define a time consistent dynamic convex risk measure $({\widetilde {\rho_t}})_{t=0{,\ldots ,}T}$ in a recursive way, starting with an arbitrary dynamic convex risk measure $({\rho_t})_{t=0{,\ldots ,}T}$, via $$\label{recurs}
\begin{aligned}
{\widetilde \rho_T}(X)&:=\rho_T(X)=-X\\
{\widetilde \rho_t}(X)&:=\rho_t(-{\widetilde \rho_{t+1}}(X)),\quad t=0{,\ldots ,}T-1,\quad X\in{L^{\infty}}.
\end{aligned}$$ The recursive construction was introduced in [@cdk6 Section 4.2], and also studied in [@Samuel; @ck6]. It is easy to see that $({\widetilde {\rho_t}})_{t=0{,\ldots ,}T}$ is indeed a time consistent dynamic convex risk measure, and each ${\widetilde {\rho_t}}$ is [continuous from above ]{}if each ${\rho_t}$ has this property.
\[cheaper\] If the original dynamic convex risk measure $({\rho_t})_{t=0{,\ldots ,}T}$ is rejection (resp. acceptance) consistent, then the time consistent dynamic convex risk measure $({\widetilde {\rho_t}})_{t=0{,\ldots ,}T}$ defined via (\[recurs\]) lies below (resp. above) $({\rho_t})_{t=0{,\ldots ,}T}$, i.e. $${\widetilde {\rho_t}}(X)\le(\text{resp.}\ge){\rho_t}(X)\quad\text{for all $t=0{,\ldots ,}T$ and all $X\in{L^{\infty}}$}.$$ This can be easily proved by backward induction using Proposition \[rejrecursiveness\], monotonicity, and . Moreover, as shown in [@Samuel Theorem 3.10] in the case of rejection consistency, $({\widetilde {\rho_t}})_{t=0{,\ldots ,}T}$ is the biggest time consistent dynamic convex risk measure that lies below $({\rho_t})_{t=0{,\ldots ,}T}$.
For all $X\in{L^{\infty}}$, the process $({\widetilde {\rho_t}}(X))_{t=0{,\ldots ,}T}$ has the following properties: ${\widetilde \rho}_T(X)\ge-X$, and $$\label{sust}
{\rho_t}({\widetilde {\rho_t}}(X)-{\widetilde \rho}_{t+1}(X))=-{\widetilde {\rho_t}}(X)+{\rho_t}(-{\widetilde \rho}_{t+1}(X))=0\qquad\forall\; t=0{,\ldots ,}T-1,$$ by definition and cash invariance. In other words, the process $({\widetilde {\rho_t}}(X))_{t=0{,\ldots ,}T}$ covers the final loss $-X$ and is sustainable with respect to the original risk measure $({\rho_t})_{t=0{,\ldots ,}T}$. The next proposition shows that $({\widetilde {\rho_t}}(X))_{t=0{,\ldots ,}T}$ is in fact the smallest process with both these properties. This result is a generalization of [@ipen7 Proposition 2.5.2 ], and, in the coherent case, related to [@d6 Theorem 6.4].
\[smallest\] Let $({\rho_t})_{t=0{,\ldots ,}T}$ be a dynamic convex risk measure such that each ${\rho_t}$ is [continuous from above]{}. Then, for each $X\in{L^{\infty}}$, the risk process $({\widetilde {\rho_t}}(X))_{t=0{,\ldots ,}T}$ defined via (\[recurs\]) is the smallest bounded adapted process $(U_t)_{t=0{,\ldots ,}T}$ such that $(U_t)_{t=0{,\ldots ,}T}$ is sustainable with respect to $({\rho_t})_{t=0{,\ldots ,}T}$ and $U_T\ge-X$.
We have already seen that ${\widetilde {\rho_t}}_T(X)\ge-X$ and $({\widetilde {\rho_t}}(X))_{t=0{,\ldots ,}T}$ is sustainable with respect to $({\rho_t})_{t=0{,\ldots ,}T}$ due to (\[sust\]). Now let $(U_t)_{t=0{,\ldots ,}T}$ be another bounded adapted process with both these properties. We will show by backward induction that $$\label{ineq}
U_t\ge {\widetilde {\rho_t}}(X)\quad{P\mbox{-a.s.}}\qquad\forall\;t=0{,\ldots ,}T.$$ Indeed, we have $$U_T\ge-X={\widetilde \rho_T}(X)\qquad {P\mbox{-a.s.}}.$$ If (\[ineq\]) holds for $t+1$, Theorem \[optdec\] yields for all $Q\in{\mathcal{Q}}_t^f$: $$\begin{aligned}
U_t&\ge E_Q\left[\,U_{t+1}-{\alpha_{t, t+1}^{\min}(Q)}\,|\,{\mathcal{F}}_t\,\right]\nonumber\\&\ge E_Q\left[\,{\widetilde \rho_{t+1}}(X)-{\alpha_{t, t+1}^{\min}(Q)}\,|\,{\mathcal{F}}_t\,\right]\quad {P\mbox{-a.s.}}.\end{aligned}$$ Thus $$\begin{aligned}
U_t&\ge\operatorname*{ess\,sup}_{Q\in{\mathcal{Q}}_{t}^f} \left(E_Q\left[{\widetilde \rho_{t+1}}(X)|{\mathcal{F}}_t\right]-{\alpha_{t, t+1}^{\min}(Q)}\right)\\
&={\rho_t}(-{\widetilde \rho_{t+1}}(X))={\widetilde \rho_{t}}(X)\quad {P\mbox{-a.s.}},\end{aligned}$$ where we have used representation . This proves (\[ineq\]).
The recursive construction can be used to construct a time consistent dynamic Average Value at Risk, as shown in the next example.
It is well known that dynamic Average Value at Risk $(AV@R_{t,\lambda_t})_{t=0{,\ldots ,}T}$ (cf. Example \[avar\]) is not time consistent, and does not even satisfy weaker notions of time consistency from Definition \[cons\]; see, e.g., [@adehk7; @ros7]. Moreover, since $\alpha_0^{\min}(P)=0$ in this case, the set ${{\mathcal Q}^{\ast}}$ in is not empty, and [@fp6 Corollary 4.12] implies that there exists no time consistent dynamic convex risk measure $({\rho_t}){_{t\in{\mathbb{T}}}}$ such that each ${\rho_t}$ is continuous from above and $\rho_0=AV@R_{0,\lambda_0}$. However, for $T<\infty$, the recursive construction can be applied to $(AV@R_{t,\lambda_t})_{t=0{,\ldots ,}T}$ in order to modify it to a time consistent dynamic coherent risk measure $(\tilde{{\rho_t}})_{t=0{,\ldots ,}T}$. This modified risk measure takes the form $$\begin{aligned}
\tilde{{\rho_t}}(X)&= \operatorname*{ess\,sup}{\left\{\,}
\newcommand{\rk}{\right\}}E_Q[-X|{\mathcal{F}}_t]{\;\big|\;}Q\in{\mathcal{Q}}_t, \frac{Z^Q_{s+1}}{Z^Q_s}\leq \lambda_s^{-1}, s=t{,\ldots ,}T-1\rk \\
&= \operatorname*{ess\,sup}{\left\{\,}
\newcommand{\rk}{\right\}}E_P\left[-X\prod_{s=t+1}^{T}L_s{\;\big|\;}{\mathcal{F}}_t\right]{\;\big|\;}L_s\in L^{\infty}_s, 0\leq L_s\leq \lambda_s^{-1}, E[L_s|{\mathcal{F}}_{s-1}]=1, s=t+1{,\ldots ,}T\rk\nonumber\end{aligned}$$ for all $t=0{,\ldots ,}T-1$, where $Z^Q_t=\frac{dQ}{dP}|_{{\mathcal{F}}_t}$. This was shown, e.g., in [@ck6 Example 3.3.1].
The dynamic entropic risk measure {#entropic}
=================================
In this section we study time consistency properties of the dynamic entropic risk measure $${\rho_t}(X)=\frac{1}{{\gamma}_t}\log E[e^{-{\gamma}_t X}|{\mathcal{F}}_t],\qquad t\in{\mathbb{T}},\qquad X\in{L^{\infty}},$$ where the risk aversion parameter $\gamma_t$ satisfies $\gamma_t>0\, P$-a.s. and $\gamma_t, \frac{1}{\gamma_t}\in{L^{\infty}_t}$ for all $t\in{\mathbb{T}}$ (cf. Example \[ex:entr\]).
It is well known (see, e.g., [@dt5; @fp6]) that the conditional entropic risk measure $\rho_{t}$ has the robust representation with the minimal penalty function $\alpha_t$ given by $$\alpha_t(Q)=\frac{1}{\gamma_t}H_t(Q|P),\quad Q\in{\mathcal{Q}}_t,$$ where $H_t(Q|P)$ denotes the conditional relative entropy of $Q$ with respect to $P$ at time $t$: $$H_t(Q|P)=E_Q{\left[}\log\frac{dQ}{dP}{\;\big|\;}{\mathcal{F}}_t {\right]},\quad Q\in{\mathcal{Q}}_t.$$
The dynamic entropic risk measure with constant risk aversion parameter $\gamma_t=\gamma_0\in{\mathbb R}$ for all $t$ was studied in [@dt5; @cdk6; @fp6; @ck6]. It plays a particular role since, as proved in [@ks9], it is the only law invariant time consistent relevant dynamic convex risk measure.
In this section we consider an *adapted* risk aversion process $(\gamma_t){_{t\in{\mathbb{T}}}}$, that depends both on time and on the available information. As shown in the next proposition, the process $(\gamma_t){_{t\in{\mathbb{T}}}}$ determines time consistency properties of the corresponding dynamic entropic risk measure. This result corresponds to [@ipen7 Proposition 4.1.4], and generalizes [@Samuel Proposition 3.13].
\[riskav\] Let $({\rho_t}){_{t\in{\mathbb{T}}}}$ be the dynamic entropic risk measure with risk aversion given by an adapted process $({\gamma}_t){_{t\in{\mathbb{T}}}}$ such that $\gamma_t>0\,P$-a.s. and $\gamma_t, 1/{\gamma}_t\in{L^{\infty}_t}$. Then the following assertions hold:
1. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is rejection consistent if ${\gamma}_t\ge{\gamma}_{t+1}\:P$-a.s. for all $t\in{\mathbb{T}}$ such that $t<T$;
2. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is acceptance consistent if ${\gamma}_t\le{\gamma}_{t+1}\:P$-a.s. for all $t\in{\mathbb{T}}$ such that $t<T$;
3. $({\rho_t}){_{t\in{\mathbb{T}}}}$ is time consistent if ${\gamma}_t={\gamma}_0\in{\mathbb R}\: P$-a.s. for all $t\in{\mathbb{T}}$ such that $t<T$.
Moreover, assertions 1), 2) and 3) hold with “if and only if”, if ${\gamma}_t\in{\mathbb R}$ for all $t$, or if the filtration $({\mathcal{F}}_t){_{t\in{\mathbb{T}}}}$ is rich enough in the sense that for all $t$ and for all $B\in{\mathcal{F}}_t$ such that $P[B]>0$ there exists $A\subset B$ such that $A\notin{\mathcal{F}}_t$ and $P[A]>0$.
Fix $t\in{\mathbb{T}}$ and $X\in{L^{\infty}}$. Then $$\begin{aligned}
{\rho_t}(-{\rho_{t+1}}(X))&=\frac{1}{{\gamma}_t}\log\left(E\left[\exp\left\{\frac{{\gamma}_t}{{\gamma}_{t+1}}\log\left(E\left[e^{-{\gamma}_{t+1}X}|{\mathcal{F}}_{t+1}\right]\right)\right\}\big|{\mathcal{F}}_t\right]\right)\nonumber\\
&=\frac{1}{{\gamma}_t}\log\left(E\left[E\left[e^{-{\gamma}_{t+1}X}|{\mathcal{F}}_{t+1}\right]^{\frac{{\gamma}_t}{{\gamma}_{t+1}}}\big|{\mathcal{F}}_t\right]\right).\end{aligned}$$ Thus ${\rho_t}(-{\rho_{t+1}})={\rho_t}$ if ${\gamma}_t={\gamma}_{t+1}$ and this proves time consistency. Rejection (resp. acceptance) consistency follow by the generalized Jensen inequality that will be proved in Lemma \[jensen\]. We apply this inequality at time $t+1$ to the bounded random variable $Y:=e^{-{\gamma}_{t+1}X}$ and the ${{\mathcal B}}\left((0,\infty)\right)\otimes{\mathcal{F}}_{t+1}$-measurable function $$u\;:\;(0,\infty)\times\Omega\;\rightarrow\;{\mathbb R},\quad\quad u(x,\omega):=x^{\frac{{\gamma}_t(\omega)}{{\gamma}_{t+1}(\omega)}}.$$ Note that $u(\cdot,\omega)$ is convex if ${\gamma}_t(\omega)\ge{\gamma}_{t+1}(\omega)$ and concave if ${\gamma}_t(\omega)\le{\gamma}_{t+1}(\omega)$. Moreover, $u(X,\cdot)\in{L^{\infty}}$ for all $X\in{L^{\infty}}$ and $u(\cdot,\omega)$ is differentiable on $(0,\infty)$ with $$|u'(x,\cdot)|=\frac{{\gamma}_t}{{\gamma}_{t+1}}x^{\frac{{\gamma}_t}{{\gamma}_{t+1}}-1}\le ax^b\quad{P\mbox{-a.s.}}$$ for some $a,b\in{\mathbb R}$ if ${\gamma}_t\ge{\gamma}_{t+1}$, due to our assumption $\frac{{\gamma}_t}{{\gamma}_{t+1}}\in{L^{\infty}}$. On the other hand, for ${\gamma}_t\le{\gamma}_{t+1}$ we obtain $$|u'(x,\cdot)|=\frac{{\gamma}_t}{{\gamma}_{t+1}}x^{\frac{{\gamma}_t}{{\gamma}_{t+1}}-1}\le a\frac{1}{x^c}\quad{P\mbox{-a.s.}}$$ for some $a,c\in{\mathbb R}$. Thus the assumptions of Lemma \[jensen\] are satisfied and we obtain $${\rho_t}(-{\rho_{t+1}})\le{\rho_t}\quad\mbox{if}\quad{\gamma}_t\ge{\gamma}_{t+1}\quad P\mbox{-a.s.\;\; for all $t\in{\mathbb{T}}$ such that $t<T$}$$ and $${\rho_t}(-{\rho_{t+1}})\ge{\rho_t}\quad\mbox{if}\quad{\gamma}_t\le{\gamma}_{t+1}\quad P\mbox{-a.s.\;\; for all $t\in{\mathbb{T}}$ such that $t<T$}.$$ The “only if” direction for constant ${\gamma}_t$ follows by the classical Jensen inequality.
Now we assume that the sequence $({\rho_t}){_{t\in{\mathbb{T}}}}$ is rejection consistent and our assumption on the filtration $({\mathcal{F}}_t){_{t\in{\mathbb{T}}}}$ holds. We will show that the sequence $({\gamma}_t){_{t\in{\mathbb{T}}}}$ is decreasing in this case. Indeed, for $t\in{\mathbb{T}}$ such that $t<T$, consider $B:=\{\gamma_t<\gamma_{t+1}\}$ and suppose that $P\left[ B \right]>0$. Our assumption on the filtration allows us to choose $A \subset B$ with $P\left[ B \right]>P\left[ A \right]>0$ and $A\notin{\mathcal{F}}_{t+1}$. We define a random variable $X:=-xI_{A}$ for some $x>0$. Then $$\begin{aligned}
\rho_{t}( -\rho_{t+1}(X))&=\frac{1}{\gamma_t}\log \left( E\left[ \exp\left( \frac{\gamma_t}{\gamma_{t+1}}\log \left( E\left[ e^{ \gamma_{t+1}x I_{A}}\big |{\mathcal{F}}_{t+1} \right] \right) \right) \big|{\mathcal{F}}_t \right] \right)\\
&=\frac{1}{\gamma_t}\log \left( E\left[ \exp\left( \frac{\gamma_t}{\gamma_{t+1}} I_{B}\log \left( E\left[ e^{ \gamma_{t+1}x I_{A}}\big |{\mathcal{F}}_{t+1} \right] \right) \right) \big|{\mathcal{F}}_t \right] \right),\end{aligned}$$ where we have used that $A\subset B$. Setting $$Y:=E\left[ e^{ \gamma_{t+1}x I_{A}}\big |{\mathcal{F}}_{t+1} \right]\\
=e^{ \gamma_{t+1}x }P\left[ A |{\mathcal{F}}_{t+1} \right]+P\left[ A^c |{\mathcal{F}}_{t+1} \right]$$ and bringing $\frac{{\gamma}_t}{{\gamma}_{t+1}}$ inside of the logarithm we obtain $$\label{a1}
\rho_{t}\left( -\rho_{t+1}\left( X \right) \right)=\frac{1}{\gamma_t}\log \left( E\left[ \exp\left( I_{B}\log \left( Y^{\frac{\gamma_t}{\gamma_{t+1}} I_{B}} \right) \right) \big|{\mathcal{F}}_t \right] \right).$$ The function $x \mapsto x^{\gamma_{t}(\omega)/\gamma_{t+1}(\omega)}$ is strictly concave for almost each $\omega\in B$, and thus $$\begin{aligned}
\label{a2}
Y^{\frac{\gamma_t}{\gamma_{t+1}} }&=\left(e^{\gamma_{t+1}x }P\left[ A |{\mathcal{F}}_{t+1} \right]+(1-P\left[ A |{\mathcal{F}}_{t+1} \right])\right)^{\frac{\gamma_t}{\gamma_{t+1}}}\nonumber\\
&\ge e^{ \gamma_{t}x }P\left[ A |{\mathcal{F}}_{t+1}\right]+(1-P\left[ A |{\mathcal{F}}_{t+1} \right])\qquad P\mbox{-a.s on}\;B,\end{aligned}$$ with strict inequality on the set $$C:=\left\{P\left[ A |{\mathcal{F}}_{t+1}\right]>0\right\}\cap \left\{P\left[ A |{\mathcal{F}}_{t+1}\right]<1\right\}\cap B.$$ Our assumptions $P[A]>0, \,A\subset B$ and $A\notin{\mathcal{F}}_{t+1}$ imply $P[C]>0$ and using $$\label{a3}
e^{ \gamma_{t}x }P\left[ A |{\mathcal{F}}_{t+1}\right]+(1-P\left[ A |{\mathcal{F}}_{t+1} \right])=E\left[e^{{\gamma}_txI_A}|{\mathcal{F}}_{t+1}\right]$$ we obtain from (\[a1\]), (\[a2\]) and (\[a3\]) $$\label{0}
\rho_{t}\left( -\rho_{t+1}\left( X \right) \right)\ge\frac{1}{\gamma_t}\log \left( E\left[ \exp\left( I_{B}\log \left(E\left[e^{\gamma_{t}x I_{A}}|{\mathcal{F}}_{t+1} \right] \right) \right) \big|{\mathcal{F}}_t \right] \right),$$ with the strict inequality on some set of positive probability due to strict monotonicity of the exponential and the logarithmic functions. For the right hand side of (\[0\]) we have $$\begin{aligned}
\lefteqn{\frac{1}{\gamma_t}\log \left( E\left[ \exp\left( I_{B}\log \left(E\left[e^{\gamma_{t}x I_{A}}|{\mathcal{F}}_{t+1} \right] \right) \right) \big|{\mathcal{F}}_t \right] \right)=}\\
&=\frac{1}{\gamma_t}\log \left( E\left[ I_{B}E\left[e^{ \gamma_{t}x I_{A} }|{\mathcal{F}}_{t+1} \right]+I_{B^c} \big|{\mathcal{F}}_t \right] \right)\\
&=\frac{1}{\gamma_t}\log \left( E\left[ \exp\left( \gamma_{t}x I_{A} \right)\big|{\mathcal{F}}_t \right] \right)\\
&=\rho_{t}\left( X \right),\end{aligned}$$ where we have used $A\subset B$ and $B\in{\mathcal{F}}_{t+1}$. This is a contradiction to rejection consistency of $({\rho_t}){_{t\in{\mathbb{T}}}}$, and we conclude that ${\gamma}_{t+1}\le{\gamma}_t$ for all $t$. The proof in the case of acceptance consistency follows in the same manner. And since time consistent dynamic risk measure is both acceptance and rejection consistent, we obtain ${\gamma}_{t+1}={\gamma}_t$ for all $t$.
The following lemma concludes the proof of Proposition \[riskav\].
\[jensen\] Let $(\Omega,{\mathcal{F}}, P)$ be a probability space and ${\mathcal{F}}_t\subseteq {\mathcal{F}}$ a $\sigma$-field. Let $I\subseteq{\mathbb R}$ be an open interval and $$u\;:\;I\times\Omega\;\rightarrow\;{\mathbb R}$$ be a ${{\mathcal B}}\left(I\right)\otimes{\mathcal{F}}_{t}$-measurable function such that $u(\cdot, \omega)$ is convex (resp. concave) and finite on $I$ for $P$-a.e. $\omega$. Assume further that $$|u_+'(x,\cdot)|\le c(x)\quad P\mbox{-a.s. with some}\;\;c(x)\in{\mathbb R}\;\,\mbox{for all}\;\;x\in I,$$ where $u_+'(\cdot,\omega)$ denotes the right-hand derivative of $u(\cdot,\omega)$. Let $X\;:\;\Omega\;\rightarrow\;[a,b]\subseteq I$ be an ${\mathcal{F}}$-measurable bounded random variable such that $E\left[\,|u(X,\, )|\,\right]<\infty$. Then $$E\left[\,u(X,\, )\,|\,{\mathcal{F}}_t\,\right]\ge u\left(E[X|{\mathcal{F}}_t],\, \right)\quad(\mbox{resp}\le)\quad{P\mbox{-a.s.}}.$$
We will prove the assertion for the convex case; the concave one follows in the same manner. Fix $\omega\in\Omega$ such that $u(\cdot, \omega)$ is convex. Due to convexity we obtain for all $x_0\in I$ $$u(x,\omega)\ge u(x_0,\omega)+u_+'(x_0,\omega)(x-x_0)\quad \mbox{for all}\quad x\in I.$$ Take $x_0=E[X|{\mathcal{F}}_t](\omega)$ and $x=X(\omega)$. Then $$\label{dk}
u(X(\omega),\omega)\ge u(E[X|{\mathcal{F}}_t](\omega),\omega)+u_+'(E[X|{\mathcal{F}}_t](\omega),\omega)(X(\omega)-E[X|{\mathcal{F}}_t](\omega))$$ for $P$-almost all $\omega\in\Omega$. Note further that ${{\mathcal B}}\left(I\right)\otimes{\mathcal{F}}_{t}$-measurability of $u$ implies ${{\mathcal B}}\left(I\right)\otimes{\mathcal{F}}_{t}$-measurability of $u_+$. Thus $$\omega\,\rightarrow\, u(E[X|{\mathcal{F}}_t](\omega),\omega)\quad\mbox{and}\quad \omega\,\rightarrow\,u_+'(E[X|{\mathcal{F}}_t](\omega),\omega)$$ are ${\mathcal{F}}_t$-measurable random variables, and $\omega\rightarrow u(X(\omega),\omega)$ is ${\mathcal{F}}$-measurable. Moreover, due to our assumption on $X$, there are constants $a,b\in I$ such that $a\le E[X|{\mathcal{F}}_t]\le b\;P$-a.s.. Since $u_+'(\cdot,\omega)$ is increasing by convexity, by using our assumption on the boundedness of $u_+'$ we obtain $$-c(a)\le u_+'(a,\omega)\le u_+'(E[X|{\mathcal{F}}_t],\omega)\le u_+'(b,\omega)\le c(b),$$ i.e. $u_+'(E[X|{\mathcal{F}}_t],\,)$ is bounded. Since $E\left[\,|u(X,\, )|\,\right]<\infty$, we can build conditional expectation on the both sides of (\[dk\]) and we obtain $$\begin{aligned}
E[\,u(X,\,)\,|\,{\mathcal{F}}_t\,]&\ge E\left[\,u(E[X|{\mathcal{F}}_t],\,)+u_+'(E[X|{\mathcal{F}}_t],\,)(X-E[X|{\mathcal{F}}_t])\,|\,{\mathcal{F}}_t\,\right]\\
&=E\left[\,u(E[X|{\mathcal{F}}_t],\,)\,|\,{\mathcal{F}}_t\,\right]\quad{P\mbox{-a.s.}},\end{aligned}$$ where we have used ${\mathcal{F}}_t$-measurability of $u(E[X|{\mathcal{F}}_t],\,)$ and of $u_+'(E[X|{\mathcal{F}}_t],\,)$ and the boundedness of $u_+'(E[X|{\mathcal{F}}_t],\,)$. This proves our claim.
[20]{}
Acciaio, Beatrice and F[ö]{}llmer, Hans and Penner, Irina (2010). Risk assessment for uncertain cash flows: Model ambiguity, and discounting ambiguity, and the role of bubbles. Submitted Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David (1997). Thinking coherently, [*RISK*]{} [**10**]{}, 68–71 Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David (1999). Coherent measures of risk, [*Math. Finance*]{} [**9/3**]{}, [203–228]{} Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David and Ku, Hyejin (2007). Coherent multiperiod risk adjusted values and Bellman’s principle, [*Ann. Oper. Res.*]{} [**152**]{}, 5–22 (2004). [Optimal derivatives design under dynamic risk measures]{}, [*Mathematics of Finance, Contemporary Mathematics (A.M.S. Proceedings)*]{}, [13–26]{} Bion-Nadal, Jocelyne (2004). Conditional risk measure and robust representation of convex conditional risk measures. CMAP preprint 557, Ecole Polytechnique Palaiseau Bion-Nadal, Jocelyne (2008). Dynamic risk measures: time consistency and risk measures from [BMO]{} martingales, [*Finance Stoch.*]{} [**12/2**]{}, 219–244 Bion-Nadal, Jocelyne (2008). Time consistent dynamic risk processes, [*Stochastic Processes and their Applications*]{} [**119**]{}, [633–654]{} Burgert, Christian (2005). Darstellungssätze fuer statische und dynamische [R]{}isikoma[ß]{}e mit [A]{}nwendungen. Universität Freiburg Cheridito, Patrick and Delbaen, Freddy and Kupper, Michael (2004). Coherent and convex monetary risk measures for bounded càdlàg processes, [*Stochastic Process. Appl.*]{} [**112/1**]{}, [1–22]{} Cheridito, Patrick and Delbaen, Freddy and Kupper, Michael (2005). Coherent and convex monetary risk measures for unbounded càdlàg processes, [*Finance Stoch.*]{} [**9/3**]{}, [369–387]{} Cheridito, Patrick and Delbaen, Freddy and Kupper, Michael (2006). Dynamic monetary risk measures for bounded discrete-time processes, [*Electron. J. Probab.*]{} [**11/3**]{}, 57–106 Cheridito, Patrick and Kupper, Michael (2006). Composition of time-consistent dynamic monetary risk measures in discrete time. Preprint Delbaen, Freddy (2000). Coherent risk measures. Cattedra Galileiana, Scuola Normale Superiore, Classe di Scienze, Pisa Delbaen, Freddy (2006). The structure of m-stable sets and in particular of the set of risk neutral measures. In: In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math. [**1874**]{}, p. [215–258]{}, [Springer]{}, [Berlin]{} Delbaen, Freddy and Peng, Shige and Rosazza Gianin, Emanuela. Representation of the penalty term of dynamic concave utilities, [*Finance Stoch.*]{}, to appear Detlefsen, Kai and Scandolo, Giacomo (2005). Conditional and dynamic convex risk measures, [*Finance Stoch.*]{} [**9/4**]{}, [539–561]{} Drapeau, Samuel (2006). Dynamics of Optimized Certainty Equivalents and $\varphi$-Divergence. Humboldt-Universität zu Berlin El Karoui, Nicole and Ravanelli, Claudia (2009). Cash subadditive risk measures and interest rate ambiguity, [*Math. Finance*]{} [**19/4**]{}, [561–590]{} F[ö]{}llmer, Hans and Penner, Irina (2006). Convex risk measures and the dynamics of their penalty functions, [*Statist. Decisions*]{} [**24/1**]{}, [61–96]{} F[ö]{}llmer, Hans and Schied, Alexander (2002). Convex measures of risk and trading constraints, [*Finance Stoch.*]{} [**6/4**]{}, [429–447]{} F[ö]{}llmer, Hans and Schied, Alexander (2004). [*Stochastic finance. An introduction in discrete time*]{}. De Gruyter Studies in Mathematics [**27**]{}, [Berlin]{} Frittelli, Marco and Rosazza Gianin, Emanuela (2002). Putting order in risk measures, [*Journal of Banking Finance*]{} [**26/7**]{}, 1473–1486 Frittelli, Marco and Scandolo, Giacomo (2006). Risk measures and capital requirements for processes, [*Math. Finance*]{} [**16/4**]{}, [589–612]{} Jobert, Arnaud and Rogers, L. C. G. (2008). Valuations and dynamic convex risk measures, [*Math. Finance*]{} [**18/1**]{}, [1–22]{} Klöppel, Susanne and Schweizer, Martin (2005). Dynamic Utility Indifference Valuation Via Convex Risk Measure. Working Paper N 209: National Centre of Competence in Research Financial Valuation and Risk Management (2009). [Representation Results for Law Invariant Time Consistent Functions]{}, [*Mathematics and Financial Economics*]{} [**2/3**]{}, [189–210]{} (2005). [Dynamic exponential utility indifference valuation]{}, [*Ann. Appl. Probab.*]{} [**15/3**]{}, [2113–2143]{} Penner, Irina (2007). Dynamic convex risk measures: time consistency, prudence, and sustainability. Humboldt-Universität zu Berlin. Riedel, Frank (2004). Dynamic coherent risk measures, [*Stochastic Process. Appl.*]{} [**112/2**]{}, [185–200]{} (2005). [Coherent acceptability measures in multiperiod models]{}, [*Math. Finance*]{} [**15/4**]{}, [589–612]{} (2007). [Time consistency conditions for acceptability measures, with an application to [T]{}ail [V]{}alue at [R]{}isk]{}, [*Insurance Math. Econom.*]{}, [**40/2**]{}, [209–230]{} (2008). [Update rules for convex risk measures]{}, [*Quant. Finance*]{} [**8**]{}, [833–843]{} (2006). [Some results on dynamic risk measures]{}. [Ludwig-Maximilians-Universität München]{} (2006). [Distribution-Invariant Risk Measures, Information, and Dynamic Consistency]{}, [*Mathematical Finance*]{} [**16/2**]{}, [419-441]{}
[^1]: Department of Economy, Finance and Statistics, University of Perugia, Via A. Pascoli 20, 06123 Perugia, Italy. Email: beatrice.acciaio@stat.unipg.it. Financial support from the European Science Foundation (ESF) “Advanced Mathematical Methods for Finance" (AMaMeF) under the exchange grant 2281, and hospitality of Vienna University of Technology are gratefully acknowledged.
[^2]: Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany. Email: penner@math.hu-berlin.de. Supported by the DFG Research Center <span style="font-variant:small-caps;">Matheon</span> “Mathematics for key technologies”. Financial support from the European Science Foundation (ESF) “Advanced Mathematical Methods for Finance" (AMaMeF) under the short visit grant 2854 is gratefully acknowledged.
|
---
abstract: |
Logit models are usually applied when studying individual travel behavior, i.e., to predict travel mode choice and to gain behavioral insights on traveler preferences. Recently, some studies have applied machine learning to model travel mode choice and reported higher out-of-sample predictive accuracy than traditional logit models (e.g., multinomial logit). However, little research focuses on comparing the interpretability of machine learning with logit models. In other words, how to draw behavioral insights from the high-performance “black-box" machine-learning models remains largely unsolved in the field of travel behavior modeling.
This paper aims at providing a comprehensive comparison between the two approaches by examining the key similarities and differences in model development, evaluation, and behavioral interpretation between logit and machine-learning models for travel mode choice modeling. To complement the theoretical discussions, the paper also empirically evaluates the two approaches on the stated-preference survey data for a new type of transit system integrating high-frequency fixed-route services and ridesourcing. The results show that machine learning can produce significantly higher predictive accuracy than logit models. Moreover, machine learning and logit models largely agree on many aspects of behavioral interpretations. In addition, machine learning can automatically capture the nonlinear relationship between the input features and choice outcomes. The paper concludes that there is great potential in merging ideas from machine learning and conventional statistical methods to develop refined models for travel behavior research and suggests some new research directions.
address:
- 'H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology'
- 'Taubman College of Architecture and Urban Planning, University of Michigan'
- 'Department of Electrical Engineering and Computer Science, University of Michigan'
author:
- Xilei Zhao
- Xiang Yan
- Alan Yu
- Pascal Van Hentenryck
bibliography:
- 'manuscript.bib'
title: |
Modeling Stated Preference for Mobility-on-Demand Transit:\
A Comparison of Machine Learning and Logit Models
---
machine learning ,mixed logit ,mobility-on-demand ,stated-preference survey ,travel mode choice ,public transit
Introduction {#sec1}
============
Emerging shared mobility services, such as car sharing, bike sharing, ridesouring, and micro-transit, have rapidly gained popularity across cities and are gradually changing how people move around. Predicting individual preferences for these services and the induced changes in travel behavior is critical for transportation planning. Traditionally, travel behavior research has been primarily supported by discrete choice models (a type of statistical models), most notably the logit family such as the multinomial logit (MNL), the nested logit model and the mixed logit model. In recent years, as machine learning has become pervasive in many fields, there has been a growing interest in its application to model individual choice behavior.
Machine learning and conventional statistical models seek to understand the data structure based on different approaches. The logit models, like many other statistical models, are based on a theoretical foundation which is mathematically proven, but this requires the input data to satisfy strong assumptions such as the random utility maximization decision rule and a particular type of error-term distribution [@ben1985discrete]. On the other hand, machine learning relies on computers to probe the data for its structure, without a theory of what the underlying data structure should look like. In other words, while a logit model presupposes a certain type of structure of the data with its behavioral and statistical assumptions, machine learning, on the other hand, “lets the data speak for itself" and hence allows forming more flexible model structures, which can often lead to higher predictive capability (i.e., higher out-of-sample predictive accuracy).
A number of recent empirical studies have verified that machine learning can outperform logit models in terms of predictive capability [e.g. @xie2003work; @zhang2008travel; @hagenauer2017comparative; @wang2018machine; @lheritier2018airline]. Intuitively, a machine-learning model that predicts well should also be able to offer good interpretation by accurately representing the underlying data structure. However, existing studies that apply machine learning for travel-mode choice modeling have mostly focused on prediction, with much less attention being devoted to interpretation. More specifically, these studies rarely apply interpretable machine learning tools such as partial dependent plots and variable importance to extract behavioral findings from machine learning and compare/validate these findings with those obtained from traditional logit models. In mode-choice modeling applications, however, the behavioral interpretation of the results is as important as the prediction problem, since it offers valuable insights for transportation planning and policy making.
Furthermore, the existing literature comparing logit models and machine learning for modeling travel mode choice has two other limitations. First, the comparisons were usually made between the MNL model, the simplest logit model, and machine-learning algorithms of different complexity. In cases where the assumption of independence of irrelevant alternatives (IIA) is violated, such as when panel data (i.e., data containing multiple mode choices made by the same individuals) are examined, more advanced logit models such as the mixed logit model should be considered. Second, existing studies rarely discuss the fundamental differences in the application of machine-learning methods and logit models to travel mode choice modeling. The notable differences between the two approaches in the input data structure and data needs, the modeling of alternative-specific attributes, and the form of predicted outputs carry significant implications for model comparison. These differences and their implications, although touched on by some researchers such as @omrani2013prediction, have not been thoroughly examined.
This paper tries to bridge these gaps: It provides a comprehensive comparison of logit models and machine learning in modeling travel mode choices and also an empirical evaluation of the two approaches based on stated-preference (SP) survey data on a proposed mobility-on-demand transit system, i.e., an integrated transit system that runs high-frequency buses along major corridors and operates on-demand shuttles in the surrounding areas [@TS2017]. The paper first discusses the fundamental differences in the practical applications of the two types of methods, with a particular focus on the implications of the predictive performance of each approach and their capabilities to facilitate behavioral interpretations. The paper then compares the performance of two logit models (MNL and mixed logit) and seven machine-learning classifiers, including Naive Bayes (NB), classification and regression trees (CART), boosting trees (BOOST), bagging trees (BAG), random forest (RF), SVM, and NN, in predicting individual choices of four travel modes and their respective market shares. Moreover, the paper compares behavioral interpretations of two logit models (MNL and mixed logit) and two machine-learning models (RF and NN). The results show that RF can produce higher out-of-sample prediction accuracy than logit models and NN. Moreover, machine learning can offer consistent behavioral interpretations compared to logit models. In particular, RF can automatically capture nonlinearities between the input data and the choice outcome.
The rest of the paper is organized as follows. The next section provides a brief review of the literature in modeling mode choices using logit and machine-learning models. Section 3 explains the fundamentals of the logit and machine-learning models, including model formulation and input data structures, model development and evaluation, and model interpretation and application. Section 4 introduces the data used for empirical evaluation and Section 5 describes the logit and machine-learning models examined and their specifications. Section 6 evaluates these models in terms of predictive capability and interpretability. Lastly, Section 7 concludes by summarizing the findings, identifying the limitations of the paper, and suggesting future research directions. Table \[tab:acy\] presents the list of abbreviations and acronyms used in this paper.
MNL Multinomial logit
------- -----------------------------------------
NB Naive Bayes
CART Classification and regression trees
RF Random forest
BOOST Boosting trees
BAG Bagging trees
SVM Support vector machines
NN Neural networks
AIC Akaike information criterion
BIC Bayesian information criterion
Min Minimum
Max Maximum
SD Standard deviation
SP Stated-preference
RP Revealed-preference
IIA Independence of irrelevant alternatives
PT Public transit
: List of Abbreviations and Acronyms.
\[tab:acy\]
Literature Review {#sec2}
=================
The logit family is a class of econometric models based on random utility maximization [@ben1985discrete]. Due to their statistical foundations and their capability to represent individual choice behavior realistically, the MNL model and its extensions have dominated travel behavior research ever since its formulation in the 1970s [@mcfadden1973conditional]. The MNL model is frequently challenged for its major assumption, the IIA property, and its inability to account for taste variations among different individuals. To address these limitations, researchers have developed important extensions to the MNL model such as the nested logit model and more recently the mixed logit model. The mixed logit model, in particular, has received much interest in recent years: Unlike the MNL model, it does not require the IIA assumption, can accommodate preference heterogeneity, and may significantly improve the MNL behavioral realism in representing consumer choices [@hensher2003mixed].
Mode-choice modeling can also be viewed as a *classification* problem, providing an alternative to logit models. A number of recent publications have suggested that machine-learning classifiers such as CART, NN, and SVM are effective in modeling individual travel behavior [@xie2003work; @zhang2008travel; @omrani2013prediction; @omrani2015predicting; @hagenauer2017comparative; @golshani2018modeling; @wang2018machine; @wong2018discriminative; @lheritier2018airline]. These studies generally found that machine-learning classifiers outperform traditional logit models in predicting travel-mode choices. For example, @xie2003work applied CART and NN to model travel mode choices for commuting trips taken by residents in the San Francisco Bay area. These machine-learning methods exhibited better performance than the MNL model in terms of prediction. Based on data collected in the same area, @zhang2008travel reported that SVM can predict commuter travel mode choice more accurately than NN and MNL. More recently, @lheritier2018airline found that the RF model outperforms the standard and the latent class MNL model in terms of accuracy and computation time, with less modeling effort.
It is not surprising that machine-learning classifiers can perform better than logit models in predictive tasks. Unlike logit models that make strong mathematical assumptions (i.e. constraining the model structure and assuming a certain distribution in the error term a priori), machine learning allows for more flexible model structures, which can reduce the model’s incompatibility with the empirical data [@xie2003work; @christopher2016pattern]. More fundamentally, the development of machine learning prioritizes predictive power, whereas advances in logit models are mostly driven by refining model assumptions, improving model fit, and enhancing the behavioral interpretation of the model results [@brownstone1998forecasting; @hensher2003mixed]. In other words, the development of logit models prioritizes parameter estimation (i.e. obtaining better model parameter estimates that underline the relationship between the input features and the output variable) and pay less attention to increasing the model’s out-of-sample predictive accuracy [@mullainathan2017machine]. In fact, recent studies have shown that the mixed logit model, despite resulting in substantial improvements in overall model fit, often resulted in poorer prediction accuracy compared to the simpler and more restrictive MNL model [@cherchi2010validation].
While recognizing the superior predictive power of machine-learning models, researchers often think that they have weak explanatory power [@mullainathan2017machine]. In other words, machine-learning models are often regarded as “not interpretable.” Machine-learning studies rarely apply model outputs to facilitate behavioral interpretations, i.e., to test the response of the output variable or to changes in the input variables in order to generate findings on individual travel behavioral and preferences [@karlaftis2011statistical]. The outputs of many machine-learning models are indeed not directly interpretable as one may need hundreds of parameters to describe a deep NN or hundreds of decision trees to understand a RF model. Nonetheless, with recent development in interpretable/explainable machine learning, a wide range of machine learning interpretation tools have been invented to extract knowledge from the black-box models to facilitate decision-making [@golshani2018modeling; @wager2018estimation; @molnar2018interpretable]. In particular, @zhao2019modeling applied interpretable machine learning to model and assess heterogeneous travel behavior. Examining these behavioral outputs from machine learning models could shed light on what factors are driving prediction decisions and also the fundamental question of whether machine learning is appropriate for behavioral analysis.
Prediction and behavioral analysis are equally important in travel behavior studies. While the primary goal of some applications is to accurately predict mode choices (and investigators are usually more concerned about the prediction of aggregate market share for each mode than about the prediction of individual choices), other studies may be more interested in quantifying the impact of different trip attributes on travel mode choices. To our knowledge, mode-choice applications that focus on behavioral outputs such as elasticity, marginal effect, value of time, and willingness-to-pay measures have received even more attention than those that focus on predicting individual mode choice or aggregate market shares in the literature. This paper thus extends current literature by comparing the behavioral findings from logit models and machine-learning methods, beyond the existing studies that primarily focus on their predictive accuracy.
Finally, this paper points out other differences in the practical applications of these two approaches that have bearings on model outputs and performance, including their input data structure and data needs, the treatment of alternative-specific attributes, and the forms of the predicted outputs. Discussions of these differences are largely absent from the current literature that compares the application of logit models and machine-learning algorithms in travel behavior research.
Fundamentals of the Logit and Machine-Learning Models
=====================================================
This section discusses the fundamentals of the logit and machine-learning models. Table \[tab:symbol\_des\] presents the list of symbols and notations used in the paper and Table \[tab:comparison\] summarizes the comparison between logit and machine-learning models from various angles. The rest of this section describes this comparison in detail.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
**Symbols & **Description\
$K$ & Total number of alternatives\
$N$ & Total number of observations\
$P$ & Total number of features\
$\boldsymbol{X}$ & Input data for logit models containing $P$ features with $N$ observations for $K$ alternatives\
$\boldsymbol X_{k,p}$ & Feature $p$ for alternative $k, k = 1, ..., K$ of $\boldsymbol X$\
$\boldsymbol{X}_{k,-p}$ & All the features except $p$ for alternative $k, k = 1, ..., K$ of $\boldsymbol X$\
$\boldsymbol{X}_{ik}$ & A row-vector for the $i$th observation for alternative $k, k = 1, ..., K$\
$\boldsymbol{X}_k$ & Input data for alternative $k$, $\boldsymbol{X}_k = [\boldsymbol{X}_{.k1}; ...; \boldsymbol{X}_{.kP}]$ where $\boldsymbol{X}_{.kp} = [X_{1kp}, ..., X_{Nkp}]$\
$\boldsymbol{X}_i$ & The $i$th observation of $\boldsymbol{X}$, $\boldsymbol{X}_i = [\boldsymbol{X}_{i.1}, ..., \boldsymbol{X}_{i.P}]$ where $\boldsymbol{X}_{i.p} = [X_{i1p}; ...; X_{iKp}]$\
$\boldsymbol{X}_p$ & The feature $p$ of $\boldsymbol{X}$, $\boldsymbol{X}_p = [\boldsymbol{X}_{1.p}, ..., \boldsymbol{X}_{N.p}]$ where $\boldsymbol{X}_{i.p} = [X_{i1p}; ...; X_{iKp}]$\
$\boldsymbol{Z}$ & Input data for machine-learning models containing $P$ features and $N$ observations\
$\boldsymbol Z_{p}$ & Feature $p$ of $\boldsymbol{Z}$\
$\boldsymbol Z_{-p}$ & All the features except $p$ of $\boldsymbol{Z}$\
$\boldsymbol{Z}_i$ & $i$th observation of $\boldsymbol{Z}, \boldsymbol{Z}_i = [Z_{i1}, ..., Z_{iP}$\]\
$U_k(\boldsymbol{X}_k|\boldsymbol{\beta}_k)$ & Utility function for mode $k$\
$\boldsymbol{\beta}_k$ & Parameter vector for alternative $k$ of MNL model\
$\boldsymbol{\beta}$ & Parameter matrix of MNL model, $\boldsymbol{\beta} = [\boldsymbol{\beta}_1,..., \boldsymbol{\beta}_K]$\
$\hat{\boldsymbol{\beta}}$ & Estimated parameter matrix of MNL model\
$\boldsymbol{\varepsilon}_{k}$ & Random error for alternative $k$ of MNL model\
$\boldsymbol{Y}$ & Output mode choice data\
$\hat{Y}_i$ & Estimated mode choice for observation $i$\
$\boldsymbol{\theta}$ & Parameter or hyperparameter vector for machine-learning models\
$\hat{\boldsymbol{\theta}}$ & Estimated parameter or hyperparameter vector\
$f(\boldsymbol{Z}|\boldsymbol{\theta})$ & Machine-learning models based on $\boldsymbol{Z}$ and $\boldsymbol{\theta}$\
$p_{ik}$ & Probability of choosing alternative $k$ of observation $i$\
$\hat{p}_{ik}$ & Predicted probability for choosing alternative $k$ of observation $i$\
$I_k(\hat{Y}_i)$ & Indicator function that equals to 1 if $\hat{Y}_i = k$\
$P_k(\boldsymbol{X}|\hat{ \boldsymbol{\beta}})$ & Aggregate level prediction for mode $k$ based on $\boldsymbol{X}$ and $\hat{ \boldsymbol{\beta}}$ for logit models\
$Q_k(\boldsymbol{Z}|\hat{ \boldsymbol{\theta}})$ & Aggregate level prediction for mode $k$ based on $\boldsymbol{Z}$ and $\hat{ \boldsymbol{\theta}}$ for machine-learning models\
$E_k(\cdot)$ & Arc elasticity for alternative $k$\
$M_k(\cdot)$ & Marginal effect for alternative $k$\
$\Delta$ & Constant\
****
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
: List of Symbols and Notations Used in the Paper
\[tab:symbol\_des\]
\[tab:comparison\]
Model Development
-----------------
Logit models and machine-learning models approach the mode choice prediction problem from different perspectives. Logit models view the mode choice problem as individuals selecting a mode from a set of travel options in order to maximize their utility. Under the random utility maximization framework, the model assumes that each mode provides a certain level of (dis)utility to a traveler, and specifies, for each mode, a utility function with two parts: A component to represent the effects of observed variables and a random error term to represent the effects of unobserved factors [@ben1985discrete]. For example, the utility of choosing mode $k$ under the MNL model can be defined as $$U_{k}(\boldsymbol{X}_k|\boldsymbol{\beta}_k) = \boldsymbol{\beta}_k^T \boldsymbol{X}_{k} + \boldsymbol \varepsilon_{k},$$ where $\boldsymbol{\beta}_k$ are the coefficients to be estimated and $\boldsymbol \varepsilon_{k}$ is the unobserved random error for choosing mode $k$. Different logit models are formed by specifying different types of error terms and different choices of coefficients on the observed variables. For instance, assuming a Gumbel distributed error term and fixed model coefficients (i.e., coefficients that are the same for all individuals) produces the MNL model [@ben1985discrete]. In the MNL, the probability of choosing alternative $k$ for individual $i$ is $$p_{ik} = \frac{\exp( {\boldsymbol{\beta}}_k^T \boldsymbol{X}_{ik}) }{\sum_{p=1}^K \exp ({\boldsymbol{\beta}}_p^T \boldsymbol{X}_{ik})},$$ Given the Beta coefficient, the MNL can be associated with the likelihood function $$\boldsymbol{L}(\boldsymbol{\beta}) = \prod_{i = 1}^N \prod_{k = 1}^K \Bigg[ \frac{\exp ({\boldsymbol{\beta}}_k^T \boldsymbol{X}_{ik})}{\sum_{p=1}^K \exp ( {\boldsymbol{\beta}}_p^T \boldsymbol{X}_{ik} )} \Bigg].$$ Maximum likelihood estimation can then be applied to obtain the “best" utility coefficients $\hat{\boldsymbol{\beta}} = \operatorname*{arg\,max}_{\boldsymbol{\beta}} \boldsymbol{L}(\boldsymbol{\beta})$. By plugging $\hat{\boldsymbol{\beta}}$ into Eqn. (2), the *choice probabilities* for each mode can be obtained. More complex logit models, such as the mixed logit and nested logit, can be derived similarly from different assumptions about the coefficients and error distribution. However, these models are more difficult to fit: They generally do not have closed-form solutions for the likelihood function and require the simulation of maximum likelihood for various parameter estimations. Observe also that logit models have a layer structure, which maps the input layer $\boldsymbol{X}_i$ to the output layer, $[p_{ik}, ..., p_{iK}]^T$.
Machine-learning models, by contrast, view mode choice prediction as a *classification* problem: Given a set of input variables, predict which travel mode will be chosen. More precisely, the goal is to learn a target function $f$ which maps input variables $\boldsymbol{Z}$ to the output target $\boldsymbol{Y}$, $\boldsymbol{Y} \in \{1, ..., K\}$, as $$\boldsymbol{Y} = f(\boldsymbol{Z}|\boldsymbol{\theta}),$$ where $\boldsymbol{\theta}$ represents the unknown parameter vector for parametric models like NB and the hyperparameter vector for non-parametric models such as SVM, CART, and RF. Unlike logit models that predetermine a (usually) linear model structure and make specific assumptions for parameters and error distributions, many machine-learning models are nonlinear and/or non-parametric, which allows for more flexible model structures to be directly learned from the data. In addition, compared to logit models that maximize likelihood to estimate parameters, machine-learning models often apply different optimization techniques, such as back propagation and gradient descent for NN, recursive partitioning for CART, structural risk minimization for SVM. Moreover, while logit models have a layer structure, machine-learning models have different model topologies for different models. For example, tree-based models (CART, BAG, BOOST, and RF) all have a tree structure, whereas NN has a layer structure.
Furthermore, since the outputs of logit models are individual choice probabilities, it is difficult to compare the prediction with the observed mode choices directly. Therefore, when evaluating the predictive accuracy of logit models at the individual level, a common practice in the literature is to assign an outcome probability to the alternative with the largest outcome probability, i.e., $$\operatorname*{arg\,max}_k (\hat{p}_{i1}, ...,\hat{p}_{iK}).$$ This produces the same type of output (i.e., the travel mode choice) as the machine-learning models. Besides the prediction of individual choices, logit models and machine-learning methods are often evaluated based on their capability to reproduce the aggregate choice distribution for each mode, i.e., the market shares of each mode. For logit models, the predicted market share of mode $k$ is $$P_k(\boldsymbol{X}_k|\hat{\boldsymbol{\beta}}_k) = \sum_i^N \hat{p}_{ik}/N,$$ and, for machine-learning methods, it is usually computed by $$Q_k(\boldsymbol{Z}|\hat{\boldsymbol{\theta}}) = \sum_i^N I_k(\hat{Y}_{i})/N.$$ However, using the proportion of predicted class labels to approximate the market share may not be ideal. Instead, similar to logit models, many machine-learning methods can directly predict class probabilities at the individual level, so in this study, we use $$Q_k(\boldsymbol{Z}|\hat{\boldsymbol{\theta}}) = \sum_i^N \hat{p}_{ik}/N,$$ to predict the market share for machine-learning models as well.
The calibration of the logit models is targeted at approximating aggregate market shares, as opposed to giving an absolute prediction on the individual choice [@ben1985discrete; @hensher2005applied]. Thus, the predictive accuracy of the models may differ at the individual level and the aggregate level: Which of them should be prioritized depends on the project at hand.
Another important difference between the two approaches lies in the input data structures. Fitting a logit model requires the data on all available alternatives. In other words, even if the attributes of non-chosen alternatives are not observed, their values need to be imputed for the model. By contrast, machine-learning algorithms require the observed (chosen) mode only and not necessarily information on the non-chosen alternatives. Some previous studies have indeed only considered attribute values of the chosen mode, e.g., travel time of the chosen mode [@xie2003work; @wang2018machine], in their machine-learning models. However, a model that leaves out the attribute values of the non-chosen alternatives does not account for the fact that a given outcome is a result of the differences in attribute values across alternatives (i.e., mode competition) rather than a result of the characteristics of the chosen alternative itself. Therefore, we believe that, like logit models, the attribute values of the non-chosen alternatives should also be included (often imputed) into a machine learning model. Figure \[fig:data\_input\] shows one observation that serves as the input to logit models and machine-learning models respectively.
![Data Structure for Logit Models and Machine-Learning Models.[]{data-label="fig:data_input"}](data_input.png){width="13cm"}
Model Evaluation
----------------
When evaluating statistical and machine-learning models, the goal is to minimize the overall prediction error, which is a sum of three terms: the bias, the variance, and the irreducible error. The bias is the error due to incorrect assumptions of the model. The variance is the error arising from the model sensitivity to the small fluctuations in the dataset used for fitting the model. The irreducible error results from the noise in the problem itself. The relationship between bias and variance is often referred to as “bias-variance tradeoff,” which measures the tradeoff between the goodness-of-fit and model complexity. Goodness-of-fit captures how a statistical model can capture the discrepancy between the observed values and the values expected under the model. Better fitting models tend to have more complexity, which may create overfitting issues and decrease the model predictive capabilities. On the other hand, simpler models tend to have a worse fit and a higher bias, causing the model to miss relevant relationships between input variables and outputs, which is also known as underfitting. Therefore, in order to balance the bias-variance tradeoff and obtain a model with low bias and low variance, one needs to consider multiple models at different complexity levels, and use an evaluation criterion to identify the model that minimizes the overall prediction error. The process is known as model selection. The evaluation criteria can be theoretical measures like adjusted $R^2$, AIC, $C_p$, and BIC, and/or resampling-based measures, such as cross validation and bootstrapping. Resampling-based measures are generally preferred over theoretical measures.
The selection of statistical models is usually based on theoretical measures. For example, when using logit models to predict individual mode choices, researchers usually calibrate the models on the entire dataset, examine the log-likelihood at convergence, and compare the resulting adjusted McFadden’s pseudo $R^2$ [@mcfadden1973conditional], AIC, and/or BIC in order to determine a best-fitting model. These three measures penalize the likelihood for including too many “useless” features. The adjusted McFadden’s pseudo $R^2$ is most commonly reported for logit models, and a value between 0.2 to 0.3 is generally considered as indicating satisfactory model fit [@mcfadden1973conditional]. On the other hand, AIC and BIC are commonly used to compare models with different number of variables.
For machine learning, cross validation is usually conducted to evaluate a set of different models, with different variable selections, model types, and choices of hyper-parameters. The best model is thus identified as the one with the highest out-of-sample predictive power. A commonly-used cross validation method is the 10-fold cross validation, which applies the following procedure: 1) Randomly split the entire dataset into 10 disjoint equal-sized subsets; 2) choose one subset for validation, the rest for training; 3) train all the machine-learning models on one training set; 4) test all the trained models on the validation set and compute the corresponding predictive accuracy; 5) repeat Step 2) to 4) for 10 times, with each of the 10 subsets used exactly once as the validation data; and 6) the 10 validation results for each model are averaged to produce a mean estimate. Cross validation allows researchers to compare very different models together with the single goal of assessing their predictive accuracy. This paper compares the logit and machine-learning models using the 10-fold cross validation in order to evaluate their predictive capabilities at individual and aggregate levels.
Finally, when applying statistical models such as the logit models, researchers often take into account the underlying theoretical soundness and the behavioral realism of the model outputs to identify a final model (in addition to relying on the adjusted McFadden’s pseudo $R^2$, AIC and/or BIC). In other words, even though balancing the bias-variance tradeoff is very important, in statistical modeling, a “worse” model may be preferred due to reasons like theoretical soundness and behavioral realism. For example, since worsening the performance of a travel mode should decrease its attractiveness, the utility coefficients of the level-of-service attributes such as wait time for transit should always have a negative sign. Therefore, when a “better” model produces a positive sign for wait time, a “worse” model with a negative sign for wait time may be preferred. On the other hand, for machine-learning models, the predictive accuracy is typically the sole criterion for deciding the best model in the past, but with the recent development of machine-learning interpretation, some researchers suggested that machine-learning models should be evaluated by both predictive accuracy and descriptive accuracy [@murdoch2019interpretable].
Model Interpretation and Application
------------------------------------
The interpretation of outputs of logit models is straightforward and intuitive. Like any other statistical model, researchers can quickly learn how and why a logit model works by examining the sign, magnitude, and statistical significance of the model coefficients. Researchers may also apply these outputs to conduct further behavioral analysis on individual travel behavior, such as deriving marginal effect and elasticity estimates, comparing the utility differences in various types of travel times, calculating traveler willingness-to-pay for trip time and other service attributes. All of these applications can be validated by explicit mathematical formulations and derivations, which allows modelers to clearly explain what happens “behind the scene.”
By contrast, machine-learning models are often criticized for being “black-box” and lacking explanation [@klaiber2011random]. The lack of interpretability is believed to be a major barrier for machine learning in many real-world applications. Recently, more attention has been paid to explaining machine-learning models, with a variety of machine-learning interpretation tools being invented [e.g. @friedman2001greedy; @goldstein2015peeking; @molnar2018interpretable]. The most commonly used machine-learning interpretation tools include variable importance and partial dependence plots [@molnar2018interpretable]. Variable importance measures show the relative importance of each input variable in predicting the response variable. Different machine-learning models have different ways to compute variable importance. For example, for tree-based models (such as CART and RF), the mean decrease in node impurity (measured by the Gini index) is commonly used to measure the variable importance. Partial dependence plots measure the influence of a variable $\boldsymbol Z_p$ on the log-odds or probability of choosing a mode $k$ after accounting for the average effects of the other variables [@friedman2001greedy]. Notably, partial dependence plots may reveal causal relationships if the machine-learning model is accurate and the domain knowledge supports the underlying causal structure [@zhao2017causal].
Arguably, the behavioral insights that one can extract from the logit models (such as marginal effects and elasticities) may also be obtained from machine-learning models by performing a sensitivity analysis. For example, for machine-learning models, the arc elasticity for feature $p$ of alternative $k$ can be obtained by $$E_k(\boldsymbol{Z}_{p}) = \frac{[Q_k(\boldsymbol{Z}_{-p}, \boldsymbol Z_{p} \cdot (1+\Delta) | \hat{ \boldsymbol{\theta}}) - Q_k(\boldsymbol{Z} | \hat{ \boldsymbol{\theta}})]/Q_k(\boldsymbol{Z} | \hat{ \boldsymbol{\theta}})}{|\Delta|},$$ and the marginal effect for feature $p$ of alternative $k$ can be computed as $$M_k(\boldsymbol{Z}_{p}) = \frac{Q_k(\boldsymbol{Z}_{-p}, \boldsymbol Z_{p} + \Delta) | \hat{ \boldsymbol{\theta}}) - Q_k(\boldsymbol{Z} | \hat{ \boldsymbol{\theta}})}{|\Delta|}.$$
In essence, all of these techniques, despite their obvious differences, measure how the output variable responds to changes in the input features. In the context of travel mode choices, they help researchers gain a better understanding of how individual choices of travel modes are impacted by a variety of different factors such as the socio-economic and demographic characteristics of travelers and the respective trip attributes for each travel mode. In the current literature, however, the behavioral findings gained from machine-learning models are rarely compared with those obtained from logit models. Since the goals of mode choice studies are often in extracting knowledge to shed light on individual travel preferences and travel behavior instead of merely predicting their mode choice, these comparisons are necessary to have a more thorough evaluation of the adequacy of machine learning. Machine-learning models that have excellent predictive power but generate unrealistic behavioral results may not be useful in travel behavior studies.
The Data for Empirical Evaluation
=================================
The data used for empirical evaluation came from a stated-preference (SP) survey completed by the faculty, staff, and students at the University of Michigan on the Ann Arbor campus. In the survey, participants were first asked to estimate the trip attributes (e.g., travel time, cost, and wait time) for their home-to-work travel for each of the following modes: Walking, biking, driving, and taking the bus. Then, the survey asked respondents to envision a change in the transit system, i.e., the situation where a new public transit (PT) system, named RITMO Transit [@RitmoTransit], fully integrating high-frequency fixed-route bus services and micro-transit services, has replaced the existing bus system (see Figure \[fig:map\]). Text descriptions were coupled with graphical illustrations to facilitate the understanding of the new system. Each survey participant was then asked to make their commute-mode choice among *Car*, *Walk*, *Bike*, and *PT* in seven state-choice experiments, where the trip attributes for *Walk*, *Bike*, and *Car* were the same as their self-reported values and the trip attributes for *PT* were pivoted from those of driving and taking the bus. A more detailed descriptions of the survey can be found in @YAN2018.
![RITMO Transit: The New Transit System Featured with Four High-Frequency Bus Routes and On-Demand Shuttles Serving Approximately 2-Mile-Radius Area of the University of Michigan Campus.[]{data-label="fig:map"}](RITMO.png){width="13cm"}
A total of 8,141 observations collected from 1,163 individuals were kept for analysis after a data-cleaning process. The variables that enter into the analysis include the trip attributes for each travel mode, several socio-demographic variables, transportation-related residential preference variables, and current/revealed travel mode choices. The travel attributes include travel time for all modes, wait time for *PT*, daily parking cost for driving, number of additional pickups for *PT*, and number of transfers for *PT*. The socio-economic and demographic variables include car access (car ownership for students and car per capita in the household for faculty and staff), economic status (living expenses for students and household income for faculty and staff), gender, and identity status (i.e., faculty vs staff vs student). The transportation-related residential preference variables are the importance of walkability/bikeability and transit availability when deciding where to live. Finally, current travel mode choices are also included as state-dependence effects (i.e., the tendency for individuals to abandon or stick with their current travel mode) are verified as important predictors of mode choice by many empirical studies. Table \[tab:var\] summarizes the descriptive statistics on these variables, including a general description of each variable, category percentages for categorical variables, and min, max, mean, and standard deviation for continuous variables.
\[tab:var\]
After extracting the data from the SP survey, we pre-processed the data and verified that all the independent variables have little multicollinearity [@farrar1967multicollinearity]. The existence of multicollinearity can inflate the variance and negatively impact the predictive power of the models. This study chose the variance inflation factor to determine which variables are highly correlated with other variables and found out that all variables had a variance inflation factor value of less than five, indicating that multicollinearity was not a concern.
Models Examined and Their Specifications {#exp.setting}
========================================
This section briefly introduces the logit and machine-learning models examined in this study. Since our dataset has a panel structure, usually a mixed logit model should be applied. However, we also fitted an MNL model as the benchmark for comparison, as previous studies generally compared machine-learning models with the MNL model only. Seven machine-learning models are examined, including simple ones like NB and CART, and more complex ones such as RF, BOOST, BAG, SVM, and NN. Most previous mode choice studies only examined a subset of these models [@xie2003work; @omrani2013prediction; @omrani2015predicting; @wang2018machine; @chen2017understanding].
Logit Models
------------
We have already introduced the MNL model formulation in detail in Subsection 3.1, so only the mixed logit model is presented here.
The mixed logit model is an extension of the MNL model, which addresses some of the MNL limitations (such as relaxing the IIA property assumption) and is more suitable for modeling panel choice datasets in which the observations are correlated (i.e., each individual is making multiple choices) [@mcfadden2000mixed]. A mixed logit model specification usually treats the coefficients in the utility function as varying across individuals but being constant over choice situations for each person [@train2009discrete]. The utility function from alternative $k$ in choice occasion $t$ by individual $i$ is $$U_{ikt} = \boldsymbol{\beta}_{ik}^T \boldsymbol{X}_{ikt} + \varepsilon_{ikt},$$ where $\varepsilon_{ikt}$ is the independent and identically distributed random error across people, alternatives, and time. Hence, conditioned on $\boldsymbol{\beta}$, the probability of an individual making a sequence of choices (i.e., $\boldsymbol{j} = \{j_1, j_2, ..., j_\tau\}$) is $$\boldsymbol{L}_{i\boldsymbol{j}}(\boldsymbol{\beta}) = \prod_{t = 1}^\tau \Bigg[ \frac{ \exp(\boldsymbol{\beta}_{ij_t}^T \boldsymbol{X}_{i j_{t} t}) }{ \sum_k \exp(\boldsymbol{\beta}_{ik}^T \boldsymbol{X}_{ikt})} \Bigg].$$ Because the $\varepsilon_{ikt}$’s are independent over the choice sequence, the corresponding unconditional probability is $$p_{ik\boldsymbol{j}} = \int \boldsymbol{L}_{i\boldsymbol{j}}(\boldsymbol{\beta}) g(\boldsymbol{\beta}) d\boldsymbol{\beta},$$ where $g(\boldsymbol{\beta})$ is the probability density function of $\boldsymbol{\beta}$. This integral does not have an analytical solution, so it can only be estimated using simulated maximum likelihood [e.g. @train2009discrete].
In this study, the MNL models can be summarized as follows: 1) The utility function of *Car* includes mode-specific parameters for TT\_Drive, Parking\_Cost, Income, CarPerCap, and Current\_Mode\_Car; 2) the utility function of *Walk* includes mode-specific parameters for TT\_Walk, Female (sharing the same parameter with *Bike*), Bike\_Walkability (sharing the same parameter with *Bike*), and Current\_Mode\_Walk; 3) the utility function of *Bike* includes mode-specific parameters for TT\_Bike, Female (sharing the same parameter with *Walk*), Bike\_Walkability (sharing the same parameter with *Walk*), and Current\_Mode\_Bike; and 4) the utility function of *PT* includes mode-specific parameters for TT\_PT, Wait\_Time, Rideshare, Transfer, PT\_Access, and Current\_Mode\_PT. We also specify three alternative-specific constants for *Walk*, *Bike*, and *PT*, respectively.
The mixed logit model has the same model specification. Moreover, in order to accommodate individual preference heterogeneity (i.e., taste variations among different individuals), coefficients on the selected level-of-service variables (i.e., TT\_PT and Parking\_Cost) are also specified as random parameters. The alternative-specific constant for *PT* is also assumed as a random parameter. These random parameters are all assessed with a normal distribution. We use 1,000 Halton draws to perform the numerical integration. Both the MNL and mixed logit models are estimated using the NLOGIT software.
Machine-Learning Models {#subsec4}
-----------------------
### Naive Bayes
The NB model is a simple machine-learning classifier. The model is constructed using Bayes’ Theorem with the naive assumption that all the features are independent [@mccallum1998comparison]. NB models are useful because they are faster and easier to construct as compared to other complicated models. As a result, NB models work well as a baseline classifier for large datasets. In some cases, NB even outperforms more complicated models [@zhang2004optimality]. A limitation of the NB model is that, in real world situations, it is very unlikely for all the predictors to be completely independent from each other. Thus, the NB model is very sensitive when there are highly correlated predictors in the model. In this study, the NB model is constructed through the R package *e1071* [@e1071].
### Tree-based Models
The CART model builds classification or regression trees to predict either a classification or a continuous dependent variable. In this paper, the CART model creates classification trees where each internal node of the tree recursively partitions the data based on the value of a single predictor. Leaf nodes represent the category (i.e., *Car*, *Bike*, *PT*, and *Walk*) predicted for that individual [@breiman2017classification]. The decision tree is sensitive to noise and susceptible to overfit [@last2002improving; @quinlan2014c4]. To control its complexity, it can be pruned. This study prunes the tree until the number of terminal nodes is 6. The CART model is obtained through the R package *tree* [@tree].
To address the overfitting issues of CART models, the tree-based ensemble techniques were proposed to form more robust, stable, and accurate models than a single decision tree [@breiman1996bagging; @friedman2001elements]. One of these ensemble methods is BOOST. For a $K$-class problem, BOOST creates a sequence of decision trees, where each successive tree seeks to improve the incorrect classifications of the previous trees. Predictions in BOOST are based on a weighted voting among all the boosting trees. Although BOOST usually has a higher predictive accuracy than CART, it is more difficult to interpret. Another drawback is that BOOST is prone to overfitting when too many trees are used. This study applies the gradient boosting machine technique to create the BOOST model [@friedman2001greedy]. 400 trees are used, with shrinkage parameter set to 0.14 and the interaction depth to 10. The minimum number of observations in the trees terminal nodes is 10. The BOOST model is created with the R package *gbm* [@gbm].
Another well-known ensemble method is BAG, which trains multiple trees in parallel by bootstrapping data (i.e., sampling with replacement) [@breiman1996bagging]. The BAG model uses all the independent variables to train the trees. For a $K$-class problem, after all the trees are trained, the BAG model makes the mode choice prediction by determining the majority votes among all the decision trees. By using bootstrapping, the BAG model is able to reduce the variance and overfitting problems of a single decision tree model. One potential drawback with the BAG model is that it assumes that all the features are independent. If the features are correlated, the variance would not be reduced with BAG. In this study, 400 classification trees are bagged, with each tree grown without pruning.
The RF model is also an ensemble method. Like BAG, RF trains multiple trees using bootstrapping [@ho1998random]. However, RF only uses a random subset of all the independent variables to train the classification trees. More precisely, the trees in RF use all the independent variables, but every node in each tree only uses a random subset of them [@breiman2001random]. By doing so, RF reduces variance between correlated trees and negates the drawback that BAG models may have with correlated variables. Similar to BAG, RF makes mode choice predictions by determining the majority voting among all the classification trees. Like other non-parametric models, RF is difficult to interpret. In this study, 500 trees are used and 12 randomly selected variables are considered for each split at the trees’ nodes. The R package used for producing the BAG and RF models is *randomForest* [@RF].
### Support Vector Machine
The SVM model is a binary classifier which, given labeled training data, finds the hyper-plane maximizing the margin between two classes. This hyperplane is a linear or nonlinear (depending on the kernel) decision boundary that separates the two classes. Since a mode choice model typically involves multi-class classification, the one-against-one approach is used [@hsu2002comparison]. Specifically, for a $K$-class problem, $K(K-1)/2$ binary classifiers are trained to differentiate all possible pairs of $K$ classes. The class receiving the most votes among all the binary classifiers is selected for prediction. SVM usually performs well with both nonlinear and linear boundaries depending on the specified kernel. However, the SVM model can be very sensitive to overfitting especially for nonlinear kernels [@cawley2010over]. In this study, a SVM with a radial basis kernel is used. The cost constraint violation is set to 8, and the gamma parameter for the kernel is set to 0.15. The SVM model is produced with the R package *e1071* [@e1071].
### Neural Network
A basic NN model has three layers of units/nodes where each node can either be turned active (on) or inactive (off), and each node connection between layers has a weight. The data is fed into the model at the input layer, goes through the weighted connections to the hidden layer, and lastly ends up at a node in the output layer which contains $K$ units for an $K$-class problem. The hidden layer allows the NN to model nonlinear relationships between variables. Although NN has shown promising results in modeling travel mode choice in some studies [@omrani2015predicting], NN models tend to be overfitting, and are difficult to interpret. In this paper, a NN with a single hidden layer of 18 units is used. The connection weights are trained by back propagation with a weight decay constant of 0.4. The R package *nnet* is used to create our NN model [@stats].
Comparison of Empirical Results
===============================
This section presents the empirical results of this study. Specifically, it compares the predictive accuracy of the logit models with that of the machine-learning algorithms. In addition, it compares the behavioral findings of two machine-learning models (i.e., RF and NN) and two logit models (i.e., MNL and mixed logit).
Predictive Accuracy
-------------------
This study applied the 10-fold cross validation approach. As discussed above, cross validation requires splitting the sample data into training sample sets and validation sample sets. One open issue is how to partition the sample dataset when it is a panel dataset (i.e., individuals with multiple observations). One approach is to treat all observations as independent choices and randomly divide these observations. The other is to subset by individuals, each with their full set of observations. This study follows the first approach, which is commonly applied by previous studies [@xie2003work; @hagenauer2017comparative; @wang2018machine].
As discussed in Subsection 3.1, the predictive power of the models may differ at the individual level (predicting the mode choice for each observation) and at the aggregate level (predicting the market shares for each travel mode). The calibration of logit models focuses on reproducing market shares whereas the development of machine-learning classifiers aims at predicting individual choices. This study compares both the mean individual-level predictive accuracy and the mean aggregate-level predictive accuracy.
### Individual-Level Predictive Accuracy
The cross validation results for individual-level predictive accuracy is shown in Table \[tab:accuracy\_ind\_out\]. The best-performing model is RF, with a mean predictive accuracy equal to 0.856. However, the accuracy of the MNL and the mixed logit model is only 0.647 and 0.631 respectively, which is much lower than the RF model.
The predictive accuracy of each model by travel mode is also presented in Table \[tab:accuracy\_ind\_out\]. All models predict [*Walk*]{} most accurately. All machine-learning models have a mean predictive accuracy value between 0.795 and 0.928, whereas the MNL model has an accuracy of 0.859 and the mixed logit model 0.797. Both logit models and three ensemble machine-learning models (i.e., BOOST, BAG, and RF) predict modes [*PT*]{} and [*Bike*]{} relatively better than mode [*Car*]{}. One possible explanation is that [*Car*]{}, with a market share of 14.888%, has fewer observations compared to other modes. The notorious class imbalance problem may cause machine-learning classifiers to have more difficulties in predicting the class with fewer observations.
Finally, it is somewhat surprising that the mixed logit model, a model that accounts for individual heterogeneity and has significantly better model fit (adjusted McFadden’s pseudo $R^2$ is 0.536) than the MNL model (adjusted McFadden’s pseudo $R^2$ is 0.365), underperformed the MNL model in terms of the out-of-sample predictive power. This finding is nonetheless consistent with the findings of @cherchi2010validation. It suggests that the mixed logit model may have overfitted the data with the introduction of random parameters, and such overfitting resulted in greater out-of-sample prediction error.
------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
Mean SD Mean SD Mean SD Mean SD Mean SD
MNL 0.647 0.016 0.440 0.044 0.859 0.018 0.414 0.033 0.698 0.029
Mixed logit 0.631 0.008 0.513 0.031 0.797 0.014 0.413 0.038 0.673 0.027
NB 0.584 0.018 0.558 0.035 0.864 0.013 0.372 0.041 0.490 0.042
CART 0.593 0.014 0.428 0.032 0.795 0.022 0.329 0.038 0.653 0.026
BOOST 0.850 0.007 0.790 0.035 0.913 0.012 0.848 0.023 0.825 0.028
BAG 0.854 0.013 0.791 0.017 0.926 0.016 0.861 0.028 0.818 0.029
RF 0.856 0.012 0.797 0.022 0.928 0.016 0.859 0.021 0.820 0.027
SVM 0.772 0.012 0.701 0.027 0.878 0.026 0.681 0.033 0.770 0.026
NN 0.646 0.016 0.434 0.045 0.853 0.025 0.451 0.051 0.679 0.024
------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
: Mean Out-of-Sample Accuracy of Logit and Machine-Learning Models (Individual Level)[]{data-label="tab:accuracy_ind_out"}
### Aggregate-Level Predictive Accuracy
We now turn to aggregate-level predictive accuracy. To quantify the sum of the absolute differences between the market share predictions and the real market shares from the validation data, we use the L1-norm, also known as the least absolute deviations. Taking machine-learning models as an example, let $Q_k^*$ and $\hat{Q}_k = Q_k(\boldsymbol{Z}|\hat{\boldsymbol \theta})$ represent the true (observed) and predicted market shares for mode $k$. The L1-norm thus is defined as $$\sum_{k = 1}^4 |Q_k^* - \hat{Q}_k|.$$
The predictive accuracy results of the logit and machine-learning models at the aggregate level are depicted in Table \[tab:accuracy\_agg\_ave\]. The results show that RF outperforms all the other models, with a prediction error of 0.0248 and a standard deviation of 0.0128. Notably, even though logit models are expected to have good performance for market share predictions, RF has lower error compared to MNL (0.0399) and mixed logit (0.0593). Again, the MNL model resulted in a higher aggregate-level predictive accuracy than the mixed logit model.
Model Mean SD
------------- -------- --------
MNL 0.0399 0.0207
Mixed logit 0.0593 0.0268
NB 0.2771 0.0363
CART 0.0463 0.0280
BOOST 0.0291 0.0151
BAG 0.0253 0.0130
RF 0.0248 0.0128
SVM 0.0362 0.0218
NN 0.0493 0.0196
: Mean L1-Norm Error for Mode Share Prediction
\[tab:accuracy\_agg\_ave\]
In summary, the results show that RF is the best model among all models evaluated and that logit models only outperform a minority of the machine learning models.
Model Interpretation
--------------------
Recent advances in machine learning make models interpretable through techniques such as variable importance and partial dependence plots. Machine-learning results can be readily applied to compute behavioral outputs such as marginal effects and arc elasticities. However, other behavioral outputs such as the value of time, willingness-to-pay, and consumer welfare measures are hard to obtain from machine-learning models, because they are grounded on the random utility modeling framework and an assumption that individual utility can be kept constant when attributes of a product substitutes each other (e.g., paying a certain amount of money to reduce a unit of time). Machine-learning models lacks the behavioral foundation required to obtain these measures.
This section interprets the results of two logit models (MNL and mixed logit) and two machine-learning models (RF and NN[^1]). For the logit models, we interpret the coefficient estimates and calculate some behavioral measures including marginal effects and arc elasticities. In the meantime, we conduct comparable behavioral analysis on the RF and NN models by applying variable importance and partial dependence plots and by performing a sensitivity analysis.
It should be noted that the behavioral analysis conducted here is far from exhaustive, as mode choice model applications often go beyond what is covered here. In particular, recent advances in mode choice modeling, such as the development of mixed logit and latent class models, are mainly concerned about deriving insights on individual preference heterogeneity. In a separate paper [@zhao2019modeling], we showed that machine learning algorithms can automatically capture individual heterogeneity and that individual conditional expectation plots can help visualize such results.
### Variable Importance and Effects
Generally speaking, for traditional statistical models, standardized Beta coefficients can represent the strength of the effect of each independent variable on the mode choice, and the variable with the largest standardized coefficient has the strongest influence. However, the utility of choosing a travel mode is a latent variable and thus unobservable, so it is not obvious how to standardize a latent variable in order to estimate the standardized Beta coefficients. If one is only interested in the rank order of the magnitude of the effects of the independent variables on the utility, the $X$-standardization is enough and easy-to-implement, by standardizing the input variables only when conducting estimation [@menard2004six]. To be specific, the $X$-standardized Beta coefficients of logit models represent the weights and direction of the input variables to show the magnitude and direction of their effects.
The outputs for the MNL and mixed logit are presented in Table \[tab:logit\_results\]. The adjusted McFadden’s pseudo $R^2$ for MNL and mixed logit are 0.365 and 0.536, which indicates satisfactory model fit. All the coefficient estimates are consistent with theoretical predictions. All the level-of-service variables carry an intuitive negative sign, and all of them are statistically significant. For both logit models, individual socio-demographic characteristics are associated with their travel mode choices. Unsurprisingly, higher-income travelers with better car access are more likely to drive than using alternative modes. Females are less likely to choose [*Walk*]{} and [*Bike*]{} than males. The model also shows that individual residential preferences and current travel mode choices are associated with their travel mode choices of [*Car*]{}, [*Walk*]{}, and [*Bike*]{}. However, people tend to have weak attachment to [*PT*]{} as shown by the small and insignificant Beta coefficient. Individuals who value walking, biking, and transit access when choosing where to live are more likely to use these modes. The model shows that travelers tend to stick to their current mode even when a new travel option is offered. Furthermore, for the mixed logit model, the random parameter standard deviations are also statistically significant.
We also presented $X$-standardized Beta coefficients for the MNL and mixed logit models, allowing researchers to assess the relative importance of the independent variables, i.e., a coefficient of larger magnitude indicate a greater impact of the corresponding independent variable on the choice outcome [@menard2004six]. For both models, the results show that the most important variable in predicting the mode choice is TT\_Bike, followed by the travel time variables for the other three modes, several revealed-preference (RP) variables (i.e., current travel modes), and some level-of-service attributes. These results are reasonable and generally consistent with findings in the existing literature.
\[tab:logit\_results\]
To extract similar interpretations of logit models, this study applies widely-used tools including variable importance measures and partial dependence plots to interpret the RF model and compare the behavioral findings obtained from the RF with those from the MNL models. Like the $X$-standardized Beta coefficients in a logit model, a variable importance measure can be used to indicate the impact of an input variable on predicting the response variable for machine-learning models. Unlike $X$-standardized Beta coefficients that can show the direction of association between the input variable and the outcome variable with a positive or negative sign, however, variable importance measures provide no such information and we need to use additional machine-learning interpretation tools, such as partial dependence plots, to extract these insights. This study uses the Gini index to measure variable importance for RF. For NN, the variable importance is computed using the method proposed by @gevrey2003review, which applies combinations of the absolute values of the weights.
Table \[tab:VarImp\] shows the ranking of variable importance for RF, NN, MNL, and mixed logit. Note that Current\_Mode\_PT is not included for RF and NN. The ranking of the input features in RF is generally consistent with that of the two logit models, but NN shows very different variable importance results compared to RF, MNL, and mixed logit in many cases. For the RF model and the two logit models, the travel times of walking, driving, biking, and transit have very high influence on their stated mode choice; on the other hand, some differences do exist: For example, PT\_Access, Bike\_Walkability, Income, and CarPerCap are more important for RF compared to MNL and mixed logit.
**Variable** **RF** **NN** **MNL** **Mixed logit**
--------------------- -------- -------- --------- -----------------
TT\_Walk 1 16 2 2
TT\_Drive 2 14 4 5
TT\_Bike 3 13 1 1
TT\_PT 4 11 3 3
Current\_Mode\_Bike 5 2 5 10
PT\_Access 6 8 13 16
Bike\_Walkability 7 6 16 13
Income 8 10 14 15
CarPerCap 9 7 11 14
Current\_Mode\_Walk 10 1 6 12
Rideshare 11 9 9 9
Transfer 12 5 8 8
Wait\_Time 13 15 10 11
Female 14 3 15 17
Parking\_Cost 15 12 12 4
Current\_Mode\_Car 16 4 7 6
Current\_Mode\_PT / / 17 7
: Ranking of Variable Importance for RF, NN, MNL, and Mixed Logit
\[tab:VarImp\]
[0.48]{} ![Partial dependence plots of variables for choosing *PT* as the travel mode[]{data-label="fig:PDP"}](PDP_TT_PT.png "fig:"){height="2.8in"}
[0.48]{} ![Partial dependence plots of variables for choosing *PT* as the travel mode[]{data-label="fig:PDP"}](PDP_Waittime.png "fig:"){height="2.8in"}
[0.48]{} ![Partial dependence plots of variables for choosing *PT* as the travel mode[]{data-label="fig:PDP"}](PDP_Rideshare.png "fig:"){height="2.8in"}
[0.48]{} ![Partial dependence plots of variables for choosing *PT* as the travel mode[]{data-label="fig:PDP"}](PDP_Transfer.png "fig:"){height="2.8in"}
Partial dependence plots are another important tool that helps interpret machine-learning models. Figure \[fig:PDP\] presents how the probability of choosing *PT* changes as the value of the selected variable changes for RF and MNL. The shape of the curves sheds light on the direction and magnitude of the changes, which is similar to the Beta coefficients (without standardization) estimated from the MNL model. However, the Beta coefficients in logit models affect the utility of mode $k$ (see Eqns. (1) and (11)) rather than the probability of choosing mode $k$ (see Eqns. (2) and (13)). Accordingly, we translate utility estimates into probability estimates for the MNL model in order to compare it with RF directly.
As shown in Figure 3(a), RF, MNL, and mixed logit share a similar decreasing trend for TT\_PT, while NN presents a different decreasing pattern. As shown in Figures 4(b)-4(d), for Wait\_Time, Rideshare, and Transfer, RF and NN also differ from two logit models. While MNL and mixed logit show a nearly linear relationship between these features and the probability of choosing *PT*, the two machine-learning models reveal some nonlinear relationships. For example, the following observations can be highlighted on the RF model: 1) For TT\_PT, RF has relative flat tails before 10 minutes and after 25 minutes, showing people tend to become insensitive to very short or very long transit times; 2) travelers are more sensitive to wait times less than 5 minutes; and 3) the choice probability of *PT* decreases more significantly from 0 to 1 rideshare compared to from 1 to 2 rideshares. Based on these observations, we specified piece-wise utility functions (i.e., specifying different coefficients for a variable in different data intervals) for the logit models (MNL and mixed logit). While not showing the model outputs here, we found that the model fit improved and that the coefficient estimates largely agreed with the nonlinearies revealed by the RF model. These results will be presented in a separated paper.
Therefore, partial dependence plots of machine-learning models readily reveal the nonlinearities of mode choice responses to level-of-service attributes. In contrast to the time-consuming hand-curating procedure required in logit models (often by introducing interactions terms) to reveal nonlinear relationships, machine-learning algorithms exhibit these nonlinearities automatically and thus can generate richer behavioral insights much more effectively. Machine-learning models can thus serve as an exploratory analysis tool for identifying better specifications for the logit models in order to enhance the predictive power and explanatory capabilities of logit models.
### Arc Elasticity and Marginal Effects
Logit models are often applied to generate behavioral outputs such as marginal effects and elasticities to gain insights on individual travel behavior. Marginal effects (and elasticities) measure the changes of the choice probability of an alternative in response to one unit (percent) change in an independent variable. This study calculates marginal effects and arc elasticities for the level-of-service variables associated with the proposed mobility-on-demand transit system, including TT\_PT, Wait\_Time, Rideshare, and Transfer.
The marginal effects and arc elasticity results for MNL, mixed logit, NN, and RF are presented in Table \[tab:marginal\_effect\_elas\]. It is notable that we use $\Delta = 2$ min to compute the marginal effects of Wait\_Time, and we present the results for RF in two ways (all: entire market; constrained: part of the market with “out-of-bound” observations removed). This is mainly because the nature of the RF model: RF consists of hundreds of decision trees which apply decision rules based on discrete values, and so they may not be sensitive enough to small marginal changes and they are unable to properly predict “out-of-bound” observations.
Table \[tab:marginal\_effect\_elas\] illustrates that the arc-elasticity and marginal-effect estimates are all negative, indicating that when the level-of-service of transit gets worse, the travelers’ preferences for transit will decrease. Moreover, for Wait\_Time, Transfer, and Rideshare, the marginal-effect estimates of logit models are larger than those of NN and RF. For TT\_PT, the marginal effects and arc elasticity estimates for MNL, NN, and RF are similar in magnitude, whereas the estimates for the mixed logit model are much smaller. For RF, removing “out-of-bound” observations increases the marginal effects and elasticity estimates.
Therefore, we find significant differences in the behavioral outputs across the four models. Without the ground truth, it is difficult to assess the validity of these results. However, one can obtain more readily interpretable behavioral insights by converting these marginal-effect estimates into relative value-of-time measures. The following value-of-time measures are obtained by dividing all marginal effects estimates with that of transit travel time. First, the penalty of a transfer is approximately equal to 5.5 min (MNL), 13.4 min (mixed logit), 2.6 min (NN), and 3.1 min (RF) of transit travel time. Also, the penalty of a rideshare stop is equivalent to 4.2 min (MNL), 8.9 min (mixed logit), 1.0 min (NN), and 2.1 min (RF) of transit travel time. Finally, the value of one min wait time is equal to 1.5 min (MNL), 3.4 min (mixed logit), 0.2 min (NN), 0.7 min (RF) of transit travel time. The results of RF seem more realistic and more consistent with logit models.
The existing literature generally finds that the penalty effects of a transfer is larger than 5 min [e.g., @garcia2018transfer], and the value of wait time is slightly larger than that of in-vehicle travel time [@abrantes2011meta]. Though the results of logit models seem more aligned with the empirical findings, the results of RF may still be sound. One reason for smaller penalties of RF is that TT\_PT consists of in-vehicle and out-of-vehicle travel times (the former has lower penalty compared to the latter), and thus using TT\_PT to construct the value-of-time measures may lead to smaller outputs. The other reason is that the new MOD system is expected to be app-based and highly synchronized, so passengers may perceive that the transfer will be much more convenient and they can actively wait at home after booking the trip, leading to smaller penalties for Transfer and Wait\_Time.
------------ ------------------ ------------ ----------- ----------- ---------- ------------- ----------
All Constrained
Wait\_Time Marginal effects 1 or 2 min $-2.93$% $-2.96$% $-0.58$% $-0.83$% $-1.16$%
Transfer Marginal effects 1 unit $-10.69$% $-11.66$% $-6.27$% $-4.60$% $-5.10$%
Rideshare Marginal effects 1 unit $-8.13$% $-7.74$% $-2.54$% $-2.08$% $-3.41$%
Marginal effects 1 min $-1.94$% $-0.87$% $-2.45$% $-1.63$% $-1.63$%
Arc elasticity 10% $-0.89$ $-0.49$ $-1.28$ $-1.07$ $-1.08$
------------ ------------------ ------------ ----------- ----------- ---------- ------------- ----------
: Marginal Effects and Arc Elasticity of *PT* Market Share with Respect to Transfer, Rideshare, TT\_PT, and Wait\_Time.
\[tab:marginal\_effect\_elas\]
Discussion and Conclusion {#sec7}
=========================
The increasing popularity of machine learning in transportation research raises questions regarding its advantages and disadvantages compared to conventional logit-family models used for travel behavioral analysis. The development of logit models typically focuses on parameter estimation and pays little attention to prediction (i.e., lack of a procedure to validate out-of-sample prediction accuracy). On the other hand, machine-learning models are built for prediction but are often considered as difficult to interpret and are rarely used to extract behavioral findings from the model outputs.
This paper aims at improving the understanding of the relative strengths and weaknesses of logit models and machine learning for modeling travel mode choices. It compared logit and machine-learning models side by side using cross validation to discover their predictive and interpretability capabilities. The results showed that the best-performing machine-learning model, the RF model, significantly outperforms the logit models both at individual and aggregate levels. In fact, most machine learning models outperform the logit models. Somewhat surprsingly, the mixed logit model underperformed the MNL in terms of the out-of-sample predictive accuracy, which may result from overfitting. Moreover, to interpret the machine-learning models, we applied three techniques, including variable importance, partial dependence plots, and sensitivity analysis, to extract behavioral insights from the model outputs.
Some of the results were illuminating. First, machine learning and logit models largely agree on variable importance and the direction of impact that each variable has on the choice outcome. However, there are some differences in the behavioral outputs (marginal effects and arc elasticities) between machine learning and logit models. Moreover, we find that the RF model can automatically capture the nonlinear effects of an independent variable on the choice outcome. This indicates that machine learning can, at minimum, serve as an exploratory analysis tool to reveal nonlinearities; researchers can then apply such information to specify logit models that can better represent behavioral preferences and have better predictive capabilities, which should be much more efficient than a hand-curating procedure typically done with statistical models.
Overall, these results are encouraging and identify many new research directions in applying machine learning to model travel behavior and forecast travel demand. Prediction and interpretation are two major topics in modeling individual choice behavior. Traditionally, each approach has focused on one aspect and ignored the other. We have demonstrated that both approaches can be applied to make predictions and infer behavior. Nonetheless, there are several major topics in travel-behavior research that we have not examined in depth. The first topic is concerned with preference heterogeneity. The development of the mixed logit model has mostly been driven by its capability to capture both observed and unobserved preference heterogeneity among individuals. We have not addressed this important topic in this paper. The second topic is on mechanisms to correct the reporting bias associated with the SP data. The SP data are generally considered as containing reporting bias due to their hypothetical nature. Logits models using joint RP and SP data have been proposed to correct for this bias [@train2009discrete] but, to our knowledge, no machine learning algorithms allow such a joint estimation process.
We believe that there is great potential in merging important ideas from machine learning and logit models to develop more refined models for the research of travel behavior modeling. Besides addressing the limitations mentioned above, other possible research directions include: 1) examining which machine-learning models are more suitable than others for behavioral analysis; and 2) imposing behavioral constraints to the risk functions of machine-learning models to improve their interpretability.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was partly funded by the Michigan Institute of Data Science (MIDAS) and by Grant 7F-30154 from the Department of Energy.
[^1]: The reasons for choosing these two machine-learning models are: 1) RF is the best-performing model among the seven machine-learning classifiers; and 2) NN is one of the most popular machine-learning classifier used for travel mode choice modeling.
|
---
abstract: |
We have used $R_C I_C$ CCD photometry from the Isaac Newton telescope and intermediate resolution spectroscopy from the Gemini North telescope to identify and characterise low-mass ($0.15<M/M_{\odot}<1.3$) pre-main sequence stars in the young open cluster NGC 2169. Isochrone fitting to the high- and low-mass populations yields an intrinsic distance modulus of $10.13^{+0.06}_{-0.09}$ mag and a model-dependent age of $9\pm
2$Myr. Compared with the nearby, kinematically defined groups of a similar age, NGC 2169 has a large low-mass population which potentially offers a more precise statistical investigation of several aspects of star formation and early stellar evolution. By modelling the distribution of low-mass stars in the $I_C$ versus $R_C-I_C$ diagram we find that any age spread among cluster members has a Gaussian full width at half maximum $\leq
2.5$Myr. A young age and small age spread ($<10$Myr) are supported by the lack of significant lithium depletion in the vast majority of cluster members. There is no clear evidence for accretion or warm circumstellar dust in the low-mass members of NGC 2169, bolstering the idea that strong accretion has ceased and inner discs have dispersed in almost all low-mass stars by ages of 10Myr.
bibliography:
- 'iau\_journals.bib'
- 'master.bib'
date: Submitted 5 September 2006
nocite:
- '[@burningham05]'
- '[@landolt92]'
- '[@stetson00]'
- '[@schaller92]'
- '[@lejeune01]'
- '[@rastorguev99]'
- '[@meynet93]'
- '[@houdebeine96]'
- '[@white03]'
- '[@cutri03]'
- '[@jth01]'
- '[@jeffries03]'
- '[@naylor02]'
- '[@naylor98]'
- '[@burningham03]'
- '[@baraffe02]'
- '[@baraffe98]'
- '[@dantona97]'
- '[@leggett96]'
- '[@bessell88]'
- '[@kirkpatrick93]'
- '[@zapatero02]'
- '[@siess00]'
- '[@stauffer98]'
- '[@stauffer99]'
- '[@chiosi92]'
- '[@meynet97]'
- '[@meynet00]'
- '[@barradoldb04]'
- '[@barrado03]'
- '[@muzerolle98]'
- '[@rieke85]'
- '[@piskunov04]'
- '[@taylor86]'
- '[@naylor06]'
- '[@mouschovias06]'
- '[@maclow04]'
- '[@mohanty05]'
- '[@sicilia05]'
- '[@jayawardhana06]'
- '[@song02]'
- '[@white05]'
- '[@dahm05]'
- '[@jeffries05]'
- '[@delgado92]'
- '[@sagar76]'
- '[@abt77]'
- '[@perry78]'
- '[@kroupa01]'
- '[@pinfield98]'
- '[@king62]'
- '[@soderblom99]'
- '[@jeffrieslireview06]'
- '[@dean78]'
- '[@lyra06]'
- '[@muzerolle00]'
- '[@jenniskens94]'
- '[@bailer-jones04]'
- '[@randich01]'
- '[@soderblom93pleiadesli]'
- '[@jones96pleiades]'
- '[@oliveira03]'
- '[@pena94]'
- '[@harris76]'
- '[@jerzykiewicz03]'
- '[@liu89]'
- '[@cuffey56]'
- '[@hoag61]'
- '[@duquennoy91]'
- '[@fischer92]'
- '[@schlesinger72]'
- '[@palla00]'
- '[@palla02]'
- '[@palla05]'
- '[@hartmann01]'
- '[@hartmann03]'
- '[@littlefair04]'
title: 'The Keele-Exeter young cluster survey: I. Low mass pre-main sequence stars in NGC 2169'
---
\[firstpage\]
stars: pre-main sequence – stars: abundances – stars: late-type – open clusters and associations: individual: NGC 2169
Introduction
============
There are notably few well-studied clusters in the literature with ages between about 5 and 30Myr. Yet investigating the coeval populations in such clusters is vital for our understanding of: (i) the lifetimes and subsequent evolution of high mass (8–20$M_{\odot}$) main sequence stars which contribute the majority of metal enrichment to the universe; (ii) the evolution of circumstellar material, formation of planetary systems and loss of angular momentum in lower mass stars.
In this paper we report the first results from the Keele-Exeter young cluster survey (KEY clusters) to find and characterise clusters with age 5–30Myr. NGC 2169 ($=$C0605+139) is a concentrated, but sparsely populated young open cluster (class I3p – Ruprecht 1966) in the constellation of Orion, with an age of $\simeq
10$Myr and distance of $\sim 1$kpc (see Perry, Lee & Barnes 1978 and Section \[previous\]). Using CCD photometry and intermediate resolution spectroscopy we have uncovered the low-mass ($0.1<M/M_{\odot}<1.3$) pre-main sequence (PMS) population of NGC 2169 and used isochronal fits to the high- and low-mass stellar populations and measurements of photospheric Li depletion to test evolutionary models, determine the cluster age and investigate the possibility of any age spread within the cluster. Armed with an age, we have investigated timescales for the dispersal of gas and dust discs by comparing the low mass stars of NGC 2169 with those in younger and older clusters.
The paper is organised as follows. Section 2 summarises previous work on this cluster; Section 3 describes a new photometric survey (the results of which are provided in electronic form) used to identify candidate low-mass cluster members; Section 4 describes intermediate resolution spectroscopy from the Gemini North telescope which is used to confirm candidate membership and study Li depletion; Section 5 compares constraints on the cluster age imposed by low-mass isochrones, Li depletion models and the evolutionary status of high-mass stars in the cluster; Section 6 deals with the spatial structure and mass function of the newly discovered low-mass population; Section 7 discusses the evidence for age spreads within the cluster and the evolution of circumstellar accretion. Our conclusions are presented in Section 8.
----------------------- --------------- -------------------- ------- ---------------------------
Authors $E(B-V)$ Intrinsic Distance Age Notes
(mag) Modulus (mag) (Myr)
Sagar (1976) 0.18 9.60 $<9$ $UBV$ photoelectric
Harris (1976) $<12$ MK spectra
Abt (1977) $0.17\pm0.03$ $10.9\pm0.3$ MK spectra
Perry et al. (1978) $0.20\pm0.01$ $10.2\pm0.1$ $<23$ $ubvy\beta$ photoelectric
Delgado et al. (1992) 0.20 10.05 $<16$ $ubvy\beta$ photoelectric
Peña & Peniche (1994) $0.25\pm0.02$ $9.7\pm 0.3$ $<50$ $ubvy\beta$ photoelectric
----------------------- --------------- -------------------- ------- ---------------------------
\[clusterparams\]
{width="150mm"}
Previous studies of NGC 2169 {#previous}
============================
There are several previous studies of the high mass population in NGC 2169. Photoelectric photometry of the brighter candidate cluster members is presented in Cuffey & McCuskey (1956, in the $PV$ system), Hoag et al. (1961, $UBV$), Sagar (1976, $UBV$), Perry et al. (1978, $uvby\beta$), Delgado et al. (1992, $uvby\beta$) and Peña & Peniche (1994, $uvby\beta$). Harris (1976) and Abt (1977) have published spectral types on the Morgan-Keenan system for the brightest stars in the cluster and more recently, Liu, Janes & Bania (1989) conducted a radial velocity (RV) survey of 9 bright A and B stars in the cluster, finding several spectroscopic binaries. A number of surveys for peculiar or variable stars have been conducted (see Jerzykiewicz et al. 2003 and references therein) and at least two candidate beta Cepheids and an A0V Si star have been identified.
For this paper, the important parameters are the cluster age, distance and reddening. Table \[clusterparams\] gives a summary of the conclusions reached by other authors. There is good agreement between photometric and spectroscopic determinations of the reddening and very little star-to-star dispersion ($\sigma_{E(B-V)}\leq 0.02$ – Delgado et al. 1992). There is less agreement in the cluster distance and age, primarily because of disagreement over which stars should be considered members of the cluster and which stars are binary systems, but also because of different calibrations of the main sequence turn-off upon which the age estimates are based. As there are no obviously evolved stars apart from the binary system Hoag 1 (B2III – Abt 1977) these ages are inevitably upper limits. We prefer the estimates provided by Perry et al. (1978) and Delgado et al. (1992), which attempt to weed out field interlopers before estimating the cluster parameters. For the moment we assume the cluster is less than 23Myr old, at a distance of $\simeq 1000$pc and has $E(B-V)=0.20$, although these parameters are re-assessed in Section \[age\].
![A colour-magnitude diagram for unflagged objects with uncertainties $<0.1$ mag in $I_C$ and $R_C-I_C$ seen in CCD 4 (see Fig. \[gaiaplot\]). The solid line shows a theoretical 10Myr PMS (from Baraffe et al. 2002) at an intrinsic distance modulus of 10.13, and with a reddening/extinction corresponding to $E(R_C-I_C)=0.14$ (see Section \[age\]). []{data-label="cmd1"}](fig2.ps){width="75mm"}
----- ---------------- ------------------------------ ------------------------------ ------------------------------ ---------------------------- -- -- -- -- -- -- -- -- -- -- --
CCD Colour range 1 0.340 $< R_C-I_C < $ 1.839 2 0.318 $< R_C-I_C < $ 1.750 3 0.342 $< R_C-I_C < $ 2.323 4 0.207 $< R_C-I_C < $ 2.314
----- ---------------- ------------------------------ ------------------------------ ------------------------------ ---------------------------- -- -- -- -- -- -- -- -- -- -- --
: The range of colours for the photometric standards observed by each CCD detector.
\[stand\_col\]
Gemini Spectroscopy
===================
Observations {#gmosobs}
------------
------ ------------ -------------- --------- ----- ---------------- --------
Mask RA Dec Date N Exp Time Seeing
\# (JD-2453000) (s) (arcsec)
1 06 08 34.8 $+13$ 55 12 672.108 18 $3\times 1800$ 0.6
2 06 08 42.0 $+14$ 00 00 706.973 15 $3\times 1800$ 0.5
3 06 08 24.4 $+14$ 00 00 674.100 15 $3\times 1800$ 0.5
------ ------------ -------------- --------- ----- ---------------- --------
\[slitmask\]
{width="71mm"} {width="71mm"} {width="71mm"}
{width="71mm"} {width="71mm"} {width="71mm"}
The Gemini Multi-Object Spectrograph (GMOS) was used in conjunction with slit masks at the Gemini North telescope to observe 47 candidate low-mass members of NGC 2169 with $14.6<I_C<19.3$. This corresponds to a mass range (for an assumed distance of 1000 pc and age of 10Myr) of $0.14<M/M_{\odot}<1.3$ according to the models of Baraffe et al. (1998). Stars were targeted based on their location in the colour-magnitude diagram and with the aim of maximising the number of targets that could be included in three separate slit mask designs (see Table \[slitmask\] and Fig. \[gaiaplot\]). Table \[targetlist\] gives the coordinates and photometry of the targets, which are split according to which of the three masks they were observed in. One target (number 28) was observed in both masks 2 and 3.
We used slits of width 0.5 arcsec and with lengths of approximately 8–10 arcsecs. The R831 grating was used with a long-pass OG515 filter to block second order contamination. The resolving power was 4400 and simultaneous sky subtraction of the spectra was possible. The spectra covered $\sim 2000$Å, with a central wavelength of 6200Å– 7200Å depending on the location of the slits within the 5.5 arcminute field of view.
Observations were taken in queue mode (program number GN-2005B-Q-30) through three separate masks during October and December 2005 (see Table \[slitmask\]). For each mask we obtained $3\times1800$s exposures bracketed by observations of a CuAr lamp for wavelength calibration and a quartz lamp for flat-fielding and slit location. The spectra were recorded on three $2048\times4068$ EEV chips leading to two $\simeq 16$Å gaps in the coverage. The CCD pixels were binned $2\times2$ before readout, corresponding to $\sim 0.67$Å per binned pixel in the dispersion direction and 0.14 arcsec per binned pixel in the spatial direction. Conditions were clear with seeing of 0.5–0.6 arcsec (FWHM measured from the spectra).
The data were reduced using version 1.8.1 of the GMOS data reduction tasks running with version 2.12.2a of the Image Reduction and Analysis Facility (IRAF). The data were bias subtracted, mosaiced and flat-fielded. A two-dimensional wavelength calibration solution was provided by the arc spectra and then the target spectra were sky-subtracted and extracted using 2 arcsec apertures. The three individual spectra for each target were then combined using a rejection scheme which removed obvious cosmic rays. The instrumental wavelength response was removed from the combined spectra using observations of a white dwarf standard to provide a relative flux calibration. The same calibration spectrum was used to construct a telluric correction spectrum. A scaled version of this was divided into the target spectra, tuned to minimise the RMS in regions dominated by telluric features.
The SNR of each combined extracted spectrum was estimated empirically from the RMS deviations of straight line fits to segments of “pseudo-continuum” close to the 6708Å features (see below). As small unresolved spectral features are expected to be part of these pseudo-continuum regions, these SNR estimates, which range from $\sim 10$–20 in the faintest targets to $>200$ in the brightest, should be lower limits. Examples of the reduced spectra are shown in Fig. \[specplot\]. All the reduced spectra are available in “fits” format from the “Cluster” Collaboration’s home page (see footnote 1).
Analysis
--------
Each spectrum was analysed to yield a spectral type, equivalent widths of the 6708Å and H$\alpha$ lines and a heliocentric RV. Each of these analyses is described below. The results are given in Table \[specresults\].
### Spectral Types {#spt}
![Spectral types deduced from the TiO(7140Å) index as a function of $R_C-I_C$. Also shown are results from the spectra of standard stars where $R_C-I_C$ is available (see text).[]{data-label="sptri"}](fig4.ps){width="75mm"}
Index Spectral Type
------- ---------------
0.99 K5
1.13 K7
1.26 M0
1.40 M1
1.53 M2
1.74 M3
2.08 M4
2.61 M5
3.38 M6
: The relationship between TiO(7140Å) index and spectral type
\[sptindex\]
Spectral types were estimated from the strength of the TiO(7140Å) narrow band spectral index (see Oliveira et al. 2003). This index is quite temperature sensitive and can be calibrated for spectral type using spectra of well known late K and M-type field dwarfs. We constructed a polynomial relationship between spectral type and the TiO(7140Å) index that was used to estimate the spectral type of our targets, based on a numerical scheme where M0–M6$=$0–6, K5$=-2$ and K7$=-1$. Table \[sptindex\] gives the adopted relationship between the TiO(7140Å) index and spectral type. The scatter around the polynomial indicates that these spectral types are good to $\pm$ half a subclass for stars of type M0 and later, but to only a subclass at earlier type stars as the molecular band is very weak at spectral type K5.
A plot of spectral type derived from the TiO(7140Å) index versus $R_C-I_C$ colour reveals a smooth relationship (see Fig. \[sptri\]) with little scatter. We expect this even if there is some contamination by field interlopers in the sample. These would have similar colours and spectral types to the cluster members, but would be foreground objects with lower luminosities. A comparison of the positions of standard stars on this plot ($R_C-I_C$ colours where available are from Leggett 1992) reveals an average redward offset of $\simeq 0.05$ mag in the $R_C-I_C$ values of our targets at a given spectral type. Of course we expect cluster members to have suffered a reddening $E(R_C-I_C)\simeq 0.14$ mag (corresponding to $E(B-V)=0.20$ mag – Taylor 1986). This comparison demonstrates that the photometric calibration for these red stars is reasonable, although there is a hint that the $R_C-I_C$ values may be too blue at the reddest colours. We have to temper this conclusion with the probability that the relationship between colour and spectral type is slightly different in very young stars.
### Lithium measurements {#li}
The 6708Å resonance feature should be strong in cool young stars with undepleted Li – with an equivalent width (EW) of 0.5Å to 0.6Å according to the curves of growth presented by Zapatero-Osorio et al. (2002). Insets in Fig. \[specplot\] show the Li region in a number of our targets. Lithium is an ephemeral element in the atmospheres of very cool stars, and its presence in the photospheres of late K and M-type stars is a strong indicator of youth.
The EW of the 6708Å feature was estimated by fitting it with a Gaussian function. We preferred this to direct integration because in lower signal-to-noise spectra we eliminate the subjectivity involved in choosing the integration limits and we get a straightforward indication of rapid rotation (see below). The “pseudo-continuum” was estimated using straight line fits to the regions immediately around the Li feature, excluding regions beyond 6712Å which contain a strong Ca line and which are noisy due to the subtraction of a strong S[ii]{} sky line. None of the Li lines in our spectra show any strong evidence for a non-Gaussian shape or double lines; target 44 (shown in Fig. \[specplot\]) has the most “non-Gaussian” appearance, but even here the Gaussian fit is only rejected at 93 per cent confidence and in any case the EW estimated by direct integration would not differ significantly from the Gaussian result. The EW and Gaussian FWHM of the Li lines are given in Table \[specresults\]. Uncertainties in the EW are estimated using the formula $\delta\,{\rm EW} = \sqrt{2fp}/{\rm SNR}$, where $2f$ is twice the Gaussian FWHM of the line (approximately the range over which the EW is integrated) and $p$ is the pixel size (0.67Å).[^1] In 8 cases there was no obvious Li feature to measure, in which case a $3\sigma$ upper limit is quoted. In 3 cases the Li feature fell in a gap between the detectors and no EW could be measured.
For the majority of the sample there are clear detections of the Li feature with EW$>0.3$Å. Comparisons with Li-depletion patterns in open clusters of known age (e.g. see Fig. 10 of Jeffries et al. 2003) place empirical age constraints on these stars. Li EWs of $>0.3$Å are not seen for any stars of spectral type cooler than K5 in the Pleiades or Alpha Per clusters (with ages of 120Myr and 90Myr respectively and excepting the very low luminosity stars beyond the lithium depletion boundary where Li remains unburned – see Soderblom et al. 1993; Jones et al. 1996). Nor can strong Li lines be seen in M dwarfs of the 35–55Myr open clusters NGC 2547, IC 2391 and IC 2602 (see Randich et al. 2001; Jeffries et al. 2003; Barrado y Navascués et al. 2004; Jeffries & Oliveira 2005, and again excepting the very cool M dwarfs beyond the lithium depletion boundary). In summary we assume that objects with EW\[Li\]$>0.3$Å are all younger than 100Myr and younger than 50Myr if they have spectral type $\geq$ M0. These Li-rich objects are therefore very likely to be members of NGC 2169 and this conclusion is supported by RV measurements (see below). However the converse may not be true – a lack of Li is not used as the sole criterion for excluding a candidate member, as one of the aims of this paper is to investigate possible instances of anomalously large Li depletion.
In a number of cases the FWHM of the 6708Å line is significantly broader than the 1.7Å expected from the intrinsic width of the doublet and the instrumental resolution. In these cases rotational broadening is suspected, which implies projected rotational velocities from 25 (for a FWHM of 2Å) up to about 60 for the broadest lines.
![H$\alpha$ EW as a function of spectral type. Objects with Li, without Li or where the Li status is unknown are indicated. Two lines are shown that have previously been used to separate accretion-generated H$\alpha$ emission from a lower level of emission that could be attributable to a chromosphere (Barrado y Navascués & Martín 2003; White & Basri 2003). []{data-label="haspt"}](fig5.ps){width="75mm"}
### H$\alpha$ measurements and circumstellar material {#accrete}
{width="71mm"}
{width="71mm"}
H$\alpha$ EWs were measured for all our targets (except two where the feature fell in a detector gap) by direct integration above (or below) a pseudo-continuum. The main uncertainty here is the definition of the pseudo-continuum as a function of spectral type and probably results in uncertainties of order 0.2Å, even for the bright targets.
H$\alpha$ emission is ubiquitous from young stars. It arises either as a consequence of chromospheric activity or is generated by accretion activity in very young objects (e.g. Muzerolle, Calvet & Hartmann 1998). The H$\alpha$ emission from accreting “classical” T-Tauri stars (CTTS) is systematically stronger and broader (velocity widths $>270$– White & Basri 2003) than the weak line T-Tauri stars (WTTS) where the emission is predominantly chromospheric.
An empirical division between CTTS and WTTS can be made either on the basis of H$\alpha$ EW or the width of the H$\alpha$ emission line (e.g. Barrado y Navascués & Martín 2003; White & Basri 2003). Fig. \[haspt\] shows the H$\alpha$ EW of our targets (where available) versus spectral type along with the empirical dividing line between CTTS and WTTS defined by Barrado y Navascués & Martín and by White & Basri. On the basis of these plots none of our targets are clear examples of CTTS. A caveat here is that we have a single epoch spectrum. Accretion or chromospheric activity can be variable phenomena and multiple observations are preferable for a secure classification. Large variations in H$\alpha$ strength have been seen in some PMS stars (e.g. Littlefair et al. 2004), however in a recent paper by Jayawardhana et al. (2006), multiple H$\alpha$ spectra did not reveal variability that would change the classification of a significant fraction of young objects. The most likely error would be a chromospheric flare leading to a CTTS classification for a WTTS.
The profiles of the H$\alpha$ line were inspected for all targets and apart from targets 8 and 13, none show evidence for velocity widths (at 10 per cent of maximum) in excess of 300. Target 8 is discussed in Section \[member\], it does not show a 6708Å line and is probably not a cluster member. Target 13 (shown in Fig. \[specplot\]) has a narrow, strong line and shows a blue H$\alpha$ emission wing extending to $\sim 200$. The H$\alpha$ EW lies below the accretion thresholds in Fig. \[haspt\]. It is possible that either accretion at a low level or chromospheric flaring could explain this observation. There are a number of other objects with widths at about the 270 threshold advocated by White & Basri (2003) as an accretion discriminator. However, we note that our spectral resolution (70 FWHM) is relatively poor compared with that used by White & Basri. That and the fact that some objects appear to have rotationally broadened photospheric Li lines, mean that this threshold should be raised considerably in some cases. None of the targets show any evidence for other emission lines that are often (but not always) associated with young accreting low-mass stars, such as He[i]{} 5876, 6678Å.
In addition we have checked for any emission from warm circumstellar dust by plotting the $H-K$ versus $I_C-J$ colour-colour diagram (see Fig. \[ijhk\]a). $JHK$ photometry for our targets was taken from the 2MASS point source catalogue (Cutri et al. 2003). An excess would show up as an anomalously large $H-K$ colour with respect to the photospheres of dwarf stars with the same $I_C-J$, although $H-K$ is nowhere near as sensitive to warm dust as excesses at longer wavelengths. None of our targets are significantly($>2$ sigma) discrepant from the main sequence dwarf locus reddened according to $E(B-V)=0.2$. The same is true of the more conventional $J-H$ versus $H-K$ diagram (Fig. \[ijhk\]b). In summary we find no strong evidence for accretion or warm circumstellar dust in any of our targets.
### Radial velocities
![Radial velocities as a function of spectral type. Stars with and without Li and those with unknown Li status are indicated.[]{data-label="rvspt"}](fig7.ps){width="75mm"}
Our observations were not optimised for measuring RVs – minimal arc calibrations were performed and no RV standards were observed. Nevertheless we have been able to estimate RVs of all the targets relative to [*one*]{} of the targets and then estimate a zeropoint based on archival spectra.
Relative RVs were determined by cross-correlation against target 5. We chose this star to act as a template because it has a high SNR and with a spectral type of M2, it has spectral features in common with targets of both earlier and later spectral types.
A correlation wavelength range of 6600–7400Å was used for stars of spectral type M0 and later. For earlier spectral types where molecular features become small then the range 6000–6500Å was used. Raw correlation lags were adjusted to the same heliocentric reference frame. A further correction to the RV zero point was estimated by cross-correlating sky emission lines between the target spectra and the spectrum of target 5 [*prior*]{} to the sky subtraction data reduction step. The typical size of this correction was $\pm 3$, reflecting inaccuracies in the wavelength calibration, possibly due to flexure in the spectrograph during the 90 minutes of observation for each mask. A heliocentric RV zeropoint was estimated by cross-correlating stars of type M4 or later with archival VLT UVES spectra of the M4V–M6V stars GL402, GL406 and GL876 for which precise heliocentric RVs are known (see Bailer-Jones 2004 for details).
Heliocentric RVs versus spectral type are plotted in Fig. \[rvspt\]. Typical internal uncertainties are of order a few . Most objects are closely clustered in this diagram. However, there is a clear upward trend towards later spectral types that seems to be a consequence of spectral type mismatch between target and template, probably due to our reliance on broad molecular bands rather than sharp atomic lines in the later-type stars. Our best estimate for the true heliocentric RV of the cluster is $+16.8\pm
1.1$ from the Li-rich stars of spectral type M4 or later. The quoted error includes the scatter in our measurements and an estimate of the external uncertainty judged from the variance of results using the three different standard star templates. Our result agrees with (though is much more precise than) the $+16.6\pm 6.0$ quoted by Rastorguev et al. (1999).
Cluster Membership {#member}
------------------
![Spectra for two targets with questionable cluster membership (see Section \[member\]). The inserts show normalised regions around the H$\alpha$ and Li[i]{} features. []{data-label="strangespectra"}](fig8a.ps "fig:"){width="75mm"} ![Spectra for two targets with questionable cluster membership (see Section \[member\]). The inserts show normalised regions around the H$\alpha$ and Li[i]{} features. []{data-label="strangespectra"}](fig8b.ps "fig:"){width="75mm"}
The presence of a strong (EW$>0.3$Å) 6708Å feature is taken as a positive indicator of membership and is entirely supported by the chromospheric H$\alpha$ emission exhibited by all these Li-rich objects. The Li-rich candidates also have RVs with a very small intrinsic dispersion (less than a few after taking account of the broad trend with spectral type) that is consistent with cluster membership.
All three of the M dwarfs with no Li measurement have H$\alpha$ emission lines consistent with the Li-rich targets. However H$\alpha$ emission is no guarantee of extreme youth among M dwarfs and one of these three objects has an RV inconsistent with cluster membership, unless it is a short period binary system. For now we regard all three objects as questionable members.
In order to investigate the possibility of anomalously large Li depletion, we also consider those targets where Li was undetected. Of eight such targets, five have an H$\alpha$ line in absorption or barely in emission. These are extremely unlikely to be young objects and all have an RV which is inconsistent with cluster membership. These five are all classified as definite non members. One object has no H$\alpha$ measurement and an RV inconsistent with cluster membership and we also classify this as a definite non-member. The remaining two objects have unusual spectra (see Fig. \[strangespectra\]) which are discussed below.
Target 8. This star shows double peaked central H$\alpha$ emission within a broad absorption feature. The spectrum is dominated by several strong absorption lines at 5780.5Å, 5796.9Å, 6283.9Å, 6613.7Åand 7224.0Å, which can be identified with diffuse interstellar absorption bands (DIBS). There is no Li at 6708Å although there is an unidentified line at 6703Å. The EWs of the DIBS imply quite a high reddening. Using the relationships given by Jenniskens & Désert (1994), we estimate $E(B-V)\simeq 0.9$. From this, and also from the position of the star in the $J-H$ versus $H-K$ diagram, we infer that it has a spectral type of late A or early F and that the spectral type of K5 assigned in Section \[spt\] must be in error. The star lies blueward of the best fitting cluster isochrone in the $I_C$ versus $R_C-I_C$ CMD and the presence of additional reddening would increase this discrepancy. However, given that the H$\alpha$ emission profile strongly suggests circumstellar material and possibly an accretion disc, we cannot be sure that the intrinsic optical colours would be representative of the photosphere of a late A star in any case. Hence this object could be a young Herbig Ae star either within NGC 2169 or more likely at a much greater distance. Given this uncertainty, we will not consider the star as a cluster member in what follows.
Target 14. This object has an upper limit to its Li EW which could just be consistent with the presence of significant Li in the photosphere. Indeed, there is a hint of an Li line at the 1-2 sigma level. Its EW\[H$\alpha$\] is the largest in our sample and close to the empirical border between CTTS and WTTS. Its RV is consistent with cluster membership. The width of the H$\alpha$ line is only $\simeq
180$ at 10 per cent of maximum. The status of this star is questionable. It is either a cluster member or a very active foreground dMe star. A better spectrum of the 6708Å line is needed and it is not considered as a cluster member in the analysis that follows.
Age estimates {#age}
=============
Absolute ages for open clusters are usually model dependent. This is especially true in very young open clusters and star forming regions. A valuable exercise is to compare cluster ages derived from techniques that rely on different aspects of stellar physics. Agreement would instil confidence in the accuracy of stellar ages whilst discrepancies would highlight potential weaknesses in our understanding of stellar evolution.
Previous work has taken ages determined by fitting model isochrones to the positions of high-mass stars that have undergone nuclear evolution in the Hertzsprung-Russell (H-R) diagram, and compared them with ages determined from isochrones which trace the descent of low-mass PMS stars contracting towards the hydrogen burning main sequence. In clusters with ages of 50–700Myr good agreement has been claimed (e.g. Lyra et al. 2006), but in younger clusters ($<30$Myr) discrepancies are more common (e.g. Piskunov et al. 2004). Here, ages from evolved high-mass stars are often more uncertain because there are fewer such objects and the ages are dependent on details such as rotation and the degree of mixing in the convective core (Chiosi, Bertelli & Bressan 1992; Meynet & Maeder 1997, 2000). The ages obtained from fitting low-mass isochrones also become more model dependent – the details of the stellar atmospheres, interior convection and even the initial conditions become increasingly important (see Baraffe et al. 2002).
A third technique has begun to be added to these comparisons. The abundance of Li in convective envelopes and atmospheres is sensitively dependent to the temperature at the base of the convection zone (or stellar centre in the case of fully convective PMS stars). Once at a threshold temperature of about $3\times10^{6}$K, Li is burned rapidly in (p,$\alpha$) reactions. The mass and hence luminosity and temperature at which Li burning commences is age dependent and isochrones of Li depletion can be used to estimate the ages of clusters (see Jeffries 2006 for a review). This technique has been used to obtain precise ages for several open clusters. Isochronal ages from high-mass stars agree with the lithium depletion ages for open clusters in the range 50–150Myr providing that some extra mixing (caused by core overshoot or rotation?) extends the main sequence lifetimes of 5–8$M_{\odot}$ stars (Stauffer, Schultz & Kirkpatrick 1998; Stauffer et al. 1999; Barrado y Navascués, Stauffer & Jayawardhana 2004). Jeffries & Oliveira (2005) have also shown that the lithium depletion age of a 35Myr cluster agrees with an age derived from isochronal fits to low-mass PMS stars in the same cluster. However, discrepancies have also been reported. Song, Bessell & Zuckerman (2002) and White & Hillenbrand (2005) have reported on individual PMS stars which appear to show much more Li depletion than expected for their isochronal ages in the H-R diagram.
Isochrone matches to high mass stars {#himassage}
------------------------------------
{width="71mm"}
{width="71mm"}
{width="71mm"}
The high-mass population of NGC 2169 is relatively sparse with only one clearly evolved B2III binary star (Abt 1977). Age constraints from these stars will therefore be quite poor, but we can still get valuable constraints on the cluster distance that will be useful when considering low-mass isochronal fits. Perry et al. (1978) based their age on what they consider to be the brightest, unevolved, non-binary member of NGC 2169 – namely Hoag 6, with a spectral type of B3V (Abt 1977). A calibration of the duration of main sequence burning and absolute magnitude due to Schlesinger (1972) was used, leading to an upper age limit of 23Myr. Delgado et al. (1992) claim a similar status for Hoag 4 with a spectral type of B2.5IV (Abt 1977) or B2V (Harris 1976) and hence determine a slightly lower upper age limit of 16Myr.
We have re-evaluated the reddening, distance and age of NGC 2169 using the $UBV$ data set published by Sagar (1976) together with the solar metallicity evolutionary models of Schaller et al. (1992) which have been transformed into isochrones in the Johnson $UBV$ system by Lejeune & Schaerer (2001 – henceforth known as the Geneva models and isochrones). The Sagar (1976) dataset is the largest homogeneous set of photoelectric data for NGC 2169. We reviewed the cluster membership using information in Sagar (1976), Perry et al. (1978) and Delgado et al. (1992). There was generally consensus over membership, but in three cases (Hoag 2, 5, and 15) membership is disputed.
The intrinsic $U-B$ vs $B-V$ colour-colour diagram, $V$ vs $B-V$ and $V$ vs $U-B$ CMDs are shown in Fig. \[ubvplot\]. We find that a reddening of $E(B-V)=0.20\pm 0.01$ applied to the 10Myr Geneva isochrone satisfactorily models the colour-colour diagram. Note that we only consider the age-independent main sequence portion of the $U-B$ vs $B-V$ diagram below the “blue turn-off” and exclude the bluest (evolved) star from the fit. As this portion of the colour-colour locus is age independent for ages $\leq 20$Myr, then the reddening estimate is also independent of assumed age in this range (see below). Then, assuming that $A_V/E(B-V)$=3.10, the $V$ vs $B-V$ CMD can be matched (by eye) with an intrinsic distance modulus of $10.2\pm0.2$. From this CMD the age of the cluster is certainly less than 20Myr based on the two brightest undisputed members and could be less than 10Myr based only on the brightest object. The brightest star in the cluster is actually a close-to-equal mass binary system but this does not affect the age limit because the 10Myr isochrone is almost vertical at this magnitude. In principle $V$ vs $U-B$ could offer better distance precision because the ZAMS locus has a shallower gradient for B stars in this CMD. However, we find the photometry appears more scattered, with points lying well outside a plausible band that could be explained by equal mass binary systems. In this CMD a less certain distance modulus of $10.2\pm 0.3$ and an age of $\simeq 10$Myr seem appropriate. An age of $\geq 20$Myr or more is still ruled out by the brightest pair of undisputed members.
Low mass isochrones {#lowmassage}
-------------------
The Li depletion age {#limassage}
--------------------
{width="71mm"}
{width="71mm"}
{width="71mm"}
{width="71mm"}
That all of the firm cluster members have EW\[Li\]$>0.3$Å puts strong constraints on the age of NGC 2169. For clusters with ages$>30$Myr, there is observational evidence that an “Li depletion chasm” opens up, starting with stars at spectral type $\simeq$M2 and widening with age to encompass spectral types either side (see Jeffries 2006). The cool side of this chasm, the so-called Lithium Depletion Boundary, is sharp and almost model independent. It occurs when the cores of contracting, fully convective stars reach a temperature sufficient to burn Li. The warm side of the chasm exhibits a more gradual roll-off with temperature (see Fig. \[plotliewvsri\]). Here the depletion takes place at the boundary of a radiative core and a receding convective envelope. The amount of depletion is sensitively dependent to details of the convection treatment, interior opacities and chemical composition.
In principle, isochrones of Li depletion can be used as an alternative way to estimate the age of a cluster. The difficulties in doing so are converting the data in the observational plane (spectral type/colour and a spectrum or EW\[Li\]) into the quantities predicted by models (i.e. $T_{\rm eff}$ and a Li abundance) or vice versa. A further slight complication is that models predict Li depletion rather than Li abundance. The initial Li abundance must be assumed, although a value of A(Li) ($= 12 + \log {\rm N(Li)/N(H)}$) of $3.3\pm 0.1$ appears to agree with observational constraints from meteorites and very young (presumably undepleted) stars (Soderblom et al. 1999).
We chose to perform a comparison of data and models in the observational plane of EW\[Li\] versus $R_C-I_C$ colour. Model temperatures were transformed using the same colour-$T_{\rm eff}$ relations required to make the same models match the Pleiades CMD (see Section \[lowmassage\]). The predicted abundance was transformed into EW\[Li\] using $T_{\rm eff}$ and the relationship between abundance and EW\[Li\] derived from cool stellar atmospheres and synthetic spectra by Zapatero-Osorio et al. (2002) and extended to warmer temperatures and lower abundances by Jeffries et al. (2003). This latter relationship was derived to interpret spectra with a spectral resolution of 1.68Åand predicts “pseudo EWs” with respect to a local continuum. In this respect it is ideal for interpreting our measured EWs.
The comparisons with four models are shown in Fig. \[plotliewvsri\]. These are the isochrones arising from the Li depletion predicted by: the Baraffe et al. (1998, 2002) evolutionary tracks, using a mixing length of either 1.0 or 1.9 pressure scale heights respectively; the Siess et al. (2000) models with no convective overshoot and a mean metallicity of $Z=0.02$ and; the grey atmosphere models of D’Antona & Mazzitelli (1997) featuring the full spectrum turbulence treatment of convection.
Taking the observations at face value, the majority of targets appear to possess Li at undepleted levels, with one or two objects showing some evidence of depletion (amounting at most to about a factor of 10 in target 32). On the other hand, some of the coolest objects appear to show significantly enhanced Li with respect to the assumed initial abundance. One of these (target 28) has two mutually consistent measurements taken on different nights. Looking at the bulk of the objects, the Li measurements can only give upper limits to the cluster age. On the redward side ($R_C-I_C>1.5$), the lack of a clear cut lithium depletion boundary implies a cluster age of $<15$Myr for all the models. For $R_C-I_C<1.5$ there is some model dependence due to the increasing efficiency of convection as we move from the top-left to bottom-right of Fig. \[plotliewvsri\]. The Baraffe models with mixing length of 1.0 scale heights suggest an age $<10$Myr for most stars with the possibility of a couple of objects as old as 15Myr. Models with increased convective efficiency suggest progressively younger ages with the bulk of objects consistent with ages of $\leq 5$Myr and maximum ages of about 12, 10 and 8Myr respectively for the Baraffe et al. model with larger mixing length, the Siess et al. model and the D’Antona & Mazzitelli model respectively.
The ages inferred from the Li depletion are in reasonable agreement with those inferred from the low-mass isochronal fits using the same models. We do not find strong evidence for examples of anomalously rapid Li depletion amongst young stars that might imply a problem with the evolutionary models as suggested by Song et al. (2002) and White & Hillenbrand (2005).
Cluster structure and total mass {#clustermass}
================================
![A colour magnitude diagram for the area covered by the three GMOS fields (see Fig. \[gaiaplot\]). Stars with Li (members), those without Li (non-members) and those with unknown Li (questionable membership) according to Section \[member\] are indicated. The dashed polygon indicates a region we have chosen for photometric selection of cluster members with $15.5<I_{c}<19.0$. Also shown are an isochrone and binary sequence from Baraffe et al. (2002 – with mixing length set at 1.0 pressure scale height) corresponding to an age of 10Myr and intrinsic distance modulus of 10.13 (see Section \[age\]). []{data-label="plotiri"}](fig15.ps){width="80mm"}
![Radial surface density profile for photometric cluster candidates with $15.5<I_C<19.0$. The best fitting King profile is shown. []{data-label="plotradial"}](fig16.ps){width="80mm"}
Our spectroscopic membership determination can be used to define a selection region in the $I_C$ versus $R_C-I_C$ CMD that preferentially includes cluster members and excludes contaminating objects. A larger sample of probable cluster members can then be chosen from our entire photometric survey and used to investigate the spatial distribution of low mass stars in NGC 2169.
Figure \[plotiri\] shows a polygon that we have used to select photometric candidates with $15.5<I_C<19.0$. The appearance of Fig. \[plotiri\] and also Fig. \[cmd1\] suggest that few cluster members could lie redward or blueward of the polygon boundaries. We have applied this photometric selection criterion to our entire catalogue (Table \[ccd\_catalogue\]) and the resulting cluster candidates are plotted in Fig. \[gaiaplot\] to show their spatial distribution. There are 64 photometric candidates in the $\sim
70$ arcmin$^{2}$ defined by the overlapping GMOS fields, 38 of which were among our spectroscopic targets. Of these, 33 are Li-rich cluster members, 2 are non-members without Li and 3 have uncertain membership because of the lack of an EW\[Li\] measurement. From this we deduce that the spatial density of contaminating non-members in our selection box is $\simeq0.05$ stars arcmin$^{-2}$. Incompleteness due to any lack of sensitivity in the photometric survey can be neglected at these magnitudes, although our choice of photometric selection boundaries, which deliberately exclude non-members, may have biased the non-member density downwards.
Figure \[gaiaplot\] shows a spatial concentration of photometric cluster candidates. By fitting a Gaussian plus a constant term to the spatial distribution along the RA and Dec axes we find the centre of this concentration to be at RA$=$ 06h08h26$(\pm
4)$s, Dec$= +$13d58m23$(\pm18)$s. The radial distribution of the candidates about this centre is shown in Fig. \[plotradial\]. We have corrected for incompleteness in coverage (e.g. due to proximity of bright stars), the azimuthal asymmetry in the survey extent and gaps in the CCD mosaic by normalising with a similar distribution for stars with $15.5<I_C<19.0$ which are blueward of the candidate cluster members. The radial distribution has been modelled with a King profile (see King 1962) plus a constant background density. The tidal radius of a cluster in the solar vicinity is given by $r_t =
1.46\,M_c^{1/3}$ (see Pinfield, Hodgkin & Jameson 1998). We assume that the mass of the cluster, $M_c = 300\,M_{\odot}$ (see below), yielding $r_t = 9.8$pc, equivalent to 32.2arcmin for a distance modulus of 10.1 (see Section \[age\]). The best fit core radius, normalisation and background density are quite insensitive to $r_t$ and hence very insensitive to the assumed cluster mass. The best fit King profile (shown in Fig. \[plotradial\]) has a core radius of $(2.8\pm
1.0)$arcmin (equivalent to 0.85pc at a distance modulus of 10.1), a normalisation of $(1.7\pm0.6)$ stars arcmin$^{-2}$, a constant background of $(0.23\pm0.03)$ stars arcmin$^{-2}$ and a total of 87 cluster stars integrated out to the tidal radius. This background density is far in excess of that deduced from the spectroscopy above.
If we believe that the cluster spatial distribution [*is*]{} well represented by this King profile, then the majority (approximately 220) of the 302 photometric candidates plotted in Fig. \[gaiaplot\] must be field stars and we would have been fortunate to observe only 2 non-members among 35 GMOS targets with Li measurements, rather than the 9 that would be predicted on the basis of the fitted background described above.
We can also check whether the mass function (MF) of the cluster is close to what has been seen in other clusters and the field. The limits of $15.5<I_C<19.0$ correspond closely to mass limits of $1.0> M/M_{\odot}>0.15$ using a 10Myr isochrone from Baraffe et al. (1998, 2002) and a distance modulus of 10.13 (see Section \[age\]). Choice of evolutionary model and variations of age and distance within the allowed uncertainties discussed in Section \[age\] can change these limits by $\sim 10$ per cent, but do not change the basic argument set out below.
Using the universal piecewise power law MF advocated by Kroupa (2001), which matches data from the field and many open clusters, then 87 stars with $0.15<M/M_{\odot}<1.0$ should be found in a cluster with 3.9 stars with $2.5<M/M_{\odot}<15.0$, corresponding to $1.2>M_{V}>-4.1$ according to the 10Myr Geneva isochrone used in Section \[himassage\]. Fig. \[ubvplot\] shows there are at least 12-15 such stars in NGC 2169. Therefore either: (i) most of these high mass stars do not belong to NGC 2169; (ii) the cluster IMF is deficient by a factor of $\simeq 3$ in low-mass stars; or (iii) the good match of the King profile to the spatial distribution is coincidental and most of the “background” in Fig. \[plotradial\] is a population of low-mass cluster members extending out to the tidal radius and which is [*much*]{} more widely dispersed than the high-mass population. A King profile with the background fixed at the 0.05 stars arcmin$^{-2}$ implied by spectroscopy in the GMOS fields is a poor fit (rejected at $>99.99$ per cent confidence) to the data.
In our view both (i) and (ii) are unlikely, but confirming that the widely dispersed low-mass cluster candidates are genuine members would require further spectroscopy. For scenario (ii) the total mass of the cluster (for $0.15<M/M_{\odot}<15.0$) is $\simeq 150\,M_{\odot}$, whereas for scenario (iii) the total mass is about $300\,M_{\odot}$ if the Kroupa (2001) mass function is assumed.
Discussion
==========
The age spread in NGC 2169
--------------------------
The possibility of age [*spreads*]{} in young clusters and star forming regions has been vigorously debated. Whether molecular clouds can sustain star formation for long periods of time ($\sim 10$Myr) or whether star formation is a rapid process that is essentially completed in a free-fall time ($\sim 1$Myr) is related to whether magnetic fields or supersonic turbulence regulate the gravitational collapse of the clouds (e.g. Mac Low & Klessen 2004; Mouschovias, Tassis & Kunz 2006).
The age distribution of stars can be deduced from their position in H-R diagrams. Based on several nearby star forming regions including Taurus and the Orion Nebula cluster (ONC), the claim has been made that star formation accelerates over the course of $\sim 10$Myr in a typical molecular cloud (Palla & Stahler 2000, 2002). These claims are disputed by Hartmann (2001, 2003) who explain apparent age spreads and accelerating formation rates in the H-R diagram as due to binarity, variability, dispersion in extinction and accretion properties and contamination by foreground non-members (see also the discussion of variability in Burningham et al. 2005). Palla et al. (2005) have recently bolstered the idea of a significant age spread with the observation of several objects in the ONC that may have depleted their photospheric Li by factors of 3–10. They argue that such depletion could not occur unless these stars were at least 10Myr old and that such ages are roughly in agreement with their positions in the H-R diagram. The possible problem we see here is that Palla et al. (2005) needed to “unveil” their spectra prior to determining the Li abundances, implying that the objects were heavily accreting. It is not clear that a plane parallel, LTE model will satisfactorily yield Li abundances in these circumstances. The 6708Å resonance doublet forms close to the top of the atmosphere and is vulnerable to NLTE effects such as overionisation by a non-photospheric UV continuum that could weaken EW\[Li\] (e.g. Houdebeine & Doyle 1996)
At an age of $9\pm 2$Myr NGC 2169 is a fascinating cluster with which to test some of these ideas. First, accretion appears to have ceased (see Section \[accrete\]), but the cluster is young enough that small changes in age still result in significant changes in luminosity for low-mass stars. Hence the scatter of stars in the H-R diagram will contain information on any age spread if it can be separated from scatter caused by binarity, intrinsic variability and differential reddening. Second, Li depletion only begins in low mass stars after about 5–10Myr, so for a given age spread any spread in Li depletion should become much more pronounced in NGC 2169 than in a younger cluster like the ONC.
In Section \[lowmassage\] we found that a magnitude-independent uncertainty of 0.03 mag needed to be added to the uncertainty in each photometric band in order for the Baraffe et al. (1998, 2002) models to provide a reasonable fit to the low-mass PMS. There could be several contributions to the requirement for this additional uncertainty: (i) that the isochrone shapes do not represent the data very well; (ii) variability due to chromospheric activity and starpots; (iii) differential reddening; (iv) incorrect assumptions about the binary frequency or mass ratio distribution; (v) an age spread. Of these, (i) appears not to be an issue (see Fig. \[bestfits\]a), (iii) is probably limited to less than 0.014 mag scatter in $E(R_{C}-I_{C})$ (Delgado et al. 1992) and (iv) has little effect when changed within reasonable limits. Instead it seems that there is additional scatter about the best fitting isochrone (especially towards the low-mass end) that could be caused by a combination of (ii) and (v), but also includes a contribution of 0.01 mag from systematic photometric uncertainties (see Section \[ccdphotom\]).
The additional scatter corresponds to [*at most*]{} $\pm0.04$ mag (1-sigma) in $R_C-I_C$. Over the mass range of our cluster members, isochrones over a small age range are nearly parallel and a 0.04 mag dispersion is equivalent to only $\pm 1.2$Myr when translated into a shift in age (independent of which models are chosen). As young stars are known to be variable this must represent an upper limit to how much of the scatter in the CMD can actually be attributed to an age spread. Of course one could relax some of the (we believe very reasonable) assumptions about the binary frequency and mass ratio distributions to increase this, but even without any binary systems, the total age spread would be less than 10Myr. A further concern might be that our spectroscopic target selection has prevented the inclusion of older cluster members that lie below the targeted cluster members in the CMD. Figure \[plotiri\] shows that there is a significant gap between the cluster members and what are presumably objects unassociated with the cluster that lie some way below the cluster main sequence. If they were cluster members, they would need to be at least 30Myr old, which seems an unrealistically large spread. Our conclusion is that we do not require age spreads beyond a Gaussian FWHM of 2.5Myr to explain the photometric data in NGC 2169.
Supporting evidence for a small age spread comes from the lack of any large dispersion in the Li abundances. This evidence has the additional merit of being independent of assumptions about binary frequencies, differential extinction or variability. Although we have already expressed our reservations about using the 6708Å line to derive Li abundances, it is likely that interpretation and modelling problems could only serve to increase the observed dispersion. Indeed, Fig. \[plotliewvsri\] shows several cooler objects that appear to have enhanced Li abundances. The Li observations imply age spreads of less than 10Myr for all the models and as the dispersion must include a significant component from uncertainties in the Li abundances then this must be very much an upper limit. There is just one object (target 32) that may be significantly Li-depleted and have an age that is 5–10Myr older than all the other cluster members. However, an age $>5$Myr older than the majority of cluster members for target 32 is not supported by its position in the CMD. Hence we do not find evidence of Li depletion similar to that found in the ONC by Palla et al. (2005) that might support an age spread as large as 10Myr.
Accretion disc lifetimes
------------------------
![The fraction of low-mass stars in NGC 2169 that have accretion signatures (according to the White & Basri \[2003\] EW\[H$\alpha$\] criteria) compared with other clusters and associations as a function of age (derived from the Baraffe et al. \[1998, 2002\] or Siess et al. \[2000\] models). The data come from Mohanty et al. (2005) for Taurus-Auriga, IC 348, Chamaeleon I and Upper Sco, from Dahm (2005) for NGC 2362, from Sicilia-Aguilar et al. (2005) for Tr 37 and NGC 7160, and from Jayawardhana et al. (2006) for the $\eta$ Cha group. The relative precisions for the cluster ages are typically $\pm 1$–2Myr. []{data-label="plotaccrete"}](fig17.ps){width="75mm"}
The timescale for the disappearance of accretion signatures in young low-mass stars is a valuable probe of infalling gas and the evolution of the inner discs in protoplanetary systems. Broad and strong H$\alpha$ emission is the most readily available signature of a strong accretion process. Finding the fraction of stars which exhibit such a signature as a function of age has been the goal of several recent investigations (e.g. Mohanty, Jayawardhana & Basri 2005; Sicilia-Aguilar et al. 2005). NGC 2169 occupies an important position, because at a similar age, studies of accretion disc evolution have largely been confined to sparsely populated nearby moving groups like $\eta$ Chamaeleontis and TW Hya (e.g. Jayawardhana et al. 2006).
All of our NGC 2169 targets have EW\[H$\alpha$\] below the empirical accretion thresholds proposed by White & Basri (2003) and Barrado y Navascués & Martín (2003). None of the targets for which we have the necessary data show a $K$-band near infrared excess indicative of warm circumstellar material. However, a low-accretion rate or unfortunate geometric arrangement of the accretion disc and/or flow could result in a small EW\[H$\alpha$\] or broadened H$\alpha$ emission with an EW below these thresholds (e.g. Muzerolle et al. 2000). Only one object in our sample (target 13) shows a broadened H$\alpha$ emission line that might indicate a low-level of accretion. Comparing H$\alpha$ EWs with mass accretion rates derived from $U$-band excesses (Sicilia-Aguilar et al. 2005) suggests that the White & Basri (2003) H$\alpha$ EW criterion corresponds to mass accretion rates of $\simeq
10^{-9}\,M_{\odot}$year$^{-1}$ for late K and early M stars, although variations in system geometry, viewing angle and the stellar mass will blur this boundary.
If we strictly adopt the White & Basri EW\[H$\alpha$\] accretion criterion then the 95 per cent upper limits to the fraction of stars exhibiting accretion or $K$-band near infrared excess in NGC 2169 are 8 (0/36) and 10 (0/30) per cent respectively. The fraction of accretors in NGC 2169 is compared to other clusters in Fig. \[plotaccrete\]. We have chosen clusters from Mohanty et al. (2005), Sicilia-Aguilar et al. (2005), Dahm (2005) and Jayawardhana et al. (2006) where the fraction of accretors has been (re)determined based on the White & Basri (2003) EW\[H$\alpha$\] criteria, where all the cluster ages have been determined using the Baraffe et al. (1998) or Siess et al. (2000) models and where the mass range of the stars considered is similar to those in NGC 2169. The data for NGC 2169 strongly reinforce the view that significant gas accretion ($\ga 10^{-9}\,M_{\odot}$year$^{-1}$) has ceased at ages of 10Myr in the vast majority of low-mass stars.
Summary
=======
The main findings of this paper can be summarised as follows:
1. We have uncovered the low-mass population of NGC 2169, spectroscopically confirming 36 objects with $0.15<M/M_{\odot}<1.3$ as cluster members on the basis of their Li abundances, H$\alpha$ emission and radial velocities. We provide a catalogue of these spectroscopic members and a full catalogue of $R_{C}I_{C}$ photometry covering 880 arcmin$^2$ around the cluster which contains several hundred other photometric candidates (see below).
2. The high mass population of the cluster has been used to estimate an intrinsic distance modulus of $10.13^{+0.06}_{-0.09}$mag. At this distance, isochrone fitting with several low-mass evolutionary models yields ages from 5 to 11Myr. The age from the Baraffe et al. (1998, 2002) and Siess et al. (2000) models, which provide the best description of the low-mass data, is $9\pm 2$Myr. Age constraints from the main sequence turn-off and from estimates of Li depletion in the low-mass stars are consistent with this age.
3. Using reasonable assumptions for the binary frequency and mass ratio distribution, the low-mass isochronal fits do not require any age spread in the cluster population beyond a Gaussian FWHM of 2.5Myr. The observed levels of Li depletion are also consistent with a small age spread ($<10$Myr) and only one M-type cluster member shows any evidence of significant Li depletion that might indicate it is $>5$Myr older than the rest of the cluster. Hence the observations do not support scenarios where significant star formation in a cluster proceeds over extended periods of time ($\ga 5$Myr).
4. On the basis of the strength and width of their H$\alpha$ emission lines and the lack of any $K$-band near infrared excesses, we find no strong evidence of accretion activity or warm circumstellar material in the confirmed cluster members. Comparison with younger clusters reinforces the idea that significant levels of gas accretion cease on timescales $<10$Myr for the vast majority of low-mass stars.
5. Informed by the spectroscopically confirmed cluster members we have photometrically selected several hundred other low-mass cluster candidates. A consideration of the number and spatial distribution of these candidates suggest either that the cluster has a “top-heavy” mass function or that the cluster’s low-mass stars are much more widely distributed than the high-mass stars – out to radii of 20 arcminutes. The total cluster mass for stars of $0.15<M/M_{\odot}<15$ is 150–300$M_{\odot}$.
Although much further away, the low-mass population of NGC 2169 is larger than those in the kinematically defined groups in the solar vicinity (e.g. $\eta$ Cha, TW Hya) which have so far provided the focus for investigations of the early evolution of stars and planetary systems at a similar age. Nearby moving groups will continue to provide the best targets for programmes such as direct imaging, where spatial resolution is crucial, but clusters like NGC 2169 offer much more potential for precise statistical investigations of low-mass stellar properties such as spectral energy distributions, rotation rates and X-ray activity. The greater distance will frequently be offset (as in this paper) by the multiplexing capability of multi-object or wide-field instruments that can observe many low-mass objects simultaneously.
Acknowledgments {#acknowledgments .unnumbered}
===============
Based on observations obtained at the Gemini Observatory (program GN-2005B-Q-30), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil) and CONICET (Argentina). Also based on observations made with the Isaac Newton Telescope which is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
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. NJM acknowledges the receipt of a studentship funded by the UK Particle Physics and Astronomy Research Council.
\[lastpage\]
[^1]: This formula arises from adding the uncertainties in each pixel flux estimate in quadrature, assuming that these uncertainties are given by the average signal-to-noise ratio and that the line is integrated over a range of $2f$. The additional statistical uncertainty due to the continuum level estimate is small in comparison.
|
---
abstract: |
Extracting shape information from object boundaries is a well studied problem in vision, and has found tremendous use in applications like object recognition. Conversely, studying the space of shapes represented by curves satisfying certain constraints is also intriguing. In this paper, we model and analyze the space of shapes represented by a $3$D curve (space curve) formed by connecting $n$ pieces of quarter of a unit circle. Such a space curve is what we call a *Tangle*, the name coming from a toy built on the same principle.
We provide two models for the shape space of $n$-link open and closed tangles, and we show that tangles are a subset of trigonometric splines of a certain order. We give algorithms for curve approximation using open/closed tangles, computing geodesics on these shape spaces, and to find the deformation that takes one given tangle to another given tangle, i.e., the Log map. The algorithms provided yield tangles upto a small and acceptable tolerance, as shown by the results given in the paper.
author:
- 'Aditya Tatu [^1]'
bibliography:
- 'tanglebib.bib'
title: Tangled Splines
---
**Keywords:** Tangles, Trigonometric splines, Implicitly defined manifolds, Shape, Space curves, Geodesics, Exponential map, Log map.
Introduction
============
The shape of an object is of tremendous interest to the computer vision community. It has several applications like object recognition, summarizing shape of collection of objects (mean shape), object retrieval and others. There are various aspects and inquiries with respect to shape of an object: shape statistics, shape matching, shape based registration, modeling shape variability and others.
Shape is typically defined as whatever is left in the object representation after eliminating variability due to rigid transformations[@Kendall1989]. Factoring out these variations depends on the mode of object representation chosen. One popular choice is to represent the object by its boundary, which in turn can be represented using one of the following: landmark points, Fourier descriptors, smooth curves, level set functions, medial axis, and others.
The set of all possible object representation chosen forms a space to work with, from which factoring out the variabilities not of interest gives us a *shape space*. Using tools of differential geometry, one then works with metrics, geodesic distances, parallel transport to answer queries related to shape as listed in the first paragraph.
Interesting shape (or curve) spaces have been proposed by putting restrictions on variabilities of an object or limiting the object representation, for example, cubic splines, Fourier descriptors, and the space of Bicycle-chain shape models. In this paper, we are interested in studying and analyzing what we call the *Tangle shape space*. Tangle is a toy (and much more) created by Richard X. Zawitz’s Tangle creations ([www.tanglecreations.com](www.tanglecreations.com)), consisting of connected copies of a planar curve (arc) to form a closed space curve. Few examples are shown in Figure \[fig:1\] below. Each arc is rigid and each joint can be subjected to local rotations which may have global effects on the shape. In this paper, we are interested in the mathematical modeling of the tangle shape space, and developing shape related computations on it. We consider two cases, open tangles and closed tangles; in the former the initial point of the first link and final point of the last link need not be the same, while in the latter, the points and the respective tangents must be the same.
We develop mathematical models for tangles as parametric space curves, and show that the set of all possible tangle curves is a subset of all trigonometric splines of a specific order. Apart from this, some useful characteristics of tangles, curvature and torsion for instance, are also derived. We also show that the set of all possible open tangles can be modeled as an appropriate dimensional torus, while the set of all possible closed tangles is a subset of such a torus. We also provide algorithms for space curve approximation using tangles, and for computing geodesics and Log map on the tangle shape space. Exploring applications like protein shape modeling is left to another paper.
$
\begin{array}{cc}
\includegraphics[width=1.75in]{T18c} &
\includegraphics[width=1.75in]{T8c}\\
\includegraphics[width=1.75in]{T18o} &
\includegraphics[width=1.75in]{T8o}
\end{array}$
The paper is organized as follows: We summarize related work in the next section, and develop models for shape spaces of tangles in Section \[sec:modeltangle\]. In the same section, we also show that tangles form a subset of trigonometric splines. Thereafter we provide computational algorithms for curve approximation, computing geodesics and Log map on tangle shape space in Section \[sec:comptools\]. We demonstrate the performance of these algorithms in Section \[sec:expts\], and conclude with some discussions in Section \[sec:discon\].
Related Work
============
The first proper study of shape is credited to D’Arcy Thompson for his work *On Growth and Forms*[@Thompson1917]. The modern treatment of shape is mainly due to Kendall[@Kendall1984] and Bookstein[@Bookstein1986], wherein it was proposed to represent an object by some landmark points located on the boundary of the object in ${{\mathbb R}}^2$. Kendall used the quotient of this space with the group of translation, scaling and rotation to obtain a *shape space* on which he defined the dissimilarity of two shapes by computing the geodesic distance between the two shapes on the shape space.
Planar curves have received much more attention than space curves. The space of planar smooth curves is an important model for shape space (though not strictly a shape space). In a more mathematical treatment of this subject, Michor and Mumford[@Michor2007] study the set of embeddings and immersions of $S^1$ into ${{\mathbb R}}^2$, where $S^1$ is the unit circle in ${{\mathbb C}}$ and represents the values that a parameter used to define a closed curve can take. The space they consider is then the quotient of the above space with the group of diffeomorphisms from and to $S^1$ representing the group of reparameterizations. This forms an infinite dimensional space on which geodesics are obtained for various metrics. In [@Younes1998], Younnes also works on the space of smooth planar curves, and defines the distance between two curves as the energy needed to deform one curve to another. The energy is based on the length of path on the group of infinitesimal deformation field that deforms one curve to another. Klassen et.al.[@Klassen2004] use Fourier descriptors of the tangent to the planar curve to represent the curve. A $2$D closed curve can also be represented by its *medial axis*[@Blum1973]. A more comprehensive representation called the shock graph has been explored for representation and shape distance computation in [@Giblin1999; @Sebastian2001]. Sommer et.al. introduced the Bicycle chain shape models[@Sommer2009] in which, every landmark on the curve is constrained to be equidistant from its neighbors. Glaunes et al.[@Glaunes2006] consider a shape as an interior of a planar curve, while the shape space is built by looking at all possible diffeomorphic deformations of the interior of closed curves.
Srivastava et.al. [@Srivastava2011] represent curves in ${{\mathbb R}}^n$ using the *square root velocity* function, using which they devise shape analysis tools. Another interesting representation of space curves is presented by Kessler[@Kessler2007], in which a space curve is represented via segments of planar curves called $E$-sets, and has been used for shape analysis of protein molecules. While there exists generalizations of models like the Fourier descriptors to space curves[@Reti1993], splines can be used for space curves without any modification. Splines were introduced by Schoenberg[@Schoenberg1964] and de Boor [@Boor1962] as interpolation methods, and have found use in signal & image processing[@Unser1993], computer graphics[@Bartels1987], manufacturing processes[@Weiss1997], and others. Several types and variations of splines have been developed based on choice of basis and constraints; B-splines, Bézier curves, Rational B-splines, Trigonometric splines, to name a few. Apart from applications several theoretical results have also been worked out[@Schumaker2007],[@Champion2009]. Following [@Schumaker2007], we briefly introduce trigonometric splines here, as the tangle shape space will be shown to be its subset later on.
Given a positive integer $m$, an $m$-dimensional space is defined as $$\mathcal{T}_m = \left\{
\begin{array}{ll}
span\{v_1,\ldots,v_m\}, \ \ m = 2r\\
span\{u_1,\ldots,u_m\}, \ \ m = 2r+1
\end{array}
\right.,$$ where $$\begin{aligned}
\{v_1,\ldots,v_m\} = & \{\cos(t/2),\sin(t/2),\ldots\\
&\ \cos[(r-1/2)t],\sin[(r-1/2)t]\}, \ \ m = 2r,\\
\{u_1,\ldots,u_m\} = & \{1,\cos(t),\sin(t),\ldots,\\
&\ \cos(rt),\sin(rt)\}\ \ m = 2r+1.\end{aligned}$$ Let $D^k, k \geq 0$ represent the derivative of order $k$ operator. Then, given a partition $\Delta = \{a = t_0 <t_1,\ldots,t_{k+1} = b\}$ of an interval $[a,b]$, and a vector $\mathcal{M} = (m_1,\ldots,m_k)$ of $k$ integers, with $1\leq m_i \leq m$, the space of trigonometric splines of order $m$, with multiplicities $\mathcal{M}$ is defined as $$\begin{aligned}
\mathcal{S}(\mathcal{T}_m,\mathcal{M},\Delta) = \left\{
\begin{array}{ll}
s\ |\ \exists \ s_0,\ldots,s_k \in \mathcal{T}_m \mbox{ with } s(t) = s_i(t)\\
\forall t \in [t_i,t_{i+1}], i = 0,\ldots,k, \mbox{ and } \\
D^{j-1}s_{i-1}(t_i) = D^{j-1}s_i(t_i)\\
j = 1,\ldots,m-m_i,\ \ i=1,\ldots,k.
\end{array}
\right\}
\label{eqn:trigspl}\end{aligned}$$ In words, a trigonometric spline is a collection of elements from $\mathcal{T}_m$, each defined on a sub-interval $[t_i,t_{i+1}]$, such that at common points of adjacent elements (also called *knots*), they satisfy continuity constraints for derivatives upto order $m-m_i-1$. Thus the space $\mathcal{S}(\mathcal{T}_m,\mathcal{M},\Delta)$ is $m + \sum_{i=1}^k m_i$ dimensional.
Of special interest to us is the space of trigonometric splines of order $m=3$, with $\mathcal{M} = (1,1,\ldots,1)$, henceforth denoted by $\mathcal{M}_1$. We get $\mathcal{T}_3 = \{1,\cos(t),\sin(t)\}$ in this case, while the constraints defined in Equation are $$\begin{aligned}
\label{eqn:T3con1}s_{i-1}(t_i) = s_i(t_i),\ \ i=1,\ldots,k\\
\label{eqn:T3con2}Ds_{i-1}(t_i) = Ds_i(t_i),\ \ i=1,\ldots,k.\end{aligned}$$ It can be seen that the space $\mathcal{S}(\mathcal{T}_3,\mathcal{M}_1,\Delta)$ is $k+3$ dimensional. In order to represent a space curve as a trigonometric spline, each coordinate is considered as a trigonometric spline function of the parameter $t$, and the space curve between a sub-interval $[t_i,t_{i+1}]$ of $\Delta$, is given by (for $i=0,\ldots,k$), $$\begin{aligned}
\label{eqn:T3}s_i(t) = a_i + b_i\cos(t-t_i) + c_i\sin(t-t_i),\ \forall t \in [t_i,t_{i+1}],\end{aligned}$$ where $a_i,b_i,c_i \in {{\mathbb R}}^3$ are the coefficient vectors that satisfy constraints given by Equations ,. The degrees of freedom available for a trigonometric spline space curve is thus $3k+9$.
In the next section, we model the Tangle shape space and show that it is a subset of $\mathcal{S}(\mathcal{T}_3,\mathcal{M}_1,\Delta)$.
Modeling Tangles {#sec:modeltangle}
================
In this work, we will represent tangle configurations using parametric space curves, formed by connecting $n$ copies of a quarter of a unit-circle (each copy called a link), translated and rotated appropriately. Depending on whether the first and last links are connected or not, we obtain a closed or an open tangle. Different configurations are obtained with the help of local rotations applied on each link. These local rotations have a global effect on the tangle, akin to the effect of changing a control point of a spline curve. In what follows, $s$ represents a parametric space curve, $s(t)$ denotes the point on the curve at parameter value $t$, while $\dot{s}(t)$ denotes the derivative of the functions $s$ at parameter value $t$, and represents the tangent to the curve at point $s(t)$. Unless stated otherwise, $(p,q,r) \in {{\mathbb R}}^3$ represents a row vector, while $(p,q,r)^T \in {{\mathbb R}}^3$ or any element $v \in {{\mathbb R}}^3$ will be assumed to be represented as column vectors.
Since each link is a translated and rotated version of a quarter of a unit-circle, without loss of generality, we assume the following canonical form of a quarter of a unit-circle which when translated and rotated appropriately, yields any given link: $$\label{eqn:can}c(t) = (\cos t,\sin t,0)^T,\ \forall t \in [0,\frac{\pi}{2}].$$ The tangent to $c(t)$ is $\dot{c}(t) = (-\sin t,\cos t,0)^T,\ \forall t \in [0,\frac{\pi}{2}]$, specifically, $\dot{c}(0) = (0,1,0)^T$ and $\dot{c}(\frac{\pi}{2}) = (-1,0,0)^T$. Let $s(t), t \in [0,n\frac{\pi}{2}]$ denote the $n$-link space curve, and let $t_i = \frac{i\pi}{2}, i = 0,\ldots,n$. Let $R_{\hat{n},\theta} \in {{\mathbb R}}^{3 \times 3}$ denote the rotation matrix with axis $\hat{n}$ and angle of rotation $\theta$. For a given space curve $s$ that represents an $n$-link tangle, let $s_i(t) = s(t),t \in [t_i,t_{i+1}]$ denote the $(i+1)^{st}$ link, where $i=0,\ldots,n-1$. Since any link is a rotated and translated form of the assumed canonical form $c$ from Equation , we have, $$\begin{aligned}
\label{eqn:sit}s_i(t) =& {R_{\hat{n}_i,\theta_i}}c(t-t_i) + T_i,\ \forall t \in [t_i,t_{i+1}],\end{aligned}$$ for some unit normal $\hat{n}_i$, angle $\theta_i$, and translation vector $T_i \in {{\mathbb R}}^3$. The tangent to the $(i+1)^{st}$ link is $$\begin{aligned}
\dot{s}_i(t) =& {R_{\hat{n}_i,\theta_i}}\dot{c}(t-t_i), \forall t \in [t_i,t_{i+1}],\end{aligned}$$ and specifically, $$\begin{aligned}
\label{eqn:1}\dot{s}_i(t_i) =& {R_{\hat{n}_i,\theta_i}}(0,1,0)^T\\
\label{eqn:2}\dot{s}_i(t_{i+1}) =& {R_{\hat{n}_i,\theta_i}}(-1,0,0)^T.\end{aligned}$$ Moreover, since ${\langle{\dot{c}(0)},{\dot{c}(\frac{\pi}{2})}\rangle} = 0$, ${\langle{\dot{s}_i(t_i)},{\dot{s}_i(t_{i+1})}\rangle} = 0$, and $||\dot{c}(0)|| = ||\dot{c}(\frac{\pi}{2})|| = ||\dot{s}_i(t_i)|| = ||\dot{s}_i(t_{i+1})|| = 1$.
Thus, one can determine ${R_{\hat{n}_i,\theta_i}}$ uniquely from an orthogonal pair of unit vectors $(\dot{s}_i(t_i), \dot{s}_i(t_{i+1}))$ by solving Equations ,, and conversely, given ${R_{\hat{n}_i,\theta_i}}$, one can determine the tangent vectors: $\dot{s}_i(t_i), \dot{s}_i(t_{i+1})$ from the same equations. Given two consecutive links $s_i$ and $s_{i+1}$, these should satisfy the following constraints: $$\begin{aligned}
\label{eqn:tancon1}s_i(t_{i+1}) =& s_{i+1}(t_{i+1})\\
\label{eqn:tancon2}\dot{s}_i(t_{i+1}) =& \dot{s}_{i+1}(t_{i+1}),\end{aligned}$$ the former stating that the end point of link $(i+1)$, $s_i(t_{i+1})$ should be the same as the starting point of link $i+2$, $s_{i+1}(t_{i+1})$, and the latter ensuring that the tangent vector at the end point of link $i+1$, $\dot{s}_i(t_{i+1})$ should be the same as the tangent vector at the starting point of link $i+2$, $\dot{s}_{i+1}(t_{i+1})$. The first constraint gives: $$\begin{aligned}
{R_{\hat{n}_i,\theta_i}}c(\frac{\pi}{2}) + T_i = R_{\hat{n}_{i+1},\theta_{i+1}} c(0) + T_{i+1}.\end{aligned}$$ Thus, if $s_i(t)$ is known and $R_{\hat{n}_{i+1},\theta_{i+1}}$ is known (using tangent vectors as discussed earlier), $T_{i+1}$ can be computed using $$\begin{aligned}
\label{eqn:ti}T_{i+1} = {R_{\hat{n}_i,\theta_i}}c(\frac{\pi}{2}) + T_i - R_{\hat{n}_{i+1},\theta_{i+1}} c(0).\end{aligned}$$ Let $V_i, V_{i+1}, V_{i+2}$ denote the tangent vectors $\dot{s}_i(t_i), \dot{s}_i(t_{i+1}) = \dot{s}_{i+1}(t_{i+1}), \dot{s}_{i+1}(t_{i+2})$ respectively. Since ${\langle{V_i},{V_{i+1}}\rangle} = {\langle{V_{i+1}},{V_{i+2}}\rangle} = 0$, one can represent the tangent vector $V_{i+2}$ as $V_{i+2} = R_{V_{i+1},\theta_i} V_i$, for some $\theta_i$. In words, given the orthonormal tangent vectors $V_i, V_{i+1}$, any unit vector $V_{i+2}$ which is orthogonal to $V_{i+1}$ can be produced via a rotation about the axis $V_{i+1}$ of the vector $V_i$.
Let $V_i = \dot{s}(t_i)$ denote the tangent vector at the initial point of $(i+1)^{st}$ link for $i = 0,\ldots,n-1$, and $V_n = \dot{s}(t_n)$ denote the tangent vector at the end-point of the $n^{th}$ link. Assuming $V_0,V_1$ to be given such that ${\langle{V_0},{V_1}\rangle} = 0, ||V_0|| = ||V_1|| = 1$, an $n$-link *open* tangle parametric space curve $s:[0,n\frac{\pi}{2}] \rightarrow {{\mathbb R}}^3$, is also uniquely represented by the set of angles $(\theta_0,\ldots,\theta_{n-2})$, where $V_{i+2} = R_{V_{i+1},\theta_i} V_i, i = 0,\ldots,n-2$. Thus, including the degrees of freedom in choosing the orthonormal vectors $V_0,V_1$ (which is three), and degrees of freedom available through a global translation (again three), the total degrees of freedom for an $n$-link open tangle is $n+5$.
A closed form expression for each $s_i(t)$ can be obtained by carefully observing Equations ,. It is not difficult to see that the first two columns of ${R_{\hat{n}_i,\theta_i}}$ in Equation are $-s_i(t_{i+1})$ or $-V_{i+1}$ and $s_i(t_i)$ or $V_i$. Substituting these in Equation and , and noting the special form of $c$ from Equation , one gets the following closed form expression: $$\begin{aligned}
\label{eqn:clsit}s_i(t) = & V_i\sin(t-t_i) - V_{i+1}\cos(t-t_i) + T_i, \forall t \in [t_i,t_{i+1}],\end{aligned}$$ for $i = 0,\ldots,n-1$, and $$\begin{aligned}
T_{i+1} =& V_i + V_{i+2} + T_i, i = 0,\ldots,n-2,\end{aligned}$$ with $T_0 = 0$. Comparing Equations , and with Equations , and , it is clear that the set of all tangle curves is a subset of $\mathcal{S}(\mathcal{T}_3,\mathcal{M}_1,\Delta)$.
In case shape as defined by Kendall[@Kendall1989] of an $n$-link open tangle is concerned, the degrees of freedom reduces (by six) to $n-1$. As far as shape representation for an $n$-link open tangle is of interest, one can factor out the global translation and rotation by fixing the first of the $n$ links to our canonical quarter circle $c$ from Equation , i.e., $s_0(t) = c(t), t \in [0,\frac{\pi}{2}]$. This implies fixing $V_0 = (0,1,0)^T$ and $V_1 = (-1,0,0)^T$. The angles $(\theta_0,\ldots,\theta_{n-2})$ are the degrees of freedom that generate $n$-link open tangles with different shapes[^2], and thus the shape space of an $n$-link open tangle is the $n-1$ dimensional torus $\mathbb{T}^{n-1}$.
Thus the set of $n$-link open tangle shapes $S_o$, can be modeled by any of the following equivalent ways: $$\begin{gathered}
S_o = \mathbb{T}^{n-1}\\
\mbox{ or }\\
S_o = \left\{(V_2,\ldots,V_n), V_i \in {{\mathbb R}}^3, i = 2,\ldots,n\ |\ {\langle{V_i},{V_{i+1}}\rangle} = 0,\right.\\
\left. i = 2,\ldots,n-1, ||V_i|| = 1, i=2,\ldots,n\right\}
\end{gathered}$$ with $V_0 = (0,1,0)^T$ and $V_1 = (-1,0,0)^T$.
$n$-link closed tangles
-----------------------
For $s:[0,n\frac{\pi}{2}]\rightarrow {{\mathbb R}}^3$ to represent a closed $n$-link tangle, it needs to satisfy two additional constraints: $$\begin{aligned}
s(t_n) =& s(t_0)
V_n =& V_0\\\end{aligned}$$ The first constraint ensures that the first and the last point coincide, while the second ensures continuity of the first derivative. The first condition can also be met by using the fact that line integral over a closed curve of its tangent vector field is zero, which in terms of $s$ can be written and simplified as, $$\begin{aligned}
\int_0^{n\frac{\pi}{2}} \dot{s}(t)\ dt & = \sum_{i=0}^{n-1} \int_{i\frac{\pi}{2}}^{(i+1)\frac{\pi}{2}} \dot{s}_i(t)\ dt\\
& = \sum_{i=0}^{n-1} {R_{\hat{n}_i,\theta_i}}\left(c(\frac{\pi}{2}) - c(0)\right)\\
& = \sum_{i=0}^{n-1} {R_{\hat{n}_i,\theta_i}}\left(\dot{c}(\frac{\pi}{2}) + \dot{c}(0)\right) = 0, \end{aligned}$$ the last equality resulting because of the special choice of $c$. Using Equations ,, and representing the tangent vectors with $V_i$, the constraint that the initial and end-point of the $n$-link tangle must coincide boils down to, $$\begin{aligned}
\sum_{i=0}^{n-1} V_i = 0.\end{aligned}$$ In case shape of the $n$-link closed tangle is of interest, we fix $V_0 = (0,1,0)^T$ and $V_1 = (-1,0,0)^T$. We thus need to ensure ${\langle{V_0},{V_{n-1}}\rangle} = 0$. With $V_{i+2} = R_{V_{i+1},\theta_i}, i = 0,\ldots,n-3$, the $n$-link closed tangle shape space is $S_c = F^{-1}(0)$, where $F:\mathbb{T}^{n-2} \rightarrow {{\mathbb R}}^4$ is defined as, $$\begin{aligned}
F(\theta_0,\ldots,\theta_{n-3}) = \left(\sum_{i=0}^{n-1} V_i,{\langle{V_0},{V_{n-1}}\rangle}\right).\end{aligned}$$ Thus, the degrees of freedom[^3] as far as shapes of $n$-link closed tangles is $n-6$, and the shape space is a subset of $\mathbb{T}^{n-2}$. The $n$-link closed tangle shapes $S_c$, can be modeled by any of the following equivalent ways: $$\begin{gathered}
S_c = \left\{(\theta_0,\ldots,\theta_{n-3}) \in \mathbb{T}^{n-2}\ \left|\right.\ F(\theta_0,\ldots,\theta_{n-3}) = 0\right\}\\
\mbox{ or }\\
S_c = \left\{(V_2,\ldots,V_{n-1}), V_i \in {{\mathbb R}}^3, i = 2,\ldots,n-1\ |\right.\\
{\langle{V_i},{V_{i+1}}\rangle} = {\langle{V_0},{V_{n-1}}\rangle} = 0, i = 1,\ldots,n-2,\\
\left.||V_i|| = 1, i=2,\ldots,n-1, \sum_{i=0}^{n-1} V_i = 0\right\},
\end{gathered}$$ with $V_0 = (0,1,0)^T$ and $V_1 = (-1,0,0)^T$. For computations on the tangle shape space, we will prefer the latter characterizations in terms of unit tangent vector sets (satisfying appropriate constraints), instead of elements of the appropriate torus, since closed form expressions relating the angle $\theta_i$ with the unit tangent vector $V_i$ become far too involved as the number of links increases. This is especially true for closed tangles.
As expected, the set of $4$-link closed tangle shape space contains one point, the circle starting with the canonical quarter $c$. It can also be shown that there are no closed tangle curves with $n=5$ links[^4].
Properties of Tangles
---------------------
We now briefly summarize the properties of tangles as space curves.
1. *Unit Speed curves:* Tangles as given by Equations , are unit-speed parametric space curves. This is obvious, since for the canonical form $||\dot{c}(t)|| = 1, \forall t$, while all links are obtained by rotating and translating this canonical form.
2. *Curvature:* The tangent at any point $t \in [t_i,t_{i+1}]$ is given by $$\begin{aligned}
\dot{s}_i(t) = V_i\cos(t-t_i) + V_{i+1}\sin(t-t_i).\end{aligned}$$ Note that the space of tangle curves was shown to be a subset of $\mathcal{S}(\mathcal{T}_3,\mathcal{M}_1,\Delta)$, hence second order derivatives may not exist at the knots/points connecting two links. But at all other points, the tangles are smooth. Since the curve is unit speed, the curvature magnitude is $|\kappa(t)| = ||\ddot{s}_i(t)||, \forall t \neq t_i$, $$\begin{aligned}
|\kappa(t)| = ||V_{i+1}\cos(t-t_i) - V_i\sin(t-t_i)|| = 1, \forall t \neq t_i\end{aligned}$$ and therefore $\kappa(t) = \pm 1, \forall t \neq t_i$.
3. *Torsion:* The unit normal at $t \in [t_i,t_{i+1}]$, is $$\begin{aligned}
\mathbf{n}(t) = V_{i+1}\cos(t-t_i) - V_i\sin(t-t_i).\end{aligned}$$ and therefore the *binormal vector* at $t \in [t_i,t_{i+1}]$, is $$\begin{aligned}
\mathbf{b}(t) = V_i \times V_{i+1},\end{aligned}$$ Since each link lies in the plane spanned by $V_i$ and $V_{i+1}$, for each $t \in (t_i,t_{i+1})$, the torsion $\tau$ is zero.
Computational Tools {#sec:comptools}
===================
We first provide an algorithm to approximate a given space curve with open tangles, followed by algorithms to compute geodesics and Log map in the open/closed tangle shape space.
Curve approximation using Tangles
---------------------------------
We assume that we are given a smooth (at least $\mathcal{C}^1$) constant speed parameterized space curve $p:\left[0,n\frac{\pi}{2}\right] \rightarrow {{\mathbb R}}^3$ which is to be approximated by an $n$-link open tangle $s$. In case only an ordered set of points in ${{\mathbb R}}^3$ is to be fit with a tangle, we fit a spline $p$ through the set of points. Our method tries to fit the tangent field of the tangle to the tangent field of the given curve in the least square sense, i.e., it minimizes the following cost: $$\begin{aligned}
\tilde{J}(s) = \int_0^{n\frac{\pi}{2}} ||\dot{p}(t) - \dot{s}(t)||^2\ dt\end{aligned}$$ The tangent field of the given space curve is either computed analytically if the parametric form is available, or computationally via a spline fit as discussed earlier. This gives, $$\begin{aligned}
\tilde{J}(s) = \sum_{i=0}^{n-1} \int_{t_i}^{t_{i+1}} ||\dot{p}(t) - V_i\cos(t-t_i) - V_{i+1}\sin(t-t_i)||^2\ dt.\end{aligned}$$ Since the tangle is uniquely specified (upto global rigid motions) once the tangent vectors at the connecting points are determined, we obtain a constrained minimization problem in terms of $V = [V_0,\ldots,V_n]$ (note here we do not fix $V_0$ and $V_1$): $$\begin{aligned}
J(V) = & \sum_{i=0}^{n-1} \int_{t_i}^{t_{i+1}} ||\dot{p}(t) - V_i\cos(t-t_i) - V_{i+1}\sin(t-t_i)||^2\ dt \\
\mbox{such that } & {\langle{V_i},{V_{i+1}}\rangle} = 0, i=0,\ldots,n-1,\\
\mbox{and } & ||V_i|| = 1, i = 0,\ldots,n,\end{aligned}$$ Sampling the tangent field of the given space curve $p$ and the tangle curve $s$ to be estimated uniformly (using methods from [@Wang2002], if required), and replacing the integral with a summation at these sample points in the above cost function $J$, gives us a constrained least square problem in the variables $V$, for which we deploy the penalty method from [@Luenberger2015][@Bertsekas2009]. Note that since only the tangent fields are aligned, an additional rigid (and global) transformation alignment is performed once the above optimization process ends. Given a closed parametric space curve, we can approximate it with a closed tangle configuration with the same method given above, with few additional constraints listed in Section \[sec:modeltangle\] ensuring that the tangle configuration is closed.
Geodesics on Tangle space
-------------------------
We now compute geodesics emanating at a given point in the tangle shape space in any given tangent direction. Geodesics on a Riemannian manifold $M$, can be computed via the *Riemannian Exponential map* (henceforth Exp map) as follows. For $p \in M$, and tangent space of $M$ at $p$ denoted by $T_pM$, the Exp map $Exp_p:T_pM \rightarrow M$, maps every $v \in T_pM$ to a point $q \in M$ obtained by moving along a geodesic starting at $p$ in the direction $\frac{v}{||v||}$, for a distance $||v||$, where $||.||$ is the Riemannian metric defined on $T_pM$. Thus given the Exp map $Exp$, one can compute segment of a geodesic $\gamma$ starting at $p$ along a tangent vector $v \in T_pM$ as $\gamma: t \mapsto Exp_p(tv)$, $t \in [0,1]$.
As discussed earlier, the $n$-link open tangle shape space $S_o$ is simply $\mathbb{T}^{n-1}$. Moreover, $\mathbb{T}^{n-1}$ is a Lie group[@Boothby2003], with $T_p(\mathbb{T}^{n-1}) = {{\mathbb R}}^{n-1}$ for any $p \in \mathbb{T}^{n-1}$. Identifying $\mathbb{T}^{n-1}$ with $S^{n-1}$, where $S$ is the unit circle in ${{\mathbb C}}$, we can use the Exp map defined on $S^{n-1}$ to compute geodesics on $\mathbb{T}^{n-1}$. For every $p = (p_0,\ldots,p_{n-2}) \in S^{n-1}$ and $\forall \theta = (\theta_0,\ldots,\theta_{n-2}) \in {{\mathbb R}}^{n-1}$, $Exp_p:{{\mathbb R}}^{n-1} \rightarrow S^{n-1}$ is defined as, $$\begin{aligned}
Exp_p(\theta) = (p_0\exp(i\theta_0),\ldots,p_{n-2}\exp(i\theta_{n-2})),
\label{eqn:expso}\end{aligned}$$ where $i = \sqrt{-1}$, and $\exp$ is the usual exponential function.
The Exp map on the space of closed tangles is more involved. We defer the discussion on whether the set of $n$-link closed tangle configurations forms a manifold or not to a later section. Here, we work under the assumption that geodesics are computed at points contained in a sufficiently large neighborhood where singularities do not occur. The geodesics can be computed in two ways, using each of the model presented in Section\[sec:modeltangle\]. For brevity, we present the method using the following model for $n$-link closed tangle shapes: $$\begin{aligned}
\nonumber S_c = \left\{(V_2,\ldots,V_{n-1}), V_i \in {{\mathbb R}}^3, i = 2,\ldots,n-1\ |\right.\\
\nonumber {\langle{V_i},{V_{i+1}}\rangle} = {\langle{V_0},{V_{n-1}}\rangle} = 0, i = 1,\ldots,n-2,\\
\label{eqn:Sc}\left.||V_i|| = 1, i=2,\ldots,n-1, \sum_{i=0}^{n-1} V_i = 0\right\},\end{aligned}$$ with $V_0 = (0,1,0)^T, V_1 = (-1,0,0)^T$ fixing the first link. Then $S_c = F^{-1}(0)$ is assumed to be an implicitly defined manifold, using the function $F$, where $F:{{\mathbb R}}^{3n-6} \rightarrow {{\mathbb R}}^{2n}$ encode the $2n$ constraints on the set of tangent vectors $v = (V_2,\ldots,V_{n-1}) \in {{\mathbb R}}^{3n-6}$ given in Equation. To simplify notations, let $p = 3n-6$, and $q = 2n$. The tangent space of $S_c$ at $v$ is given by $ker(JF(v))$, where $ker(A)$ denotes the kernel of matrix $A$, while $JF(v) \in {{\mathbb R}}^{q \times p}$ is the Jacobian matrix of $F$. Let $HF_i(v) \in {{\mathbb R}}^{p \times p}$ denote the Hessian of the constraint $F_i, i = 1,\ldots,q$. Following an optimal control strategy given by Dedieu & Nowicki [@Dedieu2005], in order to compute the exponential map at a point $v \in S_c$ of the tangent vector $u \in T_v(S_c)$, we solve the following set of differential equations in $v(t), t \in [0,1]$,: $$\begin{aligned}
\nonumber \dot{r}(t) = -\sum_{i=1}^q \mu_i HF_i(v(t)) \dot{v}(t),\\
\nonumber \dot{v}(t) = \left(Id - JF(v(t))^{\dagger} JF(v(t))\right) r(t),\\
\nonumber \mu(t) = - \left(JF(v(t))^T\right)^{\dagger} r(t),\\
v(0) = v, r(0) = u,\end{aligned}$$ where $r \in {{\mathbb R}}^p$ acts as an auxiliary variable, $A^{\dagger}$ is the pseudo-inverse of matrix $A$, and $A^T$ denotes transpose of $A$. This differential equation in the variables $(r,v)$ can be solved using explicit methods or implicit methods given in[@Dedieu2005]. We use explicit methods, since we are interested in solving the differential equations for a small interval $t \in [0,1]$, for which in practice the errors are negligible.
Log map on Tangle shape space
-----------------------------
We now compute the inverse of the Exp map, the Riemannian Log map. The Exp map is invertible only on a small open set of the manifold known as the cut locus.
For the $n$-link open tangle space $S_o$, since the Riemannian Exp map is given as Equation , it can be seen that the Log map at $p \in S^{n-1}$ for point $q \in S^{n-1}$, if it exists, is given as, $$\begin{aligned}
(\theta_0,\ldots,\theta_{n-2}) = \left(\arg\left(\frac{q_0}{p_0}\right),\ldots,\arg\left(\frac{q_{n-2}}{p_{n-2}}\right)\right) \in T_p(S_o),
\label{eqn:logso}\end{aligned}$$ For the $n$-link closed tangle shape space $S_c$, we use the shooting method [@Klassen2004],[@Mio2004] to compute the Riemannian Log map. Given $v_0 \in S_c$, in order to compute $a = Log_{v_0}(v_1) \in T_{v_0}(S_c)$ for a point $v_1 \in S_c$, the iterative method starts with an initial estimate $\hat{a}_0 = \Pi_{v_0}(v_1 - v_0)$, where $\Pi_{v_0}$ is the orthogonal projection operator to $T_{v_0}(S_c)$. Let $w_k = Exp_{v_0}(\hat{a}_k)$ be the point obtained by *shooting* with the current estimate $\hat{a}_k$. The estimate is updated using the parallel transport to $T_{v_0}(S_c)$ of the projection of the current error, i.e, $\hat{a}_{k+1} = \hat{a}_k + P\left(\Pi_{w_k}(v_1 - w_k)\right)$, where $P$ denotes the parallel transport to $T_{v_0}$ along the geodesic $\gamma(-t), t \in [0,1]$, where $\gamma(t)$ is the geodesic connecting $v_0$ and $w_k$ computed via the Exp map. This is repeated until convergence.
Let us briefly describe the parallel transport mechanism for transporting $r \in T_{w}(S_c)$ along a geodesic $\gamma \in S_c$ with $\gamma(0) = w, \gamma(1) = v_0$, to a vector $P(r) \in T_{v_0}(S_c)$. Let $r(0) = r$ and $r(1) = P(r)$. The conditions defining such a parallel transport are: $$\begin{aligned}
& r(t) \in T_{\gamma(t)}(S_c),\ t \in [0,1],\\
& \Pi_{\gamma(t)}\left(\frac{dr}{dt}(t)\right) = 0 \Rightarrow \frac{dr}{dt}(t) \in \left(T_{\gamma(t)}(S_c)\right)^{\perp},\ t \in [0,1].\end{aligned}$$ While the first equation above states that all vectors $r(t)$ should belong to the appropriate tangent space, the second states that the vector field $r$ obtained via parallel transport should be the one with a zero tangential acceleration along $\gamma$. Rewriting these two equations, we get for all $t \in [0,1]$, $$\begin{aligned}
JF(\gamma(t))r(t) = 0,\\
\frac{dr}{dt}(t) = JF(\gamma(t))^T\lambda(t),\end{aligned}$$ where $\lambda \in {{\mathbb R}}^q$. Differentiating the first equation above with respect to $t$, using the second equation above and some algebra gives us the following differential equation in $r$: $$\begin{aligned}
\label{eqn:pt} \frac{dr}{dt}(t) = -JF^T\cdot (JF^T)^{\dagger}\cdot JF^{\dagger} \cdot D^2F(\dot{\gamma},r),\end{aligned}$$ where dependence on $t$ of the right hand side has been suppressed to simplify the expression, and $D^2F(\dot{\gamma},r)$ is a vector in ${{\mathbb R}}^q$ whose $k^{th}$ component ($1\leq k \leq q$) is given by $\dot{\gamma}(t)^T HF_k(\gamma(t)) r(t)$. The above differential equation is solved using an explicit solver to compute $P(r) = r(1)$, for the given $r(0) = r$ and geodesic $\gamma$.
Apart from the shooting method, one may also use the path straightening approach[@Noakes1998] to compute the Log map.
Experiments {#sec:expts}
===========
We now demonstrate results for the computational tools developed in the last section. The algorithms described in the previous section have been implemented in MATLAB.
Curve approximation
-------------------
One may expect the tangle configurations to be able to replicate helical structures very well. As it turns out, very few helices can be represented by tangles. A helix has the following parameterization for $t \in [0,\frac{n\pi}{2}]$, $$\begin{aligned}
\label{eqn:helix}h(t) = \left(a\cos t,a\sin t,bt\right)^T. \end{aligned}$$ The tangent vector at any $t \in [0,\frac{n\pi}{2}]$, is $$\begin{aligned}
\dot{h}(t) = &\left(-a\sin t,a\cos t,b\right)^T. \end{aligned}$$ Even thought $h$ is not unit speed, it is a constant speed parameterization. At knot points $t_i = \frac{i\pi}{2}, i=0,\ldots,n$, the tangents are $$\begin{aligned}
\dot{h}(t_i) = \left(-a\frac{1-(-1)^i}{2},a\frac{1+(-1)^i}{2},b\right)^T. \end{aligned}$$ As can be seen from the above equation, in order for a helix to be a tangle curve, ${\langle{\dot{h}(t_i)
},{\dot{h}(t_{i+1})}\rangle} = 0, i = 0,\ldots,n-1$, which implies $b=0$. The same conclusion can be drawn from the fact that curvature of helix at any point is $\frac{a}{\sqrt{a^2+b^2}}$, while that of a tangle is $\pm 1$, wherever it exists. Thus for the curves to match, $b=0$. In Figure\[fig:OT\_curveapp\], we show results of our curve approximation algorithms for helices with various curvatures: $0.7514, 0.9951, 0.4$, and a straight line, using open tangles.
$
\begin{array}{cc}
\includegraphics[width=2in]{OT_CA_8} &
\includegraphics[width=2in]{OT_CA_8_good}\\
\includegraphics[width=2in]{OT_CA_8_lowk} &
\includegraphics[width=2in]{OT_CA_18_line}
\end{array}$
For curve approximation using closed tangles, we consider space curves of the form $$\begin{aligned}
h(t) = & (a\cos t + \cos(bt)\cos(ct), a\sin t + \cos(bt)\sin(ct),\\
& d\sin(ct))^T, t \in [0,2\pi],
\label{eqn:CTCA} \end{aligned}$$ where $a,b,c$ and $d$ are real constants. We show couple of examples of such curves and the closed tangles that approximate them in Figure \[fig:CT\_curveapp\]. As can be seen from this analysis, using tangles for curve approximation will find only limited applications, perhaps to model helical structures in protein molecules.
$
\begin{array}{cc}
\includegraphics[width=2in]{CT_CA_12} &
\includegraphics[width=2in]{CT_CA_27}
\end{array}$
Geodesics on Tangle space
-------------------------
As shown in Section \[sec:modeltangle\], geodesics on $n$-link open tangle space can be obtained via the Exp map on $\mathbb{T}^{n-1}$ (refer Equation ). For a few $p \in \mathbb{T}^{n-1}$ and $v \in T_{p}(\mathbb{T}^{n-1})$, samples of the geodesic $\gamma:t \mapsto Exp_{p}(tv)$ are shown in Figure \[fig:geoOT\]. Note here apart from round-off/precision errors, there are no other numerical inaccuracies, since the Exp map has an explicit form.
$
\begin{array}{cc}
\includegraphics[width=2in]{OT_geo_6} &
\includegraphics[width=2in]{OT_geo_17}
\end{array}$
Geodesics on $n$-link closed tangle space are much harder to compute as shown in Section \[sec:comptools\]. Examples of geodesics computed via the Exp map defined earlier, are shown in Figure \[fig:geoCT\]. We demonstrate two cases here: geodesics on the $n$-link closed tangle shape space $S_c$, and geodesics on the $n$-link closed tangle space. The former being different from the latter in the aspect that global rigid motions have been factored out by fixing the first link to be the canonical quarter of unit circle going from $(0,1,0)$ to $(1,0,0)$. In all examples shown in Figure \[fig:geoCT\], the maximum deviation of angles between tangent vectors of consecutive links from right angles is strictly less than $0.1$ degrees, while all tangent vectors computed are vectors with a maximum deviation less than $0.01$ from being unit norm. This shows the accuracy of the computational method.
$
\begin{array}{ccc}
\includegraphics[width=2in]{CTvres_geo_8} &
\includegraphics[width=2in]{CTvres_geo_24} &
\includegraphics[width=2in]{CTvall_geo_18}
\end{array}$
Log map
-------
Log map on $n$-link open tangle shape space has the closed form description given in Equation . We do not show results for the Log map on open tangle shape space, since both Exp map and Log map have nice closed form solutions, and we observe $Exp_p(Log_p(q)) = q$ upto round-off/precision errors. Let $v \in T_p(S_o)$ be such that $q = Exp_p(v)$, then $Log_p(q)$ may not equal $v$ depending on whether $q$ is in the cut-locus of $p$.
Results for Log map on the $n$-link closed tangle shape space are shown in Figure \[fig:Logmap\]. The algorithm stops when the Euclidean distance between the target tangle curve and the exponential of the estimated Log map is below a certain threshold. Note that for any tangle obtained while computing the Log map, the maximum deviation of angles between tangent vectors of consecutive links from right angles was strictly less than $0.1$ degrees, while all tangent vectors computed were vectors with a maximum deviation less than $0.01$ from being unit norm.
$
\begin{array}{cc}
\includegraphics[width=2in]{CTLog_8} &
\includegraphics[width=2.5in]{CTLog_18}
\end{array}$
Discussion and Conclusion {#sec:discon}
=========================
We now address the question: Is the $n$-link closed tangle shape space a manifold or not. In case $S_c = F^{-1}(0)$ is a manifold, by the constant rank level set theorem[@Lee2003], the rank of $F$ needs to be constant on $S_c$. The tangles provided in Figure \[fig:cntexample\] give a counterexample. For $6$-link closed tangles, the constraint function defined after Equation is of the kind $F:{{\mathbb R}}^{12} \rightarrow {{\mathbb R}}^{12}$. As it turns out, for the tangle shown on the left in Figure\[fig:cntexample\], the rank of $F$ (defined as rank of $JF$) is $11$, while the rank of $F$ for the tangle shown on the right is $12$. This makes the Jacobian a full rank matrix. Thus there is neighborhood in which the Jacobian remains full rank, implying that the two tangle configurations belong to two different connected components of $S_c$. The tangle on the left is part of a $1$D path on $S_c$ ($1$ D tangent space), while the latter is an isolated point ($0$ D tangent space). This confirms that $S_c$ with $6$ links is not a manifold and contains at least two connected components, one being a $1$D path (in fact a closed one), while the other being an isolated point. Few samples of the geodesic for the tangle on the left from Figure \[fig:cntexample\] along the only tangent vector is shown in Figure \[fig:Exptangle6\]. When the norm of the tangent vector is increased, the corresponding geodesic reaches the initial tangle, thus forming a closed path in $S_c$.
$
\begin{array}{cc}
\includegraphics[width=2in]{CT6_1D_col} &
\includegraphics[width=2in]{CT6_0D_col}
\end{array}$
{width="2in"}
Having said that, we conjecture that with a larger $n$, $S_c$ has only one connected component. This stems from the intuition that with more degrees of freedom available, it should be possible to deform one tangle configuration to another.
Forming complex space curves from arcs of planar circles is a fascinating aspect of tangles. This is analogous to creating $3$D meshes from planar triangles. It would be interesting to analyze $3$D meshes, where every triangle is obtained via a rigid transformation of a single fixed equilateral triangle. The expressibility of such a mesh would be limited though.
To conclude, we summarize our contribution. We have given two models of $n$-link open and closed tangle shape spaces. We have shown that tangles are a subset of trigonometric splines. We have presented algorithms for space curve approximation, computing geodesics and Log map on the space of $n$ link closed/open tangles.
[^1]: Dhirubhai Ambani Institute of Information & Communication Technology, Gandhinagar, India. E-mail: aditya\_tatu@daiict.ac.in.
[^2]: Some $n$-link tangles may have additional symmetries which we ignore in this work.
[^3]: The reason behind not using the term dimension in place of degree of freedom will be made clear in a later section.
[^4]: We avoid giving the proof here as it lacks elegance and it takes the paper in a tangential direction
|
---
abstract: '.2in We propose a new evolution equation for the gluon density relevant for the region of small $x_B$. It generalizes the GLR equation and allows deeper penetration in dense parton systems than the GLR equation does. This generalization consists of taking shadowing effects more comprehensively into account by including multigluon correlations, and allowing for an arbitrary initial gluon distribution in a hadron. We solve the new equation for fixed $\alpha_s$. We find that the effects of multigluon correlations on the deep-inelastic structure function are small.'
---
[[O]{}]{}[[O]{}]{} c ł Ł ø Ø v ${\left(}
\def$[)]{} [[Tr]{}]{} \#1[[cite\#\#1\#\#2[\#\#1]{}]{}]{} \#1[cite[\#1]{}]{}
[[tr]{}]{} [[Tr]{}]{} \#1[{\#1}]{} \#1[{\#1}]{} \#1\#2[[\#1\#2]{}]{} \#1\#2\#3[[\^2 \#1\#2 \#3]{}]{} \#1[\#1|]{} \#1[| \#1]{} \#1[\#1]{} \#1[/]{} \#1[| \#1|]{} \#1[\#1\^]{} [[12]{}]{} \#1[ ]{} \#1\#2[ ]{} \#1[[eq. (\[\#1\])]{}]{} \#1[[Eq. (\[\#1\])]{}]{} \#1\#2[[eqs. (\[\#1\])–(\[\#2\])]{}]{} \#1\#2[ $$\begin{array}{l}
#1 \\
\bentarrow #2
\end{array}$$ ]{} \#1\#2\#3[ $$\begin{array}{r c l}
#1 & \rightarrow & #2 \\
& & \bentarrow #3
\end{array}$$ ]{} \#1\#2\#3\#4[ $$\begin{array}{r c l}
#1 & \rightarrow & #2#3 \\
& & \phantom{\; #2}\bentarrow #4
\end{array}$$ ]{} \#1\#2\#3\#4\#5[ $$\begin{array}{r c l}
#1 & \rightarrow & #2#3 \\
& & \:\raisebox{1.3ex}{\rlap{$\vert$}}\raisebox{-0.5ex}{$\vert$ } \phantom{#2}\!\bentarrow #4 \\
& & \bentarrow #5
\end{array}$$ ]{} \#1\#2\#3[ [*Ann. Phys. (NY)* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Ann. Rev. Nucl. Part. Sci.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Nucl. Phys.* ]{}[**B\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Phys. Lett.* ]{}[**B\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Phys. Rev.* ]{}[**D\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Phys. Rep.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Phys. Rev. Lett.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Prog. Theor. Phys.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Rev. Mod. Phys.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Z. Phys.* ]{}[**C\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Mod. Phys. Lett.* ]{}[**A\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Nuovo Cim.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Yad. Fiz.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Sov. J. Nucl. Phys.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Sov. Phys.* ]{}[JETP ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*JETP Lett.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*(Sov. J. Nucl. Phys.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*(Sov. Phys. JETP* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[[*(JETP Lett.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Zh. ETF* ]{}[**\#1**]{}\#2 \#3]{} \#1\#2\#3[ [*Comm. Math. Phys.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2\#3[ [*Comp. Phys. Commun.* ]{}[**\#1**]{} \#2 \#3]{} \#1\#2[ [*Dissertation,* ]{}[\#1 ]{} 19\#2]{} \#1\#2\#3[ [*Diplomarbeit,* ]{}[\#1 \#2]{} 19\#3 ]{} \#1\#2\#3[ [*ibid.* ]{}[**\#1**]{} \#2 \#3]{} \#1[ [*J. Phys*]{}. [**G\#1**]{}]{}
CERN-TH/95-61\
TAUP-2226-95\
CBPF NF-012/95\
March 1995
0.7 true cm
Eric Laenen
[^1]
0.2 true cm [*CERN Theory Division\
CH-1211, Geneva 23\
Switzerland*]{} 0.6cm
Eugene Levin
[^2] [^3]
0.2 true cm [*Mortimer and Raymond Sackler Institute of Advanced Studies\
School of Physics and Astronomy, Tel Aviv University\
Ramat Aviv, 69978, Israel*]{}\
and\
[*LAFEX, Centro Brasileiro de Pesquisas Físicas / CNPq\
Rua Dr. Xavier Sigaud 150, 22290 - 180, Rio de Janiero, RJ, Brasil*]{}
.1in
Introduction
============
Our objective in this paper is to derive a new evolution equation describing the behavior of large partonic densities. The fact that parton densities increase as the Bjorken scaling variable $x_B$ decreases follows directly from linear evolutions equations such as the GLAP [@GLAP] or BFKL [@BFKL] equation, and has been experimentally observed by both HERA collaborations [@HERA]. They observe a powerlike behavior of the deep inelastic structure function \[ONE\] F\_2 (x\_B,Q\^2) \~x\^[- \_0]{}\_B, with $\omega_0 \simeq 0.3 - 0.5$. This behavior is predicted by the BFKL equation and is not inconsistent with the GLAP equation. (Such powerlike growth at small $x$ can be imitated by a solution of the GLAP equation with a distribution that is constant in $x$ at a low initial scale $Q_0$.)
From a physical point of view such behavior is inconsistent with unitarity. This fact necessitates a change in the evolution equation in the region of small $x_B$. The first attempt, more than ten years old, to write down a new equation led to the nonlinear GLR equation [@GLR]. See [@REVIEW] for a more extensive review.
We will explain in this paper that the GLR equation only includes two-gluon correlations in the parton cascade. Multigluon correlations should be essential to solving the small $x_B$ problem, at least theoretically [@TWOTW; @HIGHTW]. It is the aim of this paper to take such correlations into account.
In section 2 we derive and explain the limitations of the GLR equation. We consider multigluon correlations in section 3 by employing the relation between these correlations and high twist contributions to the deep-inelastic structure function. In section 4 we suggest the evolution equation which takes these correlations into account. Some particular solutions and a comparison with the GLR equation are discussed in section 5, while we discuss the general solution of the new evolution equation in section 6. We summarize and conclude in section 7.
The GLR Equation
================
In the region of small $x_B$ and large $Q^2$ we face a system of partons at mutually small distances (thus the QCD coupling $\alpha_s$ is still small), but dense enough that the usual perturbative QCD (pQCD) methods cannot be applied. The physics that governs this region is nonperturbative, but of a different nature than the one associated with large distances. The latter corresponds to the confinement region, and is usually analyzed using lattice field theory or QCD sum rules. In contrast, we encounter here a situation where new methods must be devised to analyze such a dense relativistic system of partons in a non-equilibrium state. We need, in fact, new quantum statistical methods to describe the behavior of such a system and to chart this unknown region. We are unfortunately only at the beginning of this exploration.
On the upside, we can approach this kinematical region in theory from the pQCD region, and assume that in a transition region between pQCD and high density QCD (hdQCD) we can study such a dense system in some detail. To illustrate what new physics one might anticipate in this transition region let us look at deep-inelastic scattering and compare with the pQCD results for this process. We can expect the following phenomena in the transition region:
\(i) As $x_B$ decreases the total cross section $\sigma (\gamma^* N)$ grows and, near the border with the hdQCD domain, becomes even comparable with the geometrical size of the nucleon $\sigma(\gamma^* N) \rightarrow \alpha_{em}
\pi R^2_N$. Here the cross section should be a smooth function of $\ln Q^2$.
\(ii) Although the parton language can be used to discuss the main properties of the process, interactions between partons become important, especially the annihilation process. This interaction induces screening (a.k.a. shadowing) corrections.
\(iii) In this particular kinematical region such screening corrections are fortunately under theoretical control. They modify however the pQCD linear evolution equation. The correct evolution equation now becomes nonlinear.
First some nomenclature. In this paper we will only deal with the gluon density in the nucleon, $x_B G(x_B,Q^2)$, which can be measured fairly directly at small $x_B$ in e.g experiments on diffraction dissociation or heavy quark production in deeply inelastic processes or even directly from the deep-inelastic structure function $F_2(x_B,Q^2)$ [@EKL]. We will thus also refer to it as the gluon structure function.
We will now give a simple derivation of this nonlinear equation, based on physical concepts. As we stated, the main new processes that we must consider at high density are parton-parton interactions. To incorporate these in our physical descriptions we must identify a new small parameter that controls the accuracy of calculations involving these interactions. Such a parameter is \[W\] W= , where $\rho$ is the gluon density in the transverse plane =. Here $R_N$ characterizes the area of a hadron which the gluons populate. It is the correlation length of gluons inside a hadron. Naturally, this radius must be smaller than the radius of a hadron (proton). Since this paper is mostly devoted to the discussion of purely theoretical questions we will not specify further the value of $R_N$. However it should be stressed that $R_N$ is nonperturbative in nature: all physics that occurs at distance scales larger than $R_N$ is nonperturbative.
The first factor in (\[W\]) is the cross section for gluon absorption by a parton in the hadron. Hence $W$ has the simple physical meaning of being the probability of parton (mainly gluon) recombination in the parton cascade. The unitarity constraint mentioned in the introduction can be represented as [@GLR] W1. Thus $W$ is indeed the small parameter sought. The parton cascade can be expressed as a perturbation expansion in this parameter. This perturbative series can in fact be resummed [@GLR] and the result understood by considering the structure of the QCD cascade in a fast hadron.
There are two elementary processes in the cascade that impact on the number of partons. ( 12); [probability]{} \_s ; $${\rm annihilation}\,\,\,\,(\, 2\,\rightarrow\,1\,);\,\,\,
\,\,{\rm probability}\,\,\propto\,\,\a^2_s\,
d^2\, \rho^2 \,\,\propto\,\,\a_s^2\,\frac{1}{Q^2}\, \rho^2\,\,,$$ where $d$ is the size of the parton produced in the annihilation process. In the case of deep-inelastic scatterinq $d^2 \sim 1/Q^2 $.
When $x_B$ is not too small only the splitting of one parton into two counts because $\rho$ is small. However as $x_B \rightarrow 0$ annihilation comes into play as $\rho$ increases.
This simple picture allows us to write an equation for the change in the parton density in a ‘phase space’ cell of volume $\D\ln\frac{1}{x_B} \D\ln Q^2$: \[EVPHYS\] = -\^2, where $N_c$ is the number of colors. In terms of the gluon structure function \[GLR\] = x\_B G (x\_B, Q\^2 )- ( x\_B G (x\_B,Q\^2 ) )\^2. This is the so-called GLR equation [@GLR]. To determine the value of $\gamma$ and the understand the kinematical range of validity of (\[GLR\]) this simple physical description does not suffice; rather one must analyze the process carefully in $W$-perturbation theory [@GLR] [@MUQI]. The result for $\gamma$ was found to be [@MUQI] $$\gamma\,\,\,=\,\,\,\frac{81}{16} \,\,\,\,\,\mbox{\rm for} \,\,N_c\,\,=\,\,3.$$ We would like to emphasize that the main assumption in the above derivation was that \[RHOSQ\] P\^[(2)]{} \~\^2, where $P^{(2)}$ denotes the probability for two gluons in the parton cascade to have the same fraction of energy $x$ and tranverse momentum (characterized by $r\simeq\ln Q^2$).
By assuming (\[RHOSQ\]) we neglect all correlations between the two gluons other than the fact that they are distributed in the hadron disc of radius $R_N$. In the large $x_B$ region this assumption is plausible because the correlations are power suppressed, and the densities involved are small. In the small $x_B$ region we cannot justify (\[RHOSQ\]), even when it holds for large $x_B$.
Induced multigluon correlations
===============================
In [@TWOTW] it was shown that the problem is oversimplified when one assumes that the probability of annihilation is simply proportional to $\rho^2$ in deriving the GLR equation. It was found that e\^, where $Q_0$ is the initial virtuality in the parton cascade. This ratio increases with decreasing $x_B$. Consequently we must take dynamical correlations into account, which could change the GLR equation crucially.
The key to the calculation of parton correlations was suggested by Ellis, Furmanski and Petronzio in [@ELPET], and was developed further in [@BULI]. It was shown that gluonic correlations are directly related to higher twist contributions in the Wilson Operator Product Expansion (OPE) arising from so called quasi partonic operators. According to the OPE, the gluon structure function can be written as \[HTSER\] x\_B G (x\_B,Q\^2 ) & =x\_B G\^[(1)]{} (x\_B,Q\^2)+ x\^2\_B G\^[(2)]{}(x\_B.Q\^2)\
& ...+... x\^[n]{}\_B G\^[(n)]{} (x\_B,Q\^2) ... where the $n$’th term results from the twist $2n$ quasi-partonic operator. The probability density $P^{(n)}$ to find $n$ gluons in the cascade with the same $x$ and $Q^2$ can be directly expressed through the $n$’th term in the above expansion ($P^{(n)} = x_B^n G^{(n)}(x_B,Q^2)/(\pi R^2_N)^n, \;\, P^{(1)} = \rho$).
We recently determined the anomalous dimensions $\gamma_{2n}$ of these high twist operators [@HIGHTW] to next-to-leading order in the number of colors $N_c$. This was done by reducing the complicated problem of gluon-gluon interactions to rescattering of colorless gluon ‘ladders’ (Pomerons) in the $t$-channel. In [@TWOTW] it was shown that this approach works for the case of the anomalous dimension of the twist 4 operator. The fact that there is no Pomeron ‘creation’ or ‘absorption’ in the $t$-channel means that we are dealing with a quantum mechanical problem (not a field theoretical one) in the calculation of the $\gamma_{2n}$ anomalous dimension. The problem then amounts to calculating the ground state energy of an $n$-particle system with an attractive interaction given by a four-particle contact term (see Fig.1) of strength $\lambda$. Its value can be calculated to be =4 , where $\delta = 1/(N_c^2-1)$ if one only takes color singlet ladders into account. Including the other color states renormalizes $\delta$ to $0.098$ [@TWOTW]. This effective theory is two-dimensional (the two dimensions corresponding to $\ln(1/x)$ and $\ln(Q^2)$) and is known as the Nonlinear Schrodinger Equation. It is well known that this model is exactly solvable. Translating the ground state energy of this model into the value of the anomalous dimension led to [@HIGHTW] \[ANDIM\] \_[2n]{} = 1+ (n\^2-1) , where $\bar \alpha_s = \alpha_s N_c/\pi$, $\omega = N-1$, $N$ being the Mellin-conjugate variable to $x_B$ \[MELLIN\] f(N) = \_0\^1 dx\_B x\_B\^[N-1]{}f(x\_B) f() = \_0\^dy e\^[-y]{}\[x\_Bf(x\_B)\], where $f$ is an arbitrary function and $y=\ln(1/x_B)$.
The answer (\[ANDIM\]) is only reliable when $\delta^2 n^2/3 \ll 1$ [@HIGHTW]. Thus to check selfconsistency we must first generalize the GLR equation based on (\[ANDIM\]) and understand what value of $n$ is important for the deep-inelastic structure function. If the answer is inconsistent with the condition $\delta^2 n^2/3 \ll 1$ we must try and find the expression for the anomalous dimension valid for any $n$.
Let us note that if we neglect the term proportional $\delta^2$ in (\[ANDIM\]) we can estimate $x_B^n G^{(n)}(x_B,Q^2)$ via the inverse Mellin transform \[EIK\] x\^n\_B G\^[(n)]{}(x\_B,Q\^2)=\_C de\^[(y +\_[2n]{} ()r)]{} M\^[(n)]{} (,Q\^2=Q\^2\_0), where $y = \ln(1/x_B)$, $r = \ln(Q^2/Q^2_0)$, $Q_0$ being the initial virtuality in the parton cascade. The contour is to the right of all singularities in M as well as to the right of the saddle point ($\omega_S$) which is given by \[SADDLE\] {y+\_[2n]{}r}|\_[= \_S]{}=0. Thus in the saddle point approximation for $\delta=0$ x\^n\_B G\^[(n)]{}(x\_B,Q\^2) \~\[ x\_B G (x\_B, Q\^2)\]\^[n]{}, in particular $P^{(2)} = \rho^2$. We also assume here the factorization of the matrix element $M^{(n)} \,=\,(\,M^{(1)}\,)^n$; this expresses the physical assumption that there are no correlations between gluons other than the fact that they are distributed in a disc of radius $R_N$. We will comment more about this assumption further on. In this sense the GLR equation is only the lowest order approximation in $\delta^2$ even if we assume the factorization of the matrix element and thus, strictly speaking, is valid only in the limit of a large number of colors. The term proportional to $\delta^2$ in (\[ANDIM\]) is clearly responsible for induced gluon correlations and we take it seriously in this paper. In next section we will therefore generalize the GLR equation.
A New Evolution Equation
========================
The first step in such a generalization is to make an [*ansatz*]{} for $P^{(n)}$ using the same approach as for the GLR equation, viz. the competition of two processes in the parton cascade. Thus, in analogy to the derivation of (\[EVPHYS\]) we write \[GENGLR\] = C\_[2n]{} P\^[(n)]{}(x\_B,Q\^2)-nP\^[(n + 1)]{}(x\_B,Q\^2), where $C_{2n}=\gamma_{2n}\,\omega$. The factor $n$ in front of the second term on the right hand side of (\[GENGLR\]) reflects the fact that in the Born approximation $n + 1$ gluons annihilate in $n$ gluons through the subprocess $ {\it gluon + gluon \,\rightarrow gluon}$, which corresponds to the $two-ladder \rightarrow one-ladder$ transition with the strength of the triple ladder vertex $\gamma$. There are $n$ such possibilities due to the time ordering of gluon emission. Note that the infinitely recursive set of equations can be cut off at any level by imposing e.g. $P^{(n)} = P^{(n-m)} P^{(m)}$. The GLR equation is the case $P^{(2)} = (P^{(1)})^2$. Since we operate under the assumption that high twists are essential for small enough $x_B$ we must however consider the whole series in (\[HTSER\]), which we now do using eqs. (\[GENGLR\]).
Let us introduce the generating function \[GEXP\] g(x\_B,Q\^2, )=\^\_[ n = 1]{} e\^[n ]{} g\^[(n)]{}, where $g^{(n)}=x^n_B G^{(n)}(x_B,Q^2)$. Comparing with (\[HTSER\]) we see that for the full structure function \[G\] x\_B G(x\_b,Q\^2) = Q\^2 R\_N\^2 g (x\_B,Q\^2, = - (Q\^2 R\_N\^2)). The recursive set of equations (\[GENGLR\]) can be summarized in one equation for $g$: \[GENGLRPAR\] = |\_s + ( - ) - \_s\^2 e\^[- (Q\^2 R\_N\^2)]{} e\^[ - ]{}( -g). To solve this linear, 4th order partial differential equation in three variables, we must impose some boundary and initial conditions, on the $Q^2$ and $x_B$ behavior respectively.
The boundary condition is straightforward , () ; g (x\_B,Q\^2,) e\^ g\_[LLA]{} (x\_B,Q\^2), where $g_{LLA} $ is the solution of the standard GLAP evolution equation for the leading twist gluon density in leading $\ln Q^2$ approximation.
The initial condition is much more difficult, because we need $g(x_B = x_B^0, Q^2, \eta)$ for solving (\[GENGLRPAR\]), whereas experimentally we can only measure the structure function, which is at fixed $\eta$. In other words, we need detailed information on the gluon distribution in a hadron at large $x_B$. We can make the following suggestion (although others are possible) \[EFOR\] g ( x\_B\^0,Q\^2,)= $$\sum^{\infty}_{n = 1} e^{n \eta} \frac{ ( - 1
)^n}{n!} \cdot [\,g_{LLA} (x_B^0,Q^2)\,]^n \,=\, 1- \exp ( -
e^{\eta} g_{LLA} (x_B^0,Q^2)\,).$$ This can be recognized as the usual eikonal approximation for the virtual gluon-hadron interaction [@MUE]. The virtues of this expression lie in the fact that it is simple, and that it has the transparent physical meaning of reflecting the assumption that there are no correlations between gluons with $x\sim 1$ other than that they are distributed in a hadron disc of radius $R_N$.
If one replaces the hadron with a nucleus, one can prove such an approach, which corresponds to the so-called Glauber Theory of shadowing corrections. In the deep-inelastic scattering case an expression of this type was discussed by A. Mueller in [@MUE].
To summarize this section, we have proposed a new evolution equation which has two new features over and above the GLR equation:
\(i) It includes induced multigluon correlations.
\(ii) It allows an arbitrary initial condition not necessarily an eikonal one, unlike the case of the GLR equation. We recall that the GLR equation has been proven only under the assumption of the factorization property of the matrix elements, which corresponds to an eikonal initial condition [@GLR]. By including all twists in our evolution equation we overcome the need for a reductive initial condition, such as (\[EFOR\]). One could e.g. try to solve (\[GENGLRPAR\]) using an initial condition with correlated gluons at large $x$ and study the consequences of its evolution, with or without $\delta$. Because such initial correlations must be very small, and because our main interest lies in comparing with the GLR equation, we do not pursue this line of inquiry here.
Approximate Solutions
=====================
In the next section we will discuss the general solution to . Here we will give approximate solutions for various special cases. In particular we try to answer the following questions: how does the nonlinear GLR equation follow from our present linear equation; what value of $n$ is typically relevant in the sum (\[GEXP\]) and how do the corrections to the ordinary GLAP evolution due to (\[GENGLR\]) differ from those due to the GLR equation?
The GLR Equation from the Generalized Equation
----------------------------------------------
The first question that arises is how the nonlinear GLR equation is contained in the linear equation (\[GENGLR\]) if we neglect the term proportional to $\delta^2$. Specifically, we would like to establish the equivalence of \[SIMPLEEX\] =|\_s -\_s\^2e\^[- (Q\^2 R\_N\^2)]{} e\^[ - ]{}(-g), to the nonlinear GLR equation. Let us parametrize the solution in the form g (x\_B,Q\^2,)= (e\^ F(y=-x\_B, r = (Q\^2 R\_N\^2))). Because we dropped the gluon-correlation term from (\[GENGLR\]) we can impose the ‘no-correlation’ initial condition at $x_B \sim 1$. This is the only initial condition the GLR equation allows [@GLR]. In the set of “fan” diagrams (Fig.3) that the GLR equation sums the initial distribution has the form \[FANIN\] g(x\_[B]{}\^0,Q\^2,)=\^\_[n = 1]{} e\^[n]{}( - 1 )\^n \^n, which gives \[FIFUN\] (t) =. The absence of the $1/n!$ in the above equation compared with (\[EFOR\]) is explained as follows. The $1/n!$ in (\[EFOR\]) enforces the correct time ordering of the produced partons in the parton cascade, related to diagrams of production of $n$ parton shadows (see Fig.4 which shows a case in which three parton shadows are produced). In the fan diagram of Fig.3 we do not have to enforce the correct time ordering because it is already included via the vertex $\gamma$ [^4]. Thus the initial condition of just corresponds to the sum of “fan" diagrams, of which Fig.3 is the lowest order example, with the assumption that there are no correlations between gluons inside the proton. We will see in section 6 that the main properties of the full solution of do not depend on the form of the initial condition. The reduction to the GLR equation does however.
With (\[FIFUN\]) it is now straightforward to check that reduces to the GLR equation to first order in $e^{\eta}$ (recall that finally we must put $\eta = -\ln(Q^2 R_N^2) \ll 1$), with $F(y,r) = x G(x,Q^2)$.
Relevant Twists in the Solution
-------------------------------
Here we try to follow the recipe mentioned below eq. (\[MELLIN\]): to see if our approach is consistent we must determine what values of $n$ are relevant in the solution to (\[GENGLRPAR\]). Should those values not be consistent with $\delta^2 n^2/3 \ll 1$ then we must generalize the expression (\[ANDIM\]) for $\gamma_{2n}$ such that it is valid for all $n$. If they are consistent we can maintain (\[ANDIM\]) and thus (\[GENGLRPAR\]).
However, to determine the relevant $n$’s we need the general solution to eq. (\[GENGLRPAR\]), which is presented in section 6. Here we will perform some rough estimates.
Let us first take a ‘worst case’ scenario by letting the term proportional to $\delta^2$ dominate, and by dropping the first term and last term on the RHS of (\[GENGLRPAR\]): $$\label{DELTAEQ}
\frac{\pa^2 g (x_B,Q^2, \eta )}{\pa \ln \frac{1}{x_B} \pa \ln Q^2 }\,\,=
\,\,\,\frac{\bar \alpha_s \delta^2}{3} (\frac{\partial^4 g}{\partial\eta^4}
- \frac{\partial g}{\partial\eta^2})$$ To solve this equation we perform a Laplace transform in the variable $\eta$ g(x\_B,Q\^2,p) \_[-]{}\^0 d e\^[p]{} g(x\_B,Q\^2,). Then \[LAPDELTA\] = p\^2(p\^2 - 1)g (y,r,p), where $ y=\ln(1/x_B), \;\;r=\ln(Q^2/Q^2_0)$. The solution to (\[LAPDELTA\]) is the Bessel function $I_0(2\sqrt{\frac{\bar\a_s\delta^2}{3}p^2(p^2 - 1)y r})$. The most general solution is then \[SOLU\] g (y,r,)=\_C e\^[- p]{} (p) I\_0( 2 ), where the contour $C$ runs to the right of all singularities in $p$. The function $\phi(p)$ is fixed by imposing an initial condition (e.g. (\[EFOR\])) g( 0,r ,) =\_C dp e\^[-p]{} (p). One may write the solution in fact directly in terms of the initial condition: \[SLTN\] g (y,r,)= \^0\_[- ]{} d ’ G (y,r,- ’) g( 0,r, ’), with the Green’s function \[GREENFUN\] G (y,r,- ’)= d p e\^[- p (-’)]{} I\_0( 2 ). From eq. (\[GEXP\]) we see that large typical $n$ corresponds to small typical $\eta$, which in turn by (\[SOLU\]) corresponds to large typical $p$ (“$p_0$”). Thus we must find $\delta^2 p_0^2 \ll 1$ to trust eq. (\[ANDIM\]).
Let us investigate (\[SLTN\]) to find the most relevant values of $\eta'$. The function $g(0,r,\eta')$ falls down monotonously for $\eta' \rightarrow -\infty$ (the first term in (\[GEXP\]) dominates) where it behaves as $\exp(\eta')$. Next, we can show that the function $G(y,r,\eta-\eta')$ has a maximum at $\eta=\eta'$ whose width is of the order of $({8\sqrt{\frac{\bar \a_s \delta^2}{3} y r}})^{(1/2)}$. To see this, evaluate (\[GREENFUN\]) using the asymptotic form for $I_0(z) \sim e^z/\sqrt{2\pi z} (1+\ldots)$. Eq. (\[GREENFUN\]) then becomes (neglecting the unimportant non-exponential prefactor) in the asymptotic form \[SADDLEDELTA\] G (y,r,- ’)= d p e\^[- p (-’) + 2 ]{}. The equation that determines the saddle point $p_S$ here is - ( -’) + =0 or 2 + = . In the dangerous case that $p_S$ is large this leads to \[TYPP\] p\_S = . With (\[SADDLEDELTA\]) this becomes G (y,r,- ’)=( 8)\^[-1/2]{} e\^[- ]{}. Thus, in summary, either or both the regions $\eta\sim 0$ and $\eta' \sim \eta$ give dominant contributions in (\[GREENFUN\]). For $\eta'\sim 0$ we have $$g(y,r,\eta) \sim
e^{-\,\frac{\eta^2}
{8\sqrt{\frac{\bar \a_s \delta^2}{3}\cdot y r }}}\,\,.$$ while for $\eta' \sim \eta$ $$g(y,r,\eta) \propto
e^{-|\eta|}\,\,.$$ The first of these can dominate the second only for a restricted range in $\eta$, namely\[ETAEST\] ||< 8, which implies, with (\[TYPP\]) p\_S 2. Outside of the range (\[ETAEST\]) the typical value of $(\eta\,-\,\eta')$ is at large $\eta$ of order $(8\sqrt{\frac{\bar \a_s \delta^2}{3}\cdot y r})^{(1/2)}$, yielding p\_S \~1 yr1. We conclude, based on the simplified model equation (\[SIMPLEEX\]) that typical values of $p$, and thus $n$, are small, and therefore we should be able to use eq. (\[ANDIM\]). We will see that this conclusion holds when we consider the full solution in section 6.
Estimates for Corrections to the GLR Equation
---------------------------------------------
In this subsection we want to estimate the possible size of corrections to the GLR equation. Recall that the GLR equation sums the contributions of fan diagrams (Fig.3) while the generalized equation includes more general graphs, which are all of the type shown in Fig.5 at large $Q^2$. To estimate the correction let us parametrize the full solution as g(x\_B,Q\^2,)= (e\^ F(y,r))+ g (y,r,), where the first term is the solution to the GLR equation found in subsection 5.1, and the second term is considered to be a small perturbation. Equation (\[GENGLRPAR\]) becomes then for $\Delta g$ \[PERT\] =|\_s + ( - ) -”\[F’\_yF’\_r- |\_s F\^2\]e\^[ 2 ]{}, where $\Phi'$ denotes $\partial\Phi/\partial t$. We neglected the contribution of the $\partial\Delta g/\partial \eta
- \Delta g$ term in since the coefficient in front of this term contains an extra power of $\as$ which we treat as a small parameter. The initial conditions for $\Delta g(y,r,\eta)$ are $\Delta g(0,r,\eta)$= $\Delta g(y,0,\eta)=0$, because we assume that all boundary conditions have been fulfilled by $\Phi$. The last term in eq. (\[PERT\]) can be neglected for two reasons: (i) at $\eta=-\ln (Q^2 R_N^2)$ this term is suppressed and (ii) the difference in brackets is small in both the semicassical and the EKL [@EKL] approach. Substitution of the explicit form of $\Phi$ in (\[FIFUN\]) yields \[PERTFIN\] =|\_s + , where $t\,=\,e^{\eta} F(y,r) $. We can simplify the equation if we keep in mind that the value of $\eta$ is large and negative in the deep-inelastic structure function. Thus $t\ll 1$ and \[PERTSIMPLE\] =|\_s - 12 F\^2 e\^[2 ]{}. Now the $\eta$ depedence of $\Delta g$ is trivial: $\Delta g = e^{2 \eta} \Delta F (y,r)$. Thus \[PERTCOR\] =4|\_s F(y,r) - 12 F\^2 . Using similar techniques as in the previous subsection, but now for the $y$ and $r$ variables, it is straightforward to show that the general solution to this equation with the initial condition $\Delta F (0,r ) \,=\,0$ has the form \[GLRDF\] F = -12 \^y\_0 d y’ e\^[y’ +f r ]{} , where $\tilde{ F^2}$ is the Laplace transform in $y$ and $r$ of the function $F^2(y,r)$. The contours for the $f$ and $\omega$ integrals lie to the right of all singularities in these variables. We now need a reasonable estimate for $\tilde{ F^2}$. Note that $\tilde{ F^2}$ corresponds to the (Laplace tranform of) the properly normalized initial gluon distribution at low virtuality and small $y$. A rough estimate can be made usine the methods of [@EKL]. \[EKLANS\] =, where $A_G$ is the normalization factor for the gluon structure function, $\omega_0$ is defined in eq. (1) and $\gamma(\omega)\,=\,\frac{\bar \a_s}{\omega}$ is the anomalous dimension for the leading twist operator. Here we consider the EKL solution to parametrize the data over a wide kinematic region, including GLR nonlinear corrections. Shortly we will discuss the case where both the GLR and multigluon corrections are considered small.
All contours in (\[GLRDF\]) are to the right of all singularities in $ \omega$ and $f$. The integrand has two poles in $\omega$, one corresponding to the initial condition ($2\omega_0$) and one from the equation ($4\bar \alpha_s/f$). We will demonstrate in the next subsection that the former is dominant for the choice $\omega_0 = 0.5$, and that restricting ourselves to its contribution is very good approximation. Under this assumption we perform the $\omega$ and $f$ integrals, and obtain g =-4 r |\_s \^2 A\^2\_G e\^[2]{} \^y\_0 d y’ e\^[2\_0y’ +2 (=\_0)r ]{}. The factor $r$ in front arises from the double pole in the $f$ variable. The answer clearly satisfies the boundary conditions. We derive further - \^2 e\^. This implies \[FINEST\] = - \^2 x\_B G(x\_B,Q\^2), if the value of the structure function is large enough in the region of low $x_B$. Recall that the correct definition of $\eta$ is $e^{\eta}\,= \,1/(Q^2 R_N^2)$. Substituting in eq. (\[FINEST\]) the value of gluon structure function from HERA data [@HERA] at $Q^2\,=\, 15\; {\rm GeV}^2$ and $x_B = 10^{-4}$ ($ x_B G \,\sim\, 30$) and a typical value of $R_N^2 = 5 GeV^{-2}$ we obtain (with $\a_s\,=\,0.25$ ) $$\frac{\Delta x_B G (x_B, Q^2 )}{ x_B G (x_B,Q^2)} \,\,\sim\,0. 4
\delta^2 \,\sim\, 6\cdot 10^{ - 3}\,\,.$$
It is more instructive to compare the above correction to the gluon structure function with the one due to the GLR equation. In this case we consider both the correction to the GLAP equation due to the GLR shadowing and due to multigluon correlations as small. Thus we try to find the solution to the GLR equation in the form: $$F(y,r)\,\,=\,\,F_{GLAP} \,\,+\,\,\Delta F_{GLR}.$$ For $\Delta F_{GLR}$ we can write the equation: =|F\_[GLR]{} -F\^2\_[GLAP]{} The solution is F\_[GLR]{} =- \^y\_0 d y’ e\^[y’+ fr]{} , where $\tilde{F}^2_{GLAP}$ is the Laplace image of function $F^2_{GLAP} (y,r)$. Again using (\[EKLANS\]) we get F\_[GLR]{}=- A\^2\_G \^y\_0 dy’ e\^[2 \_0 y’]{}\[ e\^[2 ( = \_0 )]{}-e\^[ ( = \_0 )]{}\] Assuming that the second term in the above equation is much smaller than the first we have F\_[GLR]{}=-F\^2\_[GLAP]{} Finally, we can get for the ratio = \^2 which gives a value of the order of 0.04 if $\omega_0 \sim 0.5$. We will return to a discussion of the corrections to the GLR equation in the next section where we will consider the general solution to the new equation at fixed $\as$.
The above estimates seem in contradiction with the estimates in [@BARY], where a large contribution from multigluon correlations was found. The method we use here is however quite diffent from the one in [@BARY]. There the effect of including the new (pole) singularity in $f$ (the Laplace conjugate variable to $r$) that results from the resummation of bubbles associated with the 4-Pomeron coupling was contrasted with the contribution from the two Pomeron cut at the level of the Green functions. Although the location of the singularities is quite close, their nature and residues are very different. This led to a large ratio of the contributions of these singularities to the Green functions. In the present case we include in our estimates the two-gluon source, i.e. the initial gluon distribution, and renormalize it in both cases to the same physical initial condition (here the EKL ansatz), thus absorbing the residues. As was remarked in [@BARY], one can absorb the residues alternatively in $R_N$. Further, in closing the contours involved in performing the inverse Laplace transforms we closed on the singularities of the initial condition (being the rightmost singularities), and not on the propagator poles. Therefore our renormalized $R^2_N$ has no extra dependence on $\ln(1/x)$. We believe that our method is in the above sense more physical.
Numerical Estimates
-------------------
In this subsection we will solve eq. (\[PERTCOR\]) numerically. The method we use goes as follows. We first perform a Laplace transform with respect to $y$ on (\[PERTCOR\]), which leads to \[GLRDFOM\] = F(,r) - F\^2(,r). Using various [*ansätze*]{} for $F(y,r)$ we solve this equation using Numerov’s method, and perform finally the inverse Laplace transform with respect to $\omega$ [^5]. For the various [*ansätze*]{} we fit a parametrized form to $F^2(y,r)$ of which the Laplace transform can be taken analytically.
We will investigate three cases. The first is the EKL ansatz from the previous subsection \[EKL2\] F(y,r) = A\_G e\^[\_0 y + f\_0 r]{} , where we take $\omega_0 = 0.5$, $f_0 = {\bar \as}/\omega_0$, and ${\bar \as} \simeq 0.25$. To correspond with the numbers given in the previous subsection we put $A_g \simeq 0.07$. We will use this case to check the accuracy of the estimate given earlier, and drop for this check the requirement $\D F(y=0,r) = 0$. Let us denote the numerical answer by $\D F^{\rm EKL}(num)$. Then for the pole answer we find \[EKLPOLE\] F\^[EKL]{}(pole) = r e\^[2\_0 y + 2\_0 r]{}, and for the leading term of the saddle point contribution \[EKLSADDLE\] F\^[EKL]{}(saddle) e\^[4]{} where $f_S = \sqrt{4{\bar\as}y/r}$. Note that both contributions vanish as $r\ra 0$. In Table 1 we list these three contributions for various $x$ at a fixed $Q^2 = 15$ ${\rm GeV}/c^2$. It is clear from this table that $\D F^{\rm EKL}(pole)$ is a very good approximation to $\D F^{\rm EKL}(num)$, in fact better than one might expect from $\D F^{\rm EKL}(saddle)$.
Next we treat a more realistic case. We now use for $F(y,r)$ alternatively the MRSD0’ and MRSD-’ [@MRS] gluon distribution functions, which are, at the starting scale $Q_0$, constant as function of $x$, and behave as $x^{-0.5}$ respectively. They are both parametrized by \[MRSPAR\] x G(x,Q\^2) = A\_G x\^[\_g]{} (1-x)\^[\_g]{} (1+\_g x), with the coefficients $A_G, \lambda_g, \eta_g$ and $\gamma_g$ given at $Q=Q_0$ in [@MRS]. We kept this parametrization up until $Q^2 = 15$ ${\rm GeV}/c^2$, but refitted the coefficients for every step in the Numerov procedure. The Laplace transform of this parametrization is easily determined. Following the methods described in the above we determine $\D F(y,r)$. We checked that $\D F$ vanishes for small $y$. Furthermore was solved under the condition $\D F(y,r=0) = 0$.
Similarly to the previous subsection we determined at $Q^2 = 15$ ${\rm GeV}/c^2$ for values of $x_B$ from 0.01 down to the ${\rm LEP}\otimes{\rm LHC}$ value of $10^{-5}$ the ratios \[RAT1\] = and \[RAT2\] = . These ratios are given in Table 2. We infer from this table that corrections to the gluon structure function from multigluon correlations beyond the next-to-leading twist are small, at most 5% at small $x$. We see that the MRSD-’ distribution leads to larger corrections than the MRSD0’ one. As a fraction of the GLR correction both [*ansätze*]{} are small, about 10% for the MRSD-’ case and 10-30% for the MRSD0’ case.
Thus we confirm here the rough estimates from the previous section. However we note that estimates of the gluon correlation radius $R_N$ range from $1$ fm to $0.3$ fm. The numbers in table 2 are accordingly easily adjusted.
The General Solution (for Fixed $\as$)
======================================
In this section we will discuss the general solution to and its consequences for the gluon structure function.
We start by noting that the equation (\[GENGLR\]) can be written, at fixed $\as$, as \[XIGEN\] = C\_[2n]{}x\_B\^n G\^[(n)]{} ( x\_B, r + \_0) &-&\
n\^2 e\^[-(r+\_0)]{}& x\_B\^[(n+1)]{} G\^[(n + 1)]{} ( y, r + \_0),& where we used the fact that that $x_B^nG^{(n)}$ only depends on $\ln(Q^2/Q^2_0)$ with an arbitrary $Q_0$. The above form of the equation reflects the choice $\eta_0 = -\ln Q^2_0 R_N^2$. We focus now on the hypersurface $\eta_0 = \eta$. The evolution equation (see ) can then be written in the form: \[GENGLRPARXI\] =|\_s + ( - ) - \_s\^2 e\^[- ]{}( -g), where $\xi \,=\,r\,+\,\eta$. The advantage of this form is that all explicit $\eta$ dependence such as $\exp(-\eta)$ has been removed.
We can find $ g (y, \xi = r + \eta, \eta )$ using a double Laplace transform with respect to $y$ and $\eta$, namely \[DOUBLEMEL\] g (y,,) = e\^[y+p]{} g ( , p , ) . The function $g (\omega,\xi,p)$ obeys the equation \[XIEQ\] = { |p\^2+(p\^4-p\^2)- \^2 (p-1)e\^[-]{}}g (, ,p) The solution to is: \[SOLXI\] g (, ,p )= g ( , p )e\^[(p\^2+ p\^2 ( p + 1)( p - 1 ))+( p - 1) ( e\^[-]{}-1)]{} The function $g (\omega, p )$ must be determined from the initial condition, , for $ | \eta | \,\ll\,r$, $ r\,\gg\,1$ and $ y = 0 $ where the solution looks as follows: \[INCON\] g (y = 0, r, )= g(,p)e\^[ p]{} e\^[ ( r + ) (p )+ ( p-1) (e\^[-r-]{}-1)]{} , where $$\phi(p)\,\,=\,\,\frac{\bar \as}{\omega}\,p^2\,+\,\frac{{\bar
\as}\,\delta^2}{3 \omega} \,p^2\,( p + 1)( p - 1 )\,\,.$$ We now assert that \[OMP\] g (, p )=(- f (p)) e\^ satisfies the initial condition of with $g_{LLA}
(x_{B},Q^2_0)\,=\,\delta ( y ) $ at $r\,=\,\ln(Q^2/Q^2_0)$ = 0. Here $ f(p)\,\,=\,\,p\,\,+\,\,\phi
( p ) $ and $\Gamma (- f )$ is the Euler gamma function. We will prove this shortly.
Thus, finally, the solution to looks as follows: \[FINSOL\] g (y, , )= (- f (p)) e\^[f(p)+( p - 1) e\^[-]{}-pr+y]{} or changing the integration from $p$ to $f$, \[FINSOLF\] g (, p, )= (- f ) e\^[f+( p(f) - 1) e\^[-]{}-p(f)r +y]{}, where $p (f)$ is determined by \[FP\] f=p (f)+( p(f) ). The contour of integration over $f$ is defined in a such a way that all singularities in $\Gamma( - f ) $ except the one at $f=0$ are located to the right of the contour.
We can now verify the claim made in . At $y=y_0$, $|\eta| \ll r$, $\xi\simeq r \gg 1$ we have \[CHECKINIT1\] g(y\_0,r,) e\^[y\_0]{} (-f) e\^[-r p(f) + f]{}. We now close the $f$ contour in the right half plane. Neglecting the $\delta^2$ term we have $\phi(p(n)) \simeq \bar\as n^2/\omega$, and we get indeed \[CHECKINIT2\] g(y\_0,r,) \_[n=1]{}\^ e\^[n]{} e\^[y\_0 + ]{} The structure function can be found from putting $\xi=0$ (see ). We obtain \[STRSOL\] g(y, =0,r) = (- f ) e\^[( p(f) - 1) -p(f)r +y]{}. implies that $p(f) \,\ra\, (\,\frac{3\,\omega\,f}{\bar \as \delta^2}\,)^{\frac{1}{4}}$. Therefore we can close the contour in $f$ over the singularities of $\Gamma ( - f)$. Next we must integrate over $\omega$. We evaluate this integral using the method of steepest descent and the large $f$ approximation for $p(f)$. Neglecting terms proportional to $\as$ in the exponent we find the saddlepoint $\omega = (p_0 r/4 y)^{4/3}$ where $p_0 = (3n/{\bar\as}\delta^2)^{1/4}$. Then $$g ( y, \xi = 0, r ) \,\,=\,\,\sum_{n=1}^{\infty} \,\,\frac{
(\,-\,1\,)^n}{n!}\,C (n,y)
e^{-\,4^{-\frac{1}{3}}\frac{3}{4 }\,(\,\frac{3 \,n}{\bar \as
\delta^2}\,)^{\frac{1}{3}}\cdot (\,\frac{r^4}{y}\,)^{\frac{1}{3}}}\,\,,$$ where $C (n, y)$ is a pre-exponential, smooth factor. The above series clearly converges and the typical value of $n$ in this series is of the order of unity. We thus confirm the estimates from the previous section, and see that we could trust our calculations of the anomalous dimension $\gamma_{2n}$. We note that this series has an infinite radius of convergence. Thus for the simplified model we consider and for the case of an eikonal initial condition we conclude that there is no analogy to a renormalon in the Wilson Operator Product Expansion.
Now we wish to consider the solution near $f\,\ra\,1$, to understand under which circumstances it suffices to take only this singularity into account. Note that this corresponds to the leading twist case, with GLAP evolution. In this region we can rewrite the solution in the following form. Substituting $f \,=\,1 +\,t$ we obtain: \[SOLF1\] g(y, = 0,r )= e\^ where \[PSI\] =y-r+r+tr{+-1} From this form of the exponent we see that for $\omega$ smaller than $\omega_{cr}$ where \[OMCR\] \_[cr]{} =2|(1+){1 +} \^0\_[cr]{}{ 1+ } we cannot close the contour in $t$ over singularities at positive $t$. (Here we introduce the parameter $\omega^0_{cr}$ in order to separate the effects of $\gamma$ and $\delta$.) For such $\omega$ one would need all singularities in $f$. To understand what happens in this region let us expand $\Psi$ by writing $\omega\,\,=\,\,\omega_{cr}\,\,+\,\,\D$: \[PSI2\] =\_[cr]{}y-r+r+(y -r -) Integration over $\D$ [^6] gives rise to the delta function $\delta [\, \frac{r}{\omega_{cr}}\,(t\,\,-\,\,t_0\,)\,]$ where \[TO\] t\_0=\[\^2\_[cr]{}y-|r\] Carrying out the integration gives \[SOLFINF1\] g(y, = 0,r) = e\^[\_[cr]{}y- r+r]{} Let us define the “critical line” [@GLR] by \[CRLINE\] y\_[cr]{}=r-r The gluon structure function is on this critical line \[STRFUNCRLINE\] x\_B G( x\_B, Q\^2 )=R\_N\^2Q\^2 \[+\]\^[-1]{} For $\delta\,=\,0$ this becomes \[STRFUNDO\] x\_B G( x\_B, Q\^2 )=R\_N\^2Q\^2 . Note that this is similar to the solution of the GLR equation with running coupling [@GLR], but not quite the same.
Thus, the structure of the solution to the new evolution equation with the initial condition looks as follows. In the kinematic region to the right of the critical line we can in fact solve the linear GLAP equation, but with the new initial condition on the critical line. To the left of the critical line we need the solution to the full equation. Note that if we would change the initial condition of the full evolution equation the value of the structure function on the critical line would also change.
We have thus achieved a new understanding of the role of the initial condition in the problem. In particular, we conclude that the solution on the critical line depends only on the initial condition in the region of $f\,\ra \,1$, i.e. it depends only on the initial condition for the GLAP equation. Implicitly we have used here the assumption, expressed in our initial condition (\[EFOR\]), that the multigluon correlations are sufficiently small at large $x$. We recall that the original derivation of the GLR equation was based on this same assumption. We are not restricted to such an assumption for our equation. Clearly, if there were strong correlations between gluons at large $x$ it would change the explicit form of the solution of our evolution equation. Nevetheless, the line of reasoning followed in this section would continue to hold.
For $\delta\,\neq\,0$ we have two different situations. In the first, for $$\frac{\delta^2}{3}\,\,\ll\,\,\frac{\as^2\,\gamma}{2 \bar \as \,r}$$ the solution on the critical line is the same as in . I.e. the only change that occurs in the solution of the case $\delta\,=\,0$ is that there is a new equation for the critical line, . Note that the HERA experiments correspond to this situation.
At very large values of $r$ ($Q^2 \,\gg\,Q^2_0$) , when $$\frac{\delta^2}{3}\,\,\gg\,\,\frac{\as^2\,\gamma}{2 \bar \as \,r},$$ the solution on the critical line looks as follows: \[STRFUNDNEQO\] x\_B G( x\_B, Q\^2 )=R\_N\^2Q\^2 In this case we must solve the GLAP equation using as the boundary condition.
Conclusions.
============
The main result of the paper is the new evolution equation (\[GENGLRPAR\]). It allows us to penetrate deeper into the region of high density QCD because we incorporate multigluon correlations into the evolution, and to answer questions which could not be answered before. For example one could now investigate the question of how well the Glauber theory for shadowing corrections in deep-inelastic scattering with a heavy nucleus works.
This present equation solves two theoretical problems which arise in the region of high parton density: (i) It takes induced multigluon correlations into account, which originate from parton-parton (mainly gluon-gluon) interactions at high enegy and can be calculated in the framework of perturbative QCD, and (ii) it allows for an arbitrary initial gluon distribution, which is nonperturbative in nature.
We have found the general solution to the new equation for the case of an eikonal initial condition and fixed $\alpha_s$. We found no evidence for a “renormalon” in the twist expansion.
Our numerical estimates show that the effect of multigluon correlations is rather small in the accessible region of energy. We have seen evidence for this by using approximate methods and the general solution to the new equation.
We have shown that the general solution confirms the strategy developed for the GLR equation: we have calculated the new critical line for the generalized equation and shown that to the right of this critical line we can solve the linear GLAP equation with a new boundary condition on this line. We found this boundary condition taking into account the multigluon correlations. This approach, developed in this paper simplifies also the solution to the GLR equation and allows us to understand how solutions to the GLR equation depend on the initial conditions. This is essentially a consequence of the linearization of the GLR equation in \[GENGLRPAR\].
We have not discussed here the behaviour of the solution in the region to the left of the critical line, where multigluon correlations should come more forcefully into play. We plan to do this in later publication. We hope that the solution in the latter kinematic region will have a significant impact on understanding the scale of the shadowing correction and the importance of multigluon correlations in the so-called Regge domain. This must be understood in order to provide a matching between soft and hard processes.
[**Acknowledgments**]{} E. Laenen wishes to thank the Columbia University theory group for their hospitality. E. Levin acknowledges the financial support by the Mortimer and Raymond Sackler Institute of Advanced Studies and by the CNPq.
[99]{} V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. [**15**]{} (1972) 438; L.N. Lipatov, Yad. Fiz. [**20**]{} (1974) 181; G. Altarelli and G. Parisi, Nucl. Phys. [**126**]{} (1977) 298; Yu.L. Dokshitzer, Sov.Phys. JETP [**46**]{} (1977) 641.
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov. Phys. JETP [**45**]{} (1977) 199; Ya.Ya. Balitskii and L. N. Lipatov, Sov. J. Nucl. Phys. [**28**]{} (1978) 822; L.N. Lipatov, Sov. Phys. JETP [**63**]{} (1986) 904.
A. DeRoeck (H1 Collaboration), J. Phys. [**G19**]{} (1993) 1549; S.R. Magill (ZEUS Collaboration), J. Phys. [**G19**]{}(1993) 1535; M. Besanson (H1 Collaboration), To appear in [*Proceedings of Blois Workshop on Elastic and Diffractive Scattering*]{}, Brown University, Providence, June 1993; M. Krzyzanowski (ZEUS Collaboration), To appear in [*Proceedings of Blois Workshop on Elastic and Diffractive Scattering*]{}, Brown University, Providence, June 1993.
L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep. [**C100**]{} (1983) 1.
A.D. Martin, J. Phys. [**G19**]{} (1993) 1603, and references therein; A.J. Askew, J. Kwiecinski, A.D. Martin and P. J. Sutton, Phys. Rev. [**D 47**]{} (1993) 3775; J. Kwiecinski,J. Phys. [**G19**]{} (1993) 1443, and references therein; E. Levin, J. Phys. [**G19**]{} (1993) 1453, and references therein; W.J. Stirling, Nucl. Phys. [**B423**]{} (1994) 56; J. Kwiecinski, Z. Phys. [**C29**]{} (1985) 561; E.M. Levin, [*Orsay Lectures*]{}. Orsay preprint LPTHE-91/02; A.H. Mueller, in [*Proceedings of the Workshop “QCD - 20 Years Later”*]{}, ed. P. M. Zerwas, H.A. Kastrup. Volume 1 (1992) 162. E. Laenen and E. Levin, Ann. Rev. Nucl. Part. Sci. [**44**]{} (1994) 199.
J. Bartels, Z. Phys. [**C60**]{} (1993) 471. Phys. Lett. [**B298**]{} (1993) 204; E.M. Levin, M.G. Ryskin and A.G. Shuvaev, Nucl. Phys. [**B 387**]{} (1992) 589.
E. Laenen, E. Levin and A.G. Shuvaev, Nucl. Phys. [**B419**]{} (1994) 39.
R.K. Ellis, Z. Kunszt and E.M. Levin, Nucl. Phys. [**B420**]{} (1994) 517.
A.H. Mueller and J. Qiu, Nucl. Phys. [**B268**]{} (1986) 427.
R.K. Ellis, W. Furmanski and R. Petronzio, Nucl. Phys. [**B 207**]{} (1982) 1; [**B 212**]{} (1983)29.
A.P. Bukhvostov, G.V. Frolov and L.N. Lipatov, Nucl. Phys. [**B 258**]{} (1985)601.
A.H. Mueller, Nucl. Phys. [**B335**]{} (1990) 115; Nucl. Phys. [**B317**]{} (1989) 573; Nucl. Phys. [**B307**]{} (1988) 34.
V.A. Abramovski, V.N. Gribov and O.V. Kancheli, Sov. J. Nucl. Phys. [**18**]{} (1974) 308.
J. Bartels and M.G. Ryskin, Z. Phys. [**C60**]{} (1993) 751.
A.D. Martin, R.G. Roberts and W.J. Stirling, Phys. Lett. [**306**]{} (1993) 145.
$x_B$ $\Delta F^{\rm EKL}(num)$ $\Delta F^{\rm EKL}(pole)$ $\Delta F^{\rm EKL}(saddle)$
----------- --------------------------- ---------------------------- ------------------------------
$10^{-2}$ $-3.26\cdot 10^{-2}$ $-3.41\cdot 10^{-2}$ $8\cdot 10^{-3}$
$10^{-4}$ $-3.44$ $-3.41$ $2\cdot 10^{-2}$
Table 1. Comparison of various contributions in the EKL approximation. Here $Q^2 = 15$ ${\rm GeV}^2/c^2$, $\alpha_s = 0.25$ and $R_N^2 = 5$ ${\rm GeV}^{-2}$.
Ansatz $x_B$ $\Delta (x_B G)/x_B G$ $\Delta (x_B G)/\Delta (x_B G)_{GLR}$
-------- ----------- ------------------------ ---------------------------------------
MRSD0’ $10^{-2}$ $-1.4\cdot 10^{-3}$ $0.11$
$10^{-4}$ $-8.9\cdot 10^{-3}$ $0.29$
$10^{-5}$ $-14.7\cdot 10^{-3}$ $0.31$
MRSD-’ $10^{-2}$ $-1.2\cdot 10^{-3}$ $0.084$
$10^{-4}$ $-13.5\cdot 10^{-3}$ $0.1$
$10^{-5}$ $-40.5\cdot 10^{-3}$ $0.09$
Table 2. Correction to the gluon distribution function for two different [*ansätze*]{}. Here $Q^2 = 15$ ${\rm GeV}^2/c^2$, $\alpha_s \simeq 0.21$ and $R_N^2 = 5$ ${\rm GeV}^{-2}$.
**FIGURE CAPTIONS**
[**Figure 1.**]{}
Rescattering of Pomerons in the t-channel.
[**Figure 2.**]{}
Pictorial representation of the generalized evolution equation.
[**Figure 3.**]{}
“Fan” diagram.
[**Figure 4.**]{}
Production of three gluon shadows in a parton cascade.
[**Figure 5.**]{}
Example of type of multigluon interactions that the generalized evolution equation takes into account.
[^1]: [laenen@surya11.cern.ch]{}
[^2]: [levin@lafex.cbpf.br; levin@ccsg.tau.ac.il]{}
[^3]: On leave from [*Theory Department, St. Petersburg Nuclear Physics Institute, 188350, St. Petersburg, Gatchina, Russia*]{}
[^4]: The strength of the three Pomeron vertex $\gamma$ was calculated in ref. [@MUQI] using the AGK cutting rules [@AGK], which are equivalent to time-ordering.
[^5]: We thank Keith Ellis for providing subroutines that perform this last step.
[^6]: Alternatively one may expand to $O(\D^2)$ and use steepest descent.
|
---
abstract: 'We use a new method to model fluctuations of the Lyman-Werner (LW) and Lyman-$\alpha$ radiation backgrounds at high redshift. At these early epochs the backgrounds are symptoms of a universe newly lit with its first stars. LW photons (11.5-13.6 eV) are of particular interest because they dissociate molecular hydrogen, the primary coolant in the first minihalos. By using a variation of the halo model, we efficiently generate power spectra for any choice of radiation background. We find that the LW power spectrum typically traces the matter power spectrum at large scales but turns over at the scale corresponding to the effective ‘horizon’ of LW photons ($\sim 100$ comoving Mpc), unless the sources are extremely rare. The series of horizons that characterize the Lyman-$\alpha$ flux profile shape the fluctuations of that background in a similar fashion, though those imprints are washed out once one considers fluctuations in the brightness temperature of the 21-cm signal. The Lyman-$\alpha$ background strongly affects the redshifted 21-cm signal at just about the time the LW background begins to dissociate H$_2$, so measuring that background’s properties will reveal important information about the transition from early Population III stars to more normal stars. Around this time we find that fluctuations in the LW background are weak; the fractional standard deviation is less than $\sim 0.5$ on scales $\apprge 10$ cMpc, only rising to be of order unity on scales $\apprle 1$ cMpc. This should not lead to substantial spatial fluctuations in H$_2$ content, except at the earliest times. Even then, most halos form far from other sources, so the transition from star formation in low-mass to high-mass halos is rather homogeneous across the universe.'
author:
- |
Lauren N. Holzbauer[^1] and Steven R. Furlanetto\
Department of Physics & Astronomy, University of California Los Angeles; Los Angeles, CA 90095, USA
bibliography:
- 'bibfile1.bib'
title: 'Fluctuations in the High-Redshift Lyman-Werner and Lyman-$\alpha$ Radiation Backgrounds'
---
cosmology: theory – first stars – galaxies: haloes – galaxies: high-redshift – stars: Population III
Introduction
============
An important aspect of the cold dark matter (CDM) universe is that density fluctuations exist on small scales. These small scale perturbations are superimposed on larger scale perturbations; the density reaches its highest value over the smallest region. Consequently, structure forms via hierarchical buildup. An initially smooth density distribution eventually morphs into a web of sheets and filaments. It is the overdense junctions of these filaments that we call dark matter halos. Further structure development takes place inside these halos, commencing with the first (Population III) stars.
Population III (Pop III) stars illuminated our dark universe in its cold youth and from them developed the complex environment we live in today. According to hierarchical structure formation, these stars formed out of metal-free H/He gas contained in minihalos at redshifts $z \sim 20$ - $30$ [@couchmanrees1986]. The minihalos, with masses around $10^6 M_{\odot}$, have virial temperatures less than $10^4$ K - below the threshold for atomic hydrogen cooling [@ohhaiman2002]. Consequently, the halos have to rely primarily on H$_2$ for cooling [@htl96; @tegmark97; @abel2002; @bromm2002]. This cooling takes place via collisional excitation (mainly between H$_2$ molecules and energetic H atoms) and subsequent radiative decay of the rotational transitions of H$_2$.
In the classical view of Pop III star formation, molecular cooling produces a single, massive star from cold gas that becomes trapped in the dark matter potential well of one minihalo [@bromm2009]. A dense core, or protostar, gradually emerges and grows into a massive star by accreting the surrounding gas. These first stars could theoretically grow to be several hundred solar masses [@brommloeb2004]; most were probably $\sim 100 M_\odot$ [@brommlarson2004]. But what would happen if the infalling gas became fragmented? Several studies show that a primordial protostellar cloud will most likely not violently fragment enough for a secondary clump to compete with the parent clump and form a second star [@abel2002; @yoshida2008]. However very recent simulations suggest that if a gas cloud surrounding a protostellar core has an initial degree of angular momentum, it could collapse into a dense disk, cool, and fragment, resulting in a binary or even multiple Pop III star system [@stacy2010b; @turk2009] consisting of two or more lighter stars as opposed to a single, massive star. These new studies could indicate that the formation of the first stars could be more complicated and varied than previously believed.
In these primordial star cookers, the ability to form new stars is dependent on the abundance of H$_2$. However, as the population of these luminous stars grows, the sites of star formation are increasingly irradiated by soft UV photons (the Lyman-Werner, or LW, bands: 11.2-13.6 eV) from existing stars. This LW radiation can photodissociate the H$_2$ molecules in the gas through the two-step Solomon process [@field66; @stecher67], $${\rm H}_2 + \gamma \rightarrow {\rm H}^{*}_{2} \rightarrow 2{\rm H},$$ in which an H$_2$ molecule hit by a LW photon bumps it up to an excited electronic state, H$^{*}_{2}$. A fraction of decays from this excited state end up in the vibrational continuum of the ground state, dissociating the molecule. If the LW background becomes strong enough, it can prevent further collapse and consequently stall the further formation of primordial ionizing sources by terminating the minihalos’ primary cooling supply [@hrl97]. As a result, the only halos able to cool (via atomic line cooling) and form new stars are those with $T_{\rm vir} \ga 10^4$ K, or masses above $\sim 10^8 M_\odot [(1+z_{\rm vir})/10]^{-3/2}$. Minihalos, with temperatures below the threshold for atomic cooling, will not be able to collapse past virialization without a sufficient supply of H$_2$.
Although these larger halos may still form metal-free stars, the thermodynamics of the cooling process is sufficiently different that we expect the resulting stars to differ substantially (especially in their characteristic mass). The two populations are sometimes described as Population III.1 and Population III.2 to emphasize this: both may be ‘primordial,’ but they have very different properties regulated by the LW (and other) radiation backgrounds (see, e.g., @bromm2009).
Most previous calculations of the LW background used a homogeneous approximation, in which they assumed a uniform distribution of sources [@haiman00; @ricotti2002; @yoshida2003]. But the highly clustered, discrete sources responsible for the background radiation do not generate a uniform background. If these fluctuations are large enough, the transition from H$_2$ cooling to atomic line cooling would be very patchy, potentially allowing exotic star formation to persist for long periods even after the mean background reaches the threshold value for H$_2$ suppression. @dijkstra2008 were the first to consider the inhomogeneous LW background, but only in the context of close halo pairs (using a Monte Carlo model) and only when the background was already well above threshold. [@ahn] were the first to consider the inhomogeneous background using a large-scale radiative transfer simulation of reionization.
These photons have other observable effects as well; most importantly, as they redshift into the Lyman-$\alpha$ transition they couple the excitation temperature of the 21-cm transition of hydrogen to the gas kinetic temperature via a radiative pumping mechanism known as the Wouthuysen-Field effect [@wouthuysen1952; @field1959a]. This renders the 21-cm signal visible in emission or absorption. [@barkloeb2005] showed that the fluctuations in this young Lyman-$\alpha$ background produced strong fluctuations in the 21-cm signal. Conversely, observing these fluctuations can reveal a wealth of information as to the properties of these first luminous sources. We will see that these photons begin to affect the 21-cm background at roughly the same background intensity at which they suppress H$_2$ cooling. Thus redshifted 21-cm measurements offer an excellent chance to study the transition from Population III.1 to III.2 stars as well as the inhomogeneities in the ultraviolet radiation field during the ‘cosmic dawn.’
In this paper, we present a new method with which to efficiently calculate the power spectrum of an arbitrary radiation field for any desired redshift and range in scale (in this paper we focus on the LW and Lyman-$\alpha$ backgrounds specifically; see @mesfurl2009 for an earlier application specific to the ionizing background). Using the halo model to determine the spatial distribution of halos, we can build up the radiation background by superimposing a flux profile specific to that particular background on each halo. This profile effectively replaces the mass density profile traditionally used in the halo model to calculate fluctuations in the density field. We aim to study the importance of fluctuations in these backgrounds and complement the radiative transfer simulation of [@ahn] with our simple, analytic model. Our method also takes a very different approach to calculating 21-cm fluctuations (due to perturbations in the Lyman-$\alpha$ radiation field) compared to existing work [@barkloeb2005; @pritchardfurl].
In this first exploration of the radiation background, we restrict our attention to the soft-UV background from a relatively simple model of first galaxy formation. In fact, many other physical factors contributed to the transition from Population III to Population II star formation. The most obvious is metal enrichment, which also affects the cooling and is highly inhomogeneous (see, e.g., @furlloeb2005). Also, X-rays emanating from the first sources can counteract H$_2$ destruction by increasing the free electron fraction and so catalyzing its formation [@mcdowell61; @hrl96; @haiman00]. There has been considerable debate as to which of these backgrounds is more influential. For our simple model, we will follow @machacek2003 by assuming that the enhancement of the electron density due to the X-ray background occurs too slowly to compete with photodissociation and so neglect the X-ray background.
Recently, @tseliakhovich2010 pointed out that the residual relative velocities of the baryon fluid and underlying dark matter distribution, imprinted during the recombination era by the baryons’ close coupling to photons and now visible as baryon acoustic oscillations, may have important implications for star formation in these early, fragile halos. These large-scale velocities will suppress the accretion of gas onto small dark matter halos [@tseliakhovich2010; @tseliakhovich2010b]. The actual implications for star formation are as yet unclear; the first simulations show modest effects on the reionization era itself [@maio2010; @stacy2010], but the effect on the earlier epochs is important for the LW and Lyman-$\alpha$ backgrounds [@dalalpen10]. Because the effects are as yet unclear, we will ignore these velocity corrections here, thus providing a baseline prediction for comparison with future work better incorporating them.
This paper is organized as follows: in Section \[method\] we describe our method for calculating the power spectrum of the LW and Lyman-$\alpha$ radiation background fluctuations using the halo model. In Sections \[lw\] and \[lya\] we calculate the flux profiles for the LW and Lyman-$\alpha$ backgrounds, respectively, and also present our results. We summarize our results and conclude in Section \[summary\]. We adopt a background cosmology $(\Omega_0, \Omega_\Lambda, \Omega_b, h, \sigma_8,n)=(0.26, 0.74, 0.044, 0.74, 0.8, 0.95)$ consistent with the most recent measurements [@komatsu2011].
Method
======
We are interested in modeling the power spectrum of fluctuations in the LW and Lyman-$\alpha$ radiation backgrounds using the halo model. Unlike earlier treatments of the LW background, we are specifically interested in its large-scale inhomogeneities, complementing the high resolution, large-scale N-body radiative transfer simulation of @ahn and the small-scale treatment of @dijkstra2008. Instead we will expand the model of [@mesfurl2009], who treated the inhomogeneous hydrogen-ionizing ultraviolet background using a halo model-like prescription.
The halo model, as described in [@cooraysheth], uses properties of virialized dark matter halos to calculate the effects of non-linear gravitational clustering, assuming that all mass in the universe is compartmentalized in such halos, whose properties can be parameterized purely by their mass $m$. The three ingredients of the model are the (1) halo number density, $n(m)$, (2) spatial distribution of the halos, and (3) distribution of mass within each halo, or halo density profile, $\rho(r|m)$, (where $r$ is the distance away from the center of a halo with mass $m$). Typically, a theoretically-motivated halo mass function for (1) (the classic choice being @pressschechter) allows one to calculate (2). The density profile for (3) can be calibrated by numerical simulations, such as the NFW [@nfwref] or @m99ref profiles. This model is very powerful in that it can efficiently determine the power and many other useful properties for any density field at an arbitrary epoch and scale.
Since we wish to quantify the radiation background rather than the mass density field, we simply replace the ‘halo density profile’ with the profile of the radiation field around each halo. This flux profile, $\rho_{\rm rad}(r|m)$, depends on the radiation background under consideration and will be discussed later. For a spherically symmetric profile, the normalized Fourier transform, $u(k|m)$, can be written as: $$\label{profreduced}
u(k|m) = \frac{\int_{0}^{r_c} dr4\pi r^2 [\sin (kr)/(kr)] \rho_{\rm rad}(r|m)}{\int_{0}^{r_c} dr4\pi r^2 \rho_{\rm rad}(r|m)},$$ where $r_c$ is the cutoff distance at which an observer can no longer see the radiation emanating from the source. For the case of the LW radiation, for example (described more fully in section \[lwprofile\]), this cutoff distance, or horizon, is given by $r_{\rm LW} \sim 100$ comoving Mpc (cMpc).
As described above, we are interested in modeling the fluctuations in a variety of radiation backgrounds (in this paper, the LW and Lyman-$\alpha$ backgrounds). We use the power spectrum, $P(k)$, or the dimensionless quantity $\Delta(k) \equiv k^3P(k)/2\pi^2$ to quantify these fluctuations. Following the halo model, we write the power as a sum of two terms: the first term, $P^{\rm 1h}(k)$, describes the case for which radiation at two points comes from the same source,[^2] while the second term, $P^{\rm 2h}(k)$, describes the case for which the two points are illuminated by two sources: $$\begin{aligned}
P(k) & = & P^{\rm 1h}(k) + P^{\rm 2h}(k), \mbox{where} \\
P^{\rm 1h}(k) & = & \int_{M_{\rm min}}^{\infty} dmn(m) \left(\frac{m}{\bar \rho f_{\rm coll}} \right)^2 |u(k|m)|^2 \\
P^{\rm 2h}(k) & = & \left[ \int_{M_{\rm min}}^{\infty} dmn(m) \left(\frac{m}{\bar \rho f_{\rm coll}} \right) u(k|m) b(m) \right]^2 P^{\rm lin}(k).
\label{ps_halomodel}\end{aligned}$$ Here, $\bar \rho$ is the matter density, $f_{\rm coll}$ is the collapse fraction (fraction of mass in the universe contained in galaxies, or collapsed in a halo), $b(m)$ denotes the halo bias (describing how strongly clustered the halos are; @mowhite96), and lastly $P^{\rm lin}(k)$ is the linear power spectrum. Here we have approximated the halo-halo power spectrum, for two halos with mass $m_1$ and $m_2$, as $b(m_1) b(m_2) P^{\rm lin}(k)$, which requires that the halo fluctuations remain linear on the appropriate scales (i.e., those on which $P^{\rm 2h}$ dominates). While the density fluctuations themselves are very weak at the redshifts of interest to us, the halos are also highly biased, so nonlinear corrections will be important on sufficiently small scales. We use the [@eisensteinhu] fit to the transfer function to calculate $P^{\rm lin}(k)$ and the Sheth-Tormen mass function and collapse fraction [@shethtormen].
To model the density power spectrum one must include the entire halo population, over all masses. However, we are interested in the total radiation field and so should not include the low-mass halos unable to host stars. We assume that only halos more massive than a cutoff mass, $M_{\rm min}$, host stars and so contribute to the radiation background. To motivate our choices for $M_{\rm min}$, we first consider the ‘filter mass,’ $M_{\rm filter}$, the characteristic scale over which baryonic perturbations are smoothed in linear perturbation theory or the minimum mass of a halo to accrete baryons [@gnedinhui1998; @naozbarkana2007], as a lower limit ($\sim 10^5 M_\odot$ in our redshift regime). In linear theory, the relative force balance between gravity and pressure can be characterized by the Jeans mass, $M_J$; the corresponding Jeans scale is the minimum scale on which a small perturbation will grow due to gravity. $M_J$ depends on the instantaneous value of the sound speed of the gas, consequently overestimating the characteristic mass scale by up to an order of magnitude [@gnedin2000]. In contrast, $M_{\rm filter}$, which takes into account the full thermal history of the gas, is a more accurate mass scale.
An upper limit would be the threshold for atomic cooling, $T_{\rm vir} \sim 10^8K$ [@ohhaiman2002]. Since the H$_2$ fraction, $f_{\rm H_2}$, increases with halo mass ($f_{\rm H_2} \propto T_{\rm vir}^{1.5}$, @tegmark97) the cutoff mass certainly lies somewhere between these two limits. The classical criterion that the cooling time be smaller than the dynamical time will set the redshift-dependent transition: in the absence of a LW background, these successful minihalos probably have $f_{\rm H_2} \sim 10^{-4}$ and $M_{\rm halo} \sim 10^6 M_\odot$ [@htl96; @tegmark97; @yoshida2003]. However, rather than try to model this in detail we will employ a variety of selections for $M_{\rm min}$ in order to remain most general.
The LW Background {#lw}
=================
In this section we will apply the above method to fluctuations in the LW radiation background, which determines if the sterilization of minihalos at high redshift (through the photodissociation of H$_2$) was a patchy or homogeneous transition.
The Flux Profile {#lwprofile}
----------------
\
Our LW flux profile (shown in Figure \[fmod\] for $z = 10$ and $25$) for a halo with mass, $m$, located at an effective luminosity distance, $r$, from the observer is given by: $$\rho_{\rm rad}(r|m) \propto m \frac{f_{\rm mod}(r)}{4\pi r^2},$$ where we assume for simplicity that the luminosity of each halo scales with its mass. [^3] Here we use the picket fence modulation factor, $f_{\rm mod}(r)$, from [@ahn]. This is the fraction of LW continuum radiation emitted by a source that is received by the observer without redshifting into a hydrogen Lyman series resonance line, where it will either be absorbed or scattered. An absorbed photon will either cascade to the 2$p$ level and produce a Lyman-$\alpha$ photon or cascade to the metastable 2$s$ level and decay by two photon emission [@pritchardfurl]– either way, the resulting photon will be below the LW range. On the other hand, the scattered photon will be reabsorbed until it, too, decays into a low-frequency photon (typically after just a few scatterings). So, for a given photon at observed frequency, $\nu_{\rm obs}$, we can define a maximum redshift, $z_{{\rm max,}i}$, corresponding to the maximum distance within which photons from a source remain in the LW band without redshifting into the closest Lyman line from above (located at frequency $\nu_i$): $$\label{zmax}
\frac{1+z_{{\rm max,}i}}{1+z_{\rm obs}} = \frac{\nu_i}{\nu_{\rm obs}}.$$
With each Lyman line associated with its own $z_{{\rm max,}i}$ and the spacing between them decreasing with increasing $\nu_i$, we are left with a transmission spectrum resembling a poorly fashioned picket fence, illustrated in Figure 2 from [@ahn]. The modulation factor, $f_{\rm mod}$, is defined as the fraction of the LW frequency interval, 11.5-13.6eV, that lies within the pickets, or that is successfully transmitted to the observer: $$\label{fmodsigma}
f_{\rm mod} = 1 - \sum_j \left( \frac{h\Delta\nu_{{\rm gap,}j}}{2.1 \ {\rm eV}} \right),$$ where $\Delta\nu_{{\rm gap},j}$ is the frequency interval between each picket in which there is no transmission. [^4] The profile terminates at the ‘LW horizon,’ $r_{\rm LW} = 97.39 \alpha$ cMpc, the distance at which a photon redshifts across the maximum picket spacing (between the pickets corresponding to the Ly$\delta$ and Ly$\gamma$ lines). The scaling factor, $\alpha$, is defined as: $$\label{alpha}
\alpha = \left(\frac{h}{0.7}\right)^{-1}\left( \frac{\Omega_m}{0.27}\right)^{-1/2}\left( \frac{1+z}{21}\right)^{-1/2}.$$
While $f_{\rm mod}$ can be calculated numerically, [@ahn] have devised a fitting formula: $$f_{\rm mod}(r) = 1.7\exp{\left[-(r_{\rm{cMpc}}/116.29\alpha)^{0.68}\right]} - 0.7$$ if $r_{\rm{cMpc}}/\alpha \le 97.39$ and zero otherwise, where $r_{\rm{cMpc}}$ is the distance to the source in cMpc. We have successfully reproduced $f_{\rm mod}$ using the method described by [@ahn] and have confirmed that the fitting formula is accurate to within 2 per cent error of the true numerical values.
### The Light Cone {#timedep_lw}
There is one difficulty with the halo model as usually constructed for our problem: it does not allow the sources to evolve over time. As usually constructed, the halo model takes the properties of each halo at a particular instant. This is not actually appropriate for our application, where the time delay from the finite speed of light implies that many sources will only be visible to a given point as they were long in the past, when their luminosity may have differed from the present value. For example, the light travel time across the LW horizon is 38.6 Myr at $z=10$ and 11.1 Myr at $z=25$. These values are a full $\sim 5\%$ of the Hubble time at those epochs, so a fraction of the visible sources would appear much dimmer than the above model would suggest. We next estimate how much we would expect the inclusion of such a time dependence to alter our results.
We can crudely account for halo growth by attaching a damping factor to the flux profile: $$\label{timeprofile}
\rho_{\rm rad}(r | m) = L(m)\frac{f_{\rm mod}}{4\pi r^2} \longrightarrow L(m)\frac{f_{\rm mod}e^{-r/r_{\star}}}{4\pi r^2},$$ where $r_\star = ct_\star (1+z)$ in cMpc and $t_\star$ corresponds to the typical timescale for halo growth, assuming that at these high redshifts the halos grow exponentially fast so that the luminosity of a halo $L(m) \propto \exp^{t/t_\star}$. The growth timescale we define as: $t_\star = a(z)t_{\rm H}$ (where $a$ is some proportionality factor that evolves over time and $t_{\rm H}$ is the Hubble time).
To estimate $a$, consider a population of identical halos with mass $m$, at some redshift $z$, in which the halos are conserved; no new halos are created and none are destroyed. In this simple case, the collapse fraction is given by: $$f_{\rm coll}(t) = \frac{n\cdot m(t)}{\bar\rho}.$$ Note how the halo mass is now a function of time, $t$. Taking the time derivative of this expression leads to: $$\frac{1}{f_{\rm coll}(t)}\frac{df_{\rm coll}}{dt} = \frac{1}{m}\frac{dm}{dt} \equiv \frac{1}{t_{\star}}.$$ Thus for example, $a(z=10) = 0.283, 0.167$ for $M_{\rm min} = 10^6 M_\odot, 10^8 M_\odot$ respectively and $a(z=25) = 0.062, 0.034$ for the same choices of $M_{\rm min}$.
Although this simple model overestimates the growth rate of individual sources (because in reality much of the increase in collapse fraction is driven by new halos passing the relevant mass threshold), it provides a simple conservative parameterization of the effects of growth.
Adding this rapid source evolution effectively damps the source profile whenever $r_\star \la r_{LW}$ – these effects are illustrated in Figure \[fmod\]. For example, for $M_{\rm min} = 10^8 M_\odot$, $r_\star(z=10) = 401.6$ cMpc and $r_\star(z=25) = 53.7$ cMpc while $r_{\rm LW} = 129.7$ cMpc and $84.4$ cMpc for those respective redshifts. We include this crude model for the light cone effect below but also point out where it modifies our results substantially.
The Threshold Intensity {#j21}
-----------------------
\
In order to determine when fluctuations are most important, we next compute the evolution of the LW intensity – and hence the point at which H$_2$ cooling is suppressed – in some simple models of structure formation. This section is not meant to provide a detailed model of star formation, but it should provide some context for the fluctuations we will later examine. We can estimate the mean LW intensity, $\bar{J}_{\rm LW}(z)$, with the following: $$\label{jlw}
\bar J_{\rm LW} (z) = \frac{(1+z)^2}{4\pi} \int_{z}^{z+z_{\rm LW}} \frac{c dz'}{H(z')} \bar\epsilon (z') f_{\rm mod}(z' - z),$$ where $f_{\rm mod}(z' - z)$ is part of the LW flux profile (described more fully in §\[lwprofile\]) and the mean emissivity, $\bar\epsilon(z')$, is given by: $$\label{meanemissivity}
\bar\epsilon (z) = f_{\star} \bar n_b^0 \frac{d}{dt}f_{\rm coll}(z) \epsilon_b,$$ with $f_\star$ being the star forming efficiency (fraction of baryons that actually form stars), which we take to be $10\%$ as a fiducial value, and $\bar n_b^0$ is the mean baryon number density. We can approximate the spectral distribution function (defined as the number of photons per frequency $\nu$ emitted per baryon), $\epsilon_b(\nu)$, as its mean value $\epsilon_b$ over the LW range (11.2–13.6 eV) for simplicity since we are looking at such a small range in frequency. We normalize $\epsilon_b(\nu)$ to produce $4800$ photons per baryon between Lyman-$\alpha$ and the Lyman limit for very massive Population III.1 (zero-metallicity, $M \geq 100M_\odot$) stars [@barkloeb2005]. Note that the light cone effect is inherent in this expression due to the redshift dependence of the mean emissivity.
Our results are summarized in Figure \[j21\_allz\_allm\]. The LW intensity is calculated in units of $J_{\rm LW,21} = J_{\rm LW}/(10^{-21}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$ sr$^{-1})$. The solid curves represent the very massive Population III stars. Replacing the emissivity with that of a Population II star (with metallicity equal to $1/20$ the solar value) producing $9690$ photons per baryon in our frequency range [@barkloeb2005] boosts the intensities by a factor of $\sim 2$; the background reaches threshold earlier. According to [@haiman00], background intensities of $J_{\rm LW,21} \sim 10^{-2}$ – $1$ are needed to suppress H$_2$ cooling in all minihalos over a range of redshift from $z \sim 10 - 50$. The lower value describes H$_2$ suppression in halos near the $T_{\rm vir} < 10^{2.4}$ K limit; below these low temperatures H$_2$ cooling is inefficient even in the absence of any photodissociating background, so no stars will form. Since $f_{\rm H_2}$ increases with $T_{\rm vir}$, it will be more difficult to terminate a more massive halo’s larger cooling supply. Thus, as the halo population evolves, becoming more numerous and more massive with time, the threshold intensity must increase, self-regulating star formation by shifting the minimum mass to higher values. Once halos reach $T_{\rm vir} > 10^{3.8}$ K the value of the background intensity is once again irrelevant since these large halos are able to cool via atomic line cooling and no longer rely on their fragile H$_2$ supply. In further calculations, we will take $J_{\rm LW,21} = 0.1$ as our fiducial threshold intensity.
It is evident from Figure \[j21\_allz\_allm\] that our models reach this threshold between redshifts $z \sim 15 - 35$. Of course, these models are extremely naive and ignore a host of complications (such as the evolving star formation efficiency and cooling threshold, as well as other feedback mechanisms). But they suffice to illustrate approximately when a given model reaches the H$_2$ photodissociation threshold, where the fluctuations which we will study are particularly interesting. Note that, because structure formation itself proceeds exponentially fast at high redshifts, uncertainties in the star formation parameters themselves are relatively unimportant. In any case, $J_{\rm LW,21} \propto f_\star \epsilon_b$, so it is easy to read off the appropriate intensity for such a model.
We emphasize that the fluctuations will be most important near the threshold, because that is when the transition in cooling modes actually occurs. If fluctuations are small, the transition would occur uniformly over the entire Universe. If not, the Universe could contain isolated, sparsely populated patches in which H$_2$ cooling remains possible. If a minihalo inhabits one of these ‘safe’ patches, it could continue to form very massive stars via H$_2$ cooling even after the mean intensity reaches threshold. On the other hand, even well before an average IGM point reaches threshold, regions near existing sources will be well above it, and this could strongly affect the highly-clustered early sources. We will examine this phase in §\[earlyflucs\].
Power Spectrum {#powerspec}
--------------
\
Results are depicted in Figures \[delta2lw\_z10\] and \[delta2lw\_fmass\]. In the former, we simultaneously vary $M_{\rm min}$ and redshift so that $J_{\rm LW,21} = 0.1$ is fixed, while in the latter we follow a single star formation model over redshift (varying $L_{\rm LW, 21}$).
The normalized scenarios displayed in Figure \[delta2lw\_z10\] include: $M_{\rm min} = 10^6 M_\odot$, $z=30.15$ (bottom panel) and $M_{\rm min} = 10^8 M_\odot$, $z=20.5$ (top panel). The short-dashed curves represent the light cone versions for both scenarios, using equation (\[timeprofile\]) for the flux profile. It is evident that the inclusion of the light cone effect preserves the shape of the power but modestly boosts the amplitude (by a factor of $\sim 2$). We have also separately displayed the 1-halo and 2-halo terms (bottom-most, solid lines) for the $M_{\rm min}=10^6 M_\odot$ scenario so as to gain a sense of when these terms are dominant and to show how they work in tandem to determine the shape of $\Delta^2(k)$.
It is important to note that the above prescriptions assume that all halos above the mass threshold, $M_{\rm min}$, form stars continuously. Of course, these stars have finite lifetimes, and in the classical Pop III scenario in which each halo undergoes only a short burst of star formation, not all of these stars are going to be ‘turned on’ when we take a snapshot of the fluctuations at a particular point in time. We can account for this simply by incorporating a duty cycle, $f_{\rm duty}$, into our calculation. This addition exclusively affects the one-halo term in equation \[ps\_halomodel\]; since both $n(m)$ and the effective $f_{\rm coll}$ (which in this model gives the fraction of halos hosting *active* sources) are altered by a factor of $f_{\rm duty}$, the two-halo term remains unchanged.
The two topmost, long-dashed curves in both panels of Figure \[delta2lw\_z10\] represent the $M_{\rm min} = 10^8 M_\odot$ and $10^6 M_\odot$ scenarios including the light cone effect using two different values of $f_{\rm duty}$. A reasonable estimate for $f_{\rm duty}$ would be the ratio between the average lifetime of a Pop III star, $\tau$, and the Hubble time, $t_H$. The lifetime of these massive stars is believed to be a few million years (Myrs) [@barkloeb2001; @brommlarson2004], though that remains to be directly measured. The topmost curve in both cases assumes an average lifetime of $\tau \sim 3$ Myrs ($f_{\rm duty} = 0.02$ for $M_{\rm min} = 10^6 M_\odot$ and $0.01$ for $M_{\rm min} = 10^8 M_\odot$) while the second curve assumes $\tau \sim 10$ Myrs ($f_{\rm duty} = 0.06$ and $0.04$ for $M_{\rm min} = 10^6 M_\odot$ and $10^8 M_\odot$ respectively). This greatly increases the importance of the one-halo term and so boosts the fluctuations on scales below the LW horizon. However, note that the mean background intensity also falls by a factor of $f_{\rm duty}$, so these strong fluctuations occur well *before* threshold is reached.
The most striking feature common to all power spectra is the first turnover located at $k_{\rm LW} \sim 0.06$ cMpc$^{-1}$. This is a strong signature of the LW flux profile, which terminates at $r_{\rm LW} \sim 100$ cMpc ($k_{\rm LW} = 2\pi/r_{\rm LW} \sim 0.06$ cMpc$^{-1}$). Power is smallest for the largest scales and then steadily increases until it reaches $k_{\rm LW}$. In this regime, regions are far outside the LW horizon of each source and so sample independent patches in the radiation field. The total power is therefore simply proportional to the matter power spectrum multiplied by a mean bias factor (squared). However, at $k_{\rm LW}$, $\Delta^2$ turns over and begins to fall. This is because such scales sample the variations within $r_{\rm LW}$; if the two points see the same halo populations, their radiation amplitudes will vary together and the fluctuations decrease. On the smallest scales the power turns up and increases monotonically. This indicates where the $P^{\rm 1h} (k)$ term becomes dominant, which occurs on larger scales (smaller $k$) for increasing choice of $M_{\rm min}$ because the sources become more rare.
Note how the signature shifts to slightly higher $k$ in the light cone versions, because the damping scale $r_\star < r_{\rm LW}$. The turnover is also smoothed out as $M_{\rm min}$ increases and the one-halo component begins to dominate at larger scales. This signature is further smoothed by accounting for the duty cycle. The shorter the stellar lifetime (and the smaller the duty cycle), the more amplified the one-halo term will be relative to the unchanged two-halo contribution.
\
Figure \[delta2lw\_fmass\] shows the $M_{\rm min} = 10^8 M_\odot$ scenario at several different redshifts. Curves are labeled according to their normalized $J_{\rm LW,21}$ values. The central curve (blue in the online version) corresponds to the $J_{21} \sim 0.1$ normalized version (z=20.5). Apparent in Figure \[delta2lw\_fmass\] is the washing out of the turnover at increasing $z$; at high $z$ halos are more rare and the LW background patchier – consequently the 1-halo term begins to dominate earlier on scales $k < k_{\rm LW}$, thus smoothing out the key signature.
One important caveat for our model is the assumption of linear bias when computing the 2-halo term in the power spectrum. For example, consider the $M_{\rm min}=10^8 \ M_\odot$ model, which reaches threshold at $z \sim 20$. At that time, such a halo has $b \sim 10$. Thus, even though the rms density fluctuation on $\sim 5$ Mpc scales is $\sim 0.04$, the halo fluctuations are $\sim b \sigma \sim 0.4$, where nonlinear effects are becoming important. The steep intensity profiles around these sources make clustering somewhat more important for the radiation background, as found by @mesfurl2009, and probably enhance the fluctuations on moderately small scales by a factor of a few. For example, @mesfurl2009 found from semi-numeric simulations that nonlinear clustering tends to smooth out the signature turnover in the power spectrum of the ionizing background (where it is due to the smaller attenuation length of high-$z$ ionizing photons); see also the discussion in §\[ahncomp\] below.
As discussed in § \[j21\], we assumed $f_\star = 0.1$ here. This is likely to be an upper limit, and it could be much smaller if, for example, the first star to form in each halo suppresses the formation of any others. In this case the radiation field would not reach threshold until *later*, when there are many more halos and hence smaller fluctuations. Our scenarios therefore provide upper limits to the fluctuation amplitude at threshold.
\
Given the unusual shapes of these power spectra, we next compute the real-space standard deviation in the intensity to provide better intuition for the amplitude of these fluctuations. We calculate the fractional standard deviation, $\sigma(R)$, in the following way: $$\label{variance}
\sigma^2(R) = \int dk \frac{k^2}{2\pi^2} P(k) W^{2}_R(k),$$ where $W^{2}_R(k)$ is the ‘window function’ or smoothing window over which we consider varying $P(k)$. We employ a simple Gaussian window for computational simplicity: $$W_R(k) = e^{-k^2R^2/2}.$$
We display $\sigma(R)$ in Figure \[sigma\] for the choice of $M_{\rm min}=10^8 M_\odot$. From top to bottom, the solid curve is normalized to $10^{-5} J_{21}$ ($z=30.0$), while the others have $J_{\rm LW,21} = 10^{-4},\ 10^{-3}, \ 0.01,\ 0.1,$ and $1$.
The signature turnover at $k_{\rm LW}$ (see Figures \[delta2lw\_z10\] and \[delta2lw\_fmass\]) has been lightly imprinted onto the shape of $\sigma(R)$ in the form of a gentle kink at $R \sim 100$ cMpc. It is evident that at intensity levels nearly approaching, at, and beyond the threshold value, $\sigma(R)$ is small ($\la 1$) down to very small scales ($\sim 1$ cMpc), indicating a fairly uniform background. This suggests that it is unlikely for isolated patches still harboring H$_2$ to exist and foster star-forming minihalos around the threshold.
Fluctuations in the Background at Early Phases {#earlyflucs}
----------------------------------------------
Nevertheless, Figure \[sigma\] shows that fluctuations in the background are large early on when $J_{\rm LW}$ is well below threshold; $\sigma(8 \ \rm{cMpc}) \sim 20$ for $M_{\rm min}=10^8 M_\odot$ at $z=30$. On the flip side of asking whether or not scattered H$_2$ driven star formation could persist in epochs close to or at threshold, these large fluctuations could indicate that even in epochs for which the background is substantially below threshold there will be patches that are locally at threshold in which H$_2$ cooling is suppressed.
However, in this regime the fluctuations are not gaussian, so the standard deviation $\sigma$ is not a good representation of the importance of the fluctuations. Moreover, because we primarily care about the radiation intensity at highly clustered sites of other halos – where star formation is trying to occur – simply taking a pure spatial average is not necessarily the proper approach (see also @dijkstra2008).
\
In order to delve into this new question, we follow the method presented in [@furlloeb2005] with which they calculated the probability that a collapsing halo forms in a region already enriched by galactic winds at high redshift. In contrast, we are interested in calculating the probability that a collapsing halo forms in a region with a LW background above the dissociation threshold. If the sources are very rare, this corresponds to lying within a radius $R_{\rm thres}$ of a LW emitting halo. Within this radius, the ‘new’ halo – one that has passed the threshold to form stars – is irradiated by a local LW intensity above the threshold value for suppressing H$_2$ cooling. We can start by calculating the fraction of space contained within $R_{\rm thres}$ of all LW emitting halos, $Q_{\rm thres}'(z)$, assuming that these regions do not overlap: $$Q_{\rm thres}'(z) = \int_{M_{\rm min}}^{\infty} dm \left( \frac{m}{\bar\rho} \right) \eta(m)n(m),$$ where $\eta(m)$ is the ratio of mass irradiated within $R_{\rm thres}$ of a halo with mass $m$ to that halo’s mass: $$\eta(m) = {4 \pi \bar{\rho} R_{\rm thres}^3/3 \over m}.$$ For example, for a halo of mass $10^8 M_\odot$ at $z=20$ with $M_{\rm min} = 10^8 M_\odot$ and $f_\star = 0.1$, $R_{\rm thres} \sim 3$ cMpc and $\eta \sim 5.5 \times 10^4$. The corresponding values for $z = 30$ are $R_{\rm thres} \sim 8$ cMpc and $\eta \sim 9.7 \times 10^5$. If the flux profile were a pure $1/r^2$ power law, then $R_{\rm thres} \propto L^{1/2}$, so $\eta \propto m^{1/2}$. In reality, the modulation factor steepens the flux profile, so $\eta$ is closer to flat.
In the limit in which sources are truly isolated, $Q_{\rm thres}'$ would be the total filling factor of threshold regions. However, as more sources appear, their regions will begin to overlap and – because $\eta$ is an increasing function of $m$ – grow faster. Because we are only after a crude estimate of this effect, we do not worry about overlap here and use $Q_{\rm thres}'$ as our fiducial estimate. A more sophisticated numerical or Monte Carlo model can easily incorporate this possibility [@dijkstra2008]. The model will, of course, break down when sources become common enough to sit within each other’s $R_{\rm thres}$ (indeed, $Q_{\rm thres}'$ is not limited to be less than unity). Fortunately, in this regime near threshold the halo model approach is perfectly adequate.
Since the newly forming halos are spatially biased and preferentially collapse near existing halos, this expression is not entirely correct. If we consider two halos, the first a newly formed halo and the second an established LW emitting halo, then the excess probability that the two halos live near each other is quantified by the correlation function, $\xi_{\rm gg}$. To linear order, the correlation function can be written as $\xi_{\rm gg} = b_{\rm new}\bar b_{\rm thres}\xi_{\delta\delta}$, where $b_{\rm new} = b(M_{\rm min})$ and $\bar b_{\rm thres}$ are the biases of the newly collapsed and LW threshold region respectively and $\xi_{\delta\delta}$ is the dark matter correlation function. The mean bias of the LW threshold regions surrounding established halos can be written as: $$\bar b_{\rm thres} = \frac{\int dm m\eta(m)b(m)n(m)}{\int dm m\eta(m)n(m)}.$$
With this in mind, we can approximate the corrected probability that a new halo lives within $R_{\rm thres}$ of an established halo as: $$\label{eqn:q}
Q_{\rm thres} = Q_{\rm thres}' \left[ 1 + b_{\rm new}\bar b_{\rm thres} \xi_{\delta\delta} (R_{\rm thres}) \right].$$ We found that these corrections typically boost $Q_{\rm thres}'$ by a factor of $\sim 2$ in our scenarios.
Figure \[qprob\] shows $Q_{\rm thres}$ for a choice of $M_{\rm min} = 10^8 M_\odot$ (rightmost, solid black curves) and $M_{\rm min} = 10^8 M_\odot$ (leftmost, solid red curves) with $f_\star = 0.1$. The dashed curves in Figure \[qprob\] represent the mean intensity relative to threshold, $\bar J_{\rm LW,21}/J_{\rm thres}$. $Q_{\rm thres}$ increases as redshift decreases and the LW background builds up and becomes more uniform in both scenarios. It is evident from Figure \[qprob\] that increasing the choice of $M_{\rm min}$ delays H$_2$ suppression; increasing $M_{\rm min}$ from $10^8 M_\odot$ to $M_{\rm min} = 10^9 M_\odot$ decreases $Q_{\rm thres}$, for example, by a factor of $\sim 33$ at $z \sim 20$. By increasing the mass threshold one eliminates contributions from a host of less massive potential sources, requiring more time for the background to strengthen and boost $Q_{\rm thres}$. Identical calculations using $f_\star = 0.01$ yield values of $Q_{\rm thres}$ that are up to factors of $\sim 15$ smaller.
Also, note that $Q_{\rm thres}$ increases roughly in proportion to $\bar{J}_{\rm LW}$: evidently the mean background provides a good estimate of the volume illuminated by a high intensity of LW radiation. However, note that the quantitative similarity of this filling factor and $\bar{J}_{\rm LW}/J_{\rm thres}$ shown in Fig. \[qprob\] is coincidental and does not occur if we, e.g., change our choice of threshold value or $f_\star$.
Once $J_{\rm LW,21}$ reaches threshold, $Q_{\rm thres} \sim 1.2$ (for $M_{\rm min}=10^8 M_\odot$ and $f_\star = 0.1$). At earlier times, the discrete nature of the sources does substantially increase the probability for a new halo to lie within a threshold region over and above what one might naively guess from Figure \[sigma\]; for example, when $J_{\rm LW} \sim 0.1 J_{\rm thres}$, $\sim 15\%$ of the halos lie in this regime, even though $\sigma \la 1$ down to very small scales. Nevertheless, we still find that at early times there are relatively few patches above threshold.
Comparison to Other Work {#ahncomp}
------------------------
There has been relatively little work on fluctuations in the LW background at high redshifts. The most salient comparison is to @ahn, who calculated the LW background power spectra using a large-scale radiative transfer simulation with size $R_{\rm box} \sim 50$ cMpc. They resolved halos down to $10^8 M_\odot$ and implemented a simple two-population model for galaxies, in which halos with $M<10^9 M_\odot$ had large ionizing efficiencies and larger halos had more modest efficiencies. By following the radiative transfer of ionizing photons, they also included the suppression of galaxy formation in halos with $M<10^9 M_\odot$ following reionization, so this higher-efficiency population gradually disappeared.
Although this simulation box should be sufficiently large to include a fair sample of the halo population,[^5] it is still much smaller than $r_{\rm LW}$. @ahn therefore used a periodic tiling in order to fill out the LW horizon. Unfortunately, this means that they cannot measure the turnover at $k_{\rm LW}$, nor the regime in which the two-halo component dominates and approaches the straightforward limit $b^2 P^{\rm lin}$, so interpreting their results is somewhat difficult.
We are unable to implement the self-regulated reionization model used by @ahn in our simpler analytic model, but we have nevertheless made several test calculations to compare our results, choosing a reasonable minimum mass to match theirs at the different redshifts. Fortunately, @ahn find that their simulation reaches threshold at $z \approx 16$, when the ionized fraction is less than a percent – thus we do not expect the self-regulation to be important in the regime of most interest. On the other hand, their box has only one radiation source at $z=19$, so the pre-threshold regime probably suffers from their finite box size even more than expected.
Comparing to their Figure 12, we find less power, by an order of magnitude or so, in the $k \sim 0.3$–$20$ Mpc$^{-1}$ regime probed by their simulations, from $z \sim 16$–$8$. (They provide only an upper limit at higher redshifts because of the many fewer sources in the box.) However, our model does help to explain the *shape* of their power spectrum, which (when converted to $\Delta^2$) shows relatively flat power over this range, especially at the lower redshifts. This is because it lies between the turnover at $k_{\rm LW}$ and the regime in which the one-halo term dominates (see, e.g., the lower-redshift curves in our Fig. \[delta2lw\_fmass\]). The flattening becomes more pronounced at lower redshifts as the one-halo term decreases in importance, thanks to the increased source density.
Nonlinear clustering is one likely explanation for the different amplitudes: the simulations can include the fully nonlinear clustering of these sources, while our model ignores them. @mesfurl2009 did indeed find a boost of power on comparable scales comparing a halo-model implementation of radiation fluctuations to semi-numeric simulations (in this case, the ionizing background at $z \sim 6$). However, they found a much more modest boost (a factor $\sim 2$), followed by a steepening toward much smaller scales relative to the halo model prediction. Reconciling our results with those of [@ahn] requires a much larger effect. One possible explanation is the amplitude of the source fluctuations, which is $\sim 2$ times larger in the @ahn model at $z=20$ than in the @mesfurl2009 comparison, because the much higher bias of the higher-redshift halos compensates for the smaller fluctuations in the density field.
Another possible explanation is the finite source lifetimes imposed in the numerical simulations, which decreases the number of sources in the box and increases the importance of the one-halo term [@ahn; @iliev2007], albeit in a non-uniform manner within the numerical simulation.
In any case, however, both the simulations and analytic models agree that, before the threshold is reached, fluctuations are relatively unimportant. The large-scale uniformity of the background seems robust, in the absence of large-scale modulation to the source population itself (as may be provided by relative velocities between baryons and dark matter; @tseliakhovich2010).
In principle, we can also compare our model with the detailed Monte Carlo simulations of @dijkstra2008. However, they focus exclusively on times far beyond threshold and very close halo pairs, where our crude approximation no longer applies.
The Lyman-$\alpha$ Background {#lya}
=============================
The Lyman-$\alpha$ background imprints fluctuations onto the 21-cm signal [@barkloeb2005] by way of the Wouthuysen-Field effect [@wouthuysen1952; @field1958] in which the two hyperfine states of neutral hydrogen are mixed via the absorption and reemission of a Lyman-$\alpha$ photon. Once the first sources in the universe turn on and amalgamate into a Lyman-$\alpha$ background, this effect drives the spin temperature, $T_S$, to the gas temperature, $T_K$, resulting in a nonzero brightness temperature relative to the CMB, $T_b$, and allowing the 21-cm line to become visible. We can write $T_b$ as [@fob]: $$\label{tbright}
T_b(\nu) \approx 9x_{\rm {HI}} (1+\delta)(1+z)^{1/2}\left[ 1-\frac{T_\gamma(z)}{T_S} \right] \left[ \frac{H(z)/(1+z)}{dv_\parallel/dr_\parallel} \right] \ \rm {mK},$$ where $x_{\rm HI}$ is the neutral fraction, $(1+\delta)$ is the fractional overdensity of baryons, $T_\gamma(z) = 2.73 (1+z) \ \rm K$ is the brightness temperature of the CMB, and $dv_\parallel/dr_\parallel$ is the gradient of the proper velocity along the line of sight. We can relate $T_\gamma/T_S$ to $T_\gamma/T_K$ with [@fob]: $$\label{tbrightcmb}
\left[ 1-\frac{T_\gamma(z)}{T_S} \right] = \frac{x_c+x_\alpha}{1+x_c+x_\alpha} \left[ 1-\frac{T_\gamma(z)}{T_K} \right],$$ where $x_c$ and $x_\alpha$ are the collisional and Lyman-$\alpha$ scattering coupling coefficients.
Observing the 21-cm signal could provide us a window with which to investigate properties of the exotic sources that collectively shaped the nature of this background. This means that the Lyman-$\alpha$ background is a directly observable effect of (nearly) the same photons that make up the LW background. This allows us to measure that feedback process directly rather than having to infer it from modeling star formation in halos, which is quite difficult.
\
The Flux Profile {#lyaflux}
----------------
Our construction of the Lyman-$\alpha$ flux profile follows [@pritchardfurl]. To calculate the Lyman-$\alpha$ flux originating from a particular source one must consider contributions from all Ly$n$ levels. After absorption (and ignoring recombinations directly to the ground state, which just regenerate the original photons), a fraction, $f_{\rm recycle}$ [@hirata2006], of Ly$n$ photons will be converted into Lyman-$\alpha$ photons via cascades through a series of radiative transitions and will contribute to the total observed Lyman-$\alpha$ flux. The remainder will wind up in the metastable 2$s$ configuration and decay via two-photon emission, resulting in no Lyman-$\alpha$ photon. We use the values of $f_{\rm recycle}$ presented in [@pritchardfurl]. For example, $f_{\rm recycle} = 0.2609, 0.3078$ for $n = 4, 5$ respectively. At large $n$, $f_{\rm recycle}$ asymptotes to a value of $\sim 0.36$. However we note that quantum selection rules ($\Delta L = \pm 1$) forbid a Lyman-$\beta$ photon from producing a Lyman-$\alpha$ photon.
Since a source can only be separated from the observer by a finite distance before its photons redshift into their nearest Ly$n$ transitions, a photon received at redshift $z$ as a Ly$n$ photon must have been emitted below redshift $z_{\rm max}$: $$1 + z_{\rm max}(n) = (1 + z) \frac{[1 - (n + 1)^{-2}]}{(1 - n^{-2})}.$$
This imprints a set of horizons on the flux profile; horizons become smaller and smaller as you consider higher Ly$n$ levels. The series of horizons results in a step-like structure of the overall profile. We assume the flux from one halo takes the following form: $$\label{fluxlya}
F_{\rm Ly\alpha} \propto \frac{m}{4\pi r^2} \sum_{n}^{n_{\rm max}} \epsilon_{\rm b, \alpha}(\nu_n') f_{\rm recycle}(n)$$ where $\epsilon_{\rm b, \alpha}(\nu)$ is the spectral distribution function (defined as the number of photons per baryon emitted at frequency $\nu$ per unit frequency) described by a power law $\epsilon_b(\nu) \propto \nu^{\alpha_s - 1}$. We use the values for $\alpha_s$ presented in [@barkloeb2005] for massive Population III stars and Population II stars. A photon emanating from the source at emission frequency, $\nu'_n$, located at redshift $z'$ is absorbed by level $n$ at redshift $z$: $$\nu'_n = \nu_n \frac{(1 + z')}{(1 + z)}.$$ The sum is ultimately truncated at $n_{\rm max} = 23$ to exclude levels for which the horizon lies within the HII region of a typical (isolated) galaxy [@barkloeb2005a].
The Lyman-$\alpha$ flux profiles for $z=10$ (top curve) and $25$ (bottom curve) are displayed in Figure \[lyaprofile\]. We normalize the curves arbitrarily here in order to focus on the shape as a function of redshift. The Lyman-$\alpha$ horizon distance, $r_{\rm Ly\alpha}$, corresponds to the distance over which a Lyman-$\beta$ photon would redshift into the Lyman-$\alpha$ resonance. This is the maximum range a photon can travel and become a Lyman-$\alpha$ photon; at $z=10$ and $25$, $r_{\rm Ly\alpha}\sim 390$ and $254$ cMpc respectively. In comparison, the LW horizons for those redshifts are $\sim 130$ and $84$ cMpc respectively. We can therefore expect *smaller* fluctuations in the Lyman-$\alpha$ background than in the LW background. However, note that the difference is not as large as one might otherwise expect because the delay from the light travel time already reduces the importance of distant sources.
The dotted curves represent the light cone corrected profiles; we treat the light cone effect in the same fashion that we amended the LW profile in §\[timedep\_lw\]. As can be seen in Figure \[lyaprofile\], the light cone curves begin to diverge from the original version on scales $R > 10$ cMpc. The final and largest ‘step’ (and more and more of the smaller steps as you look at higher redshift) in the profile is effectively beveled as the flux begins to prematurely slope downward until it runs into the horizon. This difference morphs the shape of the power spectrum as discussed further in §\[lyaresults\].
The Mean 21-cm Background {#mean21cm}
-------------------------
We next estimate the redshifts for which the Lyman-$\alpha$ background fluctuations are important when observing the 21-cm signal. Following [@fob] we can write the fractional variation of the brightness temperature of the 21-cm line, $\delta_{21}$, in the following way: $$\label{delta21}
\delta_{21} = \beta\delta_b + \beta_\alpha \delta_\alpha - \delta_{\partial_v},$$ where $\delta_b$ is the perturbation in the baryonic density, $\delta_\alpha$ is that for the Lyman-$\alpha$ coupling coefficient $x_\alpha$, and $\delta_{\partial_v}$ is that for the line of sight peculiar velocity gradient. The expansion coefficients, $\beta_i$, and their evolution over time determine the epochs for which the various perturbations influence the fluctuations in $T_b$. In particular, $$\label{eqn:beta}
\beta = 1 + \frac{x_c}{x_{\rm tot}(1+x_{\rm tot})}$$ and $$\label{beta}
\beta_\alpha = \frac{x_\alpha}{x_{\rm tot}(1 + x_{\rm tot})}.$$ $\beta_\alpha$ is basically the fractional contribution of the Wouthuysen-Field effect [@wouthuysen1952; @field1958] to the coupling, where $x_{\rm tot} \equiv x_c + x_\alpha$ and $x_c$ and $x_\alpha$ are the coupling coefficients for collisions and Lyman-$\alpha$ scattering. For simplicity in our calculations, and for easy comparison to earlier work [@barkloeb2005; @pritchardfurl], we ignore all fluctuations except for those due to density ($\beta \delta_b$) and the Lyman-$\alpha$ background ($\beta_\alpha \delta_\alpha$). We neglect perturbations in the neutral fraction ($\beta_x \delta_x$) and the gas kinetic temperature, $T_K$ ($\beta_T \delta_T$).
The collisional coupling coefficient was calculated as in [@furlanetto2006]. The Lyman-$\alpha$ coupling coefficient can be written as $$\label{xalpha}
x_\alpha = S_\alpha \frac{J_\alpha}{J_\alpha^c},$$ where $S_\alpha$ is a correction factor of order unity [@chenmiralda2004; @hirata2006; @pritchardfurl] that we neglect in our simple model and $J_\alpha$ is the mean Lyman-$\alpha$ intensity. In a similar fashion to equation (\[jlw\]), $J_\alpha$ is given by: $$J_\alpha(z) = \sum_{n=2}^{n_{\rm max}} \int_z^{z_{\rm max}(n)} dz' f_{\rm recycle}(n) \frac{(1 + z)^2}{4\pi} \frac{c}{H(z')} \epsilon(\nu_n', z').$$\[jla\]
The critical intensity, $J_{\alpha,21}^c = 0.66 [(1 + z)/20]$ (in the units of $J_{21}$), corresponds to the threshold level of $J_\alpha$ for which $T_S$ sticks to $T_{K}$ [@fob; @chenmiralda2004]. How does this threshold intensity compare to the LW intensity at which H$_2$ cooling is suppressed? We have displayed $J_\alpha^c$ in units of $J_{21}$ as the dot-dashed line in Figure \[j21\_allz\_allm\]. The dashed curves in Figure \[j21\_allz\_allm\] represent the calculated average Lyman-$\alpha$ intensities for scenarios with $M_{\rm min} = M_{\rm filter}(z), \ 10^6 M_\odot, \ 10^7 M_\odot,$ and $10^8 M_\odot$ and $f_\star = 0.1$. Notice how these intensities are larger than their LW counterparts (solid curves); the Lyman-$\alpha$ horizon distance is a factor of $\sim 3$ times larger than the LW horizon, which not only allows the Lyman-$\alpha$ background to build up more quickly but also allows for a more uniform background, as discussed in § \[lyaflux\].
Around the time that the LW intensity reaches threshold for H$_2$ suppression, $J_\alpha$ is also somewhat higher than that, and hence very close to $J_\alpha^c$. As a result, the 21-cm background is directly sensitive to the physics of cooling; around the time when $T_S$ sticks to $T_K$ numerous minihalos are shutting down stellar production as their H$_2$ supplies are wiped out. Conveniently, this makes the 21-cm background a nearly direct probe of this very interesting epoch in the history of galaxy formation.
\
We present $\beta_\alpha(z)$ in Figure \[betaz\] for $M_{\rm min} = 10^8 M_\odot$ and $f_\star = 0.1$. We find that, for $M_{\rm min} = 10^8 M_\odot$, $\beta_\alpha$ peaks at $z \sim 22$ and is significantly nonzero from $z \sim 15 -30$; fluctuations in the Lyman-$\alpha$ background are important over this range. For this scenario, the mean LW background reaches threshold ($J_{\rm LW,21} \sim 0.1$) by $z \sim 21$.
Results {#lyaresults}
-------
\
Results for the Lyman-$\alpha$ radiation background power spectra are displayed in Figure \[delta2lya\_threslw\]. For ease in comparison to our LW results, we use the corresponding scenarios from Fig \[delta2lw\_z10\], normalized to reach $J_{\rm LW,21} \sim 0.1$. Values for $\beta_\alpha$ in these scenarios are 0.89 for $M_{\rm min} = 10^6 M_\odot$ at $z=30.15$ (bottom panel) and 0.83 for $M_{\rm min} = 10^8 M_\odot$ at $z=20.5$ (top panel). The light cone versions are also shown (short-dashed curves; original versions are the solid curves). The two, long-dashed curves at the top of both panels represent fluctuations for these scenarios including the light cone effect and duty cycle. The topmost curve in both cases assumes an average lifetime of $\tau \sim 3$ Myrs ($f_{\rm duty} = 0.02$ for $M_{\rm min} = 10^6 M_\odot$ and $0.01$ for $M_{\rm min} = 10^8 M_\odot$) while the second curve assumes $\tau \sim 10$ Myrs ($f_{\rm duty} = 0.06$ and $0.04$ for $M_{\rm min} = 10^6 M_\odot$ and $10^8 M_\odot$ respectively).
As discussed in § \[mean21cm\], the mean Lyman-$\alpha$ intensity, $J_\alpha$, is very nearly the critical intensity, $J_\alpha^c$, around the time that the LW intensity reaches the threshold level for H$_2$ suppression (this can be seen in Figure \[j21\_allz\_allm\]). However, $J_\alpha \propto f_{\rm duty}$: stars are ‘turned on’ for a smaller fraction of the time and thus build the radiation background more slowly. Thus the duty cycle curves in this Figure are *not* at the coupling threshold. Obviously the fluctuations are boosted on small and mid-range scales (see Figure \[delta2lya\_threslw\]), but this is not surprising given that we are no longer probing the threshold epochs.
Present in the original models are a series of sequentially damped wiggles, in contrast to the smooth transition of the LW power. These result from the (Fourier transform of the) discontinuous horizon steps present in the Lyman-$\alpha$ flux profile. Unfortunately, these signature wiggles are smoothed out once the light cone effect is applied to the models. As discussed earlier in §\[lyaflux\], the light cone effect bevels out the horizon steps that give the flux profile its distinctive shape, resulting in a more featureless profile and producing a nearly featureless power spectrum.
The power turns over at roughly $k_{\rm Ly\alpha} \sim 2\pi/r_{\rm Ly\alpha}$, where $r_{\rm Ly\alpha}$ is the Lyman-$\alpha$ horizon distance (discussed above in §\[lyaflux\]), except for the light cone versions whose turnovers shift to somewhat higher $k$. In addition, the amplitude of the light cone Lyman-$\alpha$ power is roughly a factor of $2$ smaller than the corresponding amplitudes for the LW background for scales of $k \sim 0.1$ cMpc$^{-1}$. This is likely a symptom of the larger Lyman-$\alpha$ horizon distances, which are $\sim 3 r_{\rm LW}$ for these redshifts. Furthermore, the light cone effect on the Lyman-$\alpha$ power is stronger than the corresponding effects on the LW fluctuations. The Lyman-$\alpha$ models corrected for halo growth over time are boosted in amplitude by a factor of $\sim 7$, while the LW models receive a boost by a factor of $\sim 1.5$. The large Lyman-$\alpha$ horizon allows points to ‘see’ more halos, bolstering the light cone effect on the most distant sources.
The 21-cm Signal
----------------
Finally, armed with the fluctuations in the radiation background and the fluctuations in the baryon density (computed with the linear power spectrum on these scales), we can estimate the 21-cm signal itself. Referring back to equation (\[delta21\]), we consider fluctuations in $T_b$ sourced by perturbations in the matter density, Wouthuysen-Field coupling (or the Lyman-$\alpha$ flux), and radial velocity gradient of the gas. All of these fluctuations are isotropic except for the velocity fluctuation, which introduces an anisotropy to the power spectrum and can be written as $\delta_{\partial_v}(k) = -\mu^2\delta_b$ [@bharadwajali2004], where $\mu$ is the cosine of the angle between the wavenumber $\mathbf{k}$ of the Fourier mode and the line of sight. This enables the total power spectrum for $T_b$ to be separated into powers of $\mu^2$ [@barkloeb2005a]: $$P_{T_b}(\mathbf{k}) = \mu^4P_{\mu^4}(\mathbf{k}) + \mu^2P_{\mu^2}(\mathbf{k}) + P_{\mu^0}(\mathbf{k}).$$
The $\mu^2$ term, which can be written as [@barkloeb2005a] $$P_{\mu^2}(k) = 2\mu^2[ \beta P_\delta(k) + \beta_\alpha P_{\delta-\alpha}(k)],$$ contains contributions of density-induced fluctuations in the Lyman-$\alpha$ flux, where $P_{\delta-\alpha}(k)$ is the cross-power spectrum for the matter density and Lyman-$\alpha$ radiation background. In our halo model, this is very easy to calculate (c.f., @cooraysheth): $$\begin{aligned}
P_{\delta-\alpha}(k) & = & P_{\delta-\alpha}^{1h}(k) + P_{\delta-\alpha}^{2h}(k), \mbox{where} \\
P_{\delta-\alpha}^{1h}(k) & = & \int_{M_{\rm min}}^{\infty} dm n(m) \left(\frac{m}{\bar \rho f_{\rm coll}} \right)\left(\frac{m}{\bar \rho}\right) |u_\delta(k|m)| |u_{\alpha}(k|m)| \\
P_{\delta-\alpha}^{2h}(k) & = & \left[ \int_{0}^{\infty} dm n(m) \left(\frac{m}{\bar \rho} \right) u_\delta(k|m) b(m) \right] \\
& & {} \times \left[ \int_{M_{\rm min}}^{\infty} dmn(m) \left(\frac{m}{\bar \rho f_{\rm coll}} \right) u_{\alpha}(k|m) b(m) \right] P^{\rm lin}(k),
\label{crossps}\end{aligned}$$ where $u_\delta$ is the halo density profile that describes the distribution of mass within each halo.
Therefore, $P_{\mu^2}(k)$ can easily be used to investigate fluctuations in the Lyman-$\alpha$ background at high redshift. We display $P_{\mu^2}(k)$ in Figure \[ptb\] for $M_{\rm min} = 10^8 M_\odot$ at redshifts $z = 11.05, 17.0, 20.5, 23.35,$ and $25.8$ (from top to bottom) so as to correspond to the scenarios presented in Figure \[delta2lw\_fmass\]. We find that fluctuations in $T_b$ increase with decreasing redshift and decrease with scale. The increase in amplitude levels off once $J_\alpha$ reaches the critical intensity, $J_\alpha^c$ (which, for the scenario in Figure \[ptb\], occurs around $z \sim 18$). At this point the Lyman-$\alpha$ coupling is saturated ($x_{\rm tot} \gg 1$) and the 21 cm fluctuations become insensitive to the fluctuations in the Lyman-$\alpha$ background. One can also pick out the ’one-halo’ term kicking in on small scales for these later epochs (the $z=11.05$ curve in Figure \[ptb\]).
The amplitude and overall shape of our power spectra are comparable to those presented in [@pritchardfurl], who used the same model to describe the Lyman-$\alpha$ flux but calculated $P_{\mu^2}$ using a linear transfer function. The amplitudes from our model also agree with those from [@barkloeb2005], although the shape of the power differs because they neglected the effects of atomic cascading (by assuming $f_{\rm recycle} = 1$) in their calculations.
Unfortunately, these power spectra do not display any sort of tell-tale signature feature such as the distinct LW turnover as can be seen in Figures \[delta2lw\_z10\] and \[delta2lw\_fmass\] or the wiggles from the horizon steps, because the density-induced fluctuations wash them out. This agrees with previous work; recently, [@vonlanthen2011] showed with a radiative transfer numerical simulation that the horizon steps present in the Lyman-$\alpha$ flux profile left imprints in the differential brightness temperature profile just after the first luminous sources turned on. However, as time progressed and more new sources began contributing to the Lyman-$\alpha$ background, the steps were effectively wiped out.
\
Summary & Conclusions {#summary}
=====================
In this paper, we have modeled fluctuations in the LW and Lyman-$\alpha$ radiation backgrounds using a variation of the halo model. First, we calculated the LW power spectrum and found that the power is characterized by an abrupt cut-off at the LW horizon distance, $r_{\rm LW}$; the power turns over at the horizon wavenumber, $k_{\rm LW}$, unless the sources are so rare that the one-halo term dominates (i.e., correlations are determined by the flux profiles of individual sources).
We found that the fluctuations in the background are weak and should not lead to substantial spatial fluctuations in H$_2$ content. Once a population of low-mass halos produces enough stars to generate a threshold LW background large enough to destroy their own H$_2$ reservoirs used for cooling, star formation can only proceed in larger halos with more substantial reservoirs. Our model predicts that, by the time this threshold is reached, fluctuations in the intensity field will be quite small, so this transition will be rather homogeneous across the entire Universe.
Though we found fluctuations in the background to be small around threshold, on the flip side we also found them to be large in those early epochs during which the background was approaching threshold. This could indicate the presence of patches of collapsing halos that are locally above threshold and H$_2$ suppressed. Taking into account the bias of sources, we crudely approximated the probability that a new halo lives in such a patch and found that even though source clustering substantially increases this probability, it is still relatively small. Thus, at this time, the majority of newly-forming halos can continue to cool via H$_2$, even in the presence of established galaxies [@dijkstra2008].
Eventually the first stars will build up the LW background by enough to suppress their own star formation; the process will terminate when only those halos above the atomic cooling threshold can form stars. However, the mode of star formation in these halos is very different from the minihalos that produce the first stars, so the transition to higher-mass halos has important consequences for the global star formation history [@ohhaiman2002]. This transition from Population III to Population II star formation is of course extremely complex, and we have examined only a small part of it. The formation of very massive Population III stars requires two physical conditions: (1) metal-free gas and (2) a reservoir of H$_2$ that allows the gas to cool [@brommlarson2004]. We have not examined the first condition, but the slow speed of galactic winds (compared to the speed of light) guarantees that metal enrichment will be very inhomogeneous, and pristine pockets of gas could persist until very low redshifts [@scann2002; @furlloeb2005]. In contrast, the LW background is spatially uniform and so will induce a rapid, homogeneous transformation in the fundamental processes of star formation, even when metal enrichment remains inhomogeneous. This will induce a shift from very massive Population III.1 stars to less massive – but still primordial – Population III.2 stars that require atomic cooling.
Next we considered the fluctuations in the Lyman-$\alpha$ background in a similar fashion. The Lyman-$\alpha$ flux profile imprints a series of wiggles on the shape of the power, corresponding to the series of horizon steps that characterize the profile. Unfortunately, unlike the LW case, these signature wiggles are washed out once we account for halo growth over time. We found that the amplitude of the Lyman-$\alpha$ power is smaller than that for the LW background by a factor of $\sim 2$ for scales larger than $\sim 15$ cMpc. The smaller fluctuations are due to the large Lyman-$\alpha$ horizon distance, $r_{\rm Ly\alpha} \sim 3 r_{\rm LW}$, allowing the halos to ‘see’ further.
We used our model for the fluctuations in the Lyman-$\alpha$ background to generate power spectra for the brightness temperature, $T_b$, of the 21-cm signal. We find our values to be in good agreement with previous estimates [@pritchardfurl; @barkloeb2005] that used a very different approach to estimate the radiation field fluctuations. We do not see a signature feature present in the power in contrast to the distinct LW turnover.
Our relatively simple model, though convenient and efficient, has a number of caveats that compromise accuracy. Most notably, we neglect nonlinear effects on the backgrounds by relying on a linear approximation for the halo-halo correlation function. Thus, our models are only good down to the scales for which this linear approximation still holds. In addition, we assumed a uniform star forming efficiency, $f_\star$, for all minihalos in our calculations. As the halo population grows in size and complexity in the later epochs, variations in galactic properties – sourced partly by the suppression of H$_2$ cooling, but also due to a myriad of other factors – will complicate our simple treatment. After comparing our results to the previous simulation from [@ahn] we find that the shapes can be well matched even with the limited dynamic range of the Ahn values (the signature turnover is not covered in their range). Although our amplitudes disagree by a factor of $\sim 10$ at the lower redshifts, we both find gentle fluctuations in the background around the time it reaches threshold, implying that ‘safe’ patches sheltering isolated sources of H$_2$ are rare by this epoch.
Furthermore, we have neglected the effects of X-rays in our models. If the first luminous sources had hard spectra that extended out to X-rays, these X-rays would have far reaching effects due to their large mean free paths – they could, by catalyzing the formation of new H$_2$, potentially counteract H$_2$ photodissociation by the growing UV background [@mcdowell61; @hrl96; @haiman00]. This could stall the transition of Population III to Population II stars by allowing minihalos to continue forming new stars, altering the make-up of the sources responsible for reionization.
Finally, @tseliakhovich2010 argue that a new nonlinear effect must be considered in structure formation: the supersonic relative velocity between dark matter and baryons can suppress the matter power spectrum near the baryonic Jeans scale, altering the abundance and clustering properties of the first dark matter halos. This effect could be accounted for in a future version of our simple model by introducing a modulation factor to the halo mass function. In fact, @dalalpen10 argue that the resulting fluctuations in the radiation background could strongly affect the 21-cm absorption power spectrum. However, they assume that the velocity exerts a very strong effect on galaxy formation, which may be at odds with more detailed numerical simulations [@maio2010; @stacy2010]. In any case, such a large scale modulation will add more power to the radiation field and may have important implications for the homogeneity of the LW and Lyman-$\alpha$ backgrounds, since these acoustic features appear on comparable scales to the LW horizon ($\sim 100$ Mpc).\
This research was partially supported by the David and Lucile Packard Foundation, the Alfred P. Sloan Foundation, and by NASA through the LUNAR program. The LUNAR consortium (http://lunar.colorado.edu), headquartered at the University of Colorado, is funded by the NASA Lunar Science Institute (via Cooperative Agreement NNA09DB30A) to investigate concepts for astrophysical observatories on the Moon.
[^1]: Email: holzbauer@astro.ucla.edu
[^2]: In the context of the halo model, our radiation background calculation is analogous to the dark matter density power spectrum in the halo model, *not* to the (discrete) galaxy power spectrum. Thus the ‘one-halo’ term is very important to our results on small scales, even if each halo contains only one galaxy.
[^3]: A more complex relationship is likely, but any such relationship can be bracketed by our different choices of $M_{\rm min}$.
[^4]: We implicitly assume a flat photon spectrum within the LW range here, which is a reasonable approximation over this short frequency interval.
[^5]: According to the methods of @barkloeb2004, the missing large-scale modes only suppress the halo population by a few percent.
|
---
abstract: 'Evidence derived with minimal assumptions from existing published observations is presented to show that an ion-proton plasma is the source of radio-frequency emission in millisecond and in normal isolated pulsars. There is no primary involvement of electron-positron pairs. This conclusion has also been reached by studies of the plasma composition based on well-established particle-physics processes in neutron stars with positive polar-cap corotational charge density. This work has been published in a series of papers which are also summarized here. It is now confirmed by simple analyses of the observed radio-frequency characteristics, and its implications for the further study of neutron stars are outlined.'
author:
- |
P. B. Jones[^1]\
University of Oxford, Department of Physics, Denys Wilkinson Building,\
Keble Road, Oxford OX1 3RH, U.K.
title: 'Ion-proton pulsars'
---
\[firstpage\]
pulsars: general - polarization - plasmas - stars: neutron
Introduction
============
We believe that a strong case has been developed in a series of publications (Jones 2010a - 2016) for an ion-proton plasma as the source of emission in radio-loud pulsars. The initial approach was to determine the plasma composition in the open magnetosphere above the polar caps in ${\bf \Omega}\cdot{\bf B} < 0$ neutron stars from well-established particle-physics processes. (Here ${\bf \Omega}$ is the rotational angular velocity and ${\bf B}$ the polar-cap magnetic flux density: the polar-cap Goldreich-Julian charge density is positive.) This lead immediately to the recognition that the plasma must have a significant proton component arising from photo-nuclear reactions in the electromagnetic showers produced by electrons that are accelerated towards the neutron-star surface. Either pair creation (if present) or photoelectric ionization of accelerated ions are sources of these electrons. Later it was realized that the reverse electron flux from photoelectric ionization limits the electric field component antiparallel with ${\bf B}$ as would pair creation were it present. Photoelectric ionization transition rates are rapidly increasing functions of ion Lorentz factor so that particle acceleration is a self-limiting process resulting in moderately relativistic but not ultra-relativistic energies. Pair creation by single-photon magnetic conversion is not normally possible above the polar cap, but the presence of ions and protons with only moderate Lorentz factors leads to strong growth of longitudinal and quasi-longitudinal Langmuir modes. (In this paper, the term Langmuir mode is used to describe any longitudinal or quasi-longitudinal unstable mode in a system of two or more parallel particle beams of different velocities and of any charged particle type.)
The complete process is not particularly simple and, unfortunately, depends quite naturally on two parameters that are not well-known, the surface ion atomic number $Z$ and the neutron star surface temperature $T_{s}$, more precisely the temperature of the surface area of about one steradian centred on the polar cap, which is the source of the photo-ionizing black-body photons. An ion-proton plasma is formed for any $Z$ such that ions at the point of leaving the surface retain some bound electrons provided also that values of $T_{s}$ are such that the black-body radiation field has a large enough photon density to produce adequate photoelectric transition rates. Therefore, the formation of an ion-proton plasma is possible for quite wide intervals of these parameters, in particular, temperatures $T_{s} > 1-2\times 10^{5}$ K are adequate. Even so, there may be doubts that suitable values would be present in cooling neutron stars to account for the number of isolated old radio-loud pulsars with ages as large as $10^{2-3}$ Myr. For this reason, observational evidence that the emission region contains ions and protons but few if any electrons and positrons is very desirable provided it can be interpreted with minimal and uncontentious assumptions.
It was not recognized until most of the work had been completed (Jones 2016) that such evidence already existed in the pulse-longitude dependence of the integrated profiles of circularly-polarized intensity, the Stokes parameter $V$, exemplified particularly in the high-resolution and good signal-to-noise ratio measurements of Karastergiou & Johnston (2006) for a set of 17 pulsars. Section 2 of the present paper contains a survey of this, and of other less direct but still persuasive observational evidence for an ion-proton plasma. It also includes a review of estimates of the emission region altitude, which is of prime importance in the interpretation of the circular polarization data.
Early papers (Jones 2010a - 2013a) converge towards this paper’s view of the ion-proton plasma. Thus some results given in those papers have either been superseded or are of only marginal relevance. In particular, it was recognized only in Jones (2014c) that the presence of a density of electrons or positrons small compared with the Goldreich-Julian density can reduce to zero the Langmuir-mode growth rate which would in its absence exist in an ion-proton plasma. The significance of this result is that it provides a specific mechanism for the bi-stability associated with the phenomena of mode-changes and nulls. To correct this, Section 3 gives a summary of the properties of the ion-proton plasma including the case of a small background electron-positron component which is likely to be relevant to an understanding of mode-changes, nulls and subpulse drift. The implications, for both signs of ${\bf \Omega}\cdot{\bf B}$, of the ion-proton plasma in radio-loud pulsars are considered in Section 4. In particular, we believe it shows that the open magnetosphere current, which relates the spin-down torque at the neutron-star surface to that beyond the light cylinder, is set by conditions at the neutron-star surface rather than the light cylinder.
Observational evidence
======================
The characteristics of what we describe as the primary radio-frequency emission are almost universal. Excluding a small number of [*sui generis*]{} cases, spectral indices are large and negative. At low frequencies, in cases where observations exist, there is usually a turn-over below $\sim 100$ MHz. Luminosities have been estimated by Szary et al (2014) for all pulsars which have been observed at $1.4$ GHz, solely from the flux densities at that frequency. The luminosity distribution is more than two orders of magnitude wide but, remarkably, it is independent of position in the $P -\dot{P}$ plane. It has also been noted many times (Jenet et al 1998; Kramer et al 1999; Espinoza et al 2013) that the spectra of radio-loud millisecond pulsars (MSP) are broadly similar to those of normal pulsars.
Thus spectral properties appear to have no significant trends of variation over five orders of magnitude in $B$, and three orders of magnitude in rotational period $P$. It might be thought that this conclusion in itself does not seem obviously consistent with the canonical model of a secondary electron-positron plasma source. But there are more specific concerns.
Radio-frequency power generation
--------------------------------
In relation to the emission process, we believe that the radio-frequency energy generated per unit charge accelerated at the polar cap, equal to $W = L/f_{GJ}$, where $L$ is the luminosity and $f_{GJ}$ is the Goldreich-Julian rate at which unit charges leave the polar cap, is an interesting parameter. The luminosities were estimated by Jones (2014b) for a set of 29 pulsars which appeared in the spectral compilation published by Malofeev et al (1994) and also had a sufficient signal-to-noise ratio to be listed in the paper of Weltevrede, Edwards & Stappers (2006). The luminosities found there are typically one or two orders of magnitude smaller than the mean value of $\sim 10^{29}$ erg s$^{-1}$ given by Szary et al. Values of $W$ are distributed over more than two orders of magnitude. In a few cases, for example, $29$ GeV for B1642-03, they can be large enough to suggest possibly the presence of a caustic in one of the variables defining the beam. But the average value, $W = 4.3$ GeV, is also large.
How is it possible that this value could be generated by either the secondary electron-positron pairs produced by a single ultra-relativistic electron or positron or by unit charge of an ion-proton plasma? For the electron-positron case the relativistic Penrose condition (see Buschauer & Benford 1977) must be satisfied. The optimum electron-positron energy distribution which does so is given by $N_{\pm}(\delta(\gamma - \gamma_{1}) + \delta(\gamma - \gamma_{2}))$ per primary electron in the neutron-star frame, where $\gamma$ is the electron or positron Lorentz factor and the constants are usually assumed to be of the order of $\gamma_{1,2} \sim 10^{1-2}$. Here $N_{\pm}$ is the secondary-pair multiplicity. Given the absence of any accelerating field in the emission region, application of the conservation laws for energy and for the momentum component parallel with ${\bf B}$ shows that the maximum energy in photons that can be extracted is, $$\begin{aligned}
W = N_{\pm}mc^{2}\frac{(\gamma_{1} - \gamma_{2})^{2}}{(\gamma_{1} + \gamma_{2})},\end{aligned}$$ emphasizing the fact that it is the velocity difference between the beams which makes radio-frequency energy generation possible. (This excludes the generation of curvature radiation.) For $W = 4.3$ GeV, a dense plasma with large $N_{\pm} \sim 10^{2}$ would be necessary for the values of $\gamma_{1,2}$ which are usually assumed and are necessary for adequate Langmuir-mode growth rates. But there is no evidence that such a favourable distribution forms or that if formed, it would have a similar density in pulsars in all parts of the $P - \dot{P}$ plane. Our conclusion is that, even in this most favourable case, the electron-positron plasma is unlikely to produce the values of $W$ that are observed.
Owing to the differing charge-to-mass ratio of ions and protons, the $\delta$-function velocity distribution occurs naturally in that plasma. Application of the kinematic conservation laws to this case, specifically equation (14) of Jones (2014b), shows that, owing to the baryonic masses of the particles, the values of $W$ estimated from the observed luminosities can be achieved in an ion-proton plasma.
Our use of energy and momentum conservation in the emission region would be invalid if there remained any acceleration field ${\bf E}_{\parallel}$. The effect of radio-frequency energy generation is to reduce the Penrose-condition velocity difference, but a residual ${\bf E}_{\parallel}$ would have the reverse effect in the case of an ion-proton plasma owing to the differing charge-to-mass ratio of the components. This would, in principle, allow greater generation of radio-frequency energy than the conservation-law values given by Jones (2014b) and might be the explanation for the few very large values of $W$ that are seen.
The emission altitude
---------------------
Estimates of absolute radio-emission altitudes are always model-dependent. It is not appropriate to describe here the various assumptions that different authors have made, but we refer to the papers cited in Table 1. Estimates are usually given in the form of upper limits: these can be superseded by smaller upper limits. Thus the papers cited have been limited to more recent work except where actual values, with or without errors, have been given. The results are a miscellany and comparisons for a given pulsar are not possible except in the cases of B0329+54 and B1133+16. Absolute radii $r_{e}$ are given in the right-hand column. Errors, where given, are large. Noutsos et al (2015) attempted to obtain $r_{e}$ for 11 pulsars but found in 7 cases that their procedure gave values either negative or consistent with zero. They do not give details for these cases but it is possible that their inclusion would have been relevant to the estimation of an average $r_{e}$ for the whole set. Considering all the values given in Table 1 our conclusion is that it is not possible to give an average value for $r_{e}$.
The size of the emission region $\Delta r_{e}$ can be estimated with fewer assumptions and less model-dependence. Here, the results are all upper limits. Kramer et al (1997) and Hassall et al (2012) show by using profiles at different frequencies that, for normal pulsars with periods $P\sim 1$ s, the emission regions are compact. They place upper limits on the effects of aberration and retardation, and make no assumptions about radius-to-frequency mapping. (Also, for a substantial set of MSP, Kramer et al 1999 were able to quote the extremely small value $\Delta r_{e} < 2.4$ km as an upper limit to the transit-time difference between radiation at the different observed frequencies.)
The general similarity of normal pulsar and MSP radio spectra is consistent with their being produced in similar plasma conditions. Goldreich-Julian particle number densities are $\propto BP^{-1}$ and at a given radius are usually smaller in MSP than in normal pulsars but by no more than an order of magnitude, so that plasma conditions are not too dissimilar. A typical MSP light-cylinder radius is only $R_{LC}\sim 200$ km. Therefore, an emission radius of perhaps one half of this would require a very compact $\Delta r_{e}$.
On balance, and owing to the similarity of plasma density in MSP and in normal pulsars, we accept the LOFAR results of Hassall et al. Their general validity will be assumed in Sections 2.3 and 2.4.
[@llrr@]{} Authors & Pulsars & $\Delta r_{e}$ & $r_{e}$\
& & km & km\
\
von Hoensbroech & & &\
& Xilouris, 1997 & & - & $210\pm 180^{a}$\
& & & $420\pm200^{b}$\
\
Kramer et al, 1997 & 0329+54& $<320$ & -\
& 0355+54& $<50$ & -\
& 0540+23& $<120$ & -\
& 1133+16& $<310$ & -\
& 1706-16& $<210$ & -\
& 1929+10& $<110$ & -\
& 2020+28& $<160$ & -\
& 2021+51& $<360$ & -\
\
Thomas & Gangadhara, 2010 & 1839+09& - & $50^{a}$\
& & & $60^{b}$\
& 1916+14& - & $100^{a}$\
& & - & $300^{b}$\
& 2111+46& - & $500^{c}$\
& & - & $80^{d}$\
\
Karuppusamy, Stappers & & &\
& Serylak, 2011 & 1133+16& - & $ 560$\
\
Hassall et al, 2012 & 0329+54& $<128$ & $<183$\
& 0809+74& $<384$ & -\
& 1133+16& $<59$ & $<110$\
& 1919+21& $<49$ & -\
\
Noutsos et al, 2015 & 0823+26& - & $144^{+136}_{-134}$\
& 0834+06& - & $452^{+437}_{-432}$\
& 1133+16& - & $349^{+158}_{-150}$\
& 1953+50& - & $177^{+74}_{-119}$\
Frequency spectrum
------------------
Pulsar radio spectra in cases where sufficient low-frequency observations exist have a turnover, usually below $100$ MHz, and above that a power-law decrease in intensity with large negative spectral index. There may be a break, a change of index, usually below $1$ GHz and to more rapid intensity decrease. Almost all the power is emitted well below $1$ GHz. Some information about the distribution of index values can be found for those pulsars which have fluxes at $400$ and $1400$ MHz listed in the ATNF catalogue (Manchester et al 2005). This shows that the spectral index is almost uncorrelated with the parameter $X = B_{12}P^{-1.6}$, which is a measure of the capacity of a neutron star to produce pairs above the polar cap in the ${\bf \Omega}\cdot{\bf B} > 0$ case, derived from Fig. 1 of Harding & Muslimov (2002).
The emission region can be regarded only as a black box within which plasma turbulence exists. The radiation frequency is well below the threshold for cyclotron absorption and the only natural frequency present in the box is the rest-frame plasma frequency $\omega^{c}_{p}$. It is natural to attempt to relate this to the observed spectrum as in Jones (2013b). In the case of a secondary electron-positron plasma, the rest-frame plasma frequency is, $$\begin{aligned}
\omega^{c}_{p} = \left(\frac{8\pi n_{GJ}N_{\pm}{\rm e}^{2}}{m\gamma_{e}}\right)^{1/2}\end{aligned}$$ where $n_{GJ}$ is the number density of the primary beam of electrons or positrons and $\gamma_{e}$ the average secondary electron-positron Lorentz factor. It determines the typical wavenumber of the fluctuations from whose growth turbulence develops. The transfer of energy to higher rather than lower wavenumbers is a general property of developing turbulence, leading in the example of incompressible fluids to a power-law spectrum with the $-5/3$ Kolmogorov index. Recent work by Zrake & East (2016) on force-free magnetic turbulence, though not immediately applicable here, has produced further evidence for the ubiquity of the Kolmogorov law. Therefore, it is natural here to associate $\omega^{c}_{p}$ with the low-frequency turnover region of the spectrum at a typical radiation frequency, $$\begin{aligned}
\nu_{obs} = \gamma_{e}\omega^{c}_{p}/\pi \approx 25\left(N_{\pm}\gamma_{e}\frac{B_{12}}{P}\right)^{1/2} \left(\frac{R}{r_{e}}\right)^{3/2} {\rm GHz},\end{aligned}$$ where $R$ is the neutron-star radius and $B_{12}$ the polar-cap surface magnetic flux density in units of $10^{12}$ G. This is in order of magnitude disagreement with observation for acceptable values of $N_{\pm}\gamma_{e}$ and for values of $r_{e}$ favoured by Table 1. In our view, this represents a serious problem for the canonical electron-positron source model.
For an ion-proton plasma, equation (3) is replaced by, $$\begin{aligned}
\nu_{obs} = 4.2\times 10^{2}\left(\gamma_{A,Z}\frac{ZB_{12}}{AP}\right)^{1/2}\left(\frac{R}{r_{e}}\right)^{3/2} \hspace{5mm} {\rm MHz}\end{aligned}$$ in which $\gamma_{A,Z}$ is the ion Lorentz factor in the neutron-star frame. Here, a turn-over below $100$ MHz is easily possible. This conclusion was reached in Jones (2013b) specifically in the case of B1133+16.
Circular polarization
---------------------
Circular polarization provides the most definitive evidence for an ion-proton plasma. In Jones (2016) we have referred to the paper of Karastergiou & Johnston (2006) as a consequence of the number of pulsar integrated profiles observed, the signal-to-noise ration and the time resolution in that work. But the crucial feature of the circular polarization can also be seen in the papers of Han et al (1998) and of Yan et al (2011) and Dai et al (2015) on MSP.
The assumptions made here are minimal. We assume as in Section 2.3 the existence of an emission region above the polar cap whose upper boundary could be defined roughly as the surface of last absorption for radio-frequency radiation. Above this, the radiation propagates through plasma moving outward from the polar cap. Observationally, we know that the radiation generally has strong linear polarization: this must be a feature of the emission process but it is modified by propagation through the plasma at $r > r_{e}$. As it leaves the emission region, it propagates as a linear combination of the normal modes of the plasma whose polarizations for wavevector ${\bf k}$ are parallel with ${\bf k}\times({\bf k }\times {\bf B})$ and ${\bf k}\times {\bf B}$ for the O- and E-modes, respectively. The E-mode refractive index is negligibly different from unity, but the O-mode refractive index differs from unity at angular frequency $\omega$ by $$\begin{aligned}
\Delta n_{O} = - \sum_{i}\frac{\omega^{2}_{p}\sin^{2}\theta_{k}}
{2\gamma^{3}_{i}\tilde{\omega}^{2}} \approx -\sum_{i}\frac{2\omega^{2}_{p}\theta^{2}_{k}}
{\gamma^{3}_{i}\omega^{2}(\theta^{2}_{k} + \gamma^{-2}_{i})^{2}},\end{aligned}$$ in which $\tilde{\omega} = \omega - k_{z}v^{i}_{z}$ and $\omega^{2}_{p} =
4\pi n_{i}q^{2}_{i}/m_{i}$, where $q_{i}$ is the charge of particle type $i$. The local direction of ${\bf B}$ is the $z$-axis, $\theta_{k}$ is the angle between ${\bf k}$ and ${\bf B}$, $n_{i}$ and $\gamma_{i}$ are the particle number density and Lorentz factor. Equation (5) is valid for a particular particle type, provided $\gamma_{i}\tilde{\omega} \ll \omega^{i}_{B}$ (the radiation angular frequency in the particle rest frame is small compared with the cyclotron frequency $\omega^{i}_{B}$) and in this limit, the particle velocity $v^{i}_{z} = v^{i}$. The rate of change of the relative phase of O- and E-modes decreases with distance propagated as the plasma density falls and the polarization of the radiation approaches asymptotically what is observed. In terms of observer intensities, this is in general a sum of linearly and circularly polarized components. For the conditions assumed by Jones (2016), the unequal amplitudes of the O- and E-modes result in the linear component being not much changed from the original linear polarization of the emission region. Then the circularly polarized component is relatively small and its intensity and sense depend on the final relative phase of the two plasma normal modes. Along the line of sight, this is $$\begin{aligned}
\phi_{ret} = \frac{\omega}{c}\int^{\infty}_{r_{e}}\Delta n_{O}dr = \sum_{i}
\frac{r_{e}\omega^{2}_{p}}{4\gamma_{i}\omega c}.\end{aligned}$$ This is valid for small $r_{e}$ such as follow by acceptance of the Hassall et al estimates given in Table 1. In this case $\theta_{k} \sim \gamma^{-1}_{i}$, as assumed in Jones (2016). If flux-line curvature were thought to be relatively more significant, as it might be at emission radii an order of magnitude larger, $\Delta n_{O}$ would be reduced by a factor of $\gamma^{2}_{i}\theta^{2}_{k}$ but the phase retardation given by equation (6) by only a much smaller amount because most of the phase retardation occurs at radii below $2r_{e}$.
A large part of the seminal paper by Cheng & Ruderman (1979) on propagation and circular polarization was concerned with the effects of refraction in electron-positron plasmas, in particular, the conditions under which coherence between O- and E-modes would be maintained. But in an ion-proton plasma, $\Delta n_{O}$ is so small that the angular deviation between O- and E-modes produced by refraction in gradients perpendicular to ${\bf B}$ is negligible as is the relative lateral displacement of the modes.
The final relative phase which determines the polarization state must be a function of observed longitude within the profile even if only as a consequence of the variation of particle Lorentz factor across the open magnetosphere. There is a change in the sign of $V$ (left or right circular polarization) for each increment of $\pi$ change in $\phi_{ret}$, interspersed by zeros in $V$. Examination of the longitude dependence and changes of sense of the circular polarized intensity in each of the Karastergiou & Johnston profiles shows that these vary quite slowly, from which it follows that the absolute relative phase change $\phi_{ret}$ occurring after the surface of last absorption cannot be large but must be no more than of the order of $1 - 10 \pi$. This is qualitatively consistent with equation (6) for an ion-proton plasma and $r_{e}\sim 10^{2}$ km.
Components of the dielectric tensor are inversely proportional to particle mass so that even a small electron-positron number density has an overwhelming effect on the refractive index of an ion-proton plasma given by equation (5). A Goldreich-Julian current-density primary beam with $N_{\pm} \sim 10^{2}$ pairs per positron would produce phase retardation several orders of magnitude too large even at emission radii $r_{e} \sim 10^3$ km and its rate of variation would be much faster than is observed. The rapid changes of sense would render circular polarization unobservable. This has to be viewed in the context of Table 1 on which basis we do not believe that the electron-positron plasma is a possible source.
There is a further factor in favour of the ion-proton plasma. Owing to the rapid decrease of $\omega^{i}_{B}$ with altitude in both electron-positron and ion-proton plasmas, there are regions nearer the light cylinder in which cyclotron absorption occurs. In the electron-positron case, the absence of any strong observational evidence for this process in pulsars has been considered in the past to be a very real problem (see Blandford & Scharlemann 1976; Mikhailovskii et al 1982; Lyubarskii & Petrova 1998; Fussell, Luo & Melrose 2003). The explanation is that the cross-section in the case of protons is negligibly small.
Properties of the ion-proton plasma
===================================
Consequences of reverse electron flow
-------------------------------------
A reverse flux of electrons from photo-ionization (or pair creation should it occur) must certainly be a feature of polar caps. The consequent physical processes are all individually well-established and their application to polar caps was described by Jones (2010a, 2011). A summary is given here. Formation of the nuclear giant dipole state by electromagnetic shower photons in the energy interval $15 - 30$ Mev is the dominant hadronic interaction giving neutrons and protons. The photon track length in an electromagnetic shower per unit photon energy and primary electron energy is known, as are the cross-sections (integrated over photon energy) for giant dipole-state formation. From these, the proton production rate is easily obtained. The polar-cap surface, whether liquid or solid, lies under a thin atmosphere of ions, in local thermodynamic equilibrium (LTE), having a scale height of about $10^{-1}$ cm. Its column density is an approximately exponential function of surface temperature and ion work function, and therefore is difficult to estimate. Electromagnetic showers are formed either partially or entirely within this atmosphere depending on its column density. Protons emitted are quickly reduced to thermal energies and diffuse to the top of the atmosphere with negligible probability of nuclear interaction, and with characteristic time $\tau_{p} \sim 1$ s.
Atmospheric structure is important. Given LTE, it depends on the charge-to-mass ratio of its components. Thus the protons cannot be in static equilibrium within the predominantly ionic atmosphere. They pass through and are preferentially accelerated from the polar cap. Ions are accelerated only if there are insufficient protons to form a Goldreich-Julian current density. The relation between ion and proton current densities is local and time-dependent.
With regard to the effect of high static magnetic fields on shower formation, approximate cross-sections for electron-positron pair creation and bremsstrahlung have been obtained up to $B \sim B_{c} = 4.41\times10^{13}$ G (Jones 2010b). The overall influence of high fields on shower formation is complex. The Landau-Pomeranchuk-Migdal effect (see Klein 1999) is present. It represents the effect of small-angle Coulomb multiple scattering in the medium in partially removing the coherence intrinsic to the Feynman diagram description of these processes, which becomes significant at high electron energies and at high matter densities , as at the neutron-star surface. It reduces bremsstrahlung and pair-creation cross-sections significantly at $B_{c}$. But an order of magnitude below this, the conclusion, adequate for present purposes, is that proton creation per GeV shower energy is only a weakly varying function of $B$ and in the interval of interest here does not differ much from its zero-field value.
It was recognized in Jones (2012a, 2012b) that the presence of reverse electrons from photo-ionization adjusts the charge density above the polar cap so as to limit the acceleration field ${\bf E}_{\parallel}$ to values much below those predicted by the Lense-Thirring effect (Beskin 1990, Muslimov & Tsygan 1992). The effective blackbody field is not that of the polar cap but of the surface area, temperature $T_{s}$, of about a steradian centred on the polar cap, whose photons transformed to the ion rest frame can reach the photo-ionization threshold at those altitudes where the Lense-Thirring acceleration field remains significant. The number of photons reaching this threshold increases rapidly with increasing ion Lorentz factor with the consequence that ion acceleration is self-limiting. At temperatures $T_{s} > 1-2\times 10^{5}$ K, no more than moderately relativistic Lorentz factors are reached: ${\bf E}_{\parallel}$ is limited to values far below those needed for pair creation. A model polar cap was constructed on these bases in Jones (2013a).
But the problem with these papers is that, whilst the processes were in essence correctly investigated, their application to the observed classes of pulsar including the Rotating Radio Transients (RRAT) was flawed owing to the assumption that a secondary electron-positron plasma could be a source of the coherent emission. The properties of the ion-proton plasma above the polar cap as they appear in the Jones (2013a) model can be summarized as follows.
\(i) The effect of photo-ionization is local. The proton flux at a point on the polar cap is determined exclusively by the previous reverse-electron energy flux at that point.
\(ii) The state of the whole polar cap is chaotic: the rate of change of ion or proton current densities is characterized by $\tau_{p}$. Thus the fluctuations in ion and proton current densities are broadly consistent with the intensity differences observed between successive individual pulses.
\(iii) Coincident non-zero ion and proton current densities in any part of the polar cap are necessary for Langmuir-mode growth above that area. Absence of either component results in no coherent emission except that very weak emission may be observed if there is a small background of electrons or positrons in addition to the baryonic component (see Jones 2014c).
\(iv) For a certain interval of $T_{s}$, mixed current densities occur preferentially near the polar-cap perimeter, with central regions having almost entirely proton current densities for much of the time.
\(v) The ion Lorentz factors reached are approximately those for which the blackbody photons in the ion rest frame reach successive photo-ionization thresholds. Hence, very roughly, $\gamma_{i} T_{s}$ can be regarded as a constant.
\(vi) Low values, roughly $T_{s} < 2\times 10^{5}$ K, are insufficient for photo-ionization unless Lorentz factors are high. The Langmuir mode growth-rate is $\exp \Lambda$ where $\Lambda \propto \gamma^{-3/2}_{i}$, so that in the low-$T_{s}$ and presumably old-age limit, growth to a turbulent state does not occur. The neutron star is not then observable as a radio pulsar.
Multi-component Langmuir modes and bi-stability
-----------------------------------------------
The growth rate $\exp \Lambda$ for either longitudinal or for the quasi-longitudinal Langmuir modes described by Asseo, Pelletier & Sol (1990) are large for an ion-proton plasma of moderate Lorentz factor. Approximate rates were given by Jones (2012a, 2012b). The conditions for growth are ideal in that the velocity distributions for each component are essentially $\delta$-functions and the growth rate is not strongly dependent on the ratio of the two current densities. Growth rates are such that an exponent $\Lambda = 30$ can be reached at altitudes of $\sim 2R$.
The dielectric tensor component $D_{zz}$ in the high-field limit (the $z$-axis is locally anti-parallel with ${\bf B}$) for an ion-proton plasma with the addition of a continuum electron-positron component is, $$\begin{aligned}
D_{zz} = 1 - \frac{\omega^{2}_{p1}}{\gamma^{3}_{1}(\omega - k_{\parallel}v_{1})^{2}}
- \frac{\omega^{2}_{p2}}{\gamma^{3}_{2}(\omega - k_{\parallel}v_{2})^{2}} \nonumber \\
+ \frac{m\omega^{2}_{pe}}{k_{\parallel}}\int^{\infty}_{-\infty}dp \frac{\partial f}{\partial p}\frac{1}{\omega - k_{\parallel}v(p) + i\epsilon}.\end{aligned}$$ Here $k_{\parallel}$ is the wave-vector component anti-parallel with ${\bf B}$, $m$ is the electron mass and $f(p)$ its momentum distribution normalized to unity. The indices $1$ and $2$ here refer to ions and protons, respectively, and $\omega^{2}_{i} = 4\pi n_{i}q^{2}_{i}/m_{i}$ in which $n_{i}$ and $q_{i}$ are the mass and charge of particle $i$. In the high-field limit, the velocities are all anti-parallel with the local ${\bf B}$. The final term in equation (7) is identical with the expression given by Buschauer & Benford (1977). The Langmuir modes are defined by $D_{zz} = 0$ and its evaluation for an electron-positron distribution uniform between limits $p_{0}$ and $p_{m}$ shows that, dependent on $\gamma_{0}$ and $\gamma_{m}$, $D_{zz}$ is dominated by the electron-positron component even at number densities one or two orders of magnitude smaller than Goldreich-Julian. For $\gamma_{0,m} > \gamma_{1,2}$, ion-proton Langmuir modes exist (see Jones 2014c) but for $\gamma_{0} < \gamma_{1,2}$, the mode has no growth rate except at number densities two orders of magnitude smaller than Goldreich-Julian.
It is very difficult to be precise about the sources of small electron-positron number density components at altitudes below $r_{e}$ or about their Lorentz factors in the emission region. Apart from the usual magnetospheric processes, there is a small flux of backward-moving MeV photons from the photo-electron showers. Its sources are successive Compton scattering within the shower and the nuclear capture of neutrons produced by giant dipole-state decay. The magnetic conversion of these photons is very much a function of the polar-cap magnetic field. One reason why their flux can change is that the atomic numbers of ions in the LTE atmosphere in the region of the showers is subject to the medium time-scale instability described by Jones (2011). This is a consequence of the gradual change in atomic number of nuclei within the shower region as they move towards the surface, caused by continual giant dipole-state formation. Thus the surface $Z$ is much below the initial $Z_{0}$ of the undisturbed surface elsewhere on the star. The order of magnitude of the associated time is that for the emission at Goldreich-Julian current density of ions equivalent to one radiation length, $$\begin{aligned}
\tau_{rl} \approx 2.1\times 10^{5}PZ^{-1}B^{-1}_{12}\left(\ln\left(12Z^{1/2}B^{-1/2}_{12}\right)\right)^{-1} \hspace{3mm} s,\end{aligned}$$ which is also of similar order of magnitude to mode-change and null time-scales. Hence the capacity of even a small electron-positron number density to reduce ion-proton Langmuir-mode growth rates to zero is of direct interest in relation to the magnetospheric bi-stability which is apparent in the mode-changes and nulls that are observed in some pulsars. This was considered by Jones (2015a) and here we refer specifically to Fig. 1 of that paper. The Langmuir-mode growth rate exponent is $\Lambda \propto B^{1/2}_{12}P^{-1/2}$ and Fig. 1 gives this quantity as a function of $B$, extending over seven orders of magnitude, for various classes of neutron star or pulsar. It is immediately obvious that the existence of mode-changes and nulls is a function of $B$. There appears to be no evidence of them in the MSP. They begin to be present at $B\sim 10^{11}$ G and appear more frequently with increasing $B$ until the RRAT are reached at $B\sim 10^{13}$ G. To maintain the clarity of the diagram, it does not show the very large number of normal pulsars in which mode-changes or nulls are not observed. But their average value of $B$ obtained from the ATNF catalogue (Manchester et al 2005) is lower than for pulsars with nulls. Radio emission described as primary is not usually observed at $B\sim 10^{14}$ G. The conclusion of Jones (2015a) was that this sequence simply reflects the fact that the rate of background pair creation by single-photon magnetic conversion from the photon sources discussed earlier in this Section is a very strongly varying function of $B$. Hence it is negligible in the MSP but becomes progressively more likely as $B$ increases. In the intermediate-$B$ region, changes in the background electron-positron density arising from changes in the surface atomic number $Z$ are sufficient to lead to changes in Langmuir-mode growth rate over the whole or some part of the polar cap, with time-scales of the order of $\tau_{rl}$. Therefore, mode-changes and nulls are expected to be a feature of intermediate values of $B$, as is observed. High electron-positron background densities at large $B$ reduce growth rates to zero so that no primary radio emission is observed.
Further problems
----------------
The set of profiles published by Karastergiou & Johnston are of such complexity that there is some difficulty in accepting that they are not stochastic but are integrated and long-term stable. This complexity must be anchored by some fixed feature of the polar cap. If the surface were solid, inhomogeneity in $Z$ with a transverse length-scale of the order of $10^{3}$ cm would be one possibility, but there appears to be no reason to expect this. A second and perhaps more plausible possibility is that the shape of the open magnetic flux lines is complex and possibly age-dependent at the surface, but approaches dipole form at $r > r_{e}$. This kind of structure, specific to an individual neutron star, would imply that realistically, there may be limits to what can be understood in detail. Whilst a physical mechanism for magnetospheric bi-stability has been identified in this work, its detailed application to mode-changes and nulls in individual pulsars may prove impossible.
The same comment can be made with regard to subpulse drift. This was considered by Jones (2014a) where it was stated that there is little evidence for a systematic rotation pattern of plasma about the magnetic axis caused by ${\bf E}\times{\bf B}$ drift. Instead, a repetitive travelling modulation of the ion and proton current densities was proposed along arcs on the polar cap. This would permit the changes of drift direction that are observed.
The actual nature of the black-box interior is also a problem. Growth of either longitudinal or quasi-longitudinal Langmuir-mode amplitudes leads to non-linearity and, it is reasonable to expect, the development of turbulence, also the consequent movement of energy to higher wavenumbers. But in Jones (2016), no more than order-of-magnitude estimates of time-scales were given. A much more complete analysis of turbulence in the high-field limit and its coupling with the radiation field is needed.
The complete set of papers summarized here assumes corotation in the closed sector of the inner magnetosphere as in the paper of Goldreich & Julian (1969) in which the induction field of a non-aligned neutron star was neglected. Melrose & Yuen (2012, 2014) have shown recently that its inclusion is likely to lead to a departure from corotation. This has no essential effect on the conclusions of these papers (see Jones 2015b) but indicates that in detail, the structure of polar-cap current densities may be more complicated than we have assumed. There may be detail that is within the capability of new observing instruments such as the Square Kilometre Array (SKA).
Implications
============
Although there are uncertainties of detail, we believe that there is good observational evidence for the existence of an ion-proton plasma, with the possibility of at most a very small background of electrons and positrons, in radio-loud pulsars. This defines the plasma condition in the inner magnetosphere with some certainty for the ${\bf \Omega}\cdot{\bf B} < 0$ case and has implications for the further study of neutron stars which we wish to outline here.
The application of computational techniques in numerical plasma physics to the pulsar magnetosphere (see, for example, Bai & Spitkovsky 2010) assumes, following Mestel et al (1985) and Beloborodov (2008), that the open magnetosphere current density is determined by the state of the entire system and, crucially, that this is well-approximated by the outer magnetosphere. Thus complete solutions are obtained external to a sphere of radius $\sim 0.2R_{LC}$, usually under the assumption of force-free electrodynamics. Particles of either sign can pass freely through this surface and their charge and current densities ($\rho$ and ${\bf j}$) in certain regions of the solution satisfy $|{\bf j}| > |\rho c|$, which is possible only with particle counterflow requiring pair creation. The same assumption is the basis for the very recent computational paper of Philippov et al (2015) on the Lense-Thirring effect. These authors assume that the open-magnetosphere current density is set in the vicinity of the light cylinder so that near the neutron-star surface, its magnitude exceeds the corotational Goldreich-Julian charge density as modified by the Lense-Thirring effect. (This is the opposite effect as compared with, for example, Harding & Muslimov 2002, in whose work it is lower) In the work of Philippov et al, there is also electron acceleration to ultra-relativistic energies with the concomitant formation of a dense secondary electron-positron plasma.
As stated previously (Jones 2013a), we do not accept this view and regard the high-field and high energy-density region of the polar cap as setting what amount to boundary conditions for the true outer-region solution. Interpretation of the circular polarization observations with only minimal assumptions does not support the presence of such a plasma in radio-loud pulsars and there are other reasons set out in Section 2 why such a source of emission would not match observations. These arguments, we believe, show that the basis for their computations is incorrect, at least for ${\bf \Omega}\cdot{\bf B} < 0$ neutron stars.
A possible alternative approach to computational solutions for the outer magnetosphere would be, in outline, to accept the properties found for the inner magnetosphere as forming boundary conditions for the outer solution. In the ${\bf \Omega}\cdot{\bf B} < 0$ case, the conditions would be those of the ion-proton plasma. The case of ${\bf\Omega}\cdot{\bf B} > 0$ neutron stars is different because protons produced in reverse-positron showers have no effect. But there appears to be no reason why the current density should not be determined at the polar cap, as in the ${\bf \Omega}\cdot{\bf B} < 0$ case.
Involvement of the neutron-star surface as a condensed-matter system with additional degrees of freedom as well as an electromagnetic boundary condition means that the ${\bf \Omega}\cdot{\bf B} < 0$ case is very rich in possible modes of behaviour as compared with ${\bf \Omega}\cdot{\bf B} > 0$. In this respect alone it appears a very much better match for the observed range of pulsar phenomena. The drawing of a clear distinction between ${\bf \Omega}\cdot{\bf B} > 0$ and ${\bf \Omega}\cdot{\bf B} < 0$ neutron stars and the identification of the latter with normal radio-loud pulsars and MSP raises the question of how the former are observed in the electromagnetic spectrum. Insofar as incoherent emissions are concerned there should be little difference provided, as is likely, the ultra-relativistic electrons or positrons necessary for these processes are present near the light cylinder. Secondary pairs can be produced by curvature radiation in small-$P$ neutron stars or more generally in limited densities by inverse Compton scattering at the polar cap (see Hibschman & Arons 2001: Harding & Muslimov 2002). But it may be that the conditions necessary for pair creation in the outer magnetosphere can be determined with certainty only by inspection of numerical solutions. It is also unclear whether or not the electron-positron number densities present in a secondary plasma are needed or play any significant part in incoherent emission such as high-energy $\gamma$-rays. There is some magnetar radio emission (for example, the emission of J1810-197 has a small negative spectral index to 144 GHz: Camilo et al 2007, see also Serylak et al 2009) and such emission could be evidence for the ${\Omega}\cdot{\bf B} > 0$ class.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank the anonymous referee whose reading and comments have much improved the presentation of this work.
[99]{} Asseo E., Pelletier G., Sol H., 1990, MNRAS, 247, 529 Bai X. -N., Spitkovsky A., 2010, ApJ, 715, 1282 Beloborodov A. M., 2008, ApJ, 683, L41 Beskin V. S., 1990, Sov. Astron. Lett., 16, 286 Blandford R. D., Scharlemann E. T., 1976, MNRAS, 174, 59 Buschauer R., Benford G., 1977, MNRAS, 179, 99 Camilo F., et al, 2007, ApJ, 669,561 Cheng A. F., Ruderman M. A., 1979, ApJ, 229, 348 Dai S., et al, 2015, MNRAS, 449, 3223 Espinoza C. M., et al, 2013, MNRAS, 430, 571 Fussell D., Luo Q., Melrose D. B., 2003, MNRAS, 343, 1248 Goldreich P., Julian W. H., 1969, ApJ, 157, 869 Han J. L., Manchester R. N., Xu R. X., Qiao G. J., 1998, MNRAS, 300, 373 Harding A. K., Muslimov A. G., 2002, ApJ, 568, 862 Hassall T. E., et al, 2012, A&A, 543, A66 Hibschman J. A., Arons J., 2001 , ApJ, 554, 624 von Hoensbroech A., Xilouris K. M., 1997, A&A, 324, 981 Jenet F. A., Anderson S. B., Kaspi V. M., Prince T. A., Unwin S. C., 1998, ApJ, 498, 365 Jones P. B., 2010a, MNRAS, 401, 513 — , 2010b, MNRAS, 409, 1719 — , 2011, MNRAS,414, 759 — , 2012a, MNRAS, 419, 1682 — , 2012b, MNRAS, 423, 3502 — , 2013a, MNRAS, 431, 2756 — , 2013b, MNRAS, 435, L11 — , 2014a, MNRAS, 437, 4027 — , 2014b, MNRAS, 445, 770 — , 2014c, MNRAS, 445, 2297 — , 2015a, MNRAS, 450, 1420 — , 2015b, MNRAS, 453, 1941 — , 2016, MNRAS, 455, 3814 Karastergiou A., Johnston S., 2006, MNRAS, 365, 353 Karuppusamy R., Stappers B. W., Serylak M., 2011, A&A, 525, A55 Klein S., 1999, Rev. Mod. Phys., 71, 1501 Kramer M., Xilouris K. M., Jessner A., Lorimer D. R., Wielebinski R., Lyne A. G., 1997, A&A, 322, 846 Kramer M., Lange C., Lorimer D. R., Backer D. C., Xilouris K. M., Jessner A., Wielebinski R., 1999, ApJ, 526, 957 Lyubarskii Y. E., Petrova S. A., 1998, A&A, 337, 433 Malofeev V. M., Gil J. A., Jessner A., Malov I.F., Seiradakis J. H., Sieber W., Wielebinski R., 1994, A&A, 285, 201 Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2005, Astron. J., 129, 1993 Melrose D. B., Yuen R., 2012, ApJ, 745, 169 Melrose D. B., Yuen R., 2014, MNRAS, 437, 262 Mestel L., Robertson J. A., Wang Y. -M., Westfold K. C., 1985, MNRAS, 217, 443 Mikhailovskii A. B., Onishschenko O. G., Suramlishvili G. I., Sharapov S. E., 1982, Sov. Astron. Lett., 8, 369 Muslimov A. G., Tsygan A. I., 1992, MNRAS, 255, 61 Noutsos A., et al, 2015, A&A, 576, A62 Philippov A. A., Cerutti B., Tchekhovskoy A., Spitkovsky A., 2015, ApJ, 815, L19 Weltevrede P., Edwards R. T., Stappers B. W., 2006, A&A, 445, 243 Serylak M., et al, 2009, MNRAS, 394, 295 Szary A., Zhang B., Melikidze G. I., Gil J., Xu Ren-Xin, 2014, ApJ, 784:59 Thomas R. M. C., Gangadhara R. T., 2010, A&A, 515, A86 Yan W. M., et al, 2011, MNRAS, 414, 2097 Zrake J., East W. E., 2016, ApJ, 817, 89
\[lastpage\]
[^1]: E-mail: p.jones1@physics.ox.ac.uk
|
---
abstract: 'The evolution of halos consisting of weakly self-interacting dark matter particles is investigated using a new numerical Monte-Carlo N-body method. The halos initially contain kinematically cold, dense $r^{-1}$-power-law cores. For interaction cross sections $\sigma^* = \sigma/m_p \geq$ 10 - 100 cm$^2$ g$^{-1}$ weak self-interaction leads to the formation of isothermal, constant density cores within a Hubble time as a result of heat transfer into the cold inner regions. This core structure is in good agreement with the observations of dark matter rotation curves in dwarf galaxies. The isothermal core radii and core densities are a function of the halo scale radii and scale masses which depend on the cosmological model. Adopting the currently popular $\Lambda$CDM model, the predicted core radii and core densities are in good agreement with the observations. For large interaction cross sections, massive dark halos with scale radii r$_s \geq 1.4 \times 10^4$ (cm$^2$ g$^{-1}$ /$\sigma^*$) kpc could experience core collapse during their lifetime, leading to cores with singular isothermal density profiles.'
author:
- Andreas Burkert
title: 'The structure and evolution of weakly self-interacting cold dark matter halos'
---
Introduction
============
Cosmological models with a dominant cold dark matter component predict dark matter halos with strongly bound, kinematically cold cores (Dubinski & Carlberg 1991, Warren [*et al.*]{} 1992, Navarro [*et al.*]{} 1997). Within the core region, the dark matter density increases as a power-law $\rho \sim r^{-\gamma}$ with $\gamma$ in the range of 1 to 2 and the velocity dispersion $\sigma$ decreases towards the center (Carlberg 1994, Fukushige & Makino 1997). Numerous numerical simulations (e.g. Moore [*et al.*]{} 1998, Huss [*et al.*]{} 1999, Jing & Suto 2000), as well as analytical theory (Syer & White 1998, Kull 1999), have shown that such a core structure follows naturally from collisionless hierarchical merging of cold dark matter halos, independent of the adopted cosmological parameters.
It has recently become clear that on galactic scales the predictions of cold dark matter models are not in agreement with several observations. High-resolution calculations by Klypin [*et al.*]{} (1999) and Moore [*et al.*]{} (1999) have shown that the predicted number and mass distribution of galaxies in galactic clusters is consistent with the observations. However, on scales of the Local Group, roughly one thousand dark matter halos should exist as separate, self-gravitating objects, whereas less than one hundred galaxies are observed. This disagreement can be attributed to the high core densities of satellite dark halos in cosmological models which stabilize them against tidal disruption on galactic scales. Mo [*et al.*]{} (1998) and lateron Navarro and Steinmetz (2000) found that cold-dark matter models reproduce well the I-band Tully-Fisher slope and scatter. They however fail to match the zero-point of the Tully-Fisher relation as well as the relation between disk rotation speed and angular momentum. Again, this problem can be traced to the excessive central concentrations of cold dark halos. Finally, recent observations of dark matter dominated rotation curves of dwarf galaxies have indicated shallow dark matter cores which can be described by isothermal spheres with finite central densities (Moore 1994, Flores & Primack 1994, Burkert 1995, de Blok & Mc Gaugh 1997, Burkert & Silk 1999, Dalcanton & Bernstein 2000, see however van den Bosch [*et al.*]{} 1999), in contrast to the power-law cusps, expected from cosmological models. The disagreement between observations and theory indicates that a substantial revision to the cold dark matter scenario might be required which could provide valuable insight into the origin and nature of dark matter.
Motivated by these problems, Spergel & Steinhardt (1999) proposed a model where dark matter particles experience weak self-interaction on scales of kpc to Mpc for typical galactic densities. They noted that self-interaction could lead to satellite evaporation due to the dark particles within the satellites being kicked out by high-velocity encounters with dark particles from the surrounding dark halo of the parent galaxy. In order for weak interaction to be important on galactic scales, they estimate that the ratio of the collision cross section and the particle mass should be of order $\sigma_{wsi}/m_p \approx$ 1 cm$^2$ g$^{-1}$.
The Spergel and Steinhardt model has already motivated several follow-up studies. For example, Ostriker (1999) demonstrated that weak self-interaction would have the interesting side product of naturally growing black holes with masses in the range $10^6 - 10^9$ M$_{\odot}$ in galactic centers. Hogan & Dalcanton (2000) investigated analytically the effect of particle self-interactions on the structure and stability of galaxy halos. Moore et al. (2000), adopting a gas-dynamical approach, showed that in the limit of infinitely large interaction cross sections dark halos would develop singular isothermal density profiles which are not in agreement with observations. Mo & Mao (2000) and Firmani et al. (2000) investigated the affect of self-interaction on rotation curves. In addition, models of repulsive dark matter (Goodman 2000), fluid dark matter (Peebles 2000) and self-interacting warm dark matter (Hannestad & Scherrer 2000) have recently been discussed.
In this paper we will investigate the effect of weak self-interaction on the internal structure of cold dark matter halos. If the interaction cross section is not exceptionally large, the dark matter system cannot be treated as a collision dominated, hydrodynamical fluid. Section 2 therefore introduces a new numerical Monte-Carlo-N-body (MCN) method for weakly interacting particle systems. Initial conditions are discussed in section 3. Using the MCN-method, the evolution of weakly self-interacting dark matter halos is investigated in section 4. Conclusions follow in section 5.
The Monte-Carlo N-body method
=============================
Within the framework of weak self-interacting, the mean free path $\lambda$ of a dark matter particle is determined by $\lambda = (\rho \sigma^*)^{-1}$, where $\rho$ is the local dark matter mass density and $\sigma^* = \sigma_{wsi}/m_p$ is the ratio between the self-interaction collision cross section $\sigma_{wsi}$ and the particle mass $m_p$. If, within a timestep $\Delta t$, the path length $l = v \Delta t$ of a particle with velocity v is short compared to $\lambda$, the probability $P$ for it to interact with another particle can be approximated by
$$P = l/\lambda = \sigma^* \rho v \Delta t.$$
We use a Monte-Carlo approach in order to include weak self-interaction in a collisionless N-body code that utilizes the special purpose hardware GRAPE (GRAvity PipE; Sugimoto et al. 1990) in order to determine the gravitational forces between the dark matter particles by direct summation. For each particle, a list of its 50 nearest neighbors is returned by the boards which allows the determination of the local dark matter mass density $\rho$. The particle experiences an interaction with its nearest neighbor with a probability given by equation (1). Each weak interaction changes the velocities of the two interacting particles. Here, due to the lack of a more sophisticated theory, we assume that the interaction cross section is isotropic and that the interaction is completely elastic. In this case, the directions of the velocity vectors after the interaction are randomly chosen and their absolute values are completely determined by the requirement of energy and momentum conservation.
The computational timestep $\Delta t$ must be chosen small enough in order to guarantee that the evolution is independent of the numerical parameters, that is the adopted timestep and the number of particles. Otherwise, particles with large velocities could penetrate too deeply into a dense region like the core of a dark matter halo, violating the requirement $l << \lambda$. Test calculations have shown that $\Delta t \leq \eta (\sigma^* \rho v)^{-1}$ with $\eta \approx 0.1$ leads to reliable results that are independent of the numerical parameters.
Initial conditions
==================
Cold dark matter halos form on dynamical timescales. If $\sigma^*$ is small enough, the halos will achieve an equilibrium state within a few dynamical timescales that is determined by collisionless dynamics alone, before self-interaction becomes important. The structure of the halos subsequently changes due to self-interaction on longer timescales. This secular evolution is similar to the long-term evolution of globular clusters which experience core collapse due to gravitational 2-body encounters after virialization.
We start with an equilibrium model of a virialized dark matter halo and study its secular dynamical evolution due to weak self-interaction using the MCN-method. As initial condition, a Hernquist halo model (Hernquist 1990) is adopted. Its density distribution is $\rho(r) = \rho_{s}/\left(r/r_s (1+r/r_s)^3 \right)$ where $\rho_{s}$ and $r_s$ are the scale density and scale radius, respectively. The mass profile is $M(r)=M r^2/(r_s+r)^2$ with $M$ the finite total halo mass. Assuming hydrostatic equilibrium and an isotropic velocity distribution, the velocity dispersion is zero at the center and increases outwards, reaching a maximum at the inversion radius $r_i = 0.33 r_s$ outside of which it decreases again. A similar structure is seen in cosmological simulations (Carlberg 1994, Fukushige & Makino 1997). In general, within the interesting region $r \leq r_s$ the Hernquist model provides an excellent fit to the structure of cold dark matter halos that result from high-resolution cosmological models. Only in the outermost regions do the dark matter halo profiles deviate significantly from the Hernquist model, predicting a density distribution that decreases as $r^{-3}$ and a dark halo mass that diverges logarithmically (Navarro [*et al.*]{} 1997). Note, that our model neglects any clumpy substructure that might exist within dark matter halos (Moore [*et al.*]{} 1999). This should be a reasonable approximation for the inner regions where satellites are efficiently disrupted by tidal forces. The evolution of weakly self-interacting, clumpy dark halos will be presented in a subsequent paper (see also Moore et al. 2000).
In the following, we will adopt dimensionless units: G=1, $r_s=1$ and $M=1$. The total mass and the mean mass density within the inversion radius $r_i$ is $M_i=0.06$ and $\rho_i = 0.4$, respectively, leading to a dynamical timescale within $r_i$ of $\tau_{dyn} = 0.8$. Most numerical calculations have been performed adopting 80000 particles and a gravitational softening length of $\epsilon$ = 0.002$\times$r$_s$. Test calculations with 120000 particles did not change the results. N-body calculations without weak interaction have shown that the dark halo is stable and its density distribution does not change outside of r $\geq$ 0.006$\times$r$_s$ within 20 dynamical timescales.
The evolution of weakly self-interacting dark halos
===================================================
Figure 1 shows the evolution of the dark matter density distribution and the velocity dispersion profile inside the core region, adopting a collision cross section $\sigma^*$=10$\times$r$_s^2$/M$_s$. The density distribution initially has the characteristic power-law cusp and the velocity dispersion decreases towards the center for $r < r_i$. Within this region, the kinetic temperature inversion leads to heat conduction inwards. The central velocity dispersion increases with time and the core expands, resulting in a shallower density distribution. After 3 dynamical timescales, an isothermal, constant density core has formed with a radius that is of order the initial inversion radius $r_{i}$. Subsequently, weak interactions between the kinematically hotter core and the cooler envelope lead to a flow of kinetic energy outwards which causes the isothermal core to contract and heat up further due to its negative specific heat, starting a core collapse phase. The calculations are stopped after 16 dynamical timescales when the central density and the central velocity dispersion has increased further by a factor of 4 and 1.4, respectively. Note that during the core collapse phase, the system maintains an isothermal, constant density core with the core radius decreasing with time. Overall, the evolution of the dark halo is very similar to the secular evolution of particle systems with Hernquist profiles that are affected by gravitational 2-body interactions (Heggie [*et al.*]{} 1994, Quinlan 1999).
Several calculations with different interaction cross sections $\sigma^*$ have been performed. In all cases, the evolution is similar to that shown in Fig. 1, independent of the adopted collision cross section. The timescale $\tau_{iso}$ for the formation of the isothermal constant density core does however depend on $\sigma^*$ with
$$\tau_{iso} \approx \frac{30 \tau_{dyn}}{\sigma^*} \frac{r_s^2}{M_s} .$$
In agreement with the calculations of Quinlan (1999) the core collapse timescales are roughly an order of magnitude larger than $\tau_{iso}$.
Observations of dark matter dominated dwarf galaxies show a characteristic dark matter core structure that can be fitted well by the empirical density distribution (Burkert 1995) $\rho = \rho_0*(r+r_0)^{-1}(r^2+r_0^2)^{-1}$ where $\rho_0$ and $r_0$ are the isothermal core density and radius, respectively. Figure 2 compares this profile (solid line) with the core structure of weakly interacting dark halos at $t=0$ (dashed line) and after core expansion at $t = \tau_{iso}$ (points with error bars). It is well known that power-law cores do not provide a good fit to the observations. An excellent agreement can however be achieved after core expansion if one adopts the following core parameters
$$\begin{aligned}
r_0 \approx 0.6 r_i \\ \nonumber
\rho_0 \approx 1.54 M_i r_i^{-3} \end{aligned}$$
where $M_i$ is the initial dark matter mass inside the inversion radius r$_i$.
Conclusions
===========
The previous MCN calculations have shown that isothermal cores with shallow density profiles form naturally in weakly interacting dark halos. The cores have density distributions that are in excellent agreement with the observations of dark matter rotation curves in dwarf galaxies. The core size is determined by the radius r$_i$ inside which heat is conducted inwards, that is where the initial velocity dispersion decreases towards the center. Note, that this conclusion should be valid, independent of whether the density diverges as $\rho \sim r^{-1}$ or even steeper (Moore [*et al.*]{} 1998, Jing & Suto 2000) for r $\ll r_s$.
A quantitatively comparison with the observations requires the determination of the typical scale parameters r$_s$ and M$_s$ for dark matter halos. Recent cosmological $\Lambda$CDM models (Navarro & Steinmetz 2000) predict that cold dark matter halos with total masses M$_{200} \approx 10^{10} - 10^{12}$ M$_{\odot}$ should have concentrations c = r$_{200}$/r$_{s} \approx 20$, where M$_{200}$ is the total dark matter mass within the virial radius r$_{200}$ which denotes the radius inside which the averaged overdensity of dark matter is 200 times the critical density of the universe. Adopting a Hubble constant h=0.7 leads to r$_{200}$ = 0.02 (M$_{200}$/M$_{\odot})^{1/3}$ kpc $\approx$ 40 – 200 kpc and with c=20 to scale radii r$_s \approx$ 2 – 10 kpc. For a NFW-profile (Navarro [*et al.*]{} 1997) the dark matter mass inside r$_s$ is M$_s \approx$ 0.1 M$_{200}$ and the core density is M$_s$/r$_s^3 \approx$ 0.01 M$_{\odot}$ pc$^{-3}$. In contrast to the Hernquist model with $r_i = 0.33 r_s$, the inversion radius of the NFW-profiles coincides with the scale radius $r_i = r_s$ due to the shallower outer density distribution. According to equation 3, weak interaction in NFW-halos should therefore lead to isothermal cores with radii r$_0 \approx$ 0.6 r$_s \approx$ 1.2 – 6 kpc and densities $\rho_0 \approx$ 1.55 M$_s$ r$_s^{-3} \approx 1.5 \times 10^{-2}$ M$_{\odot}$ pc$^{-3}$. The observations indicate core radii r$_0 \approx$ 2 – 10 kpc with core densities $\rho_0 \approx$ 0.01 M$_{\odot}$ pc$^{-3}$ (Burkert 1995), in very good agreement with the theoretical predictions.
In order for dark matter cores to be affected by weak self-interaction, the core expansion timescale must be smaller than the age $\tau$ of the halo: $\tau_{iso} \leq \tau \approx 100 \tau_{dyn}$. With equation (2) and adopting M$_s$/r$_s^3 \approx 0.01 $M$_{\odot}$ pc$^{-3}$, this requirement leads to a minimum value of the collision cross section for weak self-interaction to be important
$$\sigma^* \geq 100 \left( \frac{kpc}{r_s} \right) \left(\frac{cm^2}{g} \right)$$
Note, that this lower limit would be a factor of 25 larger if cosmological models underestimate the scale radii of dark halos by a factor of 5.
Dark matter halos with $\tau_{iso} \approx \tau_{dyn}$ are likely to have gone through core collapse if their ages are $\tau >> \tau_{dyn}$. This condition requires the halo core radii to be larger than r$_s > 1.4 \times 10^4$ (cm$^2$ g$^{-1}$)/$\sigma^*$ kpc, indicating that more massive halos could have experienced core collapse while lower mass halos could still be in the process of core expansion.
I would like to thank Matthew Bate for providing a subroutine to find nearest neighbors using GRAPE and Paul Steinhardt, Ben Moore and Jerry Ostriker for interesting discussions and the referee for important comments. Special thanks to David Spergel for pointing out the different dependence of the inversion radius on the scale radius for NFW- and Hernquist profiles.
Burkert, A. 1995, , 447, L25. Burkert, A. & Silk 1997, , 488, L55. De Blok, W.J.G. & McGaugh, S.S. 1997, MNRAS, 290, 533. Carlberg, R. 1994, , 433, 468. Dalcanton, J.J. & Bernstein, R.A. 2000, ASP Conference Series, 197, 161. Dubinski, J. & Carlberg, R. 1991, , 378, 496. Firmani, C., D’Onghia, E., Avila-Reese, V., Chincarini, G. & Hernández, X. 2000, astro-ph/0002376. Flores, R.A. & Primack, J.R. 1994, , 427, L1. Fukushige, T. & Makino, J. 1997, , 477, L9. Goodman, J. 2000, astro-ph/0003018. Heggie, D.C., Inagaki, S. & McMillan, S.L.W. 1994, MNRAS, 271, 706. Hernquist, L. 1990, , 356, 359. Hogan, C.J. & Dalcanton, J.J. 2000, astro-ph/0002330. Huss, A., Jain, B. & Steinmetz, M. 1999, , 517, 64. Jing, Y.P. & Suto, Y. 2000, , 529, L69. Klypin, A., Kravtsov, A.V., Valenzuela, O. & Prada, F. 1999, , 522, 82. Kull, A. 1999, , 516, L5. Mo, H.J., Mao, S. & White, S.D.M. 1998, MNRAS, 295, 319. Mo, H.J. & Mao, S. 2000, astro-ph/0002451. Moore, B. 1994, Nature, 370, 629. Moore, B., Governato, F., Quinn, T., Stadel, J. & Lake, G. 1998, , 499, L5. Moore, B., Ghigna, S., Governato, F., Lake, G., Quinn, T. & Stadel, J. 1999, , 524, L19. Moore, B., Gelato, S., Jenkins, A., Pearce, F.R. & Quilis, V. 2000, astro-ph/0002308. Navarro, J.F., Frenk, C.S. & White, S.D.M. 1997, , 490, 493. Navarro, J.F. & Steinmetz, M. 2000, astro-ph/0001003. Ostriker, J.P. 1999, astro-ph/9912548. Peebles, P.J.E. 2000, astro-ph/0002495. Quinlan, G. 1999, astro-ph/9606182. Spergel, D.N. & Steinhardt, P.J. 1999, astro-ph/9909386. Sugimoto, D., Chikada, Y., Makino, J., Ito, T., Ebisuzaki, T. & Umemura, M. 1990, Nature, 345, 33. Syer, D. & White, S.D.M. 1998, , 293, 337. van den Bosch, F.C., Robertson, B.E., Dalcanton, J.J. & de Blok, W.J.G. 1999, astro-ph/9911372 Warren, S.W., Quinn,P.J., Salmon, J.K. & Zurek, H.W. 1992, , 399, 405.
|
---
abstract: 'This note is a report on the observation that the Enriques–Fano threefolds with terminal cyclic quotient singularities admit Calabi–Yau threefolds as their double coverings. We calculate the invariants of those Calabi–Yau threefolds when the Picard number is one. It turns out that all of them are new examples.'
address:
- ' Department of Mathematics Education, Hongik University 42-1, Sangsu-Dong, Mapo-Gu, Seoul 121-791, Korea '
- 'School of Mathematics, Korea Institute for Advanced Study, Dongdaemun-gu, Seoul 130-722, South Korea '
author:
- 'Nam-Hoon Lee'
title: 'Calabi–Yau double coverings of Fano–Enriques threefolds'
---
Introduction
============
The threefolds whose hyperplane sections are Enriques surfaces were studied by G. Fano in a famous paper [@Fa]. The modern proofs for the results of [@Fa] were given in [@CoMu]. Such varieties are always singular and their canonical divisors are not Cartier but numerically equivalent to Cartier divisors. We call such threefolds *Fano–Enriques threefolds* (see Definition \[def1\]). In this note, we consider Fano–Enriques threefolds whose singularities are terminal cyclic quotient ones. It is worth noting that any Fano–Enriques threefold with terminal singularities admits a -smoothing to one with terminal cyclic quotient singularities ([@Mi]). The canonical coverings (which are double-coverings) of Enriques–Fano threefolds with terminal cyclic quotient singularities are smooth Fano threefolds ([@Ba; @Sa]). Hence all the singular points of such Enriques–Fano threefolds are of type $\frac{1}{2} (1, 1, 1)$. Using the classification of smooth Fano threefolds, L. Bayle ([@Ba]) and T. Sano ([@Sa]) gave a classification of such threefolds. In this note, we observe that all those Enriques–Fano threefolds also admit some Calabi–Yau threefolds as their double covering, branched along some smooth surfaces and those eight singularities. A is a compact Kähler manifold with trivial canonical class such that the intermediate cohomology groups of its structure sheaf are trivial ($h^1 (Y, \mco_Y) = h^2(Y, \mco_Y) =0$). We calculate the invariants of those Calabi–Yau double coverings when their Picard numbers are one (Table \[t1\]). It turns out that all those Calabi–Yau threefolds are new examples. Although a number of Calabi–Yau threefolds have been constructed, those with Picard number one are still quite rare. Note that they are primitive and play an important role in the moduli spaces of all Calabi–Yau threefolds ([@Gr]).
Calabi–Yau double coverings
===========================
As the higher dimensional algebraic geometry being developed, the definition of Fano–Enriques threefolds also has evolved and has been generalized. We adapt the following version of the definition.
\[def1\] A three-dimensional normal projective variety $W$ is called a Fano–Enriques threefold if $W$ has canonical singularities, $-K_W$ is not a Cartier divisor but numerically equivalent to an ample Cartier divisor $H_W$.
Y. Prokhorov proved in [@Pr] that the generic surface in the linear system $|H_W|$ is an Enriques surface with canonical singularities and that the Enriques surface is smooth if the singularities of $W$ is isolated and $-K_W^2 \neq 2$. We refer to [@Ch1; @GLMR; @Pr1] for more systematic expositions of Fano–Enriques threefolds. In this note, we consider the case that $W$ has only terminal cyclic quotient singularities. We summarize the properties of $W$ ([@Ba; @CoMu; @Fa; @Sa]).
1. All the singularities of $W$ are the type of $\frac{1}{2}(1,1,1)$.
2. The number of singularities of $W$ is eight.
3. $-2K_W$ is linearly equivalent to $-2H_W$.
4. There is a smooth Fano threefold that covers doubly $W$, branched only at the singularities of $W$.
L. Bayle ([@Ba]) and T. Sano ([@Sa]) gave a classification of smooth Fano threefolds that double cover Fano–Enriques threefolds.
Let $\varphi:X \ra W$ be the double covering, branched along the singularities of $W$. Then $X$ is one of smooth Fano threefolds in Theorem 1.1 in [@Sa]. We want to find a Calabi–Yau threefold that double-covers $W$, using the following theorem which is a special case of Theorem 1.1 in [@Lee1].
\[leethm\] Let $W$ be a projective three-dimensional variety with singularities of type $\frac{1}{2}(1,1,1)$ such that $h^1(W, \mco_W) = h^2(W, \mco_W)=0$. Suppose that the linear system $|-2K_W|$ contains a smooth surface $S$, then there is Calabi–Yau threefold $Y$ that is a double covering of $W$ with the branch locus $S \cup \Sing(W)$.
Let $p_1, p_2$ be any points of $X$. From the description of those Fano threefolds $X$’s in Theorem 1.1 of [@Sa], one can find an effective divisor $D$ from $|-K_X|$ such that $D$ does not contain $p_1, p_2$. Let $\theta$ be the covering involution on $X$, i.e. the quotient $X/\langle \theta \rangle $ is $W$. Therefore, for any point $q \in W$ and we can find an effective divisor $D$ in the linear system $|-K_X|$ such that $D \cap \varphi^{-1}(\{q \}) = \emptyset$. Note the effective divisor $$\varphi(D)+\varphi(\theta(D))$$ belongs to the linear system $|-2K_W|$ and it does not contain the point $q$. So the linear system $|-2K_W|$ is base-point-free and we can find a smooth surface $S$ from it. Hence, by Theorem \[leethm\], there is a Calabi–Yau threefold $Y$ that covers doubly $W$, branched along $S$ and singularities of $W$.
Since $H^i(X, \mco_X)=0$ and $H^i(W, \mco_W)=0$ for $i=1,2$, we have isomorphisms $$H^2(X, \mbz) \simeq \Pic(X), H^2(W, \mbz) \simeq \Pic(W)$$ by the exponential sequences. Hence we can regard classes of Cartier divisors of $X, W$ as elements of $H^2(X, \mbz), H^2(W, \mbz)$ respectively. Let $r$ be the index of Fano threefold $X$ (i.e. the largest integer $r$ such that $-K_X = r H_X$ for some ample divisor $H_X$ of $X$).
Now we calculate the invariants of $Y$. For a double covering with dimension higher than two, it is a non-trivial task to calculate the topological invariants even in the case that the base of the covering is smooth. In our case, $S$ is an ample divisor of $W$, so it may be worth trying to apply the Lefschetz hyperplane theorem. However $W$ is not smooth, so the usual Lefschetz hyperplane theorem does not apply here. There are other versions of the Lefschetz hyperplane theorem for singular varieties but they all require that $W-S$ is smooth, which is not true for our case. We prove a type of the Lefschetz hyperplane theorem for $S \subset W$. We say that an element $\alpha$ of an additive Abelian group $G$ is divisible by an integer $k$ if $\alpha = k \alpha'$ for some element $\alpha' \in G$. $\alpha$ is said to be primitive if it is divisibly by only $\pm 1$. We denote the quotient of $G$ by its torsion part as $G_f$.
\[lem1\] The map $H^2(W, \mbq) \ra H^2(S, \mbq)$, induced by the inclusion $S \hookrightarrow W$, is injective and the image $H_W|_S$ in $ H^2(S, \mbz)_f$ of $H_W \in H^2(W, \mbz)$ is divisible by $r$.
Consider the commutative diagram:
$$\xymatrix{
X \ar[r]^{ \varphi} & W \\
S_X \ar[r]^{\varphi|_{S_X}} \ar@{^{(}->}[u] & S \ar@{^{(}->}[u]
}$$ where $S_X=\varphi^{-1}(S)$ and the vertical maps are inclusions. Note $$\varphi|_{S_X}:S_X \ra S$$ is an unramified double covering. We have an induced commutative diagram: $$\xymatrix{
H^2(X, \mbq) \ar[d] & H^2 (W, \mbq) \ar[d] \ar[l]^{ \varphi^*} \\
H^2(S_X, \mbq) & H^2(S, \mbq) \ar[l]^{{(\varphi|_{S_X})}^*}
}$$ Note the pull-backs $\varphi^*: H^2(W, \mbq) \ra H^2 (X, \mbq)$ is injective. Since $S_X$ is a smooth ample divisor of $X$, the map $H^2(X, \mbq) \ra H^2(S_X, \mbq)$ is injective by the Lefschetz hyperplane theorem. So we have the injectivity of the map $$H^2(W, \mbq) \ra H^2(S, \mbq).$$
Consider another commutative diagram. $$\xymatrix{
H^2(X, \mbz) \ar[d] & H^2 (W, \mbz) \ar[d] \ar[l]^{ \varphi^*} \\
H^2(S_X, \mbz)_f & H^2(S, \mbz)_f \ar[l]^{{(\varphi|_{S_X})}^*}
}$$
Note $-K_X = r H_X$. We note that $\theta^* (-K_X) = -K_X$ in $H^2(X, \mbz)$. Since $H^2(X, \mbz)$ has no torsion, $\theta^* (H_X) = H_X$ in $H^2(X, \mbz)$ and so $$(\theta|_{S_X})^*(H_X|_{S_X}) = \theta^*(H_X)|_{S_X} =H_X|_{S_X}$$ in $H^2(S_X, \mbz)_f$. Hence $h':=H_X|_{S_X}$ lies in the image of the map $$H^2(S, \mbz)_f \ra H^2(S_X, \mbz)_f.$$
Note $\varphi^*(H_W) = -K_X = r H_X$. Hence $$(\varphi|_{S_X})^*(H_W|_S)= \varphi^*(H_W)|_{S_X} =rH_X|_{S_X} = r h'$$ is divisible by $r$ in $H^2(S_X, \mbz)_f$. Since the map $$(\varphi|_{S_X})^*:H^2(S, \mbz)_f \ra H^2(S_X, \mbz)_f$$ is injective, $H_W|_S$ is divisible by $r$ in $H^2(S, \mbz)_f$.
We note that $H_W$ is primitive in $H^2(W, \mbz)$. By the above lemma, $H_W|_S$ is not primitive in $H^2(S, \mbz)_f$ when $r > 1$. This is different from what the usual Lefschetz hyperplane theorem expects for smooth threefolds.
\[prop1\] We have $$h^2(Y) \leq h^2(X),$$ $$e(Y) = e(X)- 24-2(- K_X)^3$$ and $$\psi^*(H_W) \cdot c_2(Y) = (-K_X)^3 + 24,$$ where $e(Y)$ is the topological Euler characteristic of $Y$ and $c_2(Y)$ is the second Chern class of $Y$.
Consider the following fiber product of two double covers: $$\xymatrix{
& \widetilde X \ar[dr] \ar[dl] & \\
Y \ar[dr]^{\psi}& & X \ar[dl]_\varphi\\
&W&
}$$ Then it is easy to see that
- $ \widetilde X \ra Y$ is an étale double cover and
- $\widetilde X \ra X$ is the double cover branched along a member $\widetilde S $ of $| -2K_X|$.
Using the fact that $\widetilde S$ is an ample divisor of $\widetilde X$, one can show $h^2(\widetilde X) = h^2(X)$ ([@Cy]). Since $h^2(Y) \leq h^2(\widetilde X)$, we have $h^2(Y) \leq h^2(X).$
Note $e(X) = 2 e(W) - 8$ and $e(S_X) = 2 e(S)$. Note $S_X \sim -2K_X$ and by the Riemann–Roch theorem, $$1=\chi(X, \mco_X) = \frac{1}{24} c_2(X) \cdot (-K_X).$$ By the adjunction formula, we have $$e(S_X) = c_2(X) \cdot (-2K_X) +4(- K_X)^3 = 48+4(- K_X)^3 .$$ So $$e(Y) = 2 e(W) - e(S) -8 = e(X) - \frac{1}{2} e(S_X) = e(X)- 24-2(- K_X)^3.$$
Note $\psi^*(S) \sim 2 S_Y$ and $S \sim 2 H_W$, where $S_Y= \psi^{-1}(S)$. So $ \psi^*(H_W) \cdot c_2(Y) =S_Y \cdot c_2(Y)$. By the adjunction formula, $$\begin{aligned}
S_Y \cdot c_2(Y) &= - S_Y^3 + c_2(S_Y) \\
&= -\psi^*(H_W)^3 + e(S_Y) \\
&= -2 H_W^3 + e(S)\\
&= -\varphi^*(H_W)^3 + \frac{1}{2} e(S_X) \\
&= -(-K_X)^3 + 24+2(- K_X)^3 = (-K_X)^3 + 24.\end{aligned}$$ Hence $\psi^*(H_W) \cdot c_2(Y) = (-K_X)^3 + 24$.
We are interested in the case that the Calabi–Yau threefold $Y$ has Picard number one. Hence we assume that $X$ has Picard number one. There are four families of them:
1. complete intersection of a quadric and a quartic in the weighed projective space $\mbp(1,1,1,1,1,2)$, $r_1=1$, $-K_{X_1}^3=4$, $e(X_1) = -56$.
2. complete intersection of three quadrics in $\mbp^6$, $r_2=1$, $-K_{X_2}^3=8$, $e(X_2) = -24$.
3. hypersurface of degree $4$ in $\mbp(1,1,1,1, 2)$, $r_3=2$, $-K_{X_3}^3=16$, $e(X_3) = -16$.
4. compete intersection of two quadrics in $\mbp^5$, $r_4=2$, $-K_{X_4}^3=32$, $e(X_4) = 0$.
Suppose that $X$ has Picard number one, then $W$, $Y$ have Picard number one, $$H_Y^3 = \frac{1}{r^3} (-K_X^3)$$ and $$H_Y \cdot c_2(Y) = \frac{1}{r}((-K_X)^3 + 24),$$ where $H_Y$ is an ample generator of $\Pic(Y)$.
By Proposition \[prop1\], $$1 \leq h^2(W) \le h^2(Y) \le h^2(X)=1,$$ so $W$, $Y$ have Picard number one. Since $Y$ has Picard number one, $\psi^*(H_W) = k H'_Y$ for some ample generator $H'_Y$ of $\Pic(Y)$ ( $\simeq H^2(Y, \mbz)$) and a positive integer $k$. Note that $H_Y-H_Y'$ is a torsion element and that $H_Y'$ is primitive in $H^2(Y, \mbz)$. We also note that $S_Y$ is a smooth ample divisor of $Y$. By the Lefschetz hyperplane theorem, $H'_Y|_{S_Y}$ is primitive in $H^2(S_Y, \mbz)_f$. By Lemma \[lem1\], $H_W|_S$ is divisible by $r$ in $H^2(S, \mbz)_f$. So its image $(\psi|_{S_Y})^*(H_W|_S)$ in $H^2(S_Y, \mbz)_f$ is divisible by $r$. Note $$(\psi|_{S_Y})^*(H_W|_S) = \psi^*(H_W)|_{S_Y} = k (H'_Y|_{S_Y}).$$ So $k$ is divisible by $r$. Let $k=lr$ for some positive integer $l$. We will show that $l=1$. Note $$H_Y^3 = {H_Y'}^3 = \frac{1}{k^3} \psi^*(H_W)^3 = \frac{2}{k^3} H_W ^3 = \frac{1}{k^3} \varphi^*(H_W)^3 = \frac{1}{r^3 l^3} (-K_X^3)$$ and $$H_Y \cdot c_2(Y) = \frac{1}{k}\psi^*(H_W) \cdot c_2(Y) =\frac{1}{r l}((-K_X^3) + 24).$$
For $X_1, X_3, X_4$, the condition of $H_Y^3$ being a positive integer requires that $l=1$. For $X_2$, by the Riemman–Roch theorem, we have $$\chi(Y, H_{Y}) = \frac{H_{Y}^3}{6} + \frac{H_Y \cdot c_2(Y)}{12} = \frac{8}{ 6l^3} + \frac{32}{12 l}
=\frac{4+8l^2}{3l^3},$$ which should be an integer. So we have $l=1$ also in this case. Therefore, $$H_Y^3 = \frac{1}{r^3} (-K_X^3)$$ and $$H_Y \cdot c_2(Y) = \frac{1}{r}((-K_X)^3 + 24).$$
By Proposition \[prop1\] and the relation $e(Y) = 2 (h^{1,1}(Y) - h^{1,2}(Y))$, we can determine all the Hodge numbers of $Y$. We list the invariants of the Calabi–Yau threefolds $Y$’s in Table \[t1\]. It turns out that they are all new examples . See Appendix I of [@Kap] for a list of known examples of Calabi–Yau threefolds of Picard number one.
$X_d$ $H_Y^3$ $H_Y \cdot c_2(Y)$ $h^{1,1}(Y)$ $h^{1,2}(Y)$
------- --------- -------------------- -------------- --------------
$X_1$ 4 28 1 45
$X_2$ 8 32 1 33
$X_3$ 2 20 1 37
$X_4$ 4 28 1 45
: Invariants of Calabi–Yau double coverings
\[t1\]
Note that the invariants of $Y_1$ and those of $Y_4$ overlap. Consider the commutative diagram in the proof of Proposition \[prop1\]. For $X_1$, the branch locus of $\widetilde X_1 \ra X_1$ is a quadric section, thus $\widetilde X_1$ is a $(2, 2, 4)$-weighted complete intersection of $\mbp(1,1,1,1,1,1, 2)$. For $X_4$, the branch locus of $\widetilde X_4 \ra X_4$ is a quartic section, thus $\widetilde X_4$ is also a $(2, 2, 4)$-weighted complete intersection of $\mbp(1,1,1,1,1,1,2)$. Therefore, $\widetilde X_1$ and $\widetilde X_4$ are in the same family. Since $Y_1$ and $Y_4$ are étale $\mbz_2$-quotients of $\widetilde X_1$ and $\widetilde X_4$ respectively, they have the same invariants.
[9999]{}
Bayle, Lionel *Classification des variétés complexes projectives de dimension trois dont une section hyperplane générale est une surface d’Enriques.* J. Reine Angew. Math. 449 (1994), 9–63.
Chelʹtsov, I. A. *Rationality of an Enriques-Fano threefold of genus five.* Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 3, 181–194; translation in Izv. Math. 68 (2004), no. 3, 607–618
Conte, A.; Murre, J. P. *Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces.* Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 1, 43–80.
Cynk, Slawomir *Cyclic coverings of Fano threefolds.* Proceedings of Conference on Complex Analysis (Bielsko-Biała, 2001). Ann. Polon. Math. 80 (2003), 117–124.
Fano G. *Sulle varieta algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno.* Mem. Mat. Sci. Fis. Natur. Soc. Ital. Sci. 1038. V. 3(24). P. 41–66.
Giraldo, Luis; Lopez, Angelo Felice; Muñoz, Roberto *On the existence of Enriques-Fano threefolds of index greater than one.* J. Algebraic Geom. 13 (2004), no. 1, 143–166.
Gross, Mark *Primitive Calabi–Yau threefolds.* J. Differential Geom. 45 (1997), no. 2, 288–318.
Kapustka, Grzegorz *Projections of del Pezzo surfaces and Calabi–Yau threefolds.* Adv. Geom. 15 (2015), no. 2, 143–158.
Lee, Nam-Hoon *Calabi–Yau coverings over some singular varieties and new Calabi–Yau 3-folds with Picard number one.* Manuscripta Math. 125 (2008), no. 4, 531–547.
Minagawa, Tatsuhiro *Deformations of -Calabi–Yau 3-folds and -Fano 3-folds of Fano index 1.* J. Math. Sci. Univ. Tokyo 6 (1999), no. 2, 397–414.
Prokhorov, Yu. G. *On three-dimensional varieties with hyperplane sections—Enriques surfaces.* Mat. Sb. 186 (1995), no. 9, 113–124; translation in Sb. Math. 186 (1995), no. 9, 1341–1352
Prokhorov, Yu. On Fano–Enriques varieties. Mat. Sb. 198 (2007), no. 4, 117–134; translation in Sb. Math. 198 (2007), no. 3–4, 559–-574
Sano, Takeshi *On classifications of non-Gorenstein Q-Fano 3-folds of Fano index 1.* J. Math. Soc. Japan 47 (1995), no. 2, 369–380.
|
---
abstract: 'The many-body correlation effects in the spatially separated electron and hole layers in the coupled quantum wells (CQW) are investigated. The specific case of the *many-component* electron-hole system is considered, $\nu \gg 1$ being the number of components. Keeping the main diagrams in the parameter $1/\nu $ allows us to justify the $RPA$ diagram selection. The ground state of the system is found to be the *electron-hole liquid* which possesses the energy smaller than the exciton phase. The connection is discussed between the results obtained and the experiments in which the inhomogeneous state in the CQW is revealed.'
author:
- 'V.S. Babichenko'
- 'I. Ya. Polishchuk'
title: ' Many-Body Correlation Effects in the Spatially Separated Electron and Hole Layers in the Coupled Quantum Wells'
---
Introduction
============
The investigation of spatially separated electrons and holes in the coupled quantum wells (CQW) is motivated by expecting that the electron-hole pairs forming a long-living bound state, namely exciton, can experience the Bose-Einstein condensation [@Lozovik]. The interest in such systems has greatly grown in the recent years due to the increasing ability to manufacture the high quality quantum well structures in which electrons and holes are confined in the different regions between which the tunneling can be made negligible [@Review2011]. For this reason, in the CQW the exciton lifetime becomes by several orders of the magnitude longer than the lifetime of excitons in bulk materials, enabling an experimental observation of the exciton Bose-Einstein condensation.
As early as decade and a half, Lozovik and Berman, using the variational approach, predicted the existence of an electron-hole condensate phase in CQW [@Lozovik-Berman1996]. The further theoretical investigations of such systems reveal that the phase diagram of the system can be rather rich [@Balatsky].
The main goal of the paper is to find out how the many-body correlations in the electron-hole system in the CQW influence its ground state. A specific case of the *many-component* spatially separated electron-hole plasma in the CQW is studied in the current paper. It is assumed that $\nu $ different kinds of electrons are confined in one layer of the CQW, while $%
\nu $ different kinds of holes are confined in the other layer, the parameter being $\nu \gg 1.$ For the first time, the large $\nu $ approximation was proposed for the investigation of a 3D electron-hole liquid in many-valley semiconductors [@babich]. Then, this approach got a further development [@keldysh].
In the model under consideration, the thickness of the layers is supposed to be so small that the carriers are the $2D$ ones. Let $a_{B}$ be the effective Bohr radius and $l$ be the inter-layer distance. Below, we mainly consider the case $l\gg a_{B}.$ Under such condition, the isolated electron-hole pair may form an exciton possessing the *in-plane* radius $R_{ex}\sim a_{B}\left( l/a_{B}\right) ^{3/4}\gg a_{B}$ [@Lozovik]. Let $n$ be the carrier concentration. The concept of excitons has sense until the average in-plane inter-exciton distance being of the order of $%
n^{-1/2}$ is larger than $R_{ex},$ i.e. $R_{ex}^{2}n\ll 1.$ At higher density the exciton system brakes down into the electron-hole one.
It is shown in this paper that the negative contribution of the energy induced by the intra-layer carrier correlation strongly reduces the total energy. This effect facilitates *the electron-hole liquid* formation. For this phase, the equilibrium carrier density $n_{eq},$which is the same for the both layers, is found to obey the condition $R_{ex}^{2}n_{eq}\sim 1.$ Thus, the criterion for an existence of the exciton state is violated. Therewith, the *electron-hole liquid* is shown to possess the energy *lower* as compared to *the* *exciton state*. Note, that the similar conclusion was made for conventional 3D semiconductors, as well [@keldysh1; @rice].
The *homogeneous* electron-hole liquid state predicted in this paper is realized within the assumption that the carriers possess the infinite lifetime. If not, the system breaks down into the *low-density exciton gas* phase and the *dense electron-hole liquid* phase. Between these phases the dynamic equilibrium is established which is supported by the external pumping. A similar scenario was proposed by Keldysh for conventional 3D semiconductors [@keldysh1].
The diagrammatic technique is used to calculate the correlation contribution to the energy of the electron-hole state. The approach is based on the $%
1/\nu $ expansion, what allows one to justify the random phase approximation (RPA) diagram selection. It is interesting to note that, for $l\gg a_{B},$ the ground state energy as well as the equilibrium carrier concentration $%
n_{eq}$ do not depend on the parameter $\nu .$ This unexpected feature allows us to expect that the results obtained may be, at least qualitatively, extended even to the region $\nu \sim 1$.
The paper is organized as follows. First, we describe the model of the electron-hole system in the CQW. Then, the self-consistent diagrammatic approach is proposed to calculate the electron and hole Matsubara Green functions and estimate the chemical potential $\mu $ for the electrons and holes at temperature $T\ll \varepsilon _{F},$ $\varepsilon _{F}$ being the Fermi energy for both electrons and holes. This allows us to find the equilibrium *electron-hole liquid state* concentration $n_{eq}$. Though, to obtain the main results, it is assumed that $l\gg $ $a_{B},$the case $l\leq a_{B}$ is also analyzed at the end of the paper. In conclusion, we consider a possible relation between the results obtained and the experiments, in which the nonuniform state of the system in CQW is revealed [@Nature-2002-Butov; @Nature-2002-Snoke; @Butov2003; @Snoke2003; @Butov2004; @Larionov2002; @Dremin2004; @Timofeev2005].
Model and The Calculation of the Self-Energy
============================================
To describe the spatially separated electron-hole system in the CQW, it is supposed that the electrons are confined within one infinitely narrow $2D$ layer, while the holes are confined within the other layer. For the sake of simplicity, it is supposed that the effective mass of both electrons and holes are the same. Below the system of units is used with the effective electron (hole) charge $e=1,$ effective electron (hole) masses $m=1,$ and the Planck constant $\hbar =1$. Then, the effective Bohr radius is $a_{B}=1,$ and the energy is measured in the Hartree units.
Let $n$ be concentration of electrons or holes. According to the model under consideration, the Fermi momentum $p_{F}=2\pi ^{1/2}\left( n/\nu \right)
^{1/2}$ and the Fermi energy $\varepsilon _{F}=2\pi n/\nu $ are the same for both electrons and holes. We confine ourselves to the case of strongly degenerated plasma, that is, the temperature $T\ll \varepsilon _{F}.$
The Hamiltonian of the system is $\widehat{H}=\widehat{H_{0}}+\widehat{U,}$ $%
H_{0}$ being the kinetic energy and $\widehat{U}$ being the Coulomb interaction. In the second quantization one has $$\widehat{H_{0}}=\sum_{\alpha \sigma \mathbf{k}}\frac{k^{2}}{2}a_{\alpha
\sigma }^{+}\left( \mathbf{k}\right) a_{\alpha \sigma }\left( \mathbf{k}%
\right) ;\widehat{U}=\frac{1}{2V}\sum_{\alpha \alpha ^{\prime }\sigma \sigma
^{\prime }\mathbf{kk}^{\prime }\mathbf{q}}U_{\alpha \alpha ^{\prime }}\left(
\left\vert \mathbf{k}\right\vert \right) a_{\alpha \sigma }^{+}\left(
\mathbf{k}\right) a_{\alpha ^{\prime }\sigma ^{\prime }}^{+}\left( \mathbf{k}%
^{\prime }\right) a_{\alpha ^{\prime }\sigma ^{\prime }}\left( \mathbf{k}%
^{\prime }-\mathbf{q}\right) a_{\alpha \sigma }\left( \mathbf{k}+\mathbf{q}%
\right) . \label{eq1.1}$$
Here $\alpha =e$ for the electrons and $\alpha =h$ for the holes. Then, $%
a_{\alpha \sigma }^{+}\left( \mathbf{k}\right) $ and $a_{\alpha \sigma
}\left( \mathbf{k}\right) $ are the creation and annihilation operators, $%
\mathbf{k}$ is the in-plane $2D$ momentum, $\sigma =1,...\nu $ denotes the kind of the electron or the hole component. The Coulomb interaction in the momentum representation reads $$U_{\alpha \alpha ^{\prime }}\left( \left\vert \mathbf{k}\right\vert \right)
=\left\{
\begin{array}{c}
\frac{2\pi }{k},\alpha =\alpha ^{\prime } \\
-\frac{2\pi }{k}e^{-kl},\alpha \neq \alpha ^{\prime }%
\end{array}%
\right. . \label{eq2}$$
Let $G_{\alpha ,\sigma }^{0}\left( \omega ,\mathbf{p}\right) =\left( i\omega
+\mu -\varepsilon _{p}\right) ^{-1}$ be the electron (hole) free Matsubara Green function and $G_{\alpha ,\sigma }\left( \omega ,\mathbf{p}\right)
=\left( i\omega +\mu -\Sigma _{\alpha ,\sigma }\left( \omega ,\mathbf{p}%
\right) -\varepsilon _{p}\right) ^{-1}~$be the electron (hole) total Green function. Here $\omega $ is the Matsubara frequency, $\varepsilon
_{p}=p^{2}/2,$ $\Sigma _{\alpha ,\sigma }\left( \omega ,\mathbf{p}\right) $ is the mass operator, and $\mu $ is the chemical potential. Let us consider the system of self-consistent equations whose diagram representation is shown in Fig.\[Figure2\]
![The self-consistent diagrammatic equations for the Green function and mass operator. Here ${\Sigma }$ is the mass operator. The double solid line denotes the total Green function $G.~$ The wavy line represents the interaction (\[eq2\]) and the double wavy line corresponds to the renormalized interaction.[]{data-label="Figure2"}](plasma1.eps){width="40.00000%"}
The first diagrammatic equation in Fig.\[Figure2\] reads $$\begin{aligned}
\Sigma _{\alpha ,\sigma }\left( \varepsilon ,p\right) &=&\Sigma _{\alpha
,\sigma }^{(1)}\left( \varepsilon ,p\right) +\Sigma _{\alpha ,\sigma
}^{(2)}\left( \varepsilon ,p\right) , \label{eq2.01} \\
\Sigma _{\alpha ,\sigma }^{(1)}\left( \varepsilon ,p\right) &=&\frac{T}{V}%
\sum_{\alpha ^{\prime }\sigma ^{\prime }\mathbf{k}\omega }U_{\alpha \alpha
^{\prime }}\left( 0\right) G_{\alpha ^{\prime }\sigma ^{\prime }}\left(
\mathbf{k},\omega \right) , \label{eq2.02} \\
\Sigma _{\alpha ,\sigma }^{\left( 2\right) }\left( \varepsilon ,p\right) &=&-%
\frac{T}{V}\sum_{\mathbf{k}\omega }\widetilde{U}_{\alpha \alpha }\left(
\mathbf{k},\omega \right) G_{\alpha \sigma }\left( \mathbf{p}+\mathbf{k}%
,\varepsilon +\omega \right) . \label{eq2.03}\end{aligned}$$Here $T$ is the temperature, and $V$ is the square of the layers.
The renormalized interaction $\widetilde{U}_{\alpha \alpha }\left( \mathbf{p}%
,\omega \right) $ in Eq. (\[eq2.03\]) obeys the second diagrammatic equation in Fig.\[Figure2\] and reads $$\widetilde{U}_{\alpha \alpha }\left( \mathbf{p},\omega \right) =U_{\alpha
\alpha }\left( \mathbf{p}\right) +U_{\alpha \alpha }\left( \mathbf{p}\right)
\Pi _{\alpha \sigma }\left( \mathbf{p},\varepsilon \right) \widetilde{U}%
_{\alpha \alpha }\left( \mathbf{p},\omega \right) . \label{eq2.04}$$The polarization operator$$\Pi \left( \mathbf{p},\varepsilon \right) =\Pi _{\alpha \sigma }\left(
\mathbf{p},\varepsilon \right) =\frac{T}{V}\sum\limits_{\mathbf{k}\omega
}G_{\alpha \sigma }\left( \mathbf{p}+\mathbf{k,}\varepsilon +\omega \right)
G_{\alpha \sigma }\left( \mathbf{k},\omega \right) \label{eq2.3}$$does not depend on the kind of the particles.
Let us explain the rule of diagram selection. Each fermion loop contributes a factor $\nu $ to the diagrams. Among all the diagrams of the given order, only the diagrams are retained which contain the *maximal* number of the fermion loops. Thus, our approach is a $1/\nu $ expansion which results in keeping only the diagram of the *RPA kind*. The diagrams for which the bubbles are linked by more than one interaction line are small as compared with the RPA ones in the parameter $1/\nu .$For the same reason, the diagrams are small, which contain any *vertex corrections*.
The expression for the renormalized interaction follows from Eq. ([eq2.04]{}) $$\widetilde{U}_{\alpha \alpha }\left( \mathbf{p},\omega \right) =\widetilde{U}%
\left( \mathbf{p},\omega \right) =\frac{U_{\alpha \alpha }\left( \mathbf{p}%
\right) }{1-\nu \cdot U_{\alpha \alpha }\left( \mathbf{p}\right) \Pi \left(
\mathbf{p},\omega \right) }. \label{eq2.2}$$Let us note that, like the polarization operator $\Pi _{\alpha \sigma
}\left( \mathbf{p},\omega \right) $, both the mass operator $\Sigma _{\alpha
,\sigma }\left( \varepsilon ,p\right) $ and the renormalized interaction $%
\widetilde{U}_{\alpha \alpha }\left( \mathbf{p},\omega \right) $ do not depend on the kind of the particle.
Our first goal is to calculate the chemical potential $\mu $ as a function of the density $n$ using the self-consistent system of Eqs. (\[eq2.01\])-(\[eq2.2\]). Let us remind the exact relation
$$\mu -\Sigma \left( 0,p_{F}\right) =p_{F}^{2}/2. \label{eq2.21}$$
Then, the convergence method is used. If the interaction is neglected, the mass operator $\Sigma \left( 0,p_{F}\right) \approx 0$ and, therefore, $$\mu \approx \mu _{0}=p_{F}^{2}/2=2\pi n/\nu . \label{eq2.211}$$
To find the next iteration for the chemical potential, let us calculate $%
\Sigma \left( 0,p_{F}\right) $ taking $\mu =\mu _{0}$. In this case, one should replace $G$ by $G_{0}$ in the above system of equations. First, taking into account Eq. (\[eq2\]), we obtain $$\Sigma _{\alpha \sigma }^{(1)}\left( \varepsilon ,\mathbf{p}\right) =\frac{T%
}{V}\sum_{\alpha ^{\prime }\sigma ^{\prime }\mathbf{k}\omega }U_{\alpha
\alpha ^{\prime }}\left( 0\right) G_{\alpha ^{\prime }\sigma ^{\prime
}}^{0}\left( \mathbf{k},\omega \right) =2\pi ln. \label{eq2.2111}$$
Then, to find the next contribution to the mass operator $\Sigma _{\alpha
\sigma }^{2}\left( \omega ,\mathbf{p}\right) ,$ one should first estimate the renormalized interaction $\widetilde{U}_{\alpha \alpha }\left( \mathbf{k}%
,\omega \right) $. For this purpose, let us consider the polarization operator in zero order in the interaction:
$$\Pi \left( \mathbf{p},\varepsilon \right) \approx \frac{1}{\left( 2\pi
\right) ^{2}}\int d^{2}k\frac{n_{0}\left( \mathbf{k}\right) -n_{0}\left(
\mathbf{k+p}\right) }{i\hbar \varepsilon +\mathbf{k}^{2}/2-\left( \mathbf{k+p%
}\right) ^{2}/2}, \label{eq3}$$
where $n_{0}\left( \mathbf{k}\right) =\left[ \exp \frac{\left( k^{2}/2-\mu
_{0}\right) }{T}+1\right] ^{-1}$ is the Fermi distribution function.
For *small* momenta and frequencies $p\lesssim p_{F},$ $\varepsilon
\lesssim \varepsilon _{F},$ one finds $\Pi _{0}\left( \mathbf{p},\varepsilon
\right) \simeq -1/2\pi .$ Substituting this value into Eq. (\[eq2.2\]) gives the estimate $$\widetilde{U}\left( \mathbf{p},\varepsilon \right) =\frac{U_{\alpha \alpha
}\left( \mathbf{p}\right) }{1+U_{\alpha \alpha }\left( \mathbf{p}\right) \nu
\frac{1}{2\pi }}\simeq \frac{1}{\nu }\ll 1. \label{eq3.11}$$This estimation is valid if
$$n^{1/3}/\nu \ll 1, \label{eq01}$$
what is supposed below.
Let us turn to the behavior of the renormalized interaction $\widetilde{U}%
\left( \mathbf{p},\varepsilon \right) $ for * large* momenta ** **$\left\vert \mathbf{p}\right\vert \gg p_{F}.$ First, let us estimate the polarization operator (\[eq3\]) for such momenta. Consider, for example, the contribution $\Pi ^{\prime }\left( \mathbf{p},\varepsilon \right) $ associated with the factor $n_{\alpha \sigma }\left( \mathbf{p}+\mathbf{q}%
\right) .$ It is evident that momenta $\mathbf{k}$ alone contribute to the polarization operator for which $\left\vert \mathbf{p}+\mathbf{k}\right\vert
\leq p_{F}.$ For this reason, ** **$\left\vert \mathbf{k}\right\vert
\gg p_{F}.$ Therefore, $\left( \mathbf{p}+\mathbf{k}\right) ^{2}\ll \mathbf{k%
}^{2}\approx \mathbf{p}^{2}$ and $$\Pi ^{\prime }\left( \mathbf{p},\varepsilon \right) \approx -\frac{1}{\left(
2\pi \right) ^{2}}\frac{n/\nu }{i\hbar \varepsilon +\mathbf{p}^{2}/2}.$$Similarly, one can estimate the contribution $\Pi _{0}^{\prime \prime
}\left( \mathbf{p},\varepsilon \right) $ connected with the term $n_{\mathbf{%
p}}$ in Eq. (\[eq3\]). As a result, we obtain $$\Pi \left( \mathbf{p},\varepsilon \right) \approx -\frac{1}{\left( 2\pi
\right) ^{2}\nu }\frac{n\mathbf{p}^{2}}{\varepsilon ^{2}+\left( \mathbf{p}%
^{2}/2\right) ^{2}},~p\gg p_{F}. \label{eq4}$$Taking into account (\[eq4\]), one can estimate the renormalized interaction (\[eq2.2\])
$$\widetilde{U}\left( \mathbf{p},\omega \right) =\frac{\frac{2\pi }{p}}{1+%
\frac{2\pi }{p}\frac{np^{2}}{\omega ^{2}+\left( p^{2}/2\right) ^{2}}},~p\gg
p_{F}. \label{eq4.2}$$
Substituting (\[eq3.11\]) into (\[eq2.03\]) gives the estimate for the contribution to the self-energy $\Sigma _{\alpha \sigma }^{\left( 2\right)
}\left( \omega ,\mathbf{p}\right) $ coming from $\left\vert \mathbf{k}%
\right\vert \leq p_{F}.$ $$\left\vert \Sigma _{\alpha ,\sigma }^{\prime \left( 2\right) }\left(
\varepsilon ,p\right) \right\vert \approx \frac{1}{\pi }\frac{n}{\nu ^{2}}.
\label{eq4.201}$$Let us turn to the contribution to $\Sigma _{\alpha \sigma }^{\left(
2\right) }\left( \omega ,\mathbf{p}\right) $ gained from $\left\vert \mathbf{%
k}\right\vert \gg p_{F}.$ It is convenient to represent it in the form$$\begin{gathered}
\Sigma _{\alpha ,\sigma }^{\prime \prime \left( 2\right) }\left( \varepsilon
,p\right) =-\frac{T}{V}\sum_{\left\vert \mathbf{k}\right\vert \gg
p_{F},~\omega }(U\left( \mathbf{k}\right) G_{\alpha \sigma }^{0}\left(
\mathbf{p}+\mathbf{k},\varepsilon +\omega \right) - \notag \\
-\frac{T}{V}\Delta U\left( \mathbf{k}\right) G_{\alpha \sigma }^{0}\left(
\mathbf{p}+\mathbf{k},\varepsilon +\omega \right) ). \label{eq4.21}\end{gathered}$$Here $\Delta U\left( \mathbf{k}\right) =\widetilde{U}\left( \mathbf{k}%
,\omega \right) -U\left( \mathbf{k}\right) .$ The first sum on the R.H.S. in Eq. (\[eq4.21\]) can be estimated as
$$\frac{1}{\left( 2\pi \right) ^{2}}\int_{\left\vert \mathbf{k}\right\vert \gg
p_{F}}\frac{d^{2}\mathbf{k}}{\left\vert \mathbf{k}\right\vert }n_{0}\left(
\mathbf{k+p}\right) \leq \frac{1}{\pi ^{3/2}}\left( \frac{n}{\nu }\right)
^{1/2} \label{eq4.22}$$
Let us turn to the second term on the R.H.S. in Eq. (\[eq4.21\]). Since we are interested in $\Sigma _{\alpha ,\sigma }^{\prime \prime 2}\left(
0,p_{F}\right) ,$ let us substitute $\left[ G_{\alpha \sigma }^{0}\left(
\mathbf{p}+\mathbf{k},\varepsilon +\omega \right) +G_{\alpha \sigma
}^{0}\left( \mathbf{p}-\mathbf{k},\varepsilon -\omega \right) \right]
\approx $ $-\frac{k^{2}}{\omega ^{2}+\left( k^{2}/2\right) ^{2}}\left( k\gg
p_{F}\right) $ into this term. Then, one obtains the estimate for the second sum in Eq. (\[eq4.21\])
$$-\frac{1}{\left( 2\pi \right) ^{3}}\int d\omega \int_{k\gg p_{F}}kdk\frac{%
\left( \frac{2\pi }{k}\right) ^{2}n\left( \frac{k^{2}}{\omega ^{2}+\left(
k^{2}/2\right) ^{2}}\right) ^{2}}{1+\frac{2\pi }{k}\frac{nk^{2}}{\omega
^{2}+\left( k^{2}/2\right) ^{2}}}\approx -Cn^{1/3}. \label{eq4.31}$$
This estimate results from the following hint. Let us introduce new variables $\omega /n^{2/3}$ and $k=k/n^{1/3}$. The integrand is maximal in the vicinity of the region $\omega /n^{2/3}$ $\sim k/n^{1/3}\sim 1.$ Due to Eq. (\[eq01\]), one can put the lower limit equal to zero in integral ([eq4.31]{}). Then the integral depends on the concentration as $n^{1/3}.$ The proportionality constant is calculated numerically, $C\approx 0.615$.
Equation of state
=================
Taking into account condition (\[eq01\]), one can neglect the contribution of expressions (\[eq2.211\]) (\[eq4.201\]), (\[eq4.22\]) into the chemical potential as compared with (\[eq4.31\]). Thus, expressions ([eq2.2111]{}) and (\[eq4.31\]) alone contribute to the chemical potential. Therefore, $$\mu \approx 2\pi ln-Cn^{1/3}. \label{eq5}$$The chemical potential $\mu ,$ given by Eq. (\[eq5\]), is a result of the first iteration. Let us now substitute expression (\[eq5\]) for the chemical potential into Eqs. (\[eq2.01\]) - (\[eq2.03\]) instead of $\mu
_{0}$ and repeat all the calculations starting from Eq. (\[eq2.2111\]). It is easy to find that, whether $\mu _{0}$ or $\mu $ is taken as a starting point, the correlation contribution to the self-energy is of the same order of the magnitude given by (\[eq4.31\]). For this reason, the expression for the chemical potential (\[eq5\]) is the stable solution of the self-consistent system of equations, the constant $C$ being of the order of unity. This chemical potential corresponds to the energy per unit volume$$\varepsilon =\pi ln^{2}-\frac{3}{4}Cn^{4/3}. \label{eq50}$$Then, at *zero temperature* the pressure is $$P=\mu n-\varepsilon =\pi ln^{2}-\frac{C}{4}n^{4/3}. \label{eq51}$$At small density $n<$ $\left( C/6\pi l\right) ^{3/2},$ the compressibility $%
\partial P/\partial V>0,$ and the system is unstable. For higher density, in the interval $\left( C/6\pi l\right) ^{3/2}<n<\left( C/4\pi l\right) ^{3/2}$, one has $\partial P/\partial V<0.$ The system is stable and has a *negative* pressure. The liquid state is reached at the upper limit of this interval $$n_{eq}=\left( C/4\pi l\right) ^{3/2}. \label{eq6}$$where the pressure $P=0.$
Thus, for the electron-hole liquid in the equilibrium, the bound energy* per one particle* is $$\varepsilon _{eq}=-\varepsilon \left( n_{eq}\right) /n_{eq}=-\frac{C^{3/2}}{%
4\pi ^{1/2}}\frac{1}{l^{1/2}}. \label{eq7}$$At the same time, for the *isolated exciton* the bound energy $%
\varepsilon _{ex}^{0}~$is known to behave as $\varepsilon _{ex}^{0}\sim
-1/l. $ Since the excitons in the CQW experience* repulsion,.* the bound energy per particle is $\varepsilon _{ex}>-1/l.$ Comparing this energy with $\varepsilon _{eq}$ (see expression (\[eq7\])) one concludes that the electron-hole state possesses a smaller energy as compared to the exciton state for $l\gg 1$. In addition, note that the exciton in-plane radius $%
R_{ex}\sim l^{3/4}$ and, thus, for the equilibrium carrier concentration obtained, one has $n_{eq}R_{ex}^{2}\sim 1.$ Under such conditions, the exciton concept seems us to have no sense.
Discussion and Conclusion
=========================
The results above are obtained for the case $l\gg 1,$ i.e., $l\gg a_{B}.$ In this case, the ground-state of the system is the electron-hole liquid which possesses the equilibrium density (\[eq6\]). This is a stable state which has the energy smaller than that of the exciton state.
Consider, however, the case $l\ll 1.$ It follows from Eqs. (\[eq2.211\]), (\[eq2.2111\]), (\[eq4.31\]) that the kinetic energy $\varepsilon _{kin}$, the direct Coulomb interaction $\varepsilon _{eh}$ and the intra-layer correlation energy $\varepsilon _{corr}$ can be estimated as $$\varepsilon _{kin}\sim \frac{n}{\nu },\varepsilon _{e-h}\sim nl,\varepsilon
_{corr}\sim -n^{1/3},$$respectively. The equilibrium is reached at the concentration $$n_{eq}\sim \left( \frac{\nu }{1+\nu l}\right) ^{3/2}, \label{eq8-0}$$for which the total energy $\varepsilon _{eq}$ is minimal and $$\varepsilon _{eq}\sim -\left( \frac{\nu }{1+\nu l}\right) ^{1/2}\approx
\left\{
\begin{array}{c}
-\frac{1}{l},\text{if~}\nu l\gg 1 \\
-\nu ^{1/2},\text{if~}\nu l\ll 1%
\end{array}%
\right. . \label{eq8}$$For the case $l\ll 1$ considered in this place, the exciton bound energy $%
\varepsilon _{ex}\sim -1$ in the units taken in the paper. (In fact, this energy is even larger due to repulsion between the excitons). Throughout the paper it is assumed that $\nu \gg 1.$Thus, $\varepsilon _{eq}\ll \varepsilon
_{ex},$ as well as above in the case $l\gg 1$. Also, for the concentration (\[eq8-0\]), the average distance between the carriers is of the order of $%
~n_{eq}^{-1/2}\sim $ $\left( \frac{1+\nu l}{\nu }\right) ^{3/4}\ll 1.$ At the same time, for $l\ll 1$ the exciton in-plane radius $R_{ex}\sim a_{B}=1.$ Thus, the exciton in-plane radius $R_{ex}$ exceeds the average distance between the carriers, what has no sense. Thus, both in the case $l\gg 1$ and in the case $l\ll 1$ the ground state of the system of spatially separated electrons and holes in the CQW is the electron-hole liquid.
As the intermediate case $l\sim 1$ is concerned, both $\varepsilon _{eq}$ $%
\sim -1$ and $\varepsilon _{ex}\sim -1.$ However, in this case $%
n_{eq}^{-1/2}\sim R_{ex}\sim a_{B}\sim 1$ and, thus, the exciton concept seems has no sense as well.
Let us return to the case $l\gg 1.$An interesting feature of the results obtained is that both the bound energy (\[eq6\]) and the equilibrium concentration (\[eq7\]) of the electron-hole liquid in the CQW do not depend on the parameter $\nu .$ This is in spite of the fact that formally this parameter was supposed to be large. This circumstance allows us to expect that the liquid equilibrium electron-hole state in the CQW may arise even for $\nu \sim 1.$ However, for $\nu \gg 1,$ the kinetic energy, which is of the order of $\varepsilon _{F}\sim \frac{1}{\nu },$ plays an insignificant role in the formation of the equilibrium state. The equilibrium within the electron-hole system is reached due to balance between the positive direct Coulomb interaction (\[eq2.2111\]) and the negative *in-layer* correlation energy (\[eq4.31\]).
Let us pay attention to the interesting feature of the origin for the correlation energy associated with the RPA diagram. The correlation energy contribution results from the momentum transfer $p\sim n^{1/3}\gg p_{F}.$ This conclusion strongly differs from that of Gell-Mann and Brueckner who considered the many-body correlations for a $3D$ electron gas. Within their approach, the correlation energy originates from the momentum transfer $p\ll
p_{F}.$ Let us also point out additional arguments which approve the RPA approximation. While the Gell-Mann and Brueckner approach is justified for $%
n\gg 1,$ our approach holds for an arbitrary concentration $n$ provided that $\nu $ is large.
Let us investigate how the results obtained can be associated with the experiments. Below, we follow the scenario proposed by Keldysh for the electron-hole liquid formation in semiconductors[@keldysh]. Consider a non-equilibrium inhomogeneous electron-hole system in the CQW, composed of two coexisting phase: a rare exciton gas phase and a dense electron-hole liquid drops. Let $R$ be the characteristic radius of the drop and $\tau $ be the carrier (electron or hole) lifetime within this drop. Then, the loss of the carriers per unit time for the drop is $-\pi R^{2}n_{eq}/\tau .$ On the other hand, there exists an exciton flow $j$ in the gas phase, induced by an external source. This flow gains the number of carriers in the drop per unit time $+2\pi Rj.$ In the dynamic equilibrium, the loss and the gain balance each other, resulting in the dependence$$R\approx \frac{2j\tau }{n_{eq}}.$$The concept of the electron-hole drop in the CQW fails if $R<a_{B}=1.$Therefore, there exists a threshold value for the flow$$j>j_{c}=\frac{n_{eq}a_{B}}{2\tau }, \label{eq9}$$which provides the formation of the drop. Thus, if the flow $j$ is weak, the drop has no time to form.
Such a scenario may account for the appearance of luminescent rings in the CQW in the series of experiments [@Nature-2002-Butov; @Nature-2002-Snoke; @Butov2003; @Snoke2003; @Butov2004; @Larionov2002; @Dremin2004; @Timofeev2005]. These rings, sometimes, are associated with the exciton Bose condensate. However, in our opinion, they should be connected with the electron-hole liquid drop described above. First of all for these experiments, the criterion of the exciton existence $R_{ex}^{2}n\ll 1$ seems to be violated. However, suppose that the core of the ring under the experiments is illuminated by the external optical pump resulting in the creation of the exciton gas flow $j$ within the core. Then, the ring corresponds to the electron-hole liquid phase described above. It follows from Eq. (\[eq9\]) that the electron-hole liquid ring can dwell in the stationary state if the flow $j$ exceeds a certain critical value. In the experiments, the luminescent ring appears if the optical pumping exceeds a certain power. Thus, the results obtained may be the basis for the explanation of these experiments.
Let us mention that the correlation energy calculated above does not involve the effect of the exciton-like electron-hole correlation associated with the ladder diagrams in the electron-hole scattering channel. The involvement of this effect would result in the appearance of the nonvanishing anomalous exciton-like averages responsible for the creation of the dielectric gap on the Fermi surface[@KO]. However, the contribution of these diagrams into correlation energy is small in the parameters $1/\nu .$This makes insignificant their influence on the thermodynamics of the system, the main subject of the paper.
We are grateful to Yuri Lozovik for valuable discussions.
[99]{} Yu. E. Lozovik and V.I. Yudson, Zh. Exp.Theor. Fiz. **71**, 738 (1976) \[Sov. Phys. JETP **44**, 389 (1976)\].
K. Das Gupta, A.F. Croxall, J. Waldie, C.A. Nicoll, H.E. Beere, I. Farrer, D.A. Ritchie, and M. Pepper, Adv. Cond. Matt. Phys. Volume 2011, Article ID 727958.
Yu. Lozovik and O.L. Berman, JETP Lett. **64**, 573 (1996)\];JETP **84**, 1027 (1997).
Yogesh N. Joglekar, Alexander V. Balatsky, and S. Das Sarma, **74**, 233302 (2006)
E.A. Andrushin, V.S. Babichenko, L.V. Keldysh, et all. JETPh Lett, 24, 210 (1976).
L.V. Keldysh, Electron-Hole liquid in Semiconductors, in Morden Problems of Condense Matter Science, v. 6, C.D. Jeffries and L.V. Keldysh (North Holland, Amsterdam, 1987)
L. V. Keldysh, Excitones in Semiconductors, (Nauka, Moscow, 1971).
T. M. Rice, The electron-hole liquid in semiconductors, Solid State Physics V 32, ed.: H. Ehrenreich, F. Zeitz, D. Turnbull, Academic Press, INC. 1077.
L.V. Butov, A.C. Gossard, and D.S. Chemla, Nature (London) **418**, 751 (2002).
D. Snoke, S. Denev, Y. Liu, L. Pheiffer, and D.S. Chemla, Nature (London) **418**, 754 (2002).
L.V. Butov, Solid State Commun. **127**, 89 (2003).
D. Snoke, Y. Liu, S. Denev, L. Pheiffer, and K. West. Solid State Commun. **127**, 187 (2003).
L.V. Butov, L.S. Levitov, A.V. Mintsev, B. D. Simons, A.C. Gossard, and D.S. Chemla, Phys. Rev. Lett. **92**, 117404 (2004)
A.V. Larionov, V.B. Timofeev, P.A. Ni, S.V. Dubonos, I. Hvam, and K. Soerensen, Pis’ma Zh. Ekp. Theor. Fiz. **75**, 233 (2002) \[JETP Lett. **75**, 570 (2002)\].
A.A. Dremin, A.V. Larionov, and V. B. Timofeev, Fiz. Tverd. Tela (St. Peterburg) **46**, 168 (2004) \[Solid. State. Phys. **46**, 170 (2004)\].
V.B. Timofeev, Usp. Phys. Nauk **175**, 315, (2005) \[Phys. Usp. **48**, 295 (2005)\].
L.S. Levitov, B.D. Simons, and L.V. Butov, Phys. Rev. Lett, **94**, 176404 (2005).
L.V. Keldysh, and Yu.V. Kopaev, Sov. Phys. Solid State 6, 2219 (1965).
|
---
abstract: 'We give the exact solution of classical equation of motion of a quartic scalar massless field theory showing that this is massive and is represented by a superposition of free particle solutions with a discrete spectrum. Then we show that this is also a solution of the classical Yang-Mills field theory that is so proved acquiring mass by dynamical evolution with a corresponding discrete mass spectrum. Finally we develop quantum field theory starting with this solution.'
author:
- Marco Frasca
title: Exact Massive Solutions of Classical Massless Field Theories
---
There are two main reasons why getting an exact solution to a classical field theory is an important step beyond. A classical solution may give a clever understanding of the physics underlying the theory itself and secondly, but not less important, one can build a quantum field theory out of such a solution using perturbation theory. Indeed, somewhat striking may happen already at a classical level for a classical field theory. A very blatant example of this is given by spontaneous breaking of symmetry. We see, already at the classical level, a non trivial behavior of the theory that has as an important side effect mass generation for all other coupled fields.
Generally speaking, we do not have many of such exact solutions. What we do most of times is perturbation theory starting from the solutions of the free theory that, being linear, is straightforwardly solved. For a lot of interesting cases our ability is just limited by the fact that we have to cope with non-linear equations whose solutions are rarely known.
Recently, we published a paper where we were able the obtain both the spectrum and the Feynman propagator in the infrared of a scalar theory [@fra1]. Our aim here is to reconsider this theory in the classical limit to obtain the unexpected results that the behavior of the exact classical solution is the same of the quantum field theory in the infrared we derived giving in this way a consistent mathematical support to our previous conclusions.
So, we start by writing down the action for the theory we aim to solve. One has $$S = \int d^4x\left[\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}\phi^4\right].$$ We just note that for this theory holds conformal invariance having no mass term and does not have any dimensional parameter being $\lambda$, the coupling, adimensional. So, Euler-Lagrange equation reduces in this case to the equation of motion $$\label{eq:phi}
\ddot\phi-\Delta_2\phi+\lambda\phi^3=0.$$ We can write down immediately an exact solution of this equation being given by $$\label{eq:phis}
\phi(x)=\mu\left(\frac{2}{\lambda}\right)^{\frac{1}{4}}{\rm sn}(p\cdot x+\theta,i)$$ being ${\rm sn}$ the Jacobi snoidal function, $\mu$ and $\theta$ two integration constants. The first has the dimension of a mass while the other, being just a phase, is adimensional. The important point is that this solution holds only if the following dispersion relation holds $$\label{eq:disp}
p^2=\mu^2\left(\frac{\lambda}{2}\right)^{\frac{1}{2}}$$ that is we have a wave-like solution with a mass $m_0=\mu\left(\frac{\lambda}{2}\right)^{\frac{1}{4}}$ that we see going to zero with the coupling going to zero. So, we have reached a striking conclusion that, starting with a massless theory, the corresponding classical solution is indeed massive.
Let us now get into the solution (\[eq:phis\]) to understand what is going on. This solution represents a nonlinear wave solution that is generally well-known in physics. But in order to understand the physics of the field theory we use the following Fourier expansion, a standard result of Jacobi elliptic functions [@gr], $${\rm sn}(u,i)=\frac{2\pi}{K(i)}
\sum_{n=0}^\infty\frac{(-1)^ne^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}
\sin\left[(2n+1)\frac{\pi u}{2K(i)}\right]$$ being $K(i)$ the following elliptic integral $$K(i)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1+\sin^2\theta}}\approx 1.3111028777.$$ This means that the exact solution is given by a superposition of plane waves as $$\phi(x)=-\mu\left(\frac{2}{\lambda}\right)^{\frac{1}{4}}\frac{\pi}{iK(i)}
\sum_{n=0}^\infty\frac{(-1)^ne^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}
\exp\left[-i(2n+1)\frac{\pi p\cdot x}{2K(i)}\right]+c.c.$$ that is, the field is just a set of excitations typical of free particles having a discrete set of eigenvalues. Indeed, we can single out a mass spectrum. This can be easily accomplished in the rest frame setting ${\bf p}=0$. We are left with the following field $$\phi(t,0)=-\mu\left(\frac{2}{\lambda}\right)^{\frac{1}{4}}\frac{\pi}{iK(i)}
\sum_{n=0}^\infty\frac{(-1)^ne^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}
\exp\left[-i(2n+1)\frac{\pi}{2K(i)}
\left(\frac{\lambda}{2}\right)^{\frac{1}{4}}\mu t\right]+c.c.$$ where use has been made of the dispersion relation (\[eq:disp\]). So we have got the mass spectrum $$\epsilon_n=(2n+1)\frac{\pi}{2K(i)}
\left(\frac{\lambda}{2}\right)^{\frac{1}{4}}\mu$$ that coincides exactly with the one obtained through the quantum field theory presented in [@fra1]. We see that already at the classical level the theory appears trivial. We have just a superposition of plane waves with the discrete spectrum of a harmonic oscillator, notwithstanding we started with a non-trivial non-linear theory. But what is really striking is the appearing of a mass spectrum out of a classical massless field theory. This means that conformal invariance is dynamically broken. Finally we note that a relatively strong coupling is needed to break conformal invariance. The reason for this relies on the fact that the mass that appears in our solution goes to zero with the fourth root of the coupling. We note that we can identify a set of “golden numbers” through the ratio $\epsilon_n/m_0$. The first ones are given in table \[tab:gn\].
Order Value
------- ------------- -- --
0 1.198140235
1 3.594420705
2 5.990701175
3 8.386981645
: \[tab:gn\] The first set of “golden numbers”.
On the same ground we can consider a scalar theory with a broken phase. We can write down for this case as $$S = \int d^4x\left[\frac{1}{2}(\partial\phi)^2+
\frac{1}{2}m^2\phi^2-\frac{\lambda}{4}\phi^4\right]$$ giving the equation $$\label{eq:phib}
\ddot\phi-\Delta_2\phi+ \lambda\phi^3-m^2\phi=0.$$ It is straightforward to write down the exact solution as $$\phi(x)=v\cdot{\rm dn}(p\cdot x+\theta,i)$$ being ${\rm sn}$ the Jacobi snoidal function, $v=\sqrt{2m^2/3\lambda}$ and $\theta$ an integration constant. Also in this case, such a solution holds only when the following dispersion relation holds $$p^2=\frac{\lambda v^2}{2}.$$ In this case we have a zero mass excitation into the spectrum. Indeed, if we write the Fourier expansion, that is a well-known result of Jacobi elliptic functions [@gr], as $${\rm dn}(u,i)=\frac{\pi}{2K(i)}+\frac{2\pi}{K(i)}\sum_{n=1}^\infty
\frac{e^{-n\pi}}{1+e^{-2n\pi}}\cos\left(2n \frac{\pi u}{2K(i)}\right),$$ one has $$\phi(x)=\frac{v\pi}{2K(i)}+\frac{v\pi}{K(i)}\sum_{n=1}^\infty
\frac{e^{-n\pi}}{1+e^{-2n\pi}}e^{-2in \frac{\pi}{2K(i)}px}+c.c.$$ and, as above, setting ${\bf p}=0$ we get $$\phi(t,0)=\frac{v\pi}{2K(i)}+\frac{v\pi}{K(i)}\sum_{n=1}^\infty
\frac{e^{-n\pi}}{1+e^{-2n\pi}}e^{-2in
\sqrt{\frac{\lambda}{2}}\frac{\pi v}{2K(i)}t}+c.c.$$ and this gives the spectrum $$\epsilon_n=n \frac{\pi}{K(i)}\sqrt{\frac{\lambda}{2}}v$$ with $n=0,1,2,\ldots$. These results show how a classical scalar theory with a wrong mass sign recovers a proper physical mass spectrum due to the non-linear term. This result confirms our derivation given in Ref.[@fra2] In order to have an idea of the physical meaning of these classical solutions, we point out that are the same of plane waves of a free theory and so boundary conditions are those generally considered in this case. This can be seen when the Fourier series of these solutions is obtained showing a super-position of plane waves. So, identical boundary conditions should apply. Changing these conditions, we come to different physical situations and so to a different spectrum of excitations. We want to see, in the same way as done for the scalar theory, if the classical Yang-Mills theory admits a massive solution. We proceed like for the scalar field and write down the action that now is given by [@nair] $$S=\int d^4x\left[\frac{1}{2}\partial_\mu A^a_\nu\partial^\mu A^{a\nu}
+gf^{abc}\partial_\mu A_\nu^aA^{b\mu}A^{c\nu}
+\frac{g^2}{4}f^{abc}f^{ars}A^b_\mu A^c_\nu A^{r\mu}A^{s\nu}\right].$$ We can map this action on the one of the massless quartic scalar field so that a solution of the equation of motion of the scalar field is also a solution of the Yang-Mills equations of motion [@fra3; @fra4]. E.g., for SU(3), the following choice $A_\mu^a=((0,0,0,0),(0,\phi,0,0),(0,\phi,0,0),(0,0,\phi,0),(0,\phi,\phi,0),(0,0,\phi,0),(0,0,0,\phi),(0,0,0,\phi))$ is a possible mapping but the number of choices (Smilga’s choices) is truly large and dependent on the gauge group. Given these choices, Yang-Mills action reduces to the one of the scalar field (\[eq:phi\]) with the substitution $\lambda\rightarrow Ng^2=3g^2$ that is the ’t Hooft coupling. We get, as it should be, $N^2-1=8$ of such equations. So, by introducing the tensor $\eta^a_\mu=((0,0,0,0),(0,1,0,0),(0,1,0,0),(0,0,1,0),(0,1,1,0),(0,0,1,0),(0,0,0,1),(0,0,0,1))$, we can write down our solution as $$A^a_\mu(t,0)=\eta^a_\mu\phi(t)=\eta^a_\mu\Lambda
\left(\frac{2}{3g^2}\right)^{\frac{1}{4}}
{\rm sn}\left(\left(\frac{3g^2}{2}\right)^{1 \over 4}\Lambda t+\varphi,i\rm\right)$$ being now $\Lambda$ and $\varphi$ the arbitrary integration constants. After a Lorentz transformation $\Lambda^\nu_\mu$ we will get $$A^a_\mu(x)=\Lambda^\nu_\mu\eta^a_\nu\phi(x)=\hat\eta^a_\mu\Lambda
\left(\frac{2}{3g^2}\right)^{\frac{1}{4}}{\rm sn}(p\cdot x+\varphi,i\rm)$$ and the dispersion relation $$\label{eq:disp2}
p^2=\Lambda^2\left(\frac{3g^2}{2}\right)^{\frac{1}{2}}$$ that proves our assertion that the classical Yang-Mills equations admit massive solutions. Being all exactly as for the scalar field, we get again a mass spectrum of free particles given by $$\epsilon_n^{YM}=(2n+1)\frac{\pi}{2K(i)}
\left(\frac{3g^2}{2}\right)^{\frac{1}{4}}\Lambda.$$ So, by dividing for the mass as given in (\[eq:disp2\]), we get the same set of “golden numbers” as those of the scalar field (see tab. \[tab:gn\]). This mass of the Yang-Mills field is usually identified with the square root of the string tension $\sigma$ and its value is either $440$ MeV or $410$ MeV depending on the analysis carried out by different groups (e.g. [@tep; @mor]). The spectrum we obtain differs from what quantum expectations could be. Indeed, one cannot see asymptotic freedom being this originating from quantum corrections. As such these could modify the spectrum that eventually could be maintained just in the infrared, an evidence to be shown. One can recover the spectrum of a free particle at very high energies just working with the Fourier transform of the given exact solution. In order to work out a quantum field theory starting from the above exact solutions, we consider as usual the generating functional $$Z[j] = N\int[d\phi] e^{i\int d^4x
\left[\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}\phi^4+j\phi\right]}$$ being $N$ a normalization constant, and we take the substitution $\phi=\phi_c+\delta\phi+O(\delta\phi^2)$ being $\phi_c$ the classical solution given in eq.(\[eq:phis\]). We will recover the results given in [@fra1] and the first higher correction as it should be. After the substitution has done one has $$Z[j]=e^{i[S_c+\int d^4x j\phi_c]} \int[d\delta\phi] e^{i\int d^4x
\left[{1 \over 2}(\delta\phi)^2
-{3\over 2}\phi_c^2(\delta\phi)^2+j\delta\phi\right]}+O((\delta\phi)^3).$$ Before to evaluate the higher order correcion, we take a look at the term $$Z_0[j]=e^{i[S_c+\int d^4x j\phi_c]}.$$ We are able to put this term into an useful form if we are able to solve the equation $$\partial^2_x\Delta_0(x)+\lambda \Delta_0^3(x)=\delta^4(x)$$ and this can be easily done with eq.(\[eq:phis\]) and introducing Heaviside function. But for our needs, the limit $\lambda\rightarrow\infty$, it is enough to consider a small time expansion writing down the solution as [@fra5; @fra6] $$\phi_c(x)\approx\int d^4y\Delta_0(x-y)j(y)$$ yielding $$Z_0[j]\approx e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta_0(x-y)j(y)},$$ in agreement with Ref.[@fra1], reducing the theory to the Gaussian case. Finally, in order to compute the next to leading order correction, we do the following change of coordinates $$\delta\phi=\delta\phi_0+\int d^4y\Delta_1(x-y)j(y),$$ being $$\label{eq:d1}
\partial^2_x\Delta_1(x-y)+3\lambda\phi_c^2(x)\Delta_1(x-y)=\delta^4(x-y)$$ the first order propagator, and we are left with a Gaussian functional $$Z[j]\approx e^{{i\over 2}\int d^4xd^4yj(x)[\Delta_0(x-y)+\Delta_1(x-y)]j(y)}.$$ In order to compute the first order propagator, we have to solve eq.(\[eq:d1\]). This can be done noting that, in the limit $\lambda\rightarrow\infty$ one can use a WKB approximation in a gradient expansion. We get the equation $$\label{eq:d2}
\partial^2_t\Delta_1(x-y)+3\lambda\phi_c^2(x)\Delta_1(x-y)\approx\delta(t-t_y)\delta^3(x-y)$$ and the approximate solution $$\Delta_1(x-y)=\theta(t-t_y)\delta^3(x-y)\frac{1}{(3\lambda)^{1\over 4}
\phi_c^{1\over 2}(x)}e^{-i\sqrt{3\lambda}
\int_{t_y}^tdt'\phi_c({\mathbf x},t')}+
\theta(t_y-t)\delta^3(x-y)\frac{1}{(3\lambda)^{1\over 4}
\phi_c^{1\over 2}(x)}e^{i\sqrt{3\lambda}
\int_{t_y}^tdt'\phi_c({\mathbf x},t')}.$$ This solution is just an approximation that should be improved with higher order corrections to recover full Lorentz invariance. It is interesting to note that this correction goes like $1/\lambda^{1\over 4}$ showing that we obtained a strong coupling expansion that holds in the limit $\lambda\rightarrow\infty$. Some special treatment is required for the poles in this Green function due to the solutions of $\phi_c(x)=0$. These are known as caustics.
We have shown in this paper how a self-interacting massless field can generate massive excitations and we have obtained this through exact solutions of classical equations of motion. A mass pole is originating by the non-linearity of the theory and it is just a dynamical effect. This classical solution is recovered, as we have already shown [@fra1], in quantum field theory working the other way around.
[99]{} M. Frasca, Phys. Rev. D [**73**]{}, 027701 (2006); Erratum-ibid. D [**73**]{}, 049902 (2006). I. S. Gradshteyn, I. M. Ryzhik, [*Table of Integrals, Series, and Products*]{}, (Academic Press, 2000). M. Frasca, Int. J. Mod. Phys. A [**22**]{}, 5345 (2007). V. P. Nair, [*Quantum Field Theory*]{}, (Springer, New York, 2005), p.194. M. Frasca, Phys. Lett. [**B670**]{}, 73 (2008). M. Frasca, arXiv:0903.2357 \[math-ph\]. B. Lucini, M. Teper, U. Wenger, JHEP [**06**]{}, 012 (2004). Y. Chen, A. Alexandru, S. J. Dong, T. Draper, I. Horvath, F. X. Lee, K. F. Liu, N. Mathur, C. Morningstar, M. Peardon, S. Tamhankar, B. L. Young, J. B. Zhang, Phys. Rev. D [**73**]{}, 014516 (2006). M. Frasca, Mod. Phys. Lett. A [**22**]{}, 1293 (2007). M. Frasca, Int. J. Mod. Phys. A [**23**]{}, 299 (2008).
|
---
abstract: 'We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus $({\mathbb Z} / N{\mathbb Z})^d$, $d \geq 3$, until $uN^d$ time steps, $u > 0$, and the model of random interlacements recently introduced by Sznitman [@int]. In particular, we show that for large $N$, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time $uN^d$ converges to independent copies of the random interlacement at level $u$.'
author:
- David Windisch
- David Windisch
title: '**Random walk on a discrete torus and random interlacements**'
---
Departement Mathematik February 2008\
ETH Zürich\
CH-8092 Zürich\
Switzerland\
Introduction
============
The object of a recent article by Benjamini and Sznitman [@BS07] was to investigate the vacant set left by a simple random walk on the $d \geq 3$-dimensional discrete torus of large side-length $N$ up to times of order $N^d$. The aim of the present work is to study the connections between the microscopic structure of this set and the model of random interlacements introduced by Sznitman in [@int]. Similar questions have also recently been considered in the context of random walk on a discrete cylinder with a large base, see [@S08].
In the terminology of [@int], the interlacement at level $u \geq 0$ is the trace left on ${\mathbb Z}^d$ by a cloud of paths constituting a Poisson point process on the space of doubly infinite trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The parameter $u$ is a multiplicative factor of the intensity measure of this point process. The interlacement at level $u$ is an infinite connected random subset of ${\mathbb Z}^d$, ergodic under translation. Its complement is the so-called vacant set at level $u$. In this work, we consider the distribution of the local pictures of the trajectory of the random walk on $({\mathbb Z}/N{\mathbb Z})^d$ running up to time $uN^d$ in the neighborhood of finitely many points with diverging mutual distance as $N$ tends to infinity. We show that the distribution of these sets converges to the distribution of independent random interlacements at level $u$.
In order to give the precise statement, we introduce some notation. For $N \geq 1$, we consider the integer torus $$\begin{aligned}
{\mathbb T} = ({\mathbb Z}/N{\mathbb Z})^d, \quad d \geq 3. \label{def:torus}\end{aligned}$$ We denote with $P_x$, $x \in {\mathbb T}$, resp. $P$, the canonical law on ${\mathbb T}^{\mathbb N}$ of simple random walk on $\mathbb T$ starting at $x$, resp. starting with the uniform distribution $\nu$ on $\mathbb T$. The corresponding expectations are denoted by $E_x$ and $E$, the canonical process by $X_.$. Given $x \in {\mathbb T}$, the vacant configuration left by the walk in the neighborhood of $x$ at time $t \geq 0$ is the $\{0,1\}^{{\mathbb Z}^d}$-valued random variable $$\begin{aligned}
\omega_{x,t}(.) = 1 \{ X_m \neq \pi_{\mathbb T}(.) + x, \textrm{ for all } 0 \leq m \leq [t] \}, \label{def:vconf}\end{aligned}$$ where $\pi_{\mathbb T}$ denotes the canonical projection from ${\mathbb Z}^d$ onto $\mathbb T$. With (2.16) of [@int], the law ${\mathbb Q}_u$ on $\{0,1\}^{{\mathbb Z}^d}$ of the indicator function of the vacant set at level $u \geq 0$ is characterized by the property $$\begin{aligned}
{\mathbb Q}_u [\omega(x) = 1, \textrm{ for all } x \in K ] = \exp \{-u {\textup{ cap}(K)} \}, \textrm{ for all finite sets $K \subseteq {\mathbb Z}^d$,}\label{eq:int}\end{aligned}$$ where $\omega(x)$, $x \in {\mathbb Z}^d$, are the canonical coordinates on $\{0,1\}^{{\mathbb Z}^d}$, and $\textup{cap}(K)$ the capacity of $K$, see (\[def:cap\]) below. In this note, we show that the joint distribution of the vacant configurations in $M \geq 1$ distinct neighborhoods of distant points $x_1, \ldots, x_M$ at time $uN^d$ tends to the distribution of $M$ vacant sets of independent random interlacements at level $u$. This result has a similar flavor to Theorem 0.1 in [@S08], which was proved in the context of random walk on a discrete cylinder.
\[thm:main\] Consider $M \geq 1$ and for each $N \geq 1$, $x_1, \ldots, x_M$ points in $\mathbb T$ such that $$\begin{aligned}
&\lim_N \inf_{1 \leq i \neq j \leq M} |x_i - x_j|_\infty = \infty. \textrm{ Then} \label{thm1} \\
&\textrm{$(\omega_{x_1,uN^d}, \ldots , \omega_{x_M, uN^d})$ converges in distribution} \textrm{ to ${\mathbb Q}_u^{\otimes M}$ under $P$, as $N \to \infty$.} \label{thm2}\end{aligned}$$
We now make some comments on the proof of Theorem \[thm:main\]. Standard arguments show that it suffices to show convergence of probabilities of the form $P\left[ H_B > uN^d \right]$ with $B=\bigcup_{i=1}^M (x_i + K_i)$ and finite subsets $K_i$ of ${\mathbb Z}^d$, where $H_B$ denotes the time until the first visit to the set $B \subseteq {\mathbb T}$ by the random walk. Since the size of the set $B$ does not depend on $N$, it is only rarely visited by the random walk for large $N$. It is therefore natural to expect that $H_B$ should be approximately exponentially distributed, see Aldous [@A89], B2, p. 24. This idea is formalized by Theorem \[thm:exp\] below, quoted from Aldous and Brown [@AB]. Assuming that the distribution of $H_B$ is well approximated by the exponential distribution with expectation $E[H_B]$, the probability $P[H_B > uN^d]$ is approximately equal to $\exp \{ - uN^d/E[H_B] \}$. In order to show that this expression tends to the desired limit, which by (\[eq:int\]) and (\[thm2\]) is given by $\prod_{i=1}^M \exp \{ -u \textup{ cap}(K_i) \}$, one has to show that $N^d/E[H_B]$ tends to $\sum_{i=1}^M \textup{ cap}(K_i)$.
This task is accomplished with the help of the variational characterizations of the capacity of finite subsets of ${\mathbb Z}^d$ given by the Dirichlet and Thomson principles, see (\[eq:con\]) and (\[eq:res\]). These principles characterize the capacity of a finite subset $A$ of ${\mathbb Z}^d$ as the infimum over all Dirichlet forms of functions of finite support on ${\mathbb Z}^d$ taking the value $1$ on $A$, resp. as the supremum over the reciprocal of energies dissipated by unit flows from $A$ to infinity. Aldous and Fill [@AF] show that very similar variational characterizations involving functions and flows on $\mathbb T$ hold for the quantity $N^d / E[H_A]$, see (\[eq:con’\]) and (\[eq:res’\]) below. In these two variational characterizations one optimizes the same quantities as in the Dirichlet and Thomson principles, over functions on the torus of zero mean, respectively over unit flows on the torus from $A$ to the uniform distribution. In the proof, we compare these two variational problems with the corresponding Dirichlet and Thomson principles and thus show the coincidence of $\lim_N N^d/E[H_B]$ with $\sum_{i=1}^M \textup{cap}(K_i)$. To achieve this goal, we construct a nearly optimal test function and a nearly optimal test flow for the variational problems on $\mathbb T$ using nearly optimal functions and a nearly optimal flow for the corresponding Dirichlet and Thomson principles.
In the case of the Dirichlet principle, this construction is rather simple and only involves shifting the functions on ${\mathbb Z}^d$ whose Dirichlet forms are almost $\textup{cap}(K_i)$ to the points $x_i$ on the torus, adding and rescaling them. In the Thomson principle, we identify the torus with a box in ${\mathbb Z}^d$ and consider the unit flow from $B$ to infinity on ${\mathbb Z}^d$ with dissipated energy equal to $\textup{cap}(B)^{-1}$. To obtain a flow on ${\mathbb T}$, we first restrict the flow to the box. The resulting flow then leaves charges at the boundary. In order to obtain a nearly optimal flow from $B$ to the uniform distribution for the variational problem (\[eq:res’\]) on the torus, these charges need to be redirected such that they become uniformly distributed on $\mathbb T$, with the help of an additional flow of small energy.
The article is organized as follows:
In section \[sec:pre\], we state the preliminary result on the approximate exponentiality of the distribution of $H_B$ and introduce the variational characterizations required.
In section \[sec:proof\], we prove Theorem \[thm:main\].
Finally, we use the following convention concerning constants: Throughout the text, $c$ or $c'$ denote positive constants which only depend on the dimension $d$, with values changing from place to place. Dependence of constants on additional parameters appears in the notation. For example, $c(M)$ denotes a constant depending only on $d$ and $M$.
#### Acknowledgments.
The author is grateful to Alain-Sol Sznitman for proposing the problem and for helpful advice.
Preliminaries {#sec:pre}
=============
In this section, we introduce some notation and results required for the proof of Theorem \[thm:main\]. We denote the $l_1$ and $l_\infty$-distances on $\mathbb T$ or ${\mathbb Z}^d$ by $|.|_1$ and $|.|_\infty$. For any points $x, x'$ in $\mathbb T$ or ${\mathbb Z}^d$, we write $x \sim x'$ if $x$ and $x'$ are neighbors with respect to the natural graph structure, i.e. if $|x-x'|_1 = 1$. For subsets $A$ and $B$ of ${\mathbb T}$ or ${\mathbb Z}^d$, we write $d(A,B)$ for their mutual distance induced by $|.|_\infty$, i.e. $d(A,B) = \inf \{|x-x'|_\infty: x \in A, x' \in B\}$, $\textup{int} A = \{x \in A: x' \in A \textrm{ for all } x' \sim x \}$, as well as $\partial_{int} A$ for the interior boundary, i.e. $\partial_{int} A = A \setminus \textup{int} A$, and $|A|$ for the number of points in $A$.
We obtain a continuous-time random walk $(X_{\eta_t})_{t \geq 0}$ by defining the Poisson process $(\eta_t)_{t \geq 0}$ of parameter 1, independent of $X$. We write $P^{{\mathbb Z}^d}$ for the law of the simple random walk on ${\mathbb Z}^d$ and also denote the corresponding canonical process on ${\mathbb Z}^d$ as $X_.$, which should not cause any confusion. For $t \geq 0$, the set of points visited by the random walk until time $[t]$ is denoted by $X_{[0,t]}$, i.e. $X_{[0,t]} = \{X_0, X_1, \ldots, X_{[t]} \}$. For any subset $A$ of ${\mathbb T}$ or of ${\mathbb Z}^d$, we define the discrete- and continuous-time entrance times $H_A$ and ${\bar H}_A$ as $$\begin{aligned}
H_A = \inf \{n \geq 0: X_n \in A \} \quad \textrm{and} \quad {\bar H}_A = \inf \{t \geq 0: X_{\eta_t} \in A \}, \label{def:ent}\end{aligned}$$ as well as the hitting time $$\begin{aligned}
{\tilde H}_A = \inf \{n \geq 1: X_n \in A \}. \label{def:hit}\end{aligned}$$ Note that by independence of $X$ and $\eta$, one then has $$\begin{aligned}
E[{\bar H}_A] &= \sum_{n=0}^\infty P[H_A=n] E[\inf \{ t \geq 0: \eta_t = n \} ] = \sum_{n=0}^\infty P[H_A=n]n= E[H_A]. \label{eq:expent} \end{aligned}$$ The Green function of the simple random walk on ${\mathbb Z}^d$ is defined as $$\begin{aligned}
g(x,x') = E_x^{{\mathbb Z}^d} \biggl[ \sum_{n=0}^\infty 1\{X_n = x'\} \biggr], \quad \textrm{for } x, x' \in {\mathbb Z}^d. \label{def:g}\end{aligned}$$ In order to motivate the remaining definitions given in this section, we quote a result from Aldous and Brown [@AB], which estimates the difference between the distribution of ${\bar H}_A$ and the exponential distribution. The following theorem appears as Theorem 1 in [@AB] for general irreducible, finite-state reversible continuous-time Markov chains and is stated here for the continuous-time random walk $(X_{\eta_t})_{t \geq 0}$ on $\mathbb T$, cf. the remark after the statement.
\[thm:exp\] For any subset $A$ of $\mathbb T$ and $t \geq 0$, $$\begin{aligned}
\left| P[{\bar H}_A > t E[H_A]] - \exp \{-t\} \right| \leq cN^2/E[H_A]. \label{eq:exp}\end{aligned}$$
The main aim in the proof of Theorem \[thm:main\] will be to obtain the limit as $N$ tends to infinity of probabilities of the form $P[{\bar H}_A> uN^d]$. In view of (\[eq:exp\]), it is thus helpful to understand the asymptotic behavior of expected entrance times. To this end, we will use variational characterizations of expected entrance times involving Dirichlet forms and flows, which we now define. For a real-valued function $f$ on $E={\mathbb T}$ or ${\mathbb Z}^d$, we define the Dirichlet form ${\mathcal E}_E$ as $$\begin{aligned}
{\mathcal E}_E (f,f) = \frac{1}{2} \sum_{x \in E} \sum_{x' \sim x} \left( f(x) - f(x') \right)^2 \frac{1}{2d}. \label{def:dir}\end{aligned}$$ We write $C_c$ for the set of real-valued functions on ${\mathbb Z}^d$ of finite support and denote the supremum norm of any function $f$ by $|f|_\infty$. The integral of a function $f$ on $\mathbb T$ with respect to the uniform distribution $\nu$ is denoted by $\nu(f)$ (i.e. $\nu(f) = N^{-d} \sum_{x \in {\mathbb T}} f(x)$). A flow $I = (I_{x,x'})$ on the edges of $E={\mathbb T}$ or ${\mathbb Z}^d$ is a real-valued function on $E^2$ satisfying $$\begin{aligned}
I_{x,x'} = \left\{ \begin{array}{cl} -I_{x',x} & \textrm{if } x \sim x', \\
0 & \textrm{otherwise.} \end{array} \right. \label{def:flow}\end{aligned}$$ Given a flow $I$, we write $|I|_\infty = \sup_{x,x' \in E} |I_{x,x'}|$ and define its dissipated energy as $$\begin{aligned}
(I,I)_E = \frac{1}{2} \sum_{x \in E} \sum_{x' \in E} I_{x,x'}^2 2d. \label{def:energy}\end{aligned}$$ The set of all flows on the edges of $E$ with finite energy is denoted by $F(E)$. For a flow $I \in F(E)$, the divergence $\textup{div} I$ on $E$ associates to every point in $E$ the net flow out of it, $$\begin{aligned}
\textup{div} I (x)= \sum_{x' \sim x} I_{x,x'}, \quad x \in E. \label{def:div} \end{aligned}$$ The net flow out of a finite subset $A \subseteq E$ is denoted by $$\begin{aligned}
I(A) = \sum_{x \in A} \sum_{x' \sim x} I_{x,x'} = \sum_{x \in A} \textup{div} I (x).\end{aligned}$$ From Aldous and Fill, Chapter 3, Proposition 41, it is known that $N^d/E[H_A]$ is given by the infimum over all Dirichlet forms of functions on $\mathbb T$ of zero mean and equal to $1$ on $A$, and by the supremum over the reciprocals of energies dissipated by unit flows from $A$ to the uniform distribution $\nu$: $$\begin{aligned}
N^d / E[H_A] &= \inf \bigl\{ {\mathcal E}_{\mathbb T}(f,f): f=1 \textrm{ on } A, \nu (f) =0 \bigr\} \label{eq:con'} \\
& = \sup \bigl\{ 1/(I,I)_{\mathbb T}: I \in F({\mathbb T}), I(A)= 1- |A| N^{-d}, \label{eq:res'} \\
& \qquad \qquad \textup{div} I(x) = - N^{-d} \textrm{ for all } x \in {\mathbb T} \setminus A \bigr\}. \nonumber\end{aligned}$$ These variational characterizations are very similar to the Dirichlet and Thomson principles characterizing the capacity of finite subsets of ${\mathbb Z}^d$, to which we devote the remainder of this section. A set $A \subseteq {\mathbb Z}^d$ has its associated equilibrium measure $e_A$ on ${\mathbb Z}^d$, defined as $$\begin{aligned}
e_A(x) = \left\{ \begin{array}{ll} P^{{\mathbb Z}^d}_x[{\tilde H}_A = \infty] & \textrm{if } x \in A, \\
0 & \textrm{if } x \in {\mathbb Z}^d \setminus A. \end{array} \right.\end{aligned}$$ The capacity of $A$ is defined as the total mass of $e_A$, $$\begin{aligned}
\textup{cap}(A) = e_A ({\mathbb Z}^d). \label{def:cap}\end{aligned}$$ For later use, we record that the following expression for the hitting probability of $A$ is obtained by conditioning on the time and location of the last visit to $A$ and applying the simple Markov property: $$\begin{aligned}
P^{{\mathbb Z}^d}_x[H_A < \infty] = \sum_{x' \in A} g(x,x') e_A(x'), \quad \textrm{for } x \in {\mathbb Z}^d. \label{eq:heq}\end{aligned}$$ The Dirichlet and Thomson principles assert that $\textup{cap}(A)$ is obtained by minimizing the Dirichlet form over all functions on ${\mathbb Z}^d$ of compact support equal to $1$ on $A$, or by maximizing the reciprocal of the energy dissipated by so-called unit flows from $A$ to infinity:
\[thm:di\] $$\begin{aligned}
\textup{ cap}(A) &= \inf \left\{{\mathcal E}_{{\mathbb Z}^d}(f,f): f \in C_c, \textrm{ } f = 1 \textrm{ on } A \right\} \label{eq:con} \\
&= \sup \{1/(I,I)_{{\mathbb Z}^d}: I \in F({\mathbb Z}^d), I(A)=1, \textup{div} I(x) = 0, \textrm{ for all } x \in {\mathbb Z}^d \setminus A \}. \label{eq:res} \end{aligned}$$ Moreover, the unique maximizing flow $I^A$ in the variational problem (\[eq:res\]) satisfies $$\begin{aligned}
I^A_{x,x'} = - (2d \textup{ cap}(A))^{-1} ( P^{{\mathbb Z}^d}_{x'}[H_A < \infty] - P^{{\mathbb Z}^d}_x [H_A < \infty] ), \quad x \sim x' \in {\mathbb Z}^d. \label{eq:maxflow} \end{aligned}$$
By collapsing the set $A$ to a point (see for example [@AF], Chapter 2, Section 7.3), it suffices to consider a general transient graph $G$ instead of ${\mathbb Z}^d$ and $A = \{a\}$, for a vertex $a$ in $G$. The proof for this case can be found in [@S94]: Theorem 3.41 shows (\[eq:con\]) above and Theorem 3.25 with $\iota = 1_{\{a\}}$ (in the notation of [@S94]; allowed by Theorem 3.30 and transience of the simple random walk in dimension $d \geq 3$), combined with Corollary 2.14, yields the above claims (\[eq:res\]) and (\[eq:maxflow\]).
Proof {#sec:proof}
=====
With the results of the last section, we are now ready to give the proof of Theorem \[thm:main\].
Take any finite subsets $K_1, \ldots K_M$ of ${\mathbb Z}^d$ and, using the notations of the theorem, set $B = \bigcup_{i=1}^M (x_i + K_i)$. Note that the collection of events $\{\omega(x) = 1 \textrm{ for all } x \in K\}$ as $K$ varies over finite subsets of ${\mathbb Z}^d$ forms a $\pi$-system generating the canonical product $\sigma$-algebra on $\{0,1\}^{{\mathbb Z}^{d}}$. By compactness of the set of probability measures on $(\{0,1\}^{{\mathbb Z}^{d}})^M$, our claim will follow once we show that $$\begin{aligned}
\lim_N P[H_B>uN^d] = \prod_{i=1}^M e^{-u \textup{ cap}(K_i)}. \label{eq:thm1} \end{aligned}$$ As we now explain, we can replace $H_B$ by its continuous-time analog ${\bar H}_B$ in (\[eq:thm1\]). Indeed, assume (\[eq:thm1\]) holds with $H_B$ replaced by ${\bar H}_B$. By the law of large numbers, one has $\eta_t/t \to 1$ $a.s.$, as $t$ tends to infinity (see, for example [@durrett], Chapter 1, Theorem 7.3), and it then follows that, for $0<\epsilon <u$, $$\begin{aligned}
\limsup_{N} P[H_B>uN^d] &= \limsup_N P[X_{[0,uN^d]} \cap B = \emptyset] \\
&\leq \limsup_N P[X_{[0,\eta_{(u-\epsilon)N^d}]} \cap B = \emptyset] \\
&= \limsup_N P[{\bar H}_B > (u-\epsilon) N^d] = \prod_{i=1}^M e^{-(u-\epsilon) \textup{ cap}(K_i)},\end{aligned}$$ and similarly, $$\begin{aligned}
\liminf_N P[H_B>uN^d] &\geq \liminf_N P[X_{[0,\eta_{(u+\epsilon)N^d}]} \cap B = \emptyset] \\
&= \liminf_N P[{\bar H}_B > (u+\epsilon)N^d] = \prod_{i=1}^M e^{-(u+\epsilon) \textup{ cap}(K_i)}.\end{aligned}$$ Letting $\epsilon$ tend to $0$ in the last two estimates, one deduces the desired result. By the above observations and (\[eq:exp\]) with $A=B$ and $t=uN^d/E[H_B]$, all that is left to prove is that $$\begin{aligned}
\lim_N \frac{N^d}{E[H_B]} = \sum_{i=1}^M \textup{cap}(K_i). \label{eq:thm3}\end{aligned}$$ The claim (\[eq:thm3\]) will be shown by using the variational characterizations (\[eq:con’\]), (\[eq:res’\]), (\[eq:con\]) and (\[eq:res\]). To this end, we map the torus $\mathbb T$ to a subset of ${\mathbb Z}^d$ in the following way: We choose a point $x_*$ in $\mathbb T$ as the origin and then define the bijection $\psi: {\mathbb T} \to {\mathbb T}' = \{0, \ldots, N-1\}^d$ such that $\pi_{\mathbb T}(\psi(x_*+x)) = x$ for $x \in {\mathbb T}$, where $\pi_{\mathbb T}$ denotes the canonical projection from ${\mathbb Z}^d$ onto $\mathbb T$. Since there are only $M$ points $x_i$, we can choose $x_*$ in such a way that in ${\mathbb T}' \subseteq {\mathbb Z}^d$, $\psi(B)$ remains at a distance of order $N$ from the interior boundary of ${\mathbb T}'$, i.e. such that for $N \geq c(M)$, $$\begin{aligned}
d(\psi(B), \partial_{int} {{\mathbb T}'}) \geq c'(M) N. \label{eq:thm5.1}\end{aligned}$$ We define the subsets $C$ and $S$ of $\mathbb T$ as the preimages of $\textup{int} {\mathbb T}'$ and $\partial_{int} {\mathbb T}'$ under $\psi$, i.e. $$\begin{aligned}
C= \psi^{-1}( \textup{int} {\mathbb T}'), \quad \textrm{and} \quad S = \psi^{-1}(\partial_{int} {\mathbb T}'). \label{def:cs}\end{aligned}$$ For $\epsilon >0$, we now consider functions $f_i \in C_c$ (see above (\[def:flow\])) such that $f_i = 1$ on $K_i$ and $$\begin{aligned}
{\mathcal E}_{{\mathbb Z}^d} (f_i,f_i) \leq \textup{ cap}(K_i) + \epsilon, \textrm{ for } i=1, \ldots, M, \textrm{ cf.~(\ref{eq:con}).} \label{eq:thm4}\end{aligned}$$ Defining $\tau_x: {\mathbb T} \to {\mathbb Z}^d$ by $\tau_x(x')= \psi(x')-\psi(x)$ for $x,x' \in {\mathbb T}$, we construct the function $f$ by shifting the functions $f_i$ to the points $x_i$, subtracting their means and rescaling so that $f$ equals $1$ on $B$ (for large $N$): $$\begin{aligned}
f = \frac{\sum_{i=1}^M f_i \circ \tau_{x_i} - \nu \left( \sum_{i=1}^M f_i \circ \tau_{x_i} \right)} {1-\nu \left( \sum_{i=1}^M f_i \circ \tau_{x_i} \right)}.\end{aligned}$$ Note that by the hypothesis (\[thm1\]) and our choice (\[eq:thm5.1\]) of the origin, the finite supports of the functions $f_i(. - \psi(x_i))$ intersect neither each other nor $\partial_{int} {\mathbb T}'$ for $N \geq c(M)$. One can then easily check that for $N \geq c(M, \epsilon)$ we have $f=1$ on $B$ and $\nu(f) = 0$. It therefore follows from (\[eq:con’\]) that $$\begin{aligned}
\limsup_N N^d/E[H_B] &\leq \limsup_N {\mathcal E}_{\mathbb T} (f,f) \\
&\stackrel{(f_i \in C_c, (\ref{thm1}))}{=} %\limsup_N \frac{1}{1-N^{-d} \sum_{i=1}^M \sum_{x \in {\mathbb Z}^d} f_i(x)}
\sum_{i=1}^M {\mathcal E}_{{\mathbb Z}^d} (f_i,f_i) \stackrel{(\ref{eq:thm4})}{\leq} \sum_{i=1}^M \textup{cap}(K_i) + M\epsilon. \end{aligned}$$ Letting $\epsilon$ tend to $0$, one deduces that $$\begin{aligned}
\limsup_N N^d/E[H_B] \leq \sum_{i=1}^M \textup{cap}(K_i). \label{eq:thm5}\end{aligned}$$ In order to show the other half of (\[eq:thm3\]), we proceed similarly, with the help of the variational characterizations (\[eq:res’\]) and (\[eq:res\]). We consider the flow $I^{\psi(B)} \in F({\mathbb Z}^d)$ such that $$\begin{aligned}
&I^{\psi(B)}( \psi(B))=1, \label{eq:thm5.2} \\
& \textup{div} I^{\psi(B)}(z) = 0 \textrm{ for all $z \in {\mathbb Z}^d \setminus \psi(B)$, and } \label{eq:thm5.3} \\
&1/(I^{\psi(B)},I^{\psi(B)})_{{\mathbb Z}^d} = \textup{ cap}(\psi(B)), \textrm{ cf.~(\ref{eq:res}), (\ref{eq:maxflow}).} \label{eq:thm6}\end{aligned}$$ The aim is to now construct a flow of similar total energy satisfying the conditions imposed in (\[eq:res’\]). To this end, we first define the flow $I^{*} \in F({\mathbb T})$ by restricting the flow $I^{\psi(B)}$ to ${\mathbb T}'$, i.e. we set $$\begin{aligned}
I^{*}_{x,x'} = I^{\psi(B)}_{\psi(x),\psi(x')} \textrm{ for } x,x' \in {\mathbb T}. \label{eq:thm6.0}\end{aligned}$$ We now need a flow $J \in F({\mathbb T})$ such that $I^{*} + J$ is a unit flow from $A$ to the uniform distribution on $\mathbb T$. Essentially, $J$ has to redirect some of the charges $(\textup{div} I^*) 1_{S}$ left by $I^{*}$ on the set $S$, such that these charges become uniformly distributed on the torus, and the energy dissipated by $J$ has to decay as $N$ tends to infinity. The following proposition yields the required flow $J$:
\[pr:j\] There is a flow $J \in F({\mathbb T})$ such that $$\begin{aligned}
& \textup{div} J(x) + (\textup{div} I^{*}) 1_{S}(x) = - N^{-d}, \textrm{ for any $x \in {\mathbb T}$, and} \label{eq:j1} \\
& |J|_\infty \leq c(M) N^{1-d}. \label{eq:j2}\end{aligned}$$
Before we prove Proposition \[pr:j\], we show how it enables to complete the proof of Theorem \[thm:main\]. Let us check that for large $N$, the flow $I^{*} + J$ satisfies the hypotheses imposed in (\[eq:res’\]) with $A=B$. Since by (\[eq:thm5.1\]), $\psi (B)$ is contained in $\textup{int} {\mathbb T}'$ for $N \geq c(M)$, one has for such $N$, $$\begin{aligned}
(I^{*}+J)(B) \stackrel{(\ref{eq:thm6.0})}{=} I^{\psi(B)}(\psi(B)) + J(B) \stackrel{(\ref{eq:thm5.2}), (\ref{eq:j1})}{=} 1 - |B|N^{-d}.\end{aligned}$$ Moreover, for any $N \geq c(M)$ and $x \in {\mathbb T} \setminus B$, $$\begin{aligned}
\textup{div} (I^{*} + J)(x) &\stackrel{(\ref{eq:thm6.0})}{=} (\textup{div} I^{\psi(B)}) 1_{\textup{int} {\mathbb T}'} (\psi(x)) + (\textup{div} I^{*})1_{S}(x) + \textup{div} J(x) \\
&\stackrel{(\ref{eq:thm5.3}), (\ref{eq:j1})}{=} - N^{-d}.\end{aligned}$$ The flow $I^{*} + J$ is hence included in the collection on the right-hand side of (\[eq:res’\]) with $A=B$ and it follows with the Minkowski inequality that $$\begin{aligned}
E[H_B]N^{-d} &\leq (I^{*}+J,I^{*}+J)_{\mathbb T} \leq \left( (I^{*},I^{*})_{\mathbb T}^{\frac{1}{2}} + (J,J)_{\mathbb T}^{\frac{1}{2}} \right)^2. \label{eq:thm10}\end{aligned}$$ By the bound (\[eq:j2\]) on $|J|_\infty$, one has $(J,J)_{\mathbb T} \leq c(M) (N^{1-d})^2 N^d = c(M) N^{2-d}$. Inserting this estimate together with $$(I^{*},I^{*})_{\mathbb T} \stackrel{(\ref{eq:thm6.0})}{\leq} (I^{\psi(B)},I^{\psi(B)})_{{\mathbb Z}^d} \stackrel{(\ref{eq:thm6})}{=} 1/ \textup{cap}(\psi(B))$$ into (\[eq:thm10\]), we deduce that $$\begin{aligned}
E[H_B] N^{-d} &\leq \left( \textup{cap}(\psi(B))^{-\frac{1}{2}} + c(M) N^{-(d-2)/2} \right)^2. \label{eq:thm11}\end{aligned}$$ Finally, we claim that $$\begin{aligned}
\lim_N \textup{cap}(\psi(B)) = \sum_{i=1}^M \textup{cap}(K_i). \label{eq:thm12}\end{aligned}$$ Indeed, the standard Green function estimate from [@lawler], p. 31, (1.35) implies that for $d \geq 3$, $$P^{{\mathbb Z}^d}_x [H_{x'} < \infty] \leq g(x,x') \leq c|x-x'|_\infty^{2-d}, \quad x, x' \in {\mathbb Z}^d,$$ and claim (\[eq:thm12\]) follows by assumption (\[thm1\]) and the definition (\[def:cap\]) of the capacity. Combining (\[eq:thm11\]) with (\[eq:thm12\]), one infers that for $d \geq 3$, $$\begin{aligned}
\limsup_N E[H_B] N^{-d} &\leq \biggl(\sum_{i=1}^M \textup{cap}(K_i) \biggr)^{-1}.\end{aligned}$$ Together with (\[eq:thm5\]), this shows (\[eq:thm3\]) and therefore completes the proof of Theorem \[thm:main\].
It only remains to prove Proposition \[pr:j\].
The task is to construct a flow $J$ distributing the charges\
$(\textup{div}I^*)1_{S}$ uniformly on $\mathbb T$, observing that we want the estimate (\[eq:j2\]) to hold. To this end, we begin with an estimate on the order of magnitude of $\textup{div} I^*(x)$, for $x \in S$ and $N \geq c(M)$, where we sum over all neighbors $z$ of $\psi(x)$ in ${\mathbb Z}^d \setminus {\mathbb T}'$: $$\begin{aligned}
&\bigl|\textup{div} I^{*}(x) \bigr| \stackrel{(\ref{eq:thm6.0})}{=} \biggl|\textup{div} I^{\psi(B)}(\psi(x)) - \sum_{z} I^{\psi(B)}_{\psi(x),z} \biggr| \stackrel{(\ref{eq:thm5.1}), (\ref{eq:thm5.3})}{\leq} \sum_{z} \bigl| I^{\psi(B)}_{\psi(x),z} \bigr| \label{eq:j5}\\
&\quad \stackrel{(\ref{eq:maxflow})}{\leq} c \sum_{z} \textup{ cap}(\psi(B))^{-1} \left| P_{z}^{{\mathbb Z}^d} [H_{\psi(B)}<\infty] - P^{{\mathbb Z}^d}_{\psi(x)}[H_{\psi(B)}<\infty] \right| \nonumber\\
&\quad \stackrel{(\ref{eq:heq})}{\leq} c(M) N^{1-d}, \quad \textrm{for } x \in S, \nonumber\end{aligned}$$ where we have also used the estimate on the Green function of [@lawler], Theorem 1.5.4, together with (\[eq:thm5.1\]), for the last line. The required redirecting flow $J$ will be constructed as the sum of two flows, $K$ and $L$, both of which satisfy the estimate (\[eq:j2\]). The purpose of $K$ is to redirect the charges $(\textup{div}I^*)1_{S}$, in such a way that the magnitude of the resulting charge at any given point is then bounded by $c(M)N^{-d}$, hence decreased by a factor of $N^{-1}$, cf. (\[eq:j5\]). Then, the flow $L$ will be used to distribute the resulting charges uniformly on $\mathbb T$. The existence of the flow $L$ will be a consequence of the following lemma (recall our convention concerning constants described at the end of the introduction and that $\nu$ denotes the uniform distribution on $\mathbb T$, cf. above (\[def:flow\])):
\[lem:f\] For any function $h: {\mathbb T} \to {\mathbb R}$, there is a flow $L^h \in F({\mathbb T})$, such that $$\begin{aligned}
&(\textup{div} L^h + h)(x) = \nu(h), \textrm{ for any } x \in {\mathbb T}, \textrm{ and} \label{f3} \\
&|L^h|_\infty \leq cN|h|_\infty. \label{f4}\end{aligned}$$
We construct the flow $L^h$ by induction on the dimension $d$, and therefore write ${\mathbb T}_d$ rather than $\mathbb T$ throughout this proof. Furthermore, we denote the elements of ${\mathbb T}_d$ using the coordinates of ${\mathbb T}'_d$ as $\{ (i_1, \ldots, i_d): 0 \leq i_j \leq N-1 \}$.
In order to treat the case $d=1$, define the flow $L^h$ by letting the charges defined by $h$ flow from $0$ to $N-1$, such that the same charge is left at any point. Precisely, we set $L^h_{N-1, 0} = 0$ and $L^h_{i,i+1} = \sum_{j=0}^i (\nu(h) - h(j))$ for $i = 0, \ldots, N-2$ (the values in the opposite directions being imposed by the condition (\[def:flow\]) on a flow). The flow $L^h$ then has the required properties (\[f3\]) and (\[f4\]).
Assume now that $d \geq 2$ and that the statement of the lemma holds in any dimension $<d$. Applying the one-dimensional case on every fiber $\{(0,y), \ldots, (N-1,y)\} \cong {\mathbb T}_1$, $y \in {\mathbb T}_{d-1}$, with the function $h^1$ defined by $h^1(.,y) = h(.,y)$, one obtains the flows $L^y$ supported by the edges of $\{(0,y), \ldots, (N-1,y)\}$, such that for any $i \in {\mathbb T}_1$, $$\begin{aligned}
&(\textup{div}L^y + h) (i,y) = N^{-1} \sum_{j=0}^{N-1} h(j,y) \textrm{ and} \label{eq:f7} \\
&|L^y|_\infty \leq cN|h|_\infty. \label{eq:f8}\end{aligned}$$ We now apply the induction hypothesis on the slices ${\mathbb S}_i = \{(i,y): y \in {\mathbb T}_{d-1}\} \cong {\mathbb T}_{d-1}$, $i \in {\mathbb T}_1$, with the function $h^2$ given by $h^2(i,.) = N^{-1} \sum_{j=0}^{N-1} h(j,.)$. For any $0 \leq i \leq N-1$, we thus obtain a flow $L^i$ supported by the edges of ${\mathbb S}_i$, such that for any $y \in {\mathbb T}_{d-1}$, $$\begin{aligned}
&\textup{div}L^i(i,y) + N^{-1} \sum_{j=0}^{N-1} h(j,y) = N^{-(d-1)} \sum_{y' \in {\mathbb T}_{d-1}} h^2(i,y') = \nu(h) \textrm{ and} \label{eq:f11} \\
&|L^i|_\infty \leq cN|h|_\infty. \label{eq:f12}\end{aligned}$$ Then equations (\[eq:f7\])-(\[eq:f12\]) imply that the flow $L^h = \sum_{i=0}^{N-1} L^i + \sum_{y \in {\mathbb T}_{d-1}} L^y $ has the required properties. Indeed, the flows $L^y$ have disjoint supports, as do the flows $L^i$, and therefore the estimate (\[f4\]) on $|L^h|_\infty$ follows from (\[eq:f8\]) and (\[eq:f12\]). Finally, for any $x=(i,y) \in {\mathbb T}_1 \times {\mathbb T}_{d-1} = {\mathbb T}_d$, (\[eq:f7\]) and (\[eq:f11\]) together yield $$\begin{aligned}
(\textup{div}L^h + h)(x) = \textup{div}L^i(i,y) + \textup{div}L^y (i,y) + h(i,y) = \nu(h),\end{aligned}$$ hence (\[f3\]). This concludes the proof of Lemma \[lem:f\].
We now complete the proof of Proposition \[pr:j\]. To this end, we construct the auxiliary flow $K$ described above Lemma \[lem:f\]. Set $g = (\textup{div} I^*) 1_S$. Writing $e_1, \ldots, e_d$ for the canonical basis of ${\mathbb R}^d$, choose a mapping $e: S \to \{\pm e_1, \ldots, \pm e_d\}$ such that ${\mathbb F}'_x \stackrel{(\textrm{def.})}{=} \{\psi(x), \psi(x) + e(x), \ldots, \psi(x)+(N-1)e(x) \} \subseteq {\mathbb T}'$ (whenever there are more than one possible choices for $e(x)$, take one among them arbitrarily), and define the fiber ${\mathbb F}_x = \psi^{-1} ({\mathbb F}'_x)$.
Observe that any point $x \in \mathbb T$ only belongs to the $d$ different fibers $x + [0,N-1] e_i$, $i=1,\ldots,d$. Moreover, we claim that for any ${\mathbb F} \in \{{\mathbb F}_x\}_{x \in S}$, there are at most $2$ points $x \in S$ such that ${\mathbb F}_x = {\mathbb F}$. Indeed, suppose that ${\mathbb F}_x = {\mathbb F}_{x'}$ for $x, x' \in S$. Then $\psi({\mathbb F}_x) = \psi({\mathbb F}_{x'})$ implies that $\psi(x') = \psi(x) + ke(x)$ for some $k \in \{0, \ldots, N-1\}$ and that either $e(x) = e(x')$ or $e(x) = -e(x')$. If $e(x) = e(x')$, then for $\psi({\mathbb F}_{x'}) = \{ \psi(x) + ke(x), \psi(x) + (k+1)e(x), \ldots, \psi(x) + (k+N-1)e(x) \}$ to be a subset of ${\mathbb T}'$, we require $k=0$ (since $\psi(x) + Ne(x) \notin {\mathbb T}'$). Similarly, if $e(x) = -e(x')$ one needs $k=N-1$. Hence, $x'$ can only be equal to either $x$ or $x+(N-1)e(x)$. The above two observations on the fibers ${\mathbb F}_x$ together imply the crucial fact that any point in $\mathbb T$ belongs to a fiber ${\mathbb F}_x$ for at most $2d$ points $x \in S$.
We then define the flow $K^x$ from $x$ to $x+(N-1)e(x)$ distributing the charge $g(x)$ uniformly on the fiber ${\mathbb F}_{x}$. That is, the flow $K^x \in F({\mathbb T})$ is supported by the edges of ${\mathbb F}_{x}$, and characterized by $K^x_{x +(N-1)e(x), x} = 0$, $K^x_{x+i e(x), x+(i+1)e(x)} = -g(x) (N-(i+1))/N$ for $i = 0, \ldots, N-2$. Observe that then $|K^x|_\infty \leq |g|_\infty$ and $|\textup{div} K^x + g1_{\{x\}}|_\infty = |g(x)|/N \leq |g|_\infty/N$. Moreover, any point in $\mathbb T$ belongs to at most $2d$ fibers ${\mathbb F}_{x}$, hence to the support of at most $2d$ flows $K^x$. If we define the flow $K \in F({\mathbb T})$ as $K = \sum_{x \in S} K^x,$ then we therefore have $$\begin{aligned}
|K|_\infty \leq c \max_{x \in S} |K^x|_\infty \leq c |g|_\infty, \label{eq:f1}\end{aligned}$$ as well as, for $x \in {\mathbb T}$, $$\begin{aligned}
|(\textup{div} K + g)(x)| & \leq \sum_{x' \in S} |(\textup{div} K^{x'} + g1_{\{x'\}})(x)| \label{eq:f2} \\
& \leq |\textup{div} K^x + g1_{\{x\}}|_\infty 1_{S}(x) + \sum_{x' \neq x: x \in {\mathbb F}_{x'}} |\textup{div} K^{x'}(x)| \nonumber\\
& \leq c |g|_\infty/N. \nonumber\end{aligned}$$ We claim that the flow $J = K+L^{\textup{div} K +g}$ has the required properties (\[eq:j1\]) and (\[eq:j2\]). Indeed, using the fact that $\nu (\textup{div} I) = 0$ for any flow $I \in F({\mathbb T})$, $$\begin{aligned}
\textup{div} J + g &= \textup{div} L^{\textup{div} K +g} + \textup{div} K + g \\
&\stackrel{(\ref{f3})}{=} \nu (\textup{div} K + g) = \nu((\textup{div} I^*) 1_S) = - \nu( (\textup{div} I^*) 1_{C} ) \\
&\stackrel{(\ref{eq:thm6.0})}{=} -N^{-d} \sum_{z \in \textup{int} {\mathbb T}'} \textup{div} I^{\psi(B)}_z \stackrel{(\ref{eq:thm5.3})}{=} -N^{-d} I^{\psi(B)}(\psi(B)) \stackrel{(\ref{eq:thm5.2})}{=} -N^{-d}.\end{aligned}$$ Finally, the estimates (\[f4\]), (\[eq:f1\]) and (\[eq:f2\]) imply that $$\begin{aligned}
|J|_\infty \leq |K|_\infty + |L^{\textup{div} K +g}|_\infty \leq c|g|_\infty \stackrel{(\ref{eq:j5})}{\leq} c(M) N^{1-d}.\end{aligned}$$ The proof of Proposition \[pr:j\] is thus complete.
[99]{}
D.J. Aldous. *Probability Approximations via the Poisson Clumping Heuristic.* Springer-Verlag, 1989.
D.J. Aldous and M. Brown. Inequalities for rare events in time-reversible Markov chains I. *Stochastic Inequalities.* M. Shaked and Y.L. Tong, ed., IMS Lecture Notes in Statistics, volume 22, 1992.
D.J. Aldous and J. Fill. *Reversible Markov chains and random walks on graphs.* http://www.stat.Berkeley.EDV/users/aldous/book.html.
I. Benjamini, A.S. Sznitman. Giant component and vacant set for random walk on a discrete torus. *J. Eur. Math. Soc. (JEMS)*, 10(1):133-172, 2008.
R. Durrett. *Probability: Theory and Examples.* (third edition) Brooks/Cole, Belmont, 2005.
G.F. Lawler. *Intersections of random walks.* Birkhäuser, Basel, 1991.
L. Saloff-Coste. *Lectures on finite Markov chains*, volume 1665. Ecole d’Eté de Probabilités de Saint Flour, P. Bernard, ed., Lecture Notes in Mathematics, Springer, Berlin, 1997
P.M. Soardi. *Potential Theory on Infinite Networks.* Springer-Verlag, Berlin, Heidelberg, New York, 1994.
A.S. Sznitman. Vacant set of random interlacements and percolation, preprint, available at http://www.math.ethz.ch/u/sznitman/preprints and http://arxiv.org/abs/0704.2560.
A.S. Sznitman. Random walks on discrete cylinders and random interlacements, preprint, available at http://www.math.ethz.ch/u/sznitman/preprints.
|
---
abstract: 'Quantum many-body systems realise many different phases of matter characterised by their exotic emergent phenomena. While some simple versions of these properties can occur in systems of free fermions, their occurrence generally implies that the physics is dictated by an interacting Hamiltonian. The interaction distance has been successfully used to quantify the effect of interactions in a variety of states of matter via the entanglement spectrum \[Nat. Commun. [**8**]{}, 14926 (2017), arXiv:1705.09983\]. The computation of the interaction distance reduces to a global optimisation problem whose goal is to search for the free-fermion entanglement spectrum closest to the given entanglement spectrum. In this work, we employ techniques from machine learning in order to perform this same task. In a supervised learning setting, we use labelled data obtained by computing the interaction distance and predict its value via linear regression. Moving to a semi-supervised setting, we train an auto-encoder to estimate an alternative measure to the interaction distance, and we show that it behaves in a similar manner.'
author:
- Samuel Spillard
- 'Christopher J. Turner'
- Konstantinos Meichanetzidis
title: |
Machine Learning Entanglement Freedom Or:\
How I Learned to Stop Worrying and Love Linear Regression
---
Introduction
============
The interaction distance is a diagnostic measure of a pure state’s non-Gaussianity as it manifests in its entanglement structure. In essence, it performs pattern recognition on entanglement spectra, where the pattern is dictated by the Fermi-Dirac statistics obeyed by free fermions. When applied to ground states of quantum many-body systems, the results can be counter-intuitive and surprising, even for well known and extensively studied systems, such as the quantum Ising chain [@Schultz], parafermion chains [@FradkinKadanoff; @Fendley:2012hw], and string-net models [@WenBook].
Computing $D_\mathcal{F}$ involves a non-convex optimisation. Even if it is *in principle efficient* [@Turner2017], its computation runtime still remains impractical for entanglement spectra which are accessible efficiently with current numerical methods [@Verstraete]. Then a natural question arises; can we train a model to *predict* the interaction distance so that one does not need to perform a global optimisation for every input entanglement spectrum. In principle, the answer is affirmative. However, the caveat is that the complexity of the training set is too high if we want it to work accurately for *any* spectrum. With this motivation, we approach case studies found in previous works from a machine learning perspective. As we focus on particular models, the learning model and the training set can be chosen to be simple, at the cost of generality. We find that in these special cases, the machine learning methods reproduce results for the interaction distance with a noticeable performance improvement.
The Problem of Computing the Interaction Distance {#IntDist}
=================================================
Entanglement Spectrum
---------------------
The entanglement spectrum is defined as the spectrum of a mixed state $\rho$ obtained after biparitioning the domain of a pure state $\ket{\psi}$ into regions $R,\bar R$ and performing a partial trace over $\bar R$. We denote the eigenvalues of $\rho$ by $P_k \in [0,1]$, as it corresponds to a probability distribution $P$. We also recall the definition of the entanglement energies [@Haldane08] as ${E_k} = -\log{P_k} \in [0,\infty)$. An equivalent definition of the entanglement spectrum invokes the Schmidt decomposition of a pure state $\psi = \sum_k \xi_k \ket{\psi}^R_k \ket{\psi}^{\bar R}_k$ onto independent orthonormal bases supported on each complementary region. Then we have that $P_k = \xi_k^2$.
Interaction Distance
--------------------
To quantify the non-Gaussianity of a mixed state $\rho$, we define the interaction distance [@Turner2017] as $D_\mathcal{F}(\rho) = \min_{\sigma\in\mathcal{F}} D(\rho,\sigma)$. This is the minimal trace distance, $D(\rho,\sigma)$, between $\rho$ and the manifold ${\cal F}$, which contains all free-fermion reduced density matrices, $\sigma$. In other words, the interaction distance quantifies the non-applicability of Wick’s theorem for the state $\rho$.
It was proven [@Turner2017] that since relative rotations between matrices can only increase the trace distance, the interaction distance can be expressed exclusively in terms of the spectra of those matrices, i.e. the entanglement spectra. In other words, the trace distance reduces to the $1$-norm $D(X,Y)=\sum_i \frac{1}{2}\left| X_i-Y_i \right|$, with $0 \leq D(X,Y)\leq 1$, for probability distributions $X,Y$.
For the purposes of this work, it is useful to formulate the interaction distance in two equivalent ways, distinguished by the space in which the minimisation takes place, i.e. probability- versus energy-space, \[eq:DF\] D\_(P)&=&\_[s]{} \_k | P\_k - P\^\_k (s) | ,\
D\_(E)&=&\_ \_k | e\^[-E\_k]{} - e\^[-E\_k\^()]{} | , where $\mathrm{f}$ indicates a spectrum with a free fermion structure generated by the *polynomially large* single-body sets $s$ and $\epsilon$, to be defined in Eq., and the spectra are rank ordered, $E_k\leq E_k+1$ and $E^\mathrm{f}_k\leq E^\mathrm{f}_k+1$ or $P_k\geq P_k+1$ and $P^\mathrm{f}_k\geq P^\mathrm{f}_k+1$.
Intuitively, $D_\mathcal{F}$ is dominated by the low-lying part of the entanglement Energy spectum and it reveals the correlations between the effective quasiparticles emerging from interactions [@Haldane08]. Hence, $D_\mathcal{F}$ is expected to be stable under perturbations that do not cause phase transitions [@Turner2017].
Interaction Distance as an Inverse Problem {#sec:expand}
------------------------------------------
In order to study $D_\mathcal{F}$ with machine learning methods, we formulate its estimation as an inverse problem. A free fermion spectrum is defined to be one that is created by the expand map $\mathcal{E}$, which takes as input a set of single-body probabilities or energies and outputs a probability or energy spectrum which obeys the free fermion structure, [@Haldane08; @PeschelEisler] \_ &:& s\^N\_[<]{} P\^ (s)\^[2\^N]{}\_[>]{},\
\_ &:& \^N\_[<]{} E\^() \^[2\^N]{}\_[<]{} . The free-fermion spectra in probability space and entanglement energy space have the forms \[eq:freespec\] P\^(s)&=&\_ \_[i=1]{}\^N (+s\_i,-s\_i)\
E\^()&=&\_ E\_0+\_[i=1]{}\^N {0,\_i} with $0\leq s_i\leq\frac{1}{2}$, and $E_0=-\sum_i \log Z_i$ with $Z_i=1+e^{-\epsilon_i}$. Equivalently we can write $P^\mathrm{f}=\mathrm{sort}_{\mathrm{desc}}~\otimes_{i=1}^N \frac{1}{Z_i}(1,p_i)$ with $p_i=e^{-\epsilon_i}$. However, the $s$-parametrisation of free-fermion probability spectra in Eq. produces inherently normalised $P^\mathrm{f}$ spectra and makes convenient the computation of $D_\mathcal{F}(P)$.
The solution to the problem of computing $D_\mathcal{F}$ amounts to finding the weak inverse of the expand map, which minimises the trance distance for input outside the image of expand. We denote this generalised inverse as $\mathcal{E}^g$. If such an inverse exists it has the properties \^g \^g &=& \^g ,\
\^g &= &. From this perspective, we can write Eq. as \[eq:DF\_expand\] D\_ (P) &=& \_[\^g\_P ]{} || \_P ( \_P\^g(P)) - P ||\_1\
D\_ (E) &=& \_[\^g\_E ]{} || e\^[- \_P ( \_E\^g(E)) ]{} - e\^[- E]{} ||\_1 .
AutoEncoder Perspective {#sec:AEPerspective}
-----------------------
Ideally, one would train a deep neural network, for example an AutoEncoder (AE) [@Hinton504] that learns the entanglement freedom structure from a dataset of free states. The map $\mathcal{E}$ is in general non-linear in the single-body input set. In an autoencoder setup, we have the Coding process $\mathcal{C}$, which maps an input $X$ to the latent layer, and the Decoding $\mathcal{D}$ process which maps from the latent layer to the output, \[eq:autoencoder\] X (X) ((X)). The goal of the training step for the AE is then to minimize $D(\mathcal{D}(\mathcal{C}(X)),X)$ for all $X$, where $X$ are free states, by updating $\mathcal{C}$ and $\mathcal{D}$ appropriately. Thus, $\mathcal{C}$ would learn to implement $\mathcal{E}^g$ and $\mathcal{D}$ would learn $\mathcal{E}$. Then for a new input spectrum $Y$, the surrogate for the interaction distance is defined as \[eq:DF\_AE\] D\^\_=D(((Y)),Y). Intuitively, the interaction distance in this case measures how wrong an autoencoder is when it recognises freedom in a spectrum it has not encountered during training. However, obtaining a representative training set of all free states $X\in\mathcal{F}$ is a hard task. In the next section we do some elementary data analysis on the classes of spectra we consider and in the following sections we employ simple regression methods that estimate the interaction distance in specific contexts.
PCA of spectra
==============
Here we present a data analysis of entanglement spectra we consider throughout this work. Collecting all such spectra of a certain size into a matrix and performing PCA, we represent this dataset in the space spanned by the first three principal vectors. In Fig.\[fig:PCAspectra\] we see how spectra with different properties are clustered, and we observe that it is reasonable to expect that a freedom classifier can be defined.
Linear Approaches for Specific Case Studies
===========================================
We now focus our study to specific sets of entanglement spectra arising from condensed matter systems of interest to the community, such as the the paradigmatic quantum Ising chain and Abelian topological models. To make progress, we consider reducing our learning model’s complexity and thus its generality for the sake of lowering the complexity of the training set needed. To this end, we make a linear approximation in the inverse problem of computing the interaction distance by viewing the expand map as a linear transformation, \[eq:linearexpand\] \_X \~\_X , whose form is specified by the problem at hand.
Random States {#sec:randomstates}
-------------
We begin by considering random entanglement spectra, which are generated as described in Appendix \[app:spectracreation\]. The linear approximation to the expand map is done by ignoring the distinction between sets and ordered sets (multisets) in the entanglement energy space, \[eq:linearexpand\] \_[E]{} : \^N E\^()\^[2\^N]{}. We purposefully choose to work in $E$-space, as the linear approximation is not viable in $P$-space. In the parameterisation of $\mathcal{F}$ by the single-particle energies, we may take linear combinations of them and up to reorderings get another single-particle spectrum. For the many-body spectrum, which is the expansion of some single-body energy levels, the expansion assigns an occupation pattern to each many-body level. If you restrict to the subset of single-particle energies that expand to spectra with the same ordering of these occupation patterns with respect to the energy ordering, then expand is a linear map and accordingly has a linear weak inverse. Geometrically, in the energy parameterisation the manifold $\mathcal{F}$ is piecewise linear, whereas in terms of the energy variables, $\mathcal{F}$ is locally a hyperplane. This is our main justification for using the energy-space parametrisation for the linear regression problem. For the probabilities, on the other hand, the manifold $\mathcal{F}$ is curved. Thus, attempts to contain it within a hyperplane, or to find a hyperplane contained within it are likely to form gross approximations.
The least-squares linear regression in energy-space has a number of flaws. Firstly, as discussed the model can’t capture the ordering structure. Secondly, the least squares cost function weights all deviations in energy equally. Howerver, for the trace distance cost function, large variations in high entanglement energy levels, which are highly penalised by the least-squares solution, would be an insignificant variation in probability. Similarly, a small energy variation in the low lying energy levels would be more important.
Since now $\slashed{\mathcal{E}}_{E}$ is a linear transformation, it has a $ 2^N \times N $ matrix representation with columns carrying bit-strings corresponding to occupation patterns labelling the Fock basis states on the independent fermionic modes. This matrix acts on a column vector containing the single-body energies $\epsilon_i$, $i=1,\dots,N$ and results in a column vector containing the many-body energies $E_k$, $k=1,\dots,2^N$.
The map $\slashed{\mathcal{E}}_E$ is linear and has full column-rank, it therefore has a Moore-Penrose pseudoinverse with the property $\slashed{\mathcal{E}}_{E}^g\slashed{\mathcal{E}}_{E}=\1$. We then apply linear regression to infer a design matrix $F$, which is identified with the linear map $\slashed{\mathcal{E}}_{E}^g$, such that $\epsilon = F E + \delta$. We use the least squares method so that $\left|\left|\epsilon - F E \right|\right|_2 =\delta^2$ is minimised. Here the set $\epsilon$ contains the single-body energies that are computed by our algorithm [@Turner2017] for $D_\mathcal{F}$ for the entanglement spectra $E$ from random states, $\epsilon=\mathrm{argmin} D(E,E^\mathrm{f}(\epsilon))$. The algorithm first builds an initial guess for the free spectrum by examining the input spectrum. Then a local optimisation is performed, for example via the Nelder-Mead algorithm. Finally, a basin-hopping Monte Carlo technique can be introduced to ensure that enough local minima are visited and that the global one is reached with high probability. The number of basins is a parameter left free. Here, the basin-hopping is turned off and the minimisation is local, with the initial guess constructed as described in Ref . This approach can thus be called supervised, as $D_\mathcal{F}$ is the label of each $E$. The matrix $F$ is expected to implement $\mathcal{E}^g$, i.e. the linear approximation to the weak inverse of the expand map. Then the estimated interaction distance is $D_\mathcal{F}^\mathrm{est}(E) = D( e^{-E}, e^{-E^\mathrm{f}( FE )} )$.
The distribution of $D_\mathcal{F}$ over random states which is fitted well by a log-normal distribution, $P_\text{LN}(X,a,b)=\frac{1}{ x b \sqrt{2\pi} } e^{- \frac{(\ln{X}-a)^2}{2 b^2}}$, and the accuracy of this linear method are shown in Fig.\[fig:LR\_RandomStates\]. We observe that the linear regression estimation is able to perform unexpectedly better than the local minimisation version of our current algorithm. Note that direct computation of $D_\mathcal{F}$ corresponds to a global optimisation problem over polynomially many parameters in an exponentially large space. On the contrary, a least squares regression with $T$ training points, each with its own label, is asymptotically $\mathcal{O}(T^{3})$ in complexity. However, in order to perform the regression, a training set of the random entanglement spectra of size $2^N$, needs to be created as described in Appendix \[app:spectracreation\]. The complexity of this is $\mathcal{O} ( {2^{3N}} )$. Labelling the training points requires performing the minimisation in Ref whose runtime scales as the time required to perform a local optimisation over the $\mathcal{O}(N)$ optimisation parameters multiplied with the number of basins visited. After training, the prediction of $D_\mathcal{F}$ corresponds to matrix multiplication, whose complexity is cubic in the matrix dimension.
![ (Top) $P(D_\mathcal{F})$ over $5\cdot 10^3$ states for $q={8}$ and $\psi_j$ drawn from the power-law distribution $\delta x^{-\delta}$. Amplitudes sampled as $\psi_j\in\mathbb{R}$ (Left) and $\psi_j\in\mathbb{C}$ (Right), and ${P(D_\mathcal{F})}$ is fitted with ${P_\text{LN}}$, with dots corresponding to ${\delta=2}$ (blue) and ${\delta=20}$ (red). (Middle) $P(D_\mathcal{F})$ for $10^3$ states on $q=10$ qubits with $\psi_j\in\mathbb{R}$ (Left) and $\psi_j\in\mathbb{C}$ (Right) drawn from $\mathcal{N}(0,1)$. Again, $P(D_\mathcal{F})$ is fitted well with ${P_\text{LN}}$. The most common value for the interaction distance is comparable in all cases. (Bottom) Linear regression accuracy $\delta D^\mathrm{est}_\mathcal{F}=\left| D^\mathrm{est}_\mathcal{F}-D_\mathcal{F}\right|$ as the difference of its prediction from direct computation of the interaction distance. []{data-label="fig:LR_RandomStates"}](PDFvsDF "fig:"){width="\columnwidth"} ![ (Top) $P(D_\mathcal{F})$ over $5\cdot 10^3$ states for $q={8}$ and $\psi_j$ drawn from the power-law distribution $\delta x^{-\delta}$. Amplitudes sampled as $\psi_j\in\mathbb{R}$ (Left) and $\psi_j\in\mathbb{C}$ (Right), and ${P(D_\mathcal{F})}$ is fitted with ${P_\text{LN}}$, with dots corresponding to ${\delta=2}$ (blue) and ${\delta=20}$ (red). (Middle) $P(D_\mathcal{F})$ for $10^3$ states on $q=10$ qubits with $\psi_j\in\mathbb{R}$ (Left) and $\psi_j\in\mathbb{C}$ (Right) drawn from $\mathcal{N}(0,1)$. Again, $P(D_\mathcal{F})$ is fitted well with ${P_\text{LN}}$. The most common value for the interaction distance is comparable in all cases. (Bottom) Linear regression accuracy $\delta D^\mathrm{est}_\mathcal{F}=\left| D^\mathrm{est}_\mathcal{F}-D_\mathcal{F}\right|$ as the difference of its prediction from direct computation of the interaction distance. []{data-label="fig:LR_RandomStates"}](LinearRegressionRandomStates_edited "fig:"){width=".87\columnwidth"}
Abelian Topological States {#sec:AbelianTopoStates}
--------------------------
We now turn to topological models at their renormalisation flow fixed point. In Ref. , parafermionic chains (1D) and abelian string-nets ($2$ or higher D) are studied due to the specific structure of their ground state entanglement. Before continuing, we reiterate these results, as they are important for motivating the linear learning model we introduce.
### Optimal Free Spectra for Flat Spectra
The entanglement spectra obtained from such states comprise of only one eigenvalue with some degeneracy dictated by the instance of the model. In particular, for $\mathbb{Z}_N$ parafermions, the entanglement spectrum from an equipartition of the chain in its topological phase consists of an $N$-fold degenerate eigenvalue [@Fendley:2012hw], equal to $\frac{1}{N}$, where we denote with overbar such flat spectra, $\bar P_N$. For $\mathbb{Z}_N$ Abelian string-nets, the flat spectrum arising from bipartitioning the system into a connected region and its complement has degeneracy $N^{|\partial|-1}$, where $|\partial|$ is the length of the partition boundary [@Alex]. Thus determining the interaction distance reduces to determining the interaction distance for a parafermion case of the appropriate order.
It is conjectured that the free-fermion spectrum $P^\mathrm{f}_N$ closest to a flat spectrum $\bar P_N$ is constructed by exactly reproducing as many of the highest probability eigenvalues of the flat spectrum as possible. First, we pad $\bar P_N$ with zeros so that its size is equal to $2^{n+1}$, where $n$ is the greatest integer such that $2^n \le N$, so that the two spectra can be compared since in general their rank can be different. Padding with zeros does not alter the entanglement and can be understood as introducing redundant unentangled degrees of freedom [@Turner2017]. The most that can be reproduced exactly are $2^n$, Then there exists one non-trivial fermion mode whose gap is fixed by normalisation. In particular, in terms of Eq., for $n$ of the modes we set $s_i=0$ and one mode acquires gap $s_{n+1} = N^{-1}2^n - 1 / 2$. Then, the optimal free fermion spectrum for an $N$-fold degenerate spectrum $$\label{eq:optimalguess}
P^\mathrm{f}_N = \left(N^{-1},\dots,N^{-1},p,\dots,p \right),$$ where there are $2^n$ entries for each value $N^{-1}$, with $p = 2^{-n} - N^{-1}$ such that $\sum_k {P^\mathrm{f}_N}_k=1$.
Then evaluating $D_\mathcal{F}$ for such a choice of free spectra is straightforward. There are two contributions. The first is from the entanglement levels with index $2^n + 1 \le k \le N$ for which the probability difference is between $N^{-1}$ and $p$.The second is from levels with index $N + 1 \le k \le 2^{n+1}$ for which the probability difference is between $0$ and $p$.Thus we obtain $$D_\mathcal{F}(\bar P_N) \le 3 - \frac{N}{2^n} - \frac{2^{n+1}}{N}.
\label{eq:upperbound}$$ This result constitutes an upper bound, since the construction of the free spectrum is a conjecture. However, numerical evidence partly shown in Fig.\[fig:hills\] and fully supported in Ref. supports that the equality holds and thus we will assume this is the case.
The interpretation of the minimum and maximum value of $D_\mathcal{F}(\bar P_N)$ is as follows. In the trivial case where $N=2^n$ for some $n\in\mathbb{N}$, then this flat spectrum can be reproduced by $n$-many gapless fermion modes, $\bar P_{2^n}=P^\mathrm{f}_{2^n}$, and it is free with $D_\mathcal{F}=0$. This result has important implications in studying free-fermion parent Hamiltonians of these topological models [@KonMei18]. By analytical continuation of $\mathbb{N}$ we set $N \rightarrow \alpha 2^n$ with $\alpha\in[1,2]$ such that we densely cover the interval between consecutive powers of two between which $N$ lies. Then, maximising Eq. we find $D_\mathcal{F}^\text{max}=3 - 2\sqrt{2}$ with $\mathrm{argmax} \left( D_\mathcal{F}(\bar P{(\alpha)}) \right) = \sqrt{2}$. The fact that the maximum occurs at an irrational value means that no flat spectrum can instantiate the maximal value of $D_\mathcal{F}$. However, it can be approximated arbitrarily by the appropriate choice of $N$. By the exhaustive numerical maximisation $\max_{P} D_\mathcal{F}(P)$ for random spectra of size up to $2^8$ we have not found states with interaction distance larger than $D_\mathcal{F}^\text{max}$. Hence, this appears to be the maximum possible value of the interaction distance *for any spectrum*.
### Supervised Linear AutoEncoder
We now turn to supervised linear method inspired by autoencoders for learning optimal free spectra $P^\mathrm{f}_N$ for flat spectra $\bar P_N$. Again, we make a linear approximation to the expand map as in Eq.. First, we define a linear regression problem which qualifies as a supervised learning protocol. We fix an orthonormal basis $e_n$, with $n=1,\dots,n_\mathrm{max}$, viewed as carrying labels of *free* flat spectra $\bar P_{2^n}$ all of which are padded with zeros so that all of them are of size $2^{n_\mathrm{max}}$ and sorted in descending order. Thus, we can consider a design matrix $F=(\bar P_{2^1},\bar P_{2^2},\dots,\bar P_{2^{n_\mathrm{max}}})$ for which $\left|\left| e_n - F \bar P_{2^n} \right|\right|_2 =0$. This matrix is full column rank and its Moore-Penrose pseudoinverse ${F}^g$ is the solution of this linear regression problem. This problem can be viewed as one of Independent Component Analysis [@ICA], where the uncorrelated sources that need to be discerned correspond to the single-particle levels and the mixing matrix corresponds to the expand map.
We then define a supervised linear autoencoder (SLAE) as a modification of Eq., where we fix a basis $e_n$ for the latent layer $\mathcal{C}(X)$. Furthermore, the Coding and Decoding processes correspond to solutions of a linear regression problem $F X = e_n$, whose design matrix effectively implements $\mathcal{C}=F$ and $\mathcal{D}=F^g$. In this respect, the corresponding interaction distance for an entanglement spectrum $Y$ from a test set is defined as \[eq:DF\_SLAE\] D\^\_=D(F\^g F Y,Y).
For the particular problem of computing $D_\mathcal{F}(\bar P_{N})$, the matrix $F$ contains free flat spectra $\bar P_{2^n}$ and is identified with the Moore-Penrose inverse of the linear version of the expand map, $\slashed{\mathcal{E}}^g$. Note that $F$ here is full column rank. Then the SLAE predicts the interaction distance for all $N$-rank flat spectra $\bar P_N$ as $D^{SLAE}_\mathcal{F} (\bar P_N)= D( \mathcal{D}\mathcal{C} \bar P_N , \bar P_N )$, shown in Fig. \[fig:hills\]. The fact that $D^{SLAE}_\mathcal{F}=D_\mathcal{F}$ means the guess for the optimal free spectrum corresponding to a flat spectrum of Eq. can be viewed as a linear combination of free flat spectra. In Appendix \[sec:SLAEFuzzyFlat\] we show that the prediction is robust for almost flat spectra. Since the training set for this specific case (with $n_{\mathrm{max}} = 10$) is limited to 10 basis states, the calculation of the design matrix and its pseudoinverse is almost instantaneous. A massive speedup from the general $D_{\mathcal{F}}$ algorithm which takes $\sim$3 hours for $n_{\mathrm{max}} = 10$ and $100$ basins visited for each $\bar P (N)$.
Quantum Ising Chain {#sec:IsingSLAE}
-------------------
Finally, we relate the extensive results of Ref. on the interaction distance for ground states of the antiferromagnetic (AFM) quantum Ising chain with our supervised linear autoencoder method. The Hamitlonian $H(h_z,h_x)$, explicitly written in Eq. of Appendix \[sec:IsingApp\], has a parameter line $h_x=0$ on which the model $H(h_z,0)$ maps to a free-fermions with a quantum critical point at $h_z=1$.
By equipartitioning the chain, we obtain the entanglement spectrum from the ground state. The ground state is obtained by exact diagonalisation. One can employ matrix-product states to access larger chain lengths. We denote the entanglement probability spectra obtained by equipartitioning the chain in its ground state as $P(h_z,h_x)$, with corresponding entanglement energies $E(h_z,h_x)=-\log{P(h_z,h_x)}$. On the free line we have $D_\mathcal{F}(P(h_z,0))=0$, $\forall h_z$. In Ref. it is demonstrated by a detailed scaling analysis that in the thermodynamic limit the model is almost free everywhere in its phase diagram. For a finite system size we have finite $D_\mathcal{F}$ on the critical line. Finally, we map each $P(h_z,h_x)$ on the phase diagram to the free line by minimising its distance from that line, $\min_{h_z^\mathrm{f}}D(P(h^\mathrm{f}_z,0),P(h_z,h_x))$. The field value $h^\mathrm{f}_z$ for which the minimum occurs characterises isofree spectra.
We implement the SLAE, as it is defined in Eq., in order to quantify the distance between any $P(h_z,h_x)$ spectrum from the closest spectrum $P(h^\mathrm{f}_z,0)$ on the free line. The size $n_\mathrm{max}$ of the basis $e_n$ for the latent layer is in this case a parameter we must determine. This is done by analysing how well the entire free line is reproduced by each size of basis by minimising the absolute error of the free line over $n_\mathrm{max}$ and adding a penalty for increased $n_\mathrm{max}$ as $\min_{n_\mathrm{max}} \sum_{n=1}^{n_\mathrm{\max}} D(\mathcal{D}\mathcal{C}P(h_z(n),0), P(h_z(n),0)) + \alpha n $. Where $\alpha$ is a constant of the order of the absolute error. The basis elements correspond to labels for free spectra $P(h_z (n),0)$ that are chosen as representatives for the free line for $n$-many values of $h_z$. These free spectra then enter the columns of the design matrix of the linear regression problem of the SLAE. Then, the predicted distance from the free line is $D^{SLAE} (P(h_z,h_x))= D( \mathcal{D}\mathcal{C} P(h_z,h_x) , P(h_z,h_x) )$. In Fig. \[fig:SLAE\_Ising\] we demonstrate that the results correspond to the equivalent results of Ref. .
Autoencoder
===========
Finally, we present an alternative method of producing results equivalent to those obtained with SLAE for the flat spectra and the AFM Quantum Ising Chain in Sections \[sec:AbelianTopoStates\] and \[sec:IsingSLAE\]. The AutoEncoder (AE) in this case is semi-supervised and is trained on free spectra relevant to the particular problem at hand. Following the discussion in Section \[sec:randomstates\] on the different behaviour of a linear method on the probability and energy spaces, we are motivated by the fact that a promising proposal for a neural network that would outperfrom the linear approaches we take here should be designed to overcome these limitations. It should both capture the geometry of $\mathcal{F}$ and have an appropriate surrogate for the trace distance cost function. It should also naturally operate on multisets such that it is not confused by reorderings in the dataset.
Our autoencoder is built in TensorFlow and consists of an input/output of $\mathcal{I} = 2^N$ neurons, with three hidden layers, $h_{1},h_{2},h_{3}$. The latent vector, $h_2$, is chosen to have size $\mathcal{L} = N$, such that a free system can be fully described by these neurons. Various activation functions were tested on all layers, however the best results were found when only a softmax [@bishop2006pattern] activation function was applied to the output layer. This has the benefit of enforcing normalisation of the output.
In the case of flat spectra, obtained from Abelian topological states, the AE is trained on flat and almost flat spectra as they are defined in Appendix \[sec:SLAEFuzzyFlat\], that is flat spectra with disordered eigenvalues. This is done to increase the training set’s size, and for low disorder amplitudes it is expected to not affect the performance of the AE. We observe in Fig.\[fig:AE\_Hills\_AFM\] (top) that we reproduce a log-periodic function for the interaction distance. The shape of the curve need not be that of Eq. .
For the quantum Ising Chain, entanglement spectra $P^\Delta(h_z,0)$ are sampled from the free line $h_x=0$ and comprise the training set. The sample size is increased by introducing disorder of amplitude $\Delta$ in the couplings of the chain as described in Appendix \[sec:IsingApp\]. For this problem, the network was seen to overfit to the free line and therefore identifies the majority of the phase diagram as strongly interacting. To battle this, dropout regularization [@Dropout] was applied to $h_3$, with a dropout probability of $P_\mathrm{D} = 0.5$. This significantly improved the fitting of our AE to the majority of the phase diagram, which is shown in Fig. \[fig:AE\_Hills\_AFM\] (bottom).
The goal of the training is for the AE to effectively learn the map $\mathcal{E}\mathcal{E}^g$ as described in Sec. \[sec:AEPerspective\], from the dataset of free states we provide for the specific case studies at hand. Note that in contrast to the SLAE, we do not fix the basis of the vector space corresponding to the latent layer of the AE; the AE learns an input-output relation by trying to reproduce the input it is given as the output of a compression-decompression process implemented by a deep neural network. Limiting the number of neurons in the hidden layer of the neural network and forcing a dimensionality reduction forces the network to learn *features* of the training data that are robust to variation.
![ (Top) Autoencoder estimation $D^\mathrm{AE}_\mathcal{F}(\bar P_N)$ trained on almost flat spectra $P^\Delta_N$ (see Appendix \[sec:SLAEFuzzyFlat\]). We obtain a log-periodic curve which is viewed as a surrogate for $D_\mathcal{F}(\bar P_N)$. Here $\mathcal{I}=1024$ and $\mathcal{L}=10$. (Bottom) $D^\mathrm{AE}_\mathcal{F}(P(h_z,h_x))$ for AFM Ising, trained on the free spectra $P^\Delta(h_z,0t)$ with $\Delta=0.1$. Here, chain length is $L=10$, and we sampled $10^4$ values of $h_z$. Here $\mathcal{I}=32$ and $\mathcal{L}=5$.[]{data-label="fig:AE_Hills_AFM"}](AE_Hills "fig:"){width=".88\columnwidth"} ![ (Top) Autoencoder estimation $D^\mathrm{AE}_\mathcal{F}(\bar P_N)$ trained on almost flat spectra $P^\Delta_N$ (see Appendix \[sec:SLAEFuzzyFlat\]). We obtain a log-periodic curve which is viewed as a surrogate for $D_\mathcal{F}(\bar P_N)$. Here $\mathcal{I}=1024$ and $\mathcal{L}=10$. (Bottom) $D^\mathrm{AE}_\mathcal{F}(P(h_z,h_x))$ for AFM Ising, trained on the free spectra $P^\Delta(h_z,0t)$ with $\Delta=0.1$. Here, chain length is $L=10$, and we sampled $10^4$ values of $h_z$. Here $\mathcal{I}=32$ and $\mathcal{L}=5$.[]{data-label="fig:AE_Hills_AFM"}](AE_AFM-crop "fig:"){width=".9\columnwidth"}
Conclusions and Outlook
=======================
We have applied machine learning techniques to compute the interaction distance. By choosing to do so, we exchange generality with performance. In particular, the algorithm presented in Ref. for $D_\mathcal{F}$ can be applied to any spectrum. Our linear methods estimate $D_\mathcal{F}$ considerably faster but with the loss of generality due to the need of retraining for each model. Regarding the autoencoder, it can in principle be trained to perform a task analogue of estimating $D_\mathcal{F}$. The estimation step is indeed faster than our original brute force optimisation. However, the training set required is of high complexity. We leave it as future work to explore sampling gMPS [@gMPS], gMERA [@Fishman; @MERAwavelets], gRandomTensorNets [@EvenblyWhite], and eigenstates of random free-fermion Hamiltonians on random graphs, in order to create the dataset.
As future work, we can formulate the problem of estimating the interaction distance in terms of the correlation matrix of the state, that is its two-point correlations [@Gertis:2016wj]. In this case, the training dataset consists of correlation matrices obtained for free fermion systems. Again, even if the generation of a free-fermion correlation matrix is efficient ($\mathcal{O}(N^3)$), the free fermion systems to which they correspond need to be good representatives of $\mathcal{F}$. The approach in this case is different, however. Correlation matrices can be used as input to pattern recognition algorithms acting on images, and we expect that for an appropriate dataset, the free-fermion structure is learned. In any case, a significant improvement would be to have a method which is insensitive to the sizes of the input spectra or correlation matrices. In the case of entanglement spectra we can always pad with zeros to reach any dimension required.
Finally, we briefly comment on the distinction between classical and quantum random spectra, shown as distinct clusters in Fig.\[fig:PCAspectra\]. Actually, the r-statistics of entanglement spectra comming from the former and the latter procedures obey the Poisson and Wigner Dyson GUE distributions, respectively[@Chamon]. GUE is conjectured to signify universality of the quantum process that generated that state. Poisson is conjectured to signify non-universality. For the purposes of this work, we call the former spectra classical since the conjectures about the entanglement r-statistics are results of studies on Clifford circuits which are classically simulable and universal circuits. However, there are examples of non-universal quantum processes that generate states whose probability distributions are hard to sample from classically. We leave it for future work to investigate whether states that are output of such non-universal but classically hard to simulate processes manifest as a cluster between the two aforementioned clusters. One example would be boson sampling, where one would use the best known classical algorithm for computing permanents [@Neville2017]. The entanglement r-statistics are also an open question; that is whether another distribution Poisson or GUE.\
[**Acknowledgements :**]{} We thank Zlatko Papic for helpful discussions. C.J.T. acknowledges financial support by the EPSRC grant EP/M50807X/1. K.M. acknowledges financial support by the EPSRC Doctoral Prize EPSRC1001. Preprint of an article submitted for consideration in the International Journal of Quantum Information (2018) copyright World Scientific Publishing Company (http://www.worldscientific.com/worldscinet/ijqi).
[**APPENDIX** ]{}
Here we describe the procedures of obtaining the spectra we study in the main text. Our constructions are either direct construction of the spectrum, or sampling ground states of specific Hamiltonians. Furthermore, we show evidence that the SLAE is stable under input of almost flat spectra.
Constrution of Entanglement Spectra {#app:spectracreation}
===================================
We describe in detail how we construct random free, generic random, abelian topological, and free Ising entanglement spectra. The PCA of these spectra is shown in Fig.\[fig:PCAspectra\] in the main text.
Random Free Spectra
-------------------
A random free entanglement spectrum are sampled as follows. First sample single-body $s_i$ or $\epsilon_i$ form a probability distribution of our choice and obtaining a many-body spectrum of the form of Eq. via the expand map.
Random Spectra
--------------
We call classical spectra random probability distributions constructed by sampling random real numbers $P_k\in [0,1]$ and normalising them as $P_k \rightarrow P_k / {\sum_k P_k}$. On the other hand, we construct quantum random spectra as follows. A random matrix of complex elements $M_{ij} \in \mathbb{C}$ is created, with $i=1,\dots,2^n$ and $j=1,\dots,2^m$. It corresponds to the entanglement matrix of a random $(n+m)$-qubit state with amplitudes $\psi_k$ defined when one attempts the Schmidt decomposition of said state, $\ket{\psi} = \sum_{ij} M_{ij} \ket{i}\ket{j}$. Then the entanglement spectrum is obtained by $\xi=\mathrm{svd}(M)$.
In particular, we sample random complex numbers for the elements of $M$ as $M_{ij}=\alpha_{ij} e^{i \beta_{ij} 2\pi}$, where $\alpha_{ij}\in P$ random numbers from some probability distribution of our choice and $\beta_{ij}\in[0,1]$ uniform random numbers. We restrict to real states by setting $\beta_{ij}=0$. Then we normalise $M_{ij}\rightarrow M_{ij} / \tr{M^\dagger M}$.
![ $D^{SLAE}_\mathcal{F}(\bar P(N))$ for almost flat spectra with disorder of amplitude $\Delta \approx 0.01$ []{data-label="fig:SLAEFuzzyFlat"}](LR_Hills_Fuzzy-crop){width=".9\columnwidth"}
Parafermion Spectra
-------------------
The 1D Ising model can be mapped to the Majorana chain by means of a Jordan-Wigner transformation [@KitaevChain]. Similarly, $\mathbb{Z}_{N>2}$ generalisations of the Ising model known as the clock Potts model can be expressed in terms of parafermions [@FradkinKadanoff; @Fendley].
The chains are described by the Hamiltonian $$\label{eqn:parafermions}
H_{\mathbb{Z}_N} = - e^{i\phi}\sum_j \alpha^\dagger_{2j} \alpha_{2j+1} - f e^{i\theta}\sum_j \alpha^\dagger_{2j-1} \alpha_{2j} + \text{h.c.},$$ where $f$ is real and $\phi$ and $\theta$ are the chiral phases of the model. The parafermion operators satisfy the generalised commutation relations $\alpha_j \alpha_k = \omega \alpha_k \alpha_j $ for $k>j$, where $\omega = e^{i 2\pi/N}$ and $(\alpha_j)^N=1$. Here we focus on the gapped regime away from the critical points or critical phases [@Elitzur1979].
Fixed point: $f=0$ with $\phi=\theta=0$ and we place the bipartition between regions $A$ and $B$ at a $(2j,2j+1)$-link. The obtained spectrum is an $N$-fold degenerate eigenvalue understood as the number of parafermionic parity admissible to each partition and compatible with the global fixed parity of the chain. Our analysis is non-trivial as the fixed-point results are robust off-the-fixed point ($f>0$), as well as for finite chirality $\phi,\theta\neq 0$ [@KonMei18].
Ising Spectra {#sec:IsingApp}
-------------
We obtain entanglement spectra from equipartitioning ground states of the quantum Ising chain with periodic boundary conditions whose Hamiltonian is \[eq:IsingHam\] H (h\_z,h\_x)=-\_[j=1]{}\^L J X\_j X\_[j+1]{} +h\_z Z\_j + h\_x X\_j , where $J>0$ and $j$ runs over sites on which $\frac{1}{2}$-spins are defined and $X_j,Z_j$ are Pauli matrices that act on those spins. The boundary condition $L+1=1$ ensures that the ground state is unique.
In this work we set $J=1$. To sample free Ising spectra we set $h_x=0$. On this free line of the phase diagram the chain can be mapped to free fermions via the Jordan-Wigner transformation. Note that this free-fermion model is the well studied Kitaev chain which corresponds to the $\mathbb{Z}_2$ parafermion chain. To further increase the sample size we can also introduce disorder in the couplings, $J \rightarrow J+r$ and $h_{x,z}\rightarrow h_{x,z}+r$, where $r$ is a Gaussian random number $\mathcal{N}(0,\Delta)$. We call $\Delta$ the disorder amplitude. To sample from off-the-free-line we simply turn on the longitudinal field $h_x>0$. Disorder can be introduced in the same way to increase the sample size.
Supervised Linear Autoencoder on Almost Flat Spectra {#sec:SLAEFuzzyFlat}
====================================================
The predicted interaction distance of the supervised linear autoencoder is robust under perturbations of the input flat spectrum. Such perturbed spectra we refer to as almost flat, denoted $\bar P^\Delta_N$. Each non-zero eigenvalue is of the form $\Delta \frac{1}{N} $ and normalisation is ensured by dividing by their sum. For almost flat input spectra the prediction $D^{SLAE}_\mathcal{F}$ is shown in Fig.\[fig:SLAEFuzzyFlat\].
[10]{}
T. D. Schultz, D. C. Mattis and E. H. Liev, [*Rev. Mod. Phys.*]{} [**36**]{} (Jul 1964) 856.
E. Fradkin and L. P. Kadanoff, [*Nuclear Physics B*]{} [**170**]{} (1980) 1 .
P. Fendley, [*Journal of Statistical Mechanics: Theory and Experiment*]{} [ **2012**]{} (2012) p. P11020.
X.-G. Wen, [*Quantum Field Theory of Many-Body Systems*]{} (Oxford Graduate Texts, 2007).
C. J. Turner, K. Meichanetzidis, Z. Papic and J. K. Pachos, [*Nat. Commun.*]{} [**8**]{} (2017) p. 14926.
F. Verstraete, V. Murg and J. Cirac, [*Advances in Physics*]{} [**57**]{} (2008) 143.
H. Li and F. D. M. Haldane, [*Phys. Rev. Lett.*]{} [**101**]{} (Jul 2008) p. 010504.
I. Peschel and V. Eisler, [*Journal of Physics A: Mathematical and Theoretical*]{} [**42**]{} (2009) p. 504003.
G. E. Hinton and R. R. Salakhutdinov, [*Science*]{} [**313**]{} (2006) 504.
K. Meichanetzidis, C. J. Turner, A. Farjami, Z. Papi ć and J. K. Pachos, [*Phys. Rev. B*]{} [**97**]{} (Mar 2018) p. 125104.
A. Bullivant and J. K. Pachos, [*Phys. Rev. B*]{} [**93**]{} (Mar 2016) p. 125111.
P. Comon, [*Signal Processing*]{} [**36**]{} (1994) 287 . Higher Order Statistics.
C. Bishop, [*Pattern Recognition and Machine Learning*]{}Information Science and Statistics, Information Science and Statistics (Springer, 2006).
N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever and R. Salakhutdinov, [**15**]{} (06 2014) 1929.
N. Schuch, M. M. Wolf and J. I. Cirac (2012)
M. T. Fishman and S. R. White, [*Phys. Rev. B*]{} [**92**]{} (Aug 2015) p. 075132.
G. Evenbly and S. R. White, [*Phys. Rev. Lett.*]{} [**116**]{} (Apr 2016) p. 140403.
G. Evenbly and S. R. White, Representation and design of wavelets using unitary circuits (2016).
J. Gertis, M. Friesdorf, C. A. Riofrio and J. Eisert, [*ArXiv e-prints*]{} (2016)
D. Shaffer, C. Chamon, A. Hamma and E. R. Mucciolo, [*Journal of Statistical Mechanics: Theory and Experiment*]{} [**2014**]{} (2014) p. P12007.
A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. M. Birchall, A. Montanaro and A. Laing, [*Nature Physics*]{} [**13**]{} (Oct 2017) 1153 EP .
A. Y. Kitaev, [*Physics-Uspekhi*]{} [**44**]{} (2001) p. 131.
J. Alicea and P. Fendley, [*Annual Review of Condensed Matter Physics*]{} [ **7**]{} (2016) 119.
S. Elitzur, R. B. Pearson and J. Shigemitsu, [*Phys. Rev. D*]{} [**19**]{} (Jun 1979) 3698.
|
---
abstract: 'Resource-constrained systems are prevalent in communications. Such a system is composed of many components but only some of them can be allocated with resources such as time slots. According to the amount of information about the system, algorithms are employed to allocate resources and the overall system performance depends on the result of resource allocation. We do not always have complete information, and thus, the system performance may not be satisfactory. In this work, we propose a general model for the resource-constrained communication systems. We draw the relationship between system information and performance and derive the performance bounds for the optimal algorithm for the system. This gives the expected performance corresponding to the available information, and we can determine if we should put more efforts to collect more accurate information before actually constructing an algorithm for the system. Several examples of applications in communications to the model are also given.'
author:
- 'Albert Y.S. Lam, Yanhui Geng, and Victor O.K. Li, [^1]'
bibliography:
- 'references.bib'
title: 'Information-Theoretic Bounds for Performance of Resource-Constrained Communication Systems'
---
Algorithms, communication system performance, entropy, resource management.
Introduction
============
many communication systems, we desire to allocate limited resources effectively so as to maximize the system performance. Such a system usually has a large number of target objects to be served. However, we have a limited amount of resources which can only be given to a small number of objects. In this way, the chosen objects with resources granted become active and perform while the rest are idle (inactive). Resources here can refer to time slots, storage space, energy, channels, access rights, etc. For example, scheduling of transmissions in a wireless mesh network considers how to assign channels (resources) to routers’ radio interfaces (objects) for maximizing the network throughput (performance) [@wmn]. Depending on the system specification, the performance depends on one, some or all of the active objects. One of the key questions is how to select the correct objects to be active. To do this, we design optimal algorithms aiming to achieve the best performance. Given the amount of system uncertainty, it is very useful if we can tell how well the optimal resource allocation algorithm for the resource-constrained system with uncertain behavior performs. In this way, we can forecast the system performance for given uncertainty before actually developing the optimal algorithm. Suppose we are not satisfied with the performance of even the optimal algorithm for the current system uncertainty, then we should reduce the uncertainty instead of wasting effort on developing an optimal algorithm for the system. In this paper, we aim to characterize the performance bounds of resource-constrained communication systems in terms of uncertainty without explicitly developing any algorithms.
Resource-constrained systems are very common in communications and networking design. They refer to any systems with limited resources and the design objective is to allocate resources to the system components to meet the performance requirement. In wireless sensor networks, energy and bandwidth are limited and should be properly allocated to exploit spatial diversity [@cooperative]. In an Orthogonal Frequency Division Multiplexing relay network [@ofdma], the number of subcarriers is limited and they are assigned to the users. In a cognitive radio system [@cognitive], we allocate the limited radio spectrum to the secondary users for utilization and fairness optimization. [@compression] gives a survey on the compression and communication algorithms for multimedia in energy-constrained mobile systems. Resource-constrained systems can also be found in other engineering disciplines. For example, in an MPEG-2 streaming decoding system [@mpeg], the decoder cannot decode all the frames due to limited processing time and power. Most of the previous work focuses on allocating resources in one time instance. When extended in the time horizon, scheduling [@scheduling] and evolutionary computation [@ec] can also be cast under our framework. In this paper, we study resource-constrained communication systems, focusing on one time instance. Our results will be illustrated with more examples in Section \[sec:apps\].
Entropy measures the uncertainty of a random variable and it is one of the key elements in information theory [@shannon]. We are interested in determining the probability distributions with maximum and minimum entropies, respectively, subject to some constraints. Maximum entropy has been widely used in image processing [@image] and natural language modeling [@language] while minimum entropy has been applied to pattern recognition [@pattern]. An information measure based on maximum and minimum entropies was proposed in [@minmax]. Analytical expressions for maximum and minimum entropies with specific moment constraints were studied in [@minmax] and [@minentropy]. In this paper, we investigate the relationship between knowledge of systems and performance of algorithms, with respect to maximum and minimum entropy. We proposed a simple model for resource-constrained systems in [@smc] and applied it to opportunistic scheduling in wireless networks [@globecom]. We try to extend our previous work and our contributions in this paper include: 1) correcting a flaw in a published lower bound of the error probability; 2) determining the minimum entropy with the resource constraints; 3) developing a model of resource-constrained communication systems; 4) deriving a new upper bound of the error probability; 5) introducing merit probability; 6) deriving the lower and upper bounds of merit probability; 7) generalizing the results to systems with more general performance requirement; and 8) identifying several examples of applications of the model.
This work is motivated by the prefetching problem in [@Entropy:OnlineAlgo] which studies the performance bounds in terms of error probability of missing one webpage in the cache. We find that the lower bound stated in [@Entropy:OnlineAlgo] does not always hold. We corrected this lower bound. Moreover, an upper bound is given in [@Entropy:OnlineAlgo] but it only holds for a sequence of events generated by a stationary ergodic process. In this paper, we also obtain an upper bound without the assumption of an ergodic process and generalize the results so that they are applicable to general resource-constrained communication systems. Besides the error probability which is the focus of [@Entropy:OnlineAlgo], we propose the merit probability which allows us to extend the results to systems where merit is of interest. Most importantly, our results are more general as they allow multiple system components while only one missing webpage in the cache is studied in [@Entropy:OnlineAlgo]. The rest of this paper is organized as follows. We describe the system model of resource-constrained system in Section \[sec:model\]. In Section \[sec:optimum\_entropies\], we formulate the optimization problems of maximum and minimizing the entropy subject to the resource constraints. Section \[sec:performance\_analysis\] explains how to utilize the results of entropy optimization to derive the performance bounds of algorithms for the system model. In Section \[sec:apps\], we apply our results to several examples of communication applications and we conclude this paper in Section \[sec:conclusion\].
System Model {#sec:model}
============
An abstract model of the relationship among various elements in a resource-constrained communication system is given in Fig. \[fig:system\]. We denote the system and the resource allocation algorithm with $S$ and $A$, respectively. $S$ specifies the set of objects that we can select to activate. We employ $A$ to provide the strategy of selecting the active objects. $A$ interacts with $S$ by allocating system resources to components in $S$ based on the given system information. Usually we only have incomplete knowledge of the system and cannot tell the exact system behavior. We call the uncertain behavior of the system the *uncertain information*. This uncertainty may be due to our lack of knowledge of the system (objects), and/or the fact that the system contains some intrinsic randomness. We model this uncertainty with entropy $H(X)$, where $X$ is a discrete random variable describing behavioral outcomes of the system objects. If the algorithm is probabilistic, it has its own randomness as well and we model this uncertainty as $H(A)$. Then the joint entropy $H(X,A)$ is the total uncertainty resulted from the uncertain input and the uncertain algorithm. However, if the algorithm is deterministic, then $H(X,A)=H(X)$. The performance is the result of $H(X,A)$ and $S$.[^2] We describe the system performance in terms of error probability $\pi$ and merit probability $\psi$, whose definitions will be provided later.
![Relationship among different elements in a communication system[]{data-label="fig:system"}](system2.pdf){width="3.3in"}
![Model of system performance.[]{data-label="fig:model"}](model.pdf)
The model of system performance is illustrated in Fig. \[fig:model\]. Consider that $S$ contains a set of objects $A=\{a_1,a_2,\ldots,a_N\}$ with size $|A|=N$, where $N \geq 1$. Assume that each $a_i \in A$ is independent such that its contribution to the system performance can be solely evaluated with $f(a_i)$. In other words, the performance evaluation function $f$ is a mapping $A \rightarrow C \subset R$, where $C$ is the set of performance values.[^3] We can further classify each $c_i \in C$ into two subsets $C_1$ and $C_2$, where $C_1\cup C_2 \equiv C$ and $C_1\cap C_2 \equiv \emptyset$ (In this case, we have two performance levels). Suppose $C_1$ and $C_2$ represent good and bad performance, respectively. Assume that we have enough knowledge to distinguish between good and bad performance. Thus, we have a threshold of system performance $\theta$, such that an object $a_i$ is considered good, if $f(a_i) \geq \theta$, or $f(a_i) \in C_1$. Otherwise, it is said to have bad performance, or $f(a_i) \in C_2$.
Since the system involves randomness and we are uncertain about $f$ and do not know which $a \in A$ with $c=f(a) \in C_1$, the performance is probabilistic in nature. We model the performance evaluation of each object with $q(a_i)=\Pr\{a_i|f(a_i)\in C_1\}$. We define $p(a_i)=\frac{q(a_i)}{\sum_j{q(a_j)}}$, and thus, $\sum_{i=1}^N{p(a_i)}=1$. $p(a_i)$ is the probability of $a_i$ being mapped to $c_i\in C_1$ relative to all $a_j \in A$. In other words, it is the relative probability of $a_i$ in $A$ having good performance. Note that both $p(a_i)$ and $q(a_i)$ depend on our knowledge of the system. After gaining more information from experience or side information, the probability values may need to be updated, and the joint performance of the objects may become dependent. Only those objects allocated with resources can be activated and its performance can be evaluated. Due to the resource constraint, we cannot evaluate every object in $A$. Suppose the resources only allow us to select $M$ objects from $A$ for evaluation and they form the set $B\subseteq A$ with size $|B|=M$, where $1\leq M \leq N$. Thus, $A\setminus B$ is the set of objects which we have not selected. Consider that we are interested in $C_1$. In other words, we aim at including objects (says $a_i$) with performance $C_1$ (i.e. $f(a_i)\in C_1$) in $B$. We define the following two performance measures of system performance.
Error probability $\pi$ is defined as the probability of error in the selection process. It is the total probability of those objects, which result in the desirable performance level, e.g., $C_1$, but which have not been selected.
Merit probability $\psi$ is defined as the probability of merit in the selection process. It is the total probability of those objects, which result in the desirable performance level, e.g., $C_1$, and which have already been selected.
With the above definitions, we have $\pi = \sum_{i|a_i\in A\setminus B}{p(a_i)}$ and $\psi = 1-\sum_{i|a_i\in B}{p(a_i)} = 1-\pi$. For some systems, we are more interested in error than merit in the selection process, but in some other systems, we have the opposite. We will give examples of systems favoring merit and error, respectively, in Section \[sec:apps\]. Moreover, the requirement on the number of objects with desirable performance changes for different systems. In one extreme, one out of $M$ objects in $B$ with performance $C_1$ is already good enough for some systems At the other extreme, we require all $M$ objects in $B$ to have performance $C_1$. The requirement on some other systems may fall in between. Thus, we formally define performance requirement as follows:
\[def:k\] Performance requirement $k$ is defined as the number of objects with the desirable performance level selected, $1 \leq k \leq M$.
The algorithm $A$ undergoes a selection process (see Fig. \[fig:system\]) and $B$ is the result of $A$. Therefore, $\pi$ and $\psi$ characterize the performance of $A$ with respect to the system $S$. In particular, we are interested in the best algorithm.
The optimal strategy is the algorithm with the highest probability in generating results with the desirable performance level among all possible algorithms. It can do so by selecting the $M$ $a_i$ with the highest $p(a_i)$ out of $A$.
We will derive the performance bounds of the optimal strategy in the next section.
Optimum Entropies {#sec:optimum_entropies}
=================
In this section, we will first define some terminologies and then formulate the maximum and minimum entropies.
Preliminaries
-------------
**Symbol**
---------------------------- ---------------------------------------------------------
*X* Discrete random variable
*P* Probability distribution of *X*
*N* Number of states of *X*
*M* Number of selected states of *X*
*p(i)* Probability of state *i*
*$\pi$* Error probability
*$\pi_{max}(\pi_{min})$* Maximum (minimum) error probability
*$\pi_{UB}(\pi_{LB})$* Upper (lower) bound of error probability
*H(X) / H(P)* Entropy of distribution *P* of *X*
*$H_{max}(H_{min})$* Maximum (minimum) entropy
*$P_{max}(P_{min})$* Distribution with maximum (minimum) entropy
*$P_{i\rightarrow j}$* Partial distribution of *X* from state *i* to state *j*
*$H(P_{i\rightarrow j})$* Partial entropy of $P_{i\rightarrow j}$
*A* Set of all states of *X*
*$a_i$* The $i$th state of *X*
*B* Selected set from *A*
*$b_i$* The $i$th state of *X* in *B*
*$|\cdot|$* Cardinality
*f* System performance function
*C* Set of all performance states
*$C_1$* Subset of *C*
*$C_2$* Complement of $C_1$
***P*** Set of distributions
*$\psi$* Merit probability
*$\psi_{max}(\psi_{min})$* Maximum (minimum) merit probability
*$\psi_{UB}(\psi_{LB})$* Upper (lower) bound of merit probability
*R* Set of real numbers
: Definitions of Notations
\[table:notations\]
We list the frequently used notations and their definitions in Table \[table:notations\]. Consider a probability distribution of a discrete random variable $X$ with $N$ possible states. Let $p(i)=Pr\{X=i\}$ for $i\in \{1,2,\ldots,N\}$. A probability distribution $P$ is represented by a vector of $N$ numbers, i.e. $P=[p(1),\ldots,p(N)]$ satisfying $\sum_{i=1}^N{p(i)}=1$. Without loss of generality, we assume $$\begin{aligned}
\label{eq:prob_order}
p(i) \geq p(i+1), \qquad \text{for } i=1,2,...,N-1.\end{aligned}$$ Let $\pi$ be the sum of the probabilities of the last $(N-M)$ states, where $0\leq \pi \leq 1$ and $1\leq M \leq N$, i.e., $$\label{eq:sum1toM}
\sum_{i=1}^M{p(i)}=1-\pi$$ and $$\label{eq:sumM+1toN}
\sum_{i=M+1}^N{p(i)}=\pi.$$ A probability distribution $P(\pi)$ looks like
$$\label{eq:distribution}
[\underbrace{p(1),\ldots ,p(M)}_{\sum =1-\pi} ,\underbrace{p(M+1),\ldots ,p(N)}_{\sum =\pi}].$$
Let $\bar{p}_1$ and $\bar{p}_2$ be the means of the first $M$ terms and the last $(N-M)$ terms, respectively, i.e., $\bar{p}_1=\frac{1-\pi}{M}$ and $\bar{p}_2=\frac{\pi}{N-M}$ To have a feasible probability distribution satisfying (\[eq:prob\_order\]), (\[eq:sum1toM\]), and (\[eq:sumM+1toN\]), we have $$\label{eq:mean}
\bar{p}_1\geq \bar{p}_2.$$
Assume $0\log_2 0=0$. Unless stated otherwise, we take the logarithm to the base 2. The entropy of (\[eq:distribution\]) is given by $$\label{eq:entropy}
H(X)=-\sum_{i=1}^N p(i)\log p(i).$$
To facilitate the proofs of later results, we investigate the properties of the function $$\label{eq:fe}
f_e (x) = -x\log x,$$ for $x\in [0,1]$. It is easy to check that $f_e$ is strictly concave. We also have the following lemma[^4]:
\[lemma:use\_concave\] Consider any two points $x_1$ and $x_2$ in interval $[0,1]$ with $x_1 \geq x_2$, and an arbitrary positive number $\delta$ satisfying $x_1+\delta\leq 1$ and $x_2-\delta\geq 0$, the inequality $$\label{eq:base}
f_e(x_1+\delta)+f_e(x_2-\delta)<f_e(x_1)+f_e(x_2)$$ always holds.
Next we will consider two optimization problems, namely, entropy maximization and minimization. The solutions of these two problems will help us derive the performance bounds.
Maximum Entropy {#subsec:maxH}
---------------
Our aim is to find a probability distribution with the maximum entropy amongst all feasible distributions $P(\pi)$. Mathematically, given $M$, $N$, and $\pi$ where $1\leq M\leq N$ and $0 \leq \pi \leq 1$, we consider
\[eq:max\] $$\begin{aligned}
\underset{P}{\text{maximize}}\quad & H(X)=-\sum_{i=1}^N p(i)\log p(i) \label{eq:maxc1} \\
\text{subject to}\quad & \sum_{i=1}^M{p(i)}=1-\pi,\\
& \sum_{i=M+1}^N{p(i)}=\pi,\\
& p(1)\geq p(2)\geq \ldots \geq p(M)\geq \ldots \geq p(N)\geq 0. \label{eq:maxc5}
$$
We can see that (\[eq:entropy\]) is separable, broken down into $N$ independent terms of (\[eq:fe\]), each of which is concave. Since the entropy function is a concave function and the constraints are linear, we can follow the Kuhn-Tucker conditions to obtain the unique and simple distribution with maximum entropy. According to the principle of maximum entropy [@Entropy:Elements_of_IT], the solution of is given by *Theorem \[theorem:max\]*.
\[theorem:max\] The distribution with maximum entropy $P_{max}(\pi)$ subject to Constraints – is given by $$\begin{aligned}
[\underbrace{\bar{p}_1,\ldots,\bar{p}_1}_{M \textit{ terms}},\underbrace{\bar{p}_2,\ldots,\bar{p}_2}_{(N-M) \textit{ terms}}].
\end{aligned}$$
[@Entropy:OnlineAlgo] gave the maximum entropy and a lower bound of $\pi$. For completeness, we include the results below:
\[corollary:max\] The maximum entropy of a probability distribution subject to Constraints – is given by $$\begin{aligned}
H_{max}= (1-\pi)\cdot \log(\frac{M}{1-\pi})+\pi\cdot \log(\frac{N-M}{\pi}).
\end{aligned}$$
\[corollary:pi\_lower\_bound\] $\pi$ is bounded by $$\begin{aligned}
\pi \geq \frac{H-1-\log M}{\log (\frac{N}{M}-1)}.
\end{aligned}$$
However, there exists a flaw in deducing Corollary \[corollary:pi\_lower\_bound\]. The key assumption that $\log(\frac{N}{M}-1)$ is positive may not always hold as it depends on the values of $N$ and $M$. Corollary \[corollary:pi\_lower\_bound\] is correct only when $1\leq M<\frac{N}{2}$. For the case of $\frac{N}{2}\leq M \leq N$, no precise theoretical lower bound is available and hence we adopt zero for completeness.
\[corollary:pi\_lower\_bound\_revised\] The lower bound of $\pi$ with given entropy value $H$ subject to Constraints – is given by
----------------------------------------------------- -----------------------------
$\pi \geq \frac{H-1-\log M}{\log (\frac{N}{M}-1)}$, $1\leq M<\frac{N}{2}$,
$\pi \geq 0$, $\frac{N}{2}\leq M \leq N$.
----------------------------------------------------- -----------------------------
Minimum Entropy {#subsec:minH}
---------------
Similarly, for minimization, we have $$\label{eq:min}
\underset{P}{\text{minimize}} \quad H(X)=-\sum_{i=1}^N p(i)\log p(i)$$ subject to –. The solution of depends on $N$, $M$, and $\pi$. The entropy function and constraints form a polyhedron with multiple minimums. Those distributions with minimum entropy are the extremal points of the polyhedron.
When $M=1$, we have *Lemma \[lemma:min\]* [@Entropy:Relations].
\[lemma:min\] When $M=1$, the probability distribution with minimum entropy is achieved by $P_{min}(\pi)=[p(1),\ldots,p(N)]$ where
------------------------------------------ ------------------------------------------------
$p(1)=1-\pi$, $\quad p(2)=\pi$,
$p(3)=\cdots =p(N)=0$, $0\leq \pi \leq \frac{1}{2}$,
$p(1)=p(2)=1-\pi$, $\quad p(3)=2\pi -1$,
$p(4)=\cdots =p(N)=0$, $\frac{1}{2}\leq \pi \leq \frac{2}{3}$,
$p(1)=\cdots =p(N-1)=1-\pi$,
$p(N)=1-(N-1)(1-\pi)$, $\frac{N-2}{N-1} \leq \pi \leq \frac{N-1}{N}$.
------------------------------------------ ------------------------------------------------
We are going to determine the distribution with the minimum entropy for $M\geq 2$. We can divide $P$ into two separate segments, $P_{1\rightarrow M}$ and $P_{(M+1)\rightarrow N}$. $P_{1\rightarrow M}$ takes the first $M$ terms from $P$. Suppose that the value of $p(M)=\min (p(i),i\in \{1,\ldots,M\})$ is pre-determined and equals $p'$. It is trivial to see that $$\frac{1-\pi}{M}\geq p'\geq \frac{\pi}{N-M}.$$ With $p(M)=p'$, we can construct $P_{1\rightarrow M}$ as follows to minimize $-\sum_{i=1}^M p(i)\log p(i)$. Contrary to maximum entropy, the principle of constructing minimal entropy distribution is to allocate probabilities as less random as possible. In other words, we try to allocate large probabilities to a few states and to assign as small as possible probabilities to other states. For example, distribution $[0.6,0.4,0,0,0]$ has smaller entropy than distribution $[0.3,0.3, 0.2, 0.1, 0.1]$. We have *Lemma \[lemma:min1toM\]*.
\[lemma:min1toM\] With the smallest (also the last) element of $P_{1\rightarrow M}$ fixed at $p'$, the optimal $[p(1),\ldots,p(M)]$ which minimizes $-\sum_{i=1}^M p(i)\log p(i)$ is given by $$\begin{aligned}
\left\{
\begin{array}{l} \label{eq:min1toM}
p(1)=(1-\pi)-(M-1)\times p',\\
p(2)=\cdots =p(M)=p'.\\
\end{array} \right.
\end{aligned}$$
$P_{(M+1)\rightarrow N}$ takes the last $(N-M)$ terms from $P$. Suppose the value of $p(M+1)=\max (p(i),i\in \{M+1,\ldots,N\})$ is pre-determined and equals $p''$. It is trivial to see that $$\label{eq:p_range}
\frac{1-\pi}{M}\geq p'\geq p'' \geq \frac{\pi}{N-M}.$$ With the value of $p(M+1)$ fixed at $p''$, we can construct $P_{(M+1)\rightarrow N}$ as before to minimize $-\sum_{i=M+1}^N p(i)\log p(i)$. We have *Lemma \[lemma:minM+1toN\]*.
\[lemma:minM+1toN\] With the largest (also the first) element of $P_{(M+1)\rightarrow N}$ fixed at $p''$, the optimal $[p(M+1),\ldots,p(N)]$ which minimizes $-\sum_{i=M+1}^N p(i)\log p(i)$ is given by $$\begin{aligned}
\left\{
\begin{array}{l} \label{eq:minM+1toN}
p(M+1)=\cdots=p(M+\lfloor\frac{\pi}{p''}\rfloor)=p'',\\
p(M+\lceil\frac{\pi}{p''}\rceil)=\pi\bmod p'',\\
p(M+\lceil\frac{\pi}{p''}\rceil+1)=\cdots =p(N)=0.\\
\end{array} \right.
\end{aligned}$$
By combining $P_{1\rightarrow M}$ and $P_{(M+1)\rightarrow N}$, we have *Lemma \[lemma:pequalp\]*.
\[lemma:pequalp\] $P_{min}$ must have $p'=p''$. Let $\hat{p}=p'=p''$. We have $\hat{p}\in[\frac{\pi}{N-M},\frac{1-\pi}{M}]$.
Let $$\label{eq:y}
y\stackrel{\Delta}{=}\lceil \frac{N-M-N\pi}{1-\pi} \rceil,$$ $$\begin{aligned}
\label{eq:H1}
H_1(\hat{p})\stackrel{\Delta}{=}&-(M-1)\hat{p} \log\hat{p} - [(1-\pi)-(M-1)\hat{p}] \nonumber \\
&\times \log [(1-\pi)-(M-1)\hat{p}]\end{aligned}$$ and define $H_2(\hat{p})$ according to
.
[rCl]{} H\_2() {
[ll]{} \[eq:H2\] -(N-M), & =\
-(N-M-1) -\[-((N-M-1))\], & <\
-(N-M-2) -\[-((N-M-2))\], & <\
&\
-(N-M-y) -\[-((N-M-y))\], & <\
}
With $p(i)$ specified in (\[eq:min1toM\]) and (\[eq:minM+1toN\]), we can transform the multi-variable optimization problem (\[eq:min\]) to the single variable optimization as: $$\label{eq:single_min}
\min_{\hat{p} \in [\frac{\pi}{N-M},\frac{1-\pi}{M}]} \qquad H(\hat{p})=H_1(\hat{p})+H_2(\hat{p})$$ subject to –.
![Plot of $H(\hat{p})$ for $N=15$, $M=5$, and $\pi=0.4$.[]{data-label="fig:typical"}](typical.pdf)
$H(\hat{p})$ is a continuous function with a piecewise continuous derivative. Its leftmost and rightmost limits of $\hat{p}$ are $\frac{\pi}{N-M}$ and $\frac{1-\pi}{M}$, respectively. It is composed of a certain number of concave segments and every pair of consecutive concave segments join at a non-differentiable point. The function connecting all those non-differentiable points is $$\label{eq:min_curve}
\tilde{H}(\hat{p})=\pi \log (\hat{p}).$$ $H(\hat{p})$ and $\tilde{H}(\hat{p})$ must meet at $\hat{p}=\frac{\pi}{N-M}$, but they may or may not assemble at $\hat{p}=\frac{1-\pi}{M}$. The plots of $H(\hat{p})$ and $\tilde{H}(\hat{p})$ for an example with $N=15$, $M=5$, and $\pi=0.4$ are shown in Fig. \[fig:typical\]. The shape of $\tilde{H}(\hat{p})$ depends on the values of $N$, $M$, and $\pi$. It may be monotonically increasing, monotonically decreasing, monotonically increasing and then decreasing, etc. No matter which shape it is, $\hat{p}^*$ constituting the minimum $H(\hat{p})$ must be one of the non-differentiable points or $\frac{1-\pi}{M}$ (since the rightmost limit of $H(\hat{p})$ may not join the curve of $\tilde{H}(\hat{p})$). Let $$\begin{aligned}
\hat{P}^* \stackrel{\Delta}{=}\{\frac{\pi}{N-M},\frac{\pi}{N-M-1},\ldots ,\frac{\pi}{N-M-y+1},\frac{1-\pi}{M}\}.\end{aligned}$$ We have $$\begin{aligned}
|\hat{P}^*| = \begin{cases} \frac{N-M-N\pi}{1-\pi} &\mbox{if } y= \frac{N-M-N\pi}{1-\pi}, \\
(\frac{N-M-N\pi}{1-\pi}+1) & \mbox{otherwise.}\end{cases} \end{aligned}$$ We can further reduce the original multi-variable optimization with a continuous solution set given by (\[eq:min\]) to a single variable optimization with a discrete set, given by $$\label{eq:discrete_min}
\min_{\hat{p} \in \hat{P}^*} \qquad H(\hat{p})=H_1(\hat{p})+H_2(\hat{p}).$$ Moreover, $H(\hat{p})$ now becomes
.
[rCl]{} H()= {
[ll]{} \[eq:discrete\_H\] -(N-1) - \[(1-)-(M-1)\], & =\
-(N-2) - \[(1-)-(M-1)\], & =\
&\
-(N-y) - \[(1-)-(M-1)\], & =\
-(N-y) -\[-(N-M-y)\], & =\
}
Define $$\begin{aligned}
\Omega\stackrel{\Delta}{=} \left\{
\begin{array}{l}
\frac{(N-1)\pi}{N-M}\log \frac{N-M}{N(N-M)}, \\
\frac{(N-2)\pi}{N-M-1}\log \frac{N-M}{N(N-M-1)}, \\
\qquad\qquad\qquad\vdots\\
\frac{(N-y)\pi}{N-M-y+1}\log \frac{N-M}{N(N-M-y+1)}, \\
\frac{(N-y)(1-\pi)}{M}\log M \\
\end{array} \right\}. \\
\end{aligned}$$
Then we have *Theorem \[theorem:entropy\_lower\_bound\]*.
\[theorem:entropy\_lower\_bound\] A lower bound of the entropy is given by $\min(\Omega)$.
Let $\pi_{min}(H)$ be the value specified by *Corollary \[corollary:pi\_lower\_bound\_revised\]*. The bounds of $\pi$ are stated in the following theorem:
\[corollary:pi\_bounds\] $\pi$ is bounded by $$\begin{aligned}
\pi_{min}(H) \leq \pi \leq \pi_{max}(H),
\end{aligned}$$ where $\pi_{max}(H)$ is given by $$\begin{aligned}
\max \left\{
\begin{array}{l}
\frac{H\cdot (N-M)}{(N-1)\log \frac{N(N-M)}{N-M}},\\
\frac{H\cdot (N-M-1)}{(N-2)\log \frac{N(N-M-1)}{N-M}},\\
\qquad\qquad \vdots \\
\frac{H}{M\log \frac{N}{N-M}},\\
\frac{H}{\log \frac{1}{M}}+1\\
\end{array} \right\}.\end{aligned}$$
To evaluate the correctness and tightness of our derived bounds, we perform a series of simulations with different combinations of $N$ and $M$ with different scales. The detailed data are presented in Figs. \[fig:bounds1\] and \[fig:bounds2\]. Each of the plots contains 100 scenarios, each of which represents a probability distribution. From a distribution, we can determine its entropy and $\pi$. With our results, we can find the corresponding upper and lower bounds. The simulation results verify our theoretical bounds. The bounds are actually quite tight, especially when $\frac{N}{2}\leq M \leq N$ (e.g. Figs. \[fig:30-20(100times)\], \[fig:100-60(100times)\], \[fig:500-300(100times)\], and \[fig:1500-1000(100times)\]). The result is even better with larger value of entropy, which means that our theory can predict performance more precisely in systems with higher degrees of uncertainty. This trend can be easily observed in Figs. \[fig:20-6(100times)\], \[fig:50-15(100times)\], \[fig:200-40(100times)\], and \[fig:1000-400(100times)\].
To summarize, we have considered two optimization problems and determined the maximum and minimum entropies $H_{max}$ and $H_{min}$ of feasible distributions for specific $M$, $N$, and $\pi$. In other words, given $M$, $N$, and $\pi$, we can construct a probability distribution $P$ whose entropy $H$ is bounded by $H_{max}$ and $H_{min}$, i.e., $H_{min}\leq H \leq H_{max}$. Here $P$ and $H$ are variables while $H_{max}$ and $H_{min}$ (expressed in terms of $N$, $M$, and $\pi$) are constants. Then we consider that $P$ and $H$ are fixed and $\pi$ is a variable. This allows us to derive bounds for $\pi$, i.e., $\pi_{min}\leq \pi \leq \pi_{max}$.
Performance Analysis {#sec:performance_analysis}
====================
In this section, we will explain how to analyze performance of resource-constrained system based on the results obtained in Section \[sec:optimum\_entropies\].
Recall that there are $N$ objects in the system $S$. Based on our knowledge of the objects’ behavior in terms of $p(a_i)$, we can develop an algorithm to assign limited resources to some of the objects, i.e., select $a_i$ into set $B$. Among all possible selection strategies, we are interested in the optimal strategy, which will include the $M$ highest objects in $B$. Error probability $\pi$ (or merit probability $\psi$) characterizes the performance of the algorithm. Maximizing and minimizing the entropy allow us to give upper and lower bounds of entropy $H$, from which we can further derive upper and lower bounds of $\pi$. Since $H$ is the result of the optimal algorithm, the bounds of $\pi$ characterize the performance of the algorithm. For the merit probability, we can understand in a similar way. In the following, we consider different performance requirements $k$ with respect to error probability $\pi$ and merit probability $\psi$, respectively. We are going to derive lower and upper bounds for each case.
Evaluating One Object ($k=1$)
-----------------------------
Recall that there are $M$ objects in $B$ selected from $N$ objects in $A$. In this case, we are interested in evaluating one representative object (e.g., the object with the best performance) in $B$ only. Thus, we have $k=1$. If we are only interested in having $M$ objects in $B$ and the choice of objects in $B$ is not important, there are $N \choose M$ different possible choices of $B$. Let $X$ be the random variable describing the behavior of system objects and let its associated probability distribution be represented by $P=[p(a_1),\ldots,p(a_N)]$.
### Error Probability
For each object (e.g., $a_i$) in $B$, the probability of getting desirable performance is $p(a_i)$. Since only one object in $B$ with performance $C_1$ is enough, we have $M$ chances to meet the target, i.e., any one $a_i \in B$ with performance $C_1$. Thus, the error probability is given by $$\label{eq:sum1}
\pi(1)=\sum_{a_i\in A \setminus B}{p(a_i)},$$ where the “1” in $\pi(1)$ specifies $k=1$.
As shown in [@Entropy:Relations] and [@Entropy:OnlineAlgo], given $\pi$, we can bound the entropy of any selection process by $$\label{eq:H_bound_relationship}
\min_{P \in \textbf{P}_{\pi}} H(P) \leq H(X) \leq \max_{P \in \textbf{P}_{\pi}} H(P),$$ where $\textbf{P}_{\pi}$ is the set of all vectors $P$ such that $p(i) \geq 0,\forall i$, and they satisfy (\[eq:sum1\]). Moreover, given $H$, we also have $$\label{eq:pi_bound_relationship}
\underline{\pi}(1) \leq \pi \leq \overline{\pi}(1),$$ where $\underline{\pi}(1)$ and $\overline{\pi}(1)$ are the lower and upper bounds derived from $\max_{P \in \textbf{P}_{\pi}} H(P)$ and $\min_{P \in \textbf{P}_{\pi}} H(P)$, respectively, with $k=1$.
The optimal strategy will include those $a_i$ with highest $p(a_i)$ in $B$. If we adopt the optimal strategy, the corresponding error probability defined in (\[eq:sum1\]) is minimum, denoted $\pi_{min}$. This enforces and the results determined in Sections \[subsec:maxH\] and \[subsec:minH\] follow. Hence we get upper and lower bounds of error probability of the optimal strategy, denoted by $\overline{\pi}_{min}(1)$ and $\underline{\pi}_{min}(1)$, respectively. We have $$\label{eq:pi_min_bound_relationship}
\underline{\pi}_{min}(1) \leq \pi_{min}(1) \leq \overline{\pi}_{min}(1).$$ By applying Theorem \[corollary:pi\_bounds\], we get the closed forms of $\overline{\pi}_{min}(1)$ and $\underline{\pi}_{min}(1)$.
Note that the entropy $H$ is the result of the evaluating algorithm. Its value can be estimated through certain trial runs of the algorithm with the system or from side information.
### Merit Probability
According to the definitions, we have $$\label{eq:psi1}
\psi(1)=1-\pi(1).$$
Similarly, if we adopt the optimal strategy to select objects from $A$ to $B$, the corresponding merit probability defined in (\[eq:psi1\]) is maximum, denoted $\psi_{max}$. Therefore, we have *Theorem \[theorem:merit\_1\_bounds\]*.
\[theorem:merit\_1\_bounds\] The maximum merit probability $\psi_{max}$ is bounded, given by $$\begin{aligned}
\label{eq:merit_1_bounds}
\min \left\{
\begin{array}{l}
1-\frac{H\cdot (N-M)}{(N-1)\log \frac{N(N-M)}{N-M}},\\
1-\frac{H\cdot (N-M-1)}{(N-2)\log \frac{N(N-M-1)}{N-M}},\\
\qquad\quad \vdots \\
1-\frac{H\cdot (N-M-y+1)}{(N-y)\log \frac{N(N-M-y+1)}{N-M}},\\
\frac{H\cdot M}{(N-y)\log \frac{1}{M}}\\
\end{array} \right\} \nonumber \\
\leq \psi_{max}(1) \leq \frac{\log (N-M)-H+1}{\log (\frac{N}{M}-1)}.\end{aligned}$$
Evaluating Multiple Objects ($1\leq k\leq M$)
---------------------------------------------
We try to generalize the previous results to the cases when more than one object in $B$ with the desirable properties are required. We can accomplish the analysis for $1\leq k\leq M$ by transforming the sets $A$ and $B$. If the evaluations of the objects are conducted by independent entities, they may refer to the same objects in the evaluation. Depending on the system specifications, some of the $k$ objects may be identical in the evaluation. Thus we have two kinds of transformation, $T_{u}$ and $T_{r}$, for the case with unique objects and that with repeat objects, respectively. For the unique (repeated) case, we obtain the new sets $A_u' (A_r')$ and $B_u' (
B_r')$ by $A \xrightarrow{T_u} A_u'$ and $B \xrightarrow{T_u} B_u'$ ($A \xrightarrow{T_r} A_r'$ and $B \xrightarrow{T_r} B_r'$). We describe how the transformations are done as follows.
### The unique case {#subsubsec:unique}
Any $a'_{u} \in A'_{u}$ is, in fact, a $k$-combination[^5] of distinct $a \in A$. Since the order of the objects in the combination is not important, each $a'_{u}$ is a set of $k$ objects taken from $A$. For example, when $k=2$, $a'_{u}$ is a set $\{a_i,a_j\},i\neq j$. $A'_{u}$ is the set of all possible combinations of $\{a_i,a_j\},\forall a_i,a_j\in A,i\neq j$. Similar to $N$ and $M$, the numbers of objects in in the transformed sets $A_u'$ and $B_u'$ can be obtained by $$\label{eq:new_N_u}
N'_{u}=|A'_{u}|= {N \choose k}$$ and $$\label{eq:new_M_u}
M'_{u}=|B'_{u}|= {M \choose k}.$$ Let $\Gamma (a'_{u})$ be the permutation[^6] set of $a'_{u}$ with $|a'_{u}|=k$. We have $|\Gamma(a'_{u})|=k!$. Let $\gamma=[a_1,\ldots,a_k]\in \Gamma(a'_{u})$. Then the probability of each $\gamma$ with the desirable properties is ${\Pr}_{u}\{\gamma\}={\Pr}_{u}\{a_k|a_1,\ldots,a_{k-1}\}\cdot\ldots\cdot \cdot {\Pr}_{u}\{a_2|a_1\}\cdot{\Pr}_{u}\{a_1\}
=\frac{p(a_k)}{\sum_{a_i\in A\setminus \{a_1,a_2,\ldots,a_{k-1}\}}{p(a_i)}} \cdot\ldots\cdot \frac{p(a_2)}{\sum_{a_i\in A\setminus \{a_1\}}{p(a_i)}} \cdot p(a_1)$. Hence the probability of $a_u'$ having good performance is given by $$\begin{aligned}
\label{eq:new_p_u}
p(a'_{u})=\sum_{\gamma\in \Gamma(a'_{u})}{{\Pr}_u\{\gamma\}}.\end{aligned}$$ Moreover, $B'_{u}$ contains all those $a'_{u}$ satisfying the condition that every $a_i\in a'_{u}$ also belongs to $B$.
### The repeated case {#subsubsec:repeated}
In this case, some of the $k$ selections are allowed to refer to the same objects. Any $a'_{r} \in A'_{r}$ is, in fact, a multiset [@multiset] of cardinality $k$, with objects taken from $A$. For example, when $k=2$ and $A=\{a_1,a_2,a_3\}$, $A'_{r}$ is $$\begin{aligned}
\{&\{a_1,a_1\},\{a_1,a_2\},\{a_1,a_3\},\{a_2,a_2\},\{a_2,a_3\},\{a_3,a_3\}\}.\end{aligned}$$ $B'_{r}$ contains all those $a'_{r}$ satisfying the condition that every $a_i\in a'_{r}$ also belongs to $B$. Therefore, the numbers of objects in the transformed sets $A_r'$ and $B_r'$ can be obtained by $$\label{eq:new_N_r}
N'_{r}=|A'_{r}|= \left({N \choose k}\right) = \binom{N+k-1}{k}$$ and $$\label{eq:new_M_r}
M'_{r}=|B'_{r}|= \left( {M \choose k} \right) = \binom{M+k-1}{k},$$ where $((\cdot))$ is the multiset coefficient resembling the notation of binomial coefficients for a multiset.[^7] We define $k$-ordered-repeat-combination $\phi_k(A)$ as an ordered collection of elements which are allowed to repeat, of prescribed size $k$ and taken from $A$. For example, all possible 2-ordered-repeat-combinations of the set $\{a_1,a_2,a_3\}$ are $$\begin{aligned}
\{&[a_1,a_1],[a_1,a_2],[a_1,a_3],[a_2,a_1],[a_2,a_2],\\
&[a_2,a_3],[a_3,a_1],[a_3,a_2],[a_3,a_3]\}.\end{aligned}$$ Consider $\phi=[a_1,a_2,\ldots,a_k]$. Then we have ${\Pr}_{r}\{\phi\}={\Pr}_{r}\{a_k|a_1,\ldots,a_{k-1}\}\cdot\ldots\cdot{\Pr}_{r}\{a_3|a_1,a_2\}\cdot {\Pr}_{r}\{a_2|a_1\}\cdot{\Pr}_{r}\{a_1\}= p(a_k) \cdot \ldots \cdot p(a_3) \cdot p(a_2) \cdot p(a_1)$. We say $\phi \in a'_{r}$ if $\phi$ is a permutation of $a'_{r}$. Hence, $$p(a'_{r})=\sum_{\phi \in a'_{r}}{{\Pr}_{r}\{\phi\}}.$$
Let $N'$, $M'$ and $p(a')$ be the numbers of objects in the transformed sets $A'$ and $B'$, and the probability of $a'$ with good condition, for either the unique or repeated case (for example, $N'$, $M'$ and $p(a')$ are replaced by $N_u$, $M_u'$ and $p(a_u')$, respectively for the unique case). No matter which case we have, we can determine $N'$, $M'$ and $p(a')$ from $N$, $M$, $p(a)$, according to $k$.
\[theorem:error\_k\_bounds\] The minimum error probability with performance requirement $1\leq k \leq M$ is bounded, given by $$\begin{aligned}
\label{eq:pi_k_bounds}
&\frac{H'-1-\log M'}{\log (\frac{N'}{M'}-1)} \leq \pi_{min}(k) \nonumber \\
& \leq \max \left\{
\begin{array}{l}
\frac{H'\cdot (N'-M')}{(N'-1)\log \frac{N'(N'-M')}{N'-M'}},\\
\frac{H'\cdot (N'-M'-1)}{(N'-2)\log \frac{N'(N'-M'-1')}{N'-M'}},\\
\qquad\qquad \vdots \\
\frac{H'\cdot (N'-M'-y'+1)}{(N'-y')\log \frac{N'(N'-M'-y'+1)}{N'-M'}},\\
\frac{H'\cdot M'}{(N'-y')\log \frac{1}{M'}}+1\\
\end{array} \right\},\end{aligned}$$ where $N'$ and $M'$ are $N_u'$ and $M_u'$ ($N_r'$ and $M_r'$) given in and ( and ) for the unique (repeated) case.
\[theorem:merit\_k\_bounds\] The maximum merit probability with performance requirement $1\leq k \leq M$ is bounded, given by $$\begin{aligned}
\label{eq:psi_k_bounds}
\min \left\{
\begin{array}{l}
1-\frac{H'\cdot (N'-M')}{(N'-1)\log \frac{N'(N'-M')}{N'-M'}},\\
1-\frac{H'\cdot (N'-M'-1)}{(N'-2)\log \frac{N'(N'-M'-1)}{N'-M'}},\\
\qquad\quad \vdots \\
1-\frac{H'\cdot (N'-M'-y'+1)}{(N'-y')\log \frac{N'(N'-M'-y'+1)}{N'-M'}},\\
\frac{H'\cdot M'}{(N'-y')\log \frac{1}{M'}}\\
\end{array} \right\} \nonumber \\
\leq \psi_{max}(k) \leq \frac{\log (N'-M')-H'+1}{\log (\frac{N'}{M'}-1)},\end{aligned}$$ where $N'$ and $M'$ are $N_u'$ and $M_u'$ ($N_r'$ and $M_r'$) given in and ( and ) for the unique (repeated) case.
Applications {#sec:apps}
============
In this section, we identify several communication applications where our results can be applied.
Cache System with Focus on One Webpage
--------------------------------------
![An example for error probability with $k=1$.[]{data-label="fig:cache1computer1page"}](cache1computer1page.pdf)
The model can be applied to the cache pre-fetch problem introduced in [@Entropy:OnlineAlgo]. When browsing webpages from the Internet, we employ web proxy servers to increase the efficiency of delivering the contents to a group of local users. This may reduce the amount of data needed to be transferred from remote web servers to the users’ local computers. To do this, the proxy server pre-fetches a certain number of webpages from various remote servers and stores them in its memory. Due to the resource constraints of the proxy server, e.g. the size of the memory, the number of pages stored in the proxy must be much smaller than the total number on the Internet. The pre-fetched webpages are chosen according to the relative probability that its users are likely to request the webpages in the near future. When a user requests a webpage, it first contacts the proxy server to check if the webpage is stored locally. If so, the page is directly transmitted to the user through the local network and we say that there is a “hit” at the proxy server. If not, it becomes a “miss” and the page will be requested from the corresponding remote server instead. The situation is depicted in Fig. \[fig:cache1computer1page\]. Assume that every webpage is of the same size. There are $N$ distinct webpages in the Internet (i.e. $A$) and the cache in the proxy server (i.e. $B$) can store $M$ pages, where $M \ll N$. We model the situation that a user requests one particular webpage (i.e. $k=1$) from the cache. For the page $a_i$ stored in the cache, let $p(a_i)$ be the relative probability that $a_i$ is the requested page. For the cache pre-fetch problem, we are interested in the probability of having a miss. The total probability that the page will be missed in the proxy is given by $\pi_{min}(1) =\sum_{a_i\in A\setminus B}p(a_i)$. Eq. (\[eq:pi\_min\_bound\_relationship\]) gives the lower and upper bounds of the minimum probability of having a miss. This gives the performance of the best online algorithm for the webpage caching at the proxy server.
Cache System with Focus on Multiple Webpages
--------------------------------------------
![An example of unique case for error probability with $1\leq k \leq M$.[]{data-label="fig:cache1computerkpages"}](cache1computerkpages.pdf)
We can generalize the previous caching example to scenarios with performance evaluation on multiple webpages. Instead of one webpage, we aim at evaluating the performance of the pre-fetch algorithm for $k$ ($1\leq k \leq M$) webpages. We have two cases here, depending on how many users the system is serving.
Consider the situation that a user requests $1\leq k \leq M$ pages from the proxy server in a certain period of time (see Fig. \[fig:cache1computerkpages\]). The $k$ requested pages are unique because the requests are from a single user. This matches the conditions for $k$ unique objects discussed in Section \[subsubsec:unique\]. Similarly, we are interested in missing webpages for performance evaluation. Then the performance of the pre-fetch algorithm for this single-user multiple-page system is evaluated by the error probability bounded by (\[eq:pi\_k\_bounds\]) with $N_u$, $M_u$, and $p(a'_u)$.
![An example of repeated case for error probability with $1\leq k \leq M$.[]{data-label="fig:cachekcomputerkpages"}](cachekcomputerkpages.pdf)
Next we consider the circumstance when $k$ ($1\leq k \leq M$) users request webpages from the proxy server at the same time (see Fig. \[fig:cachekcomputerkpages\]), where each user requests one page. Since the $k$ users are independent, the pages requested may be identical. This matches the conditions for $k$ possibly repeated objects discussed in Section \[subsubsec:repeated\]. Then the performance of the pre-fetch algorithm for this multiple-user single-page system is evaluated by the error probability bounded by (\[eq:pi\_k\_bounds\]) with $N_r$, $M_r$, and $p(a'_r)$.[^8]
Opportunistic Scheduling
------------------------
![An example for merit probability with $k=M$.[]{data-label="fig:scheduling"}](scheduling.pdf)
The result can be applied to opportunistic scheduling in cellular data networks [@Entropy:scheduling]. Consider that a cellular network consists of a base station and $N$ mobile clients (see Fig. \[fig:scheduling\]). There are $M$ ($M\leq N$) channels for communication. Suppose that each client requires continuous communication with the base station and needs to secure a unique channel for successful data transfer. The scheduling is done by the base station. In other words, the base station assigns the channels to the users. In order to maximize the system throughput, the base station tries to select $M$ clients with high potential of acquiring good channel conditions. For example, users who are closer to the base station are less likely to suffer from interference and thus their adopted channels are more likely to have high data rates. Since the clients are not static, we model the relative probability of client $a_i$ having good channel condition with $p(a_i)$ [^9]. We are interested in merit more than error, as merit is more related to the common performance metrics (e.g. throughput) in computer networks. Since each channel can only sustain one user, this matches the conditions for $k$ unique objects discussed in Section \[subsubsec:unique\]. The performance of the scheduling algorithm is evaluated by the merit probability bounded by (\[eq:psi\_k\_bounds\]) with $N_u$, $M_u$, and $p(a'_u)$.
Conclusion {#sec:conclusion}
==========
Resource-constrained communication systems are common in engineering. In this paper, we propose a model to describe such systems, which forms a framework to evaluate the performance of resource allocation algorithms. These algorithms attempt to make good use of the resources in order to achieve better system performance. However, tailoring the optimal algorithm to suit a particular system configuration best is extremely difficult. Moreover, we do not have complete information about the system due to lack of knowledge and/or the random nature of the system. We can, at best, describe current information of the system with probability and entropy. Based on the entropy, we derive the upper and lower bounds of the performance of the optimal algorithm. The bounds give us hints on whether we should put additional efforts on developing an algorithm with respect to the existing knowledge or on collecting more accurate information about the system. To demonstrate the usability of our results, we have given several examples of resource-constrained communication systems, including various cache pre-fetching scenarios and opportunistic scheduling. Our contributions include: 1) correcting a flaw in a published lower bound of the error probability, 2) determining the minimum entropy with the resource constraints, 3) proposing a model of resource-constrained communication systems, 4) deriving an upper bound of the error probability, 5) introducing the merit probability with its upper and lower bounds, 6) generalizing the results to systems with more general performance requirements, and 7) identifying several applications.
*A. Proof of Lemma \[lemma:use\_concave\]* Define $\alpha \stackrel{\Delta}{=}\frac{x_1-x_2}{x_1-x_2+\delta}$. We can then write $x_1=\alpha(x_1+\delta)+(1-\alpha)x_2$ and $x_2=\alpha(x_2+\delta)+(1-\alpha)x_1$. By the strict concavity of $f_e(\cdot)$, we can write, $$\begin{aligned}
f_e(x_1)& >\alpha f_e(x_1+\delta)+(1-\alpha)f_e(x_2), \label{concaveproof1}\\
f_e(x_2)& >\alpha f_e(x_2+\delta)+(1-\alpha)f_e(x_1). \label{concaveproof2}\end{aligned}$$ Then we can get by summing and . *B. Proof of Theorem \[theorem:max\]* The result can be simply derived by applying the Kuhn-Tucker conditions to (\[eq:max\]) and –.
*C. Proof of Lemma \[lemma:min1toM\]* Suppose there is a $P_{1\rightarrow M}$, whose sum is equal to $(1-\pi)$ and $p(1)\geq \ldots \geq p(M)=p'$. Consider $p(M-1)>p'$ and let $\beta=p(M-1)-p'>0$. By Lemma \[lemma:use\_concave\], we can always assign $p(M-1)$ to $p'$ and $p(1)$ to $p(1)+ \beta$ and the resulting entropy becomes smaller. Similarly, we apply Lemma \[lemma:use\_concave\] to $p(M-2),...,p(2)$, we get (\[eq:min1toM\]) and its entropy is minimum.
*D. Proof of Lemma \[lemma:minM+1toN\]* It can be easily proved by following the same logic as in the proof of Lemma \[lemma:min1toM\]. Moreover, this theorem can also be proved by straightforward verification of the Karush-Kuhn-Tucker conditions.
*E. Proof of Lemma \[lemma:pequalp\]* As $p(M)\geq p(M+1)$, inequality $p'\geq p''$ must hold.
Consider a distribution $P_1$ with $p'>p''$. By Lemma \[lemma:use\_concave\], we can always find a positive real number $\lambda \in (0,p'-p'']$, such that we can produce $P'_1$, which is identical to $P_1$ except $p'_1(1)=p_1(1)+\lambda$ and $p'_1(M)=p'-\lambda$, with lower entropy.
Similarly, consider a distribution $P_2$ with $p'>p''$. Let $p_2(k)$ be the last non-zero element in $P_2=[p'',p(M+2),\ldots ,p_2(k),0,\ldots ,0]$. We can always find a positive real number $\xi \in (0,p'-p'']$, such that we can produce $P'_2$, which is identical to $P_2$ except $p'_2(M+1)=p''+\xi$ and $p'_2(k)=p_2(k)-\xi$, with lower entropy.
By combining the effects on $\lambda$ and $\xi$, we can deduce that a distribution with $p'=p''$ has smaller entropy than another with $p'\neq p''$. The one with the lowest entropy is $P^{min}$, and thus, $P^{min}$ must have $\hat{p}=p'=p''$. With (\[eq:p\_range\]), $\hat{p}\in[\frac{\pi}{N-M},\frac{1-\pi}{M}]$. *F. Proof of Theorem \[theorem:entropy\_lower\_bound\]* Since $0\leq (1-\pi)-(M-1)\hat{p} \leq 1$ and $0\leq \pi-(N-M-y)\hat{p} \leq 1$, with $\hat{p} \in \hat{P}^*$, we have $$\begin{aligned}
0\leq - [(1-\pi)-(M-1)\hat{p}]\log [(1-\pi)-(M-1)\hat{p}] \leq 1\end{aligned}$$ and $$\begin{aligned}
0\leq -[\pi-((N-M-y)\hat{p})]\log[\pi-((N-M-y)\hat{p})] \leq 1.\end{aligned}$$ By relaxing (\[eq:discrete\_H\]), we have $$\begin{aligned}
H(\hat{p}) &\geq \min \left\{
\begin{array}{ll}
-(N-1)\hat{p}\log\hat{p}, & \hat{p}=\frac{\pi}{N-M},\\
-(N-2)\hat{p}\log\hat{p}, & \hat{p}=\frac{\pi}{N-M-1},\\
\qquad\qquad \vdots & \qquad \vdots \\
-(N-y)\hat{p}\log\hat{p}, & \hat{p}=\frac{\pi}{N-M-y+1},\\
-(N-y)\hat{p}\log\hat{p}, & \hat{p}=\frac{1-\pi}{M},\\
\end{array} \right\} \nonumber\\
&= \min \left\{
\begin{array}{l}
-\frac{(N-1)\pi}{N-M}\log\frac{\pi}{N-M},\\
-\frac{(N-2)\pi}{N-M-1}\log\frac{\pi}{N-M-1},\\
\qquad\qquad \vdots \\
-\frac{(N-y)\pi}{N-M-y+1}\log\frac{\pi}{N-M-y+1},\\
-\frac{(N-y)(1-\pi)}{M}\log\frac{1-\pi}{M} \\
\end{array} \right\} \label{eq:simplified_lower_bounds}
\end{aligned}$$ By (\[eq:mean\]), we have $0\leq \pi \leq 1-\frac{M}{N}$. We can further relax (\[eq:simplified\_lower\_bounds\]) by replacing $\pi$ in the log functions with $(1-\frac{M}{N})$. Hence, $$\begin{aligned}
H(\hat{p}) &\geq \min \left\{
\begin{array}{l}
\frac{(N-1)\pi}{N-M}\log \frac{N-M}{N(N-M)},\\
\frac{(N-2)\pi}{N-M-1}\log \frac{N-M}{N(N-M-1)},\\
\qquad\qquad \vdots \\
\frac{(N-y)\pi}{N-M-y+1}\log \frac{N-M}{N(N-M-y+1)},\\
\frac{(N-y)(1-\pi)}{M}\log M -\frac{(N-y)(1-\pi)}{M}\log (1-\pi) \\
\end{array} \right\}. \\
\end{aligned}$$ Since $ 0 \leq -(1-\pi)\log (1-\pi) \leq 1$, $$\begin{aligned}
\label{eq:entropy_lower_bound}
H(\hat{p}) \geq \min(\Omega).
\end{aligned}$$
*G. Proof of Theorem \[corollary:pi\_bounds\]* From Theorem \[theorem:entropy\_lower\_bound\], $H$ is no smaller than the minimum of $\Omega$. By rearranging the expressions, $$\begin{aligned}
\label{eq:proofbound1}
\pi\leq \max \left\{
\begin{array}{l}
\frac{H\cdot (N-M)}{(N-1)\log \frac{N(N-M)}{N-M}},\\
\frac{H\cdot (N-M-1)}{(N-2)\log \frac{N(N-M-1)}{N-M}},\\
\qquad\qquad \vdots \\
\frac{H\cdot (N-M-y+1)}{(N-y)\log \frac{N(N-M-y+1)}{N-M}},\\
\frac{H\cdot M}{(N-y)\log \frac{1}{M}}+1\\
\end{array} \right\}.\end{aligned}$$ Since $0\leq \pi\leq 1$, from , we have $$\begin{aligned}
\label{eq:proofbound2}
-\infty\leq y\leq N-M.\end{aligned}$$ Relaxing with , together with Corollary \[corollary:pi\_lower\_bound\_revised\], gives the result.
*H. Proof of Theorem \[theorem:merit\_1\_bounds\]* By combining (\[eq:pi\_min\_bound\_relationship\]) and (\[eq:psi1\]), we have $$\begin{aligned}
1- \max \left\{
\begin{array}{l}
\frac{H\cdot (N-M)}{(N-1)\log \frac{N(N-M)}{N-M}},\\
\frac{H\cdot (N-M-1)}{(N-2)\log \frac{N(N-M-1)}{N-M}},\\
\qquad\quad \vdots \\
\frac{H\cdot (N-M-y+1)}{(N-y)\log \frac{N(N-M-y+1)}{N-M}},\\
\frac{H\cdot M}{(N-y)\log \frac{1}{M}}+1\\
\end{array} \right\} \\
\leq \psi_{max}(1) \leq 1-\frac{H-1-\log M}{\log (\frac{N}{M}-1)}.\end{aligned}$$ Simplification gives the result.
*I. Proof of Theorem \[theorem:error\_k\_bounds\]* For the unique case, in (\[eq:pi\_min\_bound\_relationship\]), we can substitute $N$ and $M$ with (\[eq:new\_N\_u\]) and (\[eq:new\_N\_u\]), respectively. $H$ is composed of $p(a_i)$ and we can find $H'$ with (\[eq:new\_p\_u\]). $y'$ can be produced with $N'$ and $M'$ according to (\[eq:y\]). The repeated case works similarly.
*J. Proof of Theorem \[theorem:merit\_k\_bounds\]* The proof is similar to that of *Theorem \[theorem:error\_k\_bounds\]*, but using (\[eq:merit\_1\_bounds\]) instead.
[^1]: A.Y.S. Lam is with the Department of Computer Science, Hong Kong Baptist University (e-mail: albertlam@ieee.org). Y. Geng is with Huawei Noah’s Ark Lab (e-mail: geng.yanhui@huawei.com). V.O.K. Li is with the Department of Electrical and Electronic Engineering, The University of Hong Kong (e-mail: vli@eee.hku.hk).
[^2]: For the sake of simplicity, we assume all uncertainty is due to the system. We will consider $H(X)$ hereafter.
[^3]: Assume that the larger the value of $c\in C$, the better the performance.
[^4]: The proofs of all the lemmas, theorems, and corollaries are included as an appendix.
[^5]: A $k$-combination is an un-ordered collection of distinct elements, of prescribed size $k$ and taken from a given set.
[^6]: A permutation is an ordered collection of distinct elements taken from a given set.
[^7]: $\left({N \choose k}\right)$ means “$N$ multichoose $k$”. Consider a multiset of cardinality $k$ with elements chosen from a set of cardinality $N$, $\left({N \choose k}\right)$ is the number of available combinations [@multiset].
[^8]: Our results can be further generalized to the multiple-user multiple-page system. Since the ideas are similar, we do not repeat the discussion here.
[^9]: The probability can be estimated from the mobility model used for the clients.
|
---
abstract: 'We report measurements of an intensity-field correlation function of the resonance fluorescence of a single trapped $^{138}$Ba$^+$ ion. Detection of a photon prepares the atom in its ground state and we observe its subsequent evolution under interaction with a laser field of well defined phase. We record the regression of the resonance fluorescence source field. This provides a direct measurement of the field of the radiating dipole of a single atom and exhibits its strong non-classical behavior. In the setup an interference measurement is conditioned on the detection of a fluorescence photon.'
author:
- 'S. Gerber$^1$, D. Rotter$^1$, L. Slodička$^1$, J. Eschner$^{1,4}$, H. J. Carmichael$^3$, and R. Blatt$^{1,2}$'
title: 'Intensity-field correlation of single-atom resonance fluorescence '
---
Resonance fluorescence of atoms, in particular individual atoms, has been the subject of quantum optical measurements for many years [@Mandel; @Loudon]. For example, resonance fluorescence is routinely used as a tool to simply detect atoms, for spectroscopy purposes, for creating photons, including for single and twin-photon sources, and its quadrature components have been used to create non-classical states of light [@Loudon]. In short, the observation of resonance fluorescence is a technology ubiquitous in experimental physics. While its features are well investigated and understood, for spectroscopy and in quantum optics, a direct observation of the time evolution of the source field at the single-atom, single-photon level has not been made.
The measurement of correlation functions, sensitive to the source-field, has been proposed in the seminal paper by W. Vogel [@Vogel]. In a first and pioneering experiment, G. T. Foster et al. have been able to report a wave-particle correlation function of the field that emanates out of a cavity and corresponds on average to only a fraction of a photon excitation [@OroCar; @CarOro; @OroPRA]. Correlation functions of the fields comprised of many photons have been observed with lasers and other light sources [@Mandel; @Loudon; @Twiss2] and were approached theoretically [@Laserphys]. However, to the best of our knowledge, the time evolution of the field that corresponds to a single resonance fluorescence photon has not been recorded so far. Moreover, the only previous observation of the intensity-field correlation [@OroCar; @OroPRA] made use of a strong local oscillator with photocurrent detection. The reported measurement employs a weak local oscillator and photon counting and thus operates in a different regime.
Motivation for this investigation is the development of a tool (i) to investigate and monitor the emission of a single-photon field and at a later stage (ii) to detect the influence of boundary conditions, such as walls, mirrors, other atoms and quite generally of a different (engineered) bath on the dynamics of the emission process. For all of these tasks it will be necessary to monitor the resonance fluorescence field and possibly even feed-back [@Eschner; @Bushev1; @Bushev2; @Uwe] on the radiating dipole.
In this letter, we report on the measurement of a third-order correlation function of a single radiating atom, using standard photon counting techniques. Using a homodyne detection scheme, we record the resonance fluorescence field conditioned on the detection of an initial resonance fluorescence photon that prepares the atom in its ground state. Since the correlation function of two fields is termed $g^{(1)}$ and that of two intensities is termed $g^{(2)}$ [@Twiss], we accordingly coin the name $g^{(1.5)}$ for this third-order intensity-field correlation.
In this work the correlation measurement is triggered by detecting a fluorescence photon from a single trapped $^{138}$Ba$^+$ ion, which projects the ion into its ground state. Stop events are obtained from a homodyne detector, where the fluorescence interferes with a local oscillator (LO) of well-controlled phase relative to the exciting laser. The experimental setup is interferometrically stabilized and the phase of the LO can be adjusted to anywhere within \[0, 2$\pi$\]. The measurement is repeated many times for different phases of the LO- field, such that the integrated signal records the average conditional time evolution of the (fluctuating) amplitude of the electromagnetic wave that constitutes the emission of a single resonance fluorescence photon.
The schematic experimental setup and the level scheme of the $^{138}$Ba$^+$ ion are shown in Fig. \[fig1\]. A single Ba$^+$ ion is loaded in a linear Paul trap using photo-ionization with laser light near 413 nm [@Rotter_diss]. The ion is confined in the harmonic pseudo-potential of the trap with radial (axial) oscillation frequency $\sim$ 1.7 MHz ($\sim$1 MHz). Micromotion is minimized using 3 pairs of dc electrodes. The ion is continuously laser-cooled by two narrow-band (laser linewidth of a few tens of kHz) linearly polarized tunable lasers at 493 nm (green) and 650 nm (red) exciting the S$_{1/2}$–P$_{1/2}$ and P$_{1/2}$–D$_{3/2}$ transitions, respectively. The green laser intensity is adjusted to give mostly elastically scattered photons [@Mandel]. After Doppler cooling, the ion is left in a thermal motional state with a mean number of vibrational excitation $<\widehat{n}>\,\approx15$. A weak magnetic field defines a quantization axis perpendicular to the laser polarization $\overrightarrow{E}$ and vector. Including the Zeeman substates, the internal structure of the atom is described as an 8-level system with the lasers exciting $\Delta m \pm 1$ transitions [@Toschek].
Resonance fluorescence is detected in channels aligned along the quantization axis in both directions. About 4$\%$ of the green fluorescence is collected with two custom-made lenses (HALO (LINOS), NA=0.4) placed about 1 cm from the trap center to the left and right side of the trap.
![ (Color online) Sketch of the experimental setup: A single $^{138}$Ba$^{+}$ ion in a linear Paul trap is continuously laser excited. Two detection channels, left and right, allow for visual observation of the ion (CCD), or for recording correlations in the emitted light. The LO is coupled out by $\rm M_1$ in front of the trap, attenuated (Att.), and its polarization is adjusted with a $\lambda/2$ plate to match the polarization of the fluorescence beam. The inset shows the relevant electronic levels of $^{138}$Ba$^{+}$. []{data-label="fig1"}](figure1new.jpg){width="8.75cm"}
The left beam can either be sent to a PMT (PMT-start) or to a CCD camera. On the right hand side the fluorescence beam is collimated with a telescope and then mixed with the LO field on a mirror $\rm
M_2$ with 99$\%$ reflectivity. After coupling to a single mode optical fiber for mode matching, the interfering fields are detected at another PMT (PMT-stop) leaving a count rate of about 10 kcps for the fluorescence after the fiber. In both detection channels a quarter wave-plate and a Glan-Thompson polarizer select $\Delta m =+1$ photons and filter out the $\Delta m =-1$ transition. The phase $\Phi$ of the interferometer is controlled with a Piezo mounted mirror in the LO path by monitoring the count rate of the homodyne signal. Thus the error in the phase of the LO is given by the shot noise of this signal and is estimated to about $\pm$10 degrees ($\phi=\pi/2$) and $\pm$24 degrees ($\phi=0$ or $\pi$) for a typical integration time of 0.1 s. Phase locking by keeping the homodyne count rate constant is continuous with a time constant of several seconds and does not affect the contrast of our data within the limits set by the shot noise. Correlations between the PMT start and the PMT stop-counts are obtained by recording single photon arrival times with a Time Acquisition Card (Correlator) with up to 100 ps resolution.
For a theoretical analysis we consider a frame rotating at the (green) laser frequency, $\omega_L$. Thus, the green source part of the radiated field by the ion reads $$\widehat{E}(t)=\xi
e^{-i\omega_Lt}\hat\sigma^-(t),\label{eq1}$$ where $t$ is in the long-time limit after the exciting laser is turned on, $\xi$ represents a constant amplitude, and $\hat\sigma^-$ is the Pauli lowering operator from $|P_{1/2},m=-1/2>$ to $|S_{1/2},m=+1/2>$, associated with a creation of a single $\Delta m = +1$ photon. With the LO path blocked we measure the conventional normalized second order (intensity) correlation, $g^{(2)}(\tau)\propto\langle\hat E^\dagger(0)\hat E^\dagger(\tau)\hat
E(\tau)\hat E(0)\rangle\equiv\lim_{t\to\infty}\langle\hat E^\dagger(t)\hat
E^\dagger(t+\tau)\hat E(t+\tau)\hat E(t)\rangle$. In terms of atomic operators it reads $$g^{(2)}(\tau)=\frac{\langle\hat\sigma^+(0)\hat\sigma^+
(\tau)\hat\sigma^-(\tau)\hat\sigma^-(0)\rangle}
{\langle\hat\sigma^+(0)\hat\sigma^-(0)\rangle^2}.\label{eq2}$$ Figure \[fig6\] depicts a measurement of this quantity. It exhibits the characteristic anti-bunching at short time, with a null rate of coincidences, $g^{(2)}(0)= 0.042(2)$ (without background subtraction). Aside from this minor offset, it is well reproduced by our 8-level Bloch simulations. Fitting parameters are the laser powers and detunings. Thus, the $g^{(2)}(\tau)$ is used for calibrating the laser settings for the $g^{(1.5)}(\tau)$ measurement.
![ (Color online) Obtained correlation function with the LO path blocked. The solid line shows the theoretical prediction using experimental parameters and 8-level optical Bloch equations. []{data-label="fig6"}](figure2new.jpg){width="8.75cm"}
With the LO arm unblocked, we measure the homodyne signal conditioned on a photon emission from the ion, where the phase $\Phi$ of the LO can be adjusted. We now write the detected fields in units of the square root of photon flux. $\gamma_1\langle\hat\sigma^+\hat\sigma^-\rangle$ represents the mean photon flux into the PMT-start, where $\gamma_1$ is the product of the radiative decay rate and the overall collection and detection efficiency of the PMT-start. We similarly denote the fluorescence field at the PMT-stop by $\sqrt{\gamma_2}\hat\sigma^-(0)$, where $\gamma_2$ is the product of radiative decay rate and the respective collection and detection efficiency of a photon at the PMT-stop. Then representing the local oscillator field by the complex amplitude $\mathcal{E} e^{i\Phi}$ the field after the interferometer reads $$X_\Phi(t)=[\mathcal{E} e^{i\Phi}+ \sqrt{\gamma_2}\hat\sigma^-(t)],$$ and for positive $\tau$ we measure a total unnormalized second-order correlation $$G^{\rm total}_\Phi(t,t+\tau)=\langle\sqrt{\gamma_1}\hat\sigma^+(t)X_{\Phi}^\dag(t+\tau)
X_{\Phi}(t+\tau)\sqrt{\gamma_1}\hat\sigma^-(t)\rangle, \label{eq4}$$ which expands out to $$G^{\rm total}_\Phi(\tau)=F\big\{(1-V)[(1-r)+rg^{(2)}(\tau)]+V g^{(1.5)}_\Phi(\tau)\big\}
.\label{eq5}$$ Here, $g^{(2)}(\tau)$ is the intensity correlation function given by Eq.(\[eq2\]), and for the third-order correlation function at a given LO phase we write $$g^{(1.5)}_\Phi(\tau)=\frac{\langle
\hat\sigma^+(0)[e^{i\Phi}\hat\sigma^+(\tau)+e^{-i\Phi}\hat\sigma^-(\tau)]\hat\sigma^-(0)\rangle}
{\langle\hat\sigma^+
+\hat\sigma^-\rangle\langle\hat\sigma^-\hat\sigma^+\rangle }.
\label{eq6}$$ For later convenience we define the LO phase relative to the asymptotic phase of the resonance fluorescence field, such that $g^{(1.5)}_{\pi/2}(\tau \rightarrow \infty)=0$. The pre-factor $F$ in Eq.(\[eq5\]) is $$F=\gamma_1\langle\hat\sigma^+\hat\sigma^-\rangle(\mathcal{E}^2 +\mathcal{E}\sqrt{\gamma_2}
\langle\hat\sigma^+ +\hat\sigma^-\rangle+ \gamma_2 \langle\hat\sigma^+ \hat\sigma^-\rangle),$$ while $$V=\frac{\mathcal{E}\sqrt{\gamma_2}\langle\hat\sigma^+ +\hat\sigma^-\rangle}
{\mathcal{E}^2+\mathcal{E}\sqrt{\gamma_2}\langle\hat\sigma^+ +\hat\sigma^-\rangle+\gamma_2
\langle\hat\sigma^+ \hat\sigma^-\rangle}$$ is the visibility of the interference part in $G^{\rm total}_\Phi(\tau)$ and $$r=\frac{\gamma_2\langle\hat\sigma^+\hat\sigma^-\rangle}{\mathcal{E}^2+\gamma_2\langle\hat\sigma^+\hat\sigma^-\rangle}\\$$ is the ratio of the florescence intensity to the total intensity at the PMT-stop.
According to Eq.(\[eq5\]), the expected correlation function consists of three parts. A $\Phi$-dependent part with visibility $V$ reveals the $g^{(1.5)}(\tau)$ correlation due to the interference of the fluorescence with the LO. The remaining non-interfering part with weight $1-V$ consists of a nor- mal second-order correlation function $g^{(2)}(\tau)$ (both start and stop counts from fluorescence photons) weighted by $r$ and a constant offset (stop counts from LO photons) weighted by $1-r$. Normalizing $G^{\rm total}_\Phi(\tau)$ by $F(1-V)$ we obtain $$\begin{aligned}
g^{\rm total}_\Phi(\tau)=(1-r)+rg^{(2)}(\tau)
+\frac{V}{1-V}g^{(1.5)}_\Phi(\tau).\label{eq10}\end{aligned}$$ This normalization is chosen such that at a phase $\Phi=\pi/2$ of the LO, when $g^{(1.5)}(\tau)$ vanishes asymptotically, $g^{\rm total}(\tau)$ yields an asymptotic value of 1.
Figure \[fig2\] shows the measured correlations between PMT-start and PMT-stop with the LO phase adjusted to $\Phi=0,\pi/2$ and $\pi$. Data are acquired after 30 minutes of accumulation for each curve and presented with a 1 ns resolution. The corresponding variance is obtained from shot noise, i.e. assuming Poisson statistics at all times $\tau$. The solid curves show the theoretical prediction using Eq.(\[eq10\]) with a visibility $V\sim18\%$ and an intensity ratio $r=0.31$. The measured correlations are well reproduced by superposition of the three contributions described by Eq.(\[eq10\]).
![ (Color online) Measured normalized correlation function $g^{\rm
total}_\Phi(\tau)$ obtained a) at $\Phi=0$; i.e. the fluorescence being in-phase with the LO , c) at $\Phi=\pi$; i.e. the fluorescence being out-of-phase with the LO and b) at $\Phi=\pi/2$. The solid curves show the theoretical predictions using Eq.(\[eq10\]). []{data-label="fig2"}](figure3new.jpg){width="8.75cm"}
All curves contain a constant contribution and a scaled $g^{(2)}(\tau)$ correlation. In addition, curve b), where the LO phase is set to $\Phi=\pi/2$, contains the imaginary part of the atomic polarization whose asymptotic contribution is zero. In contrast, curves a) and c), where the LO phase is set to 0 and $\pi$, respectively, reveal the real part of the polarization which adds constructively or destructively to the other two contributions. All curves show the same coincidence rate at $\tau=0$. Since both the $g^{(2)}(\tau)$ contribution and the $g^{(1.5)}_\Phi(\tau)$ contribution are identically zero at $\tau=0$, the measured coincidence rate at this point is solely determined by the offset of non- interfering LO photons at PMT-stop (and background counts). Calibration of the LO phase is obtained by looking for the maximum and minimum asymptotic values of $G^{\rm total}_\Phi(\tau)$ and assigning to them the LO phases $\Phi=0$ and $\pi$, respectively.
Determining the full complex $g^{(1.5)}(\tau)$ intensity field correlation function requires its measurement for two orthogonal phases. We deduce it from the data in Fig. \[fig2\] and using Eq.(\[eq10\]) in the following way: $$g^{(1.5)}_0(\tau)=\frac{1-V}{2V}( g^{\rm total}_0(\tau)-g^{\rm total}_\pi(\tau))
\label{eq8}$$ and $$g^{(1.5)}_{\pi/2}(\tau)=\frac{1-V}{2V}\Big( 2g^{\rm total}_{\pi/2}(\tau)-
(g^{\rm total}_0(\tau)+g^{\rm total}_\pi(\tau))\Big).
\label{eq11}$$ The result is shown in Fig. \[fig4\] together with the theoretical prediction of Eq.(\[eq10\]) (solid line). The data reveal the time evolution of the fluorescence field as it evolves from its initialization by an emitted photon into steady-state via damped Rabi oscillations. Comparing Fig. \[fig4\] a) with Fig. \[fig6\] we see that the $g^{(1.5)}(\tau)$ grows linearly with $\tau$ around the point $\tau=0$ while the $g^{(2)}(\tau)$ grows quadratically with $\tau$ showing clearly that the field rather than the intensity is measured.
![(Color online) The intensity-field correlation function $g^{(1.5)}_\Phi(\tau)$ with the LO phase adjusted to a) $\Phi=0$ and b) $\Phi=\pi/2$, obtained from the data in Fig. \[fig2\] using Eq.(\[eq8\]) and Eq.(\[eq11\]), respectively. The solid lines represent the theoretical model using Eq.(\[eq6\]). []{data-label="fig4"}](figure4new.jpg){width="8.75cm"}
The limitation for the visibility in the homodyne part of the setup is determined by the temporal overlap of two photon wave packets impinging at the mixing mirror. It is limited by the coherence time and the flux of the fluorescence photons with respect to the LO photons. Assuming elastically scattered photons, the coherence time is given by the 493 nm laser bandwidth with $T=1/\Delta \nu\approx\,$50$\,\mu s$ [@Raab]. The fluorescence count-rate of $10$kcps is predetermined by the collected fraction of solid angle and the fiber-coupling efficiency. While the temporal overlap would benefit from a higher LO intensity (smaller $r$), the visibility of the interferometer would decrease and locking the interferometer would get more involved due to a larger shot-noise in the LO arm. Thus, optimizing the experiment resulted in a reduced visibility of $V\sim18\,\%$ and an intensity ratio of $r=0.31$.
In summary, we have successfully measured an intensity-field correlation function of the resonance fluorescence from a single $^{138}$Ba$^+$ ion. In our setup a photon detection from the ion starts the correlation measurement in a well defined state that evolves to steady-state via Rabi oscillations. The correlation function $g^{(1.5)}$ was obtained from the measured data points recorded with the LO being in and out-of phase with the fluorescence.
This measurement clearly shows the dynamical behavior of the atomic dipole. In principle, these measurements now allow for a detailed investigation of the fluctuating dipole and its non-classical statistics that leads to the fact that resonance fluorescence produces inherently squeezed light [@Mand]. It is possible in principle to observe this effect using the third-order correlation function [@Vogel; @OroPRA]. However, for this, the single atom must be only weakly excited which was not the case in the present experiment. Observation of the squeezing of the single-ion resonance fluorescence will be subject to further investigations. Finally, the current procedure will allow us to investigate the radiating dipole field under the influence of direct backaction [@Eschner] and in the presence of boundary conditions [@Uwe; @Dubin] and active feedback [@Bushev2].
We thank L. Orozco for valuable discussions. This work has been partially supported by the Austrian Science Fund FWF (Project No. SFB F-015) and by the Institut f$\ddot{\rm u}$r Quanteninformation GmbH. J.E. acknowledges support by the European Commission (“EMALI”, MRTN-CT-2006-035369).
[99]{} L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
R. Loudon, The Quantum Theory of Light, Clarendon Press, Oxford, 1983.
W. Vogel, Phys. Rev. Lett. **67**, 2450 (1991).
H. J. Carmichael et al., Phys. Rev. Lett **85**, 1855 (2000).
G. T. Foster et al., Phys. Rev. Lett **85**, 3149 (2000).
G. T. Foster et al., Phys. Rev. A, **66**, 033807 (2002).
R. Hanbury-Brown and R. Q. Twiss, Nature **178**, 1046 (1956).
E. R. Marquina-Cruz and H. M. Castro-Beltran, Laser Physics **18**, 157 (2008).
U. Dorner et al., Phys. Rev. A **66**, 023816 (2002).
J. Eschner et al., Nature **413**, 495 (2001).
P. Bushev et al., Phys. Rev. Lett. **92**, 223602 (2004).
P. Bushev et al., Phys. Rev. Lett. **96**, 043003 (2006).
R. Hanbury-Brown and R. Q. Twiss, Nature **177**, 27 (1956).
Daniel Rotter, Dissertation, Innsbruck 2008.
M. Schubert et al., Phys. Rev. A **52**, 2994 (1995).
Ch. Raab et al., Phys. Rev. Lett. **85**, 538 (2000).
L. Mandel, Phys. Rev. Lett. **49**, 136 (1982).
F. Dubin et al., Phys. Rev. Lett **98**, 183003 (2007).
|
---
abstract: 'We study a reflectionless $\mathcal{PT}$-symmetric quantum system described by the pair of complexified Scarf II potentials mutually displaced in the half of their pure imaginary period. Analyzing the rich set of intertwining discrete symmetries of the pair, we find an exotic supersymmetric structure based on three matrix differential operators that encode all the properties of the system, including its reflectionless (finite-gap) nature. The structure we revealed particularly sheds new light on the splitting of the discrete states into two families, related to the bound and resonance states in Hermitian Scarf II counterpart systems, on which two different series of irreducible representations of $sl(2,\mathbb{C})$ are realized.'
author:
- |
[**Francisco Correa${}^{a}$ and Mikhail S. Plyushchay${}^{b}$**]{}\
\[4pt\] [*${}^{a}$ Centro de Estudios Científicos (CECs), Casilla 1469, Valdivia, Chile*]{}\
[*${}^{b}$ Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile* ]{}\
*[ ]{}*
title: ' [**Self-isospectral tri-supersymmetry in ${\cal PT}$-symmetric quantum systems with pure imaginary periodicity** ]{}'
---
[**Keywords:**]{} *Supersymmetric quantum mechanics; ${\cal PT}$-symmetry; non-Hermitian Hamiltonians; self-isospectrality; finite-gap systems; non-linear supersymmetry.*
Introduction
============
Fourteen years ago, Bender and Boettcher discovered a huge and remarkable class of non-Hermitian quantum Hamiltonians which exhibit an entirely real spectrum . One of the key points of the observation was the requirement of a $\mathcal{PT}$-*symmetry* generated by the product of the parity, ${\cal P}$, and time, ${\cal
T}$, inversion operators, $$\label{PTdef}
{\cal P}\,x\,{\cal P}=-x,\qquad
{\cal T}\,x\,{\cal T}=x,\quad
{\cal T}\,i\,{\cal T}=-i\,,$$ that substitutes the usual quantum mechanical property of Hermiticity. This condition, however, is necessary but not sufficient for the reality of the spectrum; it is also required that eigenfunctions of a ${\cal PT}$-symmetric Hamiltonian must be simultaneously eigenfunctions of the $\mathcal{PT}$ operator. In this case we say that the ${\cal PT}$ symmetry is unbroken, otherwise the eigenvalues are not real, a part or all of them appear in complex conjugate pairs, and the ${\cal PT}$ symmetry is broken .
Nowadays, quantum mechanics with non-Hermitian Hamiltonians transformed into an independent line of research where, specifically, the notion of the ${\cal PT}$-symmetry was generalized into the condition of pseudo-Hermiticity [@Mpseudoh]. Non-Hermitian Hamiltonians appear in physics in diverse areas including quantum optics, cosmology, atomic and condensed matter physics, magnetohydrodynamics, among others. For a good review of the developments in the area and applications, see [@BenderReview; @ReviewM] and references therein.
On the other hand, there is a wide class of modern techniques and methods which prove their effectiveness in the study of quantum mechanical systems. One of them is supersymmetric quantum mechanics (SUSYQM), introduced initially by Witten as a toy model to study the spontaneous supersymmetry breaking [@witten]. Over the past few decades, SUSYQM transformed into a powerful tool in quantum physics [@susyrev1] which turns out to be useful, for example, in the spectral analysis as well as in searching for new solvable systems. SUSYQM in its usual form is based on a Darboux transformation that relates two Hamiltonians by means of an intertwining, linear differential operator, and leads, as a result, to a complete or almost complete isospectrality of both systems [@MatSal].
The concept of Darboux transformations and SUSYQM was generalized in different aspects that lead to the discovery of the classes of quantum systems that reveal non-linear [@nSUSY; @MPhidnon; @KlMP], bosonized [@MPhidnon; @bosonized; @boso1; @boso2] and self-isospectral [@Dunne; @Fetal; @FerNegNie; @self; @trisusy] supersymmetries. Particularly, a certain class of potentials was found in which all the mentioned specific types of supersymmetries were brought to light in a form of a peculiar structure that was coined in [@trisusy] as “*tri-supersymmetry*". Such a structure was shown to underlie special properties of some physical quantum systems [@trisusydelta; @ABtrisusy; @newself1; @newself2; @newself3]. Among the principal characteristics of tri-supersymmetry, which will be described below, is the existence of three integrals of motion in the form of supercharges which encode the main properties of the corresponding systems. An example of the systems that reveal a tri-supersymmetric structure is provided by the Hermitian *finite-gap* periodic potentials [@trisusy], which in the limit when their real period tends to infinity are known as *reflectionless* potentials.
The idea of supersymmetry in some of its versions was adapted in the context of non-Hermitian Hamiltonians [@susypt1; @susypt2; @susypt3; @susypt4; @Znojil2000; @susypt5; @Dorey; @Mpseudosusy]. Although the non-linear and self-isospectral supersymmetries were studied before in the systems with non-Hermitian Hamiltonians [@susynonlinearpt], the presence in them of a supersymmetric structure that would unify the mentioned types of supersymmetry remains to be unknown. One can wonder therefore if there exist non-Hermitian potentials of finite-gap nature that display the properties to be similar to those of the Hermitian counterpart, and what are the peculiarities of the associated supersymmetric structure.
The purpose of the present article is to report the observation of an exotic tri-supersymmetric structure in a broad class of ${\cal
PT}$-symmetric extended finite-gap systems. This allows us, on the one hand, to understand and explain the main features of their spectra by analyzing the properties of the supercharges; on the other hand, we explicitly trace out the differences that appear in comparison with the Hermitian case, at the level of potentials as well as in the generic properties of tri-supersymmetry. The importance of a hidden, pure imaginary period in non-periodic on a real line finite-gap systems is clarified, particularly, in the light of tri-supersymmetric structure. We achieve all this by considering the pairs of ${\cal PT}$-symmetric complexified Scarf II potentials mutually displaced in the half of their unique imaginary period. For special values of the parameters these potentials become perfectly transparent, i.e. reflectionless. It is exactly for such special parameter values the tri-supersymmetric structure arises and allows us to clarify its nontrivial interplay with various discrete symmetries of the potentials. The supersymmetric structure we reveal gives us a new insight on the splitting of discrete states in such a class of $\mathcal{PT}$-symmetric quantum systems into two distinct families, on which two different representations of the $sl(2,\mathbb{C})$ algebra are realized, the fact that was established initially by Bagchi and Quesne by using group theoretical methods [@GrNe]. Note here that the complexified Scarf II potential was extensively studied in the literature in the context of ${\cal PT}$-symmetric and pseudo-Hermitian Hamiltonians; the updated summary on these investigations can be found in ref. [@BagQues]. Recently, this potential also attracted the attention in different areas of physics such as quantum field theory in curved spacetimes [@Maloney], soliton theory in nonlinear integrable systems [@Wadati] and also the physics of optical solitons [@Optics].
The plan of the article is as follows. In sections \[sec1\] and \[eigen\] the basic properties of the pair of complexified mutually conjugated Scarf II potentials are reviewed. Specifically, in the next section we study various discrete symmetries of the potentials, describe spectral properties of the systems in dependence on the parameter values, and describe the relation with the other known potentials. In section \[eigen\] we first present the explicit expressions for the two families of the singlet states in the spectrum, and then discuss the continuous spectrum and its relation with that of the free particle by means of Darboux-Crum transformations that underlie the non-linear supersymmetry. The tri-supersymmetric structure is described in section \[susysec\]. Section \[examples\] provides two concrete nontrivial examples of the systems with two and three bound states to illustrate the general results. In section \[discus\] we present the discussion and concluding remarks.
Discrete symmetries and relations {#sec1}
=================================
Consider a pair of complexified Scarf II potentials $$\label{po1}
V_{l,m}^{\pm}(x)=-\frac{l^2+m(m+1)}{\cosh^2x}\pm i l
(2m+1)\frac{\sinh x}
{\cosh^2 x} \,\, .$$ We suppose that $l$ and $m$ are real parameters, and $x\in\R$. Potentials (\[po1\]) are then free of singularities on the real line, and their real and imaginary parts vanish for $x\rightarrow
\pm \infty$, see Fig. \[FigaSc\].
![Plot of real and imaginary parts of the complexified Scarf II potential $V^+_{1,1}$, which here as well as in a generic case of (\[po1\]) are even and odd functions, respectively.[]{data-label="FigaSc"}](figure1.eps)
Pair (\[po1\]) can be transformed into a pair of original, real Scarf II potentials [@Scarf2] by a substitution $l\rightarrow
il$, see Fig. \[Scarf\].
![Examples of original (real) Scarf II potentials.[]{data-label="Scarf"}](figure2.eps)
Both the original and complexified potentials have a pure imaginary period $2i\pi$.
Parity inversion, ${\cal P}$, and time reversal, ${\cal T}$, operators intertwine the completely isospectral potentials (\[po1\]), $$\label{pt1}
{\cal P}V_{l,m}^{\pm}=V_{l,m}^{\mp}{\cal P},
\qquad {\cal
T}V_{l,m}^{\pm}=V_{l,m}^{\mp}{\cal
T} .$$ As $V^\pm_{-l,m}=V^\pm_{l,-m-1}=V^\mp_{l,m}$, another pair of intertwiners is provided by operators $\mathcal{R}_l$ and $\mathcal{R}_m$, which act on the parameters $l$ and $m$, $\mathcal{R}_l:\,(l,m)\rightarrow (-l,m)$, $\mathcal{R}_m : (l,m) \rightarrow (l,-m-1)$, $$\label{Rlm}
\mathcal{R}_l V_{l,m}^{\pm}=V_{l,m}^{\mp}
\mathcal{R}_l,
\qquad
\mathcal{R}_m V_{l,m}^{\pm}=
V_{l,m}^{\mp}\mathcal{R}_m\,.$$ Yet another pair of intertwining operators is given by pure imaginary translations for the half of the period $2i\pi$, $T_+:\
x\rightarrow x+i\pi$ and $T_-:\ x\rightarrow x-i\pi$, $$\label{T+-}
T_+ V_{l,m}^{\pm}=V_{l,m}^{\mp}
T_+,
\qquad
T_- V_{l,m}^{\pm}=
V_{l,m}^{\mp}T_-\,.$$ The product of any two of the listed intertwining operators, except of $T_+=\exp(\pi\frac{d}{dx})$ and $T_-=\exp(-\pi\frac{d}{dx})$, $T_+T_-=1$, is a nontrivial discrete symmetry of each of potentials (\[po1\]). Particularly, we find that each potential of the pair is ${\cal PT}$-symmetric, $$\label{PTsym}
[{\cal PT},V_{l,m}^{\pm}]=0\,.$$ The potentials satisfy also the relation $$\label{dis1}
V_{l,m}^\pm=V_{-l,-m-1}^\pm
\,,$$ which is produced by a composition $\mathcal{R}_l\mathcal{R}_m$ of the intertwining generators (\[Rlm\]).
We also have a symmetry relation $$\label{dis1+}
V_{l,m}^\pm=V_{m+ \frac{1}{2},\,
l-\frac{1}{2}}^\pm\,,$$ that will play a key role in the analysis below. The composition of (\[dis1\]) and (\[dis1+\]) produces yet another symmetry $$\label{dis2}
V_{l,m}^\pm=V_{-m-\frac{1}{2},
-l-\frac{1}{2}}^\pm \,\, .$$
The nature of relations (\[Rlm\]) and (\[dis1\])–(\[dis2\]) takes a somewhat more transparent form if to redefine the parameters [@LeZno]: $(l,m)\rightarrow (\alpha_+,\alpha_-$), $\alpha_\pm=m\pm l+\frac{1}{2}$. Then the coefficients in (\[po1\]) are transformed into $l^2+m(m+1)=\frac{1}{2} [\alpha_+^2
+\alpha_-^2-\frac{1}{2}]$ and $l(2m+1)=\frac{1}{2}(\alpha_+^2
-\alpha_-^2)$, and symmetry (\[dis1+\]) corresponds to a reflection $\mathcal{P}_{\alpha_-}:\,(\alpha_+,\alpha_-)\rightarrow
(\alpha_+,-\alpha_-)$ in the plane of $\alpha$-parameters. Intertwining relations (\[Rlm\]) are given by the products of reflection $\mathcal{P}_{\alpha_-}$ and $\alpha$-rotations for $\pm\pi/2$, $\mathcal{R}_{+\pi/2}:\,(\alpha_+,\alpha_-)\rightarrow
(\alpha_-,-\alpha_+)$ and $\mathcal{R}_{-\pi/2}:\,(\alpha_+,\alpha_-)\rightarrow
(-\alpha_-,\alpha_+)$, $\mathcal{R}_l=\mathcal{P}_{\alpha_-}\mathcal{R}_{+\pi/2}$, $\mathcal{R}_m=\mathcal{P}_{\alpha_-}\mathcal{R}_{-\pi/2}$, while symmetries (\[dis1\]) and (\[dis2\]) correspond, respectively, to a $\pi$-rotation $(\alpha_+,\alpha_-)\rightarrow
(-\alpha_+,-\alpha_-)$ and to a reflection in $\alpha_+$, $\mathcal{P}_{\alpha_+}:\,(\alpha_+,\alpha_-)\rightarrow
(-\alpha_+,\alpha_-)$.
It will be more convenient for us, however, to work in terms of the parameters $l$ and $m$. Relations (\[dis1\])–(\[dis2\]) allow us to interchange integer values with half-integer ones as well as positive with negative values. By this reason, without loss of generality we can suppose that $l$ and $m$ are non-negative integers; sometimes, however, negative and half-integer values will be important as well.
The described discrete relations and symmetries form the base for a rich exotic supersymmetric structure that we will reveal in the extended system ${\cal H}_{l,m}=diag(H^{+}_{l,m},H^{-}_{l,m})$, $$\label{hamil}
H^{\pm}_{l,m}=-\frac{d^2}{dx^2}+V_{l,m}^{\pm}\,,$$ with pseudo-Hermitian Hamiltonians [@Mpseudoh][^1], $$H^{\pm}_{l,m}={\cal P}
(H^{\pm}_{l,m})^\dagger\,{\cal P}\, ,$$ where we have taken into account the relation ${\cal P}^{-1}={\cal
P}$.
The spectra of $H^{\pm}_{l,m}$ display different features in dependence on the values of the parameters $l$ and $m$. The eigenfunctions of Hamiltonians (\[hamil\]) may or may not be simultaneous eigenstates of the ${\cal PT}$ operator. The situation is known as unbroken ${\cal PT}$ symmetry in the former case, or spontaneously broken in the latter case. As was shown in , the spectrum of a ${\cal PT}$ symmetric Hamiltonian is real when its eigenfunctions are simultaneously the eigenstates of the ${\cal PT}$ operator. In the case of the $H^{\pm}_{l,m}$, this situation holds when $$l^2+\left(m+1/2\right)^2\geq |2l\left(m+1/2\right)|\,,$$ that is always valid for real $l$ and $m$ [@Ahmed]. For real $l$ and $m$, the spectra of $H^{+}_{l,m}$ and $H^{-}_{l,m}$ have a finite number of bound states of non-degenerate energies, while the probability flux is conserved in a scattering sector. It was pointed out in [@LevaiCanVen2] that an imaginary displacement in the spatial coordinate $x \rightarrow x+i\delta$, where $\delta \in
\mathbb{R}$, breaks in general the conservation of probability, i.e. $|T|^2+|R|^2\neq1$. There is, however, a case that shows several special properties for this class of shifted potentials. When $l$ and $m$ take simultaneously non-negative integer values, the reflection coefficient vanishes, $|R|=0$, and the probability is conserved independently of any displacement $\delta\neq \pm \pi/2$, see below. As a result potentials (\[po1\]) belong to the class of $ \emph{reflectionless}$ potentials [@reflectionless]. In this case both potentials have the same spectrum with $m +l+1$ singlet states, where $m+l$ of them are bound states of negative energies, and there is a singlet state of zero energy at the bottom of the continuous part of the spectrum. We discuss the spectral characteristics of (\[hamil\]) in more details in Section \[eigen\].
Within a framework of $\mathcal{PT}$-symmetric quantum mechanics [@BenderReview], or in a more general framework of the systems with pseudo-Hermitian Hamiltonians [@ReviewM], it was shown that the $\mathcal{PT}$ integral can be supplied with yet another nontrivial (nonlocal) integral of motion that has a nature of the charge conjugation operator, $\mathcal{C}$ [@Coperator]. This allows finally to define a positive definite scalar product to extend a probabilistic interpretation for the case of $\mathcal{PT}$-symmetric systems. Here, we just have in mind this general picture when discuss the bound and scattering states by referring to original papers on the subject.
The double degeneracy in the spectrum for the scattering sector together with the reflectionless property and the finite number of singlet states indicate that each system $H^{+}_{l,m}$ and $H^{-}_{l,m}$ possesses a hidden, bosonized non-linear supersymmetry [@MPhidnon; @bosonized] as this happens for Hermitian reflectionless Hamiltonians [@boso1; @trisusy]. In fact for integer values of $l$ and $m$, potentials (\[po1\]) are solutions of the KdV hierarchy: they satisfy the stationary $s$-KdV$_{n}$, $n=l+m$, non-linear equations [@KdV]. This means that there should exist non-trivial integrals of motion ${\mathbb
A}^{+}_{2n+1}$ and ${\mathbb A}^{-}_{2n+1}$ for Hamiltonians (\[hamil\]) in the form of differential operators of order $2n
+1$. These integrals should satisfy relations $$\label{Adef}
[{\mathbb A}^{+}_{2n+1},H^{+}_{l,m}]=[{\mathbb A}^{-}_{2n
+1},H^{-}_{l,m}]=0,
\qquad ({\mathbb A}^{\pm}_{2n
+1})^2=P(H^{\pm}_{l,m}),\qquad n=l+m\,,$$ where $P(H^{\pm}_{l,m})$ is a polynomial of order $2n+1$, i.e. ${\mathbb A}^{+}_{2n+1}$ and $H^{+}_{l,m}$, as well as ${\mathbb
A}^{-}_{2n+1}$ and $H^{-}_{l,m}$, should compose a Lax pair [@Lax; @Belo].
There are particular cases for which $H^{\pm}_{l,m}$ are completely isospectral to the reflectionless Pöschl-Teller Hamiltonians [@PT][^2] $$\label{PT}
H^{PT}_{\lambda}=-\frac{d^2}{dx^2}-
\frac{\lambda(\lambda+1)}{\cosh^2 x},
\qquad \lambda \in \mathbb{Z}\, .$$ Namely, the Pöschl-Teller and the complexified Scarf II potentials are related by a transformation of an imaginary displacement and a rescaling of the variable, $$\label{ptrela}
H^{PT}_{\lambda}(x\pm i\pi/4)=4H^{\pm}_{l,m}(2x)\,,\quad \text{for}
\quad
\left\{
\begin{array}{ccc}
m=l-1, \quad \quad \lambda=m+1\,, \\
\text{and}\\
m=l, \quad \quad \lambda=m\,.
\end{array}
\right.$$ Another interesting relation is with a generalized, singular Pöschl-Teller potential [@susyrev1] $$\label{gpt}
V_{l,m}^{GPT}(x)=\frac{l^2+m(m +1)}{\sinh^2 x}+l(2m+1)\frac{\cosh x}
{\sinh^2 x} \, .$$ The singularity can be removed by complex shifting [@GrNe]. For special values of such a shifting we get the ${\cal PT}$-symmetric pair of potentials (\[po1\]), $$V_{l,m}^{GPT}(x\pm i\pi/2)=V_{l,m}^{\pm}(x)\, .$$
It is worth to note that the potentials (\[po1\]) and corresponding Hamiltonians admit a representation that generalizes Eq. (\[ptrela\]), $$\label{h1}
H_{l,m}^+=\frac{1}{4} \left(-
\frac{d^2}{d\xi^2}-\frac{r(r+1)}{\cosh^2 \xi}- \frac{s(s+1)}{\cosh^2
(\xi+i\frac{\pi}{2})} \right) \,,$$ $$\label{h2}
H_{l,m}^-=\frac{1}{4} \left(-
\frac{d^2}{d\xi^2}-\frac{s(s+1)}{\cosh^2 \xi}- \frac{r(r+1)}{\cosh^2
(\xi+i\frac{\pi}{2})} \right)\,,$$ where $\xi=\frac{x}{2}+i\frac{\pi}{4}$, and $$r=l+m, \qquad s=l-m-1 \,.$$ Representations (\[h1\]) and (\[h2\]) correspond, on the one hand, to the relation (\[T+-\]) between potentials (\[po1\]) generated by the shift in the half of their imaginary period, $$\label{imaginary}
H_{l,m}^-(\xi)=H_{l,m}^+\left(\xi+i\frac{\pi}{2}\right) \,.$$ On the other hand, the generators of intertwining relations (\[Rlm\]) correspond here to $\mathcal{R}_l:\,(r,s)\rightarrow
(-s-1,-r-1)$ and $\mathcal{R}_m:\,(r,s)\rightarrow (s,r)$.
We will return to (\[imaginary\]) in the discussion of the supersymmetric structure, but here we note that in the context of periodic (elliptic) finite-gap potentials, a so called tri-supersymmetry appears when the superpartner potentials are the associated Lamé potentials shifted mutually in the half of the real period [@trisusy]. As we will show below, the extended system ${\cal H}_{l,m}$ constructed from (\[hamil\]) also possesses a tri-supersymmetry in which the imaginary period, the ${\cal PT}$- symmetry, the discrete symmetries (\[dis1\]) and (\[dis2\]), and the non-linear supersymmetry together play a fundamental role to form altogether a unified structure.
As a final comment on relation of (\[po1\]) with periodic finite-gap potentials, we note that there is a generalization of the family of the associated Lamé potentials known as the Darboux-Treibich-Verdier potentials [@Veselov]. In terms of the (double periodic) Jacobi elliptic functions they read $$\begin{aligned}
V^{DTV}
&=& n_{1}(n_{1}+1)k^{2}\mathrm{sn}^{2}x
+n_{2}(n_{2}+1)k^{2}\mathrm{sn}^{2}(x+iK')\nonumber\\
&+&n_{3}(n_{3}+1)k^{2}\mathrm{sn}^{2}(x+K+iK')+
n_{4}
(n_{4}+1)
k^{2}\mathrm{sn}^{2}(x+K) \label{dtv}\\
&=&
n_{1}(n_{1}+1)k^{2}\mathrm{sn}^{2}x
+n_{2}(n_{2}+1)\frac{1}
{\mathrm{sn}^{2}x }
+n_{3}(n_{3}+1)\frac{\mathrm{dn}^{2}x}{\mathrm{cn}^{2}x}+
n_{4}
(n_{4}+1) \frac{k^{ 2}
\mathrm{cn}^{2}x}{\mathrm{dn}^{2}x}\,.\nonumber\end{aligned}$$ Here $0<k<1$ is the modular parameter, and (\[dtv\]) has a real, $2K$, and an imaginary, $2iK'$, periods; $K=K(k)$ is the elliptic complete integral of the first kind and $K'=K(k')$, $k'=\sqrt{1-k^2}$ [@specialfunctions]. The finite-gap nature of (\[dtv\]) appears when parameters $n_i$ take integer values. Particularly, when $n_2=n_3=0$, potential (\[dtv\]) reduces to the finite-gap associated Lamé potential [@trisusy; @asso]. When the modular parameter takes the limit $k\rightarrow 1$, the real period tends to infinity, $2K \rightarrow
\infty$, while $2iK'\rightarrow i\pi$, and the potential transforms into $$V^{DTV}\xrightarrow[k\rightarrow{}1]\,
-\frac{n_1(n_1+1)}{\cosh^2
x}-\frac{n_2(n_2+1)}{\cosh^2 (x+i\frac{\pi}{2})}+const\,,$$ that has the form of potentials (\[h1\]) and (\[h2\]). 0.1cm
In the next section we will study the states of the ${\cal PT}$-symmetric systems $H^{\pm}_{lm}$ in the light of the discrete symmetries and relations that we have discussed.
Wavefunctions and differential intertwiners {#eigen}
===========================================
The group theoretical methods and supersymmetry are the powerful tools in the study of quantum mechanical systems. The Hamiltonians (\[hamil\]) provide a good example of the systems for which these techniques work effectively, particularly, to analyze the spectrum and eigenfunctions. In this direction, using irreducible representations of the $sl(2,\mathbb{C})$ algebra, it was found in [@GrNe] that the non-degenerate parts of the spectra of potentials (\[po1\]) are described by the two sets of eigenfunctions, one of which is[^3] $$\label{ps1}
\Psi^{\pm}_{n,m}={\rm sech}^{m}\, x \,\, \exp{[\mp i l\arctan (\sinh
x)]} P_{n}^{l- m-1/2,-l-m-1/2}(\pm i \sinh x) \,,$$ where $P_{n}^{\,\alpha,\,\beta}(x)$ are the Jacobi polynomials [@specialfunctions]. The corresponding energy levels and the values of the parameter $n$ are $$\label{e1}
E_{n,m}=-(m-n)^2, \quad n=0,1,2...\leq m \,.$$ The eigenfunctions $\Psi^{\pm}_{n,m}$ for $n<m$ describe $m$ bound states, while $n=m$ corresponds to the singlet zero energy state at the bottom of the continuous spectrum. With the discrete symmetry (\[dis1+\]) of the potentials, $V_{l,m}^\pm=V_{m+
\frac{1}{2},l-\frac{1}{2}}^\pm$, to which corresponds a transformation $$\begin{aligned}
\label{dis3}
&(l,m) \rightarrow \left(m+\frac{1}{2}, l-
\frac{1}{2} \right)\,,&\end{aligned}$$ it is possible to write down the another set of the bound states, $$\label{ps2}
\Psi^{\pm}_{n,l}={\rm sech}^{l-1/2}\, x \,
\, \exp{[\mp i (m+1/2)\arctan (\sinh x)]}
P_{n}^{m
+1/2-l,-m-1/2-l}( \pm i \sinh x) \,,$$ with energies $$\label{e2}
E_{n,l}=-(l-n-1/2)^2,\quad n=0,1,2...<l-1/2 \,,$$ so that (\[ps1\]) and (\[ps2\]) represent together the $l+m+1$ singlet states of the systems (\[hamil\]).
As we will see, the separation of singlet states of each subsystem $H^+_{l,m}$ and $H^-_{l,m}$ into the two subsets is reflected by a specific nonlinear supersymmetry of the extended system $\mathcal{H}_{l,m}$. This supersymmetry is related to the (imaginary here) mutual half-period shift of the subsystems. To the best of our knowledge a similar kind of supersymmetric structure, that we shall discuss in the next section, was discussed till the moment only for finite-gap systems with Hermitian Hamiltonians [@trisusy]. To explain this structure and its origin, we present below some further comments on the properties of the Hamiltonians (\[hamil\]) and their eigenfunctions.
States (\[ps1\]) and (\[ps2\]) can also be obtained from a supersymmetry approach, by means of the Darboux-Crum transformations (for details we refer to [@MatSal]), so that it is possible to link the Hamiltonians with different values of $l$ and $m$ between them. Particularly, it is possible to relate the free particle, for which $H^{\pm}_{0,0}=-\frac{d^2}{dx^2}\equiv H_{0}$, with the generic case $H^{\pm}_{l,m}$. Note that the free particle system is presented equivalently here also by $H^{\pm}_{0,-1}$, $H^{\pm}_{1/2,-1/2}$ and $H^{\pm}_{-1/2,-1/2}$. The bound states described above can be computed from the appropriate non-physical states of the free particle, whereas the states from the continuous part of the spectrum are obtained from the plane wave states of the free system. To illustrate this picture we will show how the scattering sector of $H_{l,m}^{+}$ can be obtained having in mind that all the results for $H_{l,m}^{-} $ can be reproduced then with the help of symmetries (\[pt1\]), or by the shift for the half of the imaginary period. The construction of the bound states from the non-physical states of the free particle will be illustrated in Section \[examples\].
Let us define the first order differential operators $$\label{aop}
A_{l,m}^{\pm}=\frac{d}{dx}\pm\left[ \left(l-\frac{1}{2}
\right)\tanh x+i\left(m+
\frac{1}{2} \right){\rm sech} \, x \right] \, .$$ They generate the intertwining relations $$\begin{aligned}
\label{Ainter}
&A_{l,m}^+H_{l,m}^{+}= H_{l-1,m}^{+}A_{l,m}^+ \,,\qquad
A_{l,m}^-H_{l-1,m}^+= H_{l,m}^+A_{l,m}^- \,.&\end{aligned}$$ Now, by applying, as in the singlet states case, the discrete symmetry (\[dis3\]) to (\[aop\]), we obtain the operators $$\label{bop}
B_{l,m}^{\pm}=\frac{d}{dx}\pm\left(m \tanh x+i l
\,{\rm sech} \, x \right) \, .$$ Application of symmetry (\[dis3\]) to (\[Ainter\]) results then in the intertwining relations $$\begin{aligned}
\label{Binter}
&B_{l,m}^+H_{l,m}^+= H_{l,m-1}^+B_{l,m}^+ \, ,\qquad
B_{l,m}^-H_{l,m-1}^+= H_{l,m}^+B_{l,m}^- \, .&\end{aligned}$$ The operators $A_{l,m}^{\pm}$ are related between themselves by $A_{l,m}^{\pm}={\cal P}(A_{l,m}^{\mp})^\dagger{\cal P}^{-1}$. Similarly, for the operators $B_{l,m}^{\pm}$ we have $B_{l,m}^{\pm}={\cal P} (B_{l,m}^{\mp})^\dagger{\cal P}^{-1}$. Such relations of conjugation underly the pseudo-supersymmetry discussed in the literature for non-Hermitian systems [@Mpseudosusy], particularly, a ${\cal PT }$-symmetric one.
Coherently with the discrete symmetry (\[dis3\]), operators (\[aop\]) and (\[bop\]) allow us to factorize, up to an additive constant term, the same Hamiltonian in two different ways [@bifurcationComment], $$H_{l,m}^{+}=-A_{l,m}^{-}A_{l,m}^{+}-
\left(l-\frac{1}{2} \right)^2=-
B_{l,m}^{-}B_{l,m}^{+}-m^2 \, .$$ Since $A_{l,m}^{\pm}$ as well as $B_{l,m}^{\pm}$ are $\mathcal{PT}$-antisymmetric ($\mathcal{PT}$-odd), $\{\mathcal{PT},A_{l,m}^{\pm}\}=\{\mathcal{PT},B_{l,m}^{\pm}\}=0$, the Hamiltonians $H_{l,m}^{\pm}$ are the $\mathcal{PT}$-symmetric ($\mathcal{PT}$-even) operators.
In the next section we will see further implications of existence of these two related types of factorization operators. For the moment, it is worth to note that the action of the operators $A_{l,m}^{\pm}$ and $B_{l,m}^{\pm}$ on the two-parametric family of Hamiltonians (\[hamil\]) is quite simple: while the former act by lowering and raising the parameter $l$, the latter play the same role for $m$. Using this fact, we are able now to connect, for any $l$ and $m$, the Hamiltonian $H_{l,m}^+$ with the free particle Hamiltonian $H_{0}$ in two different ways by making use of the operators $$\begin{aligned}
\label{freecrum1}
{\cal D}_{l,m}^{-}&=&B_{l,m}^-B_{l,m-1}^-\ldots
B_{l,2}^-B_{l,1}^-A_{l,0}^
-A_{l-1,0}^-\ldots A_{2,0}^-A_{1,0}^- \, , \\
\tilde{{\cal D}}_{l,m}^{-}&=&B_{-l,-m}^+B_{-l,-m+1}^+\ldots B_{-l,-1}^+B_{-l,0}^+A_{-l+1,0}^+A_{-l+2,0}^+\ldots A_{-1,0}^+A_{0,0}^+ \, .
\label{freecrum2}\end{aligned}$$ The differential operator ${\cal D}_{l,m}^{-}$ has here the order $m+l$, meanwhile $\tilde{\cal D}_{l,m}^{-}$ has the order $m+l+1$. These operators intertwine Hamiltonian $H^+_{l,m}$ with the free particle Hamiltonian, $$\label{HlmH0}
{\cal D}_{l,m}^{-}H_{0}=
H_{l,m}^+{\cal D}_{l,m}^{-}\,,
\qquad \tilde{{\cal D}}_{l,m}^{-}H_{0}=
H_{l,m}^+\tilde{{\cal D}}_{l,m}^{-} \, .$$ From (\[HlmH0\]) the inverse intertwining relations are easily obtained by defining ${\cal D}_{l,m}^{+}={\cal P}({\cal
D}_{l,m}^{-})^\dagger{\cal P}^{-1}$ and $\tilde{{\cal
D}}_{l,m}^{+}=-{\cal P} (\tilde{{\cal D}}_{l,m}^{-})^\dagger{\cal
P}^{-1}$, that yields $$\label{freecrum3}
{\cal D}_{l,m}^{+}H_{l,m}^+=
H_{0}{\cal D}_{l,m}^{+}\,, \qquad
\tilde{{\cal D}}_{l,m}^{+}H_{l,m}^+=
H_{0}\tilde{{\cal D}}_{l,m}^{+} \, .$$
Several comments are in order here. First we note that the difference of orders of the two intertwining operators (\[freecrum1\]) and of their conjugate ones are related to the fact that the system $H^+_{l,m}$ can be presented alternatively by the equivalent Hamiltonian $H^+_{l,-m-1}$. Another point is worth to note is that since operators $A_{l,m}^{\pm}$ and $B_{l,m}^{\pm}$ do not commute, and the path that connects the points $(l,m)$ and $(0,0)$ in the parameter plane can be chosen in different ways, the corresponding intertwining operators have not a unique form. It is the operators ${\cal D}_{l,m}^{\pm}$ and $\tilde{\cal
D}_{l,m}^{\pm}$ that together with the corresponding pair of Hamiltonians $H^\pm_{l,m}$ will form a basis for the construction of the tri-supersymmetric structure in the present $\mathcal{PT}$-symmetric case, see Figure \[red\].
![The family of complexified reflectionless Scarf II systems may be presented on integer or half-integer lattices in the $(l,m)$ parameters plane \[in axes $\alpha_\pm=m\pm l+\frac{1}{2}$, the lattices for $H^\pm_{l,m}$ have half-integer coordinates\]. Four different points correspond to the same system, two of which, shown here as an example by the filled circles, are on the integer lattice; another two points, shown by the unfilled circles, are on the half-integer lattice. The equivalent points on the same lattice are related by the symmetry transformation (\[dis1\]), while equivalent points on different lattices are related by the discrete symmetries (\[dis1+\]) and (\[dis2\]). The filled and unfilled triangles correspond to a system shifted in the half of the pure imaginary period; two such mutually displaced systems are related by the intertwining discrete transformations (\[pt1\]) and (\[Rlm\]). The two filled and two unfilled squares correspond to a free particle system. Any two systems represented on the same lattice may be related between themselves by differential intertwining operators. Particularly, any nontrivial complexified reflectionless Scarf II system may be intertwined with the free particle. Two of such (of many possible) “intertwining paths” shown for the system $H_{2,1}^+=H^+_{-2,-2}$ correspond to the action of the operators (\[freecrum1\]) and (\[freecrum2\]) with $l=2$, $m=1$.[]{data-label="red"}](figure3.eps)
On the other hand, the fact that two different Darboux-Crum transformations can relate two different quantum mechanical systems is known for the case of Hermitian operators and was exploited in [@trisusy] to reveal a peculiar, tri-supersymmetric structure in some periodic and non-periodic finite-gap systems. Particularly, it was shown in [@newself1] that the two non-trivial Darboux-Crum transformations encode the existence of a Lax pair in reflectionless Pöschl-Teller systems. As we noted above on Eq. (\[ptrela\]), there are cases in which the potentials (\[po1\]) are reduced exactly to the shifted Hermitian Pöschl-Teller potentials, so that operators (\[freecrum1\]) and (\[freecrum2\]) match in those particular cases the corresponding operators in [@trisusy; @ptads].
One of the direct applications of the constructed intertwining operators is that we can use them to map the plane wave states of the free particle, $$\label{pw}
H_{0}\psi_{\pm k}=k^2\psi_{\pm k},\qquad
\psi_{\pm k}=e^{\pm i k
x}\,,$$ into the scattering eigenstates of $H_{l,m}^+$, $$\label{conti}
H_{l,m}^+\Psi^{+}_{\pm k}=k^2\Psi^{+}_{\pm k},
\qquad \Psi^{+}_{\pm
k}={\cal D}_{l,m}^{-}e^{\pm i k x}=
c_1(k) \tilde{{\cal D}}_{l,m}^{-}e^{\pm i k
x}\,,$$ where $k\geq 0$ and $c_1(k)$ is some ($k$-dependent) constant factor. For $k=0$, the action of the operator ${\cal D}_{l,m}^{-}$ produces the unique singlet state of the continuous spectrum of $H_{l,m}^+$, and then (\[conti\]) coincides with the eigenfunction (\[ps1\]) with $n=m$, i.e. $\Psi^{+}_{0}=\Psi^{+}_{n,n}$. On the other hand, $\tilde{{\cal D}}_{l,m}^{-}$ annihilates the singlet state located at the bottom of the continuos spectrum of the free particle, i.e. $c_1(0)=0$ in (\[conti\]). 0.1cm
This picture of the Darboux-Crum transformations explains, as in the case of Hermitian finite-gap systems [@trisusy], the reflectionless properties of the Hamiltonians (\[hamil\]) for integer (and half-integer) values of $l$ and $m$.
Tri-supersymmetric structure {#susysec}
============================
In this section we will show how the ${\cal PT}$-symmetry, originated from the intertwining relations (\[pt1\]), the discrete symmetries behind (\[Rlm\]), the self-isospectrality based on (\[T+-\]), the Lax integrals ${\mathbb A}^\pm_{2n+1}$, and the non-linear supersymmetry form altogether a peculiar structure.
To reveal and describe such an unusual extended nonlinear supersymmetric structure, we will show first that the extended $\mathcal{PT}$-symmetric Hamiltonian $$\label{sH}
{\cal H}_{l,m}=\left( \begin{array}{cc}
H^{+}_{l,m} & 0 \\
0 & H^{-}_{l,m}
\end{array}\right) \,$$ for $l\neq0$ has three *mutually commuting* non-trivial basic *integrals* of motion, anti-diagonal ${\cal X}_{l,m}$ and ${\cal Y}_{l,m}$, and diagonal ${\cal Z}_{l,m}=diag\,
({\mathbb A}^+_{2n+1},{\mathbb A}^-_{2n+1})$, $n=l+m$, $$\label{com}
[ {\cal X}_{l,m}, {\cal H}_{l,m}]=0, \quad [ {\cal Y}_{l,m},
{\cal H}_{l,m}]=0, \quad [ {\cal Z}_{l,m},
{\cal H}_{l,m}]=0 \, .$$ These are the matrix non-linear differential operators of the orders $|{\cal X}_{l,m}|=2l$, $|{\cal Y}_{l,m}|=2m+1$ and $|{\cal
Z}_{l,m}|=2(l+m)+1$, connected between themselves by a factorization relation $$\label{XYZ}
{\cal Z}_{l,m}={\cal X}_{l,m}{\cal Y}_{l,m}=
{\cal Y}_{l,m}{\cal X}_{l,m}\,.$$ It is due to these three basic nontrivial integrals the corresponding supersymmetric structure is referred to as a *tri-supersymmetry*. We will see that it reflects coherently the peculiar properties of the extended complexified Scarf II system $\mathcal{H}_{l,m}$, including the existence of two types of the discrete energy levels in its spectrum.
The case $l= 0$ is particular since the extended Hamiltonians (\[sH\]) just reduce to the two copies of the Hermitian Pöschl-Teller systems with coupling parameter $\lambda=m$ in (\[PT\]), i.e. $H^{+}_{0,\lambda}=H^{-}_{0,\lambda}=H^{PT}_m$. One integral then reduces to the Pauli sigma matrix, ${\cal
X}_{0,m}=\sigma_1$. The remaining two integrals are related in accordance with (\[XYZ\]) as ${\cal Y}_{0,m}=\sigma_1{\cal Z}_{0,m}$, where the diagonal matrix elements of ${\cal Z}_{0,m}$ coincide, ${\mathbb A}^+_{2m+1}={\mathbb A}^-_{2m+1}$, and generate the hidden bosonized nonlinear supersymmetry of the reflectionless Pöschl-Teller system $H^{PT}_m$, see [@boso1].
Before passing over to the construction of the nontrivial integrals, we note that the extended system (\[sH\]) is formed by the two self-isospectral Hamiltonians displaced mutually by the half of their imaginary period. As a result, the degeneracy of the spectrum of $ {\cal H}_{l,m}$ is twice that of its corresponding diagonal components. As in usual (Hermitian) quantum mechanics, by virtue of relations (\[com\]), it is natural to expect that there is a basis where all the eigenstates of the Hamiltonian (\[sH\]) are also the eigenstates of the nontrivial integrals of motion $\mathcal{X}_{l,m}$, $\mathcal{Y}_{l,m}$, and $\mathcal{Z}_{l,m}$. In accordance with this, as we will see, the $m+l$ doublet states corresponding to the set of bound states can be presented in the form $$\label{phi1}
\Phi_{n,l}^{\pm}=\left( \begin{array}{c}
\Psi_{n,l}^+ \\
\pm i \Psi_{n,l}^-
\end{array}\right), \qquad n=0,1,\ldots, <l-1/2\,,$$ and $$\label{phi2}
\Phi_{n,m}^{\pm}=\left( \begin{array}{c}
\Psi_{m,l}^+ \\
\pm \Psi_{m,l}^-
\end{array}\right), \qquad n=0,1,2...< m\,,$$ with eigenvalues given by (\[e2\]) and (\[e1\]), respectively. The scattering states can be written as $$\label{contspectr}
\Phi^{\pm}_{+k}=
\left( \begin{array}{c}
\Psi_{+k}^+ \\
\pm \Psi_{+k}^-
\end{array}\right), \qquad \Phi^{\pm}_{-k}=
\left( \begin{array}{c}
\Psi_{-k}^+ \\
\pm \Psi_{-k}^-
\end{array}\right),$$ with energies $E=k^2$, $k\geq 0$. The energy levels with $E>0$ are then four-fold degenerate, while for $E=0$ we have $\Phi^{\pm}_{0}=\Phi^{\pm}_{+k}=\Phi^{\pm}_{-k}$, and, as in the bound states case, the double degeneration. Up to a multiplicative constant, $\Phi^{\pm}_{0}$ coincide with (\[phi2\]) with $n=m$.
The intertwining relations (\[Rlm\]) for the potentials are trivially extended for the Hamiltonians (\[hamil\]), $$\label{comrlm}
\mathcal{R}_{l}H^{\pm}_{l,m}=
H^{\mp}_{l,m}\mathcal{R}_{l}, \qquad
\mathcal{R}_{m}H^{\pm}_{l,m}=
H^{\mp}_{l,m}\mathcal{R}_{m}\,,$$ where the generators $\mathcal{R}_{l}$ and $\mathcal{R}_{m}$ can be used to construct a discrete symmetry $\mathcal{R}_{l}\mathcal{R}_{m}$ for the extended Hamiltonian ${\cal
H}_{l,m}$. For the extended system ${\cal H}_{l,m}$, antidiagonal Pauli matrix $\sigma_1$ (as well as $\sigma_2$), produces the same effect of intertwining of the Hamiltonian’s components, $\sigma_1
diag\,(H^{+}_{l,m},H^{-}_{l,m})\sigma_1=
diag\,(H^{-}_{l,m},H^{+}_{l,m})$. Therefore, we can construct the matrix operators $$\hat{\mathcal{R}}_l=\sigma_1\mathcal{R}_l, \qquad
\hat{\mathcal{R}}_m=\sigma_1\mathcal{R}_m \, ,$$ which are the integrals of motion for our extended system (\[sH\]), $$[\hat{\mathcal{R}}_l,{\cal H}_{l,m}]=0, \qquad
[\hat{\mathcal{R}}_m,{\cal H}_{l,m}]=0 \, .$$ In the case of $\hat{\mathcal{R}}_l$, the commutation relation comes from the intertwining relation, which, in turn, is based on the equality $H^{\pm}_{- l,m}=H^{\mp}_{l,m}$. In the previous section we have seen that the operators $A^{\pm}_{l,m}$ acting on the Hamiltonans $H^{\pm}_{l,m}$, can lower or raise the index $l$, see Eq. (\[Ainter\]), by means of a chain of Darboux transformations. This means that the appropriate product of the operators $A^{\pm}_{l,m}$ produces exactly the same intertwining effect as the ${\cal R}_l$, which changes $(l,m)$ for $(-l,m)$. Indeed, this can be achieved by application of the $2l$-th order differential operators $$\begin{aligned}
\label{xdef}
X_{l,m}^+&\equiv& A_{-l+1,m}^+A_{-l+2,m}^+ \ldots A_{0,m}^+ \ldots
A_{l-1,m}^+A_{l,m}^+ \,, \\
X_{l,m}^-&\equiv& A_{l,m}^- A_{l-1,m}^-\ldots A_{0,m}^- \ldots A_{-l
+2,m}^- A_{-l+1,m}^-\,,\end{aligned}$$ which satisfy the intertwining relations of the same form as in (\[comrlm\]), $$\label{in1*}
X_{l,m}^{\pm}H_{l,m}^{\pm}=H_{-l,m}^{\pm}X_{l,m}^{\pm}=H_{l,m}^{
\mp}X_{l,m}^{\pm} \, .$$ The case $l=0$ reduces trivially to the operators $X_{0,m}^{\pm}=1$. On the other hand, the nontrivial analog of the discrete symmetry $\hat{{\cal R}}_l$, $\hat{{\cal R}}_l^2=1$, is provided by the matrix differential operator, $$\label{xdef*}
{\cal X}_{l,m}=\left( \begin{array}{cc}
0 &X_{l,m}^- \\
X_{l,m}^+ & 0
\end{array}\right) \,,$$ which, by virtue of (\[in1\*\]), is an integral of motion. In addition to (\[com\]), the integral (\[xdef\*\]) satisfies a superalgebraic-type relation $$\label{xh*}
\left \{ {\cal X}_{l,m},{\cal
X}_{l,m}
\right \}=2{\cal X}_{l,m}^2=2P_{{\cal X}}({\cal H}_{l,m})\,,$$ where $P_{{\cal X}}$ is a polynomial of order $l$ in Hamiltonian ${\cal H}_{l,m}$, $$\label{px}
P_{{\cal X}}({\cal H}_{l,m})=\prod^{[l-1/2]}_{s=0}({\cal
H}_{l,m}+(l- s-1/2)^2)^2 \,.$$ In the context of analogy of ${\cal X}_{l,m}$ with $\hat{{\cal
R}}_l$, Eq. (\[xh\*\]) is a generalization of the relation $\hat{{\cal R}}_l^2=1$. The polynomial $P_{{\cal X}}$ has the nature of a spectral polynomial, but which only includes the energies of one set of bound states (\[phi1\]). From here a remarkable property of ${\cal X}_{l,m}$ can be derived: acting on eigenstates of ${\cal H}_{l,m}$, it annihilates one complete set of doublets while the doublet states of another set are the eigenvectors with nonzero eigenvalues, $$\label{xphi}
{\cal X}_{l,m}\Phi_{n,l}^{\pm}=0, \quad {\cal X}_{l,m}
\Phi_{n,m}^{\pm}=\pm (-1)^n\prod^{[l-1/2]}_{s=0}(E_{n,m}+
(l-s-1/2)^2) \Phi_{n,m}^{\pm}\,,$$ for $ n=0,1,\ldots, <l-1/2$ and $ n=0,1,2,\ldots\leq m$. Therefore, the integral ${\cal X}_{l,m}$ identifies all the states which correspond to resonances of the Hermitian counterparts of the Hamiltonians $H^\pm_{l,m}$ [@LevaiCanVen2]. The action of the operator ${\cal X}_{l,m}$ on the scattering states is characterized by the property that it does not distinguish the waves coming from the left or from the right, but separates the states with distinct values of the upper index, $$\begin{aligned}
{\cal X}_{l,m}^{\pm}\Phi^{\pm}_{+k}&=& \pm (-1)^l
\prod^{[l-1/2]}_{s=0}(k^2+(l-
s-1/2)^2)\Phi^{\pm}_{+k}\,,\nonumber\\
{\cal X}_{l,m}^{\pm}\Phi^{\pm}_{-k}&=& \pm (-1)^l\prod^{[l-1/2]}_{s=0}
(k^2+(l-
s-1/2)^2)\Phi^{\pm}_{-k} \, .\label{Xcont}\end{aligned}$$
We can construct also differential operators of order $2m+1$, $$\begin{aligned}
\label{ydef}
Y_{l,m}^+&\equiv& B_{l,-m}^+B_{l,-m+1}^+ \ldots B_{l,0}^+ \ldots
B_{l,m-1}^+B_{l,m}^+ \,, \\
Y_{l,m}^-&\equiv& B_{l,m}^-B_{l,m-1}^- \ldots B_{l,0}^- \ldots B_{l,-m+1}^-B_{l,-m}^-\,,\end{aligned}$$ which generate the Darboux-Crum transformations similar to the intertwining relations produced by $\mathcal{R}_m$, $$\label{iny1}
Y_{l,m}^{\pm}H_{l,m}^{\pm}=
H_{l,-m-1}^{\pm}Y_{l,m}^{\pm}=H_{l,m}^{\mp}Y_{l,m}^{\pm} \,.$$ With their help, we find that the matrix differential operator $$\label{Yint}
{\cal Y}_{l,m}=i\left( \begin{array}{cc}
0 &Y_{l,m}^- \\
Y_{l,m}^+ & 0
\end{array}\right) \,$$ is the another nontrivial integral of motion for the extended system $\mathcal{H}_{l,m}$. Like the $\hat{\mathcal{R}}_{m}$ commutes with the $\hat{\mathcal{R}}_l$, the nontrivial integrals (\[Yint\]) and (\[xdef\*\]) also commute, $$[{\cal X}_{l,m},{\cal Y}_{l,m}]=0 \, .$$ The integral ${\cal Y}_{l,m}$ generates a relation $$\label{py}
\left \{ {\cal Y}_{l,m},{\cal
Y}_{l,m}
\right \}=2P_{{\cal Y}}({\cal H}_{l,m})=
2{\cal H}_{l,m}\prod^{m-1}_{r=0}({\cal H}_{l,m}+
(m-r)^2)^2$$ to be of the form similar to (\[xh\*\]). The roots of the spectral polynomial $P_{{\cal Y}}({\cal H}_{l,m})$ are complementary to those of the polynomial $P_{{\cal X}}({\cal H}_{l,m})$: they coincide with the energies (\[e1\]) of the eigenstates (\[phi2\]). In correspondence with this property, the second non-trivial integral of motion, ${\cal Y}_{l,m}$, annihilates the remaining set of discrete eigenstates of $\mathcal{H}_{l,m}$, not annihilated by the integral $\mathcal{X}_{l, m}$, which correspond to the doubly degenerate energy levels, while the zero modes of the latter integral are the eigenstates of ${\cal Y}_{l,m}$ of the nonzero eigenvalues, $$\label{yphi}
{\cal Y}_{l,m}\Phi_{n,l}^{\pm}=\pm i(-1)^n |E_{n,l}|^{1/2}
\prod^{m-1}_{r=0}(E_{n,l}+(m-
r)^2)\Phi_{n,l}^{\pm} ,
\quad {\cal Y}_{l,m}\Phi_{n,m}^{\pm}=0 \, .$$ The appearance of imaginary eigenvalues in the spectrum of the integral ${\cal Y}_{l,m}$ will be discussed later. The action of ${\cal Y}_{l,m}$ on the states of the continuous spectrum (\[contspectr\]) is given by $$\begin{aligned}
\label{Yc1}
{\cal Y}_{l,m}\Phi^{\pm}_{+k}&=& \mp (-1)^m k\prod^{m-1}_{r=0}(k^2+(m-r)^2) \Phi^{\pm}_{+k}\, ,\\
{\cal Y}_{l,m}\Phi^{\pm}_{-k}&=& \pm (-1)^m k\prod^{m-1}_{r=0}(k^2+(m-r)^2) \Phi^{\pm}_{-k} \, ,
\label{Yc2}\end{aligned}$$ i.e. this integral, unlike the ${\cal X}_{l,m}$, see (\[Xcont\]), distinguishes the waves coming from the left and from the right, and as ${\cal X}_{l,m}$, detects a difference between the states with distinct values of the upper (sign) index.
As we have seen, behind the existence of the integrals of motion $
{\cal X}_{l,m}$ and $ {\cal Y}_{l,m}$ is the fact that there are two *different* Darboux-Crum transformations, which intertwine the Hamiltonians $H^+_{lm}$ and $H^-_{l,m}$. In the case of the discrete operators ${\cal R}_l$ and ${\cal R}_m$ (one can also consider the intertwining operators ${\cal P}$, ${\cal T}$ and $T_{\pm}$, see the discussion in Section \[sec1\]), their composition transforms into a symmetry operation for the Hamiltonians $H^\pm_{l,m}$, $$\label{rlmh}
[{\cal R}_l{\cal R}_m, H^{\pm}_{l,m}]=0\,.$$ For extended system (\[sH\]), this composition corresponds to the integral $\hat{{\cal R}}=\hat{{\cal R}}_l\hat{{\cal
R}}_m=diag({\cal R}_l{\cal R}_m,{\cal R}_l{\cal R}_m)$, $$[\hat{\mathcal{R}}, {\cal H}_{l,m}]=0\,.$$ The composition of the intertwining relations generated by $X^{\pm}_{l,m}$ and $Y^{\mp}_{l,m}$, (\[in1\*\]) and (\[iny1\]) respectively, yields $$(Y_{l,m}^{\mp}X_{l,m}^{\pm})H_{l,m}^{\pm}=H_{l,m}^{\pm}
(Y_{l,m}^{\mp}X_{l,m}^{\pm}), \qquad
(X_{l,m}^{\mp}Y_{l,m}^{\pm})H_{l,m}^{\pm}=
H_{l,m}^{\pm}(X_{l,m}^{\mp}Y_{l,m}^{\pm})
\, .$$ The intertwining relations transform therefore into commutation relations, and corresponding integral of motion appears for each Hamiltonian, analogously to (\[rlmh\]). The resulting integrals are the differential operators of the order $2n+1$ with $n=m+l$, and these are nothing else as the Lax integrals ${\mathbb A}^\pm_{2n+1}$ in (\[Adef\]), which we rename here as $$Z^{\pm}_{l,m}={\mathbb A}^{\pm}_{2n+1}=Y_{l,m}^{\mp}X_{l,m}^{\pm}=
X_{l,m}^{\mp}Y_{l,m}^{\pm}, \qquad [Z^{\pm}_{l,m},H_{l,m}^{\pm}]=0
\,.$$ For the extended system, these integrals of motion can be joined to form a diagonal operator, ${\cal Z}_{l,m}$, which is generated by the anticommutator of the previous conserved quantities, $$\label{zdef}
{\cal Z}_{l,m}=i\left( \begin{array}{cc}
Z_{l,m}^+ & 0 \\
0 & Z_{l,m}^-
\end{array}\right)=\frac{1}{2}\{{\cal X}_{l,m},{\cal Y}_{l,m} \}
\,.$$
The origin of the Lax integrals in the present extended Hamiltonian from the the intertwining operators $X_{l,m}^{\pm}$ and $Y_{l,m}^{\pm}$ is illustrated on Fig. [\[red5\]]{}
![The “intertwining paths", which correspond to the action of the operators $X_{2,1}^{\pm}$ and $Y_{2,1}^{\pm}$ between the mutually shifted, self-isospectral Hamiltonians (presented by circles and triangles), are shown for the case of the extended $\mathcal{PT}$-symmetric system $\mathcal{H}_{2,1}=\mathcal{H}_{\frac{3}{2},\frac{3}{2}}$. The effect produced by these intertwining operators on the integer-valued lattice is the same as that of the discrete operators ${\cal R}_l$ and ${\cal R}_m$, respectively. On the half-integer-valued lattice, the horizontal and vertical distances between the corresponding systems are interchanged, and in this sense, the action of the operators $X_{2,1}^{\pm}$ and $Y_{2,1}^{\pm}$ on the half-integer-valued lattice is dual to that on the integer-valued lattice. Starting from any point, the composition of horizontal and vertical paths to the equivalent point on the same lattice produces the Lax integrals $Z_{2,1}^{\pm}$ and the symmetry operator ${\cal R}_l{\cal R}_m$. []{data-label="red5"}](figure4.eps)
It is straightforward to check from the above relations that ${\cal
Z}_{l,m}$ commutes with the Hamiltonian (\[com\]) and with the integrals ${\cal X}_{l,m}$ and ${\cal Y}_{l,m}$, $$[{\cal Z}_{l,m},{\cal X}_{l,m}]=0,
\qquad [{\cal Z}_{l,m},{\cal Y}_{l,m}]=0 \, .$$ Its square produces a polynomial in ${\cal H}_{l,m}$, $$\label{pz}
{\cal P}_{{\cal Z}}({\cal H}_{l,m})={\cal H}_{l,m}\prod^{m-1}_{r=0}
\prod^{[l-1/2]}_{s=0}({\cal H}_{l,m}+(m-r)^2)^2({\cal
H}_{l,m}+(l-s-1/2)^2)^2 \,.$$ whose roots are all the discrete doubly degenerate energies of the extended system. Note that the roots of the bound states are of degree two, while the zero energy state root has degree one. All the corresponding energy eigenstates are the zero modes of the integral ${\cal Z}_{l,m}$, $$\label{zphi}
{\cal Z}_{l,m}\Phi_{n,l}^{\pm}=0,
\qquad {\cal Z}_{l,m}\Phi_{n,m}^{\pm}=0,$$ which also detects the direction of propagation of the waves of the scattering sector, $$\begin{aligned}
{\cal Z}_{l,m}\Phi^{\pm}_{+k}&=& -(-1)^{m+l}
k\prod^{m-1}_{r=0}\prod^{[l-1/2]}_{s=0}(k^2+(m-r)^2)(k^2+(l-
s-1/2)^2)\Phi^{\pm}_{+k}\,,\\
{\cal Z}_{l,m}\Phi^{\pm}_{-k}
&=& (-1)^{m+l} k\prod^{m-1}_{r=0}
\prod^{[l-1/2]}_{s=0}((k^2+(m-r)^2)k^2+(l-
s-1/2)^2)\Phi^{\pm}_{-k} \, .\end{aligned}$$
Because of the $\mathcal{PT}$-odd nature of the operators $A^\pm_{l,m}$ and $B^\pm_{l,m}$, from which ${\cal X}_{l,m}$, ${\cal Y}_{l,m}$ and $ {\cal Z}_{l,m}$ are composed, all this triplet of the integrals is $\mathcal{PT}$-even. Thus, instead to be Hermitian operators, all the conserved quantities commute with the $ {\cal PT}$ operator, $$\label{compt}
[{\cal H}_{l,m},{\cal PT}]=0, \quad
[{\cal X}_{l,m},{\cal PT}]=0, \quad [{\cal
Y}_{l,m},{\cal PT}]=0, \quad
[{\cal Z}_{l,m},{\cal PT}]=0 \, .$$ It is this property of the $\mathcal{PT}$-symmetry that requires the presence of the imaginary unit as a multiplicative factor in the definition of ${\cal Y}_{l,m}$ in Eq. (\[Yint\]). In turn, the factor $i$ in Eq. (\[Yint\]) emphasizes then the existence of the splitting of the discrete eigenstates into two different families, and reveals an additional specific feature of the whole supersymmetric configuration we have here. The simultaneous requirement of a common basis of eigenstates for all the integrals of motion in addition to the relation (\[compt\]) fixes that only the integral $ {\cal Y}_{l,m}$ has imaginary eigenvalues for the set of doublet states $
\Phi_{n,l}^{\pm}$ in Eq. (\[phi1\]) (see Eq. (\[yphi\]) and also the examples in the next section). As all the integrals are mutually commuting operators, the picture here is different from that in a usual quantum mechanics where Hermitian (self-adjoint) mutually commuting operators possess a common basis of eigenstates with real eigenvalues. It is instructive to look in more detail what happens here. Explicit form of the states in (\[phi1\]) shows that they are not the eigenstates of the ${\cal PT}$ operator. Indeed, they satisfies the relation $${\cal PT}\Phi_{n,l}^{\pm}=
\Phi_{n,l}^{\mp} \, .$$ Remembering that ${\cal X}_{l,m}$ and ${\cal Z}_{l,m}$ annihilate the set of doublet states (\[phi1\]), while the extended Hamiltonian (\[sH\]) has an entire real spectrum \[and the states (\[phi1\]) are its eigenstates\], one concludes that the ${\cal
PT}$-symmetry has a *broken* nature just for the integral ${\cal Y}_{l,m}$ \[we remind parenthetically here that the eigenstates from the continuous part of the spectrum have real eigenvalues for ${\cal Y}_{l,m}$, see Eqs. (\[Yc1\]) and (\[Yc2\])\]. Taking into account independently only the Hamiltonian operator, one can find another basis where these states are simultaneously the eigenstates of the Hamiltonian and the ${\cal
PT}$ operator; therefore, for the ${\cal H}_{l,m}$ the ${\cal
PT}$-symmetry is *unbroken*.
We have identified the nontrivial integrals of the extended system and discussed their properties. Now we consider the related nonlinear supersymmetric structure. The diagonal matrix $\sigma_3$ is a trivial integral of motion for $\mathcal{H}_{l,m}$. Nevertheless it allows us to double the set of the nontrivial integrals of motion since the multiplication of any of them by $\sigma_3$ gives a new, linear independent nontrivial matrix integral of motion. So, in this way we obtain the set of six linearly independent nontrivial matrix integrals of motion $$\begin{aligned}
&\mathcal{Q}^{(1)}_{l,m}= {\cal X}_{l,m}\,,\qquad
\mathcal{Q}^{(2)}_{l,m}= \sigma_3\mathcal{Q}^{(1)}_{l,m}\,,&\label{QQ}\\
&\mathcal{S}^{(1)}_{l,m}= {\cal Y}_{l,m}\,,\qquad
\mathcal{S}^{(2)}_{l,m}=\sigma_3\mathcal{S}^{(1)}_{l,m}\,,\label{SS}&\\
&\mathcal{L}^{(1)}=\mathcal{Z}_{l,m}\,,\qquad
\mathcal{L}^{(2)}=\sigma_3\mathcal{L}^{(1)}_{l,m}\,.\label{LL}&\end{aligned}$$ Notice the absence of the imaginary factor $i$ in the definition of the second anti-diagonal supercharges in comparison with the usual SUSYQM approach with a Hermitian Hamiltonian. This guarantees that all the three new integrals are also ${\cal PT}$-symmetric operators.
The square of the matrix integral $\sigma_3$ equals $1$, and it can be identified as the grading operator, $\Gamma=\sigma_3$. This grading operator classifies then the Hamiltonian $\mathcal{H}_{l,m}$ and integrals $\mathcal{L}^{(a)}_{l,m}$, $a=1,2$, as bosonic operators, while the integrals of the antidiagonal matrix form, $\mathcal{Q}^{(a)}_{l,m}$ and $\mathcal{S}^{(a)}_{l,m}$, are classified as fermionic operators. In correspondence with this, we get a nonlinear superalgebra with the following set of nontrivial (anti)-commutation relations: $$\begin{aligned}
&\{\mathcal{Q}^{(a)}_{l,m},
\mathcal{Q}^{(b)}_{l,m}\}=(-1)^{a+1}2\delta_{ab}
P_{{\cal X}},\,\,\{\mathcal{S}^{(a)}_{l,m},\mathcal{S}^{(b)}_{l,m}\}
=(-1)^{a+1}2\delta_{ab}P_{{\cal Y}},\,\,
\{ \mathcal{Q}^{(a)}_{l,m}, \mathcal{S}^{(b)}_{l,m} \}=-2
\delta_{ab} \mathcal{L}^{(1)}_{l,m}
\, , &\nonumber \\
& [\mathcal{Q}^{(a)}_{l,m},\mathcal{L}^{(2)}_{l,m}]
=(-1)^{a}2\epsilon_{ab} \mathcal{S}^{(b)}_{l,m}P_{{\cal X}},
\quad [\mathcal{S}^{(a)}_{l,m}, \mathcal{L}^{(2)}_{l,m}]
=(-1)^{a}2\epsilon_{ab} \mathcal{Q}^{(b)}_{l,m}
P_{{\cal Y}} \, , & \label{susyalg}\end{aligned}$$ where $P_{{\cal X}}=P_{{\cal X}}(\mathcal{H}_{l,m})$ and $P_{{\cal
Y}}=P_{{\cal Y}}(\mathcal{H}_{l,m})$ are the polynomials defined in (\[px\]), (\[py\]) and (\[pz\]), respectively. Note that the integral $\mathcal{L}^{(1)}_{l,m}$ commutes with all the other integrals and, so, plays here the role of the bosonic central charge.
The choice of $\sigma_3$ as the grading operator is, however, not unique. Another possibility corresponds, for instance, to the choice $\Gamma=\mathcal{P}T_+$ (or, $\Gamma=\mathcal{P}T_-$). Indeed, this operator is a (nonlocal) integral of motion, whose square is equal to $1$. Such a grading operator classifies the integrals $\mathcal{Q}^{(a)}_{l,m}$ as bosonic integrals, while $\mathcal{S}^{(a)}_{l,m}$ and $\mathcal{L}^{(a)}_{l,m}$ are classified as fermionic integrals. The corresponding superalgebraic relations can be computed then by making use of the relations described above. In this case we have, particularly, a relation $\{\mathcal{L}^{(a)}_{l,m},\mathcal{L}^{(b)}_{l,m}\}=2\delta_{ab}
P_{{\cal Z}}(\mathcal{H}_{l,m})$. This corresponds to the fact that each of the unextended $\mathcal{PT}$-symmetric systems $H^+_{l,m}$ and $H^-_{l,m}$ is characterized by the bosonized supersymmetry, in which the $\mathcal{PT}$-symmetric integrals $iZ^+_{l,m}$ and $iZ^-_{l,m}$, respectively, are treated as the $\Gamma=\mathcal{P}T_+$-odd supercharges.
We summarize the whole picture on which the tri-supersymmetric structure is based on Fig. \[tablita\].
![The basic blocks of the tri-supersymmetric structure of the extended $\mathcal{PT}$-symmetric Hamiltonian $\mathcal{H}_{l,m}$. Horizontal arrows inside a table correspond to the duality transformation induced by the symmetry $(l,m) \rightarrow
\left(m+\frac{1}{2}, l-
\frac{1}{2} \right)$.[]{data-label="tablita"}](figure5.eps)
Examples
========
To illustrate different properties of the systems $H^\pm_{l,m}$ and the tri-supersymmetric structure of $\mathcal{H}_{l,m}$, here we present some examples for specific values of $l$ and $m$. Before doing this, we first note that according to the relation (\[ptrela\]), the simplest nontrivial case of $H^+_{1,0}(x)=H^-_{1,0}(x+i\pi)$ reduces, up to rescaling, just to the displaced reflectionless Pöschl-Teller system with one bound state in the spectrum. The unique bound state corresponds to a resonance with a complex energy value in the spectrum of the Hermitian Hamiltonian with real Scarf II potential $V(x)={\rm
sech}^2(x)-\sinh x\,{\rm sech}^2x$, which is depicted on Fig. \[Scarf\]. So we will consider more rich cases of reflectionless $\mathcal{PT}$-symmetric systems with two and three bound states.
Systems with two bound states
-----------------------------
Without loss of generality, the family of reflectionless potentials $V^\pm_{l,m}$ with two bound states in the discrete part of the spectrum can be presented by non-negative integer values of the parameters $l$ and $m$ subjected to the condition $l+m=2$. The case $ (l=0,m=2)$ corresponds here to the well known Hermitian reflectionless Pöschl-Teller potential $V_{0,2}^+=V_{0,2}^-
=-6/\cosh^2 x$. By virtue of (\[ptrela\]), this potential shares, up to rescaling, the same spectrum as complexified Scarf II potentials with $ (l=1,m=1)$. Then the remaining case $(l=2,m=0)$, $$\label{h20}
H^{\pm}_{2,0}=
-\frac{d^2}{dx^2}-\frac{4}{\cosh^2x}\pm2 i\frac{ \sinh
x} {\cosh^2x} \,,$$ provides a first nontrivial example which is not related to a Hermitian (reflectionless Pöschl-Teller) counterpart by means of shifting and rescaling of the coordinate. The potentials $V=V_{2,0}^{\pm}$ are solutions of the nonlinear $s$-KdV$_2$ equation $$V^{(5)}-10V'''V-20V'V''+30V^2V'-10(V'''-3V V')+9V'=0 \,,$$ where $V^{(5)}=d\,^5V/dx^5$. Notice that in contrast with the complexified case $V_{2,0}^{\pm}$, the Hermitian version of the potential, $V_{2i,0}$, which is plotted on Fig. \[Scarf\], is not a solution of the $s$-KdV$_2$ equation. The Hamiltonians $H^{\pm}_{2,0}$ fall into the class of systems studied in [@Wadati] in the context of the $\mathcal{PT}-$symmetric nonlinear integrable systems; the corresponding extended Hamiltonian $\mathcal{H}_{2,0}$ appears as a particular case of the diagonalized squared Dirac equation for a free spin-1/2 field in de Sitter space, see Ref. [@Maloney].
The degeneracy of the spectrum of ${\cal H}_{2,0}$ is twice that for each system in (\[h20\]), and we have two doublets of bound states and two zero energy states at the very bottom of the four-fold degenerate continuous part of the spectrum. The bound states correspond here to the complex energy resonances in the Hermitian version with real Scarf II potential, see Fig. 2. Their eigenfunctions, $$\Phi_{0,l}^{\pm}=\left( \begin{array}{c}
\displaystyle \frac{e^{- \frac{i}{2}\arctan
\sinh x}}{\cosh^{3/2}x} \\
\\ \displaystyle \pm i \frac{e^{ \frac{i}{2}\arctan
\sinh x}}{\cosh^{3/2}x}
\end{array}\right), \quad \Phi_{1,l}^{\pm}=\left( \begin{array}{c}
\displaystyle \frac{e^{ -\frac{i}{2}\arctan \sinh x}}{\cosh^{3/2}x}(1-
2i\sinh x) \\ \\
\displaystyle \pm i \frac{e^{\frac{i}{2}\arctan \sinh x}}{\cosh^{3/2}x}(1+
2i\sinh x)
\end{array}\right), \label{phi0}$$ satisfy equations $$\label{h20ener}
{\cal H}_{2,0}\Phi_{0,l}^{\pm}=
-\frac{9}{4}\Phi_{0,l}^{\pm}, \qquad {\cal
H}_{2,0} \Phi_{1,l}^{\pm}=
-\frac{1}{4} \Phi_{1,l}^{\pm} \,,$$ and correspond to wavefunctions (\[phi1\]).
These solutions can be obtained from the non-physical states of the free particle, by applicacion of the operators (\[freecrum1\]) or (\[freecrum2\]) with $l=2$ and $m=0$ in the case of the subsystem $H^{+}_{2,0}$. For generic values of $l$ and $m$, the Darboux-Crum transformations that map the free particle eigenstates into those for the lower Hamiltonian $H^{-}_{l,m}$ are realized in correspondence with intertwining relations (\[freecrum3\]) by means of the operators $$\begin{aligned}
\label{freecrum3}
{\cal D}_{l,m}^{\sharp}&=& {\cal P}
({\cal D}_{l,m}^{-}){\cal P} ={\cal T}
({\cal D}_{l,m}^{-}){\cal T} \, , \\
\tilde{{\cal D}}_{l,m}^{\sharp}&=& -{\cal P}
(\tilde{{\cal D}}_{l,m}^{-}){\cal
P} =-{\cal T} (\tilde{{\cal D}}_{l,m}^{-}){\cal T}\,.
\label{freecrum4}\end{aligned}$$ Note that in correspondence with relation $H^{-}_{l,m}(x)=H^{+}_{l,m}(x+i\pi)$, operators (\[freecrum3\]) are obtained equivalently from the intertwining operators ${\cal
D}_{l,m}^{-}$ and $\tilde{{\cal D}}_{l,m}^{-}$ by the half-period shift. The non-physical states which are transformed into bound states by means of the Darboux-Crum transformations (\[freecrum1\]), (\[freecrum2\]), (\[freecrum3\]) and (\[freecrum4\]) are $$\label{9/4}
\phi_{-9/4}^{+}=\cosh \left( \frac{3x}{2}
\right), \qquad \phi_{-9/4}^{-}=\sinh
\left( \frac{3x}{2} \right) \, ,$$ and $$\label{1/4}
\phi_{-1/4}^{+}=\cosh
\left( \frac{x}{2} \right), \qquad \phi_{-1/4}^{-}=\sinh
\left( \frac{x}{2} \right) \, ,$$ which obey the same Schrödinger equations as in (\[h20ener\]), $$\begin{aligned}
\label{nonphysicalenergies}
H_{0}\, \phi_{-9/4}^{\pm}=
-\frac{9}{4}\phi_{-9/4}^{\pm}, \quad H_{0}\,
\phi_{-1/4}^{\pm}=-\frac{1}{4}\phi_{-1/4}^{\pm} \, .\end{aligned}$$ One can choose solutions of different parity with respect to ${\cal
P}$ in (\[9/4\]) and (\[1/4\]) to obtain the bound states (\[phi0\]). Choosing the functions with positive ${\cal P}$-parity we have, $$\begin{aligned}
\Phi_{0,l}^{\pm}&=&\frac{2}{3}\left( \begin{array}{c}
{\cal D}_{2,0}^{-}
\\ \pm i\, {\cal D}_{2,0}^{\sharp}
\end{array}\right) \phi_{-9/4}^{+}=\frac{4i}{9}
\left( \begin{array}{c}
\tilde{{\cal D}}_{2,0}^{-}
\\ \mp i\, \tilde{{\cal D}}_{2,0}^{\sharp}
\end{array}\right) \phi_{-9/4}^{+} \, , \label{r1} \\
\Phi_{1,l}^{\pm}&=&-2\left( \begin{array}{c}
{\cal D}_{2,0}^{-}
\\ \pm i\, {\cal D}_{2,0}^{\sharp}
\end{array}\right) \phi_{-1/4}^{+}=4i\left( \begin{array}{c}
\tilde{{\cal D}}_{2,0}^{-}
\\ \mp i\, \tilde{{\cal D}}_{2,0}^{\sharp}
\end{array}\right) \phi_{-1/4}^{+} \,, \label{r2}\end{aligned}$$ where ${\cal D}_{2,0}^{-}=
A^-_{2,0}A^-_{1,0}$, $\tilde{{\cal
D}}_{2,0}^{-}=B^+_{-2,0}A^+_{-1,0}A^+_{0,0}$, ${\cal D}_{2,0}^{\sharp}=
A^-_{0,-1}A^-_{-1,1}$, $\tilde{{\cal
D}}_{2,0}^{\sharp}=
B^+_{2,0}A^-_{2,0}A^-_{1,0}$. Expressions (\[r1\]) and (\[r2\]) with the non-physical states of ${\cal P}$-negative parity, i.e. $\phi_{-9/4}^{-}$ and $\phi_{-1/4}^{-}$, remain almost identical up to multiplicative constant factors. One can choose these factors pure imaginary to produce, for each entry of (\[r1\]) and (\[r2\]), a state of definite ${\cal PT}$-parity by starting from $\phi_{-9/4}^{\pm}$ or $\phi_{-1/4}^{\pm}$. This can be understood by taking into account that the intertwining operators have no definite ${\cal P}$-parity, but they have a definite ${\cal PT}$-parity. In general case, while the ${\cal D}_{l,m}^{-}$ and ${\cal D}_{l,m}^{\sharp}$ are ${\cal
PT}$-even, the operators $\tilde{{\cal D}}_{l,m}^{-}$ and $\tilde{{\cal D}}_{l,m}^{\sharp}$ are ${\cal PT}$-odd.
Using the same procedure as with bound states, we can construct the eigenstates in the scattering sector (\[contspectr\]) by applying the intertwining Darboux-Crum operators to the plane waves (\[pw\]) of $H_{0}$, $$\label{sca20}
\Phi_{+k}^{\pm}=\left( \begin{array}{c}
{\cal D}_{2,0}^{-}
\\ \pm \, {\cal D}_{2,0}^{\sharp}
\end{array}\right)e^{ikx}, \qquad \Phi_{-k}^{\pm}=\left( \begin{array}{c}
{\cal D}_{2,0}^{-}
\\ \pm \, {\cal D}_{2,0}^{\sharp}
\end{array}\right)e^{-ikx}\,,$$ $${\cal H}_{2,0}\Phi_{+k}^{\pm}=
k^2\Phi_{+k}^{\pm}\,,\qquad {\cal
H}_{2,0}\Phi_{-k}^{\pm}=k^2\Phi_{-k}^{\pm} \, .$$ The eigenfunctions (\[sca20\]) for $H_{2,0}^+$ and $H_{2,0}^-$ may also be obtained by applying, instead, the operators $\tilde{{\cal
D}}_{2,0}^{-}$ and $\tilde{{\cal D}}_{2,0}^{\sharp}$, respectively, to the same plane wave eigenstates. In the case of the zero energy eigenstates we have $$\Phi_{0}^{\pm}=
\left( \begin{array}{c}
{\cal D}_{2,0}^{-}
\\ \pm \, {\cal D}_{2,0}^{\sharp}
\end{array}\right)1=\left( \begin{array}{c}
\tilde{{\cal D}}_{2,0}^{-}
\\ \pm \, \tilde{{\cal D}}_{2,0}^{\sharp}
\end{array}\right)x=-\frac{3}{4}\left( \begin{array}{c}
e^{-2i\arctan \sinh x } \\
\pm e^{2i\arctan \sinh x }
\end{array}\right),$$ i. e. the operators $\tilde{{\cal D}}_{2,0}^{-}$ and $\tilde{{\cal
D}}_{2,0}^{\sharp}$ should act on the non-physical zero energy solutions of $H_{0}$ which are proportional to $x$. Another solution of zero energy of $H_{0}$, which is a constant, is annihilated by the Darboux-Crum operators, $$\label{annihi}
\tilde{{\cal D}}_{2,0}^{-} 1=
\tilde{{\cal D}}_{2,0}^{\sharp} 1=0 \, .$$
The extended Hamiltonian ${\cal H}_{2,0}$ possesses three basic conserved quantities in the form of the matrix differential operators. One of these integrals, ${\cal X}_{2,0}$, is given by $$\label{x20}
{\cal X}_{2,0}=\left( \begin{array}{cc}
0 &X_{2,0}^- \\
X_{2,0}^+ & 0
\end{array}\right)=\left( \begin{array}{cc}
0 & A_{2,0}^-A_{1,0}^-A_{0,0}^-A_{-1,0}^- \\
A_{-1,0}^+A_{0,0}^+A_{1,0}^+A_{2,0}^+ & 0
\end{array}\right) \,.$$ The explicit form of the higher order differential operators is $$\begin{aligned}
&X^\pm_{2,0}=\frac{d^4}{dx^4}\pm
\frac{2i}{\cosh x}\frac{d^3}
{dx^3}+\frac{1}{\cosh^2 x}\left(6\mp3i\sinh x-
\frac{5}{2}\cosh^2x \right)
\frac{d^2}{dx^2}&\\
&-\frac{1}{\cosh^3 x}\left(12\sinh x
\pm i\left[\frac{1}{2}\cosh^2 x-3\right]
\right)\frac{d}{dx}-\frac{13}{\cosh^4x}
\left(13 \mp 3 i\sinh x\right)\left(1\pm i
\sinh x\right)^3\,.& \notag\end{aligned}$$ This integral acts on the physical states of the Hamiltonian ${\cal
H}_{2,0}$ as follows, $${\cal X}_{2,0}\Phi_{0,l}^{\pm}=0,
\quad {\cal X}_{2,0}\Phi_{1,l}^{\pm}=0,
\quad
{\cal X}_{2,0}\Phi_{0}^{\pm}=
\pm \frac{9}{16}\Phi_{0}^{\pm}, \quad$$ $${\cal X}_{2,0}\Phi_{+k}^{\pm}=
\pm \left(k^2+\frac{1}{4}\right)\left(k^2+
\frac{9}{4}\right)\Phi_{+k}^{\pm},
\quad {\cal X}_{2,0}\Phi_{-k}^{\pm}=\pm
\left(k^2+\frac{1}{4}\right)
\left(k^2+\frac{9}{4}\right)\Phi_{-k}^{\pm}\,.$$ The operator ${\cal X}_{2,0}$ does not distinguish the waves coming from the left or the right, but recognizes the states which correspond to resonances in the Hermitian Scarf II potential spectrum, by annihilating all of them. Another integral of motion is also an anti-diagonal matrix differential operator, $${\cal Y}_{2,0}=i\left( \begin{array}{cc}
0 &Y_{2,0}^- \\
Y_{2,0}^+ & 0
\end{array}\right)=i\left( \begin{array}{cc}
0 & B_{2,0}^- \\
B_{2,0}^+ & 0
\end{array}\right) \,,$$ where $$Y_{2,0}^\pm=B_{2,0}^\pm=\frac{d}{dx} \pm i\frac{2}{\cosh x} \,.$$ The integral of motion ${\cal Y}_{2,0}$ commutes with ${\cal
X}_{2,0}$ and encodes the information to be complementary to that provided by the latter. This can be seen from its action on the physical states, $${\cal Y}_{2,0}\Phi_{0,l}^{\pm}=
\pm\frac{3i}{2}\Phi_{0,l}^{\pm}, \qquad {\cal
Y}_{2,0}\Phi_{1,l}^{\pm}=
\pm\frac{i}{2}\Phi_{1,l}^{\pm}, \qquad {\cal
Y}_{2,0}\Phi_{k}^{\pm}=0, \quad$$ $${\cal Y}_{2,0}\Phi_{+k}^{\pm}=
\mp k \Phi_{+k}^{\pm}, \qquad {\cal
Y}_{2,0}\Phi_{-k}^{\pm}=\pm k\Phi_{-k}^{\pm}\,.$$ The remaining doublet states are annihilated by ${\cal Y}_{2,0}$, which in this case correspond to the states of the zero energy at the bottom of the continuous spectrum. The waves coming from the left or the right are recognized by it, and like (\[x20\]), the ${\cal Y}_{2,0}$ detects also the upper index of eigenstates. Note that the bound eigenstates here that correspond to resonances in the Hermitian Scarf II systems spectra have pure imaginary eigenvalues of ${\cal Y}_{2,0}$.
Coherently with the properties of the displayed antidiagonal integrals, the diagonal operator $ {\cal
Z}_{2,0}=\mathcal{X}_{2,0}\mathcal{Y}_{2,0}$ annihilates all the doublet states, and separates scattering states coming from different directions, $${\cal Z}_{2,0}\Phi_{0,l}^{\pm}=0,
\qquad {\cal Z}_{2,0}\Phi_{1,l}^{\pm}=0,
\qquad
{\cal Z}_{2,0}\Phi_{0}^{\pm}=0\,,$$ $${\cal Z}_{2,0}\Phi_{+k}^{\pm}=
- k\left(k^2+\frac{1}{4}\right)\left(k^2+
\frac{9}{4}\right)\Phi_{+k}^{\pm},
\quad {\cal Z}_{2,0}\Phi_{-k}^{\pm}= k
\left(k^2+\frac{1}{4}\right)
\left(k^2+\frac{9}{4}\right)\Phi_{-k}^{\pm}\,.$$
### Systems with three bound states
Reflectionless systems with three bound states are constrained to fullfill the relation $l+m=3$. Potentials (\[po1\]) with $(l=0,m=3)$ and $ (l=2,m=1)$ are related by Eq. (\[ptrela\]). When $(l= 3,m=0)$, the systems $H^\pm_{3,0}$ have three bound states, all of which correspond to resonance states in the Hermitian version. So, all the mentioned three bound states systems have corresponding analogs in more simple cases we have discussed above. A more non-trivial example with three bound states is given by the Hamiltonians $$H^{\pm}_{1,2}=-\frac{d^2}{dx^2}
-\frac{7}{\cosh^2x}\pm5 i\frac{ \sinh x}
{\cosh^2x}\,.$$ The Hermitian counterpart potential for $H^{+}_{1,2}$ is shown on Fig. 2. Here the potentials $V=V^{\pm}_{1,2}$ satisfy the $s$-KdV$_3$ equation $$\begin{aligned}
&_{}&V^{(7)}-14V(V^{(5)}-20V''V')-
42V^{(4)}V'-70V'''V''+70V'''V^2+70V'^3-140V'V
^3+ \notag\\
&_{}&-21(V^{(5)}-10V'''V-20V'V''+
30V^2V')+84(V'''-6V V')-64V'=0\,.\end{aligned}$$ The extended Hamiltonian ${\cal H}_{1,2}$ has six bound states, where four correspond to the bound state from the set (\[ps1\]), $$\Phi_{0,m}^{\pm}=\left( \begin{array}{c}
\displaystyle \frac{e^{ - i\arctan \sinh x}}{\cosh^2 x} \\ \\
\pm \displaystyle \frac{e^{ i\arctan \sinh x}}{\cosh^2 x}
\end{array}\right), \quad \Phi_{1,m}^{\pm}=\left( \begin{array}{c}
\displaystyle \frac{e^{ \mp i\arctan \sinh x}}{\cosh^2 x}(2-3i\sinh x) \\ \\
\displaystyle \pm \frac{e^{ \mp i\arctan \sinh x}}{\cosh^2 x}(2+3i\sinh
x)
\end{array}\right),$$ $$\label{energies12}
{\cal H}_{1,2}\Phi_{0,m}^{\pm}=
-4\Phi_{0,m}^{\pm}, \qquad {\cal
H}_{1,2}\Phi_{1,m}^{\pm}=-\Phi_{1,m}^{\pm}\,.$$ These solutions are analogs of the bound states for the Hermitian counterpart system, which, in addition, admits resonances for a complex value of energy. Those resonances correspond to two extra bound states solutions in the spectrum of the Hamiltonian ${\cal
H}_{1,2}$, $$\Phi_{0,l}^{\pm}=\left( \begin{array}{c}
\displaystyle \frac{e^{ -\frac{5i}{2}
\arctan \sinh x}}{\cosh^{1/2} x} \\ \\
\displaystyle \pm i \frac{e^{
\frac{5i}{2}\arctan \sinh x}}{\cosh^{1/2} x}
\end{array}\right)\, ,\qquad
{\cal H}_{1,2} \Phi_{0,l}^{\pm}=-
\frac{1}{4} \Phi_{0,l}^{\pm} \, .$$ All the bound states described above, $\Phi_{0,m}$, $\Phi_{1,m}$ and $\Phi_{0,l} $, can be derived from the non-physical states of the free particle, $ \phi_{-4}^{\pm}$, $\phi_{-1}^{\pm}$ and $\phi_{-1/4}^{\pm}$, respectively. Here, the unphysical solutions of $H_{0}$ are $$\label{4}
\phi_{-4}^{+}=\cosh 2x ,
\qquad \phi_{-4}^{-}=\sinh 2x \, ,$$ and $$\label{1}
\phi_{-1}^{+}=\cosh x ,
\qquad \phi_{-1}^{-}=\sinh x\,;$$ they have the same eigenvalues (of $H_0$) as the bound states $\Phi_{0,m}^{\pm}$ and $ \Phi_{1,ml}^{\pm}$ in (\[energies12\]). The mapping between the states is given by $$\begin{aligned}
&\Phi_{0,m}^{\pm}=\frac{2i}{15}
\left( \begin{array}{c}
{\cal D}_{1,2}^{-}
\\ \mp \, {\cal D}_{1,2}^{\sharp}
\end{array}\right) \phi_{-4}^{+} , \quad \Phi_{1,m}^{\pm}=-\frac{2}
{3}\left( \begin{array}{c}
{\cal D}_{1,2}^{-}
\\ \pm \, {\cal D}_{1,2}^{\sharp}
\end{array}\right) \phi_{-1}^{+}, \quad \Phi_{0,l}^{\pm}=-\frac{8}
{15}\left( \begin{array}{c}
{\cal D}_{1,2}^{-}
\\ \pm i \, {\cal D}_{1,2}^{\sharp}
\end{array}\right) \phi_{-1/4}^{-}\,,& \nonumber\end{aligned}$$ where we use the definition of the operators (\[freecrum1\]) and (\[freecrum3\]). Similar expressions can be found by making use of the operators (\[freecrum2\]) and (\[freecrum4\]); it is worth to note, however, that some non-physical states of the free particle cannot be mapped properly because are annihilated, $$\begin{aligned}
{\cal D}_{1,2}^{-} \phi_{-4}^{-}=
{\cal D}_{1,2}^{\sharp} \phi_{-4}^{-}=
{\tilde{\cal D}}_{1,2}^{-} \phi_{-4}^{+}
={\tilde{\cal D}}_{1,2}^{\sharp}
\phi_{-4}^{+}=0 \, , \\
{\cal D}_{1,2}^{+} \phi_{-1}^{-}=
{\cal D}_{1,2}^{\sharp} \phi_{-1}^{+}=
{\tilde{\cal D}}_{1,2}^{-} \phi_{-1}^{-}
={\tilde{\cal D}}_{1,2}^{\sharp}
\phi_{-1}^{-}=0 \, .\end{aligned}$$ The situation is quite similar to the previous case (\[annihi\]). In fact, in this case the constant state of the free particle is also annihilated by the operators ${\tilde{\cal D}}_{1,2}^{-}1={\tilde{\cal
D}}_{1,2}^{\sharp}1=0$.
The wave functions of the continuum are obtained by the same method from the free plane waves, $$\label{sca12}
\Phi_{+k}^{\pm}=\left( \begin{array}{c}
{\cal D}_{1,2}^- \\
\pm {\cal D}_{1,2}^\sharp
\end{array}\right)e^{ikx}, \qquad \Phi_{-k}^{\pm}=
\left( \begin{array}{c}
{\cal D}_{1,2}^- \\
\pm {\cal D}_{1,2}^\sharp
\end{array}\right)e^{-ikx} \, ,$$ and have energies $E=k^2$.
The anti-diagonal, mutually commuting basic integrals of motion for this case, ${\cal X}_{1,2}$ and ${\cal Y}_{1,2}$, have differential orders $|{\cal X}_{1,2}|=2$ and $|{\cal Y}_{1,2}| =5$, and read $$\label{x12}
{\cal X}_{1,2}=\left( \begin{array}{cc}
0 &X_{1,2}^- \\
X_{1,2}^+ & 0
\end{array}\right)=\left( \begin{array}{cc}
0 &A_{1,2}^-A_{2,0}^- \\
A_{2,0}^+A_{1,2}^+ & 0
\end{array}\right) \,,$$ and $$\label{y12}
{\cal Y}_{1,2}=i\left( \begin{array}{cc}
0 &Y_{1,2}^- \\
Y_{1,2}^+ & 0
\end{array}\right)=i\left( \begin{array}{cc}
0 & B_{1,2}^-B_{1,1}^-B_{1,0}^-B_{1,-1}^-B_{1,-2}^- \\
B_{1,-2}^+B_{1,-1}^+B_{1,0}^+B_{1,1}^+B_{1,2}^+ & 0
\end{array}\right) \,.$$ The explicit form of differential operators that compose (\[x12\]) and (\[y12\]) are $$X_{1,2}^\pm=\frac{d^2}{dx^2}\pm
\frac{5i}{\cosh x}\frac{d}{dx}-\frac{1}{8
\cosh^2 x}\left(44+\cosh^2x\pm 20i \sinh x \right)$$ and $$\begin{aligned}
&Y_{1,2}^\pm=\frac{d^5}{dx^5}\pm
\frac{5i}{\cosh x}\frac{d^4}{dx^4}-
\frac{5}{\cosh^2x}\left( \sinh^2 x\pm
2i \sinh x\right)\frac{d^3}{dx^3}-
\frac{5}{\cosh^3x}\left( 3\sinh x\mp2i
\pm i\sinh^2 x \right)\frac{d^2}{dx^2} &
\notag \\&+
\frac{1}{\cosh^4 x}\left(4\cosh^4 x+10\cosh^2x-30\mp65i\sinh x\pm10i\sinh
x \right)\frac{d}{dx}& \nonumber \\&
-\frac{15}{\cosh^5 x}\left(2\sinh x\mp5i
\pm4i\cosh^2 x \right) \nonumber \,.&\end{aligned}$$
The action of the integrals on the doublets of the Hamiltonian is given by $${\cal X}_{1,2}\Phi_{0,m}^{\pm}=
\mp \frac{15}{4}\Phi_{0,m}^{\pm}, \quad
{\cal
X}_{1,2}\Phi_{1,m}^{\pm}=
\pm \frac{3}{4}\Phi_{1,m}^{\pm}, \quad {\cal
X}_{1,2}\Phi_{0,l}^{\pm}=0,
\quad {\cal X}_{1,2}\Phi_{0,l}^{\pm}=\pm
\frac{1}
{4}\Psi_{k,0}^{\pm},$$ $${\cal Y}_{1,2}\Phi_{0,m}^{\pm}=0,
\quad {\cal Y}_{1,2}\Phi_{1,m}^{\pm}=0,
\quad
{\cal Y}_{1,2}\Phi_{0,l}^{\pm}=
\pm \frac{45i}{32}\Phi_{0,l}^{\pm}, \quad
{\cal
Y}_{1,2}\Phi_{0}^{\pm}=0 \, .$$ Note that, again, the states $\Phi_{0,l}^{\pm}$, which correspond to resonances in the Hermitian counterpart systems, are annihilated by the integral $\mathcal{X}_{1,2}$ and are characterized by pure imaginary eigenvalues of the second integral $\mathcal{Y}_{1,2}$. The scattering states (\[sca12\]) are eigenstates of the operators ${\cal X}_{1,2}$ and ${\cal Y}_{1,2}$, $${\cal X}_{1,2}\Phi_{+k}^{\pm}=
\mp \left(k^2+\frac{1}{4}\right)
\Phi_{+k}^{\pm}, \qquad {\cal X}_{1,2}
\Phi_{-k}^{\pm}=\mp \left(k^2+\frac{1}
{4}\right)\Phi_{-k}^{\pm}\,,$$ $${\cal Y}_{1,2}\Phi_{+k}^{\pm}=
\mp k(k^2+1)(k^2+4)\Phi_{+k}^{\pm}, \qquad
{\cal Y}_{1,2}\Phi_{-k}^{\pm}=
\pm k(k^2+1)(k^2+4) \Phi_{-k}^{\pm}\,.$$ Finally, the diagonal integral ${\cal Z}_{1,2}={\cal Y}_{1,2}{\cal
X}_{1,2}={\cal X}_{1,2}{\cal Y}_{1,2}$ annihilates the whole set of doublet states, $${\cal Z}_{1,2}\Phi_{0,m}^{\pm}=0,
\quad {\cal Z}_{1,2}\Phi_{1,m}^{\pm}=0,
\quad
{\cal Z}_{1,2}\Phi_{0,l}^{\pm}=0, \quad {\cal
Z}_{1,2}\Phi_{0}^{\pm}=0 \, ,$$ and recognizes, as the integral $ {\cal Y}_{1,2}$, the waves coming from the left or the right, $${\cal Z}_{1,2}\Phi_{+k}^{\pm}=
k\left(k^2+\frac{1}{4}\right)(k^2+1)(k^2+4)
\Phi_{+k}^{\pm}, \quad {\cal Z}_{1,2}
\Phi_{-k}^{\pm}= -k\left(k^2+\frac{1}
{4}\right)(k^2+1)(k^2+4) \Phi_{-k}^{\pm}\,.$$
Discussion and outlook {#discus}
======================
In this paper, by analyzing a two-parametric family of reflectionless $ {\cal PT}$-symmetric Hamiltonians, we have revealed a new supersymmetric structure. The class of potentials studied here provides an instructive example of quantum mechanical systems with non-Hermitian Hamiltonians. In comparison with the Hermitian version of the Scarf II potential, the spectrum of its complexified counterpart contains two series of singlet states discovered earlier within a framework of the group theoretical approach. Surprisingly, this characteristic is imprinted in a tri-supersymmetric structure that is based here on the specific properties of the family of potentials: their pure imaginary period and discrete symmetries of a reflection type in the indexes. Usually, the imaginary period in both Hermitian and non-Hermitian Hamiltonians does not play explicitly an important role at the level of the spectrum, or in supersymmetric aspects. Following the original idea of Dunne and Feinberg for the case of a usual SUSYQM with a linear Lie superalgebraic structure and mutually shifted (on a real line) Hermitian Hamiltonians [@Dunne], we construct an extended $\mathcal{PT}$-symmetric system composed by two Hamiltonians with self-isospectral potentials, but now displaced mutually in the half of the *imaginary* period. The obtained composed system has three basic non-trivial integrals of motion, which in the generic case are the higher order differential operators. The importance of the splitting of the discrete states becomes clear by analyzing these integrals. Two of the anti-diagonal, supercharge-type integrals, $ {\cal X}_{l,m}$ and ${\cal Y}_{l,m}$, annihilate separately the two different sets of doublets of the extended system. These mutually commuting integrals generate a third, diagonal integral, ${\cal Z}_{l,m}$, that implies the reflectionless property of the Hamiltonian: it appears as the Lax integral, which together with the Hamiltonian forms the Lax pair. The $ {\cal PT}$-operator emerges naturally as a valuable symmetry for the integrals of motion in view of the fact that all of them appear as $ {\cal PT}$-symmetric operators, in the same way as the Hamiltonian. Nevertheless, the odd order integral ${\cal
Y}_{l,m}$ reveals a finite number of pairs of complex conjugate eigenvalues when acts on the bound states which correspond to resonances with complex conjugate energy values in the Hermitian potential counterpart. We can say therefore that for this integral the $ {\cal PT}$-symmetry is spontaneously broken.
The ${\cal PT}$-symmetry is composed of the space inversion, $\mathcal{P}$, and the time reversal, $\mathcal{T}$, operators, which play a role of the intertwiners between the mutually displaced components $H^+_{l,m}(x)$ and $H^-_{l,m}(x)=H^+_{l,m}(x+i\pi)$ of the extended Hamiltonian. In addition to them and the half-period displacement operators, there are other discrete intertwiners, the products of which produce discrete symmetries of the extended system. The commuting operators $\mathcal{R}_l:\,(l,m)\rightarrow
(-l,m)$ and $\mathcal{R}_m : (l,m) \rightarrow (l,-m-1)$ produce, particularly, the same effect as the differential intertwiners $X^\pm_{l,m}$ and $Y^\pm_{l,m}$, from which the supercharges $ {\cal X}_{l,m}$ and ${\cal Y}_{l,m}$ are composed. In addition to a usual choice for the $\Z_2$-grading operator $\Gamma=\sigma_3$, other choices are also possible. The product of the $\mathcal{P}$ and of the operator of the displacement for the half of the imaginary period is one of them, which also happens to be the grading operator for the hidden, nonlinear bosonized supersymmetries of the subsystems $H^+_{l,m}$ and $H^-_{l,m}$, where the Lax operators $Z^+_{l,m}$ and $Z^-_{l,m}$ are identified as the odd supercharges.
On the other hand, the mentioned two sets of the discrete eigenstates can be related between themselves by means of another discrete symmetry of the Hamiltonian, which interchanges the integer-valued lattice of the parameters $l$ and $m$ with the half-integer-valued lattice. It is only for such, integer or half-integer, values of the parameters the complexified Scarf II potentials are reflectionless. The indicated symmetry operation intertwines the operators $A^{\pm}_{l,m}$ and $B^{\pm}_{l,m}$, which are the building blocks for the intertwiners $X^\pm_{l,m}$ and $Y^\pm_{l,m}$, respectively. As a consequence, the role of the integrals ${\cal X}_{l,m}$ and ${\cal Y}_{l,m}$ is dually interchanged by those specific discrete symmetries. In contrast with the rest of the discrete symmetries, these duality generators have no analog in a form of differential operators.
The supersymmetric structure presented here displays several similarities with the tri-supersymmetric structure in self-isospectral Hermitian finite-gap systems with elliptic potentials studied in [@trisusy]. In that class of Hermitian systems, two distinct finite-dimensional representations of $sl(2,\R)$ are realized on periodic and antiperiodic band-edge states; like here three basic integrals of motion are present in the extended system, and different choices for the grading operators are also possible. The main difference with the present structure is that the corresponding finite-gap elliptic systems are *doubly periodic*, in addition to the imaginary period the corresponding systems have also a *real* period, and there the self-isospectral systems are shifted for the half of their real period. The corresponding tri-supersymmetric systems studied in [@trisusy] are described by the associated Lamé potentials, which constitute a subclass of the Darboux-Treibich-Verdier family (\[dtv\]). It is interesting therefore to investigate the question of existence of tri-supersymmetric structure for such a class of doubly periodic ${\cal PT}$-symmetric potentials, where the extended Hamiltonian would unify the self-isospectral partners with a mutual complex displacement.
In the definition of a physically consistent, positively definite inner product for the systems with non-Hermitian Hamiltonians, the existence of the operator ${\cal C}$ of the nature of a charge conjugation operator seems to be crucial [@BenderReview]. An open question is the existence of such a kind of the operator for the complexified Scarf II potential. One can wonder then if the supersymmetric structure discussed here can be helpful in this sense, specifically, if the ${\cal C}$ can be expressed in terms of, or related to the non-trivial integrals of motion.
Particular cases of the potential with $l\in\Z$ and $m=0$ (and equivalent cases obtained by symmetry transformations of indexes) discussed here appear in quantum field theory in curved space-times [@Maloney]. A natural question is if a general case of the complex reflectionless potential plays any role in some related problems, and if the the revealed supersymmetric structure could give some insight to these theories.
0.2cm
**Acknowledgements.** The work has been partially supported by FONDECYT Grants 1095027 (MP) and 3100123 (FC). FC acknowledges also financial support via the CONICYT grants 79112034 and Anillo ACT-91: Southern Theoretical Physics Laboratory (STPLab). MP and FC are grateful, respectively, to CECs and Universidad de Santiago de Chile for hospitality. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.
[99]{} C. M. Bender, S. Boettcher, *“Real spectra in non-Hermitian Hamiltonians having PT symmetry,”* Phys. Rev. Lett. [**80**]{}, 5243 (1998), \[arXiv:physics/9712001\]. C. M. Bender, S. Boettcher and P. Meisinger, *“PT symmetric quantum mechanics,”* J. Math. Phys. [**40**]{}, 2201 (1999), \[arXiv:quant-ph/9809072\]. P. Dorey, C. Dunning and R. Tateo, *“Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics,”* J. Phys. A [**34**]{}, 5679 (2001), \[arXiv:hep-th/0103051\].
A. Mostafazadeh, *“Pseudo-Hermiticity versus PT symmetry. The necessary condition for the reality of the spectrum,”* J. Math. Phys. [**43**]{}, 205 (2002), \[arXiv:math-ph/0107001\]; *“Pseudo-Hermiticity versus PT symmetry 2. A Complete characterization of non-Hermitian Hamiltonians with a real spectrum,”* J. Math. Phys. [**43**]{}, 2814 (2002), \[arXiv:math-ph/0110016\]. C. M. Bender, *“Making sense of non-Hermitian Hamiltonians,”* Rept. Prog. Phys. [**70**]{}, 947 (2007), \[hep-th/0703096 \[hep-th\]\]. A. Mostafazadeh, *“Pseudo-Hermitian representation of quantum mechanics,”* Int. J. Geom. Meth. Mod. Phys. [**7**]{}, 1191 (2010), \[arXiv:0810.5643 \[quant-ph\]\]. E. Witten, *“Dynamical Breaking Of Supersymmetry,”* Nucl. Phys. B [**188**]{}, 513 (1981).
F. Cooper, A. Khare and U. Sukhatme, *“Supersymmetry and quantum mechanics,”* Phys. Rept. [**251**]{}, 267 (1995), \[arXiv:hep-th/9405029\]; G. Junker, *Supersymmetric Methods in Quantum and Statistical Physics*, (Springer, Berlin, 1996); Bagchi B.K., *Supersymmetry in quantum and classical mechanics*, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 116, (Chapman & Hall/CRC, Boca Raton, FL, 2001).
V. B. Matveev and M. A. Salle, *Darboux Transformations and Solitons*, (Springer, Berlin, 1991).
A. A. Andrianov, M. V. Ioffe and V. P. Spiridonov, *“Higher derivative supersymmetry and the Witten index,”* Phys. Lett. A [**174**]{}, 273 (1993), \[arXiv:hep-th/9303005\]; A. A. Andrianov, M. V. Ioffe and D. N. Nishnianidze, *“Polynomial SUSY in quantum mechanics and second derivative Darboux transformation,”* Phys. Lett. A [**201**]{}, 103 (1995), \[arXiv:hep-th/9404120\].
M. S. Plyushchay, *“Hidden nonlinear supersymmetries in pure parabosonic systems,”* Int. J. Mod. Phys. A [**15**]{}, 3679 (2000), \[arXiv:hep-th/9903130\]. S. M. Klishevich and M. S. Plyushchay, *“Nonlinear supersymmetry, quantum anomaly and quasi-exactly solvable systems,”* Nucl. Phys. B [**606**]{}, 583 (2001), \[arXiv:hep-th/0012023\]
M. S. Plyushchay, *“Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions,”* Annals Phys. [**245**]{}, 339 (1996), \[arXiv:hep-th/9601116\]; J. Gamboa, M. Plyushchay and J. Zanelli, *“Three aspects of bosonized supersymmetry and linear differential field equation with reflection,”* Nucl. Phys. B [**543**]{}, 447 (1999), \[arXiv:hep-th/9808062\].
F. Correa and M. S. Plyushchay, *“Hidden supersymmetry in quantum bosonic systems,”* Annals Phys. [**322**]{}, 2493 (2007), \[arXiv:hep-th/0605104\]. V. Jakubsky, L. -M. Nieto and M. S. Plyushchay, *“The origin of the hidden supersymmetry,”* Phys. Lett. B [**692**]{}, 51 (2010), \[arXiv:1004.5489 \[hep-th\]\]. G. V. Dunne and J. Feinberg, *“Self-isospectral periodic potentials and supersymmetric quantum mechanics,”* Phys. Rev. D [**57**]{}, 1271 (1998), \[arXiv:hep-th/9706012\]. D. J. Fernandez, B. Mielnik, O. Rosas-Ortiz and B. F. Samsonov, *“The phenomenon of Darboux displacements,”* Phys. Lett. A [**294**]{}, 168 (2002), \[arXiv:quant-ph/0302204\]. D. J. Fernandez, J. Negro and L. M. Nieto, “*Second-order supersymmetric periodic potentials,*” Phys. Lett. A [**275**]{}, 338 (2000). B.F. Samsonov, M.L. Glasser, J. Negro and L.M. Nieto, “*Second-order Darboux displacements,*" J. Phys. A [**36**]{}, 10053 (2003), \[arXiv:quant-ph/0307146\].
F. Correa, V. Jakubsky, L. -M. Nieto and M. S. Plyushchay, *“Self-isospectrality, special supersymmetry, and their effect on the band structure,”* Phys. Rev. Lett. [**101**]{}, 030403 (2008), \[arXiv:0801.1671 \[hep-th\]\]; F. Correa, V. Jakubsky and M. S. Plyushchay, *“Finite-gap systems, tri-supersymmetry and self-isospectrality,”* J. Phys. A [**41**]{}, 485303 (2008), \[arXiv:0806.1614 \[hep-th\]\]. F. Correa, L. -M. Nieto and M. S. Plyushchay, *“Hidden nonlinear su($2|2$) superunitary symmetry of N=2 superextended 1D Dirac delta potential problem,”* Phys. Lett. B [**659**]{}, 746 (2008), \[arXiv:0707.1393 \[hep-th\]\]. F. Correa, H. Falomir, V. Jakubsky and M. S. Plyushchay, *“Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons,”* Annals Phys. [**325**]{}, 2653 (2010), \[arXiv:1003.1434 \[hep-th\]\]. M. S. Plyushchay and L. -M. Nieto, *“Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry,”* Phys. Rev. D [**82**]{}, 065022 (2010), \[arXiv:1007.1962 \[hep-th\]\]. M. S. Plyushchay, A. Arancibia and L. -M. Nieto, *“Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit,”* Phys. Rev. D [**83**]{}, 065025 (2011), \[arXiv:1012.4529 \[hep-th\]\]. A. Arancibia and M. S. Plyushchay, *“Extended supersymmetry of the self-isospectral crystalline and soliton chains,”*, Phys. Rev. D [**85**]{}, 045018 (2012), arXiv:1111.0600 \[hep-th\]. F. Cannata, G. Junker and J. Trost, *“Schrödinger operators with complex potential but real spectrum,”* Phys. Lett. A [**246**]{}, 219 (1998), \[arXiv:quant-ph/9805085\]. A. A. Andrianov, F. Cannata, J. P. Dedonder and M. V. Ioffe, *“SUSY quantum mechanics with complex superpotentials and real energy spectra,”* Int. J. Mod. Phys. A [**14**]{}, 2675 (1999), \[arXiv:quant-ph/9806019\]. M. Znojil, *“Shape invariant potentials with PT symmetry,”* J. Phys. A [**33**]{}, L61 (2000), \[arXiv:quant-ph/9911116\]. B. Bagchi, F. Cannata and C. Quesne, *“PT symmetric sextic potentials,”* Phys. Lett. A [**269**]{}, 79 (2000), \[arXiv:quant-ph/0003085\]. M. Znojil, F. Cannata, B. Bagchi and R. Roychoudhury, *“Supersymmetry without hermiticity within PT symmetric quantum mechanics,”* Phys. Lett. B [**483**]{}, 284 (2000), \[arXiv:hep-th/0003277\]. B. Bagchi, S. Mallik and C. Quesne, *“Generating complex potentials with real eigenvalues in supersymmetric quantum mechanics,”* Int. J. Mod. Phys. A [**16**]{}, 2859 (2001), \[arXiv:quant-ph/0102093\]. P. Dorey, C. Dunning and R. Tateo, *“Supersymmetry and the spontaneous breakdown of PT symmetry,”* J. Phys. A [**34**]{}, L391 (2001), \[arXiv:hep-th/0104119\].
A. Mostafazadeh, *“Pseudo-supersymmetric quantum mechanics and isospectral pseudoHermitian Hamiltonians,”* Nucl. Phys. B [**640**]{}, 419 (2002), \[arXiv:math-ph/0203041\].
S. M. Klishevich and M. S. Plyushchay, *“Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra,”* Nucl. Phys. B [**628**]{}, 217 (2002), \[arXiv:hep-th/0112158\]; B. Bagchi, S. Mallik and C. Quesne, *“Complexified PSUSY and SSUSY interpretations of some PT symmetric Hamiltonians possessing two series of real energy eigenvalues,”* Int. J. Mod. Phys. A [**17**]{}, 51 (2002), \[arXiv:quant-ph/0106021 \[quant-ph\]\]; A. Khare and U. Sukhatme, *“Analytically solvable PT-invariant periodic potentials,”* Phys. Lett. A [**324**]{}, 406 (2004), \[arXiv:quant-ph/0402106\]; A. Sinha and P. Roy, *“PT symmetric models with nonlinear pseudosupersymmetry,”* J. Math. Phys. [**46**]{}, 032102 (2005), \[arXiv:quant-ph/0505221\]; B. F. Samsonov, *“Irreducible second order SUSY transformations between real and complex potentials,”* Phys. Lett. A [**358**]{}, 105 (2006), \[arXiv:quant-ph/0602101\]; A. Gonzalez-Lopez and T. Tanaka, *“Nonlinear pseudo-supersymmetry in the framework of N-fold supersymmetry,”* J. Phys. A [**39**]{}, 3715 (2006), \[arXiv:quant-ph/0602177\]; A. A. Andrianov, F. Cannata and A. V. Sokolov, *“Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians. I. General properties,”* Nucl. Phys. B [**773**]{}, 107 (2007), \[arXiv:math-ph/0610024\]. R. Roychoudhury and B. Roy, *“Intertwining operator in nonlinear pseudo-supersymmetry,”* Phys. Lett. A [**361**]{}, 291 (2007); *“Pseudo-supersymmetry and third order intertwining operator,”* Phys. Lett. A [**372**]{}, 997 (2008).
B. Bagchi and C. Quesne, *“sl(2,$\mathbb{C}$) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues,”* Phys. Lett. A [**273**]{}, 285 (2000), \[arXiv:math-ph/0008020\].
B. Bagchi, C. Quesne, *“An update on PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally-extended supersymmetric partners,”* J. Phys. A [**43**]{}, 305301 (2010), \[arXiv:1002.4309 \[quant-ph\]\].
P. Lagogiannis, A. Maloney, Y. Wang, *“Odd-dimensional de Sitter space is transparent,”* \[arXiv:1106.2846 \[hep-th\]\].
M. Wadati, *“Construction of parity-time symmetric potential through the soliton theory,”* Journ. of the Physical Society of Japan [**77**]{}, 074005 (2008).
Z. H. Musslimani, K. G. Makris, R. El-Ganainy and D. N. Christodoulides, *“Optical solitons in PT periodic potentials,”* Phys. Rev. Lett. [**100**]{}, 030402 (2008); *“Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,”* J. Phys. A [**41**]{}, 244019 (2008); *“PT-symmetric periodic optical potentials,”*, Int. J. Theor. Phys. [**50**]{}, 1019 (2011).
J. W. Dabrowska, A. Khare and U. P. Sukhatme, *“Explicit wave functions for shape invariant potentials by operator techniques,”* J. Phys. A [**21**]{}, L195 (1988). M. Znojil and G. Levai, *“The interplay of supersymmetry and PT symmetry in quantum mechanics: a case study for the Scarf II potential,”* J. Phys. A [**35**]{}, 8793 (2002), \[arXiv:quant-ph/0206013\]. Z. Ahmed, *“Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT invariant potential,”* Phys. Lett. A [**282**]{}, 343 (2001). G. Levai, F. Cannata and A. Ventura, *“Algebraic and scattering aspects of a PT symmetric solvable model,”* J. Phys. A [**34**]{}, 839 (2001).
P. G. Drazin and R. S. Johnson, *Solitons: An Introduction*, (Cambridge, UK: Univ. Press, 1989); A. Perelomov, Y. Zeldovich, *Quantum Mechanics: Selected Topics*, World Scientific, Singapore, 1998.
C. M. Bender, D. C. Brody and H. F. Jones, *“Complex extension of quantum mechanics,”* eConf C [**0306234**]{}, 617 (2003) \[Phys. Rev. Lett. [**89**]{}, 270401 (2002)\] \[Erratum-ibid. [**92**]{}, 119902 (2004)\], \[arXiv:quant-ph/0208076\]. F. Gesztesy and H. Holden, *Soliton Equations and their Algebro-Geometric Solutions*, (Cambridge University Press, 2003).
P. D. Lax, *“Integrals of Nonlinear Equations of Evolution and Solitary Waves,”* Commun. Pure Appl. Math. [**21**]{}, 467 (1968). E. D. Belokolos et al, *Algebro-geometric approach tononlinear integrable equations*, (Springer, Berlin, 1994).
G. Pöschl and E. Teller, *“Bemerkungen zur Quantenmechanik des anharmonischen Oszillators,”* Z. Phys. [**83**]{}, 143 (1933). F. Correa, V. Jakubsky and M. S. Plyushchay, *“Aharonov-Bohm effect on AdS(2) and nonlinear supersymmetry of reflectionless Poschl-Teller system,”* Annals Phys. [**324**]{}, 1078 (2009), \[arXiv:0809.2854 \[hep-th\]\]. A. P. Veselov, *“On Darboux-Treibich-Verdier potentials,”* Lett. Math. Phys. [**96**]{}, 209 (2011), \[arXiv:1004.5355\].
W. Magnus and S. Winkler, *HillÕs Equation* (Wiley, New York, 1966).
E. T. Whittaker and G. N. Watson, *A course of modern analysis*, (Cambridge Univ. Press, Cambridge, 1980), M. Abramowitz and I. Stegun (Eds.), *Handbook of Mathematical Functions*, Dover (1990).
Z. Ahmed, *“Addendum to ’Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT invariant potential’,”* Phys. Lett. A [**287**]{}, 295 (2001).
B. Bagchi, C. Quesne, *“Comment on ‘Supersymmetry, PT-symmetry and spectral bifurcation’,”* Annals Phys. [**326**]{}, 534 (2011), \[arXiv:1007.3870 \[math-ph\]\]. K. Abhinav and P. K. Panigrahi, *“Supersymmetry, PT-symmetry and spectral bifurcation,”* Annals Phys. [**325**]{}, 1198 (2010), \[arXiv:0910.2423 \[quant-ph\]\].
[^1]: A pair of complex potentials (\[po1\]) with $m=0$ and corresponding extended Hamiltonian ${\cal H}_{l,0}$ emerged recently in [@Maloney] under investigation of quantum field theory in de Sitter space.
[^2]: The free particle corresponds to $\lambda=0$ and $\lambda=-1$, and can be considered as a zero-gap case of the family of finite-gap, reflectionless Pöschl-Teller systems (\[PT\]) [@boso1; @ptads].
[^3]: It is worth to note that in the early stages of studying the complexified Scarf II potential, just one series of the singlet states, (\[ps1\]), that comes by analytic continuation of the Hermitian version, was considered in the literature [@susypt3; @Ahmed]. Later, the complete set was found by algebraic methods in [@GrNe] and reconfirmed in refs. [@LeZno; @LevaiCanVen2; @AhmedAd]. The second, lost set of states, (\[ps2\]), corresponds to resonances in the Hermitian counterpart potential. Within the problem for the complexified Scarf II potential this second set can be obtained from the counterparts of the singlet states of the Hermitian problem by applying the symmetry transformation of the parameters (\[dis3\]).
|
---
abstract: 'Digital Healthcare systems are very popular lately, as they provide a variety of helpful means to monitor people’s health state as well as to protect people against an unexpected health situation. These systems contain a huge amount of personal information in a form of electronic health records that are not allowed to be disclosed to unauthorised users. Hence, health data and information need to be protected against attacks and thefts. In this paper, we propose a secure distributed architecture for healthcare data storage and analysis. It uses a novel security model to rigorously control permissions of accessing sensitive data in the system, as well as to protect the transmitted data between distributed system servers and nodes. The model also satisfies the NIST security requirements. Thorough experimental results show that the model is very promising.'
author:
- '\'
bibliography:
- 'References.bib'
title: 'Security-Aware Access Model for Data-Driven EHR System'
---
Healthcare system, Electronic Health Record (EHR), Kerberos, LDAP, Symmetric Key Scheme, Public Key Scheme, Hash Function, SSL/TLS.
Introduction {#sec:Int}
============
E-Healthcare systems are very popular in recent years, as it becomes a matter of urgency to manage and control patients’ data. This data can be used to support them in bad health and emergency situations, to evaluate practitioner performance, or to operate as a health-caring consultant[^1] as well. Hence, users find the most of healthcare applications and devices helpful. In order to predict and support patients well-being, their data should be collected and stored in a form of electronic health records (EHR). However, as patients’ data is very sensitive and must be protected against leakage and attacks, the patients do not trust e-healthcare systems. For instance, Orange telecommunication provider declared that $78\%$ of users did not trust companies in the way their data was used.[^2]. Therefore, users’ trust has been raised, as their personal data must be protected and processed in a secret way against possibilities of being unexpectedly disclosed. Moreover, increasing user’s trust will surely guarantee the sustainability of the potential data-driven economy.
To protect patients’ data against any abusive use or attacks, several international regulations [@Rule15] have been issued, such as the US Health Insurance Portability and Accountability Act (HIPAA), the Health Information Technology for Economic and Clinical Health (HITECH), the European Union’s Data Protection Directive, the Australian Privacy Act and Japan’s Personal Information Protection Act (PIPA). These regulations set policies on data access to the health information in the majority regions of the world. However, they do not protect personal data directly against all vulnerabilities. Therefore, the risks of leaking patient’s personal information still exist.
Marci et al. in [@Marci2006] analysed the potential issues of security and privacy that can affect healthcare systems, such as access to data, how and when data is stored, security of data transfer, data analysis rights, and the governing policies. Among them, [@Liu2015] [@Ahmad2016] [@Huang2017] protected the data during the storage and access processes, while [@Boon2009] [@Marcos2015] [@Guo2012] proposed secure solutions for data when it is transmitted through the network.
In [@ThienAnetal], a large-scale healthcare system was built to collect and analyse the health information about obesity in children. The data is collected from patients and voluntary students in different schools, cities, countries and regions in Europe. Besides the data analytics, the question is how to protect this big data against unauthorised users. The proposed security protocol is similar to [@ThienAnetal], but with more efficient policies and mechanisms. In this paper, we propose a distributed architecture to manage healthcare data more autonomously and process data queries more rigorously and with high security.
The remainder of the paper is organised as follows. Section \[sec:RelatedWorks\] presents the related works along with their drawbacks and trade-offs. Sections \[sec:EHR\] and \[sec:Archi\] describe the adopted healthcare system and its architecture, respectively. Section \[sec:UserAUTH\] describes user authentication and authorisation mechanisms. Section \[sec:CAE\] presents controller authentication and its authorisation methods. The secure data transmission is presented in Section \[sec:SDT\]. Section \[sec:SecEval\] discusses the evaluation of the proposed model against the security attacks. Experimental results are reported in Section \[sec:Experi\]. We conclude in Section \[sec:Concl\].
Related Work {#sec:RelatedWorks}
============
Health care systems have been under development for quite a while. Their main goal is to reduce the administrative cost, operational time, and to improve the work efficiency in medical related works. In order to retrieve patients’ data more effectively and to provide richer data sets for health measurements, authors in [@Schatz15] proposed a database using mobile health monitors. They can collect a huge amount of data for further processes in predicting probabilities of chronic diseases, for instance. Whereas, authors in [@Yu15] built a healthcare system with data analytics capabilities for diagnosing diseases early from symptoms provided by patients. The authors exploited the data mining technique, namely Naive Bayes, to detect diseases from given symptoms. These works did not focus on the privacy and security issues.
Privacy and security issues in the healthcare information management systems have been emphasised for years. To address privacy risks in sharing healthcare information, [@Ostherr17] outlined how sensitive this data is and any disclosure of such data may have devastating consequences on patients. Furthermore, Kotz et al. [@Kotz16] discussed several issues from policy, regulation to technologies, such as anonymity, hash, homomorphic cryptography, and differential privacy. However, the point is that these technologies cannot guarantee a complete privacy. There is still a need for a security-aware model that combines several security and privacy technologies to ensure data confidentiality and user privacy in healthcare systems.
Some works had different approaches in designing the security-aware system framework in protecting user privacy or detecting the system risks to get back to defend users. As in [@Dubo2017], the authors proposed a privacy-aware framework for managing and sharing electronic medical record data for the cancer patient care based on block chain technology. [@Tahar2006] proposed a secure framework for a distributed network based on grid technique and encryption techniques in controlling data access through a set of policies. Another work [@Tahar2015] presented a security-aware model to forensically visualise the evidence and attack scenario in a computer system from network log files so that the system operation can avoid such attacks. Authors in [@Tahar2016] proposed an interesting security-aware framework to protect the system from the internal attackers. In this work, authors protect the system resources by creating walls, each of which owns different security properties, between users and the system resources to make users unable to recognise those resources. Another work by Liu et al. [@Liu2015] built up the access control applied Role-based Access Control (RBAC) to protect the user privacy in EHRs stored in the database from any outside access. This work can protect the data from external accesses, however, the secure data transmission has not been dealt with.
For cryptographic solutions, the authors in [@Rao15] discussed possible solutions for security and privacy issues of big data heathcare systems. They studied a solution based on de-identification, using data-centric approach that allows data to be decrypted, untokened or unmasked by only authorised objects or access control. Hence, a more detailed solution is needed for a distributed system and protect it from all risks of disclosing the patients’ information. Authors in [@Blobel2016] adopted the de-identification and proper ID management techniques to hide important information in the database. Whereas, in [@ThienAnetal], the authors focused on the privacy preservation by applying de-identification and anonymity techniques to prevent the health information to be disclosed from the data storage. Another approach in applying the encryption technique was proposed by Miao et al. [@Miao2016]. They designed a secure model for searching the health information from the encrypted database by applying encryption algorithm. In addition, some other works as [@Barbara2013] [@Barbara2014] [@Barbara2015] applied the homomorphic encryption and secure comparison algorithms to make the collaboration among parties in the system unable to violate the data of the other sides. Some other research works focus on leveraging distributed ledger technology in protecting the health information. In [@dwi2019], the authors preserved the privacy of medical data stored and transmitted among Internet of Things (IoT) devices by adopting block chain approach and symmetric lightweight cryptographic algorithm in the distributed network. In [@Zhang2010] [@Shafer2010], the authors focused on securing data transmission by applying TLS/SSL solutions and encryption algorithms such as AES and RSA, but the data storage security has not been dealt with in their solutions.
From another approach view, in this paper, we propose a solution for preserving user privacy and data security in both data transmission and storage for an effective dealt with the big health data. More specifically, we propose a distributed model for executing methods of protecting data against attacks such as replay attack, eavesdropping attack, and unauthorised spy. In order to set such objectives, we exploit the current powerful techniques, such as Kerberos, LDAP, Active Directory, SSL/TLS, etc., combined with different cryptographic algorithms AES CBC/GCM, RSA, DES, SHA-2, ECDSA, etc.
Data Access Security Settings {#sec:EHR}
=============================
In the following we define the end-user role categorisation in detail, their respective permissions to data access, and other system components’ permissions.
User Role Categorisation {#sec:user_role}
------------------------
In our system, we define different classes of users depending on their roles, and grant different permissions for each class role. Each user has a pair of [*username*]{}, [ *password*]{}, and roles in the system. A user can have more than one role in the system. Figure \[fig:EHClassification\] shows the relationship among user roles in the system. More specifically, there are three classes of users: [*school*]{}, [*clinic*]{}, and [ *admin*]{}. For each class, different access rights were defined to reflect their activities and interactions with the system. The [*admin*]{} user manages users and classes. The roles are defined in the following:
- Each class [*school*]{} has a [*school admin*]{} for managing the school users, who may have different roles. The school user roles are [*teacher*]{}, and [ *students*]{}. In the school, there are many [*group*]{}s of **students**. A group is managed by a teacher.
- Each class [*clinic*]{} has a [*clinic admin*]{} for managing clinic users. The clinic roles are [*clinician*]{} and [*patients*]{}. A clinician manages a number of patients in the clinic.
- The [*admin*]{} class includes the core system administrators, admin outside schools and clinic admin staff. In this class, a system admin, called [*global admin*]{}, manages all the users of the whole system, a [*policy maker*]{} identifies conditions associated to childhood obesity and designs effective policies, which are then applied to hospitals, schools, communities, etc.
![EHS User Categorisation.[]{data-label="fig:EHClassification"}](EHSOrganization.pdf)
Moreover, the storage system has two major databases, which are MongoDB[^3] and Cassandra[^4]. The Cassandra database stores the time-series data collected from users’ wearable devices, such as smart watch and smart sensors, and smart phone, whereas, the MongoDB database stores the most sensitive data of the system. These include user management and other private data. In this work, we apply Role-based Access Control (RBAC) technique to efficiently manage system’s users based on their roles. The admin information are stored in MongoDB. These include the user’s registration information and other information required by the system, such as [*username, password, roles and global ID number, etc*]{}. Depending on the roles, a user may have other administrative details such as [*sub-role, group ID, school ID, clinic ID, and supervisor ID*]{}. All collected and derived data of a participant, that was stored in either MongoDB or Cassandra is identified with a global ID. In other words, a unique global ID is assigned to each participant in the system to reference any piece of their data in the BigO system. Users are granted access permissions to the system based on their role. For example, the clinician can access the data of their patients only via clinic portal, the teacher can access data of their students only via school portal. More details of the user verification process are presented in Section \[sec:UserAUTH\].
Permission Settings
-------------------
The system consists of two types of components: controllers and modules. A controller includes a set of services provided to end-users or to the other modules. Each controller has limited privileges for providing access to the back-end. Therefore, all access permissions requested by the controllers are checked based on a technique called Discretionary Access Control (DAC). Each controller is evaluated for its own permissions for each collection, and it may have the permissions to [*read*]{} ([**R**]{}) a data, [ *write*]{} ([**W**]{}) a data or both [*read*]{} and [*write*]{} ([**RW**]{}). With a reading permission, for instance, a controller can make [*select*]{} or [*find*]{} request to the database ([*select*]{} query for MongoDB and [*find*]{} query for Cassandra). For writing, a controller can send an [*insert/update/remove*]{} or [*modify*]{} request to any if the two databases.
In the DAC Table (see Figure \[fig:DAC\]), a number of controllers are provided with the reading and writing permissions with the respective data, that is, collections/tables in the databases. The columns in DAC table are the system controllers, and the rows represent collections/tables. For instance, the [*portal controller*]{} has [**RW**]{} permissions on all the collections. A detailed description of the controller verification process is presented in Section \[sec:controller\_authorization\].
![DAC Table of Controllers vs Collections[]{data-label="fig:DAC"}](DAC.pdf){width="6cm" height="5cm"}
Security-Aware Architecture {#sec:Archi}
===========================
We adopted the MongoDB and Cassandra database management systems to leverage the power of each of them, especially, a combination of the both can exploit fully their strong characteristics. Specifically, Cassandra manages only user data crawled from the smart mobile devices, and its very fast writing queries. Whereas, MongoDB manages the rest of user information (e.g., users, clinics, schools, timelines, regions, statistics, etc.), and it supports a good querying performance for its feature *index settings*. Moreover, MongoDB supports authentication and authorisation using Kerberos [@Kerberos] and LDAP [@LDAP], as well as AES encryption[^5] at-rest and in-flight modes. Cassandra supports authentication and authorisation using LDAP, data transmission using Transparent Data Encryption (TDE)[^6]. The security techniques supported by MongoDB and Cassandra fulfil the security requirements against current attacks.[^7]
![Security-aware $3 \times 3$ matrix model[]{data-label="fig:architecture"}](dataflow.pdf){width="8.5cm" height="7.2cm"}
We adopted a security design architecture at every layer and between layers (see Figure \[fig:architecture\]). The layers are [*controller layer*]{}, [*access control layer*]{} and [*back-end layer*]{}. Each layer verifies the permissions of the lower layer. Therefore, it is more strict to check the data access from the bottom layer to top layer. Within each layer one needs to deal with the security of the following types of data [*original data*]{}, [*de-identified statistics data*]{}, and [*anonymised data*]{}. The security level increases from original to anonymised data, as the data gets de-identified, anonymised and perturbed respectively.
Consider the controller layer, a controller contains a set of services providing the end-users access from the external network or from another component from the system. The [*Mobile App Controller*]{} provides a set of interfaces for mobile users to access services such as inputting user’s information about meals, activities, etc. These mobile data values are saved in the mobile device memory before their transfer to Cassandra. The [*Portal controller*]{} supports a set of interfaces that can be used to access the supported web services, such as [*Clinical Analysis Services*]{} for measuring the activity rates, predicting the effect of users’ activities, etc. This data is transmitted to MongoDB or the related [*Analysis Services*]{}. Any controller access that is requested by the end-users are evaluated using RBAC as mentioned in Section \[sec:user\_role\]. The two other controllers; [*Web Portal*]{} and [*Analysis Services*]{} access the data through a Secured View.
For the access control layer, there are two investigations for evaluating its requests, that is, [*DAC*]{} and [*Secured View*]{}. These parts are executed by the authentication server. Only Mobile App Controller and Portal Controller can access DAC, while only Analysis Backend and Analysis Services can access Secured View. DAC determines weather the requesting controller is eligible to access the data in the back-end layer before performing its request. The Secured View controls and restricts access to the data by encouraging the controllers to use de-identified and anonymised data in order to protect data from possible vulnerabilities of leaking the personal information.
At the back-end layer, the database server protects data one more time against the unauthorised controllers and end-users by determining weather that controller and user can access such requested data. The [*Original Data*]{} is raw data and it is readable. The [*Identified Statistics Data*]{} contains data excluding the identity information of the users. The [*Anonymised Data*]{} stores data without leaving any trace of an inference to the user identity.
Furthermore, components of $3\times 3$ matrix are located in a physically distributed servers. In particular, databases are stored in different servers, i.e., original MongoDB data, original Cassandra data, and reference data of original data column are stored on the three separate servers. The individual aggregated data and population statistics data are stored on one server, and the de-identified data for analysis is stored on another server. The distributed database storage plays a key role in improving data management and privacy against the linkability attack.
![Double security check at each component and module.[]{data-label="fig:2_secure_check"}](twice_secure_check.pdf){width="2.5cm" height="2.5cm"}
Moreover, each component shown in Figure \[fig:architecture\] is independent. Therefore, the process of receiving a request from a component and transferring the result to the next component can cause a privilege violation, if the security between components is not enforced. Hence, the data flow between components should be rigorously verified on its path at each component hop. Each component identifies the client and evaluates its permissions before processing the received request. Then it transfers the result to the next component. The next component again evaluates the client’s permissions (c.f. Figure \[fig:2\_secure\_check\]) and so on until the last hop is reached. For example, in Figure \[fig:architecture\] only mobile app and portal controllers can access directly to Role-Based Access Control (RBAC) component of the access control layer, to query the original and statistical datasets, and only Web Applications Controller and Analysis Services can access Secured Views of the access control layer. The RBAC component stores mappings of which controller/user can access which database, while the secured views store anonymised data that remove the Quasi-identifiable (QI) data [@QI].
End-user Privilege Check {#sec:UserAUTH}
========================
A request transaction can be described a security-aware $3\times 3$ matrix model. Basically, the controller layer implements a set of REST APIs to support user queries to database. An end-user who wants to access a database needs to log into a given controller. This is the first shield of the model in protecting the database. Each end-user $u$ has their own credential (i.e., user name $un_u$ and password $p_u$). The user can have one or more roles, denoted as $lrole_u$, in the system (e.g., a teacher, a clinician, a school administrator, a clinic administrator, etc.), a unique ID $ID_u$ and their organisation ID $orgID_u$ (e.g., school ID, clinic ID, etc.). To authenticate and authorise a user, the [*Role-Based Access Control*]{} (RBAC) technique [@RBAC] is used to handle the data access permission of controllers. The role of a user or their organisation is checked for any data access. The details are given in the following.
Token-based User Authentication {#sec:tokenbasedUserAuthen}
-------------------------------
User authentication happens when a user sends a log-in service request to the controller, a long with their credential as an input. The user credentials are formally formulated as follows.
\[def:userCredential\] Let $u$ be a user of the system. Let $(un_u,p_u)$ be a pair of user name and password. Let $ID_u$, $lrole_u$, $org_u$ be $u$’s ID, role list, and organisation ID, respectively. The user credential $Cr_u$ can be defined as: $$Cr_u = [ID_u, (un_u, p_u), lrole_u, orgID_u]$$
All passwords are hashed before being stored in the database. Based on this credential, the controller authenticates the user in collaboration with a security server. To provide the user a legal authentication certification to use for their subsequent requests, the controller gets a JASON Web Token (JWT) generated by the security server. This JWT is created using the RSA algorithm with 512-bit key size. The JWT is expired after a $120$ minutes. Therefore, the user has to send a request to the server to get a new JWT. With the new JWT, the process can continue. More formally, the JWT is formulated as follows.
\[lemma:JWT\] Let $u$, $Cr_u$ be a user and their credential. $JWT_{(Cr_u,t)}$ is JWT token at time $t$ for the user $u$ with credential $Cr_u$. It is defined as follows: $$\label{equ:jwt}
\begin{split}
JWT_{Cr_u} =& [RSA512(Header_{(Cr_u, t)})|\\
& RSA512(Payload_{(Cr_u, t)})|\\
& RSA512(Sign_{(Cr_u, t)})]\\
with\ Header_{(Cr_u,t)} &= \{alg_{jwt}|type_{jwt}\}\\
Payload_{(Cr_u, t)} &= \{lrole_u | iss_u | ID_u |\\ & exp\_time_{(u, t)}| version_{u,t}| iat_u\}
\end{split}$$
where
- $'|'$ is a concatenation operator.
- $Header_{(u, t)}$ contains algorithm $alg_{jwt}$ (e.g., RSA512) and token type $type_{jwt}$.
- $Payload_{(u, t)}$ is the token body involving a user $u$’s role list $lrole_u$, the issuer $iss_u$ grants this token for $u$, an identity $ID_u$ of $u$, an expiration time $exp\_time_{(u, t)}$ computed since time $t$ to use the services, version of token, and the issuing time of token.
- $Sign_{(u, t)}$ is the signature of the header, payload and secret key using SHA-512.
\[ex:JWT\] An example of JWT token, generated from user $u$’s credential, is given in Figures \[fig:endec\_JWT\]. Figure \[fig:endec\_JWT\]a presents a code of an encoded JWT, and Figure \[fig:endec\_JWT\]b shows a structure of post-decoded JWT.
\[lemma:unique\_User\_Token\] Let $t$ denote the time and $u$ the user with credential $Cr_u$. The token $JWT_{(Cr_u,t)}$ is unique for user $u$ within a period of time $t+\Delta$.
JWT token is based on the current date time $t$ of the server. The time $t$ is unique and it is obtained automatically from the server. The user name and their credential are unique. Only one hash value using SHA512 appended into the JWT token as the hashed content is unique with the unique secret key and the unique issuing time. Therefore, there is only one $JWT_{Cr_u}$ to be generated according to one $Cr_u$.
The Lemma \[lemma:unique\_User\_Token\] guarantees that a legal user can always access the controller’s services in an interval of time $\Delta$. As the same user credential and a requested time, the generated token is unique, hence, the controller can recognise the user identity through their token uniquely.
After successful authentication, the end-user $u$ can access the REST APIs of a given controller using their $JWT_{(Cr_u, t)}$. Whenever a user $u$ requests a service, $JWT_{(Cr_u, t)}$ is embedded into that request. The called REST API checks if $JWT_{(Cr_u, t)}$ is valid, and the user’s roles in the $JWT_{(Cr_u, t)}$ cover the permission of using the called request.
Collection-based User Authorisation {#sec:CollectionbasedUserAuthorize}
-----------------------------------
Database stores user information using a specific data structure, called [ *collection*]{}. A collection is a list of documents and each document keeps the information of one user. This storage method does not check if all documents have a uniform format. To authorise a user, the controller needs to seek their relevant information in a collection called “USER” of the database where all users’ information are saved. Then, it checks the user’s permission on the requested data by checking their credential and their administrative information. Specifically, a user permission is well authenticated if it satisfies two conditions, that is, (C1) only the administrative staff can access the information of users they manage (see Definition \[def:legal\_stewardship\]), and (C2) a staff member can only read or write the information of a user in their organisation (see Definition \[def:solid\_membership\]).
\[def:legal\_stewardship\] Let $u$ and $v$ be the data user and owner, respectively. Let $lrole_u$ be the role list of $u$. Let $orgID_v$ be the organisation ID of $v$. A Legal Stewardship is established between $u$ and $v$ iff: $$\exists i | ((role_u[i] \in lrole_u) \wedge (role_u[i] \in admin(orgID_v)))$$ where $admin()$ is the function returning a list of the administrative roles of $v$’s organisation.
\[def:solid\_membership\] Let $u$, $v$ be the data user and owner, respectively. Let $orgID_u$, $orgID_v$ be the IDs of the organisations of $u$ and $v$, respectively. A Solid Membership between $u$ and $v$ exists iff: $$orgID_u = orgID_v$$
User $u$ can access the data of a patient $v$ if and only if user $u$’s permission satisfies the two conditions (C1) and (C2) as in Definitions \[def:legal\_stewardship\] and \[def:solid\_membership\], respectively.
Let consider the case where the user is a clinician $A$ of Clinic $X$. $A$ makes a query on the user $B$’s information from the database. Moreover, $B$ is a patient of Clinic $X$. Therefore, $C1$ and $C2$ are satisfied. Then, clinician $A$ is allowed to query on $B$’s information from the database.
The authorisation procedure is described in Algorithm \[alg:authorizeUser\], namely *authorizeUserPermission()*, to check the user $u$’s permission on some data of $v$ which $u$ aims to access. Basically, as mentioned in Definitions \[def:legal\_stewardship\], \[def:solid\_membership\], the $u$’s information, involving $u$’s role list $lrole_u$ and $u$’s organisation ID $orgID_u$, are achieved from their credential (cf. Definition \[def:userCredential\]). To satisfy both lemmas above, $v$’s role and organisation ID need to be obtained. $v$’s information are queried from the relevant collections through $v$’s ID.
$lrole_u$, $orgID_u$, $ID_v$ $'true'|'false'$
$orgID_v = findOrginCol("USER", ID_v)$;\[alg1:lineSeekVOrgID\] $lroleAdminOfOrg_v = findAdminOfVOrg(OrgID_v)$;\[alg1:lineSeekListAdminRoleOfV\]
$isRightRole = false$;\[alg1:lineSetVar1False\] $isSameOrg = false$;\[alg1:lineSetVar2False\]
\[alg1:lineForI\] \[alg1:lineForJ\] \[alg1:lineCheck2Role\] $isRightRole = true$;\[alg1:lineSetSuccessCheckRole\] $break$;\[alg1:lineExitLoopI\] \[alg1:lineIsRightRole\] $break$;\[alg1:lineExitLoopJ\] $return\ false$;\[alg1:lineReturnFailedCheckRole\]
\[alg1:lineCheckOrg\] $isSameOrg = false$;\[alg1:lineSetFailCheckOrg\] $return\ false$;\[alg1:lineReturnFailedCheckOrganization\]
$return\ true$;\[alg1:lineReturnSuccess\]
Specifically, the Algorithm \[alg:authorizeUser\] inputs are the list of roles of the user $u$, their organisation ID, and $v$’s ID. The algorithm outputs a boolean value $"True"$ if the authorisation is successful, and returns a boolean value $"False"$ if the authorisation failed. Initially, the necessary information of $v$ are retrieved, such as their organisation ID $orgID_v$ (cf. line \[alg1:lineSeekVOrgID\]) based on which the list of administrative roles in $v$’s organisation $lroleAdminOfOrg_v$ are obtained (cf. line \[alg1:lineSeekListAdminRoleOfV\]). Moreover, to check the two aforementioned conditions $C1$ and $C2$, we initialise the two Boolean variables, i.e., $isRightRole$ and $isSameOrg$. $isRightRole$ contains the result of $C1$ evaluation, and $isSameOrg$ contains the result of $C2$ evaluation. Initially, both are set to $"False"$ (cf. lines \[alg1:lineSetVar1False\], \[alg1:lineSetVar2False\]). Then, $u$’s role list $lrole_u$ is examined (cf. line \[alg1:lineForI\]). The administrative roles in $v$’s organisation $lroleAdminOfOrg_v$ are investigated simultaneously (cf. line \[alg1:lineForJ\]). Each element $role_i$ in $lrole_u$ is compared to each element $role_j$ in $lroleAdminOfOrg_v$. If there is a case of two matched elements (cf. line \[alg1:lineCheck2Role\]), $isRightRole$ is set to $"True"$ (cf. line \[alg1:lineSetSuccessCheckRole\]), and the two loops are stopped (cf. lines \[alg1:lineExitLoopI\], \[alg1:lineIsRightRole\], \[alg1:lineExitLoopJ\]). Otherwise, the algorithm stops and return $"False"$ (cf. line \[alg1:lineReturnFailedCheckRole\]). In the case of $C1$ is successful, function $authorizeUser()$ continues to check $C2$ through the two organisation IDs of $u$ and $v$ (i.e., $orgID_u$ and $orgID_v$) (cf. line \[alg1:lineCheckOrg\]). If both are different, the variable $isSameOrg$ is set to $"False"$ (cf. line \[alg1:lineSetFailCheckOrg\]), function $authorizeUserPermission$ is stopped and returns $"False"$ (cf. line \[alg1:lineReturnFailedCheckOrganization\]). In the case of the two conditions $C1$ and $C2$ are satisfied, the authorisation is successful, and function $authorizeUserPermission()$ returns $"True"$.
Controller Authority Evaluation {#sec:CAE}
===============================
The second part of the model protecting the database is the access control layer (cf. Figure \[fig:architecture\]) which manages the data influence by controllers. In the proposed architecture, databases in the back-end layer are not accessible by all controllers. Each database is restricted for a number of specific controllers. The controller that receives user request and makes a connection to a certain database to get the data is called *requesting controller*. The database that contains the requested data is called *requested database*. There is an authentication and authorisation process between the requested database and the requesting controller whenever they handshake for a data request/response transaction. In our work, *Role-Based Access Control* (RBAC) technique [@RBAC] is deployed to handle the data access permission of controllers. RBAC is executed based on leveraging the controller’s identity.
Single-Sign-On Controller Authentication {#sec:Cont_Aut}
----------------------------------------
To identify the requesting controller, the access control layer verifies the controller’s credential. Each controller is set with a controller name and a password by the developer. Its name must be unique in the system. Moreover, the controller has a secret key denoted as $K_c$, generated from controller’s password by using some certain hash function.
Let $c, cn_c, p_c$ be the verified controller, controller’s name, and password, respectively. Let $ID_c$ and $K_c$ be the controller’s identity and public key, respectively. Formally, the credential of a controller is defined as: $$Cr_c = [(cn_c, p_c), ID_c, K_c]$$
In order to monitor several users accessing system services, a Kerberos Center (KC) is used for controlling authentication (cf. Figure \[fig:RAC\]). Kerberos is an authentication protocol between unknown parties in a secure manner. $KC$ involves two servers, that is, Authentication Server ($AS$) and Ticket Granting Service ($TGS$). In our work, this authentication protocol is operated between system controller and $KC$. The authentication is based on a ticket generated by $AS$. Basically, the ticket is generated using time constraint that limits the ticket’s time-to-live ($TTL$), and the controller name is saved in the controller credential. This authentication ticket guarantees the “Single Sign On (SSO)” property of the log-in system. It indicates that the controller needs to do authentication with $KC$ and then it uses this ticket for subsequent communications between the controller and the data.
\[Service Ticket\] \[def:service\_ticket\] Let $ID_c$, $TS_c$ be controller $c$’s ID and a time stamp of the ticket, respectively. Let $PK_s$, $S_{K_s}$ be secret key and session key of service $s$, respectively. The service ticket of $c$, denoted $T_c$, can be generated, by an encryption algorithm: $$TK_c = Enc_{PK_s}(ID_c, TS_c, S_{K_s})$$
![Keberos-based Controller Authentication[]{data-label="fig:RAC"}](RAC.pdf){width="7.5cm" height="5cm"}
Each ticket has a time stamp that protects it against the replay attack. Once the time stamp is reached, a new ticket is generated with a new time stamp. Therefore, the adversary cannot reuse the ticket for another transaction. Moreover, each ticket has a session key used for transactions between the service and the controller. Tickets generated for the same controller are obviously different from other controllers’ tickets. In addition, the encryption of the ticket is made with the secret key of the service, hence, it can be decrypted by only the same service.
\[lemma:unique\_Controller\_Ticket\] Let $c, TK_c$ be the verified controller and ticket that TGS generates for $c$ (cf. Lemma \[def:service\_ticket\]). $TK_c$ is unique.
Given a controller $c$ and a service $s$ $c$ wants to access the data service $s$. A ticket is generated for $c$; $TK_c$. $TK_c$ is unique. Because, from Lemma \[def:service\_ticket\], a ticket is an encryption of $c$’s identity $ID_c$, a $s$’s shared key $SK_s$, and a time stamp $TS_c$, where
- The controller identity $ID_c$ is unique.
- The secret key of service $s$, called $SK_s$, is unique in the system.
- The encryption made from the two above elements with the secret key of $s$ cannot be duplicated in the system.
Therefore, tickets of controllers are unique.
Furthermore, Kerberos operates in cooperation with LDAP server. All controller’s information and their permissions are stored in the active directory of the LDAP server. For each entry in the active directory, it contains username, password, organisation, group, domain name (i.e., a link attached to the organisation), access permissions, etc.[^8]
The authentication protocol is depicted in Figure \[fig:RAC\]. The details of the protocol are given in Table \[tab:RAC\]. To start an authentication, a controller $c$ connects to $KC$ and sends its ID ($ID_c$) to $AS$ (step 1). Then, $AS$ uses $ID_c$ to lookup $c$ in its master database (step 2.1). If $c$ is not in the database, the connection is rejected. Otherwise, $AS$ continues to process the authentication request. $AS$ creates a time stamp for $c$ ($TS_c$) (step 2.2). $TS_c$ is used for checking if the ticket is still valid, otherwise, $TS_c$ needs to be renewed. $AS$ then generates a ticket, $T_c$, for $c$ that is an encryption of $c$’s identity (i.e., $ID_c$), a time stamp, ($TS_c$), and a session key of $TGS$ ($S_{K_{TGS}}$) (step 2.3). $S_{K_{TGS}}$ is used by $c$ and $TGS$ in subsequent steps. The ticket $T_c$ is used for authenticating $c$ with $TGS$ before getting the permission to access database. Then, $AS$ retrieves the password of $c$, that is $Cr_c.p_c$ (step 2.4) to re-generate the secret key of $c$ ($K_c$) (step 2.5). $AS$ creates two messages; $M_A$ and $M_B$, to be sent back to $c$ (step 2.6). $M_B=T_c$ and $M_A$ is the encryption of the session key of $TGS$ ($S_{K_{TGS}}$). So, $M_A=Enc_{K_c}(S_{K_{TGS}})$. $AS$ sends $M_A, M_B$ to $c$ (step 3). $c$ generates the secret key $K_c$ from its password $Cr_c.p_c$ (step 4.1). $c$ uses $K_c$ to decrypt $M_A$ and obtain $S_{K_{TGS}}$ (step 4.2). If $c$ fails to process the decryption, it indicates that $c$ cannot access the system as their password is invalid. If $c$ successfully decrypts the message, it creates two messages; $M_C$ and $M_D$ (step 4.3). $c$ sends $M_C$ and $M_D$ to $TGS$ (step 5). $M_C$ involves $M_B$, the identity of the requested service ($ID_{DBS}$) and the requested command is executed $Req_{DBS}$ (reading $'R'$, writing $'W'$, searching $'S'$ and/or comparing $'C'$), whereas, $M_D$ is an encryption of $c$’s identity ($ID_c$) and the current time stamp of $c$ ($TS_c$). $TGS$ decrypts $M_B$ to $M_C$ by its secret key $PK_{TGS}$ (step 6.1) to get the shared key $S_{K_{TGS}}$ and the identity of $c$ ($ID_c$). $TGS$ decrypts $M_D$ with the retrieved key $S_{K_{TGS}}$ to get the identity of $c$ ($ID_c$) (step 6.2). $TGS$ compares the two identities of $c$ obtained in steps 6.1 and 6.2. If they are equal, $c$ is authenticated to $TGS$. The controller authorisation is described in Section \[sec:Cont\_Aut\].
---- --------------------- -------------------------------------------------------------
1. $c \rightarrow AS$ - Sends [*controller’s ID*]{}, $Cr_c.ID_c$, to $KC$.
2. $AS$
2.1. - Use $Cr_c.ID_c$ to check if $c$ is listed in
$KC$’s database.
2.2. - Create a time stamp $TS_c$ and a session key
$S_{K_{TGS}}$ for the ticket.
2.3. - Generate $T_c = Enc_{PK_{TGS}}(ID_c, TS_c, S_{K_{TGS}})$.
2.4. - Retrieve $Cr_c.p_c$ from $KC$’s database.
2.5. - Generate controller’s secret key
$K_c = hash(Cr_c.p_c)$.
2.6. - Create $M_A = Enc_{K_c}(S_{K_{TGS}})$,
$M_B = T_c$.
3. $AS \rightarrow c$ $M_A$, $M_B$.
4. $c$
4.1. - Generate a secret key $K_c = hash(Cr_c.p_c)$
from $c$’s password.
4.2. - Decrypt $M_A$ by $K_c$ to obtain $S_{K_{TGS}}$.
4.3. - Create $M_C = [M_B, ID_{DBS}, Req_{DBS}]$
and $M_D = Enc_{S_{K_{TGS}}}(ID_c, TS_c)$.
5. $c \rightarrow TGS$ - Sends $M_C, M_D$.
6. $TGS$
6.1. - Decrypt $M_B$ in $M_C$ by $PK_{TGS}$ to obtain
$S_{K_{TGS}}$ and $ID_c$.
6.2. - Decrypt $M_D$ by $S_{K_{TGS}}$ to get $ID_c$.
6.3. - Compare $ID_c$ retrieved in step 6.1. and $ID_c$
retrieved in step 6.2. If the result is “equal”,
$c$ is authenticated to $TGS$.
---- --------------------- -------------------------------------------------------------
: Controller Authentication Protocol[]{data-label="tab:RAC"}
Controller Authorisation {#sec:controller_authorization}
------------------------
Two kinds of permissions were investigated: [*Read*]{} and [*Write*]{}. [*Write*]{} permission indicates the act of inserting a new data into the database. [*Write (W)*]{} permission cannot delete or change data that is already stored in the database. [*Read (R)*]{} permission is the ability to retrieve data from the database without making any changes to the data.
\[ex:controllerAuthorization\] Mobile App controller can access the Cassandra original database in the first horizontal layer (cf. Figure \[fig:architecture\]), but it cannot access the de-identified statistics data. Whereas, analysis services can access the de-identified anonymised database, but they cannot write data into the original mongoDB database.
\[def:DAC\_Authority\] Let $CT=\{Ct_1, Ct_2, \cdots, Ct_n\}$ be the set of controllers in the system. Let $CL=\{Cl_1, Cl_2,\cdots, Cl_n\}$ be the set of user collections in the system. Let $P=\{R,
W\}$ be the set permissions in the system. The set of permissions granted for each controller is defined as a 2-dimension array $DAC$. $$DAC = \{DAC_{(Ct_1,Cl_1)}, DAC_{(Ct_1, Cl_2)},DAC_{(Ct_1, Cl_3)}\cdots\}$$ where $$DAC_{(Ct_1, Cl_1)} = (Ct_1, Cl_1, P_{(Ct_1, Cl_1)})$$ $$DAC_{(Ct_1, Cl_2)} = (Ct_1, Cl_2, P_{(Ct_1, Cl_2)})$$ $$\cdots$$
According to Figure \[fig:DAC\], we have $Ct = \{portal , mobile ,
analysis\ backend, \cdots\}$, $Cl = \{clinic, school, children, \cdots\}$, $P=\{R,W\}$. A DAC authority is $DAC = \{\{portal, clinic, \{R,W\}\},
\{portal, school, \{R, W\}\}, \cdots\}$.
We use LDAP protocol for communication between the authorisation process and the active directory (AD) which stores an access control list (ACL) of permissions for each controller. The DAC table (c.f. Definition \[def:DAC\_Authority\]) is stored in the AD. Each controller’s permissions are defined in an ACL entry. Whereas, the Kerberos protocol authorises requests from the controller. The LDAP active directory can be stored in a separated LDAP server or in the same Kerberos server. The authorisation process is executed after the controller authentication was successful.
The authorisation protocol must follow the authentication process (see \[sec:Cont\_Aut\]). The authorisation protocol is described in steps 4, 5, 6 of Figure \[fig:RAC\] and presented in Table \[tab:Kerberos\_Controller\_AUthorization\]. Once $TGS$ compares the two identities of the controller $c$, $TGS$ searches in its AD for an entry that defines access permissions for $c$ to the database service $DBS$. Then $TGS$ compares the access permissions in $DAC_{(c,DBS)}$ and the request $Req_{DBS}$. If they are not equal, $TGS$ sends a reject message to $c$. Otherwise, $TGS$ issues two new messages: $M_E$ and $M_F$ (steps 1.1, 1.2) where $M_E$ is the issued ticket used by $C$ with the service $DBS$. $M_E = Enc_{PK_{DBS}}(ID_c, TS_{c-DBS}, S_{K_{DBS}})$ is an encryption made by the secrete key of service (i.e., $PK_{DBS}$), of controller ID (i.e. $ID_c$), time stamp (i.e. $TS_{c-DBS}$), and the shared key of the service (i.e., $S_{K_{DBS}}$); whereas, $M_F = Enc_{S_{K_{TGS}}}(S_{K_{DBS}})$ is an encryption of the shared key of the service (i.e., $S_{K_{DBS}}$) made by $TGS$’s shared key $S_{K_{DBS}}$. Then, $TGS$ sends $M_E$ and $M_F$ to the controller $c$ (step 2). The controller $c$ decrypts $M_F$ with $TGS$’s shared key $S_{K_{TGS}}$ to obtain the shared key of service $DBS$ (i.e., $S_{K_{DBS}}$), the time stamp of controller $c$ and service $s$ (i.e., $TS_{c-DBS}$), and the identity of controller $c$ (i.e, $ID_c$) (step 3.1). Then, $c$ creates a new message $M_G$, that is an encryption of the controller’s identity (i.e., $ID_c$) and the time stamp of controller $c$ and service $DBS$ (i.e., $TS_{c-DBS}$) (step 3.2). After that, controller $c$ sends $M_E$ and $M_G$ to the database service $DBS$ (step 4). When $DBS$ receives the two messages from $c$, it first decrypts $M_E$ using the secrete key of $s$ (i.e., $P_{K_{DBS}}$) to obtain the shared key of $DBS$ and the controller’s identity $ID_c$ (step 5.1). $DBS$ decrypts $M_G$ using the shared key $S_{K_{DBS}}$ to obtain the controller’s identity (i.e., $ID_c$) (step 5.2). Then, $DBS$ compares the two messages $M_E$ and $M_G$ (step 5.3), if the comparison results in “equal” (i.e., $M_E=M_G$), $c$ is authorised and can access the data service $DBS$. Simultaneously, $DBS$ creates a message $M_H$ that is an encryption of the time stamp $TS_{c-DBS}$ granted to be used between $c$ and $DBS$. After that, $DBS$ sends $M_H$ to $c$ (step 6). After receiving $M_H$, $c$ decrypts $M_H$ using service’s shared key (i.e., $S_{K_{DBS}}$) to obtain $TS_{c-DBS}$ (step 7.1). $c$ compares the two time stamp in the two messages retrieved from decrypting $M_F$ (step 3.1) and $M_H$ (step 7.1). Finally, $c$ sequentially sends its requests to $DBS$.
---- ------ ----------------------------------------------------------
1. [**TGS**]{}
1.1. Search in AD for $c$’s entry $DAC_{(c,DBS)}$ .
1.2. Compare access permissions in $DAC_{(c,DBS)}$ and
$req_{DBS}$ in $M_C$.
1.3. If equal, do step 1.3. Otherwise, sends a rejection
response to $c$.
1.4. Create
$M_E=Enc_{PK_s}(ID_c, TS_{c-DBS}, S_{K_{DBS}})$.
1.5. Create $M_F = Enc_{S_{K_{TGS}}}(S_{K_{DBS}})$.
2. [**TGS**]{} $ \rightarrow c \quad $ - Sends $M_E, M_F$.
3. [**c**]{}
3.1. Decrypt $M_F$ by $S_{K_{TGS}}$ to obtain $S_{K_{DBS}}$,
$TS_{c-DBS}$, and $ID_c$.
3.2. Create $M_G = Enc_{S_{K_{DBS}}}(ID_c, TS_{c-DBS})$.
4. $c \rightarrow$ [**DBS**]{} $\quad $ - Sends $M_E, M_G$.
5. [**DBS**]{}
5.1. Decrypt $M_E$ by $P_{K_{DBS}}$ to obtain $S_{K_{DBS}}$
and $ID_c$.
5.2. Decrypt $M_G$ by $S_{K_{DBS}}$ to obtain $ID_c$.
5.3. Compare $M_E$ and $M_G$. If $M_E=M_G$, has $c$
authorised, and creates
$M_H = Enc_{S_{K_{DBS}}}(TS_{c-DBS})$.
6. [**DBS**]{} $\rightarrow c \quad $ - Sends $M_H$.
7. [**c**]{}
7.1. Decrypt $M_H$ by $S_{K_{DBS}}$ to obtain $TS_{c-DBS}$.
7.2. Compare $TS_{c-DBS}$ from steps 3.1 and 7.1.
If $"="$, $DBS$ is authorised.
---- ------ ----------------------------------------------------------
: Controller Authorisation Protocol[]{data-label="tab:Kerberos_Controller_AUthorization"}
Secure Data Transmission {#sec:SDT}
========================
When a subject, such as a user or a controller, attempts to access data at the back-end, it has to go through two substantial security shields, that is, authentication process (cf. Section \[sec:UserAUTH\]) and authorisation process (cf. Section \[sec:CAE\]). Even though such a data access is rigorous, the risk of data leakage is still likely to occur, especially when storing and transmitting the data. Moreover, in our system, the data storage is distributed. the databases are located on different servers (cf. Section \[sec:Archi\]). Any request to access the data the database servers collaborate to output a response.
To protect data storage against attacks, the storage system uses encryption algorithms. MongoDB supports various encryption schemas[^9], such a default with AES-256 in CBC and GCM mode.[^10] The encryption schema can be configured to comply with FIPS 140-2.[^11] Cassandra supports Transparent Data Encryption (TDE)[^12] for a lightweight encryption of data and log files that are stored in the master database of server that contains administrative data used for monitoring and controlling the system.
Data transmission also needs to be protected against eavesdroppers. Usually the stored data are queried and transmitted in a plain form that can be read in-flight by the third party. We adopted Secure Sockets Layer (SSL)/ Transport Layer Security (TLS) [@SSL] for securing the data transmitted over the network. This protocol is applied between any two components of the system. For example, as in Figure \[fig:architecture\], the two parties needing a secure data transmission between themselves included [*Original MongoDB Database*]{} and the module [*De-identification*]{}, since data from [*Original MongoDB Database*]{} is transferred to the module [*De-identification*]{}.
---- ------ ----------------------------------------------------------------------
1. $c \rightarrow s\quad $ Send
1.1. - SSL session request.
1.2. - Supported protocol version, list of cipher suits $CS$.
2. [**s**]{}
2.1. Create the request accepted.
2.2. Check supported protocol version.
2.3. Select cipher algorithms from $CS$.
3. $s \rightarrow c\quad $ Send
3.1. - Selected cipher algorithm list.
3.2. - SSL certificate of the public key $K_s$.
3.3. - $s$’s public key $K_s$.
4. [**c**]{}
4.1. - Create session key $SK_{c-s}$.
4.2. - Encrypt $SK_{c-s}$ with $K_s$.
5. $c \rightarrow s \quad $ Send $Enc_{P_s}(SK_{c-s})$.
6. [**s**]{} Decrypt $Enc_{P_s}(SK_{c-s})$ by private key $PK_s$ to get
$SK_{c-s}$.
---- ------ ----------------------------------------------------------------------
: Secure Data Transmission Protocol[]{data-label="tab:secure_data_transmission"}
The goal of this protocol is to secure the data while it is exchanged between two nodes. A secure protocol using SSL/TLS is presented in details in Table \[tab:secure\_data\_transmission\] between [**c**]{} and [**s**]{}. It is assumed that [ **c**]{} and [**s**]{} have their own pair of asymmetric keys including a public key and a private key. We assume that all transactions made between [**c**]{} and [**s**]{} are secured by the session key, $SK_{c-s}$. Basically, SSL/TLS connection is established based on the handshake between the two sides [**c**]{} and [**s**]{}. It is assumed that each side has its own pair of public key and private key. Whenever one side, e.g. [**c**]{}, has an SSL/TLS connection request, [**c**]{} sends to [**s**]{} a connection request, its supported SSL/TLS protocol version (e.g., TLS 1.2), and a list of cipher suits that it supports (step 1). Cipher suit is a set of cryptographic algorithms that [**c**]{} supports, such as symmetric (AES-256 CBC/GCM, etc.) or asymmetric algorithms (e.g., RSA), hash function (e.g., SHA, DES) and key exchange protocols (e.g. DHE), etc. An example of cipher suit: [*“ECDHE-RSA-AES128-GCM-SHA256”*]{}. Once [**s**]{} accepts a connection request, it creates an acceptance notification (step 2.1). Then [**s**]{} checks if it also supports the [ **c**]{}’s protocol version (step 2.2). If this is the case, [**s**]{} matches the cipher suits in the list received from [**c**]{} with its supported cryptographic algorithms, and selects the suitable ones (step 2.3). [**s**]{} sends to [**c**]{} the list of selected cryptographic algorithms (step 3.1), an SSL certificate (step 3.2), and its public key (i.e., $K_s$) (step 3.3). After receiving certificate and $K_s$ from [**s**]{}, [**c**]{} creates a session key, $SK_{c-s}$ used for all communications between [**c**]{} and [**s**]{} (step 4.1). [ **c**]{} encrypts the session key with the public key $K_s$ to output $Enc_{P_s}(SK_{c-s})$ (step 4.2). Then, [**c**]{} sends the encryption $Enc_{P_s}(SK_{c-s})$ to [**s**]{} (step 5). [**s**]{} uses its private key $PK_s$ to obtain the session key $SK_{c-s}$ (step 6). Then, the session key is used for all transactions between [**c**]{} and [**s**]{}. This protocol is applied for all transactions between any of two modules or two nodes in the system.
Evaluation {#sec:SecEval}
==========
In this section, we evaluate the proposed model through the immune ability against attacks, that is, replay attack, eavesdropping attack and unauthorised spy.
#### Replay attack
Usually the adversary tries to get packets transmitted through the network, and reuses them later in the following sessions for the purpose of impersonating the sending nodes. If the attack is successful, the adversary receives the response. In case the response may contain sensitive information, the adversary can take advantage of the information for, which affects very badly the users privacy and sensitive data leakage.
In the proposed model, the adversaries cannot reuse the released packet for the replay attack purpose.
In our model, each packet is stamped with a validation time. Based on this time, the receiving nodes in the network check the packet’s expiration, by comparing its time stamp with the current time to see if the time interval is greater than the requested time threshold. If this is a case, the packet is dropped. The problem is that how long is enough for a packet’s time to live so that the system can prevent the replay attack in time.
#### Eavesdropping attack
The eavesdropper tries to stalk packets found in the network for the further purposes. They may be benign or malicious. If they are just honest-but-curious adversaries, they read the sensitive information in the delivered packet only to satisfy their curiosity. However, this type of attack can leave the backdoor in the system. Then, the more malicious adversary may be, successfully do their attack will be. With malicious adversaries, they are more harassed and dangerous. They aim to get more personal information to get more benefits, and may seriously harm user’s honour, privacy and finance, or to interfere with the data to change data integrity.
in the proposed model, Eavesdroppers cannot modify the package content or get any sensitive information from the communication tunnel.
Hence, in our proposed model, we protect user data as they are transferred through the network by adopting the SSL/TLS protocol. This protocol creates a robust tunnel to hide all transmitted data especially encrypting them right when they are released from the application and at the transport layers. Moreover, with SSL/TLS protocol, a cipher suit exploits different encryption algorithms and hash functions. So, eavesdroppers cannot get any sensitive information from the tunnel.
#### Unauthorised spy
Our model can protect data against unauthorised spies by authenticating and authorising the users and the controllers attempting to access data at the back-end. Spies can abuse the lack of security check, such as authentication and authorisation, to steal important information. After retrieving the crucial information, they turn into the previous eavesdropping adversaries, and get either benign or malicious.
In the proposed model, the user and controller cannot access database and use the data with any purpose.
In the proposed model, the granted permissions of a user and controller must be stored in the active directory centre to be able to access database. When a user or controller sends a data access request, the server will check for suitable permission in the active directory. If this is a case, the server passes the request to the back-end database server.
Experimental Results {#sec:Experi}
====================
This research is a part of a larger project which focuses on monitoring the changes of obesity-related behavioural risk factors (what and how children eat, how they move, how they sleep) to the prevalence of obesity, and to define a parsimonious behavioural model and data structure that will be free of redundant individual information and will minimise the use of sensitive information. The users of our system include the clinicians, school teachers, and public health authorities.
To achieve such objectives, first we build a software system that collects, stores and analyses the data sets of children and adolescent from different locations around Europe. The children and patients join the system on voluntarily basis.
The system collects their behaviours and relevant information (e.g., daily exercises, food consumption, junk food quantity, locations, clinics, schools, groups, etc.), and stores them in the two major databases: MongoDB and Cassandra. MongoDB is the core database where the most of data is stored. Cassandra is used for collecting the time-series data from users’ smart devices which are processed then stored in the local storage of the smart devices before they are transferred to MongoDB (c.f. Section \[sec:database\_schema\]).
To provide clinicians and developers a tool to access the above two databases, we implement a set of interfaces: Clinical Advisor (CA) services. A description of these services are presented in Appendix \[sec:AppendixA\]. Each service gives an output data through a graph which the clinicians can use for comparing situations of individual children or groups of children, to monitor the children patients, to recommend the effective treatments or needed therapies suitable to each patient.
Each user in the system must be granted permissions. User permissions are discussed in Section \[sec:UserAUTH\]. Moreover, these services can be accessed by other components in the system, such as controllers or modules (c.f. Section \[sec:CAE\]).
Database Schema {#sec:database_schema}
---------------
In order to provide a full pack of Clinical Advisor services, we design and implement a database in term of $28$ collections in MongoDB (c.f. Figure \[fig:MongoDB\_schema\]), such as regions, groups, children, students, etc., and $8$ tables in Cassandra database (c.f. Figure \[fig:Cassandra\_schema\]), such as devices, sessions by user, physical activities by user, physical activities by date, etc.
Deployment Environment {#sec:exp__dataset_environment}
----------------------
In the proposed software system, there are two main modules, that is, Application module and Analysis module. Each module includes a front-end server and a cluster of nodes. The Application module is responsible for data storage in Cassandra and MongoDB, receives the requests, and provides services for Mobile App Controller, Web App Controller, and Portals Controller. This module hosts an Apache Balancer and a Tomcat Server. TLS/SSL and Kerberos are set up in this server. Whereas, the Analysis module provides the computations for analysis services. This module also provides the interfaces for Back-end Analysis and Services. Currently, they are all deployed on the same server. For our implementation and testing, the system uses Apache Server $2.4$, Tomcat $8$, Cassandra $>3.1$, MongoDB $3.4$ (Community version), Java, Python, JSON, Flask, Apache Maven[^13], JUnit[^14], and SonarQube[^15]. The Tomcat server contains the REST APIs as interfaces of the system for controllers and end-users. They contain different APIs depending on the required functionality. The Keberos and LDAP servers are always connected to the Tomcat server to get the requests from users and controllers, then process the request and reply back to the Tomcat server.
{height="13cm" width="16cm"}
{height="1.2cm" width="12cm"}
![Cassandra Database Schema[]{data-label="fig:Cassandra_schema"}](Cassandra_schema.pdf){height="7cm" width="7cm"}
System Quality Test {#sec:system_test}
-------------------
We test the Clinical Advisor services after they were integrated into the main system. We run various types of tests. For code quality testing, we use the software SonarQube, and for code testing we use the tool JUnit. SonarQube measures the following criteria to ensure that the system has high quality code; no bugs, no vulnerabilities, no technical debts, no code duplication, and no duplicated blocks.
Figure \[fig:UCD\_Test\] presents the testing results of the whole system after executing the unit testing and the code quality testing. The code smells are pieces of code that, while technically not wrong, should be altered. As all the returned measurements are rated A that means the code is of high quality.
Conclusion {#sec:Concl}
==========
This paper, we show that designing and implementing a system for large and sensitive data-driven application is not straightforward, mainly the storage system, where the data should be stored and preserved from any unauthorised access. Moreover, this is not enough, as the data needs to be accessed for analysis purposes. This creates another challenge, as it has to be protected during its transit over the network. We can argue that these are provided by the operating system of both clients and hosts machines. However, we showed in this paper that this is not enough. Sensitive data should and must be protected independently of the underlying operating system or the system security put in place either at client or host.
The proposed architecture not only incorporates security and protection by design but also provides a framework for any system architecture that deals with data-driven applications. Our architecture separates between access permissions to original data, and its derived data for special use (statistics and analysis for instance). We also provide a set of services for the external users to access the back-end. We provide a formal model to prove the system resilience to attacks and implemented it using popular and tools.
As a future work, one needs to provide an automated security-aware system for data analysis. We will to generalise the proposed architecture to any system where the data is the main focus. We will also improve the anonymised data in the architecture (see Figure \[fig:architecture\]), to provide a set of secure interfaces for external users to do data mining on the anonymous data.
Clinical Advisor (CA) Graphs {#sec:AppendixA}
============================
We implemented the graphs to support clinicians in observing and helping the patients. The Clinical Advisor module of the system provides basic statistics via two RESTful web services interface, *Individual* and *Population*.
Individual Services {#sec:individual}
-------------------
An [*indicator*]{} is measurable quantity to provide the information about an individual’s behaviour in eating, physical activity and sleep. This service provides statistical graphs of the user. Given a time period and an indicator, these graphs show how the indicator changes during that period.
### Numeric Indicators {#sec:numeric_indicator}
There are three types of graphs for Individual services for supporting clinicians. The changes are displayed by average values.
![A graph of activity counts in Clinical Portal.[]{data-label="fig:account_activity"}](activity_account.pdf){height="5cm" width="6cm"}
- [*Daily MET.*]{} This graph shows the average MET (Metabolic equivalent of task), which expresses the energy cost in physical activities during a period of time. Higher MET corresponds to heavier activities. This graph supports clinicians in checking the effectiveness of physical activities.
- [*Daily physical activity counts.*]{} This graph shows the average activity counts during a given time period (Figure \[fig:account\_activity\]). Higher counts correspond to higher physical activities. This will support clinicians in observing the effectiveness of physical activities.
- [*Daily physical activity steps.*]{} This graph shows the average activity steps during a time period. This graph provides clinicians a tool of monitoring the most popular types of physical activities (walking, running, etc.).
### Categorical Indicators {#sec:categorical_indicator}
Categorical indicators allow to provide graphs for meal types, visited locations, and transportation modes. Changes are expressed by the distribution of values.
- [*Daily dietary.*]{} This graph shows the distribution of meals at different times during the day. With this graph, clinicians can get information about the dietary habits of children (c.f. Figure \[fig:daily\_meal\_child\]).
![A graph of daily meals of a child in Clinical Portal[]{data-label="fig:daily_meal_child"}](daily_meals_of_a_child.pdf){height="7cm" width="8cm"}
- [*Daily visited locations.*]{} This graph shows the location types a child usually visits. With this graph, clinicians can learn more about favourite locations of children and their habits. For example, if children prefer sedentary activities or dynamic activities, how often children join community activities or use recreational facilities.
- [*Daily transportation.*]{} This graph presents commuting travels and frequent travelling methods. Clinicians know their travel habits.
Population Services {#sec:population}
-------------------
![A graph of daily meals of a group in Clinical Portal[]{data-label="fig:daily_meal_group"}](daily_meals_of_a_group.pdf){height="4cm" width="6cm"}
A population can be patients of a clinic, students of a school. The population graphs provide the statistical data of a population. These graphs contain the body measurement distributions (height, weight, BMI class) that are typical indicators to consider to measure children body fat. Using these graphs, clinicians can know the prevalence of obesity in a community or population.
Mock-up Services
----------------
As a future works, we will implement more graphs to support clinicians. The planned graphs are listed below:
- [*Graphs for food consumption and body measurements*]{} shows the relationship between food consumption and body measurements (e.g., weight, BMI) over a time period.
- [*Graphs for physical activities and body measurements*]{} shows the relationship between physical activities and body measurements (e.g., weight, BMI) over a time period.
- [*Graphs for food environment and body measurements*]{} shows the relationship between food environment and body measurements (e.g., weight, BMI) over a time period.
Acknowledgment {#acknowledgment .unnumbered}
==============
The work leading to these results has received funding from the European Community’s Health, demographic change and well-being Programme under Grant Agreement No. 727688, 01/12/2016 - 30/11/2020
[^1]: How Health Care Analytics Improves Patient Care, https://healthinformatics.uic.edu/blog/how-health-care-analytics-improves-patient-care/
[^2]: Rethinking Personal Data: Trust and Context in User-Centred Data Ecosystems, http://www3.weforum.org/docs/WEF\_RethinkingPersonalData\_TrustandContext\_Report\_2014.pdf
[^3]: https: //www.mongodb.com/
[^4]: http://cassandra.apache.org
[^5]: Announcing the Advanced Encryption Standard (AES), NIST, FIPS 197, 2001
[^6]: Oracle Advanced Security Transparent Data Encryption Best Practices, https://www.oracle.com/technetwork/database/security/twp-transparent-data-encryption-bes-130696.pdf
[^7]: Security Requirements for Cryptographic Modules, Federal Information Processing Standards Publication, NIST, FIPS 142, 2001
[^8]: Access control list syntax; https://www-01.ibm.com/software/network/ directory/library/publications/ dmt/ref\_acls.htm
[^9]: Securing MongoDB Part 3: Database Auditing and Encryption https://www.mongodb.com/blog/post/securing-mongodb-part-3-database-auditing-and-encryption
[^10]: Announcing the Advanced Encryption Standard (AES), NIST, FIPS 197, 2001
[^11]: Security Requirements for Cryptographic Modules, NIST, FIPS 142, 2001
[^12]: Oracle Advanced Security Transparent Data Encryption Best Practices, https://www.oracle.com/technetwork/database/security/twp-transparent-data-encryption-bes-130696.pdf
[^13]: https://maven.apache.org/
[^14]: https://junit.org/junit5/
[^15]: https://www.sonarqube.org/
|
---
abstract: 'We compactify the M-theory proposed by Horava and Witten on a Calabi-Yau manifold with boundary $S_1/Z_2$. A no-scale-like Kähler potential, the superpotential, and the gauge kinetic function are obtained in this 4-dimensional $E_6\times E_8$ model. We also study the general phenomenological consequences of the resulting M-theory-inspired model, which may include very light gravitinos, axions, and axinos.'
---
6.0in 8.5in -0.25truein 0.30truein 0.30truein
CTP-TAMU-17/97\
DOE/ER/40717–42\
ACT-06/97\
hep-ph/9704247
1.5cm [**Compactifications of M-theory and their Phenomenological Consequences\
**]{} 2.0cm [Tianjun Li$^{1,2}$, Jorge L. Lopez$^3$, and D.V. Nanopoulos$^{1,2,4}$]{}
$^1$Center for Theoretical Physics, Department of Physics, Texas A&M University\
College Station, TX 77843–4242, USA\
$^2$Astroparticle Physics Group, Houston Advanced Research Center (HARC)\
The Mitchell Campus, The Woodlands, TX 77381, USA\
$^3$ Bonner Nuclear Lab, Department of Physics, Rice University\
6100 Main Street, Houston, TX 77005, USA\
$^4$ Academy of Athens, Chair of Theoretical Physics, Division of Natural Sciences\
28 Panepistimiou Avenue, 10679 Athens, Greece\
1.5cm
April 1997\
Introduction
============
Recently, Horava and Witten [@HW] presented a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, that is related to the strong coupling limit of the $E_8\times E_8$ heterotic string. In this novel theory, many previous successes based on conventional weakly-coupled string theory may be preserved. In addition, Witten [@Witten] offered an explanation of why Newton’s constant appears to be so small when the 4-dimensional grand unified gauge coupling ($\alpha_{\rm GUT}$) takes experimentally acceptable values. It follows that the strengths of all interactions, including gravitation, may be naturally unified at the GUT scale, unlike the case of the weakly-coupled heterotic string. Many other interesting implications have been studied, such as gluino condensation and supersymmetry breaking [@Horava], the strong CP problem [@BD], threshold scale and strong coupling effects [@AQ; @VK], and phenomenological consequences [@LLN] which include a constrained sparticle spectrum within the reach of present-generation particle accelerators.
In this paper we compactify this theory on a Calabi-Yau manifold with Hodge numbers $h_{(1,1)}=1$ and $h_{(2,1)}=0$ and boundary $S_1/Z_2$. A no-scale-like Kähler potential [@no-scale], the superpotential, and the gauge kinetic function are obtained explicitly. In four dimensions this result is related to the previous weakly-coupled string no-scale supergravity result by Witten [@Witten85] through a field transformation (Sec. 2), which means that they are equivalent in four dimensions. One might then argue that many heterotic string models obtained previously may also exist in the M-theory regime.
In addition, we consider the physical couplings and scales in the Einstein frame, the eleven-dimensional metric and fivebrane units, and give the intermediate supersymmetry-breaking scale determined by the size of the eleventh dimension of this theory [@Witten; @BD; @AQ; @VK; @LLN; @DM] for different grand unified theories (Sec. 3). We also argue that there may exist a very light gravitino in this scenario, which may explain the $ee\gamma\gamma+E_{\rm T,miss}$ event [@Park] observed by the CDF Collaboration, and may have some further implications at LEP 2. Finally, we comment on very light axions and axinos that might exist in this scenario.
Formal derivation
=================
We follow the notation of Ref. [@HW], in which the bosonic part of the eleven-dimensional supergravity Lagrangian is given by $$\begin{aligned}
L_B&=&{1\over \kappa^2}\int_{M^{11}}d^{11}x\sqrt g
\left(-{1\over 2}R -{1\over 48}G_{IJKL}G^{IJKL}
\right.\left. -{\sqrt 2\over 3456}
\epsilon^{I_1I_2\dots I_{11}}C_{I_1I_2I_3}G_{I_4\dots I_7}G_{I_8\dots
I_{11}}\right)\nonumber\\
&&\qquad\qquad-\sum_{i=1,2}
{1\over\displaystyle 2\pi (4\pi \kappa^2)^{2\over 3}}
\int_{M^{10}_i}d^{10}x\sqrt g {1\over 4}F_{AB}^aF^{aAB}\,
\label{eq:1}\end{aligned}$$ where $ G_{11\,ABC}=\left(\partial_{11}C_{ABC}\pm 23 \,\,{\rm
permutations}\right)
+{\kappa^2\over \sqrt 2 \lambda^2}\delta(x^{11})\omega_{ABC}$, $\lambda^2=2\pi (4 \pi \kappa^2)^{2/3}$, and the gauge group at the boundary is $E_8\times E_8$.
To perform the dimensional reduction to five dimensions (with the 4-dimensional boundary where the Yang-Mills fields live) under the Calabi-Yau manifold with Hodge numbers $h_{(1,1)}=1$ and $h_{(2,1)}=0$, we follow Refs. [@Witten85; @FS] and keep only the SU(3) singlets in the internal indices: $$\begin{aligned}
g_{\mu \nu} \rightarrow g_{\mu \nu} ~;~ g_{i \bar j} = e^{\sigma}
\delta_{i \bar j} \,\end{aligned}$$ $$\begin{aligned}
C_{\mu i \bar j}=iC_{\mu} \delta_{i \bar j} ~;~
C_{i j k}=C'\epsilon_{i j k} \,\end{aligned}$$ Then, performing the Weyl rescaling: $$\begin{aligned}
g_{\mu \nu} \rightarrow e^{-2\sigma} g_{\mu \nu} \,\end{aligned}$$ and just paying attention to the observable sector, which we assume is at the boundary $x^{11}=0$, we obtain the canonically normalized Einstein action: $$\begin{aligned}
L_B&=&{V\over \kappa^2}\int_{M^5}d^5x\sqrt g
\left(-{1\over 2}R -{9\over 4}(\partial_{\mu}\sigma)^2-
{1\over 48}e^{6\sigma}G_{\mu \nu \rho \sigma} G^{\mu \nu \rho \sigma}
-27f_{\mu \nu}f^{\mu \nu}-36 e^{-3\sigma}|\partial_{\mu} {\hat C}'|^2
\right.\nonumber\\&&\left.
-54\sqrt 2 \epsilon^{\mu \nu \rho \sigma \delta} C_{\mu} f_{\nu \rho}
f_{\sigma \delta} + {3\over 4} \sqrt 2 i
\epsilon^{\mu \nu \rho \sigma \delta} \bar {{\hat C}'}
\stackrel{\leftrightarrow}{\partial_{\mu}} {\hat C}'
G_{\nu \rho \sigma \delta} \right)
\nonumber\\&&
-{V\over {2 \pi (4 \pi \kappa^2)^{2\over 3}}}\int_{M^4}d^4x\sqrt g
\left(-{1\over 4} e^{3\sigma}f Tr[F_{\mu \nu}F^{\mu \nu}]
-3 D_{\mu}C_x^* D^{\mu}C^x-{8\over 3} e^{-3\sigma}
|{{\partial W'}\over{\partial C}}|^2
\right.\nonumber\\&&\left.
-{9\over 2f} e^{-3\sigma}\sum_i (C^*, \lambda^i C)^2\right)\,\end{aligned}$$ where $$\begin{aligned}
W'&=& d_{xyz} C^x C^y C^z \,\end{aligned}$$ $$\begin{aligned}
f_{5\mu} &=& \partial_5 C_{\mu} -\partial_{\mu} C_5 +
{i\over 6\sqrt 2}{\kappa^2\over \lambda^2} \delta (x^5) (C_x^*
\stackrel{\leftrightarrow}{D_{\mu}}C^x) \,\end{aligned}$$ $$\begin{aligned}
\partial_{\mu} {\hat C}' &=& \partial_{\mu} C' +
{\sqrt{2}\over 3} \delta_{\mu 5}
\delta(x^5)
{\kappa^2\over \lambda^2} W' \,\end{aligned}$$ The definitions of the $C_x$, $d_{xyz}$ and $f$ are the same as in the Ref. [@Witten85], and $V$ is the coordinate volume of the Calabi-Yau manifold, i.e., $V=\int d^6x$. In addition, the (observable) gauge group at the boundary ($x^{11} =0$ or $x^5 =0$) is now $E_6$ with spin connection embedding, as in Ref. [@Witten85].
If we define a pseudoscalar $D$ by a duality transformation: $$\begin{aligned}
{1\over 4!}e^{6\sigma} G_{\mu \nu \rho \sigma} &=&
\epsilon_{\mu \nu \rho \sigma \delta} ({\partial^{\delta}D+{3\over 4}\sqrt 2
i \bar {{\hat C}'}
\stackrel{\leftrightarrow}{\partial^{\delta}} {\hat C}'}) \,\end{aligned}$$ we have the following Kähler potential in the five-dimensional bulk: $$\begin{aligned}
K &=& -\ln\,[S+\bar S-72\bar {C'} C'] \,\end{aligned}$$ where $$\begin{aligned}
S &=& e^{3\sigma}+i24\sqrt 2 D +36\bar {C'} C' \,\end{aligned}$$ This Kähler potential parametrizes the ${SU(2,1)\over {SU(2)\times SU(1)}}$ quaternionic manifold [@FS; @CCAF]. Also, $D$ is the invisible axion.
We now compactify the above 5-dimensional-with-boundary Lagrangian on $S_1/Z_2$. For the fields with 11-dimensional origin, i.e., the fields in the bulk, we keep only the zero modes, and considering the boundary condition we can expand $\delta (x^5)$ as: $$\begin{aligned}
\delta(x^5)&=&{1\over {2\pi \rho}} + {1\over {\pi \rho}}
\sum_{n=1}^{\infty} \cos{{nx^5}\over \rho}\,\end{aligned}$$ where $\rho$ is the coordinate radius of $S_1$, i.e., $\rho ={1\over {2\pi} }\int dx^5$. Furthermore, choosing the following elfbein form: $$\begin{aligned}
e_M^A &=& \left|\matrix{e_{\mu}^a & 0 \cr
0 & \varphi \cr}\right| \,\end{aligned}$$ and performing the Weyl rescaling: $$\begin{aligned}
g_{\mu \nu} \rightarrow \varphi^{-1} g_{\mu \nu} \,\end{aligned}$$ we obtain the 4-dimensional Lagrangian in the Einstein frame (note that at the boundary $C'$=0 and $C_{\mu \nu \rho} = 0$): $$\begin{aligned}
L_B&=&{V\over \kappa^2} {2 \pi \rho} \int d^4x\sqrt g \left(
-{1\over 2}R-{1\over 12} e^{6\sigma} G_{11\mu \nu \rho} G^{11\mu \nu \rho}
-{9\over 4} (\partial_{\mu} \sigma)^2
-{3\over 4} ({{\partial_{\mu} \varphi}\over \varphi})^2
\right.\nonumber\\&&\left.
-54 \varphi^{-2}(\partial_{\mu} C_5-{i\over {6\sqrt 2}}{1\over {2\pi \rho}}
{\kappa^2 \over \lambda^2} C_x^*
\stackrel{\leftrightarrow}{D_{\mu}}C^x)^2
\right.\nonumber\\&&\left.
-{1\over {2\pi \rho}} {\kappa^2 \over \lambda^2}
[ 3\varphi^{-1}D_{\mu}C_x^*D^{\mu}C^x+
{1\over 4}fe^{3\sigma}{\rm Tr}\,[F_{\mu \nu}F^{\mu \nu}]
+{8\over 3} \varphi^{-2}e^{-3\sigma}
|{{\partial W'}\over {\partial C}}|^2
\right.\nonumber\\&&\left.
+{9\over 2f} \varphi^{-2}e^{-3\sigma}\sum_i
(C^*, \lambda^i C)^2]
-8({1\over {2\pi \rho}})^2 ({\kappa \over \lambda})^4e^{-3\sigma}
\varphi^{-3} |W'|^2\right)
\label{eq:15}\\\end{aligned}$$ Finally, if we define $g_c^2=2 \pi \rho \lambda^2/\kappa^2$ and perform the transformation: $$\begin{aligned}
A_{\mu} \rightarrow g_c A_{\mu} ~;~C_x \rightarrow g_c C_x \,\end{aligned}$$ we obtain the standard supergravity Lagrangian: $$\begin{aligned}
L_B&=&{V\over \kappa^2} {2 \pi \rho} \int d^4x\sqrt g \left(
-{1\over 2}R-{1\over 12} e^{6\sigma} G_{11\mu \nu \rho} G^{11\mu \nu \rho}
-{9\over 4} (\partial_{\mu} \sigma)^2
-{3\over 4}({{\partial_{\mu} \varphi}\over \varphi})^2
\right.\nonumber\\&&\left.
-54 \varphi^{-2}(\partial_{\mu} C_5-{i\over {6\sqrt 2}}
C_x^*
\stackrel{\leftrightarrow}{D_{\mu}}C^x)^2
\right.\nonumber\\&&\left.
-3\varphi^{-1}D_{\mu}C_x^*D^{\mu}C^x
-{1\over 4}fe^{3\sigma}Tr[F_{\mu \nu}F^{\mu \nu}]
-{8\over 3} g_c^2 \varphi^{-2}e^{-3\sigma}
|{{\partial W'}\over {\partial C}}|^2
\right.\nonumber\\&&\left.
-{9\over 2f}g_c^2 \varphi^{-2}e^{-3\sigma}\sum_i
(C^*, \lambda^i C)^2
-8g_c^2e^{-3\sigma}\varphi^{-3} |W'|^2\right)\nonumber\\\end{aligned}$$
From this expression, neglecting the overall factor ${V\over \kappa^2} 2\pi
\rho$, and defining the pseudoscalar by the duality transformation: $$\begin{aligned}
{1\over 4!} e^{6\sigma } G_{11\mu \nu \rho} &=&
\epsilon_{\mu \nu \rho \sigma} (\partial^{\sigma} D) \,\end{aligned}$$ we obtain the following Kähler potential: $$\begin{aligned}
K &=& -\ln\,[S+\bar S]-3\ln\,[T+\bar T-2 C_x^* C^x] ~,~ \,\end{aligned}$$ where $$\begin{aligned}
S=e^{3\sigma}+i24\sqrt 2 D \,\end{aligned}$$ and $$\begin{aligned}
T=\varphi -i6\sqrt 2 C_5
+ C_x^*C^x \,\end{aligned}$$ Here $D$ and $C_5$ are the pseudoscalars and the invisible axions [@Witten85]. In addition, we have the following gauge kinetic function: $$\begin{aligned}
Ref_{\alpha \beta} &=& f ReS\, \delta_{\alpha \beta} \,\end{aligned}$$ and the superpotential $W$, $$\begin{aligned}
W= 8\sqrt {2\over 3} g_c\, d_{x y z} C^x C^y C^z \,\end{aligned}$$ Furthermore, if we perform the field transformation: $$\begin{aligned}
\varphi \rightarrow \phi^{3/4} e^{\sigma} ~;~e^{\sigma} \rightarrow
\phi^{-1/4} e^{\sigma} \,\end{aligned}$$ we obtain $S, T$ fields which are similar to the previous result in Ref. [@Witten85], as noticed in the Ref. [@CCAF]. This is an interesting result, and allows us to argue that the weakly-coupled heterotic string models derived previously may exist in the M-theory proposed by Horava and Witten. The above 4-dimensional results should be related to the 4-dimensional results from the weakly-coupled heterotic string compactification by a field transformation, although we note that there may not exist a 10-dimensional effective field theory (EFT) [@VK].
Phenomenological consequences
=============================
Let us now discuss the physical couplings and the physical radius of the eleventh dimension ($\rho_p$) in the various frames. If we define the Planck mass in $d$ dimensions as $8 \pi G_{N}^{(d)}= M_d^{2-d}=\kappa^2_d$ and $M_4=M_{\rm Pl}=2.4\times 10^{18}$ GeV, from the 11-dimensional Lagrangian \[Eq. (\[eq:1\])\], we have $M_{11}=\kappa^{-2/9}$, and from the above 4-dimensional Lagrangian \[Eq. (\[eq:15\])\] in the Einstein frame, we obtain: $$\begin{aligned}
8\pi\, \left[G_{N}^{(4)}\right]_E &=& {\kappa^2 \over {2\pi \rho V}}\,\\
\left[\alpha_{\rm GUT}\right]_E &=& {1\over{2 V_p f}}\,(4\pi \kappa^2 )^{2/3}\,\end{aligned}$$ where $V_p$ is the physical volume of the Calabi-Yau manifold, i.e., $V_p = V
ReS $. Similar relations have been obtained by Witten and others [@Witten; @BD; @AQ; @VK] in the metric of the eleven-dimensional theory, these are: $$\begin{aligned}
8\pi\,\left[G_{N}^{(4)}\right]_W &=& {\kappa^2 \over {2\pi \rho_p V_p}} \ , \\
\left[\alpha_{\rm GUT}\right]_W &=&{1\over {2 V_p f}}\,(4\pi\kappa^2 )^{2/3}\,\end{aligned}$$ where $\rho_p$ is physical radius, i.e., $\rho_p={1\over {2\pi}} \int
dx^{11} {\sqrt {g_{11, 11}}}$. We have also included the constant $f$, which arises from the following normalization: if we asssume that the 4-dimensional grand unified group is $G$ (i.e., $E_8$ is broken to G) and if $T$ is a generator of $G$, ${\rm Tr}_{E_8}$ and ${\rm Tr}_{G}$ are traces in the adjoint representations of $G$ and $E_8$, then ${\rm Tr}_{E_8} T^2 = f\,
{\rm Tr}_G T^2$ [@Witten85].
Because eleven-dimensional supergravity can be derived from the eleven-dimensional supermembrane world volume action by imposing kappa symmetry [@Sezgin], and there are arguments in favor of a membrane/fivebrane duality in eleven dimensions [@Duff], it has been argued that the fivebrane units are the natural or fundamental units of M-theory [@DM]. Therefore, we also consider the above relations in the fivebrane units [@DM; @PBIN]. We continue to use our above notation, i.e., we do not use the membrane quantization condition [@DM], since we want to obtain an explicit expression for $\rho^{-1}_p$ and the eleven-dimensional fundamental constant $\kappa$; the result is the same in both approaches. We obtain the relevant 4-dimensional Lagrangian in fivebrane units [@DM; @PBIN]: $$\begin{aligned}
L_5 &=&-\int d^4 x ({1\over {2\kappa^2 }} 2 \pi \rho V e^{\sigma}
e^{2\phi /3} R + {f\over {2\pi }} (4\pi \kappa^2)^{-2/3} V e^{3\sigma}
{\rm tr}\,F_{\mu \nu} F^{\mu \nu})\,\end{aligned}$$ therefore we have: $$\begin{aligned}
8\pi\left[G_{N}^{(4)}\right]_{5B} &=&
{\kappa^2 \over {2\pi \rho_p V_p}} e^{2\sigma } \ , \\
\left[\alpha_{\rm GUT}\right]_{5B} &=& {1\over{2V_pf}}\,(4\pi \kappa^2)^{2/3}\,\end{aligned}$$ where $V_p = e^{3\sigma} V$ and $\rho_p =\rho e^{2\phi /3}$. We note that in all three cases (Einstein, Witten, and fivebrane), $\alpha_{\rm GUT}$ is the same, as the term $\sqrt g\, {\rm tr}\,F^2$ is invariant under the rescaling of $g_{\mu \nu}$ in four dimensions. Therefore, we have in general: $$\begin{aligned}
M_{11} &=& \left[2 (4\pi )^{-2/3}\, V_p f\, \alpha_{\rm GUT}\right]^{-1/6}\,\end{aligned}$$ If we define $V_p = L^d l^{6-d}$, ( $0 \leq d \leq 6$ ), where $L^{-1}$ is the compactification scale and $l$ is the small internal length, we obtain: $$\begin{aligned}
L^{-1} &=& \left[2(4\pi )^{-2/3} f \alpha_{\rm GUT}\right]^{1/d}
\left({M_{11}\over l^{-1}}\right)^{(6-d)/d}\, M_{11} ~,~ \,\end{aligned}$$ which tells us that $L^{-1}\sim M_{11}$, when $l^{-1} \sim M_{11}$.
Let us now discuss $\rho^{-1}_p$ in the eleven-dimensional metric and fivebrane units (it is not natural to think of the Lagrangian in the Einstein frame as fundamental). We obtain: $$\begin{aligned}
\left[\rho^{-1}_p\right]_W &=& 8 \pi^2 \left(2 f\alpha_{\rm GUT}\right)^{-3/2}
\left(M_{\rm Pl}^W\right)^{-2} V_p^{-1/2}\, \\
\left[\rho^{-1}_p\right]_{5B} &=& 8\pi^2\left(2 f\alpha_{\rm GUT}\right)^{-3/2}
\left(M_{\rm Pl}^{5B}\right)^{-2} V_p^{-1/2} e^{-2\sigma }\end{aligned}$$ The eleven-dimensional length $(\pi \rho_p)^{-1}$, is of great phenomenological importance because it is related to the scale of supersymmetry breaking [@Horava; @LLN]. To obtain numerical results we set $M_{Pl}^W = M_{Pl}^{5B}
= M_{Pl}= 2.4\times 10^{18}\,{\rm GeV}$, $\alpha_{\rm GUT}= {1\over 25}$, $V_p=M_{\rm GUT}^{-6}$, and $M_{\rm GUT} = 10^{16}\,{\rm GeV}$. We also set $f=1$ for simplicity and to facilitate comparison with previous papers which only consider this case. We find $[(\pi \rho_p)^{-1}]_W\sim 1.9\times 10^{14}\,{\rm GeV}$ and $[(\pi \rho_p)^{-1}]_{5B}\sim 1.2\times 10^{13}\,{\rm GeV}$ (as $e^{2\sigma}\approx16.6$ if we set $V^{-1/6}=M_{11}$).
We now relax the $f=1$ choice and consider the case of realistic grand unified groups. The results for $[(\pi \rho_p)^{-1}]_W$ and $[(\pi
\rho_p)^{-1}]_{5B}$ for $G=E_6$, SO(10), SU(5), and SU(5)$\times$U(1) are listed in Table \[Table1\]. (We have assumed $V^{-1/6} = M_{11}$ in all these cases.) We then generally conclude that the supersymmetry-breaking scale is expected to be in $(10^{12}-10^{14})\,{\rm GeV}$ range.
$G$ $f$ $[(\pi\rho_p)^{-1}]_W$ $[(\pi\rho_p)^{-1}]_{5B}$
------------------- ------ ------------------------ ---------------------------
$E_6$ 2.5 $4.9\times10^{13}$ $5.3\times10^{12}$
SO(10) 3.75 $2.6\times10^{13}$ $3.8\times10^{12}$
SU(5) 6 $1.3\times10^{13}$ $2.6\times10^{12}$
SU(5)$\times$U(1) 6 $1.3\times10^{13}$ $2.6\times10^{12}$
: Supersymmetry-breaking scales in the eleven-dimensional metric (W) and fivebrane units (5B) for various choices of (observable) unified gauge groups ($G$). The parameter $f$ defined in the text is also listed in each case. All scales in GeV.[]{data-label="Table1"}
------------------------------------------------------------------------
Before addressing further phenomenological features of this scenario, we would like to connect up with our previous phenomenologically oriented study of M-theory–inspired no-scale supergravity in Ref. [@LLN]. In that paper we assumed a supergravity model with a no-scale-supergravity–like Kähler potential (implying vanishing universal scalar masses $m_0=0$), as suggested by earlier work in Refs. [@Horava; @BD]. In the present paper we have shown explicitly that such assumption is justified. Moreover, the transmission of supersymmetry-breaking effects from the hidden to the observable sector was assumed to follow the mechanism outlined by Horava [@Horava], whereby such effects are only felt for scales below $(\pi \rho_p)^{-1}$. Our previous sampling of such scales agrees well with our present results in Table \[Table1\]. These two ingredients were shown [@LLN] to lead to a rather restricted spectrum of superparticle masses within the reach of the present generation of accelerator experiments. It is also worth pointing out another, perhaps more intuitive, explanation of the $m_0=0$ result: since in M-theory the observable sector fields live in the twisted sector of an orbifold, general arguments [@AMQ] indicate that such fields should not feel any supersymmetry-breaking effects.
Because the supersymmetry breaking scale is low, there may exist light gravitinos in the M-theory regime. As has been explored in detailed recently, such light gravitinos may explain the $ee\gamma\gamma+E_{\rm T,miss}$ event reported by the CDF Collaboration, and might have some interesting consequences at LEP2 [@JLDN]. In this connection, it was noticed some time ago that reactions such as $gg\to \tilde g\widetilde G,\widetilde G\widetilde G$ (where $g,\tilde g,\widetilde G$ stand for gauge boson, gaugino, and gravitino respectively) may exceed the tree-level unitarity limit because of the non-renormalizability of the low-energy effective gravity theory [@Roy]. The critical energy was estimated to be $E_{cr} \sim c M_{\rm Pl}\,
m_{3/2}/m_{\tilde g}$, with $c\sim10^2$. As tree-level unitarity is violated for $E>E_{cr}$, above $E_{cr}$ the theory is expected to become strongly interacting or change its structure (or both) in order to restore unitarity. The eleventh dimension threshold $(\pi\rho_p)^{-1}$ appears to fulfill such requirements, and thus one might require $(\pi\rho_p)^{-1}<E_{cr}$, implying a lower bound on $m_{3/2}$ for any given supersymmetry-breaking scale. Taken at face value, this constraint and the results in Table \[Table1\] appear to require $m_{3/2}>(10^2-10^4)\,{\rm eV}$, which may be consistent with the estimated $m_{3/2}<250\,{\rm eV}$ required for the $\chi\to\gamma+\widetilde G$ decay to occur within the detector [@JLDN]. However, as we are not sure whether the above unitarity constraints can be used directly in M-theory (as they depend on specific processes that might be forbidden in this scenario), we find the above level of consistency rather encouraging.
There also exist axions in this scenario, for example $D$: if we define $e^{6\sigma} \partial_{\mu} C_{11\nu \rho } =\epsilon_{\mu \nu \rho \sigma }
\partial^{\sigma} \theta $, we have a ($\theta F^{\mu \nu } {\tilde{F}}_{\mu
\nu }$) term from ($G_{11\mu \nu \rho} G^{11\mu \nu \rho}$). The strong CP problem may be solved by such axions [@BD]. In this scenario, because the decay constant of these axions is very high $\sim 10^{16}$ GeV [@BD; @KMCI], their axino superpartners, if they are very light, might not provide the alternative explanation to the CDF missing energy event proposed in Ref. [@HKY].
Conclusions
===========
We have constructed an explicit 4-dimensional $E_6\times E_8$ model from the M-theory of Horava and Witten. We have also calculated the physical couplings and physical eleventh dimension length, which is presumed to be related to the supersymmetry breaking scale. We have done these calculations in various metrics and shown how they are all related to each other. Finally we discussed the phenomenological consequences of such scenario, which may include light gravitinos, axions, and axinos.
Acknowledgments {#acknowledgments .unnumbered}
===============
T. Li would like to thank K. Benakli, J. X. Lu, and C. N. Pope for useful discussions. This work has been partially supported by the World Laboratory. The work of J. L. has been supported in part by DOE grant DE-FG05-93-ER-40717 and that of D.V.N. by DOE grant DE-FG05-91-ER-40633.
[99]{} 0.5mm P. Horava and E. Witten, Nucl. Phys. B [**475**]{} (1996) 94. E. Witten, Nucl. Phys. B [**471**]{} (1996) 135. P. Horava, Phys. Rev. D [**54**]{} (1996) 7561. T. Banks and M. Dine, Nucl. Phys. B [**479**]{} (1996) 173 and hep-th/9609046. I. Antoniadis and M. Quiros, Phys. Lett. B [**392**]{} (1997) 61. E. Caceres, V. S. Kaplunovsky and I. M. Mandelberg, hep-th/9606036. T. Li, J. L. Lopez and D. V. Nanopoulos, hep-ph/9702237. E. Cremmer, S. Ferrara, C. Kounnas, and D. V. Nanopoulos, Phys. Lett. B[**133**]{} (1983) 61; J. Ellis, A. Lahanas, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. B[**134**]{} (1984) 429; J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B[**241**]{} (1984) 406 and B[**247**]{} (1984) 373. For a review see A. Lahanas and D. V. Nanopoulos, Phys. Rep. [**145**]{} (1987) 1. E. Witten, Phys. Lett. B[**155**]{} (1985) 151. E. Dudas and J. Mourad, hep-th/9701048. S. Park, in [*Proceedings of the 10th Topical Workshop on Proton-Antiproton Collider Physics* ]{}, Fermilab, 1995, edited by R. Raja and J. Yoh (AIP, New York, 1995), p. 62. S. Ferrara and S. Sabharwal, Class. Quantum Grav. 6 (1989) L77. A. C. Cadavid, A. Ceresole, R. D’Auria, S. Ferrara, Phys. Lett. B [**357**]{} (1995) 76. P. Binetruy, Phys. Lett. B [**315**]{} (1993) 80. E. Bergshoeff, E. Sezgin and P. K. Townsend, Phys. Lett. B [**189**]{} (1987) 75, Ann. Phys. [**185**]{} (1988) 330. M. J. Duff, J. T. Liu and R. Minasian, Nucl. Phys. B [**452**]{} (1995) 261. I. Antoniadis, C. Munoz, and M. Quiros, Nucl. Phys. B [**397**]{} (1993) 515. S. Dimopoulos, M. Dine, S. Raby and S. Thomas, Phys. Rev. Lett. [**76**]{} (1996) 3494; S. Ambrosanio, G. L. Kane, G. D. Kribs, S. Martin, and S. Mrenna, Phys. Rev. Lett. [**76**]{} (1996) 3498 and Phys. Rev. D [**54**]{} (1996) 5395; J. L. Lopez and D. V. Nanopoulos, Mod. Phys. Lett. A11 (1996) 2473 and Phys. Rev. D [**55**]{} (1997) 4450, J. L. Lopez, D. V. Nanopoulos and A. Zichichi, Phys. Rev. Lett. [**77**]{} (1996) 5168 and hep-ph/9611437 (Phys. Rev. D, to appear). See e.g., T. Bhattacharya and P. Roy, Nucl. Phys. B [**328**]{} (1989) 469, 481. K. Choi and J. E. Kim, Phys. Lett. B[**165**]{} (1985) 71. J. Hisano, T. Kobe and T. Yanagida, Phys. Rev. D[**55**]{} (1997) 411.
|
---
bibliography:
- 'Thesis.bib'
date: September 2014
title: Collider phenomenology of the 4D composite Higgs model
---
*“...per aspera sic itur ad astra...”*\
*Lucio Anneo Seneca*
\[Bibliography\]
|
---
abstract: 'Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, <span style="font-variant:small-caps;">Noetherian</span> semirings, semiring of natural numbers) that all equivalent wta $M$ and $N$ can be joined by a finite chain of simulations. More precisely, in all mentioned cases there exists a single wta that simulates both $M$ and $N$. Those results immediately yield decidability of equivalence provided that the semiring is finitely (and effectively) presented.'
author:
- 'Zoltán Ésik [^1]'
- 'Andreas Maletti [^2]'
bibliography:
- 'extra.bib'
title: Simulations of Weighted Tree Automata
---
Introduction {#sec:Intro}
============
Weighted tree automata (or equivalently, weighted tree grammars) are widely used in applications such as model checking [@abdjonmahors02] and natural language processing [@knigra05]. They finitely represent mappings, called tree series, that assign a weight (taken from a semiring) to each tree. For example, a probabilistic parser would return a tree series that assigns to each parse tree its likelihood. Consequently, several toolkits [@klamol01; @maykni06; @cle08] implement weighted tree automata.
The notion of simulation that is used in this paper is a generalization of the simulations for unweighted and weighted (finite) string automata of [@bloesi93; @esikui01]. The aim is to relate structurally equivalent automata. The results of [@bloesi93 Section 9.7] and [@koz94] show that two unweighted string automata (i.e., potentially nondeterministic string automata over the <span style="font-variant:small-caps;">Boolean</span> semiring) are equivalent if and only if they can be connected by a finite chain of relational simulations, and that in fact *functional* and *dual functional* simulations are sufficient. Simulations for weighted string automata (wsa) are called *conjugacies* in [@bealomsak05; @bealomsak06], where it is shown that for all fields, many rings including the ring ${\ensuremath{\bbbz}}$ of integers, and the semiring ${\ensuremath{\bbbn}}$ of natural numbers, two wsa are equivalent if and only if they can be connected by a finite chain of simulations. It is also shown that even a finite chain of functional (*covering*) and dual functional (*co-covering*) simulations is sufficient. The origin of those results can be traced back to the pioneering work of <span style="font-variant:small-caps;">Schützenberger</span> in the early 60’s, who proved that every wsa over a field is equivalent to a minimal wsa that is simulated by every *trim* equivalent wsa [@berreu84]. Relational simulations of wsa are also studied in [@buc08], where they are used to reduce the size of wsa. The relationship between functional simulations and the <span style="font-variant:small-caps;">Milner</span>-<span style="font-variant:small-caps;">Park</span> notion of bisimulation [@mil80; @par81] is discussed in [@bloesi93; @buc08].
In this contribution, we investigate simulations for weighted (finite) tree automata (wta). <span style="font-variant:small-caps;">Schützenberger</span>’s minimization method was extended to wta over fields in [@aleboz89; @boz91]. In addition, relational and functional simulations for wta are probably first used in [@esi98; @esi10b; @hogmalmay07d]. Moreover, simulations can be generalized to presentations in algebraic theories [@bloesi93], which seems to cover all mentioned instances. Here, we extend the results of [@bealomsak05; @bealomsak06] to wta. In particular, we show that two wta over a ring, <span style="font-variant:small-caps;">Noetherian</span> semiring, or the semiring ${\ensuremath{\bbbn}}$ are equivalent if and only if they are connected by a finite chain of simulations. Moreover, we discuss when the simulations can be replaced by functional and dual functional simulations, which are efficiently computable [@hogmalmay07d]. Such results are important because they immediately yield decidability of equivalence provided that the semiring is finitely and effectively presented.
Preliminaries {#sec:Prelim}
=============
The set of nonnegative integers is ${\ensuremath{\bbbn}}$. For every $k \in {\ensuremath{\bbbn}}$, the set $\{i \in {\ensuremath{\bbbn}}\mid 1 \leq i \leq k\}$ is simply denoted by $[k]$. We write ${\ensuremath{\lvert A \rvert}}$ for the cardinality of the set $A$. A *semiring* is an algebraic structure ${\cal A} = (A, \mathord+,
\mathord\cdot, 0, 1)$ such that $(A, \mathord+, 0)$ and $(A,
\mathord\cdot, 1)$ are monoids, of which the former is commutative, and $\cdot$ distributes both-sided over finite sums (i.e., $a \cdot 0
= 0 = 0 \cdot a$ for every $a \in A$ and $a \cdot (b + c) = ab + ac$ and $(b + c) \cdot a = ba + ca$ for every $a, b, c \in A$). The semiring $\mathcal A$ is *commutative* if $(A, \mathord\cdot, 1)$ is commutative. It is a *ring* if for every $a \in A$ there exists an *additive inverse* $-a \in A$ such that $a + (-a) = 0$. The set $U$ is the set $\{ a \in A \mid \exists b \in A \colon ab = 1
= ba\}$ of *(multiplicative) units*. The semiring ${\cal A}$ is a *semifield* if $U = A \setminus \{0\}$; i.e., for every $a \in
A$ there exists a *multiplicative inverse* $a^{-1} \in A$ such that $aa^{-1} = 1 = a^{-1}a$. A *field* is a semifield that is also a ring. For every $B \subseteq A$ let $\langle
B\rangle_{\mathord+} = \{ b_1 + \dotsb + b_n \mid n \in {\ensuremath{\bbbn}}, {\ensuremath{b_{1}, \dotsc, b_{n}}}
\in B\}$. If $A = \langle B\rangle_{\mathord+}$, then ${\cal A}$ is *additively generated by $B$*. Finally, it is *equisubtractive* if for every $a_1, a_2, b_1, b_2 \in A$ with $a_1 + b_1 = a_2 + b_2$ there exist $c_1, c_2, d_1, d_2 \in A$ such that (i) $a_1 = c_1 + d_1$, (ii) $b_1 = c_2 + d_2$, (iii) $a_2 = c_1 +
c_2$, and (iv) $b_2 = d_1 + d_2$.
The semiring ${\cal A}$ is *zero-sum free* if $a + b = 0$ implies $0 \in \{a, b\}$ for every $a, b \in A$. Clearly, any nontrivial (i.e., $0 \neq 1$) ring is not zero-sum free. Moreover, ${\cal A}$ is *zero-divisor free* if $a \cdot b = 0$ implies $a = 0 = b$ for every $a, b \in A$. All semifields are trivially zero-divisor free. Finally, the semiring ${\cal A}$ is *positive* if it is zero-sum free and zero-divisor free. An infinitary sum operation $\mathord{\sum}$ is a family $(\mathord{\sum_I})_I$ such that $\mathord{\sum_I} \colon A^I \to A$. We generally write $\sum_{i
\in I} a_i$ instead of $\sum_I (a_i)_{i \in I}$. The semiring ${\cal A}$ together with the infinitary sum operation $\mathord{\sum}$ is *complete* [@eil74; @hebwei98; @gol99; @kar04] if
- $\sum_{i \in \{j_1, j_2\}} a_i = a_{j_1} + a_{j_2}$ for all $j_1\neq j_2$ and $a_{j_1}, a_{j_2} \in A$,
- $\sum_{i \in I} a_i = \sum_{j \in J} \bigl( \sum_{i \in I_j} a_i
\bigr)$ for every index set $I$, partition $(I_j)_{j \in J}$ of $I$, and $(a_i)_{i \in I} \in A^I$, and
- $a \cdot \bigl(\sum_{i \in I} a_i \bigr) = \sum_{i \in I} aa_i$ and $\bigl(\sum_{i \in I} a_i \bigr) \cdot a = \sum_{i \in I} a_ia$ for every $a \in A$, index set $I$, and $(a_i)_{i \in I} \in A^I$.
An ${\cal A}$-*semimodule* is a commutative monoid $(B,
\mathord{+}, 0)$ together with an action $\mathord{\cdot} \colon A
\times B \to B$, written as juxtaposition, such that for every $a, a'
\in A$ and $b, b' \in B$
- $(a + a') b = ab + a'b$ and $a(b + b') = ab + ab'$,
- $0 b = 0 = a 0$, $1 b = b$ and $(a \cdot a')b = a(a'b)$.
The semiring ${\cal A}$ is <span style="font-variant:small-caps;">Noetherian</span> if all subsemimodules of every finitely-generated ${\cal A}$-semimodule are again finitely-generated.
In the following, we often identify index sets of the same cardinality. Let $X \in A^{I_1 \times J_1}$ and $Y \in A^{I_2 \times
J_2}$ for some finite sets $I_1, I_2, J_1, J_2$. We use upper-case letters (like $C$, $D$, $E$, $X$, $Y$) for matrices and the corresponding lower-case letters for their entries. A matrix $X \in
A^{I \times J}$ is *relational* if $x_{ij} \in \{0, 1\}$ for every $i \in I$ and $j \in J$. Clearly, a relational matrix defines a relation $\rho_X \subseteq I \times J$ by $(i, j) \in \rho_X$ if and only if $x_{ij} = 1$ (and vice versa). Moreover, we call a relational matrix *functional*, *surjective*, or *injective* if its associated relation has this property. As usual, we denote the *transpose* of a matrix $X$ by $X^{\mathrm T}$, and we call $X$ *nondegenerate* if its has no rows or columns of entirely zeroes. A *diagonal* matrix $X$ is such that $x_{ij} = 0$ for every $i \neq j$. Finally, the matrix $X$ is invertible if there exists a matrix $Y$ such that $XY = I = YX$ where $I$ is the unit matrix. The <span style="font-variant:small-caps;">Kronecker</span> product $X \otimes Y \in A^{(I_1
\times I_2) \times (J_1 \times J_2)}$ is such that $(X \otimes
Y)_{(i_1, i_2), (j_1, j_2)} = x_{i_1, j_1} y_{i_2, j_2}$ for every $i_1 \in I_1$, $i_2 \in I_2$, $j_1 \in J_1$, and $j_2 \in J_2$. Clearly, the <span style="font-variant:small-caps;">Kronecker</span> product is, in general, not commutative and $(1) \in A^{[1]}$ acts both-sided as neutral element. We let $X^{0,\mathord{\otimes}} = (1)$ and $X^{i+1, \mathord{\otimes}}
= X^{i, \mathord{\otimes}} \otimes X$ for every $i \in {\ensuremath{\bbbn}}$.
Finally, let us move to trees. A *ranked alphabet* is a finite set $\Sigma$ together with a mapping $\mathord{\operatorname{rk}} \colon \Sigma \to
{\ensuremath{\bbbn}}$. We often just write $\Sigma$ for a ranked alphabet and assume that the mapping $\operatorname{rk}$ is implicit. We write $\Sigma_k = \{\sigma \in
\Sigma \mid \operatorname{rk}(\sigma) = k\}$ for the set of all $k$-ary symbols. The set of $\Sigma$-*trees* is the smallest set $T_\Sigma$ such that $\sigma({\ensuremath{t_{1}, \dotsc, t_{k}}}) \in T_\Sigma$ for all $\sigma \in \Sigma_k$ and ${\ensuremath{t_{1}, \dotsc, t_{k}}} \in T_\Sigma$. A *tree series* is a mapping $\varphi
\colon T_\Sigma \to A$. The set of all such tree series is denoted by ${\ensuremath{A \langle\!\langle T_{\Sigma}
\rangle\!\rangle}}$. For every $\varphi \in {\ensuremath{A \langle\!\langle T_{\Sigma}
\rangle\!\rangle}}$ and $t
\in T_\Sigma$, we often write $(\varphi, t)$ instead of $\varphi(t)$. Let ${\scriptstyle \Box}$ be a distinguished nullary symbol such that ${\scriptstyle \Box} \notin \Sigma$. A $\Sigma$-*context* $c$ is a tree of $T_{\Sigma \cup \{{\scriptstyle \Box}\}}$, in which the symbol ${\scriptstyle \Box}$ occurs exactly once. The set of all $\Sigma$-contexts is denoted by $C_\Sigma$. For every $c \in C_\Sigma$ and $t \in T_\Sigma$, we write $c[t]$ for the $\Sigma$-tree obtained by replacing the unique occurrence of ${\scriptstyle \Box}$ in $c$ by $t$.
A *weighted tree automaton (over ${\cal A}$)*, for short: wta, is a system $(\Sigma, Q, \mu, F)$ with
- an input ranked alphabet $\Sigma$,
- a finite set $Q$ of *states*,
- transitions $\mu = (\mu_k)_{k \in {\ensuremath{\bbbn}}}$ such that $\mu_k \colon
\Sigma_k \to A^{Q^k \times Q}$ for every $k \in {\ensuremath{\bbbn}}$, and
- a *final weight* vector $F \in A^Q$.
Next, let us introduce the semantics ${\ensuremath{\lVert M \rVert}}$ of $M$. We first define the function $h_\mu \colon T_\Sigma \to A^Q$ for every $\sigma \in
\Sigma_k$ and ${\ensuremath{t_{1}, \dotsc, t_{k}}} \in T_\Sigma$ by $$h_\mu(\sigma({\ensuremath{t_{1}, \dotsc, t_{k}}})) =
\bigl( h_\mu(t_1) \otimes \dotsm \otimes h_\mu(t_k) \bigr) \cdot
\mu_k(\sigma) \enspace,$$ where the final product $\cdot$ is the classical matrix product. Then $({\ensuremath{\lVert M \rVert}}, t) = h_\mu(t) F$ for every $t \in T_\Sigma$, where the product is the usual inner (dot) product.
Let $f \colon A \to \{0, 1\}$ be such that $f(0) = 0$ and $f(a) = 1$ for all $a \in A \setminus \{0\}$. The Boolean wta $f(M)$ (i.e., essentially an unweighted tree automaton) corresponding to $M$ is $(\Sigma, Q, \mu', F')$ where
- $\mu'_k(\sigma)_{w, q} = f(\mu_k(\sigma)_{w, q})$ for every $\sigma \in \Sigma_k$, $w \in Q^k$, and $q \in Q$, and
- $F'(q) = f(F(q))$ for every $q \in Q$.
The wta $M$ is *trim* if every state is accessible and co-accessible in $f(M)$. In other words, the wta $M$ is trim if $f(M)$ is trim.
Simulation {#sec:Sim}
==========
Simulations of automata were defined in [@bloesi93; @esikui01] in order to provide a structural characterization of equivalent automata. We will essentially follow the presentation of [@bealomsak05] here.
\[df:Conj\] Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$ be wta. Then $M$ *simulates* $N$ if there exists a matrix $X \in A^{Q
\times P}$ such that
- $F = XG$, and
- $\mu_k(\sigma) X = X^{k, \mathord{\otimes}} \cdot
\nu_k(\sigma)$ for every $\sigma \in \Sigma_k$.
The matrix $X$ is called *transfer matrix*, and we write $M
\stackrel X\to N$ if $M$ simulates $N$ with transfer matrix $X$.
![Illustration of simulation.[]{data-label="fig:Conj"}](conj.mps)
Note that $X^{k, \mathord{\otimes}}_{{\ensuremath{i_{1} \dotsm i_{k}}}, {\ensuremath{j_{1} \dotsm j_{k}}}} =
\prod_{\ell = 1}^k x_{i_\ell, j_\ell}$. We illustrate Definition \[df:Conj\] in Fig. \[fig:Conj\]. If $M \stackrel X\to
M'$ and $M' \stackrel Y\to N$, then $M \stackrel{XY}\to N$. Thus, simulations define a preorder on wta.
\[thm:Equiv\] If $M$ simulates $N$, then $M$ and $N$ are equivalent.
Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$, and let $X \in A^{Q \times P}$ be a transfer matrix. We claim that $h_\mu(t)X = h_\nu(t)$ for every $t \in T_\Sigma$. We prove this by induction on $t$. Let $t = \sigma({\ensuremath{t_{1}, \dotsc, t_{k}}})$ for some $\sigma \in
\Sigma_k$ and ${\ensuremath{t_{1}, \dotsc, t_{k}}} \in T_\Sigma$. $$\begin{aligned}
&\phantom{{}={}} h_\mu(\sigma({\ensuremath{t_{1}, \dotsc, t_{k}}})) X = \bigl( h_\mu(t_1)
\otimes \dotsm \otimes h_\mu(t_k) \bigr) \cdot \mu_k(\sigma) X \\
&= \bigl( h_\mu(t_1) \otimes \dotsm \otimes h_\mu(t_k) \bigr)
\cdot X^{k, \mathord{\otimes}} \cdot \nu_k(\sigma) = \bigl(
h_\mu(t_1)X \otimes \dotsm \otimes h_\mu(t_k)X \bigr) \cdot
\nu_k(\sigma) \\
&= \bigl( h_\nu(t_1) \otimes \dotsm \otimes h_\nu(t_k) \bigr)
\cdot \nu_k(\sigma) = h_\nu(\sigma({\ensuremath{t_{1}, \dotsc, t_{k}}}))
\end{aligned}$$ With this claim, the statement can now be proved easily. For every $t \in T_\Sigma$ $$({\ensuremath{\lVert M \rVert}}, t) = h_\mu(t)F = h_\mu(t)XG = h_\nu(t)G = ({\ensuremath{\lVert N \rVert}}, t)
\enspace. \tag*{\qed}$$
\[lm:Trim\] Let $M$ and $N$ be trim wta such $M \stackrel X\to N$. If (i) $X$ is functional or (ii) ${\cal A}$ is positive, then $X$ is nondegenerate.
Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$. Moreover, let $$J = \{ p \in P \mid \forall q \in Q \colon x_{qp} =
0\} \enspace.$$ Then $\nu_k(\sigma)_{w, j} = 0$ for every $\sigma
\in \Sigma_k$, $w \in (P \setminus J)^k$, and $j \in J$. This is seen as follows. Since $\mu_k(\sigma)X = X^{k, \mathord{\otimes}}
\cdot \nu_k(\sigma)$ we obtain $$\begin{aligned}
\label{eq:Trim1}
\sum_{q \in Q} \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} \cdot x_{qj} = 0 &=
\sum_{{\ensuremath{p_{1}, \dotsc, p_{k}}} \in P} \Bigl( \prod_{\ell = 1}^k x_{q_\ell,
p_\ell} \Bigr) \cdot \nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, j}
\end{aligned}$$ for every ${\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q$ and $j \in J$. If $X$ is functional, then $$\sum_{{\ensuremath{p_{1}, \dotsc, p_{k}}} \in P} \Bigl( \prod_{\ell = 1}^k x_{q_\ell,
p_\ell} \Bigr) \cdot \nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, j} =
\nu_k(\sigma)_{\rho_X(q_1) \dotsm \rho_X(q_k), j} = 0 \enspace,$$ which proves the claim. On the other hand, if ${\cal A}$ is positive, then implies that $\prod_{\ell = 1}^k
x_{q_\ell, p_\ell} \cdot \nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, j} = 0$ for every ${\ensuremath{p_{1}, \dotsc, p_{k}}} \in P$. Since for every $p_\ell \notin J$, there exists $q_\ell$ such that $x_{q_\ell, p_\ell} \neq 0$ and $\prod_{\ell =
1}^k x_{q_\ell, p_\ell} \neq 0$ by zero-divisor freeness, we conclude that $\nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, j} = 0$ for every ${\ensuremath{p_{1}, \dotsc, p_{k}}}
\in P \setminus J$, which again proves the claim. Consequently, all states of $J$ are unreachable. Since $N$ is trim, we conclude $J =
\emptyset$, and thus, $X$ has no column of zeroes.
If $X$ is functional, then it clearly has no row of zeroes. To prove that $X$ has no row of zeroes in the remaining case, let $I =
\{ q \in Q \mid \forall p \in P \colon x_{qp} = 0\}$. Then $F_i =
0$ and $\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} = 0$ for every $\sigma \in
\Sigma_k$, $q \in Q \setminus I$, ${\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q$, and $i \in I$ such that $q_\ell = i$ for some $\ell \in [k]$. Clearly, $F_i =
\sum_{p \in P} x_{ip} G_p = 0$ for every $i \in I$. Moreover, since $\mu_k(\sigma)X = X^{k, \mathord{\otimes}} \cdot \nu_k(\sigma)$ we obtain $$\begin{aligned}
\label{eq:Trim2}
\sum_{q \in Q} \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} \cdot x_{qp} &=
\sum_{{\ensuremath{p_{1}, \dotsc, p_{k}}} \in P} \Bigl( \prod_{\ell = 1}^k x_{q_\ell,
p_\ell} \Bigr) \cdot \nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, p} = 0
\end{aligned}$$ for every ${\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q$, $p \in P$, and $i \in I$ such that $q_\ell = i$ for some $\ell \in [k]$. Since ${\cal A}$ is positive, implies that $\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} \cdot
x_{qp} = 0$ for every $q \in Q$. However, for all $q \in Q
\setminus I$, there exists $p \in P$ such that $x_{qp} \neq 0$ because $q \notin I$. Consequently, $\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} =
0$ by zero-divisor freeness, which proves the claim. Thus, all states of $I$ are unreachable. Since $M$ is trim, we conclude $I =
\emptyset$, and thus, $X$ has no row of zeroes.
\[df:ForSim\] Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$ be wta. A surjective function $\rho \colon Q \to P$ is a *forward simulation* from $M$ to $N$ if
- $F_q = G_{\rho(q)}$ for every $q \in Q$, and
- for every $p \in P$, $\sigma \in \Sigma_k$, and ${\ensuremath{q_{1}, \dotsc, q_{k}}}
\in Q$ $$\sum_{q \in Q \colon \rho(q) = p} \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} = \nu_k(\sigma)_{\rho(q_1) \dotsm \rho(q_k), p}
\enspace.$$
Finally, we say that *$M$ forward simulates $N$*, written $M
\twoheadrightarrow N$, if there exists a forward simulation from $M$ to $N$.
\[lm:FSim\] Let $M$ and $N$ be wta such that $N$ is trim. Then $M
\twoheadrightarrow N$ if and only if there exists a functional transfer matrix $X$ such that $M \stackrel X\to N$.
Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$. First suppose that $M \stackrel X\to N$ with functional $X \in A^{Q \times
P}$. Then $\rho_X \colon Q \to P$ is a surjective function by Lemma \[lm:Trim\]. Conversely, if $M \twoheadrightarrow N$ with the forward simulation $\rho \colon Q \to P$, then $\rho$ induces a surjective functional matrix $X \in A^{Q \times P}$ such that $\rho_X = \rho$.
Let $X \in A^{Q \times P}$ be a surjective, functional matrix. It remains to prove that the conditions that (1) $X$ is a transfer matrix and (2) $\rho_X$ is a forward simulation are equivalent. We discuss the two items of Definitions \[df:Conj\] and \[df:ForSim\] separately.
- $F = XG$ if and only if $F_q = G_{\rho(q)}$ for every $q
\in Q$.
- for every $\sigma \in \Sigma_k$, ${\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q$, and $p \in P$ $$\begin{aligned}
(\mu_k(\sigma)X)_{{\ensuremath{q_{1} \dotsm q_{k}}}, p} &= \sum_{q \in Q \colon
\rho_X(q) = p} \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} \\
(X^{k, \mathord{\otimes}} \cdot \nu_k(\sigma))_{{\ensuremath{q_{1} \dotsm q_{k}}}, p}
&= \nu_k(\sigma)_{\rho_X(q_1) \dotsm \rho_X(q_k), p} \enspace.
\end{aligned}$$
Thus, $X$ is a transfer matrix if and only if $\rho_X$ is a forward simulation, which proves the statement.
\[df:BackSim\] Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$ be wta. A surjective function $\rho \colon Q \to P$ is a *backward simulation* from $M$ to $N$ if
- $\sum_{q \in Q \colon \rho(q) = p} F_q = G_p$ for every $p \in P$, and
- for every $q \in Q$, $\sigma \in \Sigma_k$, and ${\ensuremath{p_{1}, \dotsc, p_{k}}}
\in P$ $$\sum_{\substack{{\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q \\ \rho(q_1) = p_1, \dotsc,
\rho(q_k) = p_k}} \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} =
\nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, \rho(q)} \enspace.$$
Finally, we say that *$M$ backward simulates $N$*, written $M
\twoheadleftarrow N$, if there exists a backward simulation from $M$ to $N$.
\[lm:BSim\] Let $M$ and $N$ be wta such that $N$ is trim. Then $M
\twoheadleftarrow N$ if and only if there exists a transfer matrix $X$ such that $X^{\mathrm T}$ is functional and $N \stackrel
X\to M$.
Let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$. First, suppose that $N \stackrel X\to M$ with the transfer matrix $X \in
A^{P \times Q}$ such that $X^{\mathrm T}$ is functional. Let $Y =
X^{\mathrm T}$. Then $\rho_Y \colon Q \to P$ is a surjective function by Lemma \[lm:Trim\]. Conversely, if $M
\twoheadleftarrow N$ with the backward simulation $\rho \colon Q \to
P$, then $\rho$ again induces a surjective, functional matrix $X \in
A^{Q \times P}$ such that $\rho_X = \rho$.
Let $X \in A^{Q \times P}$ be a surjective, functional matrix. It remains to prove that the conditions that (1) $X^{\mathrm T}$ is a transfer matrix and (2) $\rho_X$ is a backward simulation are equivalent. We discuss the two items of Definitions \[df:Conj\] and \[df:BackSim\] separately.
- $G = X^{\mathrm T}F$ if and only if $G_p = \sum_{q \in
Q \colon \rho_X(q) = p} F_q$ for every $p \in P$.
- for every $\sigma \in \Sigma_k$, ${\ensuremath{p_{1}, \dotsc, p_{k}}} \in P$, and $q \in Q$ $$\begin{aligned}
(\nu_k(\sigma)X^{\mathrm T})_{{\ensuremath{p_{1} \dotsm p_{k}}}, q} &=
\nu_k(\sigma)_{{\ensuremath{p_{1} \dotsm p_{k}}}, \rho_X(q)} \\
((X^{\mathrm T})^{k, \mathord{\otimes}} \cdot
\mu_k(\sigma))_{{\ensuremath{p_{1} \dotsm p_{k}}}, q} &= \sum_{\substack{{\ensuremath{q_{1}, \dotsc, q_{k}}} \in
Q \\ \rho_X(q_1) = p_1, \dotsc, \rho_X(q_k) = p_k}}
\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} \enspace.
\end{aligned}$$
Thus, $X^{\mathrm T}$ is a transfer matrix if and only if $\rho_X$ is a backward simulation, which proves the statement.
\[lm:Help2\] If $A = \langle U \rangle_{\mathord{+}}$, then for every $X \in A^{Q
\times P}$ there exist matrices $C, E, D$ such that
- $X = CED$,
- $C^{\mathrm T}$ and $D$ are functional, and
- $E$ is an invertible diagonal matrix.
If (i) $X$ is nondegenerate or (ii) ${\cal A}$ has (nontrivial) zero-sums, then $C^{\mathrm T}$ and $D$ can be chosen to be surjective.
For every $q \in Q$ and $p \in P$, let $\ell_{qp} \in {\ensuremath{\bbbn}}$ and $u_{qp1}, \dotsc, u_{qp\ell_{qp}} \in U$ be such that $x_{qp} =
\sum_{i = 1}^{\ell_{qp}} u_{qpi}$. In addition, let $$J = \{ (q, i, p) \mid q \in Q, p \in P, i \in [\ell_{qp}] \}
\enspace.$$ Finally, let $\pi_1 \colon J \to Q$ and $\pi_3 \colon J
\to P$ be such that $\pi_1(\langle q, i, p\rangle) = q$ and $\pi_3(\langle q, i, p\rangle) = p$ for every $\langle q, i,
p\rangle \in J$. Then we set $C^{\mathrm T}$ and $D$ to the functional matrices represented by $\pi_1$ and $\pi_3$, respectively. Together with the diagonal matrix $E$ such that $e_{\langle q, i, p\rangle, \langle q, i, p\rangle} = u_{qpi}$ for every $\langle q, i, p\rangle \in J$, we obtain $X = CED$. For every $q \in Q$ and $p \in P$ we have $$\sum_{j_1, j_2 \in J} c_{q, j_1} e_{j_1, j_2} d_{j_2, p} =
\sum_{i = 1}^{\ell_{qp}} e_{\langle q, i, p\rangle, \langle q,
i, p\rangle} = \sum_{i = 1}^{\ell_{qp}} u_{qpi} = x_{qp}
\enspace.$$
It is clear that $C^{\mathrm T}$ and $D$ are functional matrices. Moreover, $E$ is an invertible diagonal matrix because $EE^{-1} = I
= E^{-1}E$ where $E^{-1}$ is the matrix obtained from $E$ by inverting each nonzero element. If $X$ is nondegenerate, then $C^{\mathrm T}$ and $D$ are surjective. Finally, if there are zero-sums, then for every $q \in Q$ and $p \in P$ there exist $u, v
\in U$ such that $x_{qp} = 0 = u + v$, which yields that we can choose $\ell_{qp} > 0$. This completes the proof.
\[lm:Sol\] Let ${\cal A}$ be equisubtractive. Moreover, let $R \in A^Q$ and $C
\in A^P$ be such that $\sum_{q \in Q} r_q = \sum_{p \in P} c_p$. Then there exists a matrix $X \in A^{Q \times P}$ with row sums $R$ and column sums $C$; i.e., $\sum_{q \in Q} x_{qp} = c_p$ for every $p \in P$ and $\sum_{p \in P} x_{qp} = r_q$ for every $q \in Q$.
If ${\ensuremath{\lvert Q \rvert}} \leq 1$ or ${\ensuremath{\lvert P \rvert}} \leq 1$, then the statement is trivially true. Otherwise, select $i \in Q$ and $j \in P$, and let $Q' = Q \setminus \{i\}$ and $P' = P \setminus \{j\}$. By assumption $$\sum_{q \in Q'} r_q + r_i = \sum_{p \in P'} c_p + c_j
\enspace.$$ Thus, by equisubtractivity there exist $a, c'_j, r'_i,
x_{ij} \in A$ such that $$\sum_{q \in Q'} r_q = a + c'_j \qquad r_i
= r'_i + x_{ij} \qquad \sum_{p \in P'} c_p = a + r'_i \qquad c_j =
c'_j + x_{ij} \enspace.$$ Continuing the row decomposition, we obtain $Y \in A^{Q'}$ and $R' \in A^{Q'}$ such that $r_q = r'_q +
y_q$ for every $q \in Q'$ and $\sum_{q \in Q'} r'_q = a$. In a similar manner we perform column decomposition to obtain $Y' \in
A^{P'}$ and $C' \in A^{P'}$ such that $c_p = c'_p + y'_p$ for every $p \in P'$ and $\sum_{p \in P'} c'_p = a$. Thus, by the induction hypothesis, there exists a matrix $X' \in A^{Q' \times P'}$ with row sums $R'$ and column sums $C'$ because $\sum_{q \in Q'} r'_q =
\sum_{p \in P'} c'_p$. Then the matrix $$X =
\begin{pmatrix}
\; & & \; & \\
& X' & & Y \\
& & & \\
& (Y')^{\mathrm T} & & x_{ij}
\end{pmatrix}$$ obviously has the required row and column sums $R$ and $C$, respectively.
\[lm:3\] If $X \in A^{Q \times P}$ is functional (respectively, invertible diagonal), then $X^{k, \mathord{\otimes}}$ is functional (respectively, invertible diagonal) for every $k \in {\ensuremath{\bbbn}}$.
Trivial.
\[thm:2\] Let $M$ and $N$ be wta and ${\cal A}$ be equisubtractive with $A =
\langle U \rangle_{\mathord{+}}$. Then $M \stackrel X\to N$ if and only if there exist two wta $M'$ and $N'$ such that
- $M \stackrel C\to M'$ where $C^{\mathrm T}$ is functional,
- $M' \stackrel E\to N'$ where $E$ is an invertible diagonal matrix, and
- $N' \stackrel D\to N$ where $D$ is functional.
If $M$ and $N$ are trim, then $M' \twoheadleftarrow M$ and $N'
\twoheadrightarrow N$.
Clearly, $M \stackrel C\to M' \stackrel E\to N' \stackrel D\to N$, which proves that $M \stackrel{CED}\longrightarrow N$. For the converse, let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu,
G)$. Lemma \[lm:Help2\] shows that there exist matrices $C, E, D$ such that
- $X = CED$,
- $C^{\mathrm T}$ and $D$ are functional matrices, and
- $E \in A^{I \times I}$ is an invertible diagonal matrix.
Finally, let $\varphi \colon I \to Q$ and $\psi \colon I \to P$ be the functions associated to $C^{\mathrm T}$ and $D$. It remains to determine the wta $M'$ and $N'$. We construct $M' = (\Sigma, I,
\mu', F')$ and $N' = (\Sigma, I, \nu', G')$ with
- $G' = DG$ and
- $F' = EDG$.
Then $CF' = CEDG = XG = F$. Thus, it remains to specify $\mu'_k(\sigma)$ and $\nu'_k(\sigma)$ for every $\sigma \in
\Sigma_k$. To this end, we determine a matrix $Y \in A^{I^k \times
I}$ such that $$\begin{aligned}
\label{eq:2a}
C^{k, \mathord{\otimes}} \cdot Y &= \mu_k(\sigma) CE \\
\label{eq:2b}
YD &= E^{k, \mathord{\otimes}} \cdot D^{k, \mathord{\otimes}}
\cdot \nu_k(\sigma) \enspace.
\end{aligned}$$ Given such a matrix $Y$, we then let $\mu'_k(\sigma) = YE^{-1}$ and $\nu'_k(\sigma) = (E^{k, \mathord{\otimes}})^{-1} \cdot Y$. Then $$\begin{aligned}
\mu_k(\sigma) C &= C^{k, \mathord{\otimes}} \cdot \mu'_k(\sigma)
&\quad \mu'_k(\sigma) E &= E^{k, \mathord{\otimes}} \cdot
\nu'_k(\sigma) &\quad \nu'_k(\sigma) D &= D^{k,
\mathord{\otimes}} \cdot \nu_k(\sigma) \enspace.
\end{aligned}$$ These equalities are displayed in Fig. \[fig:Squares\].
Finally, we need to specify the matrix $Y$. For every $q \in Q$ and $p \in P$, let $I_q = \varphi^{-1}(q)$ and $J_p = \psi^{-1}(p)$. Obviously, $Y$ can be decomposed into disjoint (not necessarily contiguous) submatrices $Y_{{\ensuremath{q_{1} \dotsm q_{k}}}, p} \in A^{(I_{q_1} \times
\dotsm \times I_{q_k}) \times J_p}$ with ${\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q$ and $p
\in P$. Then and hold if and only if for every ${\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q$ and $p \in P$ the following two conditions hold:
1. For every $i \in I$ such that $\psi(i) = p$, the sum of the $i$-column of $Y_{{\ensuremath{q_{1} \dotsm q_{k}}}, p}$ is $\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}},
\varphi(i)} \cdot e_{i,i}$.
2. For all ${\ensuremath{i_{1}, \dotsc, i_{k}}} \in I$ such that $\varphi(i_j) = q_j$ for every $j \in [k]$, the sum of the $({\ensuremath{i_{1}, \dotsc, i_{k}}})$-row of $Y_{{\ensuremath{q_{1} \dotsm q_{k}}}, p}$ is $\prod_{j = 1}^k e_{i_j, i_j} \cdot
\nu_k(\sigma)_{\psi(i_1) \dotsm \psi(i_k), p}$.
Those two conditions are compatible because $$\begin{aligned}
&\phantom{{}={}} \sum_{\substack{i \in I \\ \psi(i) = p}}
\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, \varphi(i)} \cdot e_{i,i} = \bigl(
\mu_k(\sigma)CED \bigr)_{{\ensuremath{q_{1} \dotsm q_{k}}}, p} = \bigl( \mu_k(\sigma)X
\bigr)_{{\ensuremath{q_{1} \dotsm q_{k}}}, p} \\
&\stackrel\dagger= \bigl( X^{k, \mathord{\otimes}} \cdot
\nu_k(\sigma) \bigr)_{{\ensuremath{q_{1} \dotsm q_{k}}}, p} = \bigl( C^{k,
\mathord{\otimes}} \cdot E^{k, \mathord{\otimes}} \cdot D^{k,
\mathord{\otimes}} \cdot
\nu_k(\sigma) \bigr)_{{\ensuremath{q_{1} \dotsm q_{k}}}, p} \\
&= \sum_{\substack{{\ensuremath{i_{1}, \dotsc, i_{k}}} \in I \\ \forall j \in [k] \colon
\varphi(i_j) = q_j}} \Bigl( \prod_{j = 1}^k e_{i_j, i_j}
\Bigr) \cdot \nu_k(\sigma)_{\psi(i_1) \dotsm \psi(i_k), p}
\enspace.
\end{aligned}$$ Consequently, the row and column sums of the submatrices $Y_{{\ensuremath{q_{1} \dotsm q_{k}}}, p}$ are consistent, which yields that we can determine all the submatrices (and thus the whole matrix) by Lemma \[lm:Sol\].
If $M$ and $N$ are trim, then either
- ${\cal A}$ is zero-sum free (and thus positive because it is additively generated by its units), in which case $X$ is nondegenerate by Lemma \[lm:Trim\], or
- ${\cal A}$ has nontrivial zero-sums.
In both cases, Lemma \[lm:Help2\] shows that the matrices $C^{\mathrm T}$ and $D$ are surjective, which yields the additional statement by Lemmata \[lm:FSim\] and \[lm:BSim\].
![Illustration of the relations between the matrices in the proof of Theorem .[]{data-label="fig:Squares"}](squares.mps)
Category of simulations {#sec:Category}
=======================
In this section our aim is to show that several well-known constructions of wta are *functorial*: they may be extended to simulations in a functorial way. Below we will only deal with the sum, <span style="font-variant:small-caps;">Hadamard</span> product, $\sigma_0$-product, and $\sigma_0$-iteration (cf. [@esi10]). Scalar OI-substition, ${}^\dagger$ [@bloesi03], homomorphism, quotient, and top-concatenation [@esi10] may be covered in a similar fashion.
Throughout this section, let $\mathcal{A}$ be commutative. Let $M =
(\Sigma, Q, \mu, F)$, $M' = (\Sigma, Q', \mu', F')$, and $M'' =
(\Sigma, Q'', \mu'', F'')$ be wta. We already remarked that, if $M
\stackrel X\to M'$ and $M' \stackrel Y\to M''$, then $M
\stackrel{XY}\to M''$. Moreover, $M \stackrel I\to M$ with the unit matrix $I \in A^{Q \times Q}$. Thus, wta over the alphabet $\Sigma$ form a category $\text{\textbf{Sim}}_\Sigma$.
In the following, let $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P,
\nu, G)$ be wta such that $Q \cap P = \emptyset$.
The sum $M + N$ of $M$ and $N$ is the wta $(\Sigma, Q \cup P,
\kappa, H)$ where $H = \langle F, G \rangle = \begin{pmatrix} F \\
G \end{pmatrix}$ and $$\kappa_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} = (\mu_k(\sigma) +
\nu_k(\sigma))_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} =
\begin{cases}
\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} & \text{if } q, {\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q \\
\nu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} & \text{if } q, {\ensuremath{q_{1}, \dotsc, q_{k}}} \in P \\
0 & \text{otherwise.}
\end{cases}$$ for all $\sigma \in \Sigma_k$ and $q, {\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q \cup P$.
It is well-known that ${\ensuremath{\lVert M + N \rVert}} = {\ensuremath{\lVert M \rVert}} + {\ensuremath{\lVert N \rVert}}$. Next, we extend the sum construction to simulations. To this end, let $M
\stackrel X\to M'$ with $M' = (\Sigma, Q', \mu', F')$, and let $N
\stackrel Y\to N'$ with $N' = (\Sigma, P', \nu', G')$.
The sum $X + Y \in A^{(Q \cup P) \times (Q' \cup P')}$ of the transfer matrices $X$ and $Y$ is $$X + Y =
\begin{pmatrix}
X & 0 \\
0 & Y
\end{pmatrix} \enspace.$$
We have $(M + N) \stackrel{X + Y}\longrightarrow (M' + N')$.
We only need to verify the two conditions of Definition \[df:Conj\]. For every $\sigma \in \Sigma_k$ we have $$\begin{aligned}
&\phantom{{}={}} \bigl(\mu_k(\sigma) + \nu_k(\sigma) \bigr) \cdot
(X + Y) = \mu_k(\sigma)X + \nu_k(\sigma)Y \\
&= X^{k, \mathord{\otimes}} \cdot \mu'_k(\sigma) + Y^{k,
\mathord{\otimes}} \cdot \mu'_k(\sigma) = (X + Y)^{k,
\mathord{\otimes}} \cdot \bigl(\mu'_k(\sigma) + \nu'_k(\sigma)
\bigr)
\end{aligned}$$ and $\langle F, G \rangle = \langle XF', YG' \rangle = (X + Y) \cdot
\langle F', G' \rangle$, which completes the proof.
The function $+$, which is defined on wta and transfer matrices, is a functor $\text{\textbf{Sim}}_\Sigma^2 \to
\text{\textbf{Sim}}_\Sigma$.
It is a routine matter to verify that identity transfer matrices are preserved and $(X + Y) \cdot (X' + Y') = XX' + YY'$ for all composable transfer matrices $X, X', Y, Y'$.
Let $\sigma_0$ be a distinguished symbol in $\Sigma_0$. The $\sigma_0$-product $M \cdot_{\sigma_0} N$ of $M$ with $N$ is the wta $(\Sigma, Q \cup P, \kappa, H)$ such that $$H = \langle F, 0\rangle = \begin{pmatrix} F\\ 0 \end{pmatrix}$$ and for each $\sigma \in \Sigma_k$ with $\sigma \neq \sigma_0$, $$\kappa_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} =
\begin{cases}
\mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} & \text{if } q, {\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q \\
\mu_0(\sigma_0)_q \cdot \sum_{p \in P} \nu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}},
p} G_p & \text{if } q \in Q \text{ and } {\ensuremath{q_{1}, \dotsc, q_{k}}} \in P \\
\nu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} & \text{if } q, {\ensuremath{q_{1}, \dotsc, q_{k}}} \in P \\
0 & \text{otherwise.}
\end{cases}$$ Moreover, $$\kappa_0(\sigma_0)_q =
\begin{cases}
\mu_0(\sigma_0)_q \cdot \sum_{p \in P} \nu_0(\sigma_0)_p G_p &
\text{if } q\in Q \\
\nu_0(\sigma_0)_q & \text{if } q \in P.
\end{cases}$$
It is known that ${\ensuremath{\lVert M \cdot_{\sigma_0} N \rVert}} = {\ensuremath{\lVert M \rVert}} \cdot_{\sigma_0}
{\ensuremath{\lVert N \rVert}}$. We extend this construction to simulations. To this end, let $M \stackrel X\to M'$ and $N \stackrel Y\to N'$. Then we define $X \cdot_{\sigma_0} Y = X + Y$. The next proposition can be verified by a routine calculation.
The function $\cdot_{\sigma_0}$, which is defined on wta and transfer matrices, is a functor $\text{\textbf{Sim}}_\Sigma^2 \to
\text{\textbf{Sim}}_\Sigma$.
The <span style="font-variant:small-caps;">Hadamard</span> product $M \cdot_{\mathrm H} N$ is the wta $(\Sigma, Q \times P, \kappa, H)$ where $H = F \otimes G$ and $\kappa_k(\sigma) = \mu_k(\sigma) \otimes \nu_k(\sigma)$ for all $\sigma \in \Sigma_k$.
We again extend the construction to simulations. If $M \stackrel X\to
M'$ and $N \stackrel Y\to N'$, then we define $X \cdot_{\mathrm H}
X \otimes Y$.
The function $\cdot_{\mathrm H}$, which is defined on wta and transfer matrices, is a functor $\text{\textbf{Sim}}_\Sigma^2 \to
\text{\textbf{Sim}}_\Sigma$.
Finally, we deal with iteration. Let $\sigma_0$ be a fixed symbol in $\Sigma_0$. Here we assume that $\mathcal{A}$ is complete. Thus, $\mathcal{A}$ comes with a star operation $a^* = \sum_{n \in {\ensuremath{\bbbn}}}
a^n$ for every $a \in A$.
The $\sigma_0$-iteration $M^{*_{\sigma_0}}$ of $M$ is the wta $(\Sigma, Q, \kappa, F)$ where $$\kappa_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} = \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} +
{\ensuremath{\lVert M \rVert}}(\sigma_0)^* \cdot \sum_{p \in Q} \mu_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}},
p} F_p$$ for all $\sigma \in \Sigma_k \setminus \{\sigma_0\}$ and $\kappa_0(\sigma_0) = \mu_0(\sigma_0)$.
If $M \stackrel X\to M'$, then we define $X^{*_{\sigma_0}} = X$.
The $\sigma_0$-iteration, which is defined on wta and transfer matrices, is a functor $\text{\textbf{Sim}}_\Sigma \to
\text{\textbf{Sim}}_\Sigma$.
Several subcategories of $\text{\textbf{Sim}}_\Sigma$ are also of interest, for example the categories formed by the relational or functional simulations and their duals. The above constructions are preserved by these special kinds of simulations.
Joint reduction {#sec:Joint}
===============
Next we will establish equivalence results using the approach called *joint reduction* in [@bealomsak06]. Let $V \subseteq A^I$ be a set of vectors for a finite set $I$. Then the ${\cal
A}$-semimodule generated by $V$ is denoted by $\langle V \rangle$. Given two wta $M = (\Sigma, Q, \mu, F)$ and $N = (\Sigma, P, \nu, G)$ with $Q \cap P = \emptyset$, we first compute $M + N = (\Sigma, Q \cup
P, \mu', F')$ as defined in Section \[sec:Category\]. Now the aim is to compute a finite set $V \subseteq A^{Q \cup P}$ such that
- $(v_1 \otimes \dotsm \otimes v_k) \cdot \mu'_k(\sigma) \in
\langle V \rangle$ for every $\sigma \in \Sigma_k$ and ${\ensuremath{v_{1}, \dotsc, v_{k}}} \in
V$, and
- $v_1F = v_2G$ for every $(v_1, v_2) \in V$ such that $v_1
\in A^Q$ and $v_2 \in A^P$.
With such a finite set $V$ we can now construct a wta $M' = (\Sigma,
V, \nu', G')$ with $G'_v = v F'$ for every $v \in V$ and $$\sum_{v \in V} \nu'_k(\sigma)_{{\ensuremath{v_{1} \dotsm v_{k}}}, v} \cdot v = (v_1 \otimes
\dotsm \otimes v_k) \cdot \mu'_k(\sigma)$$ for every $\sigma \in
\Sigma_k$ and ${\ensuremath{v_{1}, \dotsc, v_{k}}} \in V$. It remains to prove that $M'$ simulates $M + N$. To this end, let $X = (v)_{v \in V}$, where each $v \in V$ is a row vector. Then for every $\sigma \in \Sigma_k$, ${\ensuremath{v_{1}, \dotsc, v_{k}}} \in V$, and $q \in Q \cup P$, we have $$\begin{aligned}
&\phantom{{}={}} (\nu'_k(\sigma) X)_{{\ensuremath{v_{1} \dotsm v_{k}}}, q} = \sum_{v \in V}
\nu'_k(\sigma)_{{\ensuremath{v_{1} \dotsm v_{k}}}, v} \cdot v_q = \Bigl( \sum_{v \in V}
\nu'_k(\sigma)_{{\ensuremath{v_{1} \dotsm v_{k}}}, v} \cdot v \Bigr)_q \\
&= \bigl( (v_1 \otimes \dotsm \otimes v_k) \cdot \mu'_k(\sigma)
\bigr)_q = \sum_{{\ensuremath{q_{1}, \dotsc, q_{k}}} \in Q \cup P} (v_1)_{q_1} \cdot \ldots
\cdot (v_k)_{q_k} \cdot \mu'_k(\sigma)_{{\ensuremath{q_{1} \dotsm q_{k}}}, q} \\
&= \bigl( X^{k, \mathord{\otimes}} \cdot \mu'_k(\sigma)
\bigr)_{{\ensuremath{v_{1} \dotsm v_{k}}}, q} \enspace.\end{aligned}$$ Moreover, if we let $X_1$ and $X_2$ be the restrictions of $X$ to the entries of $Q$ and $P$, respectively, then we have $\nu'_k(\sigma)X_1
= X_1^{k, \mathord{\otimes}} \cdot \mu_k(\sigma)$ and $\nu'_k(\sigma)X_2 = X_2^{k, \mathord{\otimes}} \cdot \nu_k(\sigma)$. In addition, $G'_v = v F' = \sum_{q \in Q \cup P} v_q F'_q = (XF')_v$ for every $v \in V$, which proves that $M' \stackrel X\to (M + N)$. Since $v_1F = v_2G$ for every $(v_1, v_2) \in V$, we have $G'_{(v_1,
v_2)} = (v_1, v_2)F' = v_1F + v_2G = (1+1)v_1F = (1 + 1)v_2G$. Now, let $G''_{(v_1, v_2)} = v_1F = v_2G$ for every $(v_1, v_2) \in V$. Then $$\begin{aligned}
G''_v &= v_1F = \sum_{q \in Q} v_q F_q = (X_1F)_v \\
&= v_2G = \sum_{p \in P} v_p G_p = (X_2G)_v \end{aligned}$$ for every $v = (v_1, v_2) \in V$. Consequently, $M'' \stackrel{X_1}\to
M$ and $M'' \stackrel{X_2}\to N$, where $M'' = (\Sigma, V, \nu',
G'')$. This proves the next theorem.
\[thm:Joint\] Let $M$ and $N$ be two equivalent wta. If there exists a finite set $V \subseteq A^{Q \cup P}$ with properties (i) and (ii), then there exists a chain of simulations that join $M$ and $N$. In fact, there exists a single wta that simulates both $M$ and $N$.
Fields {#sec:Fields}
------
In this section, let ${\cal A}$ be a field. We first recall some notions from [@aleboz89]. Let $\varphi \in {\ensuremath{A \langle\!\langle T_{\Sigma}
\rangle\!\rangle}}$ be a tree series. The *syntactic ideal* of $\varphi$ is $$I_\varphi = \{ \psi \in {\ensuremath{A \langle\!\langle T_{\Sigma}
\rangle\!\rangle}} \mid \sum_{t \in T_\Sigma}
(\psi, t) (\varphi, c[t]) = 0 \text{ for all } c \in C_\Sigma \}
\enspace.$$ Moreover, let $\mathord\equiv$ be the equivalence relation on ${\ensuremath{A \langle\!\langle T_{\Sigma}
\rangle\!\rangle}}$ such that $\psi_1 \equiv \psi_2$ if and only if $\psi_1 - \psi_2 \in I_\varphi$. The syntactic algebra is $[{\ensuremath{A \langle\!\langle T_{\Sigma}
\rangle\!\rangle}}]_\equiv$. By [@aleboz89 Proposition 2] the tree series $\varphi$ is recognizable if and only if its syntactic algebra has finite dimension. Now, let $\varphi$ be recognizable, and let $B$ be a basis of its syntactic algebra. Finally, let $M_\varphi$ be the obtained canonical weighted tree automaton, which recognizes $\varphi$.
\[thm:ReltoMin\] Every trim wta recognizing $\varphi$ simulates $M_\varphi$.
Consequently, all equivalent trim wta $M_1$ and $M_2$ simulate the canonical wta that recognizes ${\ensuremath{\lVert M \rVert}}$. Using Theorem \[thm:2\] we can show that there exist wta $M'_1$, $M'_2$, $N'_1$, and $N'_2$ such that
- $M_1 \twoheadleftarrow M'_1$,
- $M'_1 \stackrel E\to N'_1$ with an invertible diagonal matrix $E$,
- $N'_1 \twoheadrightarrow M_\varphi$,
- $N'_2 \twoheadrightarrow M_\varphi$,
- $M'_2 \stackrel{E'}\to N'_2$ with an invertible diagonal matrix $E'$, and
- $M_2 \twoheadleftarrow M_2$.
This can be illustrated as follows: $$M_1 \xleftarrow{\text{backward}} M'_1
\xrightarrow{\text{diagonal}} N'_1
\xrightarrow{\text{forward}} M_\varphi
\xleftarrow{\text{forward}} N'_2
\xleftarrow{\text{diagonal}} M'_2
\xrightarrow{\text{backward}} M_2$$
\[thm:Field\] Every two equivalent trim wta $M$ and $N$ over the field ${\cal A}$ can be joined by a chain of simulations. Moreover, there exists a minimal wta $M_{{\ensuremath{\lVert M \rVert}}}$ such that $M$ and $N$ both simulate $M_{{\ensuremath{\lVert M \rVert}}}$.
We could have obtained a similar theorem with the help of Theorem \[thm:Joint\] because the finite set $V$ can be obtained as in [@boz91]. The approach in the next section will cover this case.
<span style="font-variant:small-caps;">Noetherian</span> semirings {#sec:Int}
------------------------------------------------------------------
Now, let ${\cal A}$ be a <span style="font-variant:small-caps;">Noetherian</span> semiring. We construct the finite set $V$ as follows. Let $V_0 = \{ \mu'_0(\alpha) \mid
\alpha \in \Sigma_0\}$ and $$V_{i + 1} = V_i \cup \bigl( \{ (v_1 \otimes \dotsm \otimes v_k)
\cdot \mu'_k(\sigma) \mid \sigma \in \Sigma_k, {\ensuremath{v_{1}, \dotsc, v_{k}}} \in V_i\}
\setminus \langle V_i\rangle \bigr)$$ for every $i \in {\ensuremath{\bbbn}}$. Then $$\{ 0 \} \subseteq \langle V_0 \rangle \subseteq \langle V_1 \rangle
\subseteq \dotsb \subseteq \langle V_k \rangle \subseteq \dotsb {}$$ is stationary after finitely many steps because ${\cal A}$ is <span style="font-variant:small-caps;">Noetherian</span>. Thus, let $V = V_k$ for some $k \in {\ensuremath{\bbbn}}$ such that $\langle V_k \rangle = \langle V_{k+1} \rangle$. Clearly, $V$ is finite and has property (i). Trivially, $V \subseteq \{ h_{\mu'}(t)
\mid t \in T_\Sigma \}$, so let $v \in V$ be such that $v = \sum_{i
\in I} (h_\mu(t_i), h_\nu(t_i))$ for some finite index set $I$ and $t_i \in T_\Sigma$ for every $i \in I$. Then $$\begin{aligned}
\Bigl( \sum_{i \in I} h_\mu(t_i) \Bigr) F = \sum_{i \in I} ({\ensuremath{\lVert M \rVert}},
t_i) = \sum_{i \in I} ({\ensuremath{\lVert N \rVert}}, t_i) = \Bigl( \sum_{i \in I}
h_\nu(t_i) \Bigr) G\end{aligned}$$ because ${\ensuremath{\lVert M \rVert}} = {\ensuremath{\lVert N \rVert}}$, which proves property (ii).
\[thm:Noeth\] Let ${\cal A}$ be a <span style="font-variant:small-caps;">Noetherian</span> semiring. For every two equivalent wta $M$ and $N$ over ${\cal A}$, there exists a chain of simulations that join $M$ and $N$. In fact, there exists a single wta that simulates both $M$ and $N$.
Follows from Theorem \[thm:Joint\].
Since ${\ensuremath{\bbbz}}$ forms a <span style="font-variant:small-caps;">Noetherian</span> ring, we obtain the following corollary.
\[cor:Int\] For every two equivalent wta $M$ and $N$ over ${\ensuremath{\bbbz}}$, there exists a chain of simulations that join $M$ and $N$. In fact, there exists a single wta that simulates both $M$ and $N$.
In fact, since $M + N$ uses only finitely many semiring coefficient, it is sufficient that every finitely generated subsemiring of ${\cal
A}$ is contained in a <span style="font-variant:small-caps;">Noetherian</span> subsemiring of ${\cal A}$. Since every finitely generated commutative ring is <span style="font-variant:small-caps;">Noetherian</span> [@lan84 Cor. IV.2.4 & Prop. X.1.4], we obtain the following corollary.
\[cor:Ring\] For every two equivalent wta $M$ and $N$ over the commutative ring ${\cal A}$, there exists a chain of simulations that join $M$ and $N$. In fact, there exists a single wta that simulates both $M$ and $N$.
Natural numbers {#sec:Nat}
---------------
Finally, let ${\cal A} = {\ensuremath{\bbbn}}$ be the semiring of natural numbers. We compute the finite set $V \subseteq {\ensuremath{\bbbn}}^{Q \cup P}$ as follows:
1. Let $V_0 = \{ \mu'_0(\alpha) \mid \alpha \in \Sigma_0\}$ and $i
= 0$.
2. For every $v, v' \in V_i$ such that $v \leq v'$, replace $v'$ by $v' - v$.
3. Set $V_{i + 1} = V_i \cup \bigl( \{ (v_1 \otimes \dotsm \otimes
v_k) \cdot \mu'_k(\sigma) \mid \sigma \in \Sigma_k, {\ensuremath{v_{1}, \dotsc, v_{k}}} \in
V_i\} \setminus \langle V_i\rangle \bigr)$.
4. Until $V_{i + 1} = V_i$, increase $i$ and repeat step 2.
Clearly, this algorithm terminates since every vector can only be replaced by a smaller vector in step 2 and step 3 only adds a finite number of vectors, which after the reduction in step 2 are pairwise incomparable. Moreover, property (i) trivially holds because at termination $V_{i+1} = V_i$ after step 3. Consequently, we only need to prove property (ii). To this end, we first prove that $V \subseteq
\langle \{ h_{\mu'}(t) \mid t \in T_\Sigma \} \rangle_{\mathord{+},
\mathord{-}}$. This is trivially true after step 1 because $\mu'_0(\alpha) = h_{\mu'}(\alpha)$ for every $\alpha \in \Sigma_0$. Clearly, the property is preserved in steps 2 and 3. Finally, property (ii) can now be proved as follows. Let $v \in V$ be such that $v = \sum_{i \in I_1} (h_\mu(t_i), h_\nu(t_i)) - \sum_{i \in I_2}
(h_\mu(t_i), h_\nu(t_i))$ for some finite index sets $I_1$ and $I_2$ and $t_i \in T_\Sigma$ for every $i \in I_1 \cup I_2$. Then $$\begin{aligned}
&\phantom{{}={}} \Bigl( \sum_{i \in I_1} h_\mu(t_i) - \sum_{i \in
I_2} h_\mu(t_i) \Bigr) F = \sum_{i \in I_1} h_\mu(t_i)F - \sum_{i
\in I_2} h_\mu(t_i)F \\
&= \sum_{i \in I_1} ({\ensuremath{\lVert M \rVert}}, t_i) - \sum_{i \in I_2} ({\ensuremath{\lVert M \rVert}}, t_i) =
\sum_{i \in I_1} ({\ensuremath{\lVert N \rVert}}, t_i) - \sum_{i \in I_2} ({\ensuremath{\lVert N \rVert}}, t_i) \\
&= \sum_{i \in I_1} h_\nu(t_i)G - \sum_{i \in I_2} h_\nu(t_i)G =
\Bigl( \sum_{i \in I_1} h_\nu(t_i) - \sum_{i \in I_2} h_\nu(t_i)
\Bigr) G\end{aligned}$$ because ${\ensuremath{\lVert M \rVert}} = {\ensuremath{\lVert N \rVert}}$.
\[cor:Nat\] For every two equivalent wta $M$ and $N$ over ${\ensuremath{\bbbn}}$, there exists a chain of simulations that join $M$ and $N$. In fact, there exists a single wta that simulates both $M$ and $N$.
For all finitely and effectively presented semirings, Theorems \[thm:Field\] and \[thm:Noeth\] and Corollaries \[cor:Ring\] and \[cor:Nat\], also yield decidability of equivalence for $M$ and $N$. Essentially, we run the trivial semi-decidability test for inequality and a search for the wta the simulates both $M$ and $N$ in parallel. We know that either test will eventually return, thus deciding whether $M$ and $N$ are equivalent. Conversely, if equivalence is undecidable, then simulation cannot capture equivalence [@esimal10].
[^1]: Partially supported by grant no. K 75249 from the National Foundation of Hungary for Scientific Research and by the TÁMOP-4.2.2/08/1/2008-0008 program of the Hungarian National Development Agency.
[^2]: Financially supported by the *Ministerio de Educación y Ciencia* (MEC) grant JDCI-2007-760 and the *European Science Foundation* (ESF) short-visit grant 2978 in the activity “Automata: from Mathematics to Applications”.
|
---
abstract: 'We present a detailed analysis of the modulated-carrier quantum phase gate implemented with Wigner crystals of ions confined in Penning traps. We elaborate on a recent scheme, proposed by two of the authors, to engineer two-body interactions between ions in such crystals. We analyze for the first time the situation in which the cyclotron ($\omega_{\mathrm{c}}$) and the crystal rotation ($\omega_{\mathrm{r}}$) frequencies do not fulfill the condition $\omega_{\mathrm{c}} = 2 \omega_{\mathrm{r}}$. It is shown that even in the presence of the magnetic field in the rotating frame the many-body (classical) Hamiltonian describing small oscillations from the ion equilibrium positions can be recast in canonical form. As a consequence, we are able to demonstrate that fast and robust two-qubit gates are achievable within the current experimental limitations. Moreover, we describe a realization of the state-dependent sign-changing dipole forces needed to realize the investigated quantum computing scheme.'
author:
- 'J. D. Baltrusch$^{1,2,3}$, A. Negretti$^1$, J. M. Taylor$^4$ and T. Calarco$^{1,5}$'
bibliography:
- 'IonPaperBib.bib'
title: Fast and robust quantum computation with ionic Wigner crystals
---
Introduction {#sec:intro}
============
Despite the huge experimental progress to cool, trap, and manipulate single particles such as atoms and molecules at the quantum level, the way to build up a quantum computing hardware working with several hundreds of quantum bits (qubits) in a coherent and controllable manner is still long. By means of quantum optimal control techniques it is possible, at least theoretically, to perform one- and two-qubit quantum gates with fidelities above the demanding thresholds of fault-tolerant quantum computation [@Charron2002; @Palao2002; @Calarco2004; @Treutlein2006; @Charron2006; @Montangero2007; @Motzoi2009; @Safaei2009; @Poulsen2010]. These thresholds fix an error between 0.01% to fractions of a percent [@Steane2003; @Knill2005]. Up to now, only with cold trapped ions quantum gates with a fidelity of 99.3% have been experimentally demonstrated [@Leibfried2003; @Benhelm2008], which is not too far from the aforementioned thresholds. Similar fidelities have been also obtained for small quantum algorithms [@Gulde2003; @Chiaverini2005].
Nowadays, however, most of the experimental efforts of the atomic and molecular physics community are concentrated in the design and fabrication of microtraps, both for ions [@Kielpinski2002; @Stick2006] and neutral atoms [@Folman2000; @Reichel1999]. Even though these efforts are important, significant technical issues related to the miniaturization and trapping methodologies arise when scaling to many particles, and therefore new strategies have to be devised. A possible solution to the problem is the separation between the qubits used as quantum memory and the ones employed to process the information [@Oskin2002] or, alternatively, the exploitation of quantum distributed networks [@Cirac1997]. Another approach, instead, consists in the use of collective states of atomic ensembles with a multilevel internal structure as qubits [@Brion2007].
Apart from these technological efforts and alternative solutions, nobody can yet say which of the various physical implementations will be the successful one. It is fair to say, however, that ions represent a good candidate to implement a multi-qubit quantum processor. Indeed, two-qubit gates with ions can be realized in about few tens of $\mu$s [@Steane2000; @Chen2006], and qubits stored in internal electronic degrees of freedom of an ion have coherence lifetimes ranging from 1 s to 100 s or more [@Chen2006].
Coulomb — also named classical Wigner — crystals confined in Penning traps are natural candidates for a quantum memory, since the separation among ions, about 10 $\mu$m, allows to individually manipulate their internal degrees of freedom. Such a trap scheme uses static electric fields to confine charge particles in the axial direction (the $z$ axis in Fig. \[fig:artistsview\]), whereas the radial confinement is provided by a strong uniform magnetic field along the axial direction. Currently, Penning traps allow to trap up to $10^8$ ions [@Anderegg2010]. An appropriate choice of the trap parameters (e.g., tight axial confinement) allows the ionic ensemble to crystallize in a two-dimensional (2D) hexagonal lattice configuration with an inter-particle spacing on the order of tens of $\mu$m [@Mitchell1998], and therefore to manipulate a large number of qubits without specific micro-trap designs. The high phonon mode density, however, does not permit to resolve single modes for sideband cooling. Hence, Doppler and sympathetic cooling are the most natural techniques to be employed; we also note that Sisyphus cooling might be an alternative methodology [@Wineland1992]. Current experiments, however, performed with Doppler cooling, can reach temperatures of few mK [@Mitchell2001], that is, a high thermal occupation number distribution of phonon modes. Nonetheless, efficient quantum computation and production of small cluster states are theoretically possible [@Porras2006; @Taylor2008], and recently full control of the qubit Bloch vector with $\sim$99.85 % fidelity for Rabi flopping has been experimentally demonstrated [@Biercuk2009].
The two-qubit gate scheme considered in the proposals of Refs. [@Porras2006; @Taylor2008] is based on the so called “pushing gate” (or its variant, the modulated-carrier gate), where a spatially inhomogeneous laser field together with an appropriate combination of polarizations and frequencies induces a state-dependent dipole force on two nearest neighbours of the 2D Coulomb crystal (see Fig. \[fig:artistsview\]). Depending on the configuration of lasers and polarizations the displacements of the ions away from their equilibrium positions can be either perpendicular to the plane of the crystal [@Porras2006] or along the in-plane separation of the ions [@Cirac2000; @Calarco2001; @Sasura2003; @Taylor2008]. The coupling between these displacements, mediated by phonons, yields entanglement of the internal states (qubits) of the ions, that is, the desired quantum gate between ions.
![(Color online). Two-dimensional Coulomb crystals of ions in a Penning trap rotating at frequency $\omega_{\mathrm r}$. To manipulate the internal states of the ions, laser beams can address single sites or multiple ions.[]{data-label="fig:artistsview"}](fig01.eps){width="2.4in"}
In addition to the confinement, the radial electric and axial magnetic fields induce a drift that causes in-plane rotation of the crystal (see Fig. \[fig:artistsview\]), whose frequency $\omega_{\mathrm{r}}/(2\pi)$ is typically on the order of few tens of kHz [@Mitchell1998]. There are two possible solutions to our quantum hardware design: either we use a co-rotating (with the crystal) laser beam in order to realize the desired two-qubit quantum gate, or we have to perform the gate in a time, $\tau_{\mathrm{g}}$, such that the crystal rotation has a negligible effect on the gate operation. The latter solution translates in the condition $\omega_{\rm r}\tau_{\mathrm{g}}/(2\pi) \ll 1$. Such a requirement is instrumental because, in order to accumulate the necessary two-ion phase for the quantum gate we aim to implement, the ions have to experience the applied light force for the entire gate operation, or else, the required phase would be achieved only partially.
While the former solution applies for all rotation frequencies, but relies on a more sophisticated experimental setup, the latter restricts the range of possible values of $\omega_{\mathrm{r}}$. Thus, in both proposals [@Porras2006; @Taylor2008], where the rotation and cyclotron frequencies fulfill $2\,\omega_{\rm r} = \omega_{\rm c}$, the aforementioned condition is satisfied when $\tau_{\mathrm{g}}$ is on the order of ns, whereas the modulated-carrier gate of Ref. [@Taylor2008] had $\tau_{\mathrm{g}}=$ 5 $\mu$s. Given the above, such a proposal requires a co-rotating laser beam. Thus, by maintaining $2\,\omega_{\rm r} = \omega_{\rm c}$ one should reduce $\omega_{\rm c}$. This approach, however, would not help since the smaller the cyclotron frequency is, the longer the gate operation. Instead, if we abandon the assumption $2\,\omega_{\rm r} = \omega_{\rm c}$ and look at moderate rotation frequencies, at the expenses of possible large modulations of the force, we are able to fulfill $\omega_{\rm r}\tau_{\mathrm{g}}/(2\pi) \ll 1$. Additionally, low rotating frequencies result in low densities and large inter-particle spacing, and therefore in an easier way to address the trapped ions with a laser field.
Thus, the main goal of this work is to analyze this regime and, at the same time, to perform robust two-qubit gates within a range of experimentally achievable temperatures.
In the following we shall we present the general theory of the modulated-carrier push two-qubit gate (Sec. \[sec:modcargate\]) with details that were briefly mentioned in Ref. [@Taylor2008]. Subsequently, in Sec. \[sec:coupling\], we investigate the situation in the presence of the magnetic field in the rotating frame of reference and the relative gate performance. In section \[sec:force\] we describe how to physically realize the state-dependent force required for the proposed quantum processor, and Sec. \[sec:conc\] summarizes our results and provides some future prospectives.
Modulated-carrier gate {#sec:modcargate}
======================
In the following we make the approximation that the Wigner crystal is a rigid body, which is a good approximation in the magnetohydrodynamic regime (one component plasma) or at equilibrium [@Dubin1999]. Hence, in a rotating frame, the Hamiltonian of a crystal with $N$ ions written in cylindrical coordinates \[$\vec{r}\equiv(r,\theta,z)$\] is given by [@Dubin1999]
$$\begin{aligned}
H_R(\omega) &=& \sum_{k=1}^N \left\{\frac{p^2_{r_k} + p^2_{z_k}}{2 m}
+ \frac{[p_{\theta_k} - m(\omega_{\mathrm c} - 2 \omega)r_k^2/2]^2}{2 m r_k^2}
\right\}
\nonumber\\
&+& \sum_{k=1}^N \left\{\Upsilon(2
z_k^2 - r_k^2) + \frac{m}{2}\omega(\omega_{\mathrm c} - \omega)r_k^2\right\}
+ V_{\mathrm{c}},\nonumber\\
\label{eq:Homega}\end{aligned}$$
with $m$ being the mass of the ion and
$$\begin{aligned}
V_{\mathrm{c}} = \frac{e^2}{4 \pi \epsilon_0} \sum_{k<j}\frac{1}{\vert
\vec{r}_k - \vec{r}_j\vert}.\end{aligned}$$
Here $\Upsilon$ is a parameter describing the trap geometry and applied voltage on the electrodes [@Gosh1995], $\epsilon_0$ the vacuum permittivity, $e$ the electron charge, and $\omega_{\mathrm c} = e B / m$ is the cyclotron frequency. We see, from the first line of Eq. (\[eq:Homega\]), that there exists a special rotating frame, $\omega = \omega_{\mathrm c}/ 2$, such that the minimal coupling disappears, and, in this section, we shall consider such a frame of reference together with $\omega_{\mathrm r} = \omega_{\mathrm c}/ 2$ (i.e., the frame of reference coincides with the crystal). We note, that with “minimal coupling” we refer to the interaction $\vec{p} \cdot\vec{A}$. Such terminology is typically used in quantum field theory [@Itzykson1980].
Finally, it is worth to remind that the gate we aim to accomplish realizes the true table $\ket{\epsilon_1,\epsilon_2}\rightarrow e^{i\theta\epsilon_1\epsilon_2}
\ket{\epsilon_1,\epsilon_2}$ with $\epsilon_{1,2} = 0,1$ and $\theta = \theta_{00} - \theta_{01} - \theta_{10} + \theta_{11}$ [@Calarco2001; @Sasura2003]. Specifically, we are interested in a phase gate with $\theta=\pi$, which, up to additional single-qubit rotations, is tantamount to a two-qubit controlled NOT gate [@Chen2006].
Normal modes and canonical quantization {#sec:phonons}
---------------------------------------
The Hamiltonian (\[eq:Homega\]) in cartesian coordinates \[$\vec{r}\equiv(x,y,z)$\] reduces to
$$\begin{aligned}
H_R\left(\frac{\omega_{\mathrm c}}{2}\right) \!=\! \sum_{k=1}^N \left\{\frac{\vec{p}^{\,2}_{k}}{2 m}
+ \frac{m}{2}\left[ \omega_z^2 z_{k}^2 + \omega_{xy}^2(x_{k}^2 +
y_{k}^2)\right] \right\} + V_{\mathrm{c}}, \nonumber\\
\label{eq:Homega-special}\end{aligned}$$
where $\omega_z = \sqrt{4\Upsilon / m}$ is the axial frequency, and $\omega_{xy} = 1/2 (\omega_{\mathrm c}^2 - 2\omega_z^2)^{1/2}$ the in-plane one.
By performing a Taylor expansion of the potential up to second order around the stable equilibrium configuration, obtained by minimizing the total crystal energy, we can express the Hamiltonian in the new coordinates $q_{n,\eta} \equiv \eta_{n} - \eta_{n}^0$, that is, the displacements from the equilibrium positions. Hence, it is possible to determine an orthogonal transformation $M$ such that[^1]
$$H_R\left(\frac{\omega_{\mathrm c}}{2}\right) \approx \sum_{n,\eta} \left\{\frac{P_{n,\eta}^2}{2m}
+ \frac{m}{2} \omega_{n,\eta}^2 Q_{n,\eta}^2\right\}$$
with $Q_{n,\eta} = \sum_{k,\mu} M_{n,\eta;k,\mu} q_{k,\mu} [ = M \bf{q}]$, and $P \equiv p$.
Now, we perform the canonical quantization and we introduce the creation (annihilation) operators ${{\hat a}^{\dagger}}_K$ (${\hat a}_K$) for each mode $K\equiv (n,\eta)$, along with the harmonic oscillator ground state size $\alpha_K = \sqrt{\hbar/m \omega_K}$. Hence, the (phononic) Hamiltonian operator reads
$$\hat H_R = \sum_{K} \hbar \omega_{K} ({{\hat a}^{\dagger}}_{K} {\hat a}_{K}^{\phantom{\dagger}} + 1/2),$$
where for the sake of simplicity we drop $\left(\frac{\omega_{\mathrm c}}{2}\right)$ in $\hat H_R$.
Adiabatic and oscillatory quantum gates {#sec:adiafastgate}
---------------------------------------
Let us consider a spatially inhomogeneous laser field appropriately detuned from the internal states such that it produces a state-dependent displacement of the ions. Then, the matter-field interaction, in the electric dipole approximation, becomes
$$\begin{aligned}
\!\hat V & = & \sum_{j=1}^N [\hat{\vec{q}}_j \cdot \vec{f}_j(t)] \hat{\sigma}^z_j
= \sum_K \frac{\alpha_K}{\sqrt{2}} \hat{f}_K(t) (\ad_K + \a_K),\end{aligned}$$
where $\vec{f}_j$ is the three dimensional force due to the gradient in the laser intensity, and $\hat{\sigma}^z_j$ is the $z$ Pauli matrix. Here the following relation for the displacement coordinate operator
$$\hat{q}_K = \sum_{K^{\prime}} M_{K^{\prime}; K} \hat{Q}_{K^{\prime}}
=\sum_{K^{\prime}} M_{K^{\prime}; K} \frac{\alpha_K}{\sqrt{2}} (\a_K +
\ad_K)
\label{eq:normalmode}$$
has been used. Thus, we have \[$K\equiv (j,\mu)$\]
$$\begin{aligned}
\hat{f}_K(t) = \sum_{j^{\prime},\mu^{\prime}} M_{K;j^{\prime},\mu^{\prime}}[\vec{f}_{j^{\prime}}(t)]_{\mu^{\prime}}\hat{\sigma}^z_{j^{\prime}},\end{aligned}$$
where $[\vec{f}_{j^{\prime}}(t)]_{\mu^{\prime}}$ is the $\mu^{\prime}=x,y,z$ component of the three-dimensional vector $\vec{f}_{j^{\prime}}(t)$. Hence, the full problem reduces to $3 N$ independent, driven oscillators.
When the temporal profile of the force fulfills the condition ${\rm lim}_{t \rightarrow \pm \infty} f(t) = 0$, the unitary time evolution operator is given by $\hat U_K(t) = e^{-i \phi_K(t)} \exp(\beta_K \ad_K - \beta_K^* \a_K)\exp(-i\omega_Kt\,\ad_K\a_K)$, where $\phi_K$ and $\beta_K$ satisfy the differential equations [@Garcia2003; @Garcia2005]
$$\dot{\beta}_K = -i \omega_K \beta_K + i \frac{\alpha_K}{\hbar
\sqrt{2}} \hat{f}_K(t), \; \dot{\phi}_K = \frac{\alpha_K}{\hbar \sqrt{2}}
\hat{f}_K(t) {\mathrm{Re}}[\beta_K(t)].
\label{eq:difeq-betaphi}$$
Given that, let us consider the adiabatic regime regime where $\hat{f}_K(t)$ varies slowly with respect to $\omega_K$ [@Calarco2001]. Adiabatic elimination, by taking $\dot{\beta}_K \rightarrow 0$, yields
$$\begin{aligned}
\beta_K \approx \frac{\alpha_K \hat{f}_K(t) }{\hbar \omega_K \sqrt{2}}, \qquad
\dot\phi_K \approx \frac{\alpha_K^2 \hat{f}_K^2(t)}{2 \hbar^2 \omega_K}.\end{aligned}$$
Thus, the displacement of a normal mode $K$ induced by the gate is proportional to the force applied, and can be made zero independent of the initial phonon state by starting and ending with zero force. This eliminates any potential error due to entanglement between phonons and the internal states of the ions. Similarly, the overall phase accumulated $\sum_K \phi_K(\tau_{\mathrm{g}})$ does not depend on the initial phonon state. However, for a gate occurring over a time interval $[0,\tau_{\mathrm{g}}]$, the final qubit state has applied $\exp(-i \sum_{nj}\phi_{nj}\hat\sigma^z_n\hat\sigma^z_j)$, where the two-particle phases arise from
$$\hat f_K^2(t) = \sum_{j,n;\mu,\eta} M_{K;j,\mu} M_{K;n,\eta}
[\vec{f}_j(t)]_{\mu}[\vec{f}_n(t)]_{\eta} \hat\sigma^z_j \hat\sigma^z_n.$$
Thus, the two-particle phase is given by
$$\phi_{nj} = \sum_{\mu,\eta} S^{(nj)}_{\mu\eta}
\int_0^{\tau_{\mathrm{g}}}\! dt\, [\vec{f}_j(t)]_{\mu}[\vec{f}_n(t)]_{\eta},
\label{eq:adiabatic}$$
where the term outside the integral is a shape independent form factor, whose specific form is given by
$$S^{(nj)}_{\mu\eta} = \sum_K \frac{\alpha_{K}^2}{2 \hbar^2\omega_{K}} M_{K;j,\mu} M_{K;n,\eta}.
\label{eq:S-adiabatic}$$
Hence, we can think about (\[eq:adiabatic\]) as a convolution of the forces on the two particles, modified by the form factor representative of the characteristic oscillator variance over its frequency, which is overall proportional to $\omega_K^{-1}$.
Now, let us consider a scheme with a force $f(t) \rightarrow \cos(\nu t) f(t)$, where the carrier frequency $\nu$ must be much larger than the modes of frequency $\omega_K$ that are coupled to the force (this averages out any net displacement). If the modulation $f(t)$ is slow as compared to $\nu$ (but with no restriction with respect to $\omega_K$), we can perform a similar adiabatic elimination as above, and get a gate with the same desirable properties that can operate non-trivially on arbitrarily in-plane vibrational modes at very high temperatures.
For adiabatic elimination with respect to $\nu$, we choose the Ansatz $\beta_K = \beta_K^+ e^{i \nu t} + \beta_K^- e^{-i \nu t}$ for each mode. By inserting this Ansatz into the differential equation (\[eq:difeq-betaphi\]) we obtain
$$\begin{aligned}
\dot{\beta}_K^+ & = & e^{-2 i \nu t}\left[ i \frac{\alpha_K}{2 \sqrt{2}\hbar}
\hat f_K(t) - \dot{\beta}_K^- - i(\omega_K - \nu) \beta_K^-\right]
\nonumber \\
& & \ \ + i \frac{\alpha_K}{2\sqrt{2}\hbar} \hat f_K(t) -i(\omega_K + \nu) \beta_K^+. \end{aligned}$$
Separate adiabatic elimination of $\beta_K^-$ and $\beta_K^+$ yields $\beta_K^{\pm} = \alpha_K \hat f_K(t)/[2 \sqrt{2} \hbar(\omega_K \pm
\nu)]$. As before, in the pure adiabatic regime, we find that the displacement of a normal mode induced by the gate is proportional to the force applied. Again, it can be made zero independent of the initial phonon state by starting and ending with zero force, and therefore eliminating any potential error due to entanglement between phonons and the internal states of the ions.
Now, we examine the two-particle phase induced in this new scenario. The time evolution of the phase is governed by [@Taylor2008]
$$\dot{\phi}_K = \frac{\alpha_K^2}{2 \hbar^2}
\frac{\omega_K}{(\omega_K^2-\nu^2)}
\cos^2(\nu t) \hat f_K^2(t),$$
where the quickly varying component $\cos^2(\nu t)$ can be replaced with $1/2$. As described in the adiabatic regime, the overall phase accumulated $\sum_K \phi_K(\tau_{\mathrm{g}})$, for a gate occurring over a time interval $[0,\tau_{\mathrm{g}}]$, does not depend on the phonon initial state. In this case the pulse-shape independent form factor is given by [@Taylor2008]
$$S^{(nj)}_{\mu\eta} = -\sum_K \frac{\alpha_{K}^2\omega_K}
{4 \hbar^2(\nu^2 - \omega_K^2)} M_{K;j,\mu} M_{K;n,\eta}.
\label{eq:Sfastall}$$
Performing a Taylor expansion in $1/\nu^2$ the first term is proportional to $\sum_K M_{K;j,\mu} M_{K;n,\eta}= \delta_{j,n}\delta_{\mu,\eta}$ ($\delta_{j,n}$ indicates the Kronecker symbol). This follows from the fact that $M$ is an orthogonal matrix. Physically, this arises due to the coherent averaging of in-phase oscillating ions without any virtual excitation of phonons—accordingly, no two-body phase should be expected. The second term of the expansion is non-zero and yields
$$\tilde{S}^{(nj)}_{\mu\eta} = - \frac{1}{4 \hbar m \nu^4}
\sum_K\omega_K^2 M_{K;j,\mu} M_{K;n,\eta}
+ O\left(\nu^{-6}\right).
\label{Sfast}$$
Compared to adiabatic push gates, the modulated-carrier gate is inverted in sign and it is multiplied (in phase) by a factor $(\omega_K/\nu)^4/2$ \[see Eq. (\[eq:S-adiabatic\])\]. In the case of a lateral operating modulated-carrier gate with $\omega_{xy}\ll\nu\ll\omega_z$, the accumulated phase is enhanced by a factor $(\omega_z/\nu)^4/2$ with respect to an adiabatic push gate with a force moving the ions in the axial ($z$) direction for the same laser parameters. Given that, the gate time needed to perform a $\pi$-phase gate is reduced. In the opposite case, that is, for an adiabatic in-plane push gate ($\omega_K \sim \omega_{xy}$), and for the same laser parameters, the lateral modulated-carrier gate is reduced in phase, and therefore a longer $\tau_{\mathrm{g}}$ is required. Thus, compared to the proposal of Ref. [@Porras2006], where the push gate operates in the axial direction, our modulated-carrier gate working with in-plane modes yields a larger two-ion phase for a given set of laser parameters, and therefore it enables to perform a larger number of quantum gates within the coherence time of the system.
Modulated-carrier gate with minimal coupling {#sec:coupling}
============================================
In this section we analyze the situation where $\omega_{\mathrm r} \ne
\omega_{\mathrm c}/2$, for which we have three reasonable choices for the rotating frame of reference:
- $F_1$ coincides with the lab frame, where the equilibrium positions of the ions in the crystal are time-dependent and the minimal coupling does not vanish;
- $F_2$ rotates with a frequency $\omega = \omega_{\mathrm c}/2$, as in the previous section, where the minimal coupling vanishes, but the equilibrium positions are time-dependent;
- $F_3$ rotates with a frequency $\omega = \omega_{\mathrm r}$, where equilibrium positions are time-independent, but the minimal coupling does not vanish.
Equilibrium configuration of the crystal {#sec:equpo}
----------------------------------------
Let us discuss which of the frames of reference $F_{1,2,3}$ is more suitable to numerically determine the equilibrium configuration of the system for a fixed (a priori) value of total canonical angular momentum $P_{\theta}
$[^2]. Since we are not concerned with relativistic velocities, the electromagnetic fields involved in the problem are the same in all frames of reference. Consequently, the angular momentum of an ion in a frame rotating with uniform angular velocity with respect to the (inertial) laboratory frame coincides with the one in the latter [@Fasano2006]. This conclusion allow us to find the equilibrium configuration of the crystal, for a given value of $P_{\theta}$, by choosing a frame of reference rotating with angular velocity $\omega = \omega_{\mathrm c}/2$ (the frame $F_2$ in the above outlined list) in such a way that the coordinate systems at the initial time $t=0$ of $F_2$ and $F_3$ do coincide. Such a choice simplifies the numerical minimization procedure, because the minimal coupling in the (classical) Hamiltonian vanishes. We underscore, however, that $F_2$ is utilized only at time $t=0$ for the determination of the equilibrium configuration of the crystal. Instead, for times $t>0$ we use $F_3$, where the equilibrium positions are time-independent. With such a choice the numerical effort in order to assess the gate performance is significantly reduced.
Besides this, we also note that not all rotation frequencies $\omega_{\mathrm{r}}$ of the crystal allow to have a stable configuration, that is, ions confined within a well-defined spatial region. Indeed, by rewriting the addend of the second sum in Eq. (\[eq:Homega\]) as
$$\begin{aligned}
\Upsilon(2z_k^2 - r_k^2)
+ \frac{m}{2}\omega_{\mathrm{r}}(\omega_{\mathrm c} - \omega_{\mathrm{r}})r_k^2
= \frac{m \omega_z^2}{2}(z_k^2 + \beta r^2_k)\nonumber\\\end{aligned}$$
we see that the potential is confining if and only if $\beta$ is positive. Here the anisotropy parameter $\beta$ is defined as
= - = ( 1 - ) - , \[eq:beta\] where $\alpha_z = \omega_z/\omega_{\mathrm{c}}$. Importantly, $\beta$ relies only on $\alpha_z$ and the ratio $\omega_{\mathrm{r}}/\omega_{\mathrm{c}}$. Thus the range of admissible frequencies is: $\omega_{\mathrm{m}} <
\omega_{\mathrm{r}} < \omega_{\mathrm{c}} - \omega_{\mathrm{m}}$, where $\omega_{\mathrm{m}} = \omega_{\mathrm{c}}/2 - \omega_{xy}$ is the magneton frequency [@Dubin1999]. Of course, the admissible regime is also constrained by the condition $\alpha_z <1/\sqrt{2}$. In order to access lower rotation frequencies, the trap parameters might be changed by increasing $\omega_{xy}$, that is, by lowering $\omega_z$. Attention has to be paid, however, when $\omega_{\mathrm{c}}$ and $\omega_z$ are changed, since due to such a manipulation different structural phase transitions may occur. In particular, we are interested in the limit $\beta\ll 1$, where a 2D hexagonal lattice structure appears [@Dubin1999].
Quadratic expansion of the Hamiltonian {#sec:quadraticExp}
--------------------------------------
Let us introduce the typical scale of length $\ell_s$, momentum $p_s$, and energy $E_s$ in our problem:
$$\begin{aligned}
\ell_s = \left(
\frac{e^2}{4\pi\epsilon_0 m \omega^2_{\mathrm{c}}}
\right)^{\frac{1}{3}}
\,\,\,\,\, p_s = \ell_s m \omega_{\mathrm{c}}
\,\,\,\,\, E_s = \frac{e^2}{4\pi\epsilon_0\ell_s}.\end{aligned}$$
Then, the Hamiltonian (\[eq:Homega\]) in cartesian coordinates becomes
$$\begin{aligned}
H_R(\omega) &=& \frac{1}{2}\sum_{k=1}^N \left[p^2_{x_k} + p^2_{y_k} + p^2_{z_k}
+(y_k p_{x_k} - x_k p_{y_k})\times\right.
\nonumber\\
&\times& \left.(1+2\alpha)\right]
+\frac{1}{4}\sum_{k=1}^N\left[\alpha^2_z(2 z_k^2 - r_k^2) + \frac{r^2_k}{2}\right] \nonumber\\
&+& \frac{1}{2}\lim_{\epsilon \rightarrow 0}\sum_{k,j=1}^N\frac{1 - \delta_{k,j}}{\vert\vec{r}_k -
\vec{r}_j + \epsilon\vert},
\label{eq:Homega-dl}\end{aligned}$$
where the substitutions $H_R(\omega)\rightarrow H_R(\omega)/E_s$, $(r, z)\rightarrow (r, z)/\ell_s$, $(p_x, p_y, p_z) \rightarrow (p_x,
p_y, p_z)/p_s$, and $\alpha = \omega/\omega_{\mathrm{c}}$ have been introduced. The expression of the Coulomb potential, third line in Eq. (\[eq:Homega-dl\]), allows to obtain more compact formulae later in the present section.
Next, given the equilibrium configuration $(\vec{r}_0,\vec{0})$ of each ion, we expand the Hamiltonian (\[eq:Homega-dl\]) to second order in the spatial displacement $\mathbf{q} = \mathbf{r} -
\mathbf{r}_0$ and $\mathbf{p}$ around zero, namely
$$\begin{aligned}
H_R({\mathbf p},{\mathbf q}) \simeq H_R({\mathbf 0},{\mathbf r}_0)
+ \frac{1}{2}\mathbf{d} \tilde{H}_R\mathbf{d}^{\mathsf{T}},
\label{eq:quadExp} \end{aligned}$$
where $\mathbf{d}^{\mathsf{T}}$ is the transpose of the row vector $\mathbf{d} \equiv
(q_{1,x},p_{1,x},q_{1,y},p_{1,y},\dots,q_{N,z},p_{N,z})$, and $\tilde{H}_R=\tilde{H}_R({\mathbf 0},{\mathbf r}_0)$ is the Hessian matrix. Its non-zero matrix elements are given by:
$$\begin{aligned}
\frac{\partial^2 H_R}{\partial p_{\eta_k}^2} = 1,\qquad
\frac{\partial^2 H_R}{\partial p_{x_k}\partial y_k} = -
\frac{\partial^2 H_R}{\partial p_{y_k}\partial x_k} = \alpha + \frac{1}{2},
\nonumber\end{aligned}$$
$$\begin{aligned}
\frac{\partial^2 H_R}{\partial \eta_k\partial \mu_j} &=&
\left[
1 - 2 \alpha_z^2 + (6 \alpha_z^2 - 1) \delta_{\eta,z}
\right]\frac{\delta_{\eta,\mu}\delta_{k,j}}{4}\nonumber\\
&+&\lim_{\epsilon \rightarrow 0}\sum_{s=1}^N
\frac{(1 - \delta_{k,s})[\delta_{s,j} + (1 -
\delta_{s,j})\delta_{|k-j|,0}]}
{\vert \vec{r}_k - \vec{r}_s + \epsilon\vert^3}\times\nonumber\\
&\times&(-1)^{\delta_{k,j}}\left[
\delta_{\eta,\mu}
-3
\frac{(\eta_k - \eta_s)(\mu_k - \mu_s)}{\vert\vec{r}_k - \vec{r}_s + \epsilon
\vert^2}
\right],
\nonumber\end{aligned}$$
where $\eta,\mu = x,y,z$, and $k,j=1,\dots,N$.
Symplectic diagonalization and canonical quantization {#sec:symplettic}
-----------------------------------------------------
Hereafter we utilize the frame $F_3$ that rotates at the frequency $\omega_{\mathrm{r}}$. Hence, we are allowed to drop $H_R({\mathbf 0},{\mathbf r}_0)$ in Eq. (\[eq:quadExp\]) and the full Hamiltonian reduces to the $6N\times6N$-matrix $H_R(\omega_{\mathrm{r}})
= \mathbf{d} \tilde{H}_R \mathbf{d}^{\mathsf{T}}/2$.
In order to perform the canonical quantization, we have first to transform the classical Hamiltonian $H_R(\omega_{\mathrm{r}})$ in canonical form. A transformation $S: ({\mathbf p},{\mathbf q})\rightarrow({\mathbf P}, {\mathbf Q})$ is canonical when the condition $S \mathbb J S^T = \mathbb J$ is satisfied, where $\mathbb J = i\bigoplus_{i=1}^{3N} \hat\sigma^y$ [@Fasano2006]. Since the Hessian matrix $\tilde{H}_R$ is real and positive definite, Williamson’s theorem [@Williamson1936] guarantees that
S \_R S\^ = W = (
[ccccc]{} \_1 & & & &\
& \_1 & & &\
& & …& &\
& & & \_[3N]{} &\
& & & & \_[3N]{}
), where $\omega_k$ are real and positive numbers $\forall k=1,\dots,3N$, and $W$ is called the “Williamson form” of $\tilde{H}_R$.
Given that, we can recast the classical Hamiltonian as
H\_R(\_) = \_[k=1]{}\^[3N]{}\_k \_[2k-1]{}\^2 + \_[k=1]{}\^[3N]{}\_k \_[2k]{}\^2, where the new coordinates are determinated by the transformation $\boldsymbol{\Lambda}^{\mathsf{T}} = (S^{-1})^{\mathsf{T}}
\mathbf{d}^{\mathsf{T}}$. For the sake of simplicity, hereafter, we use the definitions $Q_k := \Lambda_{2k-1}$ and $P_k := \Lambda_{2k}$ $\forall k=1,\dots,3N$. Thus, the Hamiltonian reduces to
H\_R(\_) = \_[k=1]{}\^[3N]{}\_k (Q\^2\_k + P\^2\_k), that is, a sum of uncoupled harmonic oscillators.
Similarly to Sec. \[sec:phonons\], we perform the canonical quantization by promoting $Q_k, P_k$ to operators such that $[\hat Q_k,\hat P_s] = i\delta_{k,s}$. Besides this, we introduce the operators $\hat a_k = (\hat Q_k + i \hat P_k)/\sqrt{2}$, $\hat a_k^\dag = (\hat Q_k - i \hat P_k)/\sqrt{2}$ with $[\hat a_k,\hat a_s^\dag] = \delta_{k,s}$. Hence, the quantized Hamiltonian is simply given by
H\_R(\_) = \_[k=1]{}\^[3N]{}\_k(a\_k\^a\_k + ), and we note that the eigenvalues $\omega_k$ are dimensionless.
Finally, we rewrite the coupling between the ions and the inhomogeneous laser field. The displacement of the ion from its equilibrium position can be written as
d\_[j]{} = \_[k=1]{}\^[3N]{} A\_[k,j]{}\^\* a\_k + A\_[k,j]{} a\_k\^with $A_{kj}=S_{2k-1,j} + i S_{2k,j}$, and where $j$ is an odd integer \[see the definition of the vector $\mathbf{d}$ after Eq. (\[eq:quadExp\])\]. Then, the matter-field interaction has the following expression
$$\begin{aligned}
\hat V \!& = &\! \sum_{j=1}^N [\vec{q}_j \cdot \vec{f}_j(t)] \hat\sigma^z_j
\!= \! \sum_{j=0}^{N-1} \hat\sigma^z_{j+1} \sum_{n=1}^3 \mathcal{F}_{3j+n}(t) \hat d_{2(n+3j)-1} \nonumber\\
& = & \!\sum_{k=1}^{3N} \alpha_k^* \hat a_k + \alpha_k \hat a_k^\dag, \end{aligned}$$
where $\boldsymbol{\mathcal{F}}=(f_{1,x},f_{1,y},f_{1,z},\dots,f_{N,x},f_{N,y},f_{N,z})$, and
\_k = \_[j=0]{}\^[N-1]{} \^z\_[j+1]{} \_[n=1]{}\^3 \_[3j+n]{}(t)A\_[k,2(n+3j)-1]{}. \[eq:alpha\] Thus the full Hamiltonian is: $\hat H = \hat H_R(\omega_{\mathrm{r}}) + \hat V = \sum_k \hat H_k$, where $\hat H_k = \omega_k\left(\hat a_k^\dag \hat a_k + \frac{1}{2}\right) + \alpha_k^* \hat a_k + \alpha_k \hat a_k^\dag$.
Two-qubit phase gate {#sec:phasegate}
--------------------
The time evolution of a phonon mode state, governed by the Hamiltonian $\hat H_k$, and a generic two-qubit state is
= e\^[-i\_k(t)]{}\[\_k(t)\]e\^[-iH\_k\^0]{}, \[eq:Ut\] where $\hat{\mathfrak{D}}[\beta_k(t)]$ is the displacement operator [@Gardiner2004], $\beta_k(t) = -i \int_0^t\mathrm{d}s \alpha_k(s) e^{i\omega_k(s-t)}$, and $\hat H_k^0 = \omega_k\left(\hat a_k^\dag \hat a_k + \frac{1}{2}\right)$.
In order to disentangle the external dynamics due to the phonons and the internal dynamics of the qubit states at the end of the gate operation, $t=\tau_{\mathrm{g}}$, the following condition has to be satisfied [@Garcia2005]
$$\begin{aligned}
\mathcal{I}_k = \frac{1}{\sqrt{\omega_k}}
\int_0^{\tau_{\mathrm{g}}}\mathrm{d} t \,e^{i\omega_k t}\alpha_k(t) = 0
\qquad \forall k.
\label{eq:condforce}\end{aligned}$$
This condition, however, is more general than the adiabatic elimination we performed in Sec. \[sec:adiafastgate\], whose aim was to highlight the difference in the accumulated two-particle phases among the most common quantum gate schemes based on pushing forces with off-resonant lasers.
The necessary lateral force on the $j$-th and $k$-th ion, $\vert\vec{f}_j \vert= \vert\vec{f}_k \vert= \mathcal{A}_P\hbar\omega_{xy}\cos(\nu t)
e^{-t^2/\tau^2_{\mathrm{g}}}/\vert \vec{r}_j^{\,0}-\vec{r}_k^{\,0}\vert$, is determined by setting the dimensionless parameter $\mathcal{A}_P$ to achieve a $\pi$ phase between the chosen pair of qubits. Then the fidelity is given by
$$\begin{aligned}
\!\!\!\!\!F = \min_{\Phi^{\prime}_{\mathrm{qbit}}}\left\{
\mathrm{Tr}_{\mathrm{ph}}\left[\bra{\Phi^{\prime}_{\mathrm{qbit}}}
\hat{\mathfrak{U}}(t)\left(\hat\rho_T(0)\otimes\ket{\Phi_{\mathrm{qbit}}}
\bra{\Phi_{\mathrm{qbit}}}\right)\hat{\mathfrak{U}}^{\dagger}(t)\ket{\Phi^{\prime}_{\mathrm{qbit}}}\right]
\right\} =
\min_{\pm}\prod_{k}\exp\left[
-\frac{\mathcal{A}_P^2}{4}\left(\frac{\vert\mathcal{I}_k^{(j_1)}\pm\mathcal{I}_k^{(j_2)}\vert^2}{1-e^{-\hbar \omega_k/k_B T}}\right)
\right],
\label{eq:fidelity}\end{aligned}$$
where $\hat{\mathfrak{U}}(t)$ is the unitary evolution operator defined through Eq. (\[eq:Ut\]), $\hat\rho_T(0)$ is the initial (canonical) density operator of the phonon modes at temperature $T$, $\ket{\Phi_{\mathrm{qbit}}}$ is the initial two-qubit state, and
$$\begin{aligned}
\ket{\Phi^{\prime}_{\mathrm{qbit}}} = \sum_{\epsilon_1,\epsilon_2=0}^1(-1)^{\epsilon_1\epsilon_2}
c_{\epsilon_1,\epsilon_2}\ket{\epsilon_1}\ket{\epsilon_2}\end{aligned}$$
is the desired logical target state we aim to attain. The integral $\mathcal{I}_k^{(j_{q})}$ for $q=1,2$ is given in Eq. (\[eq:condforce\]) where the apex $(j_{q})$ refers to the ion we are considering, that is, $j=j_q$ in the sum of Eq. (\[eq:alpha\]).
Since we aim to achieve $\tau_{\mathrm{g}}\ll2\pi/\omega_{\mathrm{r}}$, we outline the following program: Firstly, we analyze the dependence of $\omega_{\mathrm{r}}$ on the total angular momentum $P_{\theta}$. This is achieved by fixing a priori a value of $P_{\theta}$ and then by determining the equilibrium configuration of the crystal, namely the positions and momenta of each ion (the most difficult part of the program). Since the crystal is a rigid body, it holds $p_{\theta_k}=m r_k^2\dot \theta_k + e r_k A_{\theta}(r_k)
= m r_k^2(\omega_{\mathrm{r}} + \omega_{\mathrm{c}}/2)$ [@Dubin1999], and from this relation the rotation frequency is extracted. Such an analysis allows us to find the smallest value of $\omega_{\mathrm{r}}$ such that $\beta<\beta_{\mathrm{c}} = 0.665/\sqrt{N}$ is fulfilled, that is, a 2D Wigner crystal configuration [@Dubin1999]. Then we choose the value of both $\omega_{\mathrm{c}}$ and $\nu$ in order to achieve high gate fidelity for a wide range of temperatures.
The determination of the classical ground state is a multidimensional minimization constrained problem for which no deterministic and efficient algorithm is known. Here we used a variant of the Metropolis [@Metropolis1953] and the multidimensional constrained Newton algorithm like the one of Ref. [@Schweigert1995]. The first method allows us to sample randomly the relevant phase space region by choosing a slow decay of the acceptance probability and by using several annealing cycles. We then coarse-grained the obtained annealing trajectories into intervals, and we employed, for the lowest energy configuration on each interval, a Newton algorithm, which is very efficient in finding a local minimum provided that the initial value is already very close. We have checked the reliability of our numerical energy minimization for $P_{\theta}=0$, that is $\omega_{\mathrm{r}} = 2 \omega_{\mathrm{c}}$, by comparing the results of Ref. [@Schweigert1995] for the minimal excitation frequency for several numbers $N$ of ions.
We investigated the robustness of the modulated-carrier phase gate against temperature for a moderate number of ions $N=30$ and $\alpha_z = 0.70$. In Fig. \[fig:wrL\] the dependence of the crystal rotation frequency on the total canonical angular momentum is showed, whereas in Fig. \[fig:infid\] the gate infidelity for different values of the ratio $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}$ is displayed. The results of Fig. \[fig:infid\] refer to $P_{\theta} = 4000 \,\ell^2_{\mathrm{s}} m \omega_{\mathrm{c}}$, for which we obtain the smallest value of $\vert\omega_{\mathrm{r}}\vert$ in Fig. \[fig:wrL\]. Beside this, we have for such a choice $\beta = 3.4\times 10^{-4}$, whereas $\beta_{\mathrm{c}} = 0.12$, that is, a stable 2D hexagonal lattice configuration. Given that, Fig. \[fig:infid\] shows that in order to reduce by a factor 10 the ratio $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}$ the fast modulation frequency $\nu$ of the force has to be (roughly) enhanced by a factor 10 as well. We also remark, that the three lines in Fig. \[fig:infid\] show an infidelity that is smaller for large gate operation times. The goal of the plot is to show how the modulation frequency increases when the ratio $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}$ is reduced for an infidelity smaller than $10^{-4}$. Of course, by carefully tuning $\nu$ one can easily get a smaller infidelity for faster gates.
In the inset (left corner - top) the result of the gate infidelity for a cyclotron frequency 100 times higher is showed, that is, the same 7.608 MHz of the experiment of Ref. [@Mitchell2001]. Here there are two important features to be highlighted: firstly, the gate fidelity is more robust for a wide range of temperatures with respect to the previous case where $\omega_{\mathrm{c}}/(2\pi) = 76.08$ kHz has been considered. On the other hand, already for $\tau_{\mathrm{g}}/\tau_{\mathrm{r}} = 0.1$, the frequency $\nu$ is on the order of hundreds of MHz. In the inset on the right (bottom) we show the gate infidelity again for the $\omega_{\mathrm{c}}=$7.608 MHz but for a smaller modulation frequency $\nu = 2.4$ MHz that lies in the gap between the two bands of different radial modes (the so called $\vec{E}\times\vec{B}$ and cyclotron modes. See also Fig. \[fig:modes\]). In this scenario $\tau_{\mathrm{g}}/\tau_{\mathrm{r}} = 10$ and therefore a co-rotating laser beam is required. In conclusion we see that for $\omega_z\sim \omega_{\mathrm{c}}/\sqrt{2}$ if we desire to avoid the employment of a co-rotating laser beam the only possible way is to achieve very high frequencies for the modulation of the state dependent force.
Alternatively, one can consider a smaller value of $\alpha_z$, which basically shifts upwards the graph of Fig. \[fig:wrL\], that is, by displacing the minimum of the $\vert\omega_{\mathrm{r}}\vert$ closer to zero for large values of $P_{\theta}$. This is the situation depicted in Fig. \[fig:infid\_a02\] for $N=30$, and $\alpha_z = 0.02$. Here it is possible to achieve gate operation times on the order of few $\mu$s with significantly smaller values of the modulation frequency. In the figure $\nu$ lies in the gap among axial and radial modes (see Fig. \[fig:modes\]). Furthermore with $\tau_{\mathrm{g}}/\tau_{\mathrm{r}} = 6\times 10^{-3}$ we do not need a co-rotating laser beam. This result is quite interesting since it works in a range of parameters that are currently employed in experiments (e.g., [@Biercuk2009]). Finally we also note that in this scenario $\omega_{xy} \gg \omega_z$, which is opposite to the requirement we identified in the case of larger $\alpha_z$ when $2\omega_r = \omega_c$. We note, however, that $\omega_{xy}$ is not the actual radial frequency when $\omega_{\mathrm{c}}\ne2\omega_{\mathrm{r}}$. Indeed, as shown in Eq. (\[eq:Homega\]), the centrifugal potential (i.e., $-m\omega^2r^2/2$) modifies the confinement. Let us write $\omega_{\mathrm{r}}= (\omega_{\mathrm{c}} - \delta\omega)/2$, where $\delta\omega>0$. Then by substituting such definition into the second line of (\[eq:Homega\]) we obtain an effective radial frequency given by: $\omega_{xy}^{\mathrm{eff}} = \frac{1}{2}\sqrt{\omega_{\mathrm{c}}^2 - \delta\omega^2 -2\omega_z^2}$. With the parameters of Fig. \[fig:infid\_a02\] we get $\omega_{xy}^{\mathrm{eff}}/(2\pi) = 31.47$ kHz which is significantly smaller than $\nu_z$ ($\sim 152$ kHz), and therefore the 2D lattice configuration is ensured. This fact is also confirmed by $\beta = 4\times 10^{-2}<\beta_{\mathrm{c}}$.
![(Color online). Ratio $\omega_{\mathrm{r}}/\omega_{\mathrm{c}}$ vs. total angular momentum $P_{\theta}$ for $\alpha_z=0.7$ and $N=30$. For $P_{\theta}=0$ we retrieve the well-known limit $\omega_{\mathrm{r}} = 2 \omega_{\mathrm{c}}$, in which there is no magnetic field in the rotating frame.[]{data-label="fig:wrL"}](fig02.eps)
![(Color online). Infidelity vs. temperature for $\alpha_z=0.7$, $N=30$, and $P_{\theta} = 4\times 10^{3} \,\ell^2_{\mathrm{s}} m \omega_{\mathrm{c}}$. Parameters: $\nu_{\mathrm{c}} = \omega_{\mathrm{c}} / (2 \pi) = 76.08$ kHz, $\nu_{xy} = \omega_{xy}/(2\pi) = 5.38$ kHz, $\nu_z = \omega_z/(2\pi) = 53.26$ kHz, and $\nu_{\mathrm{r}} = \omega_{\mathrm{r}}/(2\pi) = 32.75$ kHz. The black (solid) line corresponds to $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}=10^{-1}$ ($\tau_{\mathrm{g}} = 3 \,\mu$s), the red (dashed) line to $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}=10^{-2}$ ($\tau_{\mathrm{g}} = 0.3 \,\mu$s), and the blue (dashdot) line to $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}=10^{-3}$ ($\tau_{\mathrm{g}} = 0.03 \,\mu$s), with $\tau_{\mathrm{r}}=2\pi/\omega_{\mathrm{r}}$. The inset (on the left corner - top) provides the infidelity for $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}=10^{-1}$ ($\tau_{\mathrm{g}} = 0.03 \,\mu$s) with $\nu_{\mathrm{c}} = 7.61$ MHz as in Ref. [@Mitchell2001], $\nu= 300$ MHz, $\nu_{xy} = 537.97$ kHz, $\nu_z = 5.33$ MHz, and $\nu_{\mathrm{r}} = 3.27$ MHz. The inset (on the right corner - bottom) illustrates, for the same trapping parameters as for the former inset but with $\nu = 2.4$ MHz, the infidelity for $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}=10$ ($\tau_{\mathrm{g}} = 3 \,\mu$s). Such modulation frequency $\nu$ lies within the gap among radial and $\vec{E}\times\vec{B}$ phonon modes (see Fig. \[fig:modes\]).[]{data-label="fig:infid"}](fig03.eps)
![(Color online). Infidelity vs. temperature for $\alpha_z=0.02$, $N=30$, and $P_{\theta} = 1.3\times 10^{5} \,\ell^2_{\mathrm{s}} m \omega_{\mathrm{c}}$. Parameters: $\nu_{\mathrm{c}} = 7.61$ MHz, $\nu_{xy} = 3.80$ MHz, $\nu_z = 152.16$ kHz, and $\nu_{\mathrm{r}} = 1.65$ kHz (see text for more details).[]{data-label="fig:infid_a02"}](fig04.eps)
{width="120mm"}
Now, let us examine what is the required laser power in order to realize the gate and investigate the influence of scattered photons on the gate performance. A pair of narrow-waist ($\le 2\,\mu$m) adjacent laser beams in the standing-wave configuration produces the necessary force to be applied to each ion. Beside this, because of the tight focusing it reduces spontaneous emission and laser power. Following the treatment of Ref. [@Sasura2003], an estimate of the needed laser power to realize the logical gate is given by
P = \_P. Here $\Delta=\omega_L - \omega_A$ is the detuning, that is, the difference between the laser and the relevant atomic transition frequencies, $\Gamma$ is the linewidth of the transition, $\kappa = 2\pi/\lambda_L$ is the wave number with the laser wavelength $\lambda_L=2\pi c/\omega_L$, $c$ is the speed of light, $w$ is the size of the beam waist, and $\gamma$ is the angle between the $\kappa$ vectors of the two laser beams (see also Fig. \[fig:artistsview\]). Additionally, we can estimate the influence of photon scattering on the gate fidelity as: $F_{\mathrm{scat}}=e^{-N_{\mathrm{phot}}}$, where the number of scattered photons in the standing-wave configuration is given by
N\_ (). As the last two formulae show, by adjusting $\gamma$ we can reduce the required laser power, but at the expenses of a larger number of scattered photons, and therefore of a worsening of the gate performance.
Modulated and state-dependent dipole force {#sec:force}
==========================================
In order to realize our quantum phase gate, $\ket{\epsilon_1,\epsilon_2}\rightarrow e^{i\theta\epsilon_1\epsilon_2}
\ket{\epsilon_1,\epsilon_2}$ with $\epsilon_{1,2} = 0,1$, we have to engineer the $\theta_{kj}$ phases in $\theta$ (see also Sec. \[sec:modcargate\]). It is natural to demand that the desired value of $\theta$ is obtained with the smallest possible value of the applied force (i.e., laser power) or, alternatively, in the shortest possible time. This is equivalent to maximize $\theta$ by maximizing the effect of each $\theta_{kj}$. This happens when the phases $\theta_{01}$ and $\theta_{10}$ have the opposite sign with respect to the phases $\theta_{00}$ and $\theta_{11}$. Such condition is met when the applied force to the $j$-th ion satisfies the relation
\_j\^ = -\_j\^. \[eq:oppforces\]
Additionally, a necessary condition for the implementation of a modulated-carrier quantum phase gate is that the mean force acting on each ion (respectively each of the modes) has to be zero over $\tau_{\mathrm{g}}$, that is, we have to fulfill Eq. (\[eq:condforce\]). Such a requirement can be accomplished by making the modulation time symmetric around the center of the envelope of the laser pulse. To this aim, we impose the further condition on the force:
\_0\^t \_j\^[k]{}(t) = 0 k=0,1, \[eq:meanzeroforce\] where $\tau=2\pi/\nu$ is one period of the modulation. With such a condition we obtain a (fast) sinusoidal modulation of the force. Experimentally, this can be achieved, for example, with an acousto-optical modulator, which can vary the frequency of the laser light very quickly.
Energy shifts
-------------
In table \[tab:ZPBregimes\] we provide for some ion species the energy splitting between the $P_{1/2}$ and $P_{3/2}$ levels in the absence of an external magnetic field together with the maximal value of magnetic field $B_Z$, under which the (normal) Zeeman limit can be applied, and the minimal value $B_{PB}$ above which we enter in the Paschen-Back regime. As we can gather from the table, the higher the atomic number of the ion (or neutral atom) is, the larger $\Delta E$ and the limits $B_Z$, $B_{PB}$. For instance, for the infidelity results we showed in the previous section, the corresponding magnetic field at $\omega_{\mathrm{c}} = 7.608$ MHz are: $B=4.5$ T and $B=12$ T for Beryllium and Magnesium, respectively. These are also the values used in current experiments. Thus, for all alkaline-earth-metal atoms the Zeeman regime applies, and therefore $\hat H_B =\frac{\mu_{\mathrm{B}}}{\hbar} g_J\hat{J}_z B_z$ well describes the interaction of an ion with the external magnetic field. Here $g_J$ is the Landé factor [@Bethe1957] and the nuclear contribution has been neglected ($g_I\sim 10^{-3}$). Besides, since the external magnetic field has a strength of few Tesla, the ionic hyperfine structure can be also neglected. Additionally, we note that in the Paschen-Back regime the transitions from the energy (split) ground state ($S$-level) to the one of the excited levels ($P$) are identical for both ground levels when the ion is illuminated with a laser beam of a given polarization and frequency. Consequently, the dipole force (see Sec. \[sec:dipoleforce\]) would be the same for both states, and therefore it would not be possible in such a regime to have state-dependent forces. Instead, this is not the case for the broken degeneracy of the $S$ and $P$ levels due to the Zeeman effect (see Fig. \[fig:ZeemanShifts\]).
Atom/Ion $\Delta E/\hbar$ (THz) $B_Z$ (T) $B_{PB}$ (T)
---------- -- -- ------------------------ -- -- ----------- -- -- -------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
Be II 1.239 7.043 28.170
Mg II 17.249 98.070 392.286
Ca II 41.985 238.711 954.845
Na I 3.242 18.410 73.651
: \[tab:ZPBregimes\] Energy splitting among the $P_{1/2}$ and $P_{3/2}$ levels without external the magnetic field (second column from the left). In the third and fourth columns (from the left) the maximal and minimal value of the external magnetic field, which fix respectively the upper and lower bound for the Zeeman and Paschen-Back regimes, are given.
Finally, we note that given the selection rule $s - s^{\prime} = 0$ on the quantum number of the spin operator, optical transitions between the two levels of $S_{1/2}$ are not allowed. This fact reduces the possibility of undesired flips among the qubit pair, and therefore decoherence and dephasing mechanisms are strongly suppressed. Henceforth, we shall consider the lower energy level of $S_{1/2}$ as the logical state $\ket{0}$, whereas the upper one as $\ket{1}$.
![(Color online). Energy shifts due to the Zeeman effect vs. the $m_j$ quantum number. The distances among the energy levels are not in scale.[]{data-label="fig:ZeemanShifts"}](fig06.eps){width="3.3in"}
Dipole force and dipole matrix elements {#sec:dipoleforce}
---------------------------------------
The dipole force is produced by an intensity gradient of the laser beam illuminating the atom, which is far detuned from the relevant atomic transition, whose levels are ac-Stark shifted. Such an energy shift creates an additional potential for the particle. For a two-level atom and in the large detuning limit $\Delta\gg\vert\Omega\vert$, the dipole force on the lower energy level reads
= -(t,)\^2, where the Rabi frequency $\Omega(t,\vec{r})$ on the atomic transition is given by
$$\begin{aligned}
\hbar \Omega &= - \vec{d}_{ab} \cdot \vec{\mathcal E}(\vec r, t)
=- \vert\vec{d}_{ab}\vert \, \mathcal E_0 \, \chi(\vec r,t),\end{aligned}$$
whereas the laser is assumed to be a classical light field. Here $\vec d_{ab}$ represents the matrix element of the dipole moment operator for the transition $\ket{a}\equiv\ket{j=1/2;m_j}\rightarrow\ket{b}\equiv\ket{j=1/2,3/2;m_j}$ for a given polarization of the electric field $\vec{\mathcal E}$ with strength $\mathcal E_0$, and $\chi(\vec r,t)$ is the spatial and temporal pulse shape.
The bare (unshifted) detunings are then defined as: $\delta_{\mathrm{D}_1}= \omega_{L_1}-\omega_{\mathrm{D}_1}$ and $\quad \delta_{\mathrm{D}_2}= \omega_{L_2}-\omega_{\mathrm{D}_2}$, where $\omega_{L_1}$ and $\omega_{L_2}$ are the laser frequencies. In addition, in order to reduce the probability of unwanted photon scattering processes, we require that
$$\begin{aligned}
{\left\vert\mu_{\mathrm{B}} \,B\right\vert} &\ll {\left\vert\delta_{\mathrm{D}_1}\right\vert} \ll \Delta E & {\left\vert\mu_{\mathrm{B}} \,B\right\vert} &\ll {\left\vert\delta_{\mathrm{D}_2}\right\vert} \ll \Delta E.\end{aligned}$$
In table \[tab:redblueforces\] we provide the expressions of the state dependent forces for all polarizations of the laser field. Since $j=1/2$ for all relevant transitions, hereafter, for the sake of simplicity, we shall denote the reduced matrix element by $\mathcal{M}_{\mathrm{D}_1} = \mathcal{M}_{1/2,1/2}$ and $\mathcal{M}_{\mathrm{D}_2} = \mathcal{M}_{1/2,3/2}$, where $\mathcal{M}_{jj^{\prime}}:=\langle j,m_j\parallel e \hat{r}\parallel j^{\prime},m_j^{\prime}\rangle$.
[cc\*[4]{}[c]{}\*[3]{}[c]{}]{} & & & & & & & &\
$\mathrm{Polarization}$ & & $\vec{f}^{\,\ket{0}}\, (\mathrm{D}_1)$ & & $\vec{f}^{\,\ket{1}} (\mathrm{D}_1)$ & & $\vec{f}^{\,\ket{0}}\, (\mathrm{D}_2)$ & & $\vec{f}^{\,\ket{1}}\, (\mathrm{D}_2)$\
& & & & & & & &\
$\sigma^-$ & & 0 & & $-\frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{2\hbar (3 \delta_{\mathrm{D}_1}+4\mu_{\mathrm{B}} B/\hbar)}$ & & $-\frac{\mathcal{M}_{\mathrm{D}_2}\mathcal E_0^2
\nabla \chi^2(\vec r,t)}{4\hbar(\delta_{\mathrm{D}_2}+\mu_{\mathrm{B}} B/\hbar)}$ & & $-\frac{\mathcal{M}_{\mathrm{D}_2}
\mathcal E_0^2 \nabla \chi^2(\vec r,t)}{4\hbar(3\delta_{\mathrm{D}_2}+5\mu_{\mathrm{B}} B/\hbar)}$\
$\pi$ & & $\frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{4\hbar(2\mu_{\mathrm{B}} B/\hbar - 3\delta_{\mathrm{D}_1})}$ & & $-\frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{4\hbar(3\delta_{\mathrm{D}_1} + 2\mu_{\mathrm{B}} B/\hbar)}$ & & $\frac{\mathcal{M}_{\mathrm{D}_2} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{2\hbar(\mu_{\mathrm{B}} B/\hbar - 3\delta_{\mathrm{D}_2})}$ & & $-\frac{\mathcal{M}_{\mathrm{D}_2} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{2\hbar(3 \delta_{\mathrm{D}_2}+\mu_{\mathrm{B}} B/\hbar)}$\
$\sigma^+$ & & $\frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{2\hbar(4\mu_{\mathrm{B}} B/\hbar - 3 \delta_{\mathrm{D}_1})}$ & & 0 & & $\frac{\mathcal{M}_{\mathrm{D}_2} \mathcal E_0^2 \nabla
\chi^2(\vec r,t)}{4\hbar(5\mu_{\mathrm{B}} B/\hbar - 3\delta_{\mathrm{D}_2})}$ & & $\frac{\mathcal{M}_{\mathrm{D}_2}
\mathcal E_0^2 \nabla \chi^2(\vec r,t)}{4\hbar(\mu_{\mathrm{B}} B/\hbar - \delta_{\mathrm{D}_2})}$\
& & & & & & & &\
Different laser configurations {#sec:laserconfigs}
------------------------------
As it is evident from the table \[tab:redblueforces\], using only a laser pulse either on the D$_1$ transition line or on the D$_2$ one it is not possible to fulfill the condition (\[eq:oppforces\]). To this aim we need a second laser pulse (see Fig. \[fig:PulseSequence\] on the left) with a different detuning from the first pulse. In this section we discuss several combinations of the laser polarization in order to satisfy both (\[eq:oppforces\]) and (\[eq:meanzeroforce\]).
![(Color online). Sketch of the pulse sequence in order to design the necessary dipole forces to implement the modulated-carrier phase gate. The upper lines are red detuned, whereas the lower lines are blue detuned. In the figures the time is in arbitrary units.[]{data-label="fig:PulseSequence"}](fig07.eps){width="3.3in"}
### Pulses with the same polarization
In this case we first generate two laser pulses with different frequencies but with the same $\sigma^-$ polarization, as in Fig. \[fig:PulseSeqEqualPol\], which corresponds to the first sequence of pulses (on the left) in Fig. \[fig:PulseSequence\]. Such a configuration of lasers yields the following state-dependent forces:
$$\begin{aligned}
\vec f^{\,\ket 1} &= -\frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_{01}^2
\nabla \chi^2(\vec r,t)}{2\hbar (3 \delta_{\mathrm{D}_1}+4\mu_{\mathrm{B}} B/\hbar)}
-\frac{\mathcal{M}_{\mathrm{D}_2}
\mathcal E_{02}^2 \nabla \chi^2(\vec r,t)}{4\hbar(3\delta_{\mathrm{D}_2}+5\mu_{\mathrm{B}} B/\hbar)},
\nonumber\\
\vec f^{\,\ket 0} &= -\frac{\mathcal{M}_{\mathrm{D}_2}\mathcal E_{02}^2
\nabla \chi^2(\vec r,t)}{4\hbar(\delta_{\mathrm{D}_2}+\mu_{\mathrm{B}} B/\hbar)}.\end{aligned}$$
![(Color online). Modulated-carrier gate with the same polarization of the laser fields: $\sigma^-$-polarized (left), and $\sigma^+$-polarized (right). The distances among the energy levels are not in scale.[]{data-label="fig:PulseSeqEqualPol"}](fig08.eps){width="3.5in"}
To simplify the notation we make the following replacements: $\mathcal X_{\mathrm{D}_i} =
\mathcal{M}_{\mathrm{D}_i}\mathcal E_{0i}^2 \nabla \chi^2(\vec r,t)$ and $\mathcal B = \mu_B B/\hbar$, where $\mathcal E_{0i}$ refers to the electric field strength of either the D$_1$ ($i=1$) or D$_2$ ($i=2$) line. Thus, in order to fulfill (\[eq:oppforces\]), we have to solve the equation
+ +=0 . This can be resolved for both the intensities or the detunings, so one of them can be considered as a given parameter. For the $\sigma^+$-polarization we obtain an analogue equation:
+ +=0 . \[eq:example\] As an example, we solve equation (\[eq:example\]), for instance, for the intensities, and be obtain
= . \[eq:solexample\] Such a solution, however, fulfills only the condition (\[eq:oppforces\]) but not the one given by Eq. (\[eq:meanzeroforce\]). To this aim we need an additional two-pulse sequence, as showed in Fig. \[fig:PulseSequence\] on the right. Such two pulses can have different strengths of intensities and detunings, but they must have the same spatial and temporal profile $\chi({\vec r,t})$ of the first sequence. Again, we get, if we solve with respect to the intensities, a solution like the one given in Eq. (\[eq:solexample\]), which in general will be different from the solution (\[eq:solexample\]) for the first sequence of pulses. With such solution we can then easily satisfy also the mean zero force condition (\[eq:meanzeroforce\]) by adjusting the ratios of either the intensities or the detunings.
In Fig. \[fig:pulses\] we display a simple example that shows how to achieve the necessary laser pulse sequence. We modulate the intensities of the blue ($b$) and red ($r$) detuned laser signals like $I^{b (r)}(t) = I^{b (r)}_0 \sin^2(\nu t)$. The sequence starts ($t=0$) with both lasers with $\sigma^+$-polarization and an intensity ratio $R_+ = I^{b}_0/I^{r}_0$ given by Eq. (\[eq:solexample\]). Then, at time $t=\pi/\nu$, the polarization of the two laser fields is changed to $\sigma^-$ with another intensity ratio given by $R_- = I^{b}_0/I^{r}_0$. The ratio $R_-$ will differ from $R_+$, since in general the dipole moments are different for the two polarizations. Thus, by changing the polarization at each minimum of the laser intensity and by choosing the proper ratio $R_{\pm}$ we are able to fulfill the condition (\[eq:meanzeroforce\]). In order to satisfy the condition (\[eq:oppforces\]) we have to design furthermore the ratio of the two successive pulses. This is done by multiplying the two intensities $I^{b (r)}(t)$ with square wave signals which are displayed in Fig. \[fig:pulses\] (top) by the black lines. The resulting pulses are showed in Fig. \[fig:pulses\] (bottom-left) whose polarization state is depicted on the right lower corner. This procedure ensures that both the conditions (\[eq:meanzeroforce\]) and (\[eq:oppforces\]) are fulfilled.
. Designed laser modulation forces in order to fulfill both (\[eq:meanzeroforce\]) and (\[eq:oppforces\]). The blue detuned laser intensity with a superimposed square wave signal (top-left) and similarly for the red detuned one (top-right). The resulting signals are depicted on the bottom-left, whereas switching of polarization is given on the bottom-right.](fig09.eps)
### Pulses with the different polarization
The situation in which the laser beams have different polarization is depicted in Fig. \[fig:PulseSeqDifferentPol\]. If we illuminate the ion with red detuned and $\sigma_+$ polarized light and with a blued detuned and $\sigma_-$ beam (Fig. \[fig:PulseSeqDifferentPol\] on the left) we have the following state-dependent forces
$$\begin{aligned}
\vec f^{\,\ket 1} &= -\frac{\mathcal{M}_{\mathrm{D}_2}
\mathcal E_0^2 \nabla \chi^2(\vec r,t)}{4\hbar(3\delta_{\mathrm{D}_2}+5\mu_{\mathrm{B}} B/\hbar)},
\nonumber\\
\vec f^{\,\ket 0} &= \frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{2\hbar(4\mu_{\mathrm{B}} B/\hbar - 3 \delta_{\mathrm{D}_1})}
-\frac{\mathcal{M}_{\mathrm{D}_2}\mathcal E_0^2
\nabla \chi^2(\vec r,t)}{4\hbar(\delta_{\mathrm{D}_2}+\mu_{\mathrm{B}} B/\hbar)}, \end{aligned}$$
whereas for the inverted polarization sequence (Fig. \[fig:PulseSeqDifferentPol\] on the right) we have
$$\begin{aligned}
\vec f^{\,\ket 1} &= \frac{\mathcal{M}_{\mathrm{D}_2}
\mathcal E_0^2 \nabla \chi^2(\vec r,t)}{4\hbar(\mu_{\mathrm{B}} B/\hbar - \delta_{\mathrm{D}_2})}
-\frac{\mathcal{M}_{\mathrm{D}_1} \mathcal E_0^2
\nabla \chi^2(\vec r,t)}{2\hbar (3 \delta_{\mathrm{D}_1}+4\mu_{\mathrm{B}} B/\hbar)},
\nonumber\\
\vec f^{\,\ket 0} &= \frac{\mathcal{M}_{\mathrm{D}_2} \mathcal E_0^2 \nabla
\chi^2(\vec r,t)}{4\hbar(5\mu_{\mathrm{B}} B/\hbar - 3\delta_{\mathrm{D}_2})}. \end{aligned}$$
![(Color online). Modulated-carrier gate with different polarization of the laser fields. The distances among the energy levels are not in scale.[]{data-label="fig:PulseSeqDifferentPol"}](fig10.eps){width="3.5in"}
Both the schemes with the same and with different polarization have the drawback that the transition to the excited level $P_{3/2}$ couples both the ground states of $S_{1/2}$, and therefore producing an additional force that has to be compensated with another laser beam. Apart from the technical difficulty of putting another laser beam, such a beam would also enhance the probability of promoting an ion to an excited level of $P_{3/2}$. Such excitation would cause an additional error during the course of the gate because of spontaneous emission. Indeed, the ion could decay either in the other qubit state or even worst, such as for the $D$-levels in calcium, in another metastable state, which would be useless for the purposes of QIP.
Given that, in order to avoid such scenario, we can make still use of the scheme illustrated in Fig. \[fig:PulseSeqDifferentPol\], but by avoiding the coupling to the $P_{3/2}$ manifold, as it is showed in Fig. \[fig:PulseSeqDifferentPolBis\]. Here, however, we couple the ground state $S_{1/2}$ to only the manifold $P_{1/2}$. The pulse sequences are then the same as previously described for the other scheme. The detuning from the $P_{1/2}$ manifold, however, has to be carefully chosen, that is, it has to be much smaller than the energy difference among the $P_{1/2}$ and $P_{3/2}$ levels and much larger than $\mu_{\mathrm{B}} B$. Hence, such variant works well for sufficiently small magnetic fields.
![(Color online). Variant of the modulated-carrier gate with different polarization of the laser fields. The distances among the energy levels are not in scale.[]{data-label="fig:PulseSeqDifferentPolBis"}](fig11.eps){width="3.5in"}
In principle there are other possible arrangements either by keeping the frequencies of the laser beams constant or by keeping constant their intensities. Such combinations rely also on the technical feasibility in an experimental setup. An important requirement for the design of such state-dependent forces is, however, the switch of the field polarization, which has to be very fast in order to fulfill the condition set by the Eq. (\[eq:meanzeroforce\]), as it is also shown in the example of Fig. \[fig:pulses\](d). This can be experimentally accomplished with Pockels cells, which can be used to manipulate the polarization and the phase of the laser.
Conclusions {#sec:conc}
===========
In this work we have analyzed in detail the implementation of the modulated-carrier gate presented for the first time in Ref. [@Taylor2008]. Firstly, we presented the underlying idea of the modulated-carrier gate and we provided details of the calculations that were only briefly mentioned in Ref. [@Taylor2008]. In that analysis the frame of reference rotates at the same frequency of the crystal rotation, whose frequency was set to $\omega_{\mathrm{c}} = 2 \omega_{\mathrm{r}}$. Within this setting the minimal coupling term in the many-body Hamiltonian vanishes. Such approach allows a straightforward canonical quantization of the many-body Hamiltonian, which reduces to a sum of $3N$ independent harmonic oscillators. Even though this situation greatly simplifies the numerical analysis it does not permit to fulfill the condition $\omega_{\rm r}\tau_{\mathrm{g}}/(2\pi) \ll 1$, which would avoid the utilization of a co-rotating laser and therefore simplifying the experimental realization of the proposed quantum hardware. We thus have analyzed the situation in which $\omega_{\mathrm{c}} \ne 2 \omega_{\mathrm{r}}$. Within this scenario it is no longer possible to remove the minimal coupling term in the Hamiltonian of the Coulomb crystal. Nevertheless, by utilizing the Williamson theorem for positive definite matrices, we were able to diagonalize the classical many-body Hamiltonian, whose normal modes are a combination of both the position and momentum variables. As a consequence, we were able to perform the canonical quantization. The resulting (quantized) Hamiltonian is again given by a sum of independent harmonic oscillators. In this new situation, however, the matter-field interaction, responsible of the push on the ion, depends on both conjugate “position" and “momentum" operators. We proceeded further on by analyzing the performance of the quantum phase gate and we showed its robustness for a wide range of experimentally accessible temperatures. Importantly, we were able to demonstrate that such robustness is also displayed for a wide range of ratios $\tau_{\mathrm{g}}/\tau_{\mathrm{r}}$, therefore allowing to reduce up to three orders of magnitude the gate operation time compared to the previous analysis [@Taylor2008]. The drawback is that one has to enhance the modulation frequency $\nu$ up to hundreds of MHz in order to speed up the gate operation. We found, however, that by reducing the ratio $\omega_z/\omega_{\mathrm{c}}$, at large values of angular momentum it is possible to achieve small rotation frequencies such that $\omega_{\rm r}\tau_{\mathrm{g}}/(2\pi) \ll 1$ is fulfilled and high fidelity, for a broad range of temperatures, can be obtained with few MHz of modulation frequency. This result is quite promising since it has been attained with a cyclotron frequency that is used in current experiments.
Finally, we have provided a complete description for the design of the necessary forces to be applied on the ions in order to accomplish the desired quantum computation scheme. To this aim, we have analyzed the experimentally relevant region of external magnetic field. For all earth-alkali-metal ion species normally used in currents experiments the normal Zeeman effect provides, with good approximation, the right description of the energy shifts of the $S$ and $P$ levels. In addition, we have also analyzed several possible laser configurations and for each one we discussed advantages as well as drawbacks and, in some cases, we suggested alternative solutions.
Further investigations of such a quantum computing proposal may rely on further optimization of both the force modulation together with a reduced gate operation time and its robustness against optimal pulse distortions [@Negretti2010]. This can be achieved by means of quantum optimal control techniques. Beside this, a detailed analysis, similar to Ref. [@Poulsen2010], in order to characterize and quantify all types of errors coming from the quantum dynamics, especially due to nonlinearities in the ion-pushing force, will be pursued in future investigations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to J. J. Bollinger for his critical reading of the manuscript. J.B. acknowledges G. De Chiara and E. Kajari for helpful discussions, and A.N. useful correspondence with E. Pagani on symplectic transformations. We acknowledge financial support from the EU Integrated Project AQUTE, PICC (T.C.), the Deutsche Forschungsgemeinschaft within the Grant No. SFB/TRR21 (A.N.,T.C.), the Marie Curie Intra European Fellowship (Proposal Nr. 236073, OPTIQUOS) within the 7th European Community Framework Programme (A.N.), the Forschungsbonus of the University of Ulm and of the Ulmer Universitätsgesellschaft (A.N.), the Spanish Ministry of Science and Innovation (Consolider Ingenio 2010 “QOIT”, QNLP FIS2007-66944), and the European Science Foundation (EUROQUAM “Cavity-Mediated Molecular Cooling") (J.B.).
[^1]: Hereafter we shall use latin symbols for index the ions and greek symbols for the cartesian coordinate of the force vector acting on the ions.
[^2]: When $\omega_{\mathrm r} \ne \omega_{\mathrm c}/2$, the total canonical angular momentum $P_{\theta}\ne 0$, but it is still a constant of motion [@Dubin1999].
|
---
abstract: 'We perform an analysis of the spatial clustering properties of HI selected galaxies from the HI Parkes All Sky Survey (HIPASS) using the formalism of the halo occupation distribution (HOD). The resulting parameter constraints show that the fraction of satellite galaxies (i.e. galaxies which are not the central member of their host dark matter halo) among HIPASS galaxies is $<20\%$, and that satellite galaxies are therefore less common in HIPASS than in optically selected galaxy redshift surveys. Moreover the lack of fingers-of-god in the redshift space correlation function of HIPASS galaxies may indicate that the HI rich satellites which do exist are found in group mass rather than cluster mass dark matter halos. We find a minimum halo mass for HIPASS galaxies at the peak of the redshift distribution of $M\sim10^{11}$M$_\odot$, and show that less than 10% of baryons in HIPASS galaxies are in the form of HI. Quantitative constraints on HOD models from HIPASS galaxies are limited by uncertainties introduced through the small survey volume. However our results imply that future deeper surveys will allow the distribution of HI with environment to be studied in detail via clustering of HI galaxies.'
author:
- |
J. Stuart B. Wyithe$^1$, Michael J. I. Brown$^2$, Martin A. Zwaan$^3$, Martin Meyer$^4$\
$^1$ School of Physics, University of Melbourne, Parkville, Victoria, Australia\
$^2$ School of Physics, Monash University, Clayton, Victoria 3800, Australia\
$^3$ European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching b. M[ü]{}nchen, Germany\
$^4$ School of Physics, University of Western Australia, Crawley, WA 6009, Australia\
Email: swyithe@unimelb.edu.au
title: The Halo Occupation Distribution of HI Galaxies
---
\[firstpage\]
cosmology: large scale structure, observations – galaxies: halos, statistics – radio lines: galaxies
Introduction
============
The cosmic star-formation rate has declined by more than an order of magnitude in the past 8 billion years (Lilly et al. 1996, Madau et al. 1996), a trend that is observed across all wavelengths (Hopkins 2004 and references therein). Why this decline has taken place, and what drives it are two of the most important unanswered questions in our current understanding of galaxy formation and evolution. One of the issues that will need to be addresses in order to answer this question is the role of environment. In cold dark matter cosmologies, gas cools and collapses to form stars within gravitationally bound [*halos*]{} of dark matter. These galaxies can then grow via continued star formation or via mergers with other galaxies. As galaxies of a given baryonic mass can only reside within dark matter halos above a particular dark matter mass, galaxies are biased tracers of the overall dark matter distribution.
In linear theory, the bias in the spatial clustering of dark matter halos relative to the underlying mass distribution is a function of halo mass but not of spatial scale. As a consequence, if the mass power-spectrum is known, the clustering of galaxies on large scales yields strong constraints on the masses of the dark matter halos in which they reside. On smaller scales, the simple relationship between galaxy clustering and halo mass breaks down. Firstly the mass power-spectrum is in the non-linear regime. More importantly, multiple galaxies can be distributed within individual halos (at separations $\la$1 Mpc), with the number of galaxies within halos of a given mass exhibiting some scatter. While this complicates the modelling of galaxy clustering, it enables measurements of spatial clustering of galaxies to determine how galaxies populate dark matter halos as a function of halo mass. Some understanding of these issues is provided by simulations and these can be (and have been) tested against observations of the spatial clustering of optically selected galaxies.
By understanding how galaxies populate dark matter halos, key insights may be obtained into how galaxies grow over cosmic time. For example, while the merger rate of dark matter halos is known, modelling the dynamical friction of sub-halos (and thus galaxies) in cosmological simulations is non-trivial and the rate of galaxy growth via merging has been uncertain as a consequence. Knowing how galaxies populate dark matter halos resolves this problem. In particular, consider a case where the timescale for dynamical friction following the merger of two dark matter halos is short compared with the Hubble time. In this case galaxies within these halos will also merge soon after, and satellite galaxies will be relatively rare. On the other hand, if the dynamical friction timescale is long, then the galaxies within these halos may remain as satellite galaxies for many Gyr. In this case satellite galaxies will be relatively common. Brown et al. (2008) show that the later scenario holds for the most massive dark matter halos, with much of the stellar mass in massive halos residing within satellite galaxies.
The way in which stellar mass populates dark matter halos has, to first order, been determined for optically selected galaxy samples. However little is known about how HI, the fuel for star-formation, populates group and cluster mass dark matter halos. HI galaxies in the Fornax region have been studied by Waugh et al. (2002), who found very few galaxies to be associated with the Fornax cluster. None of the HI detections in Waugh et al. (2002) are early-type galaxies. Moreover, only 2 of the HI detections have both Fornax redshifts and are within 1 degree ($\sim300$ kpc) of the cluster centre. These results may suggest that there is a central galaxy high mass cut-off near the Fornax cluster halo mass (which is $7\times10^{13}$M$_\odot$ according to Drinkwater et al. 2001). More recently Cortese et al. (2008) have used Arecibo to survey a 5 square degree region around Abell 1367. They find a uniform distribution of HI-selected galaxies throughout the volume (i.e. when observed in HI the Abell cluster 1367 disappears), and that HI deficiency does not vary significantly with cluster-centric distance. These authors also find no finger-of-god effect in the HI-selected galaxies (in a redshift-position diagram, rather than in a clustering analysis). Similarly, Verheijen et al. (2007) study Abell 963 and 2192 at $z=0.2$ and find only one HI-selected galaxy within 1Mpc from the centre of each cluster. On the other hand, de Blok et al. (2002) find that there are HI galaxies in Sculptor and Centaurus with HI masses of $\sim10^9$M$_\odot$. However these clusters have dynamical masses $\sim1.5$ orders of magnitude lower than that of the Fornax and Abel clusters discussed above, and the identification of these with the clusters is not definitive.
Thus there are many questions. For example, is HI stripped from galaxies entering cluster, group or lower mass halos? Is there a dark matter halo mass above which HI is heated or removed from galaxies? Is the HI content of galaxies largely a function of galaxy stellar mass or host dark matter halo mass? Do the stellar masses of HI selected galaxies grow largely via star-formation or galaxy mergers? These questions can be addressed using the observed clustering of HI selected galaxies to constrain models of how HI populates dark matter halos. A popular formalism for modeling clustering on small to large scales is termed the halo occupation distribution (HOD) model (e.g. Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001; Berlind & Weinberg 2002; Zheng 2004; Zehavi et al. 2004). The HOD model includes contributions to galaxy clustering from pairs of galaxies in distinct halos which describes the clustering in the large scale limit, and from pairs of galaxies within a single halo which describes clustering in the small scale limit. The latter contribution requires a parametrisation to relate the number and spatial distribution of galaxies within a dark matter halo of a particular mass. Measurements of the HOD of optically selected galaxies provide some insights into how galaxies evolve. For example, in the most massive dark matter halos, central galaxy stellar mass is proportional to halo mass to the power of approximately $\sim1/3$. Much of the stellar mass within these halos resides within satellite galaxies (e.g., Brown et al. 2008, Moster et al. 2009). This result implies that the mergers of dark matter halos do not always lead to mergers of galaxies, and as a consequence massive galaxy growth is slow relative to the rapid growth of dark matter halos. Whether this result is also true for lower mass star-forming galaxies is unknown at this time.
In recent years large galaxy redshift surveys such as SDSS and the 2dFGRS have enabled detailed studies of the clustering of in excess of 100000 optically selected galaxies in the nearby universe. By comparison, the largest survey of HI selected galaxies contains $\sim5000$ sources, obtained as part of the HI Parkes All Sky Survey (HIPASS, Barnes et al. 2001), a blind HI survey of the southern sky. Meyer et al. (2007) have studied the clustering of these HI galaxies. Their analysis reached the conclusion of weak clustering of HI galaxies based on parametric estimates of correlation length (see also Basilakos et al. 2007), but did not study the clustering in terms of the host dark matter halo masses of the HIPASS sample. In this paper we revisit the clustering of HIPASS galaxies using the HOD model. There are systematic uncertainties in estimation of the observed clustering amplitude, arising from the selection function and small survey volume (Meyer et al. 2007), and our results show that this limits the precision with which conclusions from clustering can be made. Nevertheless we illustrate that the clustering of HIPASS galaxies already provides interesting constraints on the distribution of HI within the dark matter halo population.
=6.9in
This paper is organised as follows. We begin by summarising the clustering of HI galaxies in the HIPASS survey § \[clustering\]. We then summarise the formalism for the real and redshift-space HOD models for the correlation function (§ \[HOD\]), which is discussed relative to the HIPASS observations in § \[HODobs\] and § \[HODobsz\] respectively. We discuss the satelite fraction in § \[discussion\] and summarise our findings in § \[summary\]. In our numerical examples, we adopt the standard set of cosmological parameters (Komatsu et al. 2008), with values of $\Omega_{\rm m}=0.24$, $\Omega_{\rm b}=0.04$ and $\Omega_Q=0.76$ for the matter, baryon, and dark energy fractional density respectively, $h=0.73$, for the dimensionless Hubble constant, and $\sigma_8=0.81$ for the variance of the linear density field within regions of radius $8h^{-1}$Mpc.
clustering of HIPASS galaxies {#clustering}
=============================
Meyer et al. (2007) computed the redshift space correlation function of HI selected galaxies from 4315 detections in the HIPASS catalogue (HICAT; Barnes et al. 2001; Meyer et al. 2004; Zwaan et al. 2004). Correlation functions were produced by weighting each galaxy pair equally (termed unweighted), and by weighting each pair in a way that corrects for the survey selection function and minimises the variance in the correlation function estimate (termed weighted). From the redshift space correlation function, Meyer et al. (2007) computed the real space correlation functions in both the weighted and unweighted cases, using inversions that were both non-parametric and which assumed a powerlaw.
In this paper we restrict our attention to non-parametric estimates of the real space correlation function. However given the sensitivity of the measured clustering to the weighting scheme adopted, we fit both the unweighted and the weighted real-space correlation function from Meyer et al. (2007). From their estimated correlation function, Meyer et al. (2007) calculate a correlation length for the HIPASS galaxies. In this paper our aim is instead to interpret the astrophysical context of the measured clustering, namely the distribution of HI galaxies within the dark-matter halo population and the typical dark matter halo mass.
Density of HIPASS galaxies
--------------------------
Constraints on HOD models are provided both by the clustering of galaxies, and by the density of galaxies via comparison with the dark-matter halo mass function. We estimate the space density of HIPASS sources from the HI mass function (Zwaan et al. 2005a), yielding $$n_{\rm gal} = \theta_\star \Gamma(1+\alpha,M_{\rm HI, lim}/M_{\rm HI,\star}),$$ where $M_{\rm HI,lim}$ is the lowest HI mass included in the calculation of space densities, and the parameters have measured values of $\alpha=-1.37$, $\theta_\star=0.0060$Mpc$^{-3}$, and $M_{\rm
HI,\star}=10^{9.8}$M$_\odot$. If all HIPASS galaxies with HI masses $>10^7$M$_\odot$ were included, the space density would be $n_{\rm gal}\sim0.15$Mpc$^{-3}$. However HI masses of $10^7$M$_\odot$ can only be detected out to very small distances in HIPASS, and so are not really represented in the calculation of the correlation function. A better estimate is obtained by looking at the peak of the redshift distribution, where the typical HI mass is $\sim10^{9.25}$M$_\odot$. The space density for HI masses larger than $10^{9.25}$M$_\odot$ is $n_{\rm gal}\sim0.0069$Mpc$^{-3}$. We estimate the error on this value to be $\sim15\%$.
Dynamical masses of HIPASS galaxies
-----------------------------------
An analysis of the observed clustering of a galaxy population based on the bias of dark-matter halos implicitly assumes a relationship between galaxy luminosity (or in this case HI mass) and the host halo mass. Before proceeding to discuss the formalism for the model of halo clustering we therefore describe the relation between HI mass and dynamical mass for galaxies in the HIPASS survey. The dynamical mass $M_{\rm dyn}$ of the HI galaxies was estimated using the circular velocity ($V_{\rm c}$) derived from the width of the HI spectrum using two methods. Firstly, based on the work of Marc Verheijen (PhD thesis), we have estimated the mass of a dark matter halo with a Hernquist (1990) profile using the relation $M_{\rm dyn}=10^{10}R(V_{\rm
c}/103.9\mbox{km}\,\mbox{s}^{-1})^2$M$_\odot$, where for the radius $R$, we have adopted the B-band Kron radius (measured in kpc). The resulting relation is shown in the left hand panel of Figure \[fig1\]. Secondly, we have also estimated $M_{\rm dyn}$ from $V_{\rm c}$ based on Kochanek & White (2001), with results plotted in the right hand panel of Figure \[fig1\]. Each panel includes a linear relationship to guide the eye, showing the upper limit on HI mass $M_{\rm HI}\la\Omega_{\rm b}/\Omega_{\rm M}M_{\rm dyn}$. These panels illustrate that while there is significant scatter, larger HI masses are found in more massive host halos. Figure \[fig1\] illustrates that the relationship between is HI and dynamical mass is shallower than linear, with $M_{\rm HI}\propto M_{\rm dyn}^\gamma$ where $\gamma\sim0.5-0.7$. These dynamical masses are defined such that they are comparable to a definition based on the volume which encloses mass at $\sim200$ times the mean density of the Universe.
The largest dynamical mass among the HIPASS sample is $\sim10^{13}$M$_\odot$. For comparison, we expect a number $N\sim
V_{\rm HIPASS}\times Mdn/dM=300$ of halos in the HIPASS volume $V_{\rm
HIPASS}$, where we calculate $dn/dM$ using the Sheth-Tormen (2002) mass function. This yields $N\sim 300$, 30 and 1 for masses of $M=10^{13}$M$_\odot$, $10^{14}$M$_\odot$ and $10^{15}$M$_\odot$ respectively. Figure \[fig1\] shows that the observed number of these massive halos is much lower than the mass function predicts, although they should be detectable throughout the HIPASS search volume. Thus it appears that the most massive halos in the HIPASS volume do not host an HI galaxy which traces the halo potential.
HOD models {#HOD}
==========
In this paper we model the clustering of HIPASS galaxies using the halo occupation distribution formalism (HOD; e.g. Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001; Berlind & Weinberg 2002; Zheng 2004). Our approach is to fit HOD parameters for the non-parametric estimate of the real space HIPASS correlation function. Based on these fits, we then calculate the redshift-space correlation function using the analytic formalism described in Tinker (2007). In this section we describe the HOD modeling formalism briefly to provide context for the particular parametrisation used, and refer the reader to the above papers for details.
The real-space HOD model
------------------------
The HOD model is constructed around the following simple assumptions. First, one assumes that there is either zero or one central galaxy that resides at the centre of each halo. Satellite galaxies are then assumed to follow the dark matter distribution within the halos. The mean number of satellites is typically assumed to follow a power-law function of halo mass, while the number of satellites within individual halos follows a Poisson (or some other) probability distribution.
The two-point correlation function can be decomposed into one-halo and two-halo terms $$\label{xi1}
\xi(r) = \left[1+\xi_{\rm 1h}(r)\right]+\xi_{2h}(r),$$ corresponding to the contributions to the correlation function from galaxy pairs which reside in the same halo and in two different halos respectively (Zheng 2004). In real space the 1-halo term can be computed using (Berlind & Weinberg 2002) $$\begin{aligned}
\label{1h}
\nonumber
1+\xi_{1h}(r)&=&\frac{1}{2\pi r^2\bar{n}_{\rm g}^2}\\
&&\hspace{-20mm}\times\int_0^\infty dM\frac{dn}{dM}\frac{\langle N(N-1)\rangle_{M}}{2}\frac{1}{2R_{\rm vir}(M)}F^\prime\left(\frac{r}{2R_{\rm vir}}\right).\end{aligned}$$ Here $\bar{n}_{\rm g}$ is the mean number density of galaxies. We assume the Sheth-Tormen (1999) mass function $dn/dM$ using parameters from Jenkins et al. (2001) throughout this paper. The distribution of multiple galaxies within a single halo is described by the function $F(x)$ which is the cumulative fraction of galaxy pairs closer than $x\equiv r/R_{\rm vir}$. The contribution to $F$ is divided into pairs of galaxies that do, and do not involve a central galaxy, and is computed assuming that galaxies follow the number-density distribution of a Navarro, Frenk & White (1997; NFW) profile (see e.g. Zheng 2004). The quantity $\langle
N(N-1)\rangle_{M}$ is the average number of halo pairs. We assume an average distribution, with $\langle N(N-1)\rangle_{M}= \langle
N\rangle_{M}^2-\langle N\rangle_{M}$.
The 2-halo term can be computed as the halo correlation function weighted by the distribution and occupation number of galaxies within each halo. The 2-halo term of the galaxy power-spectrum is $$\label{2h}
P_{\rm gg}^{\rm 2h}(k) = P_{\rm m}(k)\left[\frac{1}{\bar{n}_{\rm g}}\int_{0}^{M_{\rm max}} dM \frac{dn}{dM} \langle N\rangle_{M} b_{\rm h}(M) y_{\rm g}(k,M)\right]^2,$$ where $P_{\rm m}$ is the mass power-spectrum and $y_{\rm g}$ is the normalised Fourier transform of the galaxy distribution profile (i.e. NFW). To compute the halo bias $b(M)$ we use the Sheth, Mo and Tormen (2001) fitting formula. The quantity $M_{\rm max}$ is taken to be the mass of a halo with separation $2r$. The 2-halo term for the correlation function follows from $$\xi_{2h}(r)=\frac{1}{2\pi^2}\int_0^\infty P_{\rm gg}^{\rm 2h}(k) k^2 \frac{\sin{kr}}{kr}dk.$$
On large scales the correlation function is sensitive only to the 2-halo term, and only to the number weighted galaxy bias. However on small scales, both the 1-halo and 2-halo terms contribute to the clustering, and the detailed shape of the correlation function is sensitive to the distribution of galaxies within halos. We use the following parametrisation to describe this distribution. Halos are assumed to host a single central galaxy and a number $N_{\rm sat}$ of satellite galaxies if their mass is in excess of $M_{\rm min}$. The number of satellites is taken to be a powerlaw in mass with characteristic mass scale $M_1$ and index $\alpha$. However, motivated by the fact that HI galaxies seem to be underrepresented as satellites in galaxy clusters (Waugh et al. 2002), we also include an upper limit for the halo mass which can contain an HI satellite ($M_{\rm
1,max}$). Thus the mean occupation of a halo of mass $M$ is assumed to be $$\begin{aligned}
\nonumber
\langle N\rangle_M &=& 1 + \langle M\rangle_{\rm sat} \hspace{8mm}\mbox{if}\hspace{5mm}M>M_{\rm min}\\
\nonumber
&=&0\hspace{21.5mm}\mbox{otherwise},\end{aligned}$$ where the number of satellites is defined to be $$\begin{aligned}
\nonumber
\langle N\rangle_{\rm sat} &=& \left( \frac{M}{M_1}\right)^\alpha\hspace{10mm}\mbox{if}\hspace{5mm}M_{\rm min}<M<M_{\rm 1,max}\\
\nonumber
&=&0\hspace{21.5mm}\mbox{otherwise}.\end{aligned}$$
The redshift space HOD model {#HODz}
----------------------------
Tinker (2007) has extended the above model to calculate the redshift space correlation function for a given HOD parametrisation. The redshift space model is again computed based on the sum of 1-halo and 2-halo terms as in equation (\[xi1\]). In redshift space, the apparent recessional velocity of a galaxy is the sum of its motion in the Hubble flow (directly related to its physical distance), and of peculiar velocity (which modifies the apparent distance based on Hubbles constant). The 1-halo term is computed in analogy with equation (\[1h\]), but with an additional integral over the line-of-sight distance and a probability distribution for the line of sight peculiar velocity. The result of these peculiar motions are the so-called fingers-of-god, large line-of-sight features in redshift. Tinker (2007) suggests that the 2-halo term of the redshift space correlation function at apparent line-of-sight ($r_\sigma$) and transverse ($r_\pi$) distances is most easily computed by integrating over the 2-halo term of the real space correlation function, $$1+\xi_{\rm 2h}(r_\sigma,r_\pi)=\int_{-\infty}^\infty [1+\xi_{\rm 2h}(r)] P_{\rm 2h}(v_z|r,\phi)dv_z,$$ where $P_{\rm 2h}(v_z|r,\phi)$ is the probability density for the line-of-sight velocity between pairs in two distinct halos, and $\cos{\phi}=r_\sigma/r$. Here $z^2=r^2-r_\sigma^2$, and $v_z=H(r_\pi-z)$. Calculation of $P_{\rm 2h}$, including determination of fitting formulae to N-body simulations is complex and we refer the reader to Tinker (2007) for details.
Real Space HOD Models of HIPASS Galaxies {#HODobs}
========================================
In this section we describe fitting of HOD models to the real-space correlation function of HIPASS galaxies. Given the systematic uncertainty in the estimate of the correlation function owing to the small survey volume we follow Meyer et al. (2007) and choose to fit both the unweighted and weighted HIPASS correlation functions (although we note that the latter should more fairly estimate the correlation function that would be obtained from a larger, volume limited survey). Our HOD model has four free parameters, $M_{\rm min}$, $M_{1}$, $M_{\rm 1,max}$ and $\alpha$, for combinations of which we compute the real space correlation function, and calculate the likelihood of the model as $$\mathcal{L}(M_{\rm min},M_{1},M_{\rm 1,max},\alpha) = \exp{\left(-\chi^2/2\right)},$$ where $$\begin{aligned}
\nonumber
\chi^2&=&\sum_{i=0}^{N_{\rm obs}}\left(\frac{\log{\xi(r_i|M_{\rm min},M_{1},M_{\rm 1,max},\alpha)}-\log{\xi_{\rm obs}(r_i)}}{\sigma_{\rm obs}(r_i)}\right)^2\\
&&\hspace{5mm}+\left(\frac{\log{\bar{n}_{\rm g}(M_{\rm min},M_{1},M_{\rm 1,max},\alpha)}-\log{n_{\rm gal}}}{\sigma_{\rm gal}}\right)^2.\end{aligned}$$ Here $\xi_{\rm obs}$ is the observed correlation function measured at a number ($N_{\rm obs}$) of radii $r_i$, with uncertainty $\sigma_{\rm obs}(r_i)$ (in dex), and $n_{\rm gal}$ is the observed galaxy density with uncertainty $\sigma_{\rm gal}$ (in dex). We compute the halo density $\bar{n}_{\rm g}$ using the Sheth-Tormen (2002) mass function as part of the HOD model. The error bars on the observational estimates are not symmetric. Note that we assume the correlation function points at different radii to be independent (as should be the case for a small sample, with large Poisson dominated noise). Covariance between measurements of the correlation function at different radii can lead to unrealistically small errors on constrained HOD model parameters. We do not add this layer of sophistication to our analysis, owing to the large uncertainties already introduced into the clustering measurements via the chosen weighting scheme.
We begin by fitting our HOD model to the unweighted real-space clustering of HIPASS galaxies. The upper row of the upper set of panels in Figure \[fig2\] shows contours of the likelihood in 2-d projections of this 4-d parameter space. Here prior probabilities on $\alpha$, $\log{M_{\rm min}}$, $\log{M_{\rm 1}}$ and $\log{M_{\rm
1,max}}$ are assumed to be constant. The contours are placed at 60%, 30% and 10% of the peak likelihood (the location of which is marked by a dot). The lower row shows the corresponding marginalised likelihoods on individual parameters. Meyer et al. (2007) noted that the correlation length of HIPASS galaxies is smaller than for optical surveys. Here we quantify the clustering on large scales via the host halo mass, finding a value of $M_{\rm
min}\sim10^{11.2\pm0.2}$M$_\odot$. On smaller scales, the halo occupation modeling illustrates the requirement of a non-zero 1-halo term in order to reproduce the excess clustering of galaxies at $r\la1$Mpc. We find $M_{1}\sim10^{13.6\pm0.5}$M$_\odot$, which is two orders of magnitude larger than $M_{\rm min}$. The power-law index is tightly correlated with $M_{\rm 1}$, but loosely constrained to be $\alpha\ga1$. Since $M_1$ represents the characteristic mass where satellites outnumber the central galaxies, the large value of $M_1$ indicates that there are only a small number of satellite galaxies in the HIPASS sample.
=6.9in
=7.in
=7.in
In the lower set of panels in Figure \[fig2\] we repeat this analysis for the weighted estimate of the HIPASS real-space correlation function. Here we find best fit estimates of $M_{\rm min}\sim10^{11.5\pm0.3}$M$_\odot$, and $M_{\rm
1}\sim10^{12.2\pm0.5}$M$_\odot$. There is greater tension between the galaxy density and clustering amplitude in this case leading to larger values of $\chi^2$ for the best fit. We find $M_{\rm
1}\sim10M_{\rm min}$, smaller than the difference found in the unweighted case. However the value of the power-law slope is loosely constrained to be $\alpha\sim0.7\pm0.4$, weakly preferring satellites to be in smaller halos (but consistent with a linear relation). In this case $M_1$ and $\alpha$ are again tightly correlated, with a smaller value of $M_1$ associated with a shallower index $\alpha$ in order to produce the low amplitude of the small scale clustering.
If $\alpha$ is forced to equal unity in our analysis, then we find $M_{\rm 1}=10^{13}$M$_\odot\sim50-100M_{\rm min}$ for both the unweighted and weighted estimates. For comparison, with $\alpha=1$, the red galaxy sample (chosen to exclude gas rich galaxies with a large star formation rate) from Brown et al. (2008) has clustering described by $M_{1}\sim3$M$_{\rm min}$, while clustering of galaxies in the Sloan Digital Sky Survey (including both gas-rich and gas-poor galaxies) suggests $M_{\rm 1}\sim20M_{\rm min}$ (Zehavi et al. 2005). Thus the qualitative conclusions of both the weighted and unweighted estimates of the HIPASS correlation function are consistent; namely that HI satellites in groups and clusters are rare compared to the results of optical clustering studies. We return to quantify this point further in § \[discussion\]. The effect of satellites on the real space correlation function at small scales is illustrated in Figure 5 of Meyer et al. (2007), where it can be seen that HI selected HIPASS galaxies have a smaller correlation length than optically selected samples, but also that the difference in amplitude of the correlation function is greatest at scales less than 1Mpc, where the 1-halo term dominates. Thus, by determining the relationship between $M_{\rm min}$ and $M_{1}$, the real space HOD correlation function quantifies previous suggestions that HI galaxies are under-represented in overdense environments (Waugh et al. 2002).
The inferred values of $M_{\rm min}$ for the HIPASS galaxies are quantitatively consistent between the unweighted and weighted clustering estimates, making estimates of the halo mass for HIPASS galaxies fairly robust (we note that the estimates partly driven by the galaxy density, which is common between the two cases). Moreover, the clustering estimate of host mass from the unweighted HIPASS correlation function is easily reconciled with the dynamical estimates of HIPASS galaxy host masses shown in Figure \[fig1\], for which the logarithmic means are $\langle
\log_{10}(M/\mbox{M}_\odot)\rangle=11.1$ for both of the dynamical mass estimates presented.
The HI mass fraction in HIPASS galaxies
---------------------------------------
The halo mass estimates derived from the combination of clustering and density of HI galaxies allow the fraction of baryonic mass in galaxies that is in the form of HI ($f_{\rm HI}$) to be estimated. To this end we first assume that the hydrogen to dark-matter mass ratio is the same within galaxies as in the mean universe, so that the total hydrogen mass within a halo of mass $M$ is $\sim \Omega_{\rm
b}/\Omega_{\rm M}M$. We then assume that the baryon to dark matter mass is the same for all halos, yielding $$f_{\rm HI}\sim\frac{\Omega_{\rm M}}{\Omega_{\rm b}}\frac{M_{\rm HI,lim}}{M_{\rm min}}.$$ Including the systematic uncertainty as estimated by the differing results for $M_{\rm min}$ from the unweighted and weighted clustering measurements, we find $f_{\rm HI}\sim10^{-1.4\pm0.4}$. Thus we find that less than 10% of baryons within HI selected galaxies exists in the form of HI.
Redshift Space HOD Models of HIPASS Galaxies {#HODobsz}
============================================
The line-of-sight structure of the redshift-space correlation function is dominated by gravitational infall on large transverse scales, and by virial motions of satellites on small transverse scales. Both of these features can be seen in the unweighted HIPASS redshift space correlation function (plotted as the thin contours in the right hand panels of Figures \[fig3\] and \[fig4\]), though the fingers-of-god are less pronounced than expected based on optical galaxy redshift surveys (Meyer et al. 2007). As mentioned above in § \[clustering\], the small volume of the HIPASS survey suggests that the correlation function should be constructed using a weighting scheme so that it is not dominated by galaxy pairs near the peak of the selection function. However this weighting introduces systematic uncertainty into the determination of the correlation function. The weighted redshift space correlation functions (plotted as the thin contours in the right hand panel sets of Figure \[fig4\]) show evidence for infall, but marginal or no evidence for fingers-of-god.
Additional information on the satellite galaxy distribution is contained in the redshift space correlation function. In redshift space, the line-of-sight structure of the 2-halo term is governed by gravitational infall, while the 1-halo term is dominated by the virial motions of satellite galaxies producing the so-called fingers-of-god. In this section we turn to calculation of the redshift space correlation function using the analytic HOD model of Tinker (2007). Given the large uncertainties in the construction of the HIPASS correlation function, we do not fit the redshift space correlation function directly. Rather, based on the parameter constraints in Figure \[fig2\] we calculate examples of the redshift space correlation function for qualitative comparison with the HIPASS clustering. These examples, and their comparison with the HIPASS redshift space correlation function, offer some hints regarding the satelite distribution that are not available from the real space correlation function alone. They also indicate the way in which the full 3-dimensional shape of the correlation function could be utilized within a larger, more statistically representative sample.
Three examples are shown in each of Figures \[fig3\] and \[fig4\] for comparison with each of the unweighted and weighted determinations of the HIPASS correlation function. The chosen HOD models have parameters which adequately describe the real-space clustering, as shown in the left hand panels. In each case the models differ in the values chosen for various parameters. These values effect the occupation of dark matter halos as shown in the central panels of Figures \[fig3\] and \[fig4\]. For example, smaller values of $\alpha$ and $M_1$ preferentially place the required number of satellites in smaller halos, and so reduce the prominence of the fingers-of-god. A smaller value of $M_1$ also lowers the typical mass at which satellites become common, and so increases the fraction of galaxies that are satellites (the fractions are listed in the central panels). These two parameters are varied between the upper 2 panels of Figures \[fig3\] and \[fig4\]). In the weighted case, the models also differ in the value of $M_{\rm min}$, with decreasing values from top to bottom. Larger values of $M_{\rm min}$ (and hence larger values of bias) lead to smaller values of $\beta\equiv\Omega_{\rm m}^{0.6}/b$, and in turn to a real-space correlation function that is less compressed along the line of sight on large transverse scales (as can be seen in the correlation functions of Figures \[fig3\] and \[fig4\]). However the modeled fingers-of-god are more prominent than is the case in the HIPASS data for each of these cases.
In the lower panels of Figures \[fig3\] and \[fig4\] we show examples that impose an upper limit on the host mass containing satellite galaxies. By excluding the presence of satellites in massive halos, the values of $M_{\rm 1,max}=10^{14.3}$M$_\odot$ and $M_{\rm 1,max}=10^{13.7}$M$_\odot$ in the unweighted and weighted cases force the required number of satellites to reside in smaller halos. This reduces the prominence of the fingers-of-god, which are sensitive to the magnitude of satellite virial motions within the host halo. As a result these models yield fingers-of-god which are of comparable strength to those seen in the HIPASS data. On the other hand, these same fits to the unweighted estimate of the real-space correlation function predict line-of-sight compressions at large transverse separations \[Kaiser (1987) effect\] that appears to be too large to explain the HIPASS data[^1]. In the weighted case the correlation function amplitude is below the observed estimate owing to the tension between the density and correlation function amplitudes in this case.
Satelite fraction {#discussion}
=================
Taken together the results of our modeling suggest that HI rich satellite galaxies are not common in HIPASS, or else the 1-halo term would be more prominent in the real-space correlation function. This is quantified in Figure \[fig5\], where we show the likelihoods (per unit logarithm) for the ratio $\langle N\rangle_{\rm sat}/\langle
N\rangle_M$ obtained by marginalising over the HOD distributions shown in Figures \[fig3\] and \[fig4\] for the unweighted and weighted HIPASS correlation functions respectively. A range of HOD models can describe the HIPASS real space correlation function, and our fits include a range of values with means near $\sim 3\%$ and $\sim10\%$ for the fraction of satellites in the unweighted and weighted cases. Although unlikely, we find that the weighted estimate of the real space correlation function can be described with HOD models for which the satelite fraction is greater than 20%. However we find that HOD models which have more than 20% of the galaxies as satellites have fingers-of-god that are too prominent (e.g. see Figure \[fig4\]). The satelite fraction of $\langle N\rangle_{\rm
sat}/\langle N\rangle_M\sim0.20$ should therefore be considered an upper limit for HIPASS galaxies.
For comparison, typical fits to the halo occupation distribution of optical samples have satellite fractions that vary with galaxy luminosity and type. For example, the HOD modeling of Brown et al. (2008) implies a satelite galaxy fraction of $\langle
N\rangle_{\rm sat}/\langle N\rangle_M\sim 0.5$ among red galaxies with $0.2<z<0.4$ and a comparable space density to HIPASS. This suggests that red galaxies (which are HI poor) are more common among the satelite population than HI selected galaxies. On the other hand, for galaxies in the Sloan Digital Sky Survey with r-band absolute magnitudes in excess of -19 (again a sample with a comparable density to HIPASS galaxies) the HOD parametrization found in Zehavi et al. (2005) implies a satelite fraction of $\langle N\rangle_{\rm sat}/\langle
N\rangle_M\sim 0.25$. This value lies between the fraction we find from HIPASS, and the fraction found for red galaxies (Brown et al. 2008). Zehavi et al. (2005) divide their galaxy population into blue and red galaxies. They find that the red galaxy population has a steeper correlation function, which, when interpreted in terms of the HOD model implies that satelite galaxies are rarer among blue galaxies than among red galaxies. Thus there appears to be a sequence of satellite fractions. A sub-sample of red galaxies includes a larger proportion of satellites than does a sub-sample of blue galaxies, which in turn has a larger proportion of satellites than an HI selected sub-sample of galaxies.
=2.2in
Thus, as with observations of optical galaxy clustering, studying how HI galaxies populate dark matter halos provides important insights into how galaxies are assembled and evolve over cosmic time. For example, if massive galaxies grow largely via galaxy mergers rather than in-situ star formation, then star forming galaxies with large HI masses will be largely absent from the most massive dark matter halos. HI selected satellite galaxies will also be rare if HI galaxies merge rapidly after the merger of their host dark matter halos. Similarly, if HI is consumed or removed from satellite galaxies within dark matter halos, then satellite galaxies would be under-represented in HI surveys relative to optical surveys as seems to be the case based on our analysis of HIPASS. Although our HOD results are suggestive of these scenarios, the precision with which the HI HOD can be studied with HIPASS is limited. However the much larger volumes that will become available with the advent of deeper HI surveys such as those to be undertaken with the Australian SKA Pathfinder (ASKAP, Johnston et al. 2008) will allow more detailed comparison of the halo occupation of stars and HI. This will in turn facilitate formulation of a more detailed understanding of the growth of stellar mass in galaxies.
Summary
=======
In this paper we have analysed the clustering properties of HI selected galaxies from the HIPASS survey using the formalism of the halo occupation distribution. Use of the HOD model separates the clustering amplitude into contributions from galaxy pairs that are in the same halo (the 1-halo term) and pairs that reside in different halos (the 2-halo term). The real-space clustering amplitude is significant on scales below the virial radius associated with the halo mass required to reproduce the clustering amplitude on large scales, indicating that single halo pairs are contributing a 1-halo term. However the resulting parameter constraints show that satellite galaxies make up only $\sim10\%$ of the HIPASS sample. HI satelite galaxies are therefore less significant in number and in terms of their contribution to clustering statistics than are satellites in optically selected galaxy redshift surveys. Thus HOD modeling of HI galaxy clustering quantifies the extent to which environment governs the HI content of galaxies and confirms previous evidence that HI galaxies are relatively rare in overdense environments (Waugh et al. 2002; Cortes et al. 2008). Through our real-space modeling of HIPASS clustering we find a minimum halo mass for HIPASS galaxies at the peak of the redshift distribution of $M\sim10^{11}$M$_\odot$, and show that less than 10% of baryons in HIPASS galaxies are in the form of HI.
Our analysis reveals significant degeneracies in the HOD parameters that give acceptable fits to the real-space HI correlation function. However the extra line-of-sight dimension in the redshift-space correlation function helps to break these degeneracies because the fingers-of-god are sensitive to the typical halo mass in which satellite galaxies reside. Our analysis of the redshift space correlation function indicates that in order to get fingers-of-god in a model which are as subtle as those in the HIPASS observations, the HI rich satellites required to produce the measured 1-halo term must be preferentially in group rather than cluster mass halos. In our modeling the best representations of the fingers-of-god are obtained by imposing an upper limit on the halo mass where HI satellites are found of $\sim10^{13.7-14.3}$M$_\odot$. This finding is in accord with direct observations of rich optical clusters, which show no overdensity of HI galaxies relative to the field (Waugh et al. 2002; Cortes et al. 2008). Quantitative constraints on HOD models from the HIPASS survey are limited by the small survey volume, which makes the determination of the correlation function systematically uncertain (Meyer et al. 2007). Future deeper HI surveys with telescopes like the Australian SKA Pathfinder (ASKAP) will survey a much larger volume (Johnston et al. 2008) and allow the distribution of HI with environment to be studied in more detail via precise measurements of clustering in HI galaxies.
The cosmic star-formation rate has declined by more than an order of magnitude in the past 8 billion years (Lilly et al. 1996, Madau et al. 1996). The decline is observed across all wavelengths (Hopkins 2004 and references therein) and apparently defies observational limitations such as sample selection and cosmic variance (Westra & Jones 2008). Optical studies paint a somewhat passive picture of galaxy formation, with the stellar mass density of galaxies gradually increasing and an increasing fraction of stellar mass mass ending up within red galaxies that have negligible star-formation (e.g., Brown et al. 2008). However optical studies can only address part of the picture. Currently, the combination of direct HI observations at low redshift (Zwaan et al. 2005b; Lah et al 2007) and damped Ly$\alpha$ absorbers in the spectra of high-redshift QSOs (Prochaska et al. 2005) show that the neutral gas density has remained remarkably constant over the age of the universe. At these levels, and without replenishment, HI gas would be exhausted in a few billion years (Hopkins et al 2008). Models incorporating gas infall that balances star formation and gas outflow are therefore necessary to reproduce observed star formation densities (eg. Erb 2008). The evolutionary and environmental relationships between the neutral gas which provides the fuel for star formation and the stars that form are central to understanding these and related issues. The study of the halo occupation distribution of HI based on HIPASS galaxies presented in this paper provides the first quantitative hints of this relationship.
[**Acknowledgments.**]{} The research was supported by the Australian Research Council (JSBW).
Barnes, D. G., et al. 2001, [MNRAS]{}, 322, 486
Basilakos, S., Plionis, M., Kova[č]{}, K., & Voglis, N. 2007, [MNRAS]{}, 378, 301
Berlind, A. A., & Weinberg, D. H. 2002, [ApJ]{}, 575, 587
Brown, M. J. I., et al. 2008, [ApJ]{}, 682, 937
Cortese, L., et al. 2008, [MNRAS]{}, 383, 1519
de Blok, W. J. G., Zwaan, M. A., Dijkstra, M., Briggs, F. H., & Freeman, K. C. 2002, Astron. Astrophys., 382, 43
Drinkwater, M. J., Gregg, M. D., & Colless, M. 2001, [ApJL]{}, 548, L139
Erb, D. K. 2008, [ApJ]{}, 674, 151
Hernquist, L. 1990, [ApJ]{}, 356, 359
Hopkins, A. M. 2004, [ApJ]{}, 615, 209
Hopkins, A. M., McClure-Griffiths, N. M., & Gaensler, B. M. 2008, [ApJL]{}, 682, L13
Jenkins, A., Frenk, C. S., White, S. D. M., Colberg, J. M., Cole, S., Evrard, A. E., Couchman, H. M. P., & Yoshida, N. 2001, [MNRAS]{}, 321, 372
Johnston, S., et al. 2008, Experimental Astronomy, 22, 151
Kaiser, N., 1987, Mon. Not. R. Astron. Soc, [227]{}, 1
Kochanek, C. S., & White, M. 2001, [ApJ]{}, 559, 531
Komatsu, E., et al. 2008, ArXiv e-prints, 803, arXiv:0803.0547
Lah, P., et al. 2007, [MNRAS]{}, 376, 1357
Lilly, S. J., Le Fevre, O., Hammer, F., & Crampton, D. 1996, [ApJL]{}, 460, L1
Madau, P., Ferguson, H. C., Dickinson, M. E., Giavalisco, M., Steidel, C. C., & Fruchter, A. 1996, [MNRAS]{}, 283, 1388
Meyer, M. J., et al. 2004, [MNRAS]{}, 350, 1195
Meyer, M. J., Zwaan, M. A., Webster, R. L., Brown, M. J. I., & Staveley-Smith, L. 2007, [ApJ]{}, 654, 702
Moster, B. P., Somerville, R. S., Maulbetsch, C., van den Bosch, F. C., Maccio’, A. V., Naab, T., & Oser, L. 2009, arXiv:0903.4682
Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, [ApJ]{}, 490, 493
Peacock, J. A., & Smith, R. E. 2000, [MNRAS]{}, 318, 1144
Prochaska, J. X., Herbert-Fort, S., & Wolfe, A. M. 2005, [ApJ]{}, 635, 123
Scoccimarro, R., Sheth, R. K., Hui, L., & Jain, B. 2001, [ApJ]{}, 546, 20
Seljak, U. 2000, [MNRAS]{}, 318, 203
Sheth, R. K., & Tormen, G. 2002, [MNRAS]{}, 329, 61
Sheth, R.K., Mo, H.J. & Tormen, G., 2001, Mon. Not. R. Astron. Soc., [323]{}, 1
Tinker, J. L. 2007, [MNRAS]{}, 374, 477
Verheijen, M., van Gorkom, J., Szomoru, A., Dwarakanath, K. S., Poggianti, B., & Schiminovich, D. 2007, New Astronomy Review, 51, 90
Waugh, M., et al. 2002, [MNRAS]{}, 337, 641
Westra, E., & Jones, D. H. 2008, [MNRAS]{}, 383, 339
Zehavi, I., et al. 2004, [ApJ]{}, 608, 16
Zehavi, I., et al. 2005, [ApJ]{}, 630, 1
Zheng, Z. 2004, [ApJ]{}, 610, 61
Zwaan, M. A., et al. 2004, [MNRAS]{}, 350, 1210
Zwaan, M. A., Meyer, M. J., Staveley-Smith, L., & Webster, R. L. 2005a, [MNRAS]{}, 359, L30
Zwaan, M. A., van der Hulst, J. M., Briggs, F. H., Verheijen, M. A. W., & Ryan-Weber, E. V. 2005b, [MNRAS]{}, 364, 1467
\[lastpage\]
[^1]: Note that we have plotted the redshift space correlation function using reflections of the measured correlation in the first quadrant to fill the remaining three quadrants. As a result, features due to noise in the correlation function are repeated and could give the impression of a systematic difference between the shape of the data and model correlation functions where no statistically significant difference exists.
|
---
abstract: 'We propose a scheme to create holes in the statistical distribution of excitations of a nanomechanical resonator. It employs a controllable coupling between this system and a Cooper pair box. The success probability and the fidelity are calculated and compared with those obtained in the atom-field system via distinct schemes. As an application we show how to use the hole-burning scheme to prepare (low excited) Fock states.'
address:
- 'Universidade Paulista, Rod. BR 153, km 7, 74845-090 Goiânia, GO, Brazil.'
- 'Universidade Estadual de Goiás, Rod. BR 153, 3105, 75132-903 Anápolis, GO, Brazil.'
- 'Instituto de Física, Universidade Federal de Goiás, 74001-970 Goiânia, GO, Brazil.'
author:
- 'C. Valverde'
- 'A.T. Avelar'
- 'B. Baseia'
title: Hole burning in a nanomechanical resonator coupled to a Cooper pair box
---
,
Quantum state engineering ,Superconducting circuits ,Nanomechanical Resonator,Cooper Pair Box
03.67.Lx, 85.85.+j, 85.25.C, 32. 80. Bx ,42.50.Dv
Introduction
============
Nanomechanical resonators (NR) have been studied in a diversity of situations, as for weak force detections [@Bocko96], precision measurements [@Munro02], quantum information processing [@Cleland04], etc. The demonstration of the quantum nature of mechanical and micromechanical devices is a pursued target; for example, manifestations of purely nonclassical behavior in a linear resonator should exhibit energy quantization, the appearance of Fock states, quantum limited position-momentum uncertainty, superposition and entangled states, etc. NR can now be fabricated with fundamental vibrational mode frequencies in the range MHz – GHz [@a1; @a2; @huang]. Advances in the development of micromechanical devices also raise the fundamental question of whether such systems that contain a macroscopic number of atoms will exhibit quantum behavior. Due to their sizes, quantum behavior in micromechanical systems will be strongly influenced by interactions with the environment and the existence of an experimentally accessible quantum regime will depend on the rate at which decoherence occurs [@a3; @aaa]. One crucial step in the study of nanomechanical systems is the engineering and detection of quantum effects of the mechanical modes. This can be achieved by connecting the resonators with solid-state electronic devices [@a4; @a5; @a6; @a7; @a8], such as a single-electron transistor. NR has also been used to study quantum nondemolition measurement [@a8; @a9; @a10; @a11], quantum decoherence [a7,a13]{}, and macroscopic quantum coherence phenomena [@a14]. The fast advance in the tecnique of fabrication in nanotecnology implied great interest in the study of the NR system in view of its potential modern applications, as a sensor, largely used in various domains, as in biology, astronomy, quantum computation [@a114; @a1144], and more recently in quantum information [@Cleland04; @a15; @a16; @a16a; @a17; @a19; @a20] to implement the quantum qubit [@a16], multiqubit [@a16aa] and to explore cooling mechanisms [@a21; @a22; @a23; @a24; @a25; @a25a], transducer techniques [a26,a27,a28]{}, and generation of nonclassical states, as Fock [@a29], Schrödinger-cat [@a7; @a30; @cv], squeezed states [@a31; @a32; @a33; @a34; @a344], including intermediate and other superposition states [@cv1; @cv2]. In particular, NR coupled with superconducting charge qubits has been used to generate entangled states [@a7; @a30; @a35; @a36]. In a previous paper Zhou and Mizel [@a34] proposed a scheme to create squeezed states in a NR coupled to Cooper pair box (CPB) qubit; in it the NR-CPB coupling is controllable. Such a control comes from the change of external parameters and plays an important role in quantum computation, allowing us to set ON and OFF the interaction between systems on demand.
Now, the storage of optical data and communications using basic processes belonging to the domain of the quantum physics have been a subject of growing interest in recent years [@blais]. Concerned with this interest, we present here a feasible experimental scheme to create holes in the statistical distribution of excitations of a coherent state previously prepared in a NR. In this proposal the coupling between the NR and the CPB can be controlled continuously by tuning two external biasing fluxes. The motivation is inspired by early investigations on the production of new materials possessing holes in their fluorescent spectra [@a1a] and also inspired by previous works of ours, in which we have used alternative systems and schemes to attain this goal [@malboi; @ard1; @ard]. The desired goal in producing holes with controlled positions in the number space is their possible application in quantum computation, quantum cryptography, and quantum communication. As argued in [@ard1], these states are potential candidates for optical data storage, each hole being associated with some signal (say YES, $\left\vert 1\right\rangle $, or $\left\vert +\right\rangle $) and its absence being associated with an opposite signal (NO, $\left\vert 0\right\rangle $, or $\left\vert
-\right\rangle $). Generation of such holes has been treated in the contexts of cavity-QED [@ard] and traveling waves [@av].
Model hamiltonian for the CPB-NR system
=======================================
There exist in the literature a large number of devices using the SQUID-base, where the CPB charge qubit consists of two superconducting Josephson junctions in a loop. In the present model a CPB is coupled to a NR as shown in Fig. (\[cooper\]); the scheme is inspired in the works by Jie-Qiao Liao et al. [@a16a] and Zhou et al. [@a34] where we have substituted each Josephson junction by two of them. This creates a new configuration including a third loop. A superconducting CPB charge qubit is adjusted via a voltage $V_{1}$ at the system input and a capacitance $C_{1}$. We want the scheme ataining an efficient tunneling effect for the Josephson energy. In Fig.(\[cooper\]) we observe three loops: one great loop between two small ones. This makes it easier controlling the external parameters of the system since the control mechanism includes the input voltage $V_{1}$ plus three external fluxes $\Phi (\ell ),$ $\Phi (r)$ and $\Phi _{e}(t)$. In this way one can induce small neighboring loops*.* The great loop contains the NR and its effective area in the center of the apparatus changes as the NR oscillates, which creates an external flux $\Phi
_{e}(t)$ that provides the CPB-NR coupling to the system.
![*Model for the CPB-NMR coupling.*[]{data-label="cooper"}](cpb){width=".35\textwidth"}
In this work we will assume the four Josephson junctions being identical, with the same Josephson energy $E_{J}^{0}$, the same being assumed for the external fluxes $\Phi (\ell )$ and $\Phi (r)$, i.e., with same magnitude, but opposite sign: $\Phi (\ell )=-\Phi (r)=\Phi (x)$. In this way, we can write the Hamiltonian describing the entire system as
$$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+4E_{c}\left( N_{1}-\frac{1}{2}\right) \hat{\sigma}_{z}-4E_{J}^{0}\cos \left( \frac{\pi \Phi _{x}}{\Phi _{0}}\right) \cos \left( \frac{\pi \Phi _{e}}{\Phi _{0}}\right) \hat{\sigma}_{x},
\label{a1}$$
where $\hat{a}^{\dagger }(\hat{a})$ is the creation (annihilation) operator for the excitation in the NR, corresponding with the frequency $\omega $ and mass $m$; $E_{J}^{0}$ and $E_{c}$ are respectively the energy of each Josephson junction and the charge energy of a single electron; $C_{1}$ and $C_{J}^{0}$ stand for the input capacitance and the capacitance of each Josephson tunel, respectively $\Phi _{0}=h/2e$ is the quantum flux and $N_{1}=C_{1}V_{1}/2e$ is the charge number in the input with the input voltage $V_{1}$. We have used the Pauli matrices to describe our system operators, where the states $\left\vert g\right\rangle $ and $\left\vert
e\right\rangle $ (or 0 and 1) represent the number of extra Cooper pairs in the superconduting island. We have: $\hat{\sigma}_{z}=\left\vert
g\right\rangle \left\langle g\right\vert -\left\vert e\right\rangle
\left\langle e\right\vert $, $\hat{\sigma}_{x}=\left\vert g\right\rangle
\left\langle e\right\vert -\left\vert e\right\rangle \left\langle
g\right\vert $ and $E_{C}=e^{2}/\left( C_{1}+4C_{J}^{0}\right) .$
The magnectic flux can be written as the sum of two terms, $$\Phi _{e}=\Phi _{b}+B\ell \hat{x}\text{ }, \label{a4}$$where the first term $\Phi _{b}$ is the induced flux, corresponding to the equilibrium position of the NR and the second term describes the contribution due to the vibration of the NR; $B$ represents the magnectic field created in the loop. We have assumed the displacement $\hat{x}$ described as $\hat{x}=x_{0}(\hat{a}^{\dagger }+\hat{a})$, where $x_{0}=\sqrt{m\omega /2}$ is the amplitude of the oscillation.
Substituting the Eq.(\[a4\]) in Eq.(\[a1\]) and controlling the flux $\Phi _{b}$ we can adjust $\cos \left( \frac{\pi \Phi _{b}}{\Phi _{0}}\right)
=0$ to obtain $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+4E_{c}\left( N_{1}-\frac{1}{2}\right) \hat{\sigma}_{z}-4E_{J}^{0}\cos \left( \frac{\pi \Phi _{x}}{\Phi _{0}}\right) \sin \left( \frac{\pi B\ell \hat{x}}{\Phi _{0}}\right) \hat{\sigma}_{x}, \label{a8}$$and making the approximation $\pi B\ell x/\Phi _{0}<<1$ we find $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+\frac{1}{2}\omega _{0}\hat{\sigma}_{z}+\lambda _{0}(\hat{a}^{\dagger }+\hat{a})\hat{\sigma}_{x}, \label{a9}$$where the constant coupling $\lambda _{0}=-4E_{J}^{0}\cos \left( \frac{\pi
\Phi _{x}}{\Phi _{0}}\right) \left( \frac{\pi B\ell x_{0}}{\Phi _{0}}\right)
$ and the effective energy $\omega _{0}=8E_{c}\left( N_{1}-\frac{1}{2}\right) .$ In the rotating wave approximation the above Hamiltonian results as $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+\frac{1}{2}\omega _{0}\hat{\sigma}_{z}+\lambda _{0}(\hat{\sigma}_{+}\hat{a}+\hat{a}^{\dagger }\hat{\sigma}_{-}).$$
Now, in the interaction picture the Hamiltonian is written as, $\hat{H}_{I}=\hat{U}_{0}^{\dagger }\hat{H}\hat{U}_{0}-i\hbar \hat{U}_{0}^{\dagger }\frac{\partial \hat{U}_{0}}{\partial t},$ where $\hat{U}_{0}=\exp \left[ -i\left(
\omega \hat{a}^{\dagger }\hat{a}+\frac{\omega _{0}\hat{\sigma}_{z}}{2}\right) t\right] $ is the evoluion operator. Assuming the system operating under the resonant condition, i.e., $\omega =\omega _{0}$, and setting $\hat{\sigma}_{z}=\hat{\sigma}_{+}\hat{\sigma}_{-}-\hat{\sigma}_{-}\hat{\sigma}_{+} $ and $\hat{\sigma}_{\pm }=$ $\left( \hat{\sigma}_{x}\pm i\hat{\sigma}_{y}\right) /2\ ,$ with $\hat{\sigma}_{y}=(\left\vert e\right\rangle
\left\langle g\right\vert -\left\vert e\right\rangle \left\langle
g\right\vert )/i$ the interaction Hamiltonian is led to the abbreviated form,
$$\hat{H}_{I}=\beta \left( \hat{a}^{\dagger }\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+}\right) , \label{a11}$$
where $\beta =-\lambda _{0},$ $\hat{\sigma}_{+}$ $(\hat{\sigma}_{-})$ is the raising (lowering) operator for the CPB.
We note that the coupling constant $\beta $ can be controlled through the flux $\Phi _{x}$, which influences the mentioned small loops* *in the left and right places. Furthermore, we can control the gate charge $N_{1}$ via the gate voltage $V_{1}$ syntonized to the coupling. It should be mentioned that the energy $\omega _{0}$ depends on the induced flux $\Phi
_{x}$. So, when we syntonize the induced flux $\Phi _{x}$ the energy $\omega
_{0}$ changes. To avoid unnecessary transitions during these changes, we assume the changes in the flux being slow enough to obey the adiabatic condition.
Next we show how to make holes in the statistical distribution of excitations in the NR. We start from the CPB initially prepared in its ground state $\left\vert CPB\right\rangle =\left\vert g\right\rangle ,$ and the NR initially prepared in the coherent state, $\left\vert NR\right\rangle
=\left\vert \text{$\alpha $}\right\rangle .$Then the state $\left\vert \Psi
\right\rangle $ that describes the intire system (CPB plus NR) evolves as follows $$\left\vert \Psi _{NC}(t)\right\rangle =\hat{U}(t)\left\vert g\right\rangle
\left\vert \alpha \right\rangle , \label{h}$$where $\hat{U}(t)=\exp (-it\hat{H}_{I})$ is the (unitary) evolution operator and $\hat{H}_{I}$ is the interaction Hamiltonian, given in Eq. (\[a11\]).
Setting $\hat{\sigma}_{+}=\left\vert g\right\rangle \left\langle
e\right\vert $ and $\hat{\sigma}_{-}=\left\vert e\right\rangle
\left\langle g\right\vert $ we obtain after some algebra,
$$\begin{aligned}
\hat{U}(t) &=&\cos (\beta t\sqrt{\hat{a}^{\dagger }\hat{a}+1})\left\vert
g\right\rangle \left\langle g\right\vert \text{ }+\text{ }\cos (\beta t\sqrt{\hat{a}^{\dagger }\hat{a}})\left\vert e\right\rangle \left\langle
e\right\vert \notag \\
&&-i\frac{\sin (\beta t\sqrt{\hat{a}^{\dagger }\hat{a}+1})}{\sqrt{\hat{a}^{\dagger }\hat{a}+1}}\hat{a}\left\vert g\right\rangle \left\langle
e\right\vert \text{ }-i\frac{\sin (\beta t\sqrt{\hat{a}^{\dagger }\hat{a}})}{\sqrt{\hat{a}^{\dagger }\hat{a}}}\hat{a}^{\dagger }\left\vert e\right\rangle
\left\langle g\right\vert .\end{aligned}$$
In this way, the evolved state in Eq.(\[h\]) becomes$$\left\vert \Psi _{NC}(t)\right\rangle =e^{-\frac{\left\vert \alpha
\right\vert ^{2}}{2}}\sum_{n=0}^{\infty }\frac{\alpha ^{n}}{\sqrt{n!}}[\cos
(\omega _{n}\tau )\left\vert g,n\right\rangle \text{ }-i\sin (\omega
_{n}\tau )\left\vert e,n+1\right\rangle ], \label{h1}$$where $\omega _{n}=\beta \sqrt{n+1}$, If we detect the CPB in the state $\left\vert g\right\rangle $ after a convenient time interval $\tau _{1}$ then the state $\left\vert \Psi _{NC}(t)\right\rangle $ reads $$\left\vert \Psi _{NC}(\tau _{1})\right\rangle =\eta _{1}\sum_{n=0}^{\infty }\frac{\alpha ^{n}}{\sqrt{n!}}\cos (\omega _{n}\tau _{1})\left\vert
n\right\rangle , \label{h2}$$where $\eta _{1}$ is a normalization factor. If we choose $\tau _{1}$ in a way that $\beta \sqrt{n_{1}+1}\tau _{1}=\pi /2$, the component $\left\vert
n_{1}\right\rangle $ in the Eq.(\[h2\]) is eliminated.
In a second step, supose that this first CPB is rapidly substituted by another one, also in the initial state $\left\vert g\right\rangle$, that interacts with the NR after the above detection. For the second CPB the initial state of the NR is the state given in Eq.(\[h2\]), produced by the detection of the first CPB in $\left\vert g\right\rangle $. As result, the new CPB-NR system evolves to the state $$\left\vert \Psi _{NC}(\tau _{2})\right\rangle =\sum_{n=0}^{\infty }\frac{\alpha ^{n}}{\sqrt{n!}}[\cos (\omega _{n}\tau _{2})\cos (\omega _{n}\tau
_{1})\left\vert g,n\right\rangle -i\cos (\omega _{n}\tau _{1})\sin (\omega
_{n}\tau _{2})\left\vert e,n+1\right\rangle ]. \label{h3}$$
Next, the detection of the second CPB again in the state $\left\vert
g\right\rangle $ leads the entire system collapsing to the state $$\left\vert \Psi _{NC}(\tau _{2})\right\rangle =\eta _{2}\sum_{n=0}^{\infty }\frac{\alpha ^{n}}{\sqrt{n!}}[\cos (\omega _{n}\tau _{2})\cos (\omega
_{n}\tau _{1})\left\vert n\right\rangle ], \label{h4}$$where $\eta _{2}$ is a normalization factor. In this way, the choice $\beta
\sqrt{n_{2}+1}\tau _{2}=\pi /2$ makes a second hole, now in the component $\left\vert n_{2}\right\rangle $.
By repeating this procedure $M$ times we obtain the generalized result for the $M-th$ CPB detection as$$\left\vert \Psi _{NC}(\tau _{M})\right\rangle =\eta _{M}\sum_{n=0}^{\infty }\frac{\alpha ^{n}}{\sqrt{n!}}\prod\limits_{j=1}^{M}\cos (\omega _{n}\tau
_{j})\left\vert n\right\rangle , \label{h5}$$where $\tau _{j}$ is the $j-th$ CPB-NR interaction time. According to the Eq (\[h5\]) the number of CPB being detected coincides with the number of holes produced in the statistical distribution. In fact the Eq (\[h5\]) allows one to find the expression for the statistical distribution, $P_{n}=\left\vert \langle n|\Psi _{NC}(\tau _{M})\right\vert ^{2};$ a little algebra furnishes $$P_{n}=\frac{(\alpha ^{2n}/n!)\prod_{j=1}^{M}cos^{2}(\omega _{n}\tau _{j})}{\sum_{m=0}^{\infty }(\alpha ^{2m}/m!)\prod_{j=1}^{M}cos^{2}(\omega _{n}\tau
_{j})}, \label{buracos}$$To illustrate results we have plotted the Fig.(\[figure1\]) showing the controlled production of holes in the photon number distribution.
The success probability to produce the desired state is given by $$P_{s}=e^{-\left\vert \alpha \right\vert ^{2}}\sum_{m=0}^{\infty }(\alpha
^{2m}/m!)\prod_{j=1}^{M}cos^{2}(\omega _{n}\tau _{j}). \label{ps}$$Note that the holes exhibited in Fig.(\[figure1\])(a), \[figure1\](b), and \[figure1\](c) occur with success probability of $9\%$, $4\%$, and $0.3\%$, respectively.
We can take advantage of the this procedure applying it to the engineering of nonclassical states, e.g., to prepare Fock states [@Escher05PRA] and their superpositions [@Aragao04PLA]. To this end, we present two strategies: in the first we eliminates the components on the left and right sides of a desired Fock state $|N\rangle $, namely: $|N-1\rangle ,$ $|N-2\rangle ,...$and $|N+1\rangle ,|N+2\rangle ,...;$ in the second one, we only eliminate the left side components of a desired Fock state $|N\rangle $. In both cases, it is convenient to consider the final state of the NR as, $$\left\vert \Psi _{NC}(\tau _{M})\right\rangle ^{\prime }=\eta _{M}^{\prime
}\sum_{n=0}^{\infty }\frac{\alpha ^{n}}{\sqrt{n!}}(-i)^{M}\prod\limits_{j=1}^{M}\sin (\omega _{n+j}\tau _{j})\left\vert n+M\right\rangle ,
\label{h6}$$which is easily obtained by detecting the Cooper pair box in the state $|e\rangle $. The success probability $P_{s}^{\prime }$ to produce a Fock state $|N\rangle $ reads $$P_{s}^{\prime }=e^{-\left\vert \alpha \right\vert ^{2}}\sum_{m=0}^{\infty
}(\alpha ^{2m}/m!)\prod_{j=1}^{M}sin^{2}(\omega _{n+j}\tau _{j}).
\label{pss}$$
In the first strategy, we prepare Fock states $|N\rangle $ with $N=M$, i.e., the phonon-number $N$ coincides with the number of CPB detections $M$. The fidelity of these states is given by the phonon number distribution at $P_{M} $ associated with the state $\left\vert \Psi _{NC}(\tau
_{M})\right\rangle ^{\prime },$$$P_{M}=\frac{\prod_{j=1}^{M}\sin ^{2}(\sqrt{j}\beta \tau _{j})}{\sum_{n=0}^{\infty }(\alpha ^{2n}/n!)\prod_{n=1}^{M}\sin ^{2}(\sqrt{n+j}\beta \tau _{n})}. \label{ps1}$$
We note that, in this case the fidelity coincides with the $N-th$ component of the statistical distribution $Pn$. The Fig.(\[Number\]) shows the phonon-number distribution exhibiting the creation of Fock state $|3\rangle $, $|4\rangle $, and $|5\rangle $; all with fidelity of $99\%$, for an initial coherent state with $\alpha =0.6$.
In the second strategy, we prepare Fock states $|N\rangle $ with $N=2M$ or $2M-1$. The associated fidelity is also given by the Eq.(\[ps1\]). The Fig.(\[number2\]) shows the phonon-number distribution exhibiting the creation of Fock states $|3\rangle $, $|4\rangle $, and $|5\rangle $, all them with same fidelity $99\%,$ for an initial coherent state with $\alpha =0.6$.
Conclusion
==========
Concerning with the feasibility of the scheme, it is worth mentioning some experimental values of parameters and characteristics of our system: the maximum value of the coupling constant $\beta _{m\acute{a}x}\approx 45MHz$, with $B\approx 0,1T$, $\ell =30\mu m$, $x_{0}=500fm$ and $E_{J}^{0}=5GHz$, with $\omega _{0}=200\pi MHz$.* [huang,a16a,a33,a34,b1,b2,bb2,tr,b3]{}*. The expression choosing the time spent to make a hole, $\beta \sqrt{n_{j}+1}\tau _{j}=\pi /2,$ funishes $\tau
_{j}\simeq $ $0.3~ns,$ when assuming all the CPB previously prepared at $t=0
$. On the other hand, the decoherence times of the CPB and the NR are respectively $500~ns$ and $160$ $\mu s$ [@tr]. Accordingly, one may create about 1600 holes before the destructive action of decoherence. However, when considering the success probability to detect all CPB in the state $\left\vert g\right\rangle $, a more realistic estimation drastically reduces the number of holes. A similar situation occurs in [malboi,ard1,ard]{}*,* using atom-field system to make holes in the statistical distribution $P_{n}$ of a field state; in this case, about $1\mu
s$ is spent to create a hole whereas $1ms$ is the decoherence time of a field state inside the cavity. So, comparing both scenarios the present system is about 60% more efficient in comparison with that using the atom-field system. Concernig with the generation of a Fock state $\left\vert
N\right\rangle $, it is convenient starting with a low excited initial (coherent) state, which involves a low number of Fock components to be deleted via our hole burning procedure. According to the Eq. (\[ps1\]) when one must delete many components of the initial state to achieve the state $\left\vert N\right\rangle $ this drastically reduces the success probability. As consequence, this method will work only for small values of $N$ ( $N\lesssim 5$ ).
Acknowledgements
================
The authors thank the FAPEG (CV) and the CNPq (ATA, BB) for partially supporting this work.
[99]{} Bocko M F and Onofrio R, 1996 Rev. Mod. Phys. **68** 755
Munro W J et al. 2002 Phys. Rev. A **66** 023819
Cleland A N and Geller M R 2004 Phys. Rev. Lett. **93** 070501
Cleland A N and Roukes M L 1996 Appl. Phys. Lett. **69** 2653
Carr D W, Evoy S, Sekaric L, Craighead H G and Parpia J M 1999 Appl. Phys. Lett. **75** 920
Huang X M H, Zorman C A, Mehregany M, and Roukes M L 2003 Nature **421** 496
Bose S, Jacobs K and Knight P L 1999 Phys. Rev. A **59** 3204
Midtvedt D, Tarakanov Y and Kinaret J 2011 Nano Lett. **11** 1439.
Knobel R G and Cleland A N 2003 Nature London **424** 291
LaHaye M D , Buu O, Camarota B and Schwab K C 2004 Science **304** 74
Blencowe M 2004 Phys. Rep. **395** 159
Armour A D, Blencowe M P and Schwab K C 2002 Phys. Rev. Lett. **88** 148301
Irish E K and Schwab K 2003 Phys. Rev. B **68** 155311
Santamore D H, Goan H -S, Milburn G J and Roukes M L 2004 Phys. Rev. A **70** 052105
Santamore D H, Doherty A C and Cross M C 2004 Phys. Rev. B **70** 144301
Buks E et al. 2007 arXiv:quant-ph/0610158v4
Wang Y D, Gao Y B, Sun C P 2004 Eur. Phys. J. B **40** 321
Peano V and Thorwart M 2004 Phys. Rev. B **70** 235401
Takei S, Galitski V M and Osborn K D 2011 arXiv:1104.0029v1
Liao J Q, Kuang L M 2010 arXiv:1008.1713v1
Tian L and Zoller P 2004 Phys. Rev. Lett. **93** 266403
Zou X B and Mathis W 2004 Phys. Lett. A **324** 484
Liao J Q, Wu Q Q and Kuang L M 2008 arXiv:0803.4317v1
Xue F et al. 2007 New J. Phys. **9** 35
Geller M R and Cleland A N 2005 Phys. Rev. A **71** 032311
Tian L and Carr S M 2006 Phys. Rev. B **74** 125314
Wang Y D, Chesi S, Loss D and Bruder C 2010 Phys. Rev. B **81** 104524
Martin I, Shnirman A, Tian L and Zoller P 2004 Phys. Rev. B **69** 125339
Zhang P, Wang Y D and Sun C P 2002 Phys. Rev. Lett. **95** 097204
Wilson-Rae I, Zoller P and Imamoglu A 2004 Phys. Rev. Lett. **92** 075507
Naik A et al. 2006 Nature **443** 193
Hopkins A, Jacobs K, Habib S and Schwab K 2003 Phys. Rev. B **68** 235328
Y.D. Wang et al. 2009 Phys. Rev. B **80** 144508
Hensinger W K et al. 2005 Phys. Rev. A **72** 041405
Sun CP, Wei L F, Liu Y and Nori F 2006 Phys. Rev. A **73** 022318
Milburn G J, Holmes C A, Kettle L M and Goan H S 2007 arXiv:cond-mat/0702512v1
Siewert J, Brandes T and Falci G 2005 arXiv:cond-mat/0509735v1
Tian L 2005 Phys. Rev. B **72** 195411
Valverde C, Avelar A T and Baseia B 2011 arXiv:1104.2106v1
Rabl P, Shnirman A and Zoller P 2004 Phys. Rev. B **70** 205304
Ruskov R, Schwab K and Korotkov A N 2005 Phys. Rev. B **71** 235407
Xue F, Liu Y, Sun C P, Nori F 2007 Phys. Rev. B **76** 064305
Zhou X X and Mizel A 2006 Phys. Rev. Lett. **97** 267201
Suh J et al. 2010 Nano Lett. **10** 3990.
Valverde C, Avelar A T, Baseia B and Malbouisson J M C 2003 Phys. Lett. A **315** 213
Valverde C and Baseia B 2004 Int. J. Quantum Inf. **2** 421
Eisert J, Plenio M B, Bose S and Hartley J 2004 Phys. Rev. Lett. **93** 190402
Bose S and Agarwal G S 2005 New J. Phys. **8** 34
Blais A et al. 2004 Phys. Rev. A **69** 062320
Moerner W E, Lenth W and Bjorklund G C 1988 in: W.E. Moerner (Ed.), Persistent Spectral Hole-Burning: Science and Applications, Springer, Berlin, p. 251
Malbouisson J M C and Baseia B 2001 Phys. Lett. A **290** 214
Avelar A T and Baseia B 2004 Opt. Comm. **239** 281
Avelar A T and Baseia B 2005 Phys. Rev. A **72** 025801
Escher B M, Avelar A T, Filho T M R and Baseia B 2004 Phys Rev. A **70** 025801
Wallraff A et al. 2005 Phys. Rev. Lett. **95** 060501
Liao J Q and Kuang L M 2007 J. Phys. B: At. Mol. Opt. Phys. **40** 1845
Xue F, Zhong L, Li Y and Sun C P 2007 Phys. Rev. B **75** 033407
Chen G, Chen Z, Yu L and Liang J 2007 Phys. Rev. A. **76** 024301
Liao J Q and Kuang L M 2008 Eur. Phys. J. B **63** 79
Escher B M, Avelar A T and Baseia B 2005 Phys. Rev. A **72** 045803
Aragao A, Avelar A T, Malbouisson J M C and Baseia B 2004 Phys. Lett. A **329** 284
|
---
bibliography:
- 'LargeDBumpyHolesReferences.bib'
---
**Black Ripples, Flowers and Dumbbells**
**at large $D$**
1.6cm
**David Licht$^{a}$, Raimon Luna$^{a}$ and Ryotaku Suzuki$^{a,b}$**
0.5cm
*$^{a}$Departament de F[í]{}sica Quàntica i Astrofísica, Institut de Ciències del Cosmos,*
*Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain*
*$^{b}$Department of Physics, Osaka City University,*
*Sugimoto 3-3-138, Osaka 558-8585, Japan*
0.5cm
david.licht@icc.ub.edu, raimonluna@icc.ub.edu,
s.ryotaku@icc.ub.edu
1.cm
**Abstract**
0.2cm
We explore the rich phase space of singly spinning (both neutral and charged) black hole solutions in the large $D$ limit. We find several ’bumpy’ branches which are connected to multiple (concentric) black rings, and black Saturns. Additionally, we obtain stationary solutions without axisymmetry that are only stationary at $D\rightarrow \infty$, but correspond to long-lived black hole solutions at finite $D$. These multipolar solutions can appear as intermediate configurations in the decay of ultra-spinning Myers-Perry black holes into stable black holes. Finally, we also construct stationary solutions corresponding to the instability of such a multipolar solution.
Introduction {#sec:introduction}
============
Black hole solutions in higher dimensional gravity show a far richer behavior than their counterparts in four spacetime dimensions. In higher dimensions, the rotation plays a significant role to fertilize a variety of new solutions. Since in $D>5$, the (Newtonian) gravitational potential $\sim\frac{G M}{r^{D-3}}$ falls off more rapidly than the centrifugal barrier $\sim\frac{J^2}{M^2r^2}$, the horizon can be deformed to an extended shape at large angular momentum, and hence becomes vulnerable to a Gregory-Laflamme type instability [@Gregory:1993vy; @Gregory:1994bj]. This allows a family of non-uniform stationary solutions to branch off from the zero modes of the instabilities [@Emparan:2008eg].
The increased number of degrees of freedom in a higher dimensional theory, however, complicate the construction of black hole solutions and analysis of their dynamics. To tackle this problem, several approximation techniques have been developed. One such approximation is the [*blackfold*]{} approach [@Emparan:2007wm], which has been successful in elucidating the black hole phases in the ultra-spinning regime: for example for black (multi-)rings/Saturns in which the horizon has highly elongated shape , that allows to locally approximate them as loosely bent black strings/branes.
Another successful effective approach is the large spacetime dimension limit, or the [*large $D$ limit*]{} [@Asnin:2007rw; @Emparan:2013moa], which has been proven to be useful in various problems involving higher dimensional black holes [@Asnin:2007rw; @Emparan:2013moa; @Emparan:2014cia; @Emparan:2014aba; @Emparan:2015rva; @Emparan:2014jca; @Suzuki:2015iha; @Suzuki:2015axa; @Emparan:2015hwa; @Emparan:2015gva; @Tanabe:2015hda; @Tanabe:2015isb; @Tanabe:2016opw; @Emparan:2016sjk; @Chen:2017wpf; @Mandlik:2018wnw; @Emparan:2018bmi; @Iizuka:2018zgt; @Andrade:2018zeb; @Li:2019bqc; @Casalderrey-Solana:2018uag; @Guo:2019pte; @Andrade:2018nsz; @Andrade:2019edf; @Andrade:2018rcx; @Emparan:2019obu]. This limit allows black holes to have a simple near horizon structure decoupled from the asymptotic region [@Emparan:2013xia]. As a result, the Einstein’s equation reduces to an effective theory on the horizon surface expanded in $1/D$, namely the [*large $D$ effective theory*]{} [@Emparan:2015hwa; @Emparan:2015gva; @Bhattacharyya:2015dva; @Bhattacharyya:2015fdk]. Different to the blackfold approach, the large $D$ limit is naturally endowed with the separation of scales between gradients along and orthogonal to the horizon: the gradient orthogonal to the horizon becomes large compared to gradients along the horizon in the limit of large $D$ as a result of the steepening of the gravitational potential. This enables us to formulate an effective theory without the requirement that the gradients along the horizon have to be infinitesimal, which makes the large $D$ expansion a powerful tool to study the non-uniform ’bumpy’ phases of black holes.
In this paper, we explore the phase space of compact stationary solutions with a single spin in the large $D$ limit, specifically, we focus on the (non-)axisymmetric deformed families branching off from the Myers-Perry family. The instability of ultra-spinning MP black holes and the existence of nearby ‘rippled’ solution was first conjectured in [@Emparan:2003sy] and later, after the proof of existence of the zero modes and the instability [@Dias:2009iu; @Dias:2010eu; @Dias:2010maa; @Dias:2011jg; @Hartnett:2013fba], the rippled solutions were constructed numerically and identified as solutions that connect to black rings and black Saturns [@Emparan:2007wm; @Emparan:2014pra; @Dias:2014cia; @Figueras:2017zwa].
Because of the strong suppression of gravitational radiation at large $D$ [@Andrade:2019edf], the effective large $D$ description also admits stationary non-axisymmetric branches such as [*black bars*]{} [@Andrade:2018nsz] and other multipolar solutions. Here we apply the [*blob*]{} approximation developed in [@Andrade:2018nsz; @Andrade:2018rcx], where localized black hole solutions such as the Myers-Perry black hole are identified as stationary lumps (“blobs") on a membrane which share the same horizon topology as the black brane solution but nevertheless encode most of the physics pertaining to the localized solution.
Figure \[fig:PhaseDiagram\] shows the full phase space plot of solutions we obtain. The solutions correspond to Myers-Perry solutions and their axissymmetric ‘bumpy’ deformations leading to black rings and black Saturns. We are also including stationary solutions without axisymmetry, which only can remain stationary at large $D$ since gravitational radiation decouples. These solutions have been shown to play an important role in dynamical evolutions of the ultra-spinning instability [@Andrade:2018yqu; @Andrade:2019edf; @Shibata:2009ad; @Shibata:2010wz; @Bantilan:2019bvf]. The first solution of this kind, a dipolar solution “black bar" was found analytically in [@Andrade:2018nsz]. Here we study its stationary deformations and also find its multipolar generalizations “black flowers". To illustrate features of the found solutions, we show plots of the mass density of four examples in figure \[fig:ExampleProfiles\].[^1]
![Phase space plot of the first appearing branches of solutions with a single angular momentum (per unit mass) ${{\mathcal J}}/{{\mathcal M}}$ and angular velocity $\Omega$. In the ultra-spinning regime ${{\mathcal J}}/{{\mathcal M}}>2$ the MP-BH develops instabilities and the corresponding zero modes appear at places marked with dots or crosses. For the analytically known black bar, we also study its non-uniform deformations (’dumbbells’), whose branches are shown in different shadings of a color to make them more distinguishable. \[fig:PhaseDiagram\] ](Plots/PhaseDiagramAll "fig:"){width="75.00000%"} ![Phase space plot of the first appearing branches of solutions with a single angular momentum (per unit mass) ${{\mathcal J}}/{{\mathcal M}}$ and angular velocity $\Omega$. In the ultra-spinning regime ${{\mathcal J}}/{{\mathcal M}}>2$ the MP-BH develops instabilities and the corresponding zero modes appear at places marked with dots or crosses. For the analytically known black bar, we also study its non-uniform deformations (’dumbbells’), whose branches are shown in different shadings of a color to make them more distinguishable. \[fig:PhaseDiagram\] ](Plots/plotLegendFullphaseSpace.pdf "fig:"){width="23.00000%"}
We observe that most of bumpy deformations remain tangential to their ’parent’-branch until the deformation becomes comparable to the original solution and new blobs start to form. At some point, these blobs barely have any overlap and the branches enter a new asymptotic behavior for small $\Omega$ becoming completely separated. Some very short branches stick out non-tangentially above the parent-branch.
The paper is structured as follows: in section \[sec:SetupSection\], we outline the derivation of our large $D$ effective equations for black branes and describe how they also contain localized black hole solutions. In section \[sec:axisymmSol\], we construct perturbatively and numerically stationary ‘bumpy’ deformations of the MP black hole that lead to (multiple) black rings and Saturns. In section \[sec:multipoleDef\] and \[sec:blackBars\], we construct stationary non-axisymmetric solutions from multipolar deformations of MP black holes and deformations from black bars. Section \[sec:effCharge\] discusses effects of adding charge to obtain charged (but non-extremal) solutions. In the appendix we collect details of the perturbative calculation and describe our numerical procedure in greater detail.
![Four examples of bumpy solutions: Upper Left: Ring-like ripple. Upper Right: Saturn-like ripple Lower Left: Black flower with a quadrupolar deformation. Lower Right: Dumbbell. Plots show the mass density $m$. Coloring was chosen to highlight the important details of the solution, strictly speaking all solutions share the same horizon topology. \[fig:ExampleProfiles\]](Plots/SurfaceRing "fig:"){width="49.00000%"} ![Four examples of bumpy solutions: Upper Left: Ring-like ripple. Upper Right: Saturn-like ripple Lower Left: Black flower with a quadrupolar deformation. Lower Right: Dumbbell. Plots show the mass density $m$. Coloring was chosen to highlight the important details of the solution, strictly speaking all solutions share the same horizon topology. \[fig:ExampleProfiles\]](Plots/SurfaceSaturn "fig:"){width="49.00000%"} ![Four examples of bumpy solutions: Upper Left: Ring-like ripple. Upper Right: Saturn-like ripple Lower Left: Black flower with a quadrupolar deformation. Lower Right: Dumbbell. Plots show the mass density $m$. Coloring was chosen to highlight the important details of the solution, strictly speaking all solutions share the same horizon topology. \[fig:ExampleProfiles\]](Plots/SurfaceN1m4 "fig:"){width="54.00000%"} ![Four examples of bumpy solutions: Upper Left: Ring-like ripple. Upper Right: Saturn-like ripple Lower Left: Black flower with a quadrupolar deformation. Lower Right: Dumbbell. Plots show the mass density $m$. Coloring was chosen to highlight the important details of the solution, strictly speaking all solutions share the same horizon topology. \[fig:ExampleProfiles\]](Plots/SurfaceDumbbell "fig:"){width="45.00000%"}
Branes and localized black holes at large $D$ {#sec:SetupSection}
=============================================
Large $D$ effective equations {#sec:largeD}
-----------------------------
We study possibly charged black holes in Einstein-Maxwell theory in higher dimensions $$I=\int d^D x\sqrt{-g}{\left(}R-\frac14 F^2{\right)}\,,$$ where $$D=n+p+3\,,$$ with $n$ large and $p$ a finite number. Ref. [@Emparan:2016sjk] developed an effective theory for fluctuations of $p$-branes along their extended directions $\sigma^i$ ($i=1,\dots,p$) , $$\begin{aligned}
\label{AFch}
ds^2=2dtdr-Adt^2-\frac{2}{n} C_i d\sigma^idt+\frac1{n}G_{ij}d\sigma^id\sigma^j +r^2d\Omega_{n+1}\,,\end{aligned}$$ where ${\mathsf{R}}=r^n$ and $$\begin{aligned}
A=1-\frac{m(t,\sigma)}{{\mathsf{R}}}+\frac{q(t,\sigma)^2}{2{\mathsf{R}}^2}\,,\qquad
C_i=\left(1-\frac{q(t,\sigma)^2}{2m(t,\sigma){\mathsf{R}}}\right) \frac{p_i(t,\sigma)}{{\mathsf{R}}}\,,\end{aligned}$$ $$\begin{aligned}
G_{ij}&=\delta_{ij}+\frac1{n}\left\{\left( 1-\frac{q(t,\sigma)^2 }{2m(t,\sigma){\mathsf{R}}}\right)\frac{p_i(t,\sigma) p_j(t,\sigma)}{m(t,\sigma){\mathsf{R}}}
\right. \nonumber \\
& \hphantom{{} = \delta_{ij}+\frac1{n}\,}
\left.
-\ln{\left(}1-\frac{m_-(t,\sigma)}{{\mathsf{R}}}{\right)}\left[ 2\delta_{ij}+ \nabla_i \frac{p_j(t,\sigma)}{m(t,\sigma)} +\nabla_j \frac{p_i(t,\sigma)}{m(t,\sigma)}\right]
\right\} \,.\end{aligned}$$ The electric potential is $$A_t = -\frac{q(t,\sigma)}{{\mathsf{R}}}\,.$$
The degrees of freedom of the effective theory are the mass density $m(t,\sigma)$, the charge density $q(t,\sigma)$ and the fields $p_i(t,\sigma)$. In the presence of charge it is convenient to introduce a new field $v_i(t,\sigma)$ defined by $$p_i = m v_i+\nabla_i m\,,$$ and the abbreviation $$m_\pm =\frac{1}{2}{\left(}m\pm \sqrt{m^2-2q^2}{\right)}\,.$$
The equations of motion of the effective theory are obtained by requiring that the Einstein-Maxwell equations are solve to leading order in a $1/D$-expansion and take the form of conservation equations $$\begin{aligned}
&\partial_t m+\nabla_i (m v^i)=0\,,\\
&\partial_t (m v^i)+\nabla_j (m v^i v^j +\tau^{ij})=0\\
&\partial_t q+\nabla_i j^i=0\end{aligned}$$ where $$\begin{aligned}
\tau_{ij}&= -{\left(}m_+ -m_- {\right)}\delta_{ij} -2m_+\nabla_{(i}v_{j)}-(m_+-m_-)\,\nabla_i\nabla_j \ln m \,,\\
j_i&=q v_i -m\nabla_i{\left(}\frac{q}{m}{\right)}\,.\end{aligned}$$ These equations simplify further if we consider only stationary configurations, that satisfy $$(\partial_t+v^i\partial_i) m=0\,,\qquad (\partial_t+v^i\partial_i) q=0\,,$$ and $v^i$ is a time-independent killing vector [[*i.e.,*]{}]{}$$\partial_t v^i=0, \qquad \nabla_{(i}v_{j)}=0\,.$$ which implies the absence of dissipative effects. Absence of charge diffusion requires $$\nabla_i{\left(}\frac{q}{m}{\right)}=0\,,$$ which states that the charge density is everywhere proportional to the mass density via the proportionality constant $$\label{gothq}
{\mathfrak{q}}\equiv \frac{q}{m}\,.$$ Under these assumptions the equations of motion are reduced to a single master equation that is most elegantly formulated in terms of the area-radius $${\mathcal{R}}=\ln m\,,$$ and is given by $$\begin{aligned}
\label{eq:chargedMasterEq}
\nabla_i{\left(}\frac{v^2}{2}+\frac{m_+-m_-}{m}{\left(}\mathcal{R}+\nabla_j\nabla^j\mathcal{R}+\frac{1}{2}\nabla^j\mathcal{R}\nabla_j\mathcal{R}{\right)}{\right)}=0\,.\end{aligned}$$ Using the scale invariance of the effective equations, which manifests itself in a shift symmetry of ${\mathcal{R}}$, the above equation can be formally mapped to the uncharged equation by defining the charge rescaled velocity field $$\begin{aligned}
\label{eq:chargeRescalVel}
v^i_q=\sqrt{\frac{m}{m_+-m_-}}\,v^i=\frac{v^i}{{\left(}1-2{\mathfrak{q}}^2{\right)}^{1/4}}\,,\end{aligned}$$ and shifting ${\mathcal{R}}$ to obtain the *soap bubble equation* [@Emparan:2016sjk] $$\label{mastercharge}
\frac{v_q^2}{2}+\mathcal{R}+\nabla_j\nabla^j\mathcal{R}+\frac{1}{2}\nabla^j\mathcal{R}\nabla_j\mathcal{R}=0\,.$$ Which has the same form as the uncharged equation ([[*i.e.,*]{}]{}eq. (\[eq:chargedMasterEq\]) with ${\mathfrak{q}}=0$) but with the difference that the role of $v^2$ is now taken by the norm of the charged rescaled velocity field. Since the charged equation can be mapped to the uncharged one, solving eq. (\[mastercharge\]) for a given value of $v_q$ always gives a one parameter family of solutions, parameterized by the charge parameter ${\mathfrak{q}}$. In the case of non-vanishing charge, $v_q$ is not directly the physical velocity field and allows to study the effect of charging up the solution.
Black holes as Gaussian blobs on a membrane {#sec:MP}
-------------------------------------------
Even though these equations were initially formulated to capture the dynamics of black branes. Ref. [@Andrade:2018nsz] found that this large $D$ effective theory also contains localized black hole solutions when solved with different boundary conditions. We recapitulate here the findings of [@Andrade:2018nsz; @Andrade:2018rcx].
To capture effects of a single spin we consider the case of $p=2$ and require the stationary solutions to have a purely rotational velocity field. Choosing angular coordinates for the spatial brane directions $\sigma^i=(r,\phi)$, the only non-vanishing component of the (charge rescaled) velocity field can be set to $v^\phi =\Omega_q$ and equation (\[mastercharge\]) becomes $$\begin{aligned}
{\partial}^2_r {\mathcal{R}}+\frac{{\partial}_r {\mathcal{R}}}{r}+\frac{{\partial}_\phi^2 {\mathcal{R}}}{r^2}+\frac12 {\left(}{\left(}{\partial}_r {\mathcal{R}}{\right)}^2+\frac{{\left(}{\partial}_\phi {\mathcal{R}}{\right)}^2}{r^2}{\right)}+{\mathcal{R}}+\frac{\Omega_q^2 r^2}{2}=0\,,
\label{eq:MasterEquationRotation}\end{aligned}$$ where $\Omega_q$ is the charge rescaled angular velocity, according to eq. (\[eq:chargeRescalVel\]).
The Myers-Perry (MP) black hole solution (and its charged Kerr-Newman counterpart described in [@Andrade:2018rcx]) corresponds to the axisymmetric solution $${\mathcal{R}}_{\text {KN}}(r)=\frac{2}{1+a_q^2}{\left(}1-\frac{r^2}{4}{\right)}\,,
\label{eq:MPsol}$$ with $a_q$ defined via $$\Omega_q=\frac{a_q}{1+a_q^2}\,.$$
Since this corresponds to a Gaussian in the mass variable $m=\exp{{\mathcal{R}}}$, this solution is strongly localized in the directions $\sigma^i$, but still shares the same horizon topology as the black brane (\[AFch\]). This feature of the solution is due to the fact that the rescaling of the spatial directions $\sigma^i \rightarrow \sigma^i/\sqrt{n}$ assumed in eq. (\[AFch\]) leads for localized solutions to a magnification of the region around the center of one of its hemispheres. Since at large $D$ most of the surface of the black hole is concentrated in this region, a description of it can capture most of the physics connected to the localized black hole.
The aforementioned localization of the mass density motivates the following definition of a localized black hole: We call a solution of eq. (\[eq:MasterEquationRotation\]) a (stationary) localized black hole, if it has a finite mass ${{\mathcal M}}$ according to $${{\mathcal M}}=\int_0^{2\pi}d\phi\int_0^\infty dr\, r \, m(r,\phi)\,.
\label{eq:normCond}$$ And it has an angular momentum given by $$\begin{aligned}
{{\mathcal J}}=\int_0^{2\pi}d\phi\int_0^\infty dr\, r \, p_\phi(r,\phi)=\int_0^{2\pi}d\phi\int_0^\infty dr\, \Omega\, r^3 \, m(r,\phi)\,. \label{eq:angularMomentum}\end{aligned}$$ where we used $p_\phi = \partial_\phi m+ \Omega \, r^2 m$.
Axisymmetric sector: Black Ripples {#sec:axisymmSol}
==================================
First, we consider the axisymmetric deformation of the Myers-Perry, which leads to an infinite number of ’bumpy’ solutions, or [*black ripples*]{}.
Zero mode deformations {#sec:axisymmSolPert}
----------------------
The MP-solution (\[eq:MPsol\]) allows axisymmetric co-rotating zero mode deformations according to[^2] $${\mathcal{R}}(r) = {\mathcal{R}}_{\rm MP}(r) + \delta {\mathcal{R}}(r).$$ Plugging this into eq. (\[eq:MasterEquationRotation\]), we obtain $$\delta {\mathcal{R}}''(r) + {\frac{1}{r}} \frac{1+a^2-r^2}{1+a^2} \delta {\mathcal{R}}'(r) + \delta {\mathcal{R}}(r) = - \frac{1}{2} \delta {\mathcal{R}}'(r)^2.$$ Introducing a new radial variable $z$ via $$z := \frac{r^2}{2(1+a^2)},$$ the deformation equation becomes a Laguerre equation with a quadratic source term $${{\mathcal L}}_{(a^2+1)/2} \left[ \delta {\mathcal{R}}\right] := z \delta {\mathcal{R}}''(z) + (1-z)\delta {\mathcal{R}}'(z) + \frac{a^2+1}{2} \delta {\mathcal{R}}(z) = -\frac{z}{2} \delta {\mathcal{R}}'(z)^2\, , \label{eq:perturb-eq-axisym}$$ where we introduced the Laguerre operator ${{\mathcal L}}$. We note that, in terms of the new variable, the MP-solution is now written as $${\mathcal{R}}_{\rm MP}(z) = \frac{2}{a^2+1}- z.$$ Perturbations of this solution should be normalizable in the sense of eq. (\[eq:normCond\]), which means for the perturbed profile $m=\exp({\mathcal{R}}_{\rm MP}+\delta {\mathcal{R}})$ $$\int_0^\infty dr\, r \, m(r) \sim \int_0^\infty dz e^{-z} \exp {\left(}\delta {\mathcal{R}}(z) {\right)}<\infty, \label{eq:normCond-1},$$ which is accomplished if the perturbation grows as a polynomial at each order, not showing exponential growth $\sim e^z$ or any divergences.
At leading order, the regular and normalizable perturbations are given by Laguerre polynomials [@Andrade:2018nsz], $$\delta {\mathcal{R}}(z) = {\varepsilon}L_{N}(z) + {{\mathcal O}\left({\varepsilon}^2\right)},\label{eq:axisym-linear-sol}$$ only if $a^2+1 = 2N$, for integer $N$. Non-trivial solutions have $N\geq2$. $N$ has the interpretation of a ’radial overtone’ number, [[*i.e.,*]{}]{}it counts the number of oscillations along $r$. Since these zero modes correspond to ’bumpy black holes’ [@Emparan:2003sy; @Emparan:2014pra; @Dias:2014cia], $N$ can also be interpreted as the number of bumps in the cross-section of the corresponding solution.
Nonlinear perturbations
-----------------------
In the following, we study how to include higher order perturbations for these zero-modes obtaining better control over the phase space of stationary solutions and also to support the later numerical analysis.
The general perturbative soution to eq. (\[eq:perturb-eq-axisym\]) is written as $$\delta {\mathcal{R}}(z) = \sum_{k=0}^\infty {\varepsilon}^{k+1} f_k(z).$$ and for a leading order solution with $a^2+1=2N \, , ( N=2,3,4,\dots)$, the deformation equation (\[eq:perturb-eq-axisym\]) becomes $${{\mathcal L}}_N \left[f_k(z)\right] = {{\mathcal S}}_k(z) \label{eq:perturb-eq-axisym-k}$$ at each perturbation order $k$. As usual, the source term ${{\mathcal S}}_k(z)$ is expressed by the solution up to $(k-1)$-th order.
A similar higher order perturbation analysis has been performed in [@Suzuki:2015axa; @Emparan:2018bmi] for perturbations (non-uniformities) of black strings. It was found there, that the length of the black string has to be renormalized to avoid secular terms that would break the periodic boundary condition. Here, for spinning localized solutions, it turns out that we have to renormalize the angular velocity $\Omega$ or the corresponding spin parameter $a$ which changes the blob size, to avoid secular behavior that would break the normalization condition (\[eq:normCond-1\]).
### Resonance and secular perturbation
Secular behavior in perturbation theory is typically encountered when the dependence of some physical parameter on the perturbation parameter $\varepsilon$ is ignored. A common example for this is the slightly anharmonic oscillator $$\ddot{x}(t) + {\omega_0}^2 x(t) = -\varepsilon x(t)^3,$$ Note that if we assume $x\ll 1$ the lowest order effect of the anharmonic term $\varepsilon x^3$ is to modify the frequency: $\omega_0 \rightarrow \omega_0+\varepsilon \omega_1$. The appropriate ansatz accordingly should be $x(t)=\sin((\omega_0+\varepsilon \omega_1)t)$, but naive perturbation theory $x(t)=x_0(t)+\varepsilon x_1(t)$ leads to the solution $$\begin{aligned}
x_0(t)&=\sin(\omega_0 t)\,,\label{eq:exampleSource}\\
x_1(t)&= t \cdot\sin(\omega_0t)+\dots\,, \label{eq:exampleSecTerm}\end{aligned}$$ where the first correction grows unboundedly invalidating the perturbative ansatz and violating conservation of energy. Note here that the secular term (\[eq:exampleSecTerm\]) results from a resonance phenomenon between the zeroth order solution (\[eq:exampleSource\]) acting as a resonant source for the first order correction.
For our perturbative problem (\[eq:perturb-eq-axisym-k\]), a similar resonant behavior occurs. Assuming $S_k(z)$ can be decomposed into a linear combination of Laguerre polynomials $L_M(z)$, we have to distinguish two cases in $${{\mathcal L}}_N f(z) = L_M(z). \label{eq:perturb-eq-axisym-LM}$$ For $M \neq N$, the solution remains regular and normalizable, $$\begin{aligned}
f(z) = \frac{L_M(z)}{N-M}. \label{eq:int-laguerre-nm}\end{aligned}$$ However, for $M=N$, which we are going to call the *resonant* case, the solution is $$\begin{aligned}
f(z) = -L_N(z)\log z - \sum_{I=0}^{N-1} \frac{2}{N-I} L_I(z)+B \Psi(N,0,z)\, \label{eq:int-laguerre-nn} \end{aligned}$$ with $B$ an integration constant and $\Psi(N,0,z)$ a Laguerre function of the second kind (see eq.(\[eq:laguerre-2-PM\])). Since $\Psi(N,0,z)$ has both a logarithmic divergence at $z=0$ and exponential growth for $z\to\infty$, the solution can never be regular and normalizable at the same time. This corresponds to secular behavior because the resonant term can be eliminated by a infinitesimal shift of $a$ in eq. (\[eq:perturb-eq-axisym\]) since, $$\begin{aligned}
\left. \partial_\alpha L_\alpha(z)\right|_{\alpha=N} = \Psi(N,0,z)+ L_N(z) \log z + ({\rm polynomial \ of} \ z).\end{aligned}$$
### Recurrence formula
The perturbative solution can be obtained systematically by removing resonant terms in the sources order by order, which leads to an algebraic recurrence relation. For this, we assume both $\delta {\mathcal{R}}(z)$ and $a$ are expanded in ${\varepsilon}$, $$\begin{aligned}
\delta {\mathcal{R}}(z) = \sum_{k=0}^\infty {\varepsilon}^k f_k(z),\quad a^2+1= 2N \left(1+\sum_{k=1}^\infty {\varepsilon}^k \mu_{k} \right), \label{eq:axisym-expand-dR-a}\end{aligned}$$ where we set $$f_0(z) = L_N(z).$$ Plugging this into eq. (\[eq:perturb-eq-axisym\]) and expanding in ${\varepsilon}$, we obtain the perturbation equation for each order in ${\varepsilon}$, $$\begin{aligned}
{{\mathcal L}}_N f_k(z) = -{\frac{1}{2}} \sum_{\ell=0}^{k-1} z f'_\ell(z) f'_{k-1-\ell}(z) - N \sum_{\ell=1}^{k} \mu_\ell f_{k-\ell}(z) =: {{\mathcal S}}_k(z) .\label{eq:perturb-src}\end{aligned}$$ Assuming that $f_{\ell}(z)$ are polynomials for $\ell < k$, the source term also becomes a polynomial, and hence should be decomposed to the linear combination of the Laguerre polynomials, $$\begin{aligned}
&{{\mathcal S}}_k(z) := \sum_{K=0}^M {{\mathcal C}}_K L_K(z) - N\mu_k L_N(z),\end{aligned}$$ where $M$ is a finite positive integer. After eliminating $L_N(z)$ from the source by using $\mu_k$, $f_k(z)$ can be expressed as a polynomial as well. And we can decompose the solution at each order into a finite linear combination of Laguerre polynomials $$f_k(z) = \sum_{I} {\cal C}_{k,I} L_I(z)\,.
\label{eq:laguerre-exp}$$ The coefficients of the resonant term ${{\mathcal C}}_{k,N}$ correspond to the reparametrizations of ${\varepsilon}$, and hence can be set to $0$.
So the problem reduces to determining the coefficients ${{\mathcal C}}_{k,I}$ and $\mu_k$ at each order. Substituting eq. (\[eq:laguerre-exp\]) into the source term (\[eq:perturb-src\]), we obtain $$\begin{aligned}
& {{\mathcal S}}_k(z) = {{\mathcal L}}_N \left[ - \sum_{M\neq N} \left( \sum_{I,J} \sum_{i=0}^{k-1} {{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J}
\frac{I+J-M}{4(N-M)} {{\mathcal X}}_{I,J}^M \right)L_M(z) \right.{\nonumber\\ }&\left. \hspace{4cm}- \sum_{M\neq N} \sum_{i=1}^{k-1}\frac{N \mu_i C_{k-i,M}}{N-M} L_M(z)
\right] {\nonumber\\ }& \qquad -\left[ N \mu_k + {\frac{1}{4}} \sum_{I,J} \sum_{i=0}^{k-1}(I+J-N){{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J}{{\mathcal X}}^N_{I,J} + \sum_{i=1}^{k-1} N\mu_i {{\mathcal C}}_{k-i,N}\right] L_N(z),\label{eq:axisym-source-k}\end{aligned}$$ where ${{\mathcal X}}^K_{I,J}$ comes from the decomposition of the product of Laguerre polynomials [@Watson1938], $$\begin{aligned}
& L_I(z)L_J(z) = \sum_{K=|I-J|}^{I+J}{{\mathcal X}}^K_{I,J} L_K(z),\end{aligned}$$ which is written as $${{\mathcal X}}^K_{I,J} = \frac{(-2)^{I+J-K} K!}{(K-I)!(K-J)!(I+J-K)!} { { \, {}_{3} F {}_{2} } }\left(\begin{array}{c}K+1,{\frac{1}{2}}(K-I-J),{\frac{1}{2}}(K-I-J+1)\\
K-I+1,K-J+1\end{array};1\right).$$ The last line in eq. (\[eq:axisym-source-k\]) is proportional to the resonant term, and hence should be removed by setting
$$\mu_k = - {\frac{1}{4N}} \sum_{I,J} \sum_{i=0}^{k-1}(I+J-N){{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J}{{\mathcal X}}^N_{I,J} - \sum_{i=1}^{k-1} \mu_i {{\mathcal C}}_{k-i,N}\,.
\label{eq:recurrence-mu}$$
For non-resonant terms, the $k$-th order coefficients are determined by $$\begin{aligned}
&{{\mathcal C}}_{k,M\neq N} = - \sum_{I,J} \sum_{i=0}^{k-1} {{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J}
\frac{I+J-M}{4(N-M)} {{\mathcal X}}_{I,J}^M - \sum_{i=1}^{k-1}\frac{N \mu_i C_{k-i,M}}{N-M}.
\label{eq:recurrence-C-p}\end{aligned}$$\[eq:recurrence\]
The coefficient of $L_N(z)$ is set to zero ${{\mathcal C}}_{k,N}=0$ for $k\geq1$. With these recurrence equations, the perturbation equation can be solved algebraically.
### Perturbation solution
To solve the recurrence equation (\[eq:recurrence\]), we first set $${{\mathcal C}}_{0,M} = \delta_{N,M}.$$ Then, we have the solution for $k=1$ $$\mu_1 = -\frac{1}{4} {{\mathcal X}}^N_{N,N},\quad {{\mathcal C}}_{1,M\neq N} = - \frac{2N-M}{4(N-M)} {{\mathcal X}}^M_{N,N}.\label{eq:bumpy-perturb-sol-1}$$ Repeating the calculation, we get the result at $k=2$, $$\begin{aligned}
\mu_2
= \sum_{I\neq N}\frac{(2N-I)I}{8N(N-I)}{{\mathcal X}}^I_{N,N} {{\mathcal X}}^N_{I,N},\end{aligned}$$ and $$\begin{aligned}
& {{\mathcal C}}_{2,M\neq N}
= \sum_{I\neq N} \frac{(I+N-M)(2N-I)}{8(N-M)(N-I)} {{\mathcal X}}^M_{N,I}{{\mathcal X}}^I_{N,N} - \frac{N(2N-M)}{16(N-M)^2}
{{\mathcal X}}^N_{N,N} {{\mathcal X}}^M_{N,N} .\end{aligned}$$ Especially, the leading order shift in $a$ is given by $$\mu_1 = -\frac{1}{4}{{\mathcal X}}^N_{N,N} = -(-2)^{N-2} { { \, {}_{3} F {}_{2} } }\left[\begin{array}{c}N+1,-\frac{N}{2},-\frac{N-1}{2}\\
1,1\end{array};1\right].\label{eq:axisym-mu1}$$ Here we note that $\mu_1$ alternates in sign with $N$. For the first values of $N$, we obtain $$\mu_1\bigr|_{N=2,3,4,5} = -\frac{5}{2}\ ,\ 14\ ,\ -\frac{173}{2}\ ,\ 563.$$ Using the relation to the Franel number (see Appendix. \[sec:franel\]), one can show the amplitude of $\mu$ grows very rapidly with $N$, $$\mu_1 \sim (-1)^{N+1} \frac{2^{3N}}{N}.$$
### Phase diagram
Given the perturbative solution we can calculate the physical quantities ${{\mathcal M}},\, {{\mathcal J}}$ and the value at the origin ${\mathcal{R}}_0={\mathcal{R}}(0)$ (which is used as an initial condition in the numerical analysis) perturbatively as follows.
#### Angular velocity and center thickness
By definition, the angular velocity has the expansion $$\Omega = \frac{a}{1+a^2} = \frac{\sqrt{2N-1}}{2N}\left(1-\frac{N-1}{2N-1}\mu_1 {\varepsilon}+{{\mathcal O}\left({\varepsilon}^2\right)}\right).\label{eq:axisym-perturb-Omega-res}$$ The center thickness is given by $${\mathcal{R}}_0 = \frac{2}{1+a^2}+ {\varepsilon}+{{\mathcal O}\left({\varepsilon}^2\right)} = {\frac{1}{N}}\left(1+(N-\mu_1){\varepsilon}+ {{\mathcal O}\left({\varepsilon}^2\right)}\right).$$ Which gives the gradient on the branching point is given by $$\left.\frac{ \partial_{\varepsilon}\log \Omega}{\partial_{\varepsilon}\log {\mathcal{R}}_0}\right|_{{\varepsilon}=0} = \frac{N-1}{2N-1} \frac{\mu_1}{\mu_1-N}.\label{eq:gradient-omega-to-R0}$$ Since $\mu$ grows much faster than $N$, the gradient rapidly approaches to that of the Myers-Perry branch for the larger value of $N$. For the first few values of $N$, we obtain $$\frac{ \partial_{\varepsilon}\log \Omega}{\partial_{\varepsilon}\log {\mathcal{R}}_0} \bigr|_{N=2,3,4,5} = \frac{5}{27}\ , \quad \frac{28}{55}\ , \quad \frac{519}{1267}\ ,\quad \frac{1126}{2511}.$$ At higher order, the center thickness is given by $${\mathcal{R}}_0 = \frac{a}{1+a^2} + \sum_{k=0} {\varepsilon}^{k+1} \left(\sum_{I} {{\mathcal C}}_{k,I}\right)$$ where ${{\mathcal C}}_{k,I}$ is the coefficients of the Laguerre expansion at each order in eq. (\[eq:laguerre-exp\]). To compare with the numerical result (figure \[fig:OmegaR0\]), we calculated the formula for $({\mathcal{R}}_0,\Omega)$-space up to ${\varepsilon}^2$, $$\Omega = \frac{\sqrt{2N-1}}{2N} \left(1 + \omega_1 \bar{{\varepsilon}} + \omega_2 \bar{{\varepsilon}}^2\right),$$ where $$\bar{{\varepsilon}} := N {\mathcal{R}}_0 -1.$$ $\omega_1$ coincides with eq. (\[eq:gradient-omega-to-R0\]). Here we do not show the explicit formula for $\omega_2$, since it no longer reduces to the simple form. The coefficients for several branches are $$\begin{aligned}
& \omega_1\bigr|_{N=2,3,4,5} =\frac{5}{27},\, \frac{28}{55},\, \frac{519}{1267},\, \frac{1126}{2511},\\
& \omega_2\bigr|_{N=2,3,4,5} =\frac{118}{729},\, -\frac{172629}{66550},\, \frac{82075592}{290557309},\,
-\frac{1528095425}{4691010024}.\end{aligned}$$
#### Mass and angular momentum
Provided that the perturbation is normalizable, the mass (\[eq:normCond\]) and angular momentum (\[eq:angularMomentum\]) are easily obtained by $$\begin{aligned}
&{{\mathcal M}}= {{\mathcal M}}_{\rm MP} \int_0^\infty e^{-z} \exp {\left(}\delta {\mathcal{R}}(z) {\right)}dz,\label{eq:mass-integral}\\
&{{\mathcal J}}= 2a{{\mathcal M}}- 2a{{\mathcal M}}_{\rm MP} \int_0^\infty e^{-z} L_1(z) \exp {\left(}\delta {\mathcal{R}}(z) {\right)}dz,\label{eq:ang-integral}\end{aligned}$$ where ${{\mathcal M}}_{\rm MP}$ is the mass of the Myers-Perry of the same $a$ and $z = L_0(z) - L_1(z)$ is used. Since these integrations take the form of the inner product of the Laguerre polynomials, it is convenient to use the expansion of the perturbative solution into the Laguerre polynomials, $$\delta {\mathcal{R}}(z) = \sum_{k=0}^\infty \sum_{M} {\varepsilon}^{k+1} {{\mathcal C}}_{k,M} L_M(z),$$ where ${{\mathcal C}}_{0,M} = \delta_{M,N}$ for the $N$-branch and $M$ runs over some finite at each perturbative order $k$. Up to ${{\mathcal O}\left({\varepsilon}^2\right)}$, one can expand as $$\exp {\left(}\delta {\mathcal{R}}(z) {\right)}= 1 + {\varepsilon}L_N(z) - {\varepsilon}^2 \sum_{M\neq N} \frac{M{{\mathcal X}}_{N,N}^M}{4(N-M)}L_M(z)\, ,$$ where we made use of the second order solution (\[eq:bumpy-perturb-sol-1\]). Putting this into eqs. (\[eq:mass-integral\]) and (\[eq:ang-integral\]), we obtain $$\frac{{{\mathcal J}}}{{{\mathcal M}}} = 2a\left[1- \frac{{{\mathcal X}}^1_{N,N}}{4(N-1)}{\varepsilon}^2\right],$$ in which $a$ also should be expanded according to (\[eq:axisym-expand-dR-a\]). We see that the ratio of angular momentum to mass only differs by ${{\mathcal O}\left({\varepsilon}^2\right)}$ from the Myers-Perry branch.
Numerical construction {#sec:axisymmSolNum}
----------------------
To construct fully non-linear solutions we have to solve numerically the axisymmetric version of the soap bubble equation (\[eq:MasterEquationRotation\])
$${\mathcal{R}}'' + \frac{{\mathcal{R}}'}{r} + \frac12 {\mathcal{R}}'^2 + {\mathcal{R}}+ \frac{\Omega^2 r^2}{2} = 0\,,
\label{eqn:master}$$
which is a second order nonlinear differential equation for ${\mathcal{R}}(r)$. Since $r$ is a radial coordinate, any physical solution of eq. (\[eqn:master\]) must satisfy the regularity condition ${\mathcal{R}}'(0) = 0$. This leaves the parameter ${\mathcal{R}}_0 \equiv {\mathcal{R}}(0)$ as the initial condition that is needed to integrate the differential equation outwards radially. However, not all values of ${\mathcal{R}}_0$ will result in physical solutions. In general, as a consequence of the nonlinearity of eq. (\[eqn:master\]), ${\mathcal{R}}(r)$ will become singular at a finite value of $r=r_s$ and only a discrete set of initial conditions will allow for solutions that that extend to $r \to \infty$. To find these branches our numerical procedure consists in maximizing the value $r_s$ where the singularity appears. Solutions appear as singularities/ peaks of $r_s$ as a function of the initial conditions. See Appendix \[app:method\] for a more detailed description of the numerical method.
For fixed $\Omega \in [0, 1/2]$, the two branches (stable and unstable) of the MP black hole (\[eq:MPsol\]) correspond to two such solutions. In terms of the parameter $a$, the MP solutions describe an ellipse in the $({\mathcal{R}}_0, \Omega)$ plane as
$${\mathcal{R}}_0 = \frac{2}{1+a^2}, \qquad \Omega = \frac{a}{1+a^2}.
\label{eqn:MPellipse}$$
Apart from the MP black hole solutions, we find that multiple branches of bumpy solutions extend from every axisymmetric zero-mode. They can be represented in $({\mathcal{R}}_0, \Omega)$ plane as curves that extend from the Myers-Perry ellipse, as shown in figure \[fig:OmegaR0\].
![Branches of axisymmetric deformations (blue) of MP black hole (black) on the $({\mathcal{R}}_0,\Omega)$ plane. Branches moving towards negative ${\mathcal{R}}_0$ connect to black rings. And have a decreasing mass density at the origin. While the branches moving towards positive ${\mathcal{R}}_0$ connect to black Saturns and ${\mathcal{R}}_0$ approaches a value of the stable MP black hole. The right plot is a close-up showing good agreement with the analytic expansions (orange). The right plot also shows the very short ($-$)-branches. \[fig:OmegaR0\]](Plots/OmegaR0Axisymmetric "fig:"){width="210pt"} ![Branches of axisymmetric deformations (blue) of MP black hole (black) on the $({\mathcal{R}}_0,\Omega)$ plane. Branches moving towards negative ${\mathcal{R}}_0$ connect to black rings. And have a decreasing mass density at the origin. While the branches moving towards positive ${\mathcal{R}}_0$ connect to black Saturns and ${\mathcal{R}}_0$ approaches a value of the stable MP black hole. The right plot is a close-up showing good agreement with the analytic expansions (orange). The right plot also shows the very short ($-$)-branches. \[fig:OmegaR0\]](Plots/ComparisonAxisymmetric "fig:"){width="210pt"}
We observe that the bumpy branches fall in two distinct categories. Those branches that arise from even $N$ zero modes, as defined in eq. (\[eq:axisym-linear-sol\]), tend to ${\mathcal{R}}_0 \to -\infty$ as $\Omega \to 0$ (asymptotically like ${\mathcal{R}}_{0}\propto -\frac{1}{\Omega^2}$). This is equivalent to a rapidly decreasing mass density at the rotation axis as one moves along the branch. These bumpy branches connect the MP-branch to families of $N-1$ concentric black rings. In figure \[fig:rings\], the mass density profiles $m = e^{\mathcal{R}}$ are shown. On the other hand, for the zero modes with odd values of $N$, we have ${\mathcal{R}}_0 \to 2$, which means that the mass density at the center will closely approach that of a stable MP black hole. These branches will resemble black Saturns with $N-2$ rings, as shown in figure \[fig:saturns\].
As discussed in [@Dias:2014cia; @Emparan:2014pra], every axisymmetric branch extends in both directions from the zero mode. This corresponds to the fact that linear perturbations of the Myers-Perry black hole can be added with either a positive or a negative amplitude. According to the convention in [@Emparan:2014pra], branches adding the amplitude of the sign $(-1)^{N+1}$ on the axis are called ($+$)-branches, which deform the MP-black hole towards the black rings or black Saturns, and the opposites, ($-$)-branches, which develop a singularity on the equator of the horizon. It is so far unclear if this ($-$)-branch connects to some singly spinning black hole solution.
Agreeing with this, we find that the negative side of the branches extends only for a very short interval, after which the allowed solutions cease to exist. This behavior is to some extent expected, since our approach can not resolve singular or conical solutions in phase space. Numerically the vanishing of a solution manifests itself as a vanishing pole in $r_s$. The ($-$)-branches are shown in the close-up plot of figure \[fig:OmegaR0\], as the very short blue lines extending into the opposite site of the ($+$)-branches. From the perturbative result (\[eq:axisym-perturb-Omega-res\]), one can also see that all ($-$)-branches increase $\Omega$, and vice versa at the linear level.
![Cross-sections of the mass density $m$ for black ripples leading to black rings corresponding to the zero modes $N = 2, 4$, at different values of $\Omega$. Close to the branching points the solutions develop bumpy deformations whereas far away from it the solutions closely resemble separated black rings. The (expected) pinching of the necks as we move away from the MP-branch follows a behavior described already in [@Emparan:2014pra]: For multiple rings the pinching starts at the interior necks and later on the outer ones.\[fig:rings\]](Plots/ring1 "fig:"){width="\textwidth"} ![Cross-sections of the mass density $m$ for black ripples leading to black rings corresponding to the zero modes $N = 2, 4$, at different values of $\Omega$. Close to the branching points the solutions develop bumpy deformations whereas far away from it the solutions closely resemble separated black rings. The (expected) pinching of the necks as we move away from the MP-branch follows a behavior described already in [@Emparan:2014pra]: For multiple rings the pinching starts at the interior necks and later on the outer ones.\[fig:rings\]](Plots/ring2 "fig:"){width="\textwidth"}
![Cross-sections of the mass density $m$ for black ripples leading to black Saturns corresponding to the zero modes $N = 3, 5\,$, at different values of $\Omega$.\[fig:saturns\]](Plots/saturn1 "fig:"){width="\textwidth"} ![Cross-sections of the mass density $m$ for black ripples leading to black Saturns corresponding to the zero modes $N = 3, 5\,$, at different values of $\Omega$.\[fig:saturns\]](Plots/saturn2 "fig:"){width="\textwidth"}
The angular momentum (per unit mass) is calculated numerically according to eq. (\[eq:angularMomentum\]). The bumpy branches can then be represented on the usual $({{\mathcal J}}/{{\mathcal M}}, \Omega)$ phase diagram, as depicted in figure \[fig:PhaseDiagramAxisymmetric\].
Figures \[fig:rings\], \[fig:saturns\], \[fig:PhaseDiagramAxisymmetric\] show that the bumpy branches for black rings and black Saturns seem to extend to arbitrary angular momentum[^3] without encountering any conical singularities. For a sufficiently high angular momentum, the deformation ends up as multiple lumps/rings barely connected by exponentially thin necks. Figure \[fig:PhaseDiagramAxisymmetric\] also shows this in a change of behavior of the curves: All branches show three phases of qualitative behaviors: In the first stage the branches are nearly tangential to the MP-branch. After that in an intermediate stage new (ringlike) blobs start to form until they reach a new asymptotic phase. In this final phase the blobs are practically separated and do barely deform further but the distance between the blobs keeps increasing, the angular momentum behaves asymptotically like ${{\mathcal J}}/ {{\mathcal M}}\propto 1/\Omega$.
For solutions with multiple ripples, we find that at low $\Omega$ the radii of ringlike blobs follow two different behaviors. The innermost ring has an approximate radius growing like $\Omega^{-1}$, while the distance between the following outer rings increases slower than that and we estimate it to be $\sim\sqrt{|\log\Omega|}$. The $\Omega^{-1}$-behavior agrees with the blackfold result for multi-rings if the separations of the rings are much shorter than the ring radius [@Emparan:2010sx]. These observations on the far extended branches lead us to the expectation that our ring/Saturn-like bumpy solutions will be connected via a topology changing transition to the single bumpy rings/Saturns, not directly to multi-rings or higher Saturns. This picture is consistent with the numerical result in $D=6$ bumpy Myers-Perrys [@Emparan:2014pra].
![Phase diagram for axisymmetric solutions, we show the 10 first appearing branches: Ring-branches are shown in purple, and Saturn branches in light-blue. The Myers-Perry and black bar solutions are also plotted by the black and red curves. We do not expect the Saturn branches to terminate, but they become harder to construct for low $\Omega$. \[fig:PhaseDiagramAxisymmetric\]](Plots/PhaseDiagramAxisymmetric){width="250pt"}
Multipole deformations: Black Flowers {#sec:multipoleDef}
=====================================
In the large $D$ limit, the soap bubble equation (\[eq:MasterEquationRotation\]) also admits non-axisymmetric stationary solutions, because gravitational waves are completely decoupled as a non-perturbative effect in $1/D$ and solutions with time-dependent multipoles do not radiate.
Multipolar zero modes {#sec:multipoleDefPert}
---------------------
We study again perturbations of the MP-black hole, but this time allow for angular dependence of the perturbations $${\mathcal{R}}(z,\phi) = {\mathcal{R}}_{\rm MP}(z)+ \delta {\mathcal{R}}(z,\phi)\,.$$ Then, the deformation equation becomes $${{\mathcal L}}_{z,\phi} \delta {\mathcal{R}}(z,\phi) = {{\mathcal S}}(z,\phi),\label{eq:deformedeq-nonaxisym}$$ where we defined $$\begin{aligned}
& {{\mathcal L}}_{z,\phi} := z \partial_z^2+(1-z) \partial_z + {\frac{1}{4z}}\partial_\phi^2 +\frac{a^2+1}{2}\,, \\
& {{\mathcal S}}(z,\phi) := -\frac{1}{2} z (\partial_z \delta {\mathcal{R}}(z,\phi))^2-\frac{1}{8z}(\partial_\phi \delta {\mathcal{R}}(z,\phi))^2.\end{aligned}$$ It is convenient to expand the angular dependence as a Fourier series $$\delta {\mathcal{R}}(z,\phi) = \sum_{k=0}^\infty z^\frac{k}{2} f^{(k)}(z) \cos k \phi,$$ where each radial function is expanded in ${\varepsilon}$, $$f^{(k)}(z) = \sum_{p=0}^\infty {\varepsilon}^{p+1} f^{(k)}_p(z).\label{eq:fk-perturb-ex}$$ With the Fourier decomposition, the linear part reduces to the generalized Laguerre equation $${{\mathcal L}}_{z,\phi} \delta {\mathcal{R}}(z,\phi) = \sum_{k=0}^\infty z^\frac{k}{2} {{\mathcal L}}_{(a^2+1-k)/2}^{(k)} f^{(k)}(z) \cos(k\phi),$$ which admits normalizable solutions for $k=m$ when $$a^2+1 -m = 2N \ (N=0,1,2,\dots).$$ We also decompose the source terms into Fourier modes $${{\mathcal S}}(z,\phi) = \sum_{k=0} z^\frac{k}{2} {{\mathcal S}}^{(k)}(z)\cos k\phi,$$ where $$\begin{aligned}
& {{\mathcal S}}^{(k)}(z) = - {\frac{1}{4}} \sum_{\ell = 0}^\infty z^{\ell-1} \left( \ell(\ell+k) f^{(\ell)}(z) f^{(\ell+k)}(z) + (\ell+k) z f^{(\ell)}{}'(z) f^{(\ell+k)}(z) \right.{\nonumber\\ }&\left.\hspace{5cm}+ \ell z f^{(\ell)}(z) f^{(\ell+k)}{}'(z) + 2 z^2 f^{(\ell)}{}'(z) f^{(\ell+k)}{}'(z) \right){\nonumber\\ }& \quad- {\frac{1}{8}} \sum_{ \ell =0}^k \left( (k-\ell) f^{(\ell)}{}'(z)f^{(k-\ell)}(z) + k f^{(\ell)} (z)f^{(k-\ell)}{}' (z)+ 2 z f^{(\ell)}{}' (z)f^{(k-\ell)}{}'(z)\right).\label{eq:nonaxisym-source-k}\end{aligned}$$ Here the last line exists only for $k>0$.
Nonlinear perturbation
----------------------
For higher order perturbations, we proceed in almost the same manner as for the axisymmetric sector. The generalized Laguerre operators ${\cal L}_N^{(m)}$ also show resonant behavior if they are sourced by the corresponding resonant term $L_N^{(m)}(z)$, provided $N$ is a non-negative integer. Therefore, for the solution to be regular and normalizable, the resonant term has to be removed from the source for every mode by renormalizing the angular velocity as $$a^2 + 1 = \left(N+\frac{m}{2}\right)\left(1+\sum_{p=1}^\infty \mu_p {\varepsilon}^p\right).\label{eq:omega-renom-nonaxisym}$$ A new phenomenon we observe, is that some modes can not independently excited at linear order, otherwise the renormalization of the angular velocity becomes impossible. To show this, let us assume to the contrary that we start at linear order only with the zero mode corresponding to $a^2+1 -m= 2N$, $$f_0^{(m)}(z) = L^{(m)}_N(z). \label{eq:nonaxim-ex-lin-0}$$ Then, this mode acts as a source for the neighboring perturbations $f^{(0)}_1$ and $f^{(2m)}_1$ at next-to-leading order, $$\begin{aligned}
& {{\mathcal L}}^{(0)}_{N+m/2} f^{(0)}_1(z) = {{\mathcal S}}^{(0)}(z)\,, \label{eq:sourcedNLOneighbor1}\\
& {{\mathcal L}}^{(2m)}_{N-m/2} f^{(2m)}_1(z) = {{\mathcal S}}^{(2m)}(z)\,.\label{eq:sourcedNLOneighbor2}\end{aligned}$$ If $m$ is a even, eqs. (\[eq:sourcedNLOneighbor1\]) and (\[eq:sourcedNLOneighbor2\]) will contain resonant sources.[^4] However, since we did not include the corresponding linear order term at leading order, the parameter renormalization cannot absorb the resonant terms. This implies that we are forced to include also the neighboring overtone modes at leading order $$f_0^{(0)}(z) = \alpha_0 L^{(0)}_{N+m/2},\quad f^{(m)}_0(z) = \alpha_1 L^{(m)}_N(z) ,\quad f^{(2m)}_0(z) = \alpha_2 L^{(2m)}_{N-m/2}(z). \label{eq:nonaxim-ex-lin-1}$$ Repeating the same argument for the new linear solution, one might be concerned that now we need an infinite tower of overtone modes to regularize the secular behavior. However, if $N-(i-1) m/2 <0$ for the $i$-th overtone, the equation $${{\mathcal L}}_{N-(i-1)m/2}^{(i m)} f_1^{(im)}(z) = S^{(im)}(z)$$ ceases to produce secular behavior as long as the source term is a polynomial. Therefore, given $m$ and $N$, the linear order solution should be a linear combination of its overtone modes whose overtone number does not exceed $2N/m+1$.[^5]
### Recurrence formula
Using the expansion of the spin parameter (\[eq:omega-renom-nonaxisym\]) we can derive a recurrence formula for all orders in perturbation theory. Eq. (\[eq:deformedeq-nonaxisym\]) can be rewritten as $${{\mathcal L}}_{N+(m-k)/2}^{(k)} f^{(k)}(z) = \bar{{{\mathcal S}}}{}^{(k)}(z)\,, \label{eq:deformedeq-nonaxisym-k-bar1}$$ where $$\bar{{{\mathcal S}}}{}^{(k)}(z) = {{\mathcal S}}^{(k)}(z) - \left(N+\frac{m}{2}\right)\sum_{p=1}^\infty \mu_p {\varepsilon}^p f^{(k)}(z)\,,
\label{eq:nonaxisym-source-k-bar2}$$ and ${{\mathcal S}}^{(k)}(z)$ given through eq. (\[eq:nonaxisym-source-k\]). Under the perturbative expansion (\[eq:fk-perturb-ex\]), we also expand the source term by $$\bar{{{\mathcal S}}}{}^{(k)}(z) = \sum_{p=1}^\infty {\varepsilon}^p \bar{{{\mathcal S}}}{}_p^{(k)}(z).$$ Using an inductive argument, the regular normalizable perturbations are shown to be polynomials to all orders of the perturbation. Therefore, we expand the radial functions at each order by the associated Laguerre polynomials, $$f_p^{(k)}(z) = \sum_{I} {{\mathcal C}}^{(k)}_{p,I} L_I^{(k)}(z).$$ As discussed in the previous section, the linear order solution should include all the overtone modes with $N-im/2>0$, $${{\mathcal C}}_{0,N+m/2}^{(0)}:=\alpha_0,\quad {{\mathcal C}}_{0,N}^{(m)}:=\alpha_1,\quad {{\mathcal C}}_{0,N-m/2}^{(2m)}:=\alpha_2,\ \dots\ ,{{\mathcal C}}_{0,N-(\eta-1) m/2}^{(\eta m)}:=\alpha_\eta,$$ where $\eta:=\lfloor 2N/m \rfloor+1$. If $m$ is odd, the even overtones are turned off. Using the reparametrization of ${\varepsilon}$, we set $${{\mathcal C}}_{p,N}^{(m)}=0 \quad (\text{if }p>0).$$ Substituting this expansion into eq. (\[eq:nonaxisym-source-k-bar2\]) , the source term can be decomposed into a resonant part and a normalizable part $$\bar{{{\mathcal S}}}^{(k)}_p(z) = {\cal T}^{(k)}_p L^{(k)}_{N+(m-k)/2}(z) + {{\mathcal L}}^{(k)}_{N+(m-k)/2}\bigr[({\rm polynomial\ of \ }z)\bigl]
\label{eq:nonaxisym-source-k-bar-result}$$ where ${\cal T}^{(k)}_p = 0$ gives the normalization condition[^6]. To extract the resonant term from the source, the following decomposition formula of the product of the associated Laguerre polynomials is used $$z^\frac{i+j-k}{2} L_I^{(i)}(z) L_J^{(j)}(z) = \sum_{K=0} {\cal Y}^{(i,j,k)}_{I,J,K} L^{(k)}_K(z),$$ where the coefficients are written by the integral of the triple product of the associated Laguerre polynomials $$\begin{aligned}
&{\cal Y}^{(i,j,k)}_{I,J,K} = \frac{K!}{(K+k)!}{{\mathcal I}}^{(i,j,k)}_{I,J,K}\,\end{aligned}$$ with $$\begin{aligned}
{{\mathcal I}}^{(i,j,k)}_{I,J,K} := \int_0^\infty dz e^{-z} z^\frac{i+j+k}{2} L_I^{(i)}(z) L_J^{(j)}(z) L_K^{(k)}(z).\end{aligned}$$ This integration can be expressed through Lauricella’s generalized hypergeometric functions (see Appendix. \[sec:associated-laguerre-lauricella\]) [@Erdelyi1936]. [^7]
Since the LO-perturbation only contains the fundamental mode $m$ and its overtones, also at NLO only $m$ and its overtones are excited. To eliminate the resonant part in (\[eq:nonaxisym-source-k-bar-result\]), we require for $i=0,\dots,\eta$ (again, only odd $i$ if $m$ is odd) $$\begin{aligned}
&\left(N+\frac{m}{2}\right)\sum_{q=1}^{p} \mu_{q} {{\mathcal C}}_{p-q,N+(1-i)m/2}^{(im)} {\nonumber\\ }&= - {\frac{1}{4}}\sum_{j=0}^\infty \sum_{q=0}^{p-1} \sum_{I,J} {{\mathcal C}}^{(jm)}_{q,I} {{\mathcal C}}^{((i+j)m)}_{p-1-q,J} (I+J-N+(i+2j-1)m/2) {\cal Y}^{(jm,(i+j)m,im)}_{I,J,N+(1-i)m/2} {\nonumber\\ }&\quad -{\frac{1}{8}}\sum_{j=0}^i \sum_{q=0}^{p-1} \sum_{I,J} {{\mathcal C}}^{(jm)}_{q,I} {{\mathcal C}}^{((i-j)m)}_{p-1-q,J} (I+J-N+(i-1)m/2) {\cal Y}^{(jm,(i-j)m,im)}_{I,J,N+(1-i)m/2}, \label{eq:nonaxisym-recurrence-normalizable-con}\end{aligned}$$ where the last line only exists for $i>0$. Other than the resonant terms, we also obtain the coefficients
\[eq:nonaxisym-recurrence\] $$\begin{aligned}
& {{\mathcal C}}_{p,K}^{(im)} = -\sum_{q=1}^{p-1} \frac{N+m/2}{N+(1-i)m/2-K} \mu_q {{\mathcal C}}_{p-q,K}^{(im)} {\nonumber\\ }&\qquad - \sum_{j=0}^\infty\sum_{q=0}^{p-1} \sum_{I,J} {{\mathcal C}}^{(jm)}_{q,I} {{\mathcal C}}^{((i+j)m)}_{p-1-q,J} \frac{I+J+jm-K}{4(N+(1-i)m/2-K)} {\cal Y}^{(jm,(i+j)m,im)}_{I,J,K} {\nonumber\\ }& \qquad - \sum_{j=0}^i \sum_{q=0}^{p-1} \sum_{I,J} {{\mathcal C}}^{(jm)}_{q,I} {{\mathcal C}}^{((i-j)m)}_{p-1-q,J} \frac{I+J-K}{8(N+(1-i)m/2-K)} {\cal Y}^{(jm,(i-j)m,im)}_{I,J,K}.
\label{eq:nonaxisym-recurrence-coeff}\end{aligned}$$
Again, we do not have the last line for $i=0$.
### Comparison to the numerical results
For later comparison with the numerical result, we derive an expression for the center value of each angular Fourier mode. As in the axisymmetric sector, the center thickness is defined by
\[eq:nonaxisym-thickness\] $${\mathcal{R}}_0 = \frac{2}{1+a^2} + \sum_{i=0}^\infty{\varepsilon}^{i+1} \sum_{I}{{\mathcal C}}_{i,I}^{(0)},\label{eq:nonaxisym-thickness-0}$$ and for the multipoles, we define[^8] $${\mathcal{R}}_{k} = \sum_{i=0}^\infty {\varepsilon}^{i+1} \sum_{I}\frac{ (I+k)!{{\mathcal C}}_{i,I}^{(k)}}{(2(1+a^2))^{k/2}I!k!}.
\label{eq:nonaxisym-thickness-k}$$
### Even multipoles
The analysis for different fundamental modes $(N,m)$ differs in important aspects, so we are going to distinguish several cases in the following. Let us begin with the case $m$ even. As opposed to the axisymmetric modes, the normalization condition (\[eq:nonaxisym-recurrence-normalizable-con\]) already gives the coupled equation that determines the linear coefficients and the parameter renormalization,
$$\begin{aligned}
& \mu_{1} \alpha_0 = - {\frac{1}{4}} \sum_{j=0}^\eta {\cal A}_{0,j} \alpha_j^2,\label{eq:nonaxisym-normalize-leading-0}\\
& \mu_{1} \alpha_i = - {\frac{1}{4}} \sum_{j=0}^{\eta-i} {\cal A}_{i,j} \alpha_j \alpha_{i+j} -{\frac{1}{8}}\sum_{j=0}^i {\cal B}_{i,j} \alpha_j \alpha_{i-j} \quad (i>0),\label{eq:nonaxisym-normalize-leading-i}\end{aligned}$$
\[eq:nonaxisym-normalize-leading\]
where $$\begin{aligned}
&{\cal A}_{i,j} = {\cal Y}^{(jm,(i+j)m,im)}_{N+(1-j)m/2,N+(1-i-j)m/2,N+(1-i)m/2}\,, \\
& {\cal B}_{i,j} = {\cal Y}^{(jm,(i-j)m,im)}_{N+(1-j)m/2,N+(1-i+j)m/2,N+(1-i)m/2}\,.\end{aligned}$$ The nonlinear eq. (\[eq:nonaxisym-normalize-leading\]) is hard to solve in general and we will further distinguish different cases.
#### Even multipoles with $2N<m$
Here the leading order solution consists of only two modes $$f^{(0)}_0(z) = \alpha_0 L_{N+m/2}^{(0)}(z),\quad f_0^{(m)} (z) = \alpha_1 L_{N}^{(m)}(z)\,.$$ The normalization condition (\[eq:nonaxisym-normalize-leading\]) becomes $$\begin{aligned}
& \mu_1 \alpha_0 = -\frac{{{\mathcal I}}_0}{4} \, \alpha_0^2 -\, \frac{(N+m)!}{4N!} {{\mathcal I}}_1 \alpha_1^2\,, \\
&\mu_1 \alpha_1 = - {\frac{1}{2}} {{\mathcal I}}_1 \, \alpha_0\alpha_1 \,,\end{aligned}$$ where $${{\mathcal I}}_0 ={{\mathcal X}}^{N+m/2}_{N+m/2,N+m/2}\,,\quad {{\mathcal I}}_1 = {\cal Y}_{N+m/2,N,N}^{(0,m,m)}\,.$$ Setting $\alpha_1=0$ immediately reproduces the axisymmetric result (\[eq:axisym-mu1\]). Therefore assuming $\alpha_1\neq 0$, we obtain $$\mu_1 = - {\frac{1}{2}}{{\mathcal I}}_1 \alpha_0\,,$$ and $$\left( 2{{\mathcal I}}_1-{{\mathcal I}}_0 \right) \, \alpha_0^2 = \frac{(N+m)!}{N!}{{\mathcal I}}_1 \alpha_1^2.\label{eq:cond-nonaxim-2}$$ Which has real solutions only if $$\frac{{{\mathcal I}}_0}{{{\mathcal I}}_1} \leq 2 \label{eq:cond-nonaxim-1}\,.$$ This leads to an upper bound for $m$ (see figure \[fig:nonaxim-even-m-bound\]). Since the sign of $\alpha_1$ does not matter, we obtain $$\alpha_1/\alpha_0 = \sqrt{\frac{N!}{(N+m)!}} \sqrt{2-\frac{{{\mathcal I}}_0}{{{\mathcal I}}_1} }\,.$$
![The maximum values of $m$ in the $2N<m$ sector (blue circles), defined by the constraint (\[eq:cond-nonaxim-1\]), and in the $N<m\leq 2N$ sector (red and red empty circles), defined by the positivity of eq. (\[eq:nonaxim-even-ellipse-rad\]). The blue dashed and red dotted curves denote $m=2N$ and $m=N$, respectively. Branches in each sector should be above each curve. The maximum values below $m=N$ (which can not be realized physically) are shown by red empty circles. Gray dots denote possible branches below the maxima. \[fig:nonaxim-even-m-bound\]](Plots/Flowers_max_m.pdf){width="220pt"}
The only branches satisfying $2N<m$ and the constraint (\[eq:cond-nonaxim-1\]) are
$$\begin{aligned}
&(N,m)=(0,2): \quad \mu_1 = 1, \quad \alpha_1 = \frac{1}{\sqrt{2}} \quad ({\rm black \ bar}),\\
&(N,m)=(0,4): \quad \mu_1=-3,\quad \alpha_1 = \frac{1}{6\sqrt{2}},\\
&(N,m) = (1,4): \quad\mu_1 = 20, \quad\alpha_1 = {\frac{1}{10 \sqrt{2}}},\\
&(N,m) = (1,6): \quad \mu_1 = -\frac{175}{2}, \quad\alpha_1 = {\frac{1}{210 \sqrt{5}}},\\
&(N,m) = (2,6): \quad\mu_1 =658, \quad \alpha_1 = {\frac{1}{168}}\sqrt{\frac{19}{47}},\end{aligned}$$
where we set $\alpha_0=1$.
The right hand side in eq. (\[eq:cond-nonaxim-1\]) monotonically grows in $N$, and for $N\geq3$, the bound (\[eq:cond-nonaxim-1\]) finally starts to exclude all of $m>2N$. We will see that a similar bound appears also in the sector $N<m\leq 2N$ for $N \geq 3$. This upper bound does not mean the absence of the higher multipole deformation, but rather implies such deformation should be coupled with the lower companions even in the linear order. For example, $(N,m)=(0,6)$ can be coupled with $(N,m)=(2,2)$ (together with $(3,0)$ and $(1,4)$), which is in $\frac{2}{3}N<m\leq N$ sector.
Lastly, we evaluate the center values and angular velocity in eq. (\[eq:nonaxisym-thickness\]) up to ${{\mathcal O}\left({\varepsilon}\right)}$, $${\mathcal{R}}_0 = \frac{2}{1+a^2} + \alpha_0 {\varepsilon}= \frac{1}{N+m/2} \left(1-(\mu_1-(N+m/2)\alpha_0) {\varepsilon}\right),$$ and $${\mathcal{R}}_m = \frac{(N+m)!\alpha_1}{(4n+2m)^{m/2}N!m!} {\varepsilon}.$$ By defining $\bar{{\varepsilon}}:=(N+m/2){\mathcal{R}}_0-1$ we obtain $$\Omega = \frac{\sqrt{2N+m-1}}{2N+m} \left(1+\omega_1 \bar{\varepsilon}\right),\quad {\mathcal{R}}_m = r_1\bar{\varepsilon}.\label{eq:nonaxisym-even-1-omega-r0-grad}$$ with the expansion coefficients
$$\begin{aligned}
&(N,m) = (0,4): \quad \omega_1 = {\frac{1}{5}}, \quad r_1 = {\frac{1}{1920 \sqrt{2}}},\\
&(N,m) = (1,4): \quad \omega_1 = \frac{8}{17}, \quad r_1 = -{\frac{1}{4896 \sqrt{2}}},\\
&(N,m) = (1,6): \quad \omega_1 = \frac{25}{61}, \quad r_1 = {\frac{1}{11243520 \sqrt{5}}},\\
&(N,m) = (2,6): \quad \omega_1 = \frac{2632}{5877}, \quad r_1 = -{\frac{1}{31344000}}\sqrt{\frac{19}{47}}.\end{aligned}$$
Some of these results are compared with the numerical analysis in figure \[fig:OmegaR0Stars\].
#### Even multipoles with $N < m \leq 2N$
Here three modes have to be excited at leading order $$f^{(0)}_0(z) = \alpha_0 L_{N+m/2}^{(0)}(z),\quad f_0^{(m)} (z) = \alpha_1 L_{N}^{(m)}(z) ,\quad
f^{(2m)}_0(z) = \alpha_2 L_{N-m/2}^{(2m)}(z).$$ The normalization condition (\[eq:nonaxisym-normalize-leading\]) leads to a quadratic constraint for the relative amplitudes
\[eq:nonaxisym-even-2-eq\] $$\begin{aligned}
& \mu_1 \alpha_0 = -{\frac{1}{4}}{{\mathcal I}}_0 \alpha_0^2 - {\frac{1}{4}}{{\mathcal I}}_1' \alpha_1^2 - {\frac{1}{4}}{{\mathcal I}}_2' \alpha_2^2,\label{eq:nonaxisym-even-2-eq1}\\
&\mu_1 \alpha_1 = - {\frac{1}{2}} {{\mathcal I}}_1 \alpha_0\alpha_1 -{\frac{1}{4}}{{\mathcal I}}_3 \alpha_2 \alpha_1,\label{eq:nonaxisym-even-2-eq2}\\
&\mu_1 \alpha_2 = - {\frac{1}{2}} {{\mathcal I}}_2 \alpha_0\alpha_2 - {\frac{1}{8}} {{\mathcal I}}_3' \alpha_1^2,\label{eq:nonaxisym-even-2-eq3}\end{aligned}$$
where the coefficients are given by $$\begin{aligned}
&{{\mathcal I}}_0 ={{\mathcal X}}^{N+m/2}_{N+m/2,N+m/2}\,,&& {{\mathcal I}}_1 = {\cal Y}_{N+m/2,N,N}^{(0,m,m)}\,,\\
&{{\mathcal I}}_2 = {\cal Y}^{(2m,2m,0)}_{N-m/2,N-m/2,N+m/2}\,,&& {{\mathcal I}}_3 = {\cal Y}^{(m,2m,m)}_{N,N-m/2,N}\,,\end{aligned}$$ and $$\begin{aligned}
{{\mathcal I}}_1' = \frac{(N+m)!}{N!} {{\mathcal I}}_1,\quad {{\mathcal I}}_2' = \frac{(N+3m/2)!}{(N-m/2)!} {{\mathcal I}}_2,\quad
{{\mathcal I}}_3' = \frac{(N-m/2)!}{(N+3m/2)!} \frac{(N+m)!}{N!}{{\mathcal I}}_3.\end{aligned}$$ Setting $\alpha_1=0$ immediately reproduces the previous analysis in which $m$ is replaced by $2m$. Therefore, we consider $\alpha_1\neq 0$ and (\[eq:nonaxisym-even-2-eq2\]) reduces to $$\mu_1 = - {\frac{1}{2}} {{\mathcal I}}_1 \alpha_0 -{\frac{1}{4}}{{\mathcal I}}_3 \alpha_2.\label{eq:nonaxisym-even-2-mu1}$$ Substituting this to the rest of eq. (\[eq:nonaxisym-even-2-eq\]), we obtain two quadratic equations $$\begin{aligned}
& (2 {{\mathcal I}}_1-{{\mathcal I}}_0) \alpha_0^2 + {{\mathcal I}}_3 \alpha_2\alpha_0 - {{\mathcal I}}_2' \alpha_2^2 = {{\mathcal I}}_1' \alpha_1^2 ,\label{eq:nonaxisym-even-con-a}\\
& 4({{\mathcal I}}_1-{{\mathcal I}}_2) \alpha_0\alpha_2 +2{{\mathcal I}}_3 \alpha_2^2 = {{\mathcal I}}_3' \alpha_1^2.
\label{eq:nonaxisym-even-con-b}\end{aligned}$$ ${{\mathcal I}}_1$ and ${{\mathcal I}}_2$ (and accordingly ${{\mathcal I}}_1'$ and ${{\mathcal I}}_2'$) have to have the same sign for fixed $N$ and $m$. Thus eq. (\[eq:nonaxisym-even-con-a\]) and eq. (\[eq:nonaxisym-even-con-b\]) describe an ellipse and a hyperbola in the $(\alpha_1/\alpha_0,\alpha_2/\alpha_0)$ plane. The curves always have two (or no) intersections, which are shown to be identical by a constant shift in the angular coordinate $\phi \to \phi+\pi/m$. Therefore, we have at most one branch for each $(N,m)$ with $N<m\leq 2N$.
The radii of the ellipse from eq. (\[eq:nonaxisym-even-con-a\]) are proportional to $$\begin{aligned}
2-\frac{{{\mathcal I}}_0}{{{\mathcal I}}_1} + \frac{{{\mathcal I}}_3^2}{4{{\mathcal I}}_0{{\mathcal I}}_2'}\,. \label{eq:nonaxim-even-ellipse-rad}\end{aligned}$$ The positivity of this value is the necessary condition for the existence of the branch, which gives the upper bound for $m$ (figure \[fig:nonaxim-even-m-bound\]). Since the last term in eq. (\[eq:nonaxim-even-ellipse-rad\]) decays very quickly in $N$, the upper bound coincides with that from eq. (\[eq:cond-nonaxim-1\]) for $N\geq 3$. And for $N>11$ the upper and the lower bound can not be satisfied at the same time. Accordingly this sector only contains a finite finite number of branches, like the $m>2N$ sector.
We show the result for the lower branches
$$\begin{aligned}
&(N,m)=(1,2) :\quad \mu_1=-4.48,\quad \alpha_1 = 0.382,\quad \alpha_2 = 0.00243\,,\\
&(N,m) = (2,4) :\quad\mu_1= -132.5,\quad \alpha_1 = 0.0439,\quad \alpha_2 = -3.84 \times 10^{-8}\,,\\
&(N,m) = (3,4) :\quad \mu_1 = 903.0,\quad \alpha_1 = 0.0299,\quad \alpha_2 = -1.20\times 10^{-9}\,,\\
&(N,m) = (3,6) :\quad\mu_1 = -4851.0,\quad \alpha_1 = 0.00268,\quad \alpha_2 = -2.87\times 10^{-13}\,,\end{aligned}$$
where we set $\alpha_0=1$.
One can observe that the amplitude of the overtone mode will be strongly suppressed for larger $N$ and $m$. The gradient of the angular velocity and the center values (\[eq:nonaxisym-even-1-omega-r0-grad\]) are also evaluated for the same branches as
$$\begin{aligned}
&(N,m)=(1,2) :\quad \omega_1 = 0.230, \quad r_1 = 0.0221, \quad r_2 = -4.89\times10^{-7}\,,\\
&(N,m) = (2,4) :\quad \omega_1 = 0.416, \quad r_1 = 0.0000189, \quad r_2 = -2.56\times 10^{-18}\,,\\
&(N,m) = (3,4) :\quad \omega_1 = 0.447, \quad
r_1 = -2.92\times 10^{-6}, \quad
r_2 = 2.49\times 10^{-20}\,,\\
&(N,m) = (3,6) :\quad \omega_1 = 0.454, \quad
r_1 = 3.36\times 10^{-9}, \quad r_2 =-4.64\times 10^{-32}\,,\end{aligned}$$
where we also evaluated the amplitude of the overtone $r_2$ defined via $$\begin{aligned}
{\mathcal{R}}_{2m} = \frac{(N+m/2)!\alpha_2}{(4n+2m)^m (N-m/2)!(2m)!}{\varepsilon}=: r_2 \bar{{\varepsilon}}\,.\end{aligned}$$
### Odd multipoles with $2N<m$
For odd $m$ the leading order modes do not produce secular behavior at second order, but starting from third order it will also appear in this case. Here the LO-solution consists of a single mode, $$f^{(m)}_{0} (z)= L_N^{(m)}(z).$$ At second order the even $m$ modes have to be excited $$\begin{aligned}
& {{\mathcal C}}_{1,K}^{(0)} = - \frac{2N+m-K}{4(N+m/2-K)} {\cal Y}^{(m,m,0)}_{N,N,K},\\
& {{\mathcal C}}_{1,K}^{(2m)} = - \frac{2N-K}{8(N-m/2-K)} {\cal Y}^{(m,m,2m)}_{N,N,K},\end{aligned}$$ without any renormalization, $$\mu_1 = 0.$$ Iterating eq. (\[eq:nonaxisym-recurrence\]) reveals that there are only even $m$ modes for every odd order in $\varepsilon$, and vice versa. Which results in $\mu_k=0$ for odd $k$. At third order, the normalization condition (\[eq:nonaxisym-recurrence-normalizable-con\]) becomes $$\begin{aligned}
&\mu_{2}
=- \sum_K\left[ {{\mathcal C}}^{(0)}_{1,K} \frac{K}{2N+m} {\cal Y}^{(0,m,m)}_{K,N,N}+ {{\mathcal C}}^{(2m)}_{1,K}\frac{K+m}{2(2N+m)} {\cal Y}^{(2m,m,m)}_{K,N,N}\right] {\nonumber\\ }&=\frac{N!}{(N+m)!}\left[ \sum_{K=0}^{2N+m} \frac{K(2N+m-K)}{4(2N+m)(N+m/2-K)} \left( {{\mathcal I}}^{(0,m,m)}_{K,N,N} \right)^2\right. {\nonumber\\ }&\left. \hspace{4cm}+ \sum_{K=0}^{2N} \frac{(K+m)(2N-K)}{16(2N+m)(N-m/2-K)} \frac{K!}{(K+2m)!}\left({{\mathcal I}}^{(2m,m,m)}_{K,N,N}\right)^2\right].\label{eq:nonaxisym-odd-mu2}\end{aligned}$$ Different from the even cases, the normalization condition for the simplest odd multipoles does not lead to a bound for $m$. For the lower sector $m\leq 2N$, we will have multiple overtones at linear order, which leads to coupled equations at third order as in the even modes. This may bound $m$ as in the even modes.
In contrast to the case of $m$ even, $\Omega$ and ${\mathcal{R}}_0$ only have even powers of ${\varepsilon}$ appearing in their expansion $$\begin{aligned}
& \Omega = \frac{\sqrt{2N+m-1}}{2N+m}\left(1-\frac{N+m/2-1}{2N+m-1}\mu_2 {\varepsilon}^2\right)\,,\\
& {\mathcal{R}}_0 = {\frac{1}{N+m/2}} \left[1 + {\varepsilon}^2\left( (N+m/2)\sum_{K=0}^{2N+m} {{\mathcal C}}_{1,K}^{(0)}
- \mu_2 \right)\right]\,,\end{aligned}$$ while ${\mathcal{R}}_m$ is odd in ${\varepsilon}$, $$\begin{aligned}
{\mathcal{R}}_m = \frac{(N+m)!}{(4N+2m)^{m/2}N!m!}{\varepsilon}.\end{aligned}$$ This means that odd branches go out from the Myers-Perry branch only in one direction.[^9] The leading order corrections can be written as $$\Omega = \frac{\sqrt{2N+m-1}}{2N+m}\left(1+\omega_2 {\varepsilon}^2\right),\quad
{\mathcal{R}}_0 = {\frac{1}{N+m/2}} \left(1+\rho_0 {\varepsilon}^2\right),\quad {\mathcal{R}}_m = \rho_m {\varepsilon}.$$ And the first few branches satisfy,
$$\begin{aligned}
& (N,m)=(0,3): \, \mu_2 = 0,\,\, \omega_2 = 0,\quad \rho_0 = 36, \quad \rho_m = {\frac{1}{6\sqrt{6}}},\\
& (N,m)=(0,5): \, \mu_2 = 0,\,\, \omega_2=0, \quad \rho_0 = -6400, \quad \rho_m = {\frac{1}{100\sqrt{10}}},\\
& (N,m)=(1,3): \, \mu_2 = -6592,\,\, \omega_2 = 2472,\quad \rho_0 = 4352, \quad \rho_m = {\frac{1}{5}}\sqrt{\frac{2}{5}}\, .\end{aligned}$$
For $N=0$ branches, eq. (\[eq:nonaxisym-odd-mu2\]) gives $\mu_2=0$ for any odd $m$, $$\left. \Omega\right|_{N=0} = \frac{\sqrt{m-1}}{m}\left(1+{{\mathcal O}\left({\varepsilon}^4\right)}\right).$$ For $N>0$, for example, we have $$\left.\frac{d \ln \Omega}{d \ln {\mathcal{R}}_0}\right|_{(N,m)=(1,3)} = \frac{309}{544}.$$
Numerical construction {#sec:multipoleNum}
----------------------
To obtain the fully non-linear multipole solutions numerically, we use a Fourier decomposition corresponding to overtones of a fundamental mode $m$ $${\mathcal{R}}_m(r,\phi)=\sum_{n=0}^\infty {\mathcal{R}}^{(n m)}(r) \, r^{n m} \cos(n m \phi)\,.
\label{eq:numAnsatzFourMode}$$ Plugging this into the stationary master equation (\[eq:MasterEquationRotation\]), we obtain a countable set of coupled equations for the fundamental Fourier mode ${\mathcal{R}}^{(m)}(r)$ and its overtones ${\mathcal{R}}^{(n \cdot m)}(r)$ ($n=2,3,\dots$). From the perturbative analysis, we know that close to the MP-branch higher overtones will only be weakly excited. So we truncate the Fourier series for some $n_{\text{max}}$ to obtain a finite dimensional problem. The resulting coupled ODEs can be now solved using the shooting method described in appendix \[app:method\], where now the space of initial conditions is spanned by the amplitudes of the Fourier modes ${\mathcal{R}}^{(nm)}(r)$ close to the origin, which we will denote as ${\mathcal{R}}_{0},{\mathcal{R}}_{m},{\mathcal{R}}_{2m}, \dots, {\mathcal{R}}_{n_{\text{max}} m}$.
![Beginning of the branches for $(N,m) = (0,4)$, $(1,4)$ and $(1,6)$ on the $({\mathcal{R}}_0,\Omega)$ plane.\[fig:OmegaR0Stars\]](Plots/ComparisonMultipolar){width="210pt"}
In figure \[fig:OmegaR0Stars\], we show examples of branches extracted numerically with only the fundamental Fourier mode, [[*i.e.,*]{}]{}$n_{max}=1$, and compare them to the perturbative result. We checked that the truncation $n_{max}=1$ is consistent for the beginning of the branch we show by comparing the results to a higher truncation with $n_{max}=2$ and finding good agreement of the results. To extend the branches further overtones should be included.
The inclusion of overtones however makes our numerical procedure much less efficient (see appendix \[sec:AppNumMet\] for details), s.t. at this point we do not find conclusive results for odd multipole branches and even multipole branches corresponding to the opposite sign of the perturbation.
![Mass profiles for branches with $(N,m) = (0,4)$ (left) and $(N,m) = (1,6)$ (right).\[fig:flowerSamples\]](Plots/SurfaceN0m4 "fig:"){width="35.00000%"} ![Mass profiles for branches with $(N,m) = (0,4)$ (left) and $(N,m) = (1,6)$ (right).\[fig:flowerSamples\]](Plots/SurfaceN1m6 "fig:"){width="35.00000%"}
In figure \[fig:flowerSamples\], we show representative plots of mass densities for some of the branches. The profiles for even multipoles show a behavior that can be related to the perturbative result that modes of different $N$ and $m$ couple to each other: The black flower branches show mass profiles, which when averaged over the angular direction resemble the corresponding axisymmetric branch that starts at the same branching point, which results in a similar $({{\mathcal J}}/{{\mathcal M}},\Omega)$-curve see figure \[fig:PhaseDiagramStars\].
![Dashed lines: Branches for $(N,m) =(0,4)$, $(1,4)$ and $(1,6)$ in the $({{\mathcal J}}/{{\mathcal M}},\Omega)$ plane. Solid lines: Branches of axisymmetric solutions. It can be observed that black flower curves take a similar path to the ripple branches originating from the same zero modes. \[fig:PhaseDiagramStars\]](Plots/PhaseDiagramMultipolar "fig:"){width="210pt"} ![Dashed lines: Branches for $(N,m) =(0,4)$, $(1,4)$ and $(1,6)$ in the $({{\mathcal J}}/{{\mathcal M}},\Omega)$ plane. Solid lines: Branches of axisymmetric solutions. It can be observed that black flower curves take a similar path to the ripple branches originating from the same zero modes. \[fig:PhaseDiagramStars\]](Plots/PhaseDiagramMultipolarDiff "fig:"){width="210pt"}
Deformed black bars: Dumbbells and Spindles {#sec:blackBars}
===========================================
As already studied in the previous section the large $D$ effective equations allow for stationary solutions without axisymmetry around the rotation axis, the first (and so far only) analytically known solution is the dipolar black bar [@Andrade:2018nsz]. Like the other multipolar solutions, the black bar plays an important role in the decay of the ultra-spinning instability of MP-black holes [@Andrade:2018yqu; @Andrade:2019edf; @Bantilan:2019bvf]. At high enough angular momentum the bar gets very elongated and develops a Gregory-Laflamme type instability. In this section, we are going to study the zero mode configurations corresponding to this instability.
The black bar is best studied in Cartesian coordinates in the co-rotating frame $$x = r \cos(\phi-\Omega t),\quad y = r \sin(\phi-\Omega t),$$ where it can be written as $${\mathcal{R}}_{\rm bar}(x,y) = 1-\frac{x^2}{2\ell_\perp^2}-\frac{y^2}{2\ell_\parallel^2}$$ where $$\ell_\perp^2 = \frac{2}{1+\sqrt{1-4\Omega^2}},\quad \ell_\parallel^2 = \frac{2}{1-\sqrt{1-4\Omega^2}}.
\label{eq:ellDef}$$ Note that for small $\Omega$ the bar becomes very elongated and in the limit $\Omega \rightarrow 0$ the solution connects to a non-rotating black string along the $y$-direction.
Co-rotating zero modes {#sec:blackBarsPert}
----------------------
We deform the bar perturbatively via ${\mathcal{R}}= {\mathcal{R}}_{\rm bar}(x,y) + \delta {\mathcal{R}}(x,y)$, where the deformation $\delta {\mathcal{R}}(x,y)$ satisfies $$\left[\partial_x^2 - \frac{x}{\ell_\perp^2} \partial_x + \partial_y^2 - \frac{y}{\ell_\parallel^2} \partial_y + 1\right]
\delta {\mathcal{R}}= - \frac{1}{2} ((\partial_x \delta {\mathcal{R}})^2 +(\partial_y \delta {\mathcal{R}})^2)$$ At linear order, the regular solutions are given by Hermite polynomials $$\delta {\mathcal{R}}(x,y) = {\varepsilon}H_{n_x}\left(\frac{x}{\sqrt{2}\ell_\perp}\right) H_{n_y}\left(\frac{y}{\sqrt{2}\ell_\parallel}\right)+{{\mathcal O}\left({\varepsilon}^2\right)},$$ where $n_x,n_y$ are non-negative integers with $$\frac{n_x}{\ell_\perp^2} + \frac{n_y}{\ell_\parallel^2} = 1.$$ Together with the constraint $\ell_\perp^{-2} + \ell_\parallel^{-2} = 1$, the regular and non-trivial perturbations are available only for $$n_x=0, \quad n_y = \ell_\parallel^2 \geq 2.$$
Nonlinear perturbations
-----------------------
Considering the linear result, we can assume only $y$-dependence even in the non-linear regime. Then, by rescaling $$z = \frac{y}{\sqrt{2}\ell_\parallel},$$ the deformation equation reduces to $${{\mathcal H}}_{\ell_\parallel^2} \delta {\mathcal{R}}(z) = - \frac{1}{2} \delta {\mathcal{R}}'(z)^2,$$ where ${{\mathcal H}}_N$ is the Hermite operator defined by $${{\mathcal H}}_N := \frac{d^2}{dz^2} - 2z \frac{d}{dz}+2N. \label{eq:bar-deform-0}$$ Given the value of $\ell_\parallel$, $\Omega$ and $\ell_\perp$ is written by $$\Omega = \frac{\sqrt{\ell_\parallel^2-1}}{\ell_\parallel^2},\quad \ell_\perp = \frac{\ell_\parallel}{\sqrt{\ell_\parallel^2-1}}
= {\frac{1}{\ell_\parallel\Omega}}.$$ The corrections beyond the linear order can be derived in the same manner as the bumpy deformation of the Myers-Perry. First, we expand the deformation function by ${\varepsilon}$ $$\delta {\mathcal{R}}(z) = \sum_{k=0}^\infty {\varepsilon}^{k+1} f_k(z).$$ If we consider a branch bifurcating from the zero mode $\ell_\parallel^2 = N$ on the black bar branch, one can set $$f_0(z) = H_N(z).$$ The length of the bar $\ell_\parallel$ for the deformed branch should be expanded by ${\varepsilon}$, $$\ell_\parallel^2 = N \left(1+\sum_{k=1}^\infty \mu_k {\varepsilon}^k \right),$$ where the running coefficient $\mu_k$ is determined so that $f_k(z)$ remains to be normalizable at each order. Expanding eq. (\[eq:bar-deform-0\]) by ${\varepsilon}$, we obtain $${{\mathcal H}}_N f_k(z) = -{\frac{1}{2}}\sum_{i=0}^{k-1} f_i'(z) f_{k-1-i}'(z) - 2N\sum_{i=0}^{k-1} \mu_{k-i} f_i(z) =: {{\mathcal S}}_k(z).$$ Similar to the bumpy solutions, the higher order corrections can be solved algebraically. Assuming $f_k(z)$ is a polynomial, each order solution can be expanded by the Hermite polynomials, $$f_k(z) = \sum_{M=0} {{\mathcal C}}_{k,M}H_M(z),$$ where the linear order solution is supposed to be ${{\mathcal C}}_{0,M} = \delta_{M,N}$. Substituting this, the source term of each order becomes $${{\mathcal S}}_k(z) = - {\frac{1}{2}} \sum_{i=0}^{k-1} \sum_{I,J} {{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J} H_I'(z)H_J'(z)
- 2 N \sum_{i=0}^{k-1} \sum_{I} \mu_{k-i} {{\mathcal C}}_{i,I} H_I(z).$$ Using the properties of the Hermite polynomials, the source term can be decomposed to the resonant and non-resonant terms, $$\begin{aligned}
&S_k(z) = {{\mathcal H}}_N \left[-{\frac{1}{4}} \sum_{K \neq N} \sum_{I,J} \sum_{i=0}^{k-1} {{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J} \frac{I+J-K}{N-K} {\cal Q}_{I,J}^K H_K(z)-\sum_{K\neq N} \sum_{i=1}^{k-1} \frac{N\mu_{k-i} {{\mathcal C}}_{i,K}}{N-K} H_K(z) \right]
{\nonumber\\ }&\hspace{3cm} -\left[ {\frac{1}{2}}\sum_{I,J} \sum_{i=0}^{k-1} (I+J-N){{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J} {\cal Q}_{I,J}^N+ 2N\sum_{i=0}^{k-1} \mu_{k-i} {{\mathcal C}}_{i,N}\right] H_N(z) \,,\end{aligned}$$ where $Q^K_{I,J}$ is given by eq. (\[eq:hermite-product-cf\]). Using ${{\mathcal C}}_{0,M} = \delta_{M,N}$, the regularizing condition is given by
$$\mu_k =- \sum_{I,J} \sum_{i=0}^{k-1}\frac{I+J-N}{4N}{{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J} {\cal Q}_{I,J}^N- \sum_{i=1}^{k-1} \mu_{k-i} {{\mathcal C}}_{i,N}\,,$$
and the non-resonant coefficients, $$\begin{aligned}
&{{\mathcal C}}_{k, M \neq N} =-{\frac{1}{4}} \sum_{I,J} \sum_{i=0}^{k-1} {{\mathcal C}}_{i,I} {{\mathcal C}}_{k-1-i,J} \frac{I+J-M}{N-M} {\cal Q}_{I,J}^M - \sum_{i=1}^{k-1} \frac{N\mu_{k-i} {{\mathcal C}}_{i,M}}{N-M}.\end{aligned}$$
For the resonant term, we simply set $$\begin{aligned}
{{\mathcal C}}_{k,N} =0 \quad (k>0). \end{aligned}$$ Using induction one can show for odd branches that $f_k(z)$ has only odd (even) power for the even (odd) order, and $\mu_k$ vanishes for every odd order. Similarly, for even $N$, it can be shown that at each order only even powers appear.
### Perturbation solution
By solving the recurrence equation with ${{\mathcal C}}_{0,M} = \delta_{M,N}$, one can obtain the solution to arbitrary order. The result for ${{\mathcal O}\left({\varepsilon}^2\right)}$ is $$\begin{aligned}
\mu_1 = - {\frac{1}{4}} {\cal Q}^N_{N,N},\quad {{\mathcal C}}_{1,M\neq N} = - \frac{2N-M}{4(N-M)}{\cal Q}^M_{N,N}, \label{eq:dumbbell-perturb-res-1}\end{aligned}$$ and for ${{\mathcal O}\left({\varepsilon}^3\right)}$, $$\begin{aligned}
&\mu_2 ={\frac{1}{8}} \sum_{I} \frac{I(2N-I)}{N(N-M)} {\cal Q}^I_{N,N} {\cal Q}_{N,I}^N, \label{eq:dumbbell-perturb-res-2-mu}\\
& {{\mathcal C}}_{2,M\neq N} = {\frac{1}{8}} \sum_{I\neq N} \frac{(N+I-M)(2N-I)}{(N-M)(N-I)}{\cal Q}^M_{I,N} {\cal Q}^I_{N,N}
-\frac{N(2N-M)}{16(N-M)^2}{\cal Q}^N_{N,N} {\cal Q}^M_{N,N},
\label{eq:dumbbell-perturb-res-2-C}\end{aligned}$$ where ${\cal Q}^N_{N,N}=0$ for the odd $N$, giving $\mu_1=0$ for the odd dumbbells.
### Physical quantities
Once, given the deformation $\delta {\mathcal{R}}(z)$ as $$\delta {\mathcal{R}}(z) = \sum_{i=0}^\infty \sum_{I}{\varepsilon}^{i+1} {{\mathcal C}}_{i,I} H_I(z),$$ the physical quantities are calculated using properties of the Hermite polynomials.
#### Value at the origin
Here we evaluate the center values ${\mathcal{R}}_0 = {\mathcal{R}}(0)$ and $\bar{{\mathcal{R}}_0} = {\mathcal{R}}'(0)$, which are also used as the boundary condition for the numerical analysis. Due to the mirror symmetry in the even case, $\bar{{\mathcal{R}}_0}$ only exists for the odd branches. The center thickness ${\mathcal{R}}_0$ of the deformed bar is given by $${\mathcal{R}}_0 = 1 + \sum_{i=0}^\infty \sum_{I} {\varepsilon}^{i+1}{{\mathcal C}}_{i,I} H_I(0), \label{eq:R0-dumbbell-perturb}$$ where $$H_M(0) = \left\{ \begin{array}{cl} (-2)^{M/2} (M-1)!! & \quad (M: {\rm even}) \\ 0 & \quad (M: {\rm odd}) \end{array}
\right. .$$ For the odd branch, only odd Hermite polynomials appear at every odd order in ${\varepsilon}$, so ${\mathcal{R}}_0$ becomes the function of ${\varepsilon}^2$. Using $H_{I}'(0) = - H_{I+1}(0)$, $\bar{{\mathcal{R}}_0}$ is similarly evaluated to $$\bar{{\mathcal{R}}_0} = - \sum_{i=0}^\infty \sum_{I} {\varepsilon}^{i+1}{{\mathcal C}}_{i,I} H_{I+1}(0). \label{eq:R0bar-dumbbell-perturb}$$ With eq. (\[eq:dumbbell-perturb-res-1\]), we obtain $$\begin{aligned}
&{\mathcal{R}}_0 = 1 + {\varepsilon}H_N(0) - {\varepsilon}^2\sum_{I\neq N} \frac{4N-I}{2(N-I)} {\cal Q}_{N,N}^I H_I(0) + {{\mathcal O}\left({\varepsilon}^3\right)},\\
&\bar{{\mathcal{R}}_0} = - {\varepsilon}H_{N+1}(0) + {{\mathcal O}\left({\varepsilon}^3\right)},\end{aligned}$$ where $\bar{{\mathcal{R}}_0}$ does not have ${{\mathcal O}\left({\varepsilon}^2\right)}$ term, because ${\cal Q}^I_{N,N}$ vanishes for odd $I$. For comparison with the numerical analysis (figure \[fig:OmegaR0Bars\]), we obtain, $$\Omega = \frac{\sqrt{N-1}}{N}\left(1+\omega_1 \bar{{\varepsilon}} + \omega_2 \bar{{\varepsilon}}^2\right),\quad \bar{R_0} = \bar{\rho_0} \bar{{\varepsilon}}$$ where $$\bar{{\varepsilon}} := \left\{ \begin{array}{cl}
{\mathcal{R}}_0-1& \quad ( \rm even)\\
\sqrt{|{\mathcal{R}}_0-1|} &\quad (\rm odd)
\end{array}\right.$$ For odd branches with $N=2n+3$, ${\mathcal{R}}_0$ is given by ${\mathcal{R}}_0 = 1 + (-1)^n \bar{{\varepsilon}}^2$. The even branches have $$\begin{aligned}
&\left. \omega_1 \right|_{N=4,6,8,10} = 2,\quad -16, \quad 129,\quad -896\\
&\left. \omega_2 \right|_{N=4,6,8,10} = 52,\quad 8088,\quad \frac{4178816}{5}, \quad \frac{529505120}{7},\end{aligned}$$ and the odd branches have $\omega_1=0$ and $$\begin{aligned}
&\left. \omega_2 \right|_{N=3,5,7,9} =\frac{12}{19},\quad \frac{19200}{1969},\quad \frac{5480160}{53939}, \quad\frac{23886707712}{24551641},\\
&\left. \bar{\rho_0} \right|_{N=3,5,7,9} =-2 \sqrt{\frac{3}{19}},\quad 6 \sqrt{\frac{5}{1969}}, \quad-10 \sqrt{\frac{7}{53939}},\quad \frac{210}{\sqrt{24551641}} .\end{aligned}$$ This shows that one always need to spin up the black hole for the transition to an odd branch.
#### Mass and angular momentum
The mass (\[eq:normCond\]) and angular momentum (\[eq:angularMomentum\]) can be calculated by $${{\mathcal M}}= {{\mathcal M}}_{\rm bar} \int_{-\infty}^\infty \frac{dz}{\sqrt{\pi}} e^{-z^2} \exp(\delta {\mathcal{R}}(z)),$$ and $${{\mathcal J}}= \frac{{{\mathcal M}}}{\Omega} + 4 {{\mathcal M}}_{\rm bar} \ell_\parallel^2 \Omega \int_{-\infty}^\infty
\frac{dz}{8\sqrt{\pi}}e^{-z^2} H_2(z) \exp(\delta {\mathcal{R}}(z)),$$ where ${{\mathcal M}}_{\rm bar} = 2\pi e/\Omega$ is the mass of the bar solution for the given $\Omega$. Due to the orthogonal property of the Hermite polynomials, the integrals in ${{\mathcal M}}$ and ${{\mathcal J}}$ pick up $H_0(z)$ and $H_2(z)$ components in $\exp(\delta {\mathcal{R}}(z))$, respectively.
Using the result in the previous section, the ratio of the angular momentum to the mass is given by $$\frac{{{\mathcal J}}}{{{\mathcal M}}} = {\frac{1}{\Omega}}\left(1-\frac{2(N-1)}{N(N-2)}{\cal Q}^2_{N,N} {\varepsilon}^2+{{\mathcal O}\left({\varepsilon}^3\right)}\right),$$ where we note that $\Omega$ should also varies in ${\varepsilon}$. For the odd branch, both ${{\mathcal J}}/{{\mathcal M}}$ and $\Omega$ become a function of ${\varepsilon}^2$.
Numerical construction {#sec:blackBarsNum}
----------------------
In order to find fully nonlinear deformations of the black bar, we begin by considering equation (\[mastercharge\]) with the ansatz
$${\mathcal{R}}(x,y) = - \frac{x^2}{2 \ell^2_\perp} + {\mathcal{R}}(y)\, ,
\label{eq:barAnsatz}$$
where we imply that ${\mathcal{R}}(y) \equiv {\mathcal{R}}(0, y)$, and $\ell^2_\perp$ is defined by eq. (\[eq:ellDef\]). With this substitution, we are left with
$${\mathcal{R}}'' + \frac12 {\mathcal{R}}'^2 + {\mathcal{R}}+ \frac{\Omega^2 y^2}{2} = \ell^{-2}_\perp\, .
\label{eqn:masterDumbbells}$$
Since $y$ is no longer a radial coordinate, the condition ${\mathcal{R}}'(0) = 0$ is no longer required. We can define ${\mathcal{R}}'(0) \equiv \bar {\mathcal{R}}_0$ instead. Allowed solutions must extend regularly both to $y \to -\infty$ and $y \to \infty$ simultaneously. If we start the integration from $y=0$, the initial conditions are given by ${\mathcal{R}}_0 \equiv {\mathcal{R}}(0)$ and $\bar {\mathcal{R}}_0 \equiv {\mathcal{R}}'(0)$, which have to be tuned in order to get allowed solutions.
The branches arising from even $N$ zero modes have a $y \to -y$ symmetry, so $\bar {\mathcal{R}}_0 = 0$. These bars only require ${\mathcal{R}}_0$ to be tuned, so they can be found in the same way as the axisymmetric solutions. Nonzero values of $\bar {\mathcal{R}}_0$ give rise to the branches originating in odd $N$ zero modes. This requires a slightly more involved numerical algorithm, which is described in Appendix \[app:method\].
![Branches of black bar deformations on the $({\mathcal{R}}_0,\Omega)$ plane. The right plot is a close-up showing good agreement with the analytic expansions (orange) and also zooms in on the short branches. Different tones of green are being used for different branches for the sake of clarity.\[fig:OmegaR0Bars\]](Plots/OmegaR0Dumbbells "fig:"){width="49.00000%"} ![Branches of black bar deformations on the $({\mathcal{R}}_0,\Omega)$ plane. The right plot is a close-up showing good agreement with the analytic expansions (orange) and also zooms in on the short branches. Different tones of green are being used for different branches for the sake of clarity.\[fig:OmegaR0Bars\]](Plots/ComparisonDumbbells "fig:"){width="49.00000%"}
In figure \[fig:OmegaR0Bars\], the first branches of deformed black bars are shown in the $({\mathcal{R}}_0,\Omega)$ plane. In this case, there is a strong qualitative difference between even and odd $N$. Odd branches extend only in one direction. This is to be expected, since in this case, reversing the sign of linear perturbations is equivalent to the gauge change $\phi \to \phi + \pi$. Surprisingly, for odd $N$ branches, $\Omega$ increases as we move away from the zero modes, and these branches are also very short.
Even $N$ branches result in the bar breaking apart in $N/2$ separated blobs. In $({\mathcal{R}}_0, \Omega)$ plane, they behave in a way that is qualitatively similar to the axisymmetric case, and can therefore be classified in two types. If $N$ is a multiple of 4, ${\mathcal{R}}_0 \to 0$ and the mass density approaches zero at the origin. If $N$ is even but not a multiple of 4, then one of the blobs stays at the origin, with ${\mathcal{R}}_0 \to 2$. The profiles of the first two symmetric bars ($N = 4,6$) are depicted in figure \[fig:dumbbells\].
Similar to the axisymmetric branches, even $N$ branches can be extended far away from the black bar to the arbitrarily small $\Omega$, in which the mass profile approaches to the multiple blobs located in the almost equal interval. Again, we observe these intervals grow very slowly at the same logarithmic rate as that of ring-like blobs in the axisymmetric branches. Therefore, one can expect these branches finally would pinch off to the array of binary black holes.
{width="\textwidth"} {width="\textwidth"}
The angular momentum per unit mass is calculated using eqs. (\[eq:normCond\]) and (\[eq:angularMomentum\])
$$\frac{\mathcal{J}}{\mathcal{M}} = \frac{\int dx\,dy \, p_\phi}{\int dx\,dy \, m}\, ,$$
with
$$m(x, y) = \exp \left( {\mathcal{R}}(y) - \frac{x^2}{2 \ell^2_\perp}\right),$$
$$p_\phi(x, y) =\left[(x^2 + y^2)\,\Omega + \frac{xy}{\ell^2_\perp} + x{\mathcal{R}}'(y)\right]m(x, y)$$
The phase diagram for the deformed bars is shown in figure \[fig:PhaseDiagramDumbbells\].
![The 10 first dumbbell branches, we also plot the branching points of the odd bar perturbations marked by points that only give rise to short ‘spindle’ branches.The Myers-Perry solutions are represented by the thick black curve, and the (non-deformed) black bars by the thick red curve. Different tones of green are being used for different branches for the sake of clarity.\[fig:PhaseDiagramDumbbells\]](Plots/PhaseDiagramDumbbells){width="250pt"}
Effects of adding charge {#sec:effCharge}
========================
Following the approach of [@Andrade:2018rcx] and as already described in section \[sec:largeD\] we can easily construct the (non-extremal) charged solution corresponding to every uncharged solution. According to eq. (\[eq:chargeRescalVel\]) for a given charge parameter ${\mathfrak{q}}=\frac{Q}{M}$ and given $\Omega$, the charged solution has the profile of an uncharged solution with rotation parameter $$\begin{aligned}
\Omega_q=\frac{\Omega}{{\left(}1-2{\mathfrak{q}}^2{\right)}^{1/4}}\,.\end{aligned}$$
The $({{\mathcal J}}/{{\mathcal M}},\Omega)$ phase diagrams for $|Q|>0$ are thus the same diagrams as in the uncharged case with a rescaling of the $\Omega$-axis by the factor ${\left(}1-2{\mathfrak{q}}^2{\right)}^{-1/4}$. Accordingly the bumpy branches will appear at the same ${{\mathcal J}}/{{\mathcal M}}$ but at a lower $\Omega$. As shown in the previous sections lower values of $\Omega$ correspond to more elongated/ further separated blobs, [[*i.e.,*]{}]{}adding charge to the black holes leads to stronger deformations. This intuitively can be understood as charge repulsion deforming the horizon.
Discussion {#sec:discussion}
==========
In this paper we have demonstrated that the hydro-elastic equations [@Emparan:2016sjk] contain a whole new class of ‘rippled’ stationary solutions, besides the already known black branes, their non-uniform deformations [@Emparan:2018bmi] and the non-deformed spinning localized black holes [@Andrade:2018nsz].
We have constructed solutions that branch off from the singly spinning Myers-Perry solution directly or indirectly via the black bar branch, which has been already identified in [@Andrade:2018nsz]. We found both axisymmetric and non-axisymmetric solutions, and only the former ones can remain stationary at finite $D$, since non-axisymmetric solutions will radiate gravitational waves. However, with increasing number of dimension the emission of gravitational waves becomes weaker, which will allow the non-axisymmetric solutions to be long-lived.
The axisymmetric solutions described in this paper, we have identified as [*ring-like*]{} and [*Saturn-like*]{} bumpy black holes, or [*black ripples*]{} in short. They bifurcate from the axisymmetric zero modes of Myers-Perry in the ultra-spinning regime. As in the numerical studies in finite dimensions [@Dias:2014cia; @Emparan:2014pra], we found that all branches extend in two directions: either with a positive or a negative amplitude of the deformation. The direction that increases the angular velocity leads to a very short branch, the other direction extends indefinitely at large $D$. This implies that the former directions lead to singular solutions, as observed in previous numerical constructions [@Dias:2014cia; @Emparan:2014pra].
Multipolar deformations can not be stationary in a fixed number of dimensions, but are indicative of ultraspinning instabilities of the Myers-Perry black hole. In high enough dimension they correspond to long-lived transient objects. We generically call them [*black flowers*]{}, the simplest case among them is the black bar and it has an analytic solution.
The black bar also has an infinite number of co-rotating zero modes, from which deformed branches develop: the [*dumbbells*]{} and the [*spindles*]{}. We classify the deformed bars by the parity of their zero mode as odd and even. Similarly to the ripples, the even branches go out in two directions. In the spin-down direction, the deformation grows a dumbbell-like profile with a distinct number of blobs for each branch, and hence we call them [*dumbbells*]{}. In the opposite direction, we could find only very short branches which we call [*spindles*]{}. Odd branches turned out very short as well. Odd branches and spindles correspond to solutions with increased angular velocity. One might expect that both the spindles and the odd branches end up forming a singularity.
It is very suggestive that the spindle branches correspond to the solutions that develop sharp pointy endings, as observed dynamically in [@Bantilan:2019bvf; @Andrade:2019edf]. These sharp endings of the deformed bar would be possibly affected by the Gregory-Laflamme instability, presenting a large number of zero modes close to the end of the short branch. The sharpened tips could, in principle, pinch off producing detached small black holes.
This process of a black hole developing long arms that end up pinching off has indeed been observed in [@Bantilan:2019bvf; @Andrade:2019edf], not only for the spindles but also for higher multipole deformations. We find it likely that these dynamical solutions would correspond to the short branches described above, [[*i.e.,*]{}]{}those resulting from exciting the zero modes in the direction with increasing $\Omega$. This would apply both to the spindle solutions and to multipolar deformations leading to multiple arms. This conjecture is supported by the fact that short branches go in the direction of decreasing ${{\mathcal J}}/{{\mathcal M}}$, which should be favored in finite $D$ simulations since gravitational radiation tends to decrease the angular momentum to mass ratio of the evolving object.
The method used to identify axisymmetric solutions should be exhaustive, and thus we do not expect the ripple branches to have their own secondary axisymmetric zero modes. We expect, on the other hand, that the axisymmetric solutions will become unstable to multipolar deformations. An indication of a ring-like ripple breaking apart into four black holes via an $m=4$ deformation was already found at large $D$ in [@Andrade:2019edf]. Interestingly, black rings share the same kind of instabilities and subsequent pinch-offs [@Arcioni:2004ww; @Elvang:2006dd; @Santos:2015iua; @Figueras:2015hkb]. Such instabilities would begin at zero modes along the branches of ripples. This fact leaves open the possibility of the ‘long’ multipolar branches actually merging with the ripple branches at these zero modes. No conclusive results have been obtained about this intriguing possibility in this paper.
We have found no evidence that the long multipolar branches have bifurcations. This possibility could be analyzed in future work, possibly with an improved numerical setup. The dumbbell branches end as an array of separated black holes and thus seem unlikely to have further zero modes.
The nature of the boundary conditions that are imposed in the blob formalism, together with the nonlinearity of the large $D$ effective equations, leads to a remarkably challenging numerical problem. Ordinary relaxation and spectral techniques have not been shown to give reliable results so far. This fact is probably due to the requirement of imposing boundary conditions at spatial infinity, together with the equations of motion being numerically bad-behaved as $r \to \infty$. Additionally, the equations are nonlinear, which rules out direct eigenvalue-finding standard algorithms. Fortunately, the shooting approach used in this paper, which consists in identifying sharp peaks in the radius where the numerical solution becomes singular, seems to be enough to find the right solutions. It is remarkable that this technique works even though the numerical method is usually able to integrate only to a finite value of $r$. Axisymmetric solutions are easily found this way. For the case of multipolar deformations, one encounters a multidimensional shooting problem with a scalar-valued output function ($r_s$), which becomes increasingly difficult as one increases the number of overtones. For this reason, an alternative method, possibly based on relaxation techniques, would be desirable in the future.
In the formalism employed here, the effect of the charge is simply incorporated in the effective angular velocity $\Omega_q = \Omega/(1-2{\mathfrak{q}}^2)^{1/4}$ as in [@Andrade:2018rcx]. Therefore, with a given value of charge and $\Omega$, the corresponding charged solution is immediately obtained from the uncharged one. Due to the factor ${\left(}1-2{\mathfrak{q}}^2{\right)}^{-1/4}$, the charged deformed branches will appear for the same ${{\mathcal J}}/{{\mathcal M}}$ but for a lower $\Omega$, which corresponds to more elongated/further separated blobs. This can be interpreted as the effect of the charge repulsion. Since all the analysis is written in terms of $\Omega_q$, one can take the extremal limit ${\mathfrak{q}}^2 \to 1/2$ of all branches, keeping $\Omega_q$ finite, resulting in a smooth limit, that leads to rather strange deformed ‘extremal’ branches, both with and without rotation. The proper large $D$ limit of extremal horizons is however yet unclear, and a more careful analysis seems appropriate.
#### Fate of far extended branches
All ‘long’ branches (corresponding to bulging deformations) extend far away from the original bifurcating points in the phase space, where they develop broad thin regions. Currently, very little is known about how to interpret these nearly zero thickness regions in the large $D$ effective theory. In the case of spherical black holes the thickness falls off towards infinity as a Gaussian profile, which might be interpreted as the round tip of the black hole. Therefore, if the deformation develops a thin neck between blobs, and its size grows infinitely large, one can expect such deformation to end up as a pinch off of the horizon at finite $D$. This would correspond to a topology-changing transition.
We found that the ripple branches develop such long thin necks connecting Gaussian-shaped ring blobs (with a central blob in the case of Saturns) at their final stages of deformation. Particularly, we observed that the separation process involves two distinct length scales. From the numerical solutions, we could easily estimate that the radii of ring blobs grow like $\Omega^{-1}$ as $\Omega \to 0$. The same behavior has been derived in the blackfold approach [@Emparan:2007wm; @Emparan:2010sx], which might imply that the blackfold approximation becomes already accurate in the pinch off phase, due to the localization of gravity at large $D$. Another scaling is that of the intervals between ring blobs, which are estimated as $\sim\sqrt{|\log\Omega|}$. Due to the hierarchy in these two scales, we expect the first pinch off to occur always on the axis, indicating a first topology change to a bumpy black ring/Saturn, before transitioning to the multi-rings/Saturns, as observed in the $(+)_3$-branch of $D=6$ bumpy black holes [@Emparan:2014pra].
Dumbbell branches also extend far away from the black bar to arbitrarily small $\Omega$, where the mass profile approaches that of multiple evenly spaced blobs. As opposed to the ripples, dumbbells show only a single scaling, which has the same logarithmic growth as the intervals between the ring blobs in the case of ripples. Therefore, one can expect that these branches would finally pinch off to multiple black holes[^10].
#### Finite $D$ effects
The blob coordinate is supposed to be identified as the small patch of the $\sqrt{D}$-amplified entire coordinate.[^11] Therefore, the blob approximation will break down if the length of the thin neck reaches $\sim\sqrt{D}$, when the $1/D$ corrections are included. This breakdown will give some information on the transition in phase space. For example, the pinch off from the ripples to black rings or Saturns will take place at $\Omega \sim 1/\sqrt{D}$. Actually, black rings are already constructed by using the large D effective theory approach in the same scaling [@Tanabe:2015hda; @Tanabe:2016opw]. This implies that one can use the effective theory result as the global setup to solve the local topology-change. For other logarithmic scalings $\sim \sqrt{|\log \Omega|}$, the break down will occur at much smaller spin $\Omega \sim e^{-D}$. In the black string analysis, a similar type of breakdown is already seen after including $1/D$ corrections [@Emparan:2018bmi]. The black hole entropy is another important quantity to evaluate the stability of the solutions. Since the mass and entropy become degenerate at $D\to\infty$, we would need to know the next-to-leading order terms in $1/D$ expansion to calculate the entropy difference for a given mass.
#### Blob-Blob interactions
For the ripples and dumbbells, we observed a universal scaling of the blob distance as $\sqrt{|\log \Omega|}$ at $\Omega \to 0$, implying an effective interaction between the blobs (or ring-like blobs). This indicates the possibility to reconstruct the large $D$ effective theory as a particle-like (or soliton-like) effective description of blobs weakly interacting via very thin necks. This possibility will be pursued elsewhere.
The origin of this logarithmic dependence, though very naively, might be understood as a force balance between the centrifugal force and the attraction between the blobs at large $D$. Assuming a black hole of radius $r_{\rm H}$ and an orbiting particle, the gravitational force is approximated as $(r_{\rm H}/r)^{D}$ and the centrifugal force as $\Omega^2 r$. The equilibrium is accomplished by $r/r_{\rm H}\sim 1-2D^{-1}\log \Omega$. Therefore, the particle orbit exists very close to the horizon $\sim|\log\Omega|/D$. This introduces the $|\log \Omega|$ scaling in the near horizon region. Curiously, if we assume two adjacent black holes with the same mass, the equilibrium condition would be modified to $r/r_H \sim 2-2D^{-1} \log (e^{D/2} \Omega)$ with $e^{D/2}\Omega = {{\mathcal O}\left(1\right)}$ or $|\log\Omega| \sim D$. This coincides with the value at which the neck length between blobs reaches $\sqrt{|\log \Omega|}\sim \sqrt{D}$ and the blob approximation breaks down.
#### Towards the topology change
The topology-changing transition at large $D$ is described by the conifold metric which solves the Ricci flow equation [@Emparan:2019obu]. Especially, the black string/black hole transition is completely solved by the King-Rosenau (KR) solution for the $2D$ Ricci flow. Some of the topology-changing transitions (Saturn-like ripples, dumbbells) can be reduced to the $2D$ Ricci flow problem in the co-rotating frame, since the transition occurs in a very narrow region. Hence, they should also be solved by the KR solution, due to the rigidity in $2D$ compact ancient flow [@daskalopoulos2009classification][^12]. For the transition between ring-like ripples and black rings, we need a better understanding of the $3D$ Ricci flow.
Here we should note that, in the case of the black string/black hole transition, one just has to give the global configuration (such as the black hole (blob) radius and the compactification scale) as boundary conditions for the conifold metric, without considering the force balance condition. Now, for example, if we consider the transition between a dumbbell and binary black hole, we also have the rotation $\Omega$, which will not appear in the large $D$ conifold analysis after switching to the co-rotating frame. To relate $\Omega$ with the mass and separation, one needs to find the proper force balance condition at large $D$, as roughly estimated in the previous paragraph.
In the current formalism, we could only follow the $(-)$-ripple branches for a very short range. These $(-)$-branches are shown to develop a single-sided conical horizon on the equator when they approach the end of their branch [@Emparan:2014pra]. Therefore, it should also be possible to study the ending phase of $(-)$-branches using the large $D$ conifold metric and Ricci flow methods. Different from the usual pinch off problem, one may have to find the non-compact Ricci flow solution, in which only one side is the horizon.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Roberto Emparan for comments on an earlier draft of this paper. Work supported by ERC Advanced Grant GravBHs-692951 and MEC grant FPA2016-76005-C2-2-P. RL is supported by the Spanish Ministerio de Ciencia, Innovación y Universidades Grant FPU15/01414. RS is supported by JSPS KAKENHI Grant Number JP18K13541 and partly by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).
Numerical methods {#app:method}
=================
Axisymmetric sector
-------------------
Stationary axisymmetric black holes are regular solutions of eq. (\[eqn:master\]) that extend from 0 to $r \to \infty$. Due to singular point at $r=\infty$ from the rotation term it is particularly difficult to use of spectral and relaxation methods. For this reason, the approach used in this paper is essentially a shooting method. By regularity at the origin the ODE can be generally integrated radially outwards with the initial conditions ${\mathcal{R}}(0) = {\mathcal{R}}_0$ and ${\mathcal{R}}'(0) = 0$. The numerical solution will generally become singular at some finite $r = r_s$. In figure \[fig:singularRadius\], the values of $r_s$ are shown as a function of the initial condition parameter ${\mathcal{R}}_0$, interestingly the appearance of singularities is (semi-) continuous in the space of initial conditions which makes it possible to look for singularities/ peaks where the solution extends to infinity. These peaks correspond to (approximate) locations of the allowed solutions.
![Values of $r_s$ (radius where the solution becomes singular) for $\Omega = 0.3$. The solutions that have to be free of such singularities and extend to infinity appear as sharp peaks, which we marked here with red dots. \[fig:singularRadius\]](Plots/singularRadius){width="250pt"}
For this purpose, the $({\mathcal{R}}_0, \Omega)$ plane is not a very suitable representation. This is so because the branches of solutions become very closely packed at low $\Omega$, while the ring-like branches reach very large negative values. A numerical algorithm intended to find all these peaks with a high precision needs therefore an extremely high dynamic range of detection in ${\mathcal{R}}_0$, so it can both find widely spread peaks and resolve extremely close packed ones. This is solved by introducing the coordinates $(\alpha,\beta)$ as
$$\Omega = \frac{e^\beta}{2} \text{sech} \,\alpha \, , \qquad {\mathcal{R}}_0 = 2-e^{\alpha + \beta} \text{sech} \,\alpha\,.$$
These coordinates both range from $-\infty$ to $\infty$, and cover the $(-\infty, 2)\times(0, \infty)$ region in $({\mathcal{R}}_0, \Omega)$ plane. They are analytically invertible as
$$\alpha = \log \left( \frac{2-{\mathcal{R}}_0}{2\Omega} \right), \qquad \beta = \log \left( \frac{(2-{\mathcal{R}}_0)^2 + 4\Omega^2}{2(2-{\mathcal{R}}_0)} \right)$$
In these new coordinates, the Myers-Perry black holes lay on the vertical axis ($\beta = 0$), with the Schwarzschild black hole corresponding to $\beta = 0, \; \alpha \to -\infty$ (see figure \[fig:alphabetaPlane\]). The ripple solutions become now much more suitable to be found numerically. In particular, the ring-like branches can be parametrized by $\beta$, and the Saturn-like by a polar angle $\theta$ such that $\alpha = \rho \cos \theta$ and $\beta = - \rho \sin \theta$.
![Location of the first 10 branches of black ripples (5 ring-like and 5 Saturn-like) in the $(\alpha, \beta)$ plane.\[fig:alphabetaPlane\]](Plots/AlphaBeta){width="200pt"}
When a branch ends, as for the negative amplitude modes, the peak that represents it becomes a local maximum, with no divergence whatsoever. This requires us to define a criterion for a local maximum to be considered a proper peak, or a [*vanishing peak*]{}. The criterion that has been taken for a peak to be valid is
$$\max \left\{ r_s(\alpha, \beta) - r_s(\alpha + \delta\alpha, \beta), r_s(\alpha, \beta) - r_s(\alpha - \delta\alpha, \beta)\right\} > \Delta\, ,$$
where $\delta\alpha = 0.01$ and $\Delta = 3$. When extracting the angular momenta of the solutions, it is also important to take into account that numerical error may result in extra (unphysical) oscillations of the ${\mathcal{R}}(r)$ profiles. These oscillations appear as additional bumps, or [*fake rings*]{}. These have to be properly removed before the angular momentum integration, since they could add an erroneous contribution to the integration result.
Black bar deformations
----------------------
Deformations with even values of $N$ are found in a way which is completely analogous to the axisymmetric case. In this case it is convenient to reparameterize the $({\mathcal{R}}_0, \Omega)$ by the coordinates $(\gamma, \delta)$,
$$\gamma = -\log (2\Omega)\, , \qquad \delta = -\log (2-{\mathcal{R}}_0)$$
![Location of the first 10 branches of deformed bars in the $(\gamma, \delta)$ plane.\[fig:gammadeltaPlane\]](Plots/GammaDelta){width="200pt"}
Odd deformations of bars are described by solutions of eq. (\[eqn:masterDumbbells\]) that have a nonzero value of $\bar {\mathcal{R}}_0 = {\mathcal{R}}'(0)$. This increases the complexity of the problem, since it now requires to tune both ${\mathcal{R}}_0$ and $\bar {\mathcal{R}}_0$ in order to get a solution that extends to infinity both for the negative and positive sides of the $y$ axis. This complication can be partially circumvented by noticing that, for the deformed black bars, the change $y \to - y$ is equivalent to $\bar {\mathcal{R}}_0 \to - \bar {\mathcal{R}}_0$. This means that, if $(\Omega, {\mathcal{R}}_0, \bar {\mathcal{R}}_0)$ gives an allowed solution, then so does $(\Omega, {\mathcal{R}}_0, - \bar {\mathcal{R}}_0)$. This fact allows the right values of ${\mathcal{R}}_0$ to be found by requiring the peaks in $r_s(\Omega, {\mathcal{R}}_0, \bar {\mathcal{R}}_0)$ to be located at opposite values of $\bar {\mathcal{R}}_0$. This is done by the secant root-finding method in a few iterations. Again, [*vanishing peaks*]{} and [*fake blobs*]{} are discarded in a similar way as in the axisymmetric case.
Multipole deformations {#sec:AppNumMet}
----------------------
By using the ansatz (\[eq:numAnsatzFourMode\]) truncated at some Fourier mode $\cos(n_{\text{max}} m \phi)$, we obtain a set of $n_{\text{max}} + 1$ coupled equations for the functions ${\mathcal{R}}^{(n m)}(r)$. These equations, by imposing the regularity condition ${\mathcal{R}}^{(n m)'}(0) = 0 \; \forall n$, can be solved by specifying the values of the radial functions at the origin. The problem reduces then to finding peaks in the singular radius $r_s(\Omega, {\mathcal{R}}_{0},{\mathcal{R}}_{m},{\mathcal{R}}_{2m}, \dots, {\mathcal{R}}_{n_{\text{max}} m})$.
Identifying peaks on a function with more than one variable is in general not an easy task, especially if there is no straightforward way of reducing the problem to one variable (as in the case of odd deformations of the black bar). For this reason, in this article we restrict ourselves to the fundamental Fourier mode, [[*i.e.,*]{}]{}we maximize $r_s(\Omega, {\mathcal{R}}_{0},{\mathcal{R}}_{m})$. We use the [*Mathematica*]{} function [**NMaximize**]{} to identify the peak by incrementing $\Omega$ in small steps, and constraining the search in a small region around the result of the previous step.
Even with this method, the values of the ${\mathcal{R}}_{0},{\mathcal{R}}_{m}$ still are affected by small fluctuations (which are likely due to numerical error) around the branch. We correct this by subsampling the data points.
Matching to the entire hemisphere
=================================
In general, blob solutions are thought to be identified as a polar cap of the compact black holes, in which the polar angle is stretched by $\sqrt{D}$ to match with the radial coordinate in the blob [@Andrade:2018nsz], $$r = \sqrt{D} \theta. \label{eq:blob-hole-rtheta}$$ In ref. [@Andrade:2018nsz], the linear order deformation of the blob and the perturbation in the Myers-Perry [@Suzuki:2015iha] confirmed to be matched for $1 \ll r \ll \sqrt{D}$, $$\delta {\mathcal{R}}\propto L_N\left( \frac{r^2}{2(1+a^2)} \right)\sim r^{2N} \sim \sin^{2N}\theta.\label{eq:blob-hole-match-lin}$$ Here we show that this match is also consistent beyond the linear level, despite the increase in the degree of the polynomials in the higher perturbation order. The degree of each perturbation solution can be estimated from the recurrence formula (\[eq:recurrence-C-p\]) as $${\rm deg} [f_{k}(z)] = \underset{i} {\rm max}\left({\rm deg} [f_{i}(z)] + {\rm deg} [f_{k-1-i}(z)]\right) -1,$$ where the last $-1$ comes from $I+J-K$ factor in eq. (\[eq:recurrence-C-p\]). Starting from $f_0(z) = L_N(z)$, the induction easily follows $${\rm deg} [f_{k}(z)] = (k+1) N -k.$$ Since the coordinate match (\[eq:blob-hole-rtheta\]) leads to $$z \sim r^2 \sim D \sin^2 \theta,$$ the perturbation at each order gives the match at $1 \ll z \ll D$, $${\varepsilon}^{k+1} f_k(z) \sim {\varepsilon}^{k+1} z^{(k+1)N-k}\sim \bar{{\varepsilon}}\, {}^{k+1} D^{-2k} (\sin\theta)^{2(k+1)N-2k}.$$ where we rescaled the perturbation parameter by $\bar{{\varepsilon}} = D^{2N}{\varepsilon}$, so that the linear order remains finite at $D\to \infty$. Therefore, the linear order match (\[eq:blob-hole-match-lin\]) turns out to be correct even up to the nonlinear order, and all the nonlinear perturbation will be matched with the subleading correction in $1/D$, $$\delta {\mathcal{R}}\sim \bar{{\varepsilon}} \sin^{2N}\theta+{{\mathcal O}\left(D^{-1}\right)}.$$
Useful properties of the orthogonal polynomials
===============================================
Here, we show some useful properties of the Laguerre and Hermite polynomials used in the paper.
Product of the orthogonal polynomials
-------------------------------------
#### Product of Laguerre polynomials
It is known that the product of the Laguerre polynomials of the same second parameter can be written by the linear combination of the Laguerre polynomials of the same type [@Watson1938], $$\begin{aligned}
& L^{(n)}_I(x)L^{(n)}_J(x) = \sum_{K=|I-J|}^{I+J} \overset{(n)}{{{\mathcal X}}}{}^K_{I,J} L_K^{(n)}(x)\end{aligned}$$ where the coefficients are given by $$\overset{(n)}{{{\mathcal X}}}{}^K_{I,J} = \frac{(-2)^{I+J-K} K!}{(K-I)!(K-J)!(I+J-K)!} { { \, {}_{3} F {}_{2} } }\left(\begin{array}{c}n+K+1,{\frac{1}{2}}(K-I-J),{\frac{1}{2}}(K-I-J+1)\\
K-I+1,K-J+1\end{array};1\right).$$ For $n=0$, the coefficient becomes symmetric in $(I,J,K)$, in which case we just write ${{\mathcal X}}^K_{I,J}$.
#### Product of Hermite polynomials
The decomposition of the product of the Hermite polynomials is also known $$H_I(z) H_J(z) = \sum_{K=|I-J|}^{I+J} {\cal Q}^{K}_{I,J} H_{K}(z),$$ where the coefficients have the non-zero value only if $I+J+K$ is even, $${\cal Q}^K_{I,J} := \frac{2^\frac{I+J-K}{2}I!J! }{\left(\frac{I+K-J}{2}\right)!\left(\frac{J+K-I}{2}\right)!\left(\frac{I+J-K}{2}\right)!}. \label{eq:hermite-product-cf}$$ It is worth noting that the coefficients in the above two formula become non-zero only if $(I,J,K)$ satisfy the trigonometric inequality: any of the three cannot exceed the sum of the rest two.
### Relation to Franel number {#sec:franel}
Interestingly, the renormalization coefficient $\mu$ in eq. (\[eq:axisym-mu1\]), is related to the so called Franel number, which is known in combinatorics and number theory, $${\rm Fr}_N := \sum_{i=0}^N \binom{N}{i} ^3 = { { \, {}_{3} F {}_{2} } }\left[\begin{array}{c}-N,-N,-N\\1,1\end{array};-1\right].$$ Due to the identity, $${ { \, {}_{3} F {}_{2} } }\left[\begin{array}{c}-N,-N,-N\\1,1\end{array};-1\right] = 2^N { { \, {}_{3} F {}_{2} } }\left[\begin{array}{c}N+1,-\frac{N}{2},-\frac{N-1}{2}\\
1,1\end{array};1\right],$$ $\mu$ can be rewritten as $$\mu = - {\frac{1}{4}} {{\mathcal X}}^N_{N,N} = - {\frac{1}{4}}(-1)^N {\rm Fr}_N.$$ Using the large $N$ approximation for the binomial coefficients, we can show the rapid growth in this number with respect to $N$, $${\rm Fr}_N = \sum_{i=0}^N \binom{N}{i}^3 \sim \frac{2^{3N}}{\sqrt{N}} \int^\infty_{-\infty}
e^{-6Nx^2}dx \sim \frac{2^{3N}}{N}.$$
Integral of triple associated Laguerre polynomials {#sec:associated-laguerre-lauricella}
--------------------------------------------------
As found in [@Erdelyi1936; @Lee2000], the triple integrals are given by $$\begin{aligned}
& \int_0^\infty z^\frac{i+j+k}{2} e^{-z} L^{(i)}_I(z) L^{(J)}_J(z) L^{(k)}_K(z) dz
= \frac{(i+I)!}{i!I!}\frac{(j+J)!}{j!J!}\frac{(k+K)!}{k!K!} \left(\frac{i+j+k}{2}\right)! {\nonumber\\ }& \qquad \times \quad F_A^{(3)} \left(\frac{i+j+k}{2}+1; -I,-J,-K;i+1,j+1,k+1;1,1,1\right)\end{aligned}$$ where $F_A^{(3)}$ is one of the Lauricella’s generalized hypergeometric functions defined by $$\begin{aligned}
& F^{(n)}_A (a; b_1,\dots b_n; c_1,\dots,c_n;x_1,\dots,x_n) {\nonumber\\ }& = \sum_{m_1=0}^\infty \cdots \sum_{m_n=0}^\infty \frac{(a)_{m_1+\cdots+m_n} (b_1)_{m_1}
\cdots (b_n)_{m_n}}{(c_1)_{m_1} \cdots (c_n)_{m_n}m_1!\cdots m_n!}x_1^{m_1}\cdots x_n^{m_n}.\end{aligned}$$ If $b_i$ is a negative integer, the summation with respect to $m_i$ stops at $|b_i|$.
Derivative of Laguerre functions with respect to the parameter
==============================================================
In this section, we study the infinitesimal parameter shift in the generalized Laguerre functions from the Laguerre polynomials.
Confluent hypergeometric equation
---------------------------------
We start from reviewing the confluent hypergeometric equation, $$z f''(z) + (b-z) f'(z) - a f(z) = 0.\label{eq:confluentHG}$$ A solution is given by Kummer’s confluent hypergeometric series $${ { \, {}_{1} F {}_{1} } }(a,b,z) = \sum_{k=0}^\infty \frac{(a)_k}{(b)_k} \frac{z^k}{k!} \label{eq:hg-sol-1}$$ where $(a)_k := \Gamma(a+k)/\Gamma(a)$ is the Pochhammer symbol. If $b$ is not positive integer, the other solution is given by $$U(a,b,z) = \frac{\pi}{\sin \pi b} \left(\frac{{ { \, {}_{1} F {}_{1} } }(a,b,z)}{\Gamma(1+a-b)\Gamma(b)}-z^{1-b}\frac{{ { \, {}_{1} F {}_{1} } }(1+a-b,2-b,z)}{\Gamma(a)\Gamma(2-b)}\right).\label{eq:hg-sol-2}$$ If $b$ is a positive integer, say $b=n+1\ (n = 0,1,2,\dots)$, the other solution is given by[^13] $$\begin{aligned}
&U(a,n+1,z) = (-1)^n\frac{n!(n-1)!\Gamma(a-n)}{\Gamma(a)} z^{-n} \sum_{k=0}^{n-1} \frac{n!(a-n)_k}{(1-n)_k k!}z^k {\nonumber\\ }& \qquad - \sum_{k=0}^\infty \frac{(a)_k z^k}{(n+1)_k k! } [\psi(a+k)-\psi(1+k)-\psi(1+n+k)]{\nonumber\\ }& \qquad - { { \, {}_{1} F {}_{1} } }(a,n+1,z)(\ln z+\pi \cot(\pi a))\end{aligned}$$ where $\psi(z):=\Gamma'(z)/\Gamma(z)$ is the digamma function. For the negative value of $a$, it is convenient to rewrite this to $$\begin{aligned}
&U(a,n+1,z) = \sum_{k=1}^{n} \frac{n!(k-1)!\Gamma(1-a)}{\Gamma(k+1-a) (n-k)!}z^{-k} - { { \, {}_{1} F {}_{1} } }(a,n+1,z)\ln z{\nonumber\\ }& - \sum_{k=0}^\infty \frac{(a)_k z^k}{(n+1)_k k! } [\psi(1-a-k)-\psi(1+k)-\psi(1+n+k)] \label{eq:hg-sol-2-log}\end{aligned}$$ where we used the reflection formula for the gamma functions and digamma functions, $$\begin{aligned}
\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)},\quad \psi(z) - \psi(1-z) = -\pi \cot(\pi z).\end{aligned}$$
Laguerre functions
------------------
If $b$ is a positive integer, $b=n+1 \ (n=0,1,2,\dots)$, eq. (\[eq:confluentHG\]) is called the Laguerre equation, and the first solution (\[eq:hg-sol-1\]) is called the (generalized) Laguerre functions, $$\Phi(\alpha,n,z) := { { \, {}_{1} F {}_{1} } }(-\alpha,n+1,z),$$ or the following convention is more commonly used, $$L_{\alpha}^{(n)}(z) := \frac {\Gamma(\alpha+n+1)}{\Gamma(n+1)\Gamma(\alpha+1)} { { \, {}_{1} F {}_{1} } }(-\alpha,n+1,z).$$ These definitions are equal for $n=0$. Throughout this section, we will use the former convention for the convenience. In case of $\alpha = 0,1,2,\dots$, these functions reduce to the Laguerre polynomials.
#### Laguerre functions of the second kind
Recently, for $\alpha=N\ (N=0,1,2,\dots)$, the second solution is found to be written in the closed form [@ParkeMaximon2015; @ParkeMaximon2016], $$\Psi(N,n,z) = \frac{n!}{(N+n)!}P(N,n,z) e^z z^{-n} - \Phi(N,n,z) {\rm Ei}(z) \label{eq:laguerre-2-PM}$$ where ${\rm Ei}(z)$ is the exponential integral function. The function $P(N,n,z)$ is given by $$P(N,n,z) = \sum_{m=0}^{n-1} \left[\frac{(N+m)!(n-m-1)!}{m!}\right]z^m + z^n \sum_{m=0}^{N-1} c(N,n,m) z^{m}$$ where $$c(N,n,m) = \frac{(-1)^{m+1}N!(N+n)!}{(N-m-1)!(m+n+1)!(m+1)!} { { \, {}_{3} F {}_{2} } }\left(\begin{array}{c}1,1,-N+m+1\\2+m,2+m+n\end{array};1\right). \label{eq:laguerre-2-c}$$ In ref. [@ParkeMaximon2016], eq. (\[eq:laguerre-2-PM\]) is shown to coincide with the expression in eq. (\[eq:hg-sol-2-log\]), $$\Psi(N,n,z) = U(-N,n+1,z).$$ Using the asymptotic expansion of ${\rm Ei}(z)$ , one can obtain the asymptotic behavior at the large $z$ as $$\begin{aligned}
\Psi(N,n,z) \simeq (-1)^{N+1} N! n! z^{-N-n-1} e^z \left(1+{{\mathcal O}\left(z^{-1}\right)}\right) .\label{eq:Psi-as}\end{aligned}$$ Close to $z=0$, we obtain $$\Psi(N,n,z) \simeq \sum_{k=1}^n \frac{N!n!(k-1)!}{(N+k)!(n-k)!}z^{-k} - \log z -\gamma + H_{n}-H_N+{{\mathcal O}\left(z\right)}. \label{eq:Psi-zero}$$
Derivative with respect to the parameter
----------------------------------------
Here, we evaluate $\alpha$-derivative of $\Phi(\alpha,n,z)$ on a non-negative integer. It turns out, $\partial_\alpha \Phi(N,n,z)$ can be expressed in terms of $\Phi(N,n,z)$, $\Psi(N,n,z)$, $\log z$ and polynomials, $$\begin{aligned}
&\partial_\alpha \Phi(N,n,z) = \Psi(N,n,z)+ (\gamma-H_N+\log z)\Phi(N,n,z)+\sum_{k=0}^{N-1} \frac{2}{N-k} \Phi(k,n,z){\nonumber\\ }&\hspace{2.5cm}-\sum_{k=1}^n \frac{N!n!(k-1)!}{(k+N)!(n-k)!} z^{-k}
- \sum_{k=1}^n {\frac{1}{k}} { { \, {}_{2} F {}_{2} } }\left[\begin{array}{c}-N,k\\n+1,k+1\end{array};z\right].
\label{eq:dPhi}\end{aligned}$$
#### Proof
The above formula can be obtained through the expression in eq .(\[eq:hg-sol-2-log\]). Using the reflection formula for the gamma function, we have $$\begin{aligned}
\partial_\alpha \Phi(\alpha,n,z) = -\sum_{k=0}^\infty (\psi(\alpha+1-k)-\psi(\alpha+1)) \frac{(-\alpha)_k}{(n+1)_k k!}z^k.\end{aligned}$$ Using eq. (\[eq:hg-sol-2-log\]), one can rewrite the above equation to $$\begin{aligned}
&\partial_\alpha \Phi(\alpha,n,z) = U(-\alpha,n+1,z)+ (\psi(\alpha+1)+2\gamma+\log z)\Phi(\alpha,n,z){\nonumber\\ }&-\sum_{k=1}^n \frac{n!(k-1)!\Gamma(1+\alpha)}{\Gamma(k+1+\alpha) (n-k)!} z^{-k}-\sum_{k=0}^\infty (H_{k}+H_{k+n}) \frac{(-\alpha)_k}{(n+1)_k k!}z^k,\end{aligned}$$ where $H_n$ is the harmonic number and we used $\psi(n) = H_{n-1}-\gamma$. By setting $\alpha=N,\ (N=0,1,2,\dots)$, the last sum in the second line reduces to the $N$-th order polynomial, and then, we have the following expression, $$\begin{aligned}
&\partial_\alpha \Phi(N,n,z) = \Psi(N,n,z)+ (H_N+\gamma+\log z)\Phi(N,n,z){\nonumber\\ }&-\sum_{k=1}^n \frac{N!n!(k-1)!}{(k+N)!(n-k)!} z^{-k}-\sum_{k=0}^N (H_{k}+H_{k+n}) \frac{(-N)_k}{(n+1)_k k!}z^k.
\label{eq:dPhi-0}\end{aligned}$$ The last summation can be simplified as $$\begin{aligned}
& \sum_{k=0}^N (H_{k}+H_{k+n}) \frac{(-N)_k}{(n+1)_k k!}z^k =
2\sum_{k=1}^N H_{k} \frac{(-N)_k}{(n+1)_k k!}z^k + \sum_{k=0}^N \sum_{\ell=1}^n {\frac{1}{k+\ell}} \frac{(-N)_k}{(n+1)_k k!}z^k{\nonumber\\ }& =2 H_N \Phi(N,n,z) -\sum_{k=0}^{N-1} \frac{2}{N-k} \Phi(k,n,z) + \sum_{k=1}^n {\frac{1}{k}} { { \, {}_{2} F {}_{2} } }\left[\begin{array}{c}-N,k\\n+1,k+1\end{array};z\right].\end{aligned}$$ This leads to eq. (\[eq:dPhi\]).
[^1]: The flower branches are hard to to construct far away from their branching points, so figure \[fig:PhaseDiagram\] shows them only partially.
[^2]: For brevity of presentation we restrict to the case of zero charge for now and drop the subscript $q$. We will discuss the effects of non-zero charge in section \[sec:effCharge\].
[^3]: Saturn type solutions become harder to construct numerically, since the different Saturn-type solutions pile up in initial condition space as can be seen in figure \[fig:OmegaR0\], but we see no evidence that the corresponding poles in $r_s$ vanish.
[^4]: For odd $m$, the neighboring modes would have half integer parameters, so resonant behavior only can appear starting at third order.
[^5]: This limit is the same in the case of odd $m$, taking into account that only odd overtone modes are involved.
[^6]: If $N+(m-k)/2$ is not a non-negative integer, ${\cal T}^{(k)}_p$ becomes trivially zero.
[^7]: An English reference is found, for example, in [@Lee2000].
[^8]: Which will serve as initial conditions in the numerical setup (\[eq:numAnsatzFourMode\]).
[^9]: Changing the sign of ${\varepsilon}$ in ${\mathcal{R}}_m$ is equivalent to the constant rotation $\phi \to \phi+\pi/m$, and hence does not lead to another branch.
[^10]: Or one might say ’rotating black hole array’.
[^11]: This is only an estimate from the Myers-Perry solution, in which the exact coordinate match is known.
[^12]: A solution of the Ricci flow is called [*ancient*]{}, if it can be extended to the infinite past of the flow time (corresponding to the asymptotic infinity in the large $D$ conifold metric).
[^13]: The overall factor and the term proportional to the first solution ${ { \, {}_{1} F {}_{1} } }(a,n+1,z)$ are adjusted to give the valid formula for the non-positive integer value of $a$.
|
---
abstract: 'In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation. Applications are given to ergodic control of stochastic partial differential equations.'
author:
- |
Marco Fuhrman\
Dipartimento di Matematica, Politecnico di Milano\
piazza Leonardo da Vinci 32, 20133 Milano, Italy\
e-mail: marco.fuhrman@polimi.it\
\
Ying Hu\
IRMAR, Université Rennes 1\
Campus de Beaulieu, 35042 RENNES Cedex, France\
e-mail: ying.hu@univ-rennes1.fr\
\
Gianmario Tessitore\
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca\
Via Cozzi 53, 20135, Milano Italy\
e-mail: gianmario.tessitore@unimib.it
title: Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces
---
Introduction
============
In this paper we study the following type of (markovian) backward stochastic differential equations with infinite horizon (that we shall call *ergodic* BSDEs or EBSDEs for short): $$\label{EBSDE*}
Y^x_t=Y^x_T +\int_t^T\left[\psi(X^x_\sigma,Z^x_\sigma)-
\lambda\right]d\sigma-\int_T^T Z^x_\sigma dW_\sigma, \quad 0\le
t\le T <\infty.$$ In equation (\[EBSDE\*\]) $X^x$ is the solution of a forward stochastic differential equation with values in a Banach space $E$ starting at $x$ and $(W_t)_{t\geq 0}$ is a cylindrical Wiener process in a Hilbert space $\Xi$.
Our aim is to find a triple $(Y,Z,\lambda)$, where $Y,Z$ are adapted processes taking values in $\mathbb{R}$ and $\Xi^*$ respectively and $\lambda$ is a real number. $\psi:E\times \Xi^*\to \R$ is a given function. We stress the fact that $\lambda$ is part of the unknowns of equation (\[EBSDE\*\]) and this is the reason why the above is a new class of BSDEs.
$ $
It is by now well known that BSDEs provide an efficient alternative tool to study optimal control problems, see, e.g. [@peng93], [@ElKaMaz] or, in an infinite dimensional framework, , [@masiero]. But up to our best knowledge, there exists no work in which BSDE techniques are applied to optimal control problems with *ergodic* cost functionals that is functionals depending only on the asymptotic behavior of the state (see e.g. the cost defined in formula (\[ergodic-cost\*\]) below).
$ $
The purpose of the present paper is to show that backward stochastic differential equations, in particular the class of EBSDEs mentioned above, are a very useful tool in the treatment of ergodic control problems as well, especially in an infinite dimensional framework.
$ $
There is a fairly large amount of literature dealing by analytic techniques with optimal ergodic control problems for finite dimensional stochastic state equations. We just mention the basic papers by Bensoussan and Frehse [@BeFr] and by Arisawa and Lions [@ArLi] where the problem is treated through the study of the corresponding Hamilton-Jacobi-Bellman (HJB) equation (solutions are understood in a classical sense and in a viscosity sense, respectively).
Concerning the infinite dimensional case it is known that both classical and viscosity notions of solutions are not so suitable concepts. Maslowski and Goldys in [@GoMa] employ a mild formulation of the Hamilton-Jacobi-Bellman equation in a Hilbertian framework (see [@C] and references within for the corresponding mild formulations in the standard cases). In [@GoMa] the authors prove, by a fixed point argument that exploits the smoothing properties of the Ornstein-Uhlenbeck semigroup corresponding to the state equation, existence and uniqueness of the solution of the stationary HJB equation for discounted infinite horizon costs. Then they pass to the limit, as the discount goes to zero, to obtain a mild solution of the HJB equation for the ergodic problem (see also [@duncans]). Such techniques need to assume, beside natural condition on the dissipativity of the state equation, also non-degeneracy of the noise and a limitation on the lipschitz constant (with respect to the gradient variable) of the hamiltonian function. This last condition carries a bound on the size of the control domain (see [@FuTe-ell] for similar conditions in the infinite horizon case).
$ $
The introduction of EBSDEs allow us to treat Banach valued state equations with general monotone nonlinear term and possibly degenerate noise. Non-degeneracy is replaced by a structure condition as it usually happens in BSDEs approach, see, for instance, [@ElKaMaz], [@FuTe1]. Moreover the use of $L^{\infty}$ estimates specific to infinite horizon backward stochastic differential equations (see [@bh], [@royer], [@HuTe]) allow us to eliminate conditions on the lipschitz constant of the hamiltonian. On the other side we will only consider bounded cost functionals.
$ $
To start being more precise we consider a forward equation $$dX_t^x=(AX_t^x+F(X_t^x))dt+G dW_t,\qquad X_0=x$$ where $X$ has values in a Banach space $E$, $F$ maps $E$ to $E $ and $A$ generates a strongly continuous semigroup of contractions. Appropriate dissipativity assumptions on $A+F$ ensure the exponential decay of the difference between the trajectories starting from different points $x,x'\in E$.
Then we introduce the class of strictly monotonic backward stochastic differential equations $$\label{bsderoyer*}
{Y}^{x,\alpha}_t={Y}^{x,\alpha}_T +\int_t^T(\psi(X^{x}_\sigma,
Z^{x,\alpha}_\sigma)-\alpha Y^{x,\alpha}_\sigma)d\sigma-\int_t^T
Z^{x,\alpha}_\sigma dW_\sigma, \quad 0\le t\le T <\infty.$$ for all $\alpha>0$ (see [@bh], [@royer] or [@HuTe]) where $\psi: E\times\Xi^*\rightarrow \mathbb{R}$ is bounded in the first variable and Lipschitz in the second. By estimates based on a Girsanov argument introduced in [@bh] we obtain uniform estimates on $\alpha{Y}^{x,\alpha}$ and ${Y}^{x,\alpha}-{Y}^{x',\alpha}$ that allow us to prove that, roughly speaking, $({Y}^{x,\alpha}-{Y}^{0,\alpha}_0,
{Z}^{x,\alpha}, \alpha {Y}^{0,\alpha}_0)$ converge to a solution $(Y^x,Z^x,\lambda)$ of the EBSDE (\[EBSDE\*\]), for all $x\in E$. We also show that $\lambda$ is unique under very general conditions. On the contrary, in general we can not expect uniqueness of the solution to (\[EBSDE\*\]), at least in the non markovian case. On the other side in the markovian case we show that we can find a solution of (\[EBSDE\*\]) with $Y^x_t=v(X^x_t)$ and $Z^x_t=\zeta(X^x_t)$ where $v$ is Lipschitz and $v(0)=0$. Moreover $(v, \zeta)$ are unique at least in a special case where $\psi$ is the Hamiltonian of a control problem and the processes $X^x$ are recurrent (see Section \[sec-uniq\] where we adapt an argument from [@GoMa]).
$ $
If we further assume differentiability of $F$ and $\psi$ (in the Gateaux sense) then $v$ is differentiable, moreover $\zeta =\nabla v G$ and finally $(v,\lambda)$ give a mild solution of the HJB equation $$\mathcal{L}v(x)
+\psi\left( x,\nabla v(x) G\right) = \lambda, \quad x\in E, \label{hjb*}$$ where linear operator $\mathcal{L}$ is formally defined by $$\mathcal{L}f\left( x\right) =\frac{1}{2}Trace\left(
GG^{\ast}\nabla ^{2}f\left( x\right) \right) +\langle Ax,\nabla
f\left( x\right) \rangle_{E,E^{\ast}}+\langle F\left( x\right)
,\nabla f\left( x\right) \rangle_{E,E^{\ast}}.$$ Moreover if the Kolmogorov semigroup satisfies the smoothing property in Definition \[strongly-feller\] and $F$ is genuinely dissipative (see Definition \[gen-diss\]) then $v$ is bounded.
$ $
The above results are then applied to a control problem with cost $$\label{ergodic-cost*}
J(x,u)=\limsup_{T\rightarrow\infty}\frac{1}{T}\, \mathbb E\int_0^T
L(X_s^x,u_s)ds,$$ where $u$ is an adapted process (an admissible control) with values in a separable metric space $U$, and the state equation is a Banach valued evolution equation of the form $$dX_t^x=(AX_t^x+F(X_t^x))\, dt+G(dW_t+R(u_t)\,dt),$$ where $R: U \rightarrow \Xi$ is bounded. It is clear that the above functional depends only on the asymptotic behavior of the trajectories of $X^x$. After appropriate formulation we prove that, setting $\psi(x,z)= \inf_{u\in U} [L(x,u)+ zR(u)]$ in (\[EBSDE\*\]), then $\lambda$ is optimal, that is $$\lambda=\inf_{u}J(x,u)$$ where the infimum is over all admissible controls. Moreover $Z$ allows to construct on optimal feedback in the sense that $$\lambda=J(x,u) \hbox{ if and only if } L(X_t^x,u_t)+Z_t
R(u_t)=\psi(X_t^x,Z_t).$$
Finally, see Section \[section-heat-eq\], we show that our assumptions allow us to treat ergodic optimal control problems for a stochastic heat equation with polynomial nonlinearity and space-time white noise. We notice that the Banach space setting is essential in order to treat nonlinear terms with superlinear growth in the state equation.
$ $
The paper is organized as follows. After a section on notation, we introduce the forward SDE; in section 4 we study the ergodic BSDEs; in section 5 we show in addition the differentiability of the solution assuming that the coefficient is Gateaux differentiable. In section 6 we study the ergodic Hamilton-Jacobi-Bellman equation and we apply our result to optimal ergodic control in section 7. Section 8 is devoted to show the uniqueness of Markovian solution and the last section contains application to the ergodic control of a nonlinear stochastic heat equation.
Notation
========
Let $E,F$ be Banach spaces, $H$ a Hilbert space, all assumed to be defined over the real field and to be separable. The norms and the scalar product will be denoted $|\,\cdot\,|$, $\langle\,\cdot\,,\,\cdot\,\rangle$, with subscripts if needed. Duality between the dual space $E^*$ and $E$ is denoted $\langle\,\cdot\,,\,\cdot\,\rangle_{E^*,E}$. $L(E,F)$ is the space of linear bounded operators $E\to F$, with the operator norm. The domain of a linear (unbounded) operator $A$ is denoted $D(A)$.
Given a bounded function $ \phi: E\rightarrow \mathbb{R}$ we denote $\Vert\phi\Vert_0=\sup_{x\in E}|\phi(x)|$. If, in addition, $\phi$ is also Lipschitz continuous then $\Vert\phi\Vert_{\hbox{lip}}=\Vert\phi\Vert_0+
\sup_{x,x'\in E,\,x\ne x'}|\phi(x)-\phi(x')||x-x'|^{-1}$.
We say that a function $F:E\to F$ belongs to the class $\calg^1(E,F)$ if it is continuous, has a Gateaux differential $\nabla F(x)\in L(E,F)$ at any point $x \in E$, and for every $k\in E$ the mapping $x\to \nabla F(x) k$ is continuous from $E$ to $F$ (i.e. $x\to \nabla F(x) $ is continuous from $E$ to $L(E,F)$ if the latter space is endowed the strong operator topology). In connection with stochastic equations, the space $\calg^1$ has been introduced in [@FuTe1], to which we refer the reader for further properties.
Given a probability space $\left(
\Omega,\mathcal{F},\mathbb{P}\right) $ with a filtration $(\calf_t)_{t\ge 0}$ we consider the following classes of stochastic processes with values in a real separable Banach space $K$.
1. $L^p_{\mathcal{P}}(\Omega,C([0,T],K))$, $p\in [1,\infty)$, $T>0$, is the space of predictable processes $Y$ with continuous paths on $[0,T]$ such that $$|Y|_{L^p_{\mathcal{P}}(\Omega,C([0,T],E))}^p
= \E\, \sup_{t\in [0,T]}|Y_t|_K^p<\infty.$$
2. $L^p_{\mathcal{P}}(\Omega,L^2([0,T];K))$, $p\in [1,\infty)$, $T>0$, is the space of predictable processes $Y$ on $[0,T]$ such that $$|Y|^p_{L^p_{\mathcal{P}}(\Omega,L^2([0,T];K))}=
\E\,\left( \int_{0}^{T}|Y_t|_K^2\,dt\right)^{p/2}<\infty.$$
3. $L_{\cal P, {\rm loc}}^2(\Omega;L^2(0,\infty;K))$ is the space of predictable processes $Y$ on $[0,\infty)$ that belong to the space $L^2_{\mathcal{P}}(\Omega,L^2([0,T];K))$ for every $T>0$.
The forward equation
====================
In a complete probability space $\left(
\Omega,\mathcal{F},\mathbb{P}\right) ,$ we consider the following stochastic differential equation with values in a Banach space $E$: $$\left\{
\begin{array}[c]{l} dX_t =AX_t d t+F(X_t) dt +GdW_t ,\text{ \ \ \
} t\geq 0, \\
X_0 =x\in \, E.
\end{array}
\right. \label{sde}$$ We assume that $E$ is continuously and densely embedded in a Hilbert space $H$, and that both spaces are real separable.
We will work under the following general assumptions:
\[general\_hyp\_forward\]
1. The operator $A$ is the generator of a strongly continuous semigroup of contractions in $E$. We assume that the semigroup $\{e^{tA},\, t\geq0\}$ of bounded linear operators on $E$ generated by $A$ admits an extension to a strongly continuous semigroup of bounded linear operators on $H$ that we denote by $\{S(t),\, t\geq 0\}$.
2. $W$ is a cylindrical Wiener process in another real separable Hilbert space $\Xi$. Moreover by $\calf_{t}$ we denote the $\sigma$-algebra generated by $\{W_s,\; s\in [0,t]\}$ and by the sets of $\calf$ with $\P$-measure zero.
3. $F:E\to E$ is continuous and has polynomial growth (that is there exist $c>0, k\ge 0$ such that $|F(x)|\leq c (1+|x|^k)$, $x\in E$). Moreover there exists $\eta>0$ such that $A+F+\eta I$ is dissipative.
4. $G$ is a bounded linear operator from $\Xi$ to $H$. The bounded linear, positive and symmetric operators on $H$ defined by the formula $$Q_{t}h=\int_{0}^{t}S(s)GG^{\ast}S^*(s)h\,ds,\qquad t\geq 0,\; h\in
H,$$ are assumed to be of trace class in $H$. Consequently we can define the stochastic convolution $$W^{A}_t =\int_{0}^{t}S(t-s) GdW_s,\quad t\geq 0,$$ as a family of $H$-valued stochastic integrals. We assume that the process $\{W^{A}_t,\, t \geq 0\}$ admits an $E$-continuous version.
We recall that, for every $x\in E$, with $x\neq 0$, the subdifferential of the norm at $x$, $\partial\left( |x| \right) $, is the set of functionals $x^{\ast}\in E^{\ast}$ such that $\left\langle x^{\ast },x\right\rangle _{E^{\ast},E}=| x| $ and $|
x^{\ast}|_{E^{\ast}}=1$. If $x=0$ then $\partial\left( | x|\right)
$ is the set of functionals $x^{\ast}\in E^{\ast}$ such that $|x^{\ast}|_{E^{\ast}}\leq 1$. The dissipativity assumption on $A+F$ can be explicitly stated as follows: for $x,x'\in
D(A)\subset E$ there exists $x^{\ast } \in\partial\left( \left|
x-x' \right| \right) $ such that $$\left\langle
x^{\ast} ,A( x-x' ) +F\left( x \right) -F\left( x' \right)
\right\rangle _{E^{\ast},E}
\leq-\eta\left|
x-x' \right|.$$
We can state the following theorem, see e.g. [@DP1], theorem 7.13 and [@DP2], theorem 5.5.13.
\[teo2 forward\]Assume that Hypothesis \[general\_hyp\_forward\] holds true. Then for every $x\in E$ equation (\[sde\]) admits a unique mild solution, that is an adapted $E$-valued process with continuous paths satisfying $\mathbb{P}$-a.s. $$X_{t}=e^{ t A}x+\int_{0}^{t}e^{ (t-s ) A}F\left( X_{s}\right)
ds+\int_{0}^{t}e^{(t-s ) A}GdW_{s},\text{ \ \ \ }t\geq 0 .$$
We denote the solution by $X^x $, $x\in E$.
Now we want to investigate the dependence of the solution on the initial datum.
\[prop lip X\] Under Hypothesis \[general\_hyp\_forward\] it holds: $$\left| X_t^{x_1} -X_t^{x_2} \right| \leq e^{-\eta t }\left|
x_{1}-x_{2}\right| ,\text{ }t\ge 0, \;\; x_{1},x_{2}\in E.$$
Let $X_{1}\left( t\right) =X^{x_1}_{t} $ and $X_{2}\left(
t\right) =X^{x_2}_{t} $, $x_{1},x_{2}\in E$. For $i=1,2$ we set $X_{i}^{n}\left( t\right) =J_n X_{i}\left( t\right) $, where $J_n
=n\left( nI-A\right) ^{-1}$. Since $X_{i}^{n}\left( t\right)
\in {D}\left( A\right) $ for every $t\geq 0 $, and $$X_{i}^{n}\left( t\right) =e^{t A}J_n x_{i}+\int
_{0}^{t}e^{\left( t-s\right) A}J_n F\left( X_{i}\left( s\right)
\right) ds+\int_{0}^{t}e^{\left( t-s\right) A}J_n GdW_{s},$$ we get $$\frac{d}{dt}\left( X_{1}^{n}\left( t\right) -X_{2}^{n}\left(
t\right) \right) =A\left( X_{1}^{n}\left( t\right) -X_{2}
^{n}\left( t\right) \right) +J_n \left[ F\left( X_{1}\left(
t\right) \right) -F\left( X_{2}\left( t\right) \right) \right]
.$$ So, by proposition II.8.5 in [@S] also $\left|
X_{1}^{n}\left( t\right) -X_{2}^{n}\left( t\right) \right| $ admits the left and right derivatives with respect to $t$ and there exists $x_{n}^{\ast }\left( t\right) \in\partial\left(
\left| X_{1}^{n}\left( t\right) -X_{2}^{n}\left( t\right) \right|
\right) $ such that the left derivative of $\left|
X_{1}^{n}\left( t\right) -X_{2}^{n}\left( t\right) \right| $ satisfies the following $$\frac{d^{-}}{dt}\left| X_{1}^{n}\left( t\right) -X_{2}^{n}\left(
t\right) \right| =\left\langle x_{n}^{\ast}\left( t\right)
,\frac{d}{dt}\left( X_{1}^{n}\left( t\right) -X_{2}^{n}\left(
t\right) \right) \right\rangle _{E^{\ast},E}.$$ So we have $$\begin{array}{ll}\dis \frac{d^{-}}{dt}\left| X_{1}^{n}\left(
t\right) -X_{2}^{n}\left( t\right) \right| & =\left\langle
x_{n}^{\ast}\left( t\right) ,A\left( X_{1}^{n}\left( t\right)
-X_{2}^{n}\left( t\right) \right) +F\left( X_{1} ^{n}\left(
t\right) \right) -F\left( X_{2}^{n}\left( t\right)
\right) \right\rangle _{E^{\ast},E}\\
& \quad +\left\langle x_{n}^{\ast}\left( t\right) ,J_n F\left(
X_{1}\left( t\right) \right) -F\left( X_{1}
^{n}\left( t\right) \right) \right\rangle _{E^{\ast},E}\\
& \quad -\left\langle x_{n}^{\ast}\left( t\right) ,J_n F\left(
X_{2}\left( t\right) \right) -F\left( X_{2}
^{n}\left( t\right) \right) \right\rangle _{E^{\ast},E}\\
& \leq-\eta\left|
X_{1}^{n}\left( t\right) -X_{2}^{n}\left(
t\right) \right| +\left| \delta_{1}^{n}\left( t\right)
-\delta_{2}^{n}\left( t\right) \right| ,
\end{array}$$ where for $i=1,2$ we have set $\delta_{i}^{n}\left( t\right) =J_n
F\left( X_{i}\left( t\right) \right) -F\left( X_{i}^{n}\left(
t\right) \right) $.
Multiplying the above by $e^{\eta t}$ we get $$\frac{d^{-}}{dt}\left( e^{\eta t}\left| X_{1}^{n}\left( t\right)
-X_{2}^{n}\left( t\right) \right|\right)\leq e^{\eta t} \left|
\delta_{1}^{n}\left( t\right) -\delta_{2}^{n}\left( t\right)
\right|.$$ We note that $\delta_{i} ^{n}\left( t\right) $ tends to $0$ uniformly in $t\in \left[0,T\right] $ for arbitrary $T>0$. Indeed, $$\delta_{i}^{n}\left( t\right) =nR\left( n,A\right) \left[
F\left( X_{i}\left( t\right) \right) -F\left( X_{i}^{n}\left(
t\right) \right) \right] +\left( nR\left( n,A\right) -I\right)
F\left( X_{i}\left( t\right) \right) ,$$ and the convergence to $0$ follows by a classical argument, see e.g. the proof of theorem 7.10 in [@DP1], since $X_{i}^{n}\left( t\right) $ tends to $X_{i}\left( t\right) $ uniformly in $t\in\left[ 0,T\right] $ and the maps $t\mapsto
X_{i}\left( t\right) $ and $t\mapsto F\left( X_{i}\left(
t\right) \right) $ are continuous with respect to $t$.
Thus letting $n\rightarrow\infty$ we can conclude $$\left| X_{1}\left( t\right) -X_{2}\left( t\right) \right| \leq
e^{-\eta t } \left| x_{1}-x_{2}\right| .$$ and the claim is proved.
$ $
We will also need the following assumptions.
\[hyp\_W\_A F(W\_A)\] We have $\sup_{t\geq 0}\,
\E\,|W^A_t|^2<\infty.$
\[hyp-convol-determ\] $e^{tA}G\,(\Xi)\subset E$ for all $t>0$ and $\displaystyle \int_0^{+\infty} |e^{tA} G|_{L(\Xi,E)} dt <
\infty$.
We recall that for arbitrary gaussian random variabile $Y$ with values in the Banach space $E$, the inequality $$\E \,\phi (|Y|-\E\,|Y|)\le \E \,\phi (2\sqrt{\E\,|Y|^2}\,\gamma)$$ holds for any convex nonnegative continuous function $\phi$ on $E$ and for $\gamma$ a real standard gaussian random variable, see e.g. [@kw-woy], Example 3.1.2. Upon taking $\phi(x)=|x|^p$, it follows that for every $p\ge 2$ there exists $c_p>0$ such that $\E
\,|Y|^p\le c_p(\E \,|Y|^2)^{p/2}$. By the gaussian character of $W^A_t$ and the polynomial growth condition on $F$ stated in Hypothesis \[general\_hyp\_forward\], point 3, we see that Hypothesis \[hyp\_W\_A F(W\_A)\] entails that for every $p\ge 2$ $$\label{stimegaussunif}
\sup_{t\geq 0} \E\left[ |W^A_t|^p+ |F(W^A_t)|^p \right] <\infty.$$
\[prop-X-L\^p\] Under Hypothesis \[general\_hyp\_forward\] it holds, for arbitrary $T>0$ and arbitrary $p\geq 1$ $$\label{prop-X-L^p-1}
\E\sup_{t\in [0,T]} |X_t^x|^p \leq C_{p,T}(1+|x|^p),\qquad x\in
E.$$ If, in addition, Hypothesis \[hyp\_W\_A F(W\_A)\] holds then, for a suitable constant C $$\label{prop-X-L^p-2}\sup_{t\geq 0} \E |X_t^x| \leq C(1+|x|)
,\qquad x\in E.$$ Moreover if, in addition, Hypothesis \[hyp-convol-determ\] holds, $\gamma$ is a bounded, adapted, $\Xi$-valued process and $X^{x,\gamma}$ is the mild solution of equation $$\left\{
\begin{array}{l}
dX^{x,\gamma}_t =AX^{x,\gamma}_{t} dt+F(
X^{x,\gamma}_{t} ) dt+GdW_{t}+G\gamma_{t}\,dt ,\quad t\geq 0, \\
X^{x,\gamma}_{0} =x\in E.
\end{array} \right. \label{sde-gamma}$$ then it is still true that $$\label{rel-estimate-Xgamma}
\sup_{t\geq 0} \E |X^{x,\gamma}_t| \leq C_{\gamma}(1+|x|),\qquad x\in E,$$ for a suitable constant $C_{\gamma}$ depending only on a uniform bound for $\gamma$.
We let $Z_t=X^x_t-W^A_t$, $Z^n_t=J_n Z_t $, then $$\frac{d}{dt } Z^n_t =
AZ^n_t +J_nF(X^x_t) = AZ^n_t +\left[F(Z^n_t+J_n W^A_t) - F(J_n
W^A_t)\right]+F( W^A_t)+\delta^n_t$$ where $$\delta^n_t= J_n F(X^x_t)-F(J_n X^x_t)
+F(J_n W^A_t)-F( W^A_t).$$ Proceeding as in the proof of Proposition \[prop lip X\] observing that, for all $t>0$, $\displaystyle \int_0^{t}|\delta^n_s| ds \rightarrow 0$ as $n\rightarrow\infty$, we get: $$|Z_t|\leq e^{-\eta t}|x|+\int_0^{t} e^{-\eta (t-s)}
|F(W^A_s)|ds,\;\;\; \mathbb{P}-\hbox{a.s.}$$ and (\[prop-X-L\^p-2\]) follows from (\[stimegaussunif\]).
In the case in which $X^x$ is replaced by $X^{x, \gamma}$ the proof is exactly the same just replacing $W^A_t$ by $W^{A,\gamma}_t=W^A_t+\int_0^t e^{(t-s)A}G\gamma_s ds$.
Finally to prove (\[prop-X-L\^p-1\]) we notice that (see the discussion in [@masiero]) the process $W^A$ is a Gaussian random variable with values in $C([0,T],E)$. Therefore by the polynomial growth of $F$ we get $$\E\sup_{t\in [0,T]} \left[|W^A_t|^p + |F(W^A_t)|^p\right]\leq
C_{p,T}(1+|x|^p),$$ and the claim follows as above.
$ $
Finally the following result is proved exactly as Theorem 6.3.3. in [@DP2].
\[ergodicity\] Assume that Hypotheses \[general\_hyp\_forward\] and \[hyp\_W\_A F(W\_A)\] hold then equation (\[sde\]) has a unique invariant measure in $E$ that we will denote by $\mu$. Moreover $\mu$ is strongly mixing (that is, for all $x\in E$, the law of $X_t^x$ converges weakly to $ \mu$ as $t\rightarrow \infty$). Finally there exists a constant $C>0$ such that for any bounded Lipschitz function $\phi: E\rightarrow \mathbb{R}$, $$\left|\mathbb{E}\phi(X^x_t)-\int_E \phi\, d\mu \right|\leq C(1+|x|)
e^{-\eta t /2} \Vert\phi\Vert_{\hbox{\em lip}}.$$
Ergodic BSDEs (EBSDEs)
======================
This section is devoted to the following type of BSDEs with infinite horizon $$\label{EBSDE}
Y^x_t=Y^x_T +\int_t^T\left[\psi(X^x_\sigma,Z^x_\sigma)-
\lambda\right]d\sigma-\int_t^T Z^x_\sigma\, dW_\sigma, \quad 0\le
t\le T <\infty,$$ where $\lambda$ is a real number and is part of the unknowns of the problem; the equation is required to hold for every $t$ and $T$ as indicated. On the function $\psi: E\times \Xi^*
\rightarrow {\mathbb R}$ and assume the following:
\[hypothesisroyer\] $ $ There exists $K_x, K_z>0$ such that $$|\psi(x,z)
-\psi(x',z')|\le K_x|x-x'|+ K_z |z-z'|, \qquad
x,x'\in E,\;
z,z'\in\Xi^*.$$ Moreover $\psi(\,\cdot\,,0)$ is bounded. We denote $\sup_{x\in E
}|\psi(x,0)|$ by $M$.
We start by considering an infinite horizon equation with strictly monotonic drift, namely, for $\alpha>0$, the equation $$\label{bsderoyer}
{Y}^{x,\alpha}_t={Y}^{x,\alpha}_T +\int_t^T(\psi(X^{x}_\sigma,
Z^{x,\alpha}_\sigma)-\alpha Y^{x,\alpha}_\sigma)d\sigma-\int_t^T
Z^{x,\alpha}_\sigma dW_\sigma, \quad 0\le t\le T <\infty.$$
The existence and uniqueness of solution to (\[bsderoyer\]) under Hypothesis \[hypothesisroyer\] was first studied by Briand and Hu in [@bh] and then generalized by Royer in [@royer]. They have established the following result when $W$ is a finite dimensional Wiener process but the extension to the case in which $W$ is a Hilbert-valued Wiener process is immediate (see also [@HuTe]).
\[lemmaroyer\] Let us suppose that Hypotheses \[general\_hyp\_forward\] and \[hypothesisroyer\] hold. Then there exists a unique solution $(Y^{x,\alpha},Z^{x,\alpha})$ to BSDE (\[bsderoyer\]) such that $Y^{x,\alpha}$ is a bounded continuous process, and $Z^{x,\alpha}$ belongs to $L_{\cal P, {\rm
loc}}^2(\Omega;L^2(0,\infty;\Xi^*))$.
Moreover $|Y^{x,\alpha}_t|\leq {M}/{\alpha}$, $\mathbb{P}$-a.s. for all $t\geq 0$.
We define $$v^{\alpha}(x)=Y^{\alpha,x}_0.$$ We notice that by the above $|v^{\alpha}(x)|\leq {M}/{\alpha}$ for all $x\in E$. Moreover by the uniqueness of the solution of equation (\[bsderoyer\]) it follows that $Y^{\alpha,x}_t=v^{\alpha}(X^x_t)$
To establish Lipschitz continuity of $ v^{\alpha}$ (uniformly in $\alpha$) we use a Girsanov argument due to P. Briand and Y. Hu, see [@bh]. Here and in the following we use an infinite-dimensional version of the Girsanov formula that can be found for instance in [@DP1].
\[lemma-lip-v\] Under Hypotheses \[general\_hyp\_forward\] and \[hypothesisroyer\] the following holds for any $\alpha>0$: $$|v^{\alpha}(x) - v^{\alpha}(x')| \leq \frac{K_x}{\eta} |x-x'|,
\qquad x,x'\in E.$$
We briefly report the argument for the reader’s convenience.
We set $\tilde{Y}=Y^{\alpha,x}-Y^{\alpha,x'}$, $\tilde{Z}=Z^{\alpha,x}-Z^{\alpha,x'},$ $$\beta_t=\begin{cases}
\frac{\displaystyle \psi(X^{x'}_t,Z^{\alpha,
x'}_t)-\psi(X^{x'}_t,Z^{\alpha,x}_t)} {\displaystyle
|Z^{\alpha,x}_t - Z^{\alpha,x'}_t|_{\Xi^*}^2}\left( Z^{\alpha,x}_t
- Z^{\alpha,x'}_t\right)^*,& \hbox{ if } Z^{\alpha,x}_t \neq Z^{\alpha,x'}_t \\
0, & \hbox{ elsewhere, }
\end{cases}$$ $$f_t=\psi(X^{x}_t,
Z^{x,\alpha}_t)-\psi(X^{x'}_t, Z^{x,\alpha}_t).$$ By Hypothesis \[hypothesisroyer\], $\beta$ is a bounded $\Xi$-valued, adapted process thus there exists a probability $\tilde{\mathbb{P}}$ under which $\tilde{W_{t}}=\int_0^{t} \beta_s
ds + W_{t}$ is a cylindrical $\Xi$-valued Wiener process for ${t}\in [0,T]$. Then $(\tilde{Y},\tilde{Z})$ verify, for all $0\le t\le T <\infty$, $$\label{bsderoyer-girsanov}
\tilde{Y}_t=\tilde{Y}_T -\alpha \int_t^T \tilde{Y}_\sigma d\sigma
+\int_t^T f_{\sigma}d\sigma- \int_t^T \tilde{Z}_\sigma
d\tilde{W}_\sigma.$$ Computing $d (e^{-\alpha t}\tilde{Y}_t)$, integrating over $[0,T]$, estimating the absolute value and finally taking the conditional expectation $\tilde{\mathbb{E}}^{\mathcal{F}_t}$ with respect to $\tilde{\mathbb{P}}$ and $\mathcal{F}_t$ we get: $$|\tilde{Y}_t| \leq e^{-\alpha(T-t)} \tilde{\mathbb{E}}^{\mathcal{F}_t}
| \tilde{Y}_T |+
\tilde{\mathbb{E}}^{\mathcal{F}_t}
\int_{t}^T e^{-\alpha(s-t)} |f_s| ds$$ Now we recall that $ \tilde{Y}$ is bounded and that $|f_t|\leq
K_x |X^{x}_t-X^{x'}_t|\leq K_x e^{-\eta t}|x-x'|$ by Proposition \[prop lip X\]. Thus if $T\rightarrow \infty$ we get $
|\tilde{Y}_t| \leq K_x (\eta+\alpha)^{-1}e^{\alpha t} |x-x'| $ and the claim follows setting $t=0$.
$ $
By the above Lemma if we set $$\overline{v}^{\alpha}(x)= {v}^{\alpha}(x)- {v}^{\alpha}(0),$$ then $ | \overline{v}^{\alpha}(x)|\leq K_x \eta^{-1}|x|$ for all $x\in E$ and all $\alpha>0$. Moreover by Lemma \[lemmaroyer\] $\alpha |{v}^{\alpha}(0)|\leq M$.
Thus by a diagonal procedure we can construct a sequence $\alpha_n\searrow 0$ such that for all $x$ in a countable dense subset $D\subset E$ $$\label{def-of-lambda}
{\overline{v}}^{\alpha_n}(x)\rightarrow \overline{v}(x),\qquad
\alpha_n v^{\alpha_n}(0)\rightarrow \overline{\lambda},$$ for a suitable function $ \overline{v}: D \rightarrow
\mathbb{R}$ and for a suitable real number $\overline{\lambda}$.
Moreover, by Lemma \[lemma-lip-v\], $ | \overline{v}^{\alpha}(x)-
\overline{v}^{\alpha}(x')|\leq K_x \eta^{-1}|x-x'|$ for all $x,x'\in E$ and all $\alpha>0$. So $\overline{v}$ can be extended to a Lipschitz function defined on the whole $E$ (with Lipschitz constant $K_x \eta^{-1} $) and $$\label{def-of-v} {\overline{v}}^{\alpha_n}(x)\rightarrow
\overline{v}(x),\qquad x\in E.$$
\[main-EBSDE\] Assume Hypotheses \[general\_hyp\_forward\] and \[hypothesisroyer\] hold. Moreover let $\bar \lambda$ be the real number in (\[def-of-lambda\]) and define $\bar Y^x_t= \bar
v(X^x_t)$ (where $\overline{v}$ is the Lipschitz function with $\overline{v}(0)=0$ defined in (\[def-of-v\])). Then there exists a process $\overline{Z}^{x}\in L_{\cal P, {\rm
loc}}^2(\Omega;L^2(0,\infty;\Xi^*))$ such that $\mathbb{P}$-a.s. the EBSDE (\[EBSDE\]) is satisfied by $(\bar Y^x,\bar Z^x, \bar \lambda)$ for all $0\leq t\leq T$.
Moreover there exists a measurable function $\overline{\zeta}:
E\rightarrow \Xi^*$ such that $\overline{Z}^{x}_t=\overline{\zeta}(X^x_t)$.
Let $\overline{Y}^{x,\alpha}_t={Y}^{x,\alpha}_t-v^{\alpha}(0)=
\overline{v}^{\alpha}({X}^{x}_t)$. Clearly we have, $\mathbb{P}$-a.s., $$\label{equation-proof-main-1}
\overline{Y}^{x,\alpha}_t=\overline{Y}^{x,\alpha}_T +\int_t^T(\psi(X^{x}_\sigma,
Z^{x,\alpha}_\sigma)-\alpha \overline{Y}^{x,\alpha}_\sigma-\alpha
{v}^{\alpha}(0))d\sigma -\int_t^T Z^{x,\alpha}_\sigma dW_\sigma,
\quad 0\le t\le T <\infty.$$ Since $|\bar v^{\alpha}(x)|\leq K_x|x|/\eta $, inequality (\[prop-X-L\^p-1\]) ensures that $\mathbb{E}\sup_{t\in[0,T]}\left[\sup_{\alpha>0}
|\overline{Y}^{x,\alpha}_t|^2\right]< +\infty$ for any $T>0$. Thus, if we define $\overline{Y}^x=\overline{v}(X^x)$, then by dominated convergence theorem $$\mathbb{E} \int_0^T |\overline{Y}^{x,\alpha_n}_t -\overline{Y}^{x}_t|^2 dt
\rightarrow 0\quad \hbox{and}\quad
\mathbb{E} |\overline{Y}^{x,\alpha_n}_T-\overline{Y}^{x}_T|^2
\rightarrow 0$$ as $n\rightarrow \infty$ (where $\alpha_n \searrow 0$ is a sequence for which (\[def-of-lambda\]) and (\[def-of-v\]) hold).
We claim now that there exists $\overline{Z}^{x}\in L_{\cal P,
{\rm loc}}^2(\Omega;L^2(0,\infty;\Xi^*))$ such that $$\mathbb{E} \int_0^T |{Z}^{x,\alpha_n}_t -\overline{Z}^{x}_t|_{\Xi^*}^2 dt
\rightarrow 0$$ Let $\tilde{Y}={\bar Y}^{x,\alpha_n}-{\bar Y}^{x,\alpha_m}$, $\tilde{Z}={Z}^{x,\alpha_n}-{Z}^{x,\alpha_m}$. Applying Itô’s rule to $\tilde{Y}^2$ we get by standard computations $$\tilde{Y}^2_0+\mathbb{E}\int_0^T |\tilde{Z}_t|_{\Xi^*}^2 dt
=\mathbb{E}{\tilde Y}^2_T + 2\mathbb{E}\int_0^T \tilde \psi_t
\tilde Y_t dt -2 \mathbb{E}\int_0^T \left[\alpha_n
{Y}^{x,\alpha_n}_t - \alpha_m {Y}^{x,\alpha_m}_t\right] \tilde
Y_t\,dt$$ where $\tilde
\psi_t=\psi(X^x_t,Z^{x,\alpha_n}_t)-\psi(X^x_t,Z^{x,\alpha_m}_t)$. We notice that $|\tilde\psi_t| \leq K_z|\tilde Z _t|$ and $\alpha_n |{Y}^{x,\alpha_n}_t|\leq M$. Thus $$\mathbb{E}\int_0^T |\tilde{Z}_t|_{\Xi^*}^2 dt \leq c\left[
\mathbb{E} (\tilde Y^x_T)^2 +\mathbb{E}\int_0^T (\tilde{Y}^x_t)^2
dt +\mathbb{E}\int_0^T |\tilde{Y}^x_t| dt \right].$$ It follows that the sequence $\{{Z}^{x,\alpha_m}\}$ is Cauchy in $L^2(\Omega;L^2(0,T;\Xi^*))$ for all $T>0$ and our claim is proved.
Now we can pass to the limit as $n\rightarrow \infty$ in equation (\[equation-proof-main-1\]) to obtain $$\label{equation-proof-main-2}
\overline{Y}^{x}_t=\overline{Y}^{x}_T +\int_t^T(\psi(X^{x}_\sigma,
\overline{Z}^{x}_\sigma)-\overline{\lambda })d\sigma-\int_t^T
\overline{Z}^{x}_\sigma dW_\sigma, \quad 0\le t\le T <\infty.$$ We notice that the above equation also ensures continuity of the trajectories of $\overline{Y}$ It remains now to prove that we can find a measurable function $\bar \zeta:E\rightarrow \Xi^*$ such that $\overline{Z}^{x}_t=\bar \zeta (X^x_t)$, $\mathbb{P}$-a.s. for almost every $t\geq 0$.
By a general argument, see for instance [@Fu], we know that for all $\alpha>0$ there exists $\zeta^{\alpha}:E\rightarrow
\Xi^*$ such that ${Z}^{x,\alpha}_t=\zeta^{\alpha} (X^x_t)$, $\mathbb{P}$-a.s. for almost every $t\geq 0$.
To construct $\zeta$ we need some more regularity of the processes ${Z}^{x,\alpha}$ with respect to $x$.
If we compute $d ({Y}^{x,\alpha}_t-{Y}^{x',\alpha}_t)^2$ we get by the Lipschitz character of $\psi$: $$\begin{array} {l}
\displaystyle \mathbb{E}\int_0^T
|Z^{x,\alpha}_t-Z^{x',\alpha}_t|_{\Xi^*}^2 dt \leq \mathbb{E}
(v^{\alpha}(X^x_T)- v^{\alpha}(X^{x'}_T))^2
\\
\quad + \displaystyle \mathbb{E}\int_0^T
\left(K_x|X^x_s-X^{x'}_s|
+K_z|Z^{x,\alpha}_s-Z^{x',\alpha}_s|\right)
\left|v^{\alpha}(X^x_s)- v^{\alpha}(X^{x'}_s)\right| ds
\end{array}$$ By the Lipschitz continuity of $v^{\alpha}$ (uniform in $\alpha$) that of $\psi$ and Proposition \[prop lip X\] we immediately get: $$\label{lip-of-Z}
\mathbb{E}\int_0^T |Z^{x,\alpha}_t-Z^{x',\alpha}_t|_{\Xi^*}^2 dt \leq c |x-x'|^2.$$ for a suitable constant $c$ (that may depend on $T$).
Now we fix an arbitrary $T>0$ and, by a diagonal procedure (using separability of $E$) we construct a subsequence $(\alpha_n')\subset (\alpha_n)$ such that $\alpha_n' \searrow 0$ and $$\mathbb{E}\int_0^T |Z^{x,\alpha_n'}_t-Z^{x',\alpha_m'}_t|_{\Xi^*}^2 dt \leq 2^{-n}$$ for all $m\geq n$ and for all $x\in E$. Consequently $Z^{x,\alpha_n'}_t\rightarrow \overline{Z}^x_t$, $\mathbb{P}$-a.s. for a.e. $t\in [0,T]$. Then we set: $$\bar \zeta(x)=\left\{\begin{array}{ll} \lim_n \zeta^{\alpha_n'}(x),
& \hbox{ if the limit exists in }\Xi^*,\\
0, & \hbox{ elsewhere.}\end{array}\right.$$ Since $Z^{x,\alpha_n'}_t= \zeta^{\alpha_n'}(X^x_t)\rightarrow
\overline{Z}^{x}_t$ $\mathbb{P}$-a.s. for a.e. $t\in [0,T]$ we immediately get that, for all $x\in E$, the process $X^x_t$ belongs $\mathbb{P}$-a.s. for a.e. $t\in [0,T]$ to the set where $\lim_n \zeta^{\alpha_n'}(x)$ exists and consequently $\overline{Z}^{x}_t=\bar \zeta(X^x_t)$.
We notice that the solution we have constructed above has the following “linear growth” property with respect to $X$: there exists $c>0$ such that, $\mathbb{P}$-a.s., $$\label{growt-of-Y}
|\overline{Y}^x_t|\leq c |X^x_t| \hbox{ for all $t\geq 0$}.$$
If we require similar conditions then we immediately obtain uniqueness of $\lambda$.
\[th-uniq-lambda\] Assume that, in addition to Hypotheses \[general\_hyp\_forward\], \[hyp\_W\_A F(W\_A)\] and \[hypothesisroyer\], Hypothesis \[hyp-convol-determ\] holds as well. Moreover suppose that, for some $x\in E$, the triple $(Y',Z',\lambda')$ verifies $\mathbb{P}$-a.s. equation (\[EBSDE\]) for all $0\leq t\leq T$, where $Y'$ is a progressively measurable continuous process, $Z'$ is a process in $L_{\cal P, {\rm loc}}^2(\Omega;L^2(0,\infty;\Xi^*))$ and $\lambda'\in \mathbb{R}$. Finally assume that there exists $c_x>0$ (that may depend on $x$) such that $\mathbb{P}$-a.s. $$|Y'_t|\leq c_x (|X^x_t|+1) , \hbox{ for all $t\geq 0$}.$$ Then $\lambda'=\bar \lambda$.
Let $\tilde \lambda=\lambda'-\lambda$, $\tilde
Y=Y'-\overline{Y}^x$, $\tilde Z=Z'-\overline{Z}^x$. By easy computations: $$\tilde \lambda=T^{-1}\left[\tilde Y_T-\tilde Y_0\right]+T^{-1}\int_0^T \tilde Z_t \gamma_t dt
-T^{-1}\int_0^T \tilde Z_t dW_t$$ where $$\gamma_t:=\begin{cases} \frac{\displaystyle \psi(X^{x}_t,Z'_t)-\psi(X^{x}_t,\overline{Z}^{x}_t)}{\displaystyle |Z'_t - \overline{Z}_t|_{\Xi^*}^2}\left(Z'_t - \overline{Z}_t \right)^*,& \hbox{ if } Z'_t \neq \overline{Z}_t, \\
0, & \hbox{ elsewhere },
\end{cases}$$ is a bounded $\Xi$-valued progressively measurable process. By the Girsanov Theorem there exists a probability measure $\mathbb{P}_{\gamma}$ under which $W^{\gamma}_t=-\int_0^t \gamma_s
ds+W_t$, $t\in [0,T]$, is a cylindrical Wiener process in $\Xi$. Thus computing expectation with respect to $\mathbb{P}_{\gamma}$ we get $$\tilde \lambda=T^{-1}\mathbb{E}^{\mathbb{P}_{\gamma}}
\left[\tilde Y_T-\tilde Y_0\right].$$ Consequently, taking into account (\[growt-of-Y\]), $$\label{eq-proof-uniq-lambda}
|\tilde \lambda|\leq c T^{-1}\mathbb{E}^{\mathbb{P}_{\gamma}}
(|X^x_T|+1)+ c T^{-1}(|x|+1)$$ With respect to $W^{\gamma}$, $X^x$ is the mild solution of $$\left\{
\begin{array}{l}
dX^{x,\gamma}_t =AX^{x,\gamma}_{t} dt+F( X^{x,\gamma}_{t} )
dt+GdW^{\gamma}_{t}+G\gamma_{t}\,dt ,
\quad t\geq 0 \\
X^{x,\gamma}_{0} =x\in E.
\end{array} \right.$$ and by (\[rel-estimate-Xgamma\]) we get $\sup_{T>0}\mathbb{E}^{\mathbb{P}_{\gamma}}|X^x_T|<\infty$. So if we let $T\rightarrow\infty$ in (\[eq-proof-uniq-lambda\]) we conclude that $\tilde\lambda=0$.
*The solution to EBSDE (\[EBSDE\]) is, in general, not unique. It is evident that the equation is invariant with respect to addition of a constant to $Y$ but we can also construct an arbitrary number of solutions that do not differ only by a constant (even if we require them to be bounded). On the contrary the solutions we construct are not Markovian.*
Indeed, consider the equation: $$\label{eq:nouniqueness}
-dY_t=[\psi(Z_t)-\lambda]dt-Z_tdW_t.$$ where $W$ is a standard brownian motion and $\psi:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable bounded and has bounded derivative.
One solution is $Y=0;Z=0;\lambda=\psi(0)$ (without loss of generality we can suppose that $\psi(0)=0$).
Let now $\phi:\mathbb{R}\rightarrow \mathbb{R}$ be an arbitrary differentiable function bounded and with bounded derivative. The following BSDE on $[t,T]$ admits a solution: $$\left\{\begin{array}{rcl}
-dY_s^{x,t}&=&\psi(Z_s^{x,t})ds-Z_s^{x,t}dW_s,\\
Y_T^{x,t}&=&\phi(x+W_T-W_t).
\end{array}\right.$$ If we define $u(t,x)=Y_t^{x,t}$ then both $u$ and $\nabla u$ are bounded. Moreover if $\tilde{Y}_t=Y_t^{0,0}=u(t,W_t),\
\tilde{Z}_t=Z_t^{0,0}=\nabla u(t,W_t)$ then $$\left\{\begin{array}{rcl}
-d\tilde{Y}_t&=&\psi(\tilde{Z}_t)dt-\tilde{Z}_tdW_t,\quad
t\in [0,T],\\
\tilde{Y}_T&=&\phi(W_T).
\end{array}\right.$$ Then it is enough to extend with $\tilde{Y}_t=\tilde{Y}_T,\ \tilde{Z}_t=0$ for $t>T$ to construct a bounded solution to (\[eq:nouniqueness\]).
*The existence result in Theorem \[main-EBSDE\] can be easily extended to the case of $\psi$ only satisfying the conditions $$|\psi(x,z)
-\psi(x',z)|\le K_x|x-x'|,\quad |\psi(x,0)|\le M,
\quad |\psi(x,z)|
\le K_z(1+|z|).$$ Indeed we can construct a sequence $\{\psi_n : n\in \mathbb{N}\}$ of functions Lipschitz in $x$ and $z$ such that for all $x,x'\in
H$, $z \in \Xi^*$, $n\in \mathbb{N}$ $$|\psi^n(x,z)
-\psi^n(x',z)|\le K'_x|x-x'|;\quad |\psi^n(x,0)|\leq M';\quad
\lim_{n\rightarrow \infty}|\psi^n(x,z) -\psi(x,z)|=0.$$ This can be done by projecting $x$ to the subspaces generated by a basis in $\Xi^*$ and then regularizing by the standard mollification techniques, see [@FuTeBE]. We know that if $(\bar Y^{x,n}, \bar Z^{x,n},\lambda_n)$ is the solution of the EBSDE (\[EBSDE\]) with $\psi$ replaced by $\psi^n$ then $\bar Y^{x,n}_t=\bar v^n(X^x_t)$ with $$|\bar v^n(x)
-\bar v^n(x')|\le \dfrac{K'_x}{\eta}|x-x'|;\quad \bar v^n(0)=0
;\quad |\lambda_n|\leq M'$$ Thus we can assume (considering, if needed, a subsequence) that $\bar v^n(x) \rightarrow \bar v(x)$ and $\lambda_n \rightarrow
\lambda$. The rest of the proof is identical to the one of Theorem \[main-EBSDE\].*
Differentiability
=================
We are now interested in the differentiability of the solution to the EBSDE (\[EBSDE\]) with respect to $x$.
\[th-diff\] Assume that Hypotheses \[general\_hyp\_forward\] and \[hypothesisroyer\] hold. Moreover assume that $F$ is of class ${\cal G}^1(E,E)$ with $\nabla F$ bounded on bounded sets of $E$. Finally assume that $\psi$ is of class ${\cal G}^1(E\times
\Xi^*,E)$. Then the function $\overline{v}$ defined in (\[def-of-v\]) is of class ${\cal G}^1(E,\mathbb{R})$.
In [@masiero] it is proved that for arbitrary $T>0$ the map $x\rightarrow X^x$ is of class $\mathcal{G}^1$ from $E$ to $L^p_{\mathcal{P}}(\Omega,C([0,T],E))$. Moreover Proposition \[prop lip X\] ensures that for all $h\in E$, $$\label{proof-diff-estim-nabla-X}
|\nabla X^x_t h|\leq e^{-\eta t}|h|,\quad \hbox{$\mathbb{P}$-a.s.,
for all $t\in [0,T]$}.$$ Under the previous conditions one can proceed exactly as in Theorem 3.1 of [@HuTe] to prove that for all $\alpha >0$ the map $v^{\alpha}$ is of class $\mathcal{G}^1$.
$ $
Then we consider again equation (\[bsderoyer\]): $${Y}^{x,\alpha}_t ={Y}^{x,\alpha}_T
+\int_t ^T(\psi(X^{x}_\sigma, Z^{x,\alpha}_\sigma)-\alpha
Y^{x,\alpha}_\sigma)d\sigma-\int_t ^T Z^{x,\alpha}_\sigma
dW_\sigma, \quad 0\le t \le T <\infty,$$ we recall that ${Y}^{x,\alpha}_T={v}^{\alpha}(X^{x}_T)$, and apply again [@masiero] (see Proposition 4.2 there) and [@FuTe1] (see Proposition 5.2 there) to obtain that for all $\alpha >0 $ the map $x\rightarrow Y^{x,\alpha}$ is of class $\mathcal{G}^1$ from $E$ to $L^2_{\mathcal{P}}(\Omega,C([0,T],\mathbb{R}))$ and the map $x\rightarrow Z^{x,\alpha}$ is of class $\mathcal{G}^1$ from $E$ to $L^2_{\mathcal{P}}(\Omega,L^2([0,T],\Xi^*))$. Moreover for all $h\in E$ it holds (for all $t>0$ since $T$ was arbitrary) $$-d\nabla Y^{\alpha,x}_th=[\nabla_x\psi(X^x_t,Z_t^{\alpha,x})
\nabla X_t^xh+\nabla_z\psi(X^x_t,Z_t^{\alpha,x})\nabla
Z_t^{\alpha,x}h-\alpha\nabla Y^{\alpha,x}_th]dt
-\nabla Z^{\alpha,x}h dW_t.$$ We also know that $|Y^{\alpha,x}_t|\le {M}/{\alpha}$. Now we set $$U^{\alpha,x}_t=e^{\eta t}\nabla Y^{\alpha,x}_t h,
\quad V^{\alpha,x}=e^{\eta t}\nabla Z^{\alpha,x}_t h.$$ Then $(U^{\alpha,x},V^{\alpha,x})$ satisfies the following BSDE: $$\begin{aligned}
-dU^{\alpha,x}_t&=&[e^{\eta t}\nabla_x\psi(X^x_t,Z_t^{\alpha,x})
\nabla X_t^x-(\alpha+\eta)U^{\alpha,x}_t +\nabla_z
\psi(X^x_t,Z_t^{\alpha,x}) V^{\alpha,x}_t]dt-V^{\alpha,x}_tdW_t.\end{aligned}$$ By (\[proof-diff-estim-nabla-X\]) and the usual Girsanov argument (recall the $\nabla_x \psi$ and $\nabla_z \psi$ are bounded), $$|U^{\alpha,x}_t|\le \frac{c}{\alpha+\eta},\;
\forall t\geq 0,\; \hbox{$\mathbb P-$a.s. $\qquad$ i.e. } \qquad
|\nabla Y_t^{x,\alpha}|\le e^{-\eta t}\frac{c}{\alpha+\eta}.$$ Moreover, consider the limit equation, with unknown $(U^{x},V^{x})$, $$\label{eq:limit}
-dU^x_t=[e^{\eta t}\nabla_x\psi(X^x_t,\bar Z_t^{x}) \nabla
X_t^x-\eta U^x_t+\nabla_z\psi(X^x_t,\bar Z_t^{x}) V^x]dt-V^xdW_t,$$ which, since $|e^{\eta t}\nabla_x\psi \nabla_x X_t^x|$ is bounded, has a unique solution such that $U^x$ is bounded and $V^x$ belongs to $L_{\cal P, {\rm loc}}^2(\Omega;L^2(0,\infty;\Xi^*))$ (see [@bh] and [@royer]).
We know that for a suitable sequence $\alpha_n \searrow 0$, $$\bar v^{\alpha}(x)= Y^{x,\alpha_n}_0-Y^{0,\alpha_n}_0\rightarrow \bar{Y}^x_0,$$ and we claim now that $$\nabla \bar v^{\alpha_n}(x)=\nabla Y_0^{x,\alpha_n}=U_0^{x,\alpha_n}
\rightarrow U_0^x.$$ To prove this we introduce the finite horizon equations: for $t\in [0,N]$, $$\begin{cases}
& -dU_t^{x,\alpha,N}=[e^{\eta t}\nabla_x\psi(X^x_t,Z_t^{x,\alpha})
\nabla X_t^x-(\alpha+\eta)U_t^{x,\alpha,N}
+\nabla_z \psi (X^x_t,Z_t^{x,\alpha}) V_t^{x,\alpha,N}]dt\\
& \qquad\qquad\qquad - V^{x,\alpha,N}_tdW_t,\\
& U_N^{x,\alpha,N}=0.
\end{cases}$$ $$\begin{cases}& -dU_t^{x,N}=[e^{\eta t}\nabla_x\psi(X^x_t,\bar Z_t^{x})
\nabla X_t^x-(\alpha+\eta)U_t^{x,N}
+\nabla_z \psi (X^x_t,\bar Z^{x}_t) V_t^{x,N}]dt-V^{x,N}_tdW_t,\\
& U_N^{x,N}=0.
\end{cases}$$ Since $\displaystyle \E\int_0^N |Z^{x,\alpha_n}_s-\bar Z^{x}_s|^2
ds\rightarrow 0$ it is easy to verify that, for all fixed $N>0$, $U_0^{x,\alpha_n,N}\rightarrow U_0^{x,N}$.
On the other side a standard application of Girsanov Lemma gives see [@HuTe], $$|U_0^{x,\alpha_n,N}-U_0^{x,\alpha_n}|\le \frac{c}{\alpha_n+\eta}e^{-\eta N}, \qquad |U_0^{x,N}-U_0^{x}|\le \frac{c}{\eta}e^{-\eta N}.$$ for a suitable constant $c$.
Thus a standard argument implies $U_0^{x,\alpha_n}\rightarrow
U_0^{x}$. An identical argument also ensures continuity of $U_0^{x}$ with respect to $x$ (also taking into account \[lip-of-Z\]). The proof is therefore completed.
$ $
As usual in the theory of markovian BSDEs, the differentiability property allows to identify the process $\bar Z^x$ as a function of the process $X^x$. To deal with our Banach space setting we need to make the following extra assumption:
\[Hyp-masiero\] There exists a Banach space $\Xi_0$, densely and continuously embedded in $\Xi$, such that $G\, (\Xi_0) \subset \Xi$ and $G
:\Xi_0 \rightarrow E$ is continuous.
We note that this condition is satisfied in most applications. In particular it is trivially true in the special case $E=H$ just by taking $\Xi_0=\Xi$, since $G$ is assumed to be a linear bounded operator from $\Xi$ to $H$. The following is proved in [@masiero Theorem 3.17]:
\[theorem-identif-Z\] Assume that Hypotheses \[general\_hyp\_forward\], \[hypothesisroyer\] and \[Hyp-masiero\] hold. Moreover assume that $F$ is of class ${\cal G}^1(E,E)$ with $\nabla F$ bounded on bounded subsets of $E$ and $\psi$ is of class ${\cal G}^1(E\times \Xi^*,E)$. Then $\bar Z^x_t=\nabla \bar
v(X^x_t)G$, $\mathbb{P}$-a.s. for a.e. $t\geq 0$.
\[precision\]
We notice that $\nabla \bar v(x)G\xi$ is only defined for $\xi\in
\Xi_0$ in general, and the conclusion of Theorem \[theorem-identif-Z\] should be stated more precisely as follows: for $\xi\in
\Xi_0$ the equality $Z^x_t\xi=\nabla \bar v(X^x_t)G\xi$ holds $\mathbb{P}$-a.s. for almost every $t\geq 0$. However, since $\bar Z^x$ is a process with values in $\Xi^*$, and more specifically a process in $
L^2_{\mathcal{P}}(\Omega,L^2([0,T],\Xi^*))$, it follows that $\P$-a.s. and for almost every $t$ the operator $\xi \rightarrow \nabla \bar v(X^x_t)G\xi$ can be extended to a bounded linear operator defined on the whole $\Xi$. Equivalently, for almost every $t$ and for almost all $x\in E$ (with respect to the law of $X_t$) the linear operator $\xi \rightarrow \nabla \bar v(x)G\xi$ can be extended to a bounded linear operator defined on the whole $\Xi$ (see also Remark 3.18 in [@masiero]).
\[boundedpsibar\]
The above representation together with the fact that $\bar v$ is Lipschitz with Lipschitz constant $K_x\eta^{-1}$ immediately implies that, if $F$ is of class ${\cal G}^1(E,E)$ and $\psi$ is of class ${\cal G}^1(E\times \Xi^*,E)$, then $|\bar
{Z}^x_t|_{\Xi_0^*}\leq K_x\eta^{-1} |G|_{L(\Xi_0,E)}$ for all $x\in
E$, $\mathbb{P}$-a.s. for almost every $t\geq 0$. Consequently we can construct $\bar \zeta$ in Theorem \[main-EBSDE\] in such a way that it is bounded in the $\Xi_0^*$ norm by $K_x\eta^{-1} |G|_{L(\Xi_0,E)}$.
Once this is proved we can extend the result to the case in which $\psi$ is no longer differentiable but only Lipschitz, namely we can prove than even in this case the process $\bar
{Z}^x$ is bounded. Indeed if we consider a sequence $\{\psi_n : n\in \mathbb{N}\}$ of functions of class ${\cal G}^1(E\times \Xi^*,E)$ such that for all $x,x'\in H$, $z,z'\in \Xi^*$, $n\in \mathbb{N}$, $$|\psi_n(x,z)
-\psi_n(x',z')|\le K_x|x-x'|+ K_z |z-z'|;\quad \lim_{n\rightarrow
\infty}|\psi_n(x,z) -\psi(x,z)|=0.$$ We know that if $(\bar Y^{x,n}, \bar Z^{x,n},\lambda_n)$ is the solution of the EBSDE (\[EBSDE\]) with $\psi$ replaced by $\psi_n$ then $|\bar {Z}^{x,n}_t|_{\Xi_0^*}\leq K_x\eta^{-1}
|G|_{L(\Xi_0,E)}$. Then as we did above we can show (showing that the corresponding equations with monotonic generator converge uniformly in $\alpha$) that $\mathbb{E}\int_0^T|\bar {Z}^{x,n}_t
-\bar {Z}^{x}_t|_{\Xi_0^*}^2dt\rightarrow 0$ and the claim follows.
We also notice that by the same argument we also have $ |\bar
\zeta^{\alpha}(x)|_{\Xi_0^*}\leq K_x\eta^{-1} |G|_{L(\Xi_0,E)}$, $\forall \alpha>0$.
Now we introduce the Kolmogorov semigroup corresponding to $X$: for measurable and bounded $\phi:
E\rightarrow \mathbb{R}$ we define $$\label{def-of-p}
P_t[\phi](x)=\mathbb{E}\, \phi(X^x_t)\qquad t\ge 0,\, x\in E.$$
\[strongly-feller\] The semigroup $(P_t)_{t\geq 0}$ is called strongly Feller if for all $t>0$ there exists $k_t$ such that for all measurable and bounded $\phi: E\rightarrow \mathbb{R}$, $$|
P_t[\phi](x)- P_t[\phi](x')|\leq k_t \Vert\phi\Vert_0 |x-x'|,
\qquad x,x'\in E,$$ where $\Vert\phi\Vert_0=\sup_{x\in E}|\phi(x)|$.
This terminology is somewhat different from the classical one (namely, that $P_t$ maps measurable bounded functions into continuous ones, for all $t>0$), but it will be convenient for us.
\[gen-diss\] We say that $F$ is genuinely dissipative if there exist $\epsilon>0$ and $c>0$ such that, for all $x,x'\in E$, there exists $z^*\in \partial
|x-x'|$ such that $<z^*,F(x)-F(x')>_{E^*,E}\leq c
|x-x'|^{1+\epsilon}$.
\[lemma-SF-dissip\] Assume that Hypotheses \[general\_hyp\_forward\] and \[hyp\_W\_A F(W\_A)\] hold. If the Kolmogorov semigroup $(P_t)$ is strongly Feller then for all bounded measurable $\phi: E\rightarrow\mathbb{R}$, $$\left|P_t[\phi](x)-\int_E \phi(x)\mu (dx)\right|
\leq c e^{-\eta (t/4)}(1+|x|)\Vert\phi\Vert_0.$$ If in addition $F$ is genuinely dissipative then $$\left|P_t[\phi](x)-\int_E \phi(x)\mu (dx)\right|
\leq c e^{-\eta (t/4)}\Vert\phi\Vert_0.$$
We fix $\epsilon >0$. For $t>2$ we have, by Theorem \[ergodicity\], $$\begin{array}{r}\displaystyle\left|P_t[\phi](x)-\int_E \phi(x)\mu (dx)\right|=
\left|P_{t-1}[P_1[\phi]](x)-\int_E P_{1}[\phi](x)\mu (dx)\right|
\leq C(1+|x|)
e^{-\eta t /4} \Vert P_{1}[\phi]\Vert_{\hbox{lip}}\\
\displaystyle \leq C(1+|x|) e^{-\eta t /4} k_{1}\Vert\phi\Vert_0,
\end{array}$$ and the first claim follows since $\left|P_t[\phi](x)-\int_E
\phi(x)\mu (dx)\right|\leq 2 \Vert\phi\Vert_0$.
If now $F$ is genuinely dissipative then in [@DP2], Theorem 6.4.1 it is shown that $$\left|\mathbb{E}\phi(X^x_t)-\int_E \phi\, d\mu \right|\leq
C e^{-\eta t /2} \Vert\phi\Vert_{\hbox{lip}}$$ and the second claim follows by the same argument.
We are now able to state and prove two corollaries of Theorems \[th-diff\] and \[theorem-identif-Z\].
\[characterization of lambda\] Assume that Hypotheses \[general\_hyp\_forward\], \[hyp\_W\_A F(W\_A)\], \[hypothesisroyer\] and \[Hyp-masiero\] hold. Moreover assume that $F$ is of class $\mathcal{G}^1$ with $\nabla
F$ bounded on bounded subsets of $E$, and that $\psi$ is bounded on each set $E\times B$, where $B$ is any ball of $\Xi_0^*$. Finally assume that the Kolmogorov semigroup $(P_t)$ is strongly Feller.
Then the following holds: $$\lambda=\int_E \psi(x,\bar \zeta(x))\mu (dx),$$ where $\mu$ is the unique invariant measure of $X$.
First notice that $\overline{\psi}:=
\psi(\,\cdot\, , \bar\zeta(\,\cdot\,))$ is bounded, by Remark \[boundedpsibar\]. Then $$T^{-1}\mathbb{E}[\bar Y ^x_0-\bar Y ^x_T]=
T^{-1}\E \int_0^T\left (\psi(X^x_t,\bar \zeta( X^x_t))- \int_E
\bar \phi\, d\mu \right)dt+ \left(\int_E \bar \phi\, d\mu
-\lambda\right).$$ We know that $T^{-1}\mathbb{E}[\bar Y ^x_0-\bar
Y ^x_T]\rightarrow 0$, by the argument in Theorem \[th-uniq-lambda\]. Moreover by the first conclusion of Lemma \[lemma-SF-dissip\] $$T^{-1}\E \int_0^T\left (\psi(X^x_t,\bar \zeta( X^x_t))-
\int_E \bar \phi\, d\mu \right)dt \rightarrow 0,$$ and the claim follows.
\[boundedness of v\] In addition to the assumptions of Corollary \[characterization of lambda\] suppose that $F$ is genuinely dissipative. Then $\bar
v$ is bounded.
Let $(Y^{x,\alpha},Z^{x,\alpha})$ be the solution of (\[bsderoyer\]). We know that $Y^{x,\alpha}_t=v^{\alpha}(X^x_t)$ and $Z^{x,\alpha}_t= \zeta^{\alpha}(X^x_t)$ with $v^{\alpha}$ Lipschitz uniformly with respect to $\alpha$ and $\zeta^{\alpha}$ bounded in $\Xi^*$ uniformly with respect to $\alpha$. Let $\psi^{\alpha}=\psi(\,\cdot\,,\bar \zeta^{\alpha}(\,\cdot\,))$. Under the present assumptions we conclude that also the maps $\psi^{\alpha}$ as well are bounded in $\Xi^*$ uniformly with respect to $\alpha$.
Computing $d (e^{-\alpha t} \bar Y^{x\alpha}_t)$ we obtain, $$Y^{x,\alpha}_0=\mathbb{E} e^{-\alpha T} Y^{x,\alpha}_T+
\mathbb{E} \int_0^T e^{-\alpha t} \psi^{\alpha} (X^x_t)dt,$$ and for $T\rightarrow\infty$, $$Y^{x,\alpha}_0=
\mathbb{E} \int_0^\infty e^{-\alpha t} \psi^{\alpha} (X^x_t)dt.$$ Subtracting to both sides $\alpha^{-1}\int_E
\psi^{\alpha}(x)\mu(dx)$ we obtain $$\left|Y^{x,\alpha}_0-\alpha^{-1}\int_E \psi^{\alpha}(x)\mu(dx)\right|=
\left| \int_0^\infty e^{-\alpha t} \left[P_t[\psi^{\alpha}]
(x)-\int_E \psi^{\alpha}(x)\mu(dx)\right]dt\right|\leq 4c
\eta^{-1} \Vert \psi^\alpha\Vert_0$$ where the last inequality comes from the second conclusion of Lemma \[lemma-SF-dissip\].
Thus $\left|Y^{x,\alpha}_0-Y^{0,\alpha}_0\right| \leq 8 c
\eta^{-1} \Vert \psi^\alpha\Vert_0 $ and the claim follows since by construction $Y^{x,\alpha}_0-Y^{0,\alpha}_0 \rightarrow \bar v
(x)$.
Ergodic Hamilton-Jacobi-Bellman equations
=========================================
We briefly show here that if $\bar Y_0^x=\bar v(x)$ is of class ${\cal G}^1$ then the couple $(v,\lambda)$ is a mild solution of the following “ergodic” Hamilton-Jacobi-Bellman equation: $$\mathcal{L}v(x)
+\psi\left( x,\nabla v(x) G\right) = \lambda, \quad x\in E, \label{hjb}$$ Where linear operator $\mathcal{L}$ is formally defined by $$\mathcal{L}f\left( x\right) =\frac{1}{2}Trace\left(
GG^{\ast}\nabla ^{2}f\left( x\right) \right) +\langle Ax,\nabla
f\left( x\right) \rangle_{E,E^{\ast}}+\langle F\left( x\right)
,\nabla f\left( x\right) \rangle_{E,E^{\ast}},$$ We notice that we can define the transition semigroup $(P_t)_{t\geq 0}$ corresponding to $X$ by the formula (\[def-of-p\]) for all measurable functions $\phi:E\to\mathbb{ R}$ having polynomial growth, and we notice that $\mathcal{L}$ is the formal generator of $(P_t)_{t\geq 0}$.
Since we are dealing with an elliptic equation it is natural to consider $(v,\lambda)$ as a mild solution of equation (\[hjb\]) if and only if, for arbitrary $T>0$, $v(x)$ coincides with the mild solution $u(t,x)$ of the corresponding parabolic equation having $v$ as a terminal condition: $$\left\{
\begin{array}{l}
\dfrac{\partial u(t,x)}{\partial t}+\mathcal{L}u\left( t,x\right)
+\psi\left( x,\nabla u\left( t,x\right) G\right)
-\lambda=0, \quad t\in [0,T],\; x\in E, \\ \\
u(T,x)=v(x), \quad x\in E.
\end{array}\right. \label{hjb-parab}$$ Thus we are led to the following definition (see also [@FuTe-ell]):
\[defsolmildkolmo\] A pair $(v,\lambda)$ ($v: E\rightarrow
\mathbb{R}$ and $\lambda\in \mathbb{R}$) is a mild solution of the Hamilton-Jacobi-Bellman equation (\[hjb\]) if the following are satisfied:
1. $v\in\mathcal{G}^{1}\left( E,\R \right) $;
2. there exists $C>0$ such that $\left| \nabla v\left(
x\right)
h\right| \leq C\left| h\right| _{E}\left( 1+\left| x\right| _{E}^{k}\right) $ for every $x,h\in E$ and some positive integer $k$;
3. for $0\le t\le T$ and $x\in E$, $$v(x)=P_{T-t}\left[ v\right] \left( x\right)
+\int_{t}^{T}\left(P_{s -t }\left[ \psi(\cdot,\nabla v\left(
\cdot\right) G)\right] \left( x\right) -\lambda \right) \,ds.
\label{mild sol hjb}$$
In the right-hand side of (\[mild sol hjb\]) we notice occurrence of the term $\nabla v\left( \cdot\right) G$, which is not well defined as a function $E\to\Xi^*$, since $G$ is not required to map $\Xi$ into $E$. The situation is similar to Remark \[precision\]. In general, for $x \in E$, $\nabla \bar
v(x)G\xi$ is only defined for $\xi\in \Xi_0$. In (\[mild sol hjb\]) it is implicitly required that, $\P$-a.s. and for almost every $t$, the operator $\xi \rightarrow \nabla \bar v(X^x_t)G\xi$ can be extended to a bounded linear operator defined on the whole $\Xi$. Noting that $$P_{t }\left[ \psi(\cdot,\nabla v\left( \cdot\right) G)\right]
\left( x\right) = \E \, \psi(X^x_{t},\nabla v\left( X^x_{t}\right)
G)$$ the equation (\[mild sol hjb\]) is now meaningful.
Using the results for the parabolic case, see [@masiero], we get existence of the mild solution of equation (\[hjb\]) whenever we have proved that the function $\bar v$ in Theorem \[main-EBSDE\] is differentiable.
\[th-EHJB\] Assume that Hypotheses \[general\_hyp\_forward\], \[hypothesisroyer\] and \[Hyp-masiero\] hold. Moreover assume that $F$ is of class ${\cal G}^1(E,E)$ with $\nabla F$ bounded on bounded subsets of $E$ and $\psi$ is of class ${\cal G}^1(E\times \Xi^*,E)$.
Then $(\bar v, \bar\lambda)$ is a mild solution of the Hamilton-Jacobi-Bellman equation (\[hjb\]).
Conversely, if $(v,\lambda)$ is a mild solution of (\[hjb\]) then, setting $ Y^x_t=
v(X^x_t)$ and ${Z}^{x}_t=
\nabla v( X^x_t) G$, the triple $( Y^x, Z^x, \lambda)$ is a solution of the EBSDE (\[EBSDE\]).
Optimal ergodic control {#optcontr}
=======================
Assume that Hypothesis \[general\_hyp\_forward\] holds and let $X^x$ denote the solution to equation (\[sde\]). Let $U$ be a separable metric space. We define a control $u$ as an $(\calf_t)$-progressively measurable $U$-valued process. The cost corresponding to a given control is defined in the following way. We assume that the functions $R:U\rightarrow \Xi^*$ and $L:E\times U \rightarrow \R$ are measurable and satisfy, for some constant $c>0$, $$\label{condcosto}
|R(u)|\leq c,\quad |L(x,u)|\leq c, \quad |L(x,u)-L(x',u)|\leq
c\,|x-x'|,\qquad u\in U,\,x,x'\in E.$$ Given an arbitrary control $u$ and $T>0$, we introduce the Girsanov density $$\rho_T^u=\exp\left(\int_0^T R(u_s)dW_s
-\frac{1}{2}\int_0^T |R(u_s)|_{\Xi^*}^2 ds\right)$$ and the probability $\mathbb P_T^u=\rho_T^u\mathbb P$ on $\calf_T$. The ergodic cost corresponding to $u$ and the starting point $x\in E$ is $$\label{def-ergodic-cost}
J(x,u)=\limsup_{T\rightarrow\infty}\frac{1}{T} \mathbb
E^{u,T}\int_0^T L(X_s^x,u_s)ds,$$ where $\mathbb E^{u,T}$ denotes expectation with respect to $\mathbb P_T^u$. We notice that $W_t^u=W_t-\int_0^t R(u_s)ds$ is a Wiener process on $[0,T]$ under $\mathbb P^u$ and that $$dX_t^x=(AX_t^x+F(X_t^x))dt+G(dW_t^u+R(u_t)dt),
\quad t\in [0,T]$$ and this justifies our formulation of the control problem. Our purpose is to minimize the cost over all controls.
To this purpose we first define the Hamiltonian in the usual way $$\label{defhamiton}
\psi(x,z)=\inf_{u\in U}\{L(x,u)+z R(u)\},\qquad x\in E,\,z\in
\Xi^*,$$ and we remark that if, for all $ x,z$, the infimum is attained in (\[defhamiton\]) then there exists a measurable function $\gamma:E\times \Xi^*\rightarrow U$ such that $$\psi(x,z)=l(x,\gamma(x,z))+z R(\gamma(x,z)).$$ This follows from an application of Theorem 4 of [@McS-War].
We notice that under the present assumptions $\psi$ is a Lipschitz function and $\psi(\cdot,0)$ is bounded (here the fact that $R$ depends only on $u$ is used). So if we assume Hypotheses \[general\_hyp\_forward\] and \[hyp\_W\_A F(W\_A)\] then in Theorem \[main-EBSDE\] we have constructed, for every $x\in E$, a triple $$\label{richiamoebsde}
(\bar Y^x,\bar Z^x, \bar \lambda)= (\bar v (X^x),\bar \zeta(X^x),
\bar \lambda)$$ solution to the EBSDE (\[EBSDE\]).
\[Th-main-control\] Assume that Hypotheses \[general\_hyp\_forward\], \[hyp\_W\_A F(W\_A)\] and \[hyp-convol-determ\] hold, and that (\[condcosto\]) holds as well.
Moreover suppose that, for some $x\in E$, a triple $(Y,Z,\lambda)$ verifies $\mathbb{P}$-a.s. equation (\[EBSDE\]) for all $0\leq t\leq T$, where $Y$ is a progressively measurable continuous process, $Z$ is a process in $L_{\cal P, {\rm loc}}^2(\Omega;L^2(0,\infty;\Xi^*))$ and $\lambda\in \mathbb{R}$. Finally assume that there exists $c_x>0$ (that may depend on $x$) such that $\mathbb{P}$-a.s. $$|Y_t|\leq c_x (|X^x_t|+1) , \hbox{ for all $t\geq 0$}.$$
Then the following holds:
1. For arbitrary control $u$ we have $J(x,u)\ge \lambda=\bar\lambda,$ and the equality holds if and only if $L(X_t^x,u_t)+Z_t
R(u_t)=\psi(X_t^x,Z_t)$, $\P$-a.s. for almost every $t$.
2. If the infimum is attained in (\[defhamiton\]) then the control $\bar u_t=\gamma(X_t^x,Z_t)$ verifies $J(x,\bar u)=
\bar\lambda.$
In particular, for the solution (\[richiamoebsde\]) mentioned above, we have:
1. For arbitrary control $u$ we have $J(x,u)=\bar\lambda$ if and only if $L(X_t^x,u_t)+\bar\zeta (X_t^x) R(u_t)=\psi(X_t^x,\bar \zeta
(X_t^x))$, $\P$-a.s. for almost every $t$.
2. If the infimum is attained in (\[defhamiton\]) then the control $\bar
u_t=\gamma(X_t^x,\bar\zeta (X_t^x))$ verifies $J(x,\bar u)=
\bar\lambda.$
**
1. The equality $\lambda=\bar\lambda$ clearly follows from Theorem \[th-uniq-lambda\].
2. Points $(iii)$ and $(iv)$ are immediate consequences of $(i)$ and $(ii)$.
3. The conclusion of point $(iv)$ is that there exists an optimal control in feedback form, with the optimal feedback given by the function $x\mapsto \gamma(x,\bar\zeta (x))$.
4. Under the conditions of Theorem \[th-EHJB\], the pair $(\bar v, \bar \lambda)$ occurring in (\[richiamoebsde\]) is a mild solution of the Hamilton-Jacobi-Bellman equation (\[hjb\]).
5. It follows from the proof below that if $\limsup$ is changed into $\liminf$ in the definition (\[def-ergodic-cost\]) of the cost, then the same conclusions hold, with the obvious modifications, and the optimal value is given by $\bar\lambda$ in both cases.
As $(Y,{Z}, \bar\lambda)$ is a solution of the ergodic BSDE, we have $$\begin{aligned}
-d{Y}_t&=&[\psi(X_t^x,{Z}_t)-\bar\lambda]dt-{Z}_tdW_t\\
&=&[\psi(X_t^x,{Z}_t)- \bar\lambda]dt-{Z}_tdW_t^u-{Z}_t R(u_t)dt,\end{aligned}$$ from which we deduce that $$\begin{aligned}
\bar\lambda&=&\frac{1}{T}\mathbb E^{u,T}[Y_T-Y_0]
+\mathbb E^{u,T}\frac{1}{T}\int_0^T[\psi(X_t^x,{Z}_t)-{Z}_t r(u_t)-L(X_t^x,{Z}_t)]dt\\
& &+\frac{1}{T}\mathbb E^{u,T}\int_0^T L(X_t^x,{Z}_t)dt.\end{aligned}$$
Thus $$\frac{1}{T}\mathbb E^{u,T}\int_0^T L(X_t^x,{Z}_t)dt\ge
\frac{1}{T}\mathbb E^{u,T}[Y_0-Y_T]+\bar\lambda.$$ But by (\[rel-estimate-Xgamma\]) we have $$|\mathbb E^{u,T} Y_T|\le c\mathbb E^{u,T}(|X_T^x|+1)\le c(1+|x|).$$ Consequently $T^{-1}\mathbb E^{u,T}[Y_0-Y_T]\rightarrow 0,$ and $$\limsup_{T\rightarrow\infty } \frac{1}{T}\mathbb E^{u,T}\int_0^T L(X_t^x,{Z}_t)dt
\ge \bar\lambda.$$
Similarly, if $L(X_t^x,u_t)+ Z_t R(u_t)=\psi(X_t^x,Z_t)$, $$\frac{1}{T}\mathbb E^{u,T}\int_0^T L(X_t^x,{Z}_t)dt=
\frac{1}{T}\mathbb E^{u,T}[Y_0-Y_T]+\bar\lambda,$$ and the claim holds.
Uniqueness {#sec-uniq}
==========
We wish now to adapt the argument in [@GoMa] in order to obtain uniqueness of markovian solutions to the EBSDE. This will be done by a control thoretic interpretation the requires that the Markov process related to the state equation with continuous feedback enjoys recurrence properties. In this section we assume $$\label{addizionali}
E=H \qquad\hbox{ and }\qquad F \hbox{ is bounded.}$$
We recall here a result due to [@seid] on recurrence of solution to SDEs.
\[th-rec-seidler\] Consider $$\label{eq:u}
d{X}_t=(A{X}_t+g({X}_t))dt+GdW_t.$$ where $g: H \rightarrow H$ is bounded and weakly continuous (that if $x\rightarrow\<\xi,g(x)\>$ is continuous for all $\xi\in H$). Let $$Q_t=\int_0^t e^{sA}GG^*e^{sA^*}ds.$$ and assume the following
1. $\sup_{t\ge 0} \hbox{Trace}\,(Q_t)<\infty$;
2. $Q_t$ is injective for $t>0$;
3. $ e^{t A}(H)\subset
(Q_t)^{1/2}(H)$ for $t>0$;
4. $\int_0^t
|Q_s^{-1/2}e^{sA}|ds<\infty$ for $t>0$;
5. there exists $\beta>0$ such that $\int_0^t s^{-\beta}\,
\hbox{Trace}\,(S(s)S(s)^*)\, ds<\infty$ for $t>0$.
Then, for all $T>0$, equation (\[eq:u\]) admits a martingale solution on $[0,T]$, unique in law. The associated transition probabilities $P(t,x,T,\cdot)$ on $H$ ($0\le t\le T, x\in H$) identify a recurrent Markov process on $[0,\infty)$.
Consider now the ergodic control problem with state equation: $$d{X}^{x,u}_t=(A{X}^{x,u}_t+F({X}^{x,u}_t)+GR(u_t))dt+GdW_t, \ X_0^{x,u}=x,$$ and cost $$\limsup_{T\to\infty}\frac{1}{T}\,
\mathbb E\int_0^T l(X_s,u_s)ds$$ where $R:U\rightarrow
\Xi$ is continuous and bounded.
We restrict ourselves to the class of controls given by continuous feedbacks, i.e. given arbitrary continuous $u: H\rightarrow U$ (called feedback) we define the corresponding trajectory as the solution of $$d{X}^{x,u}_t=(A{X}^{x,u}_t+F({X}^{x,u}_t))dt+G(R(u(X_t^{x,u}))dt+dW_t),
\ X_0^{u,x}=x.$$ We notice that for all $T>0$ there exists a weak solution $X^{x,u}$ of this equation, and it is unique in law.
$ $
We set as usual $$\psi(x,z)=\inf_{u\in U}\{L(x,u)+zR(u)\},$$ and assume that $\psi$ is continuous and there exists a continuous $\gamma:H\times\Xi\rightarrow U$ such that $$\psi(x,z)=L(x,\gamma(x,z))+zR(\gamma(x,z)).$$
\[th-uniqueness\] Suppose (\[addizionali\]) and suppose that the assumptions of Theorem \[th-rec-seidler\] hold. Let $(v,\zeta,\lambda)$ with $v:H\rightarrow \mathbb{R}$ continuous, $\zeta:H\rightarrow \mathbb{R}$ continuous, and $\lambda$ a real number satisfy the following conditions:
1. $|v(x)|\le c|x|$;
2. for an arbitrary filtered probability space with a Wiener process $(\hat{\Omega},\hat{\mathcal{F}},
\{\hat{\mathcal{F}}_t\}_{t>0},\hat{\mathbb{P}},\{
\hat{W}_t\}_{t>0})$ and for any solution of $$d\hat{X}_t=(A\hat{X}_t+F(\hat{X}_t))dt+Gd\hat{W}_t,\qquad t\in [0,T],$$ setting $Y_t=v(\hat{X}_t),\
Z_t=\zeta(\hat{X}_t)$, we have $$-dY_t=[\psi(\hat{X}_t,Z_t)-\lambda]dt-Z_tdW_t\quad t\in [0,T].$$
Let $$\tau_r^T=\inf \{s\in [0,T]:|X_s^{u,x}|< r\},$$ with the convention $\tau_r^T=T$ if the indicated set is empty, and $$J(x,u)=\limsup_{r\rightarrow 0}\limsup_{T\rightarrow \infty}
\mathbb E\int_0^{\tau_r^T} [\psi(X_s^{x,u},u(X_s^{x,u}))-\lambda]ds.$$ Then $$v(x)=\inf_u J(x,u),$$ where the infimum (that is a minimum) is taken over all continuous feedbacks $u$.
Let $u:H\to U$ be continuous. We notice that $X^{x,u}$ solves on $[0,T]$: $$dX_t^{x,u}=(AX_t^{x,u}+F(X_t^{x,u}))ds+Gd\tilde{W}_t^u,\ t\in [0,T],$$ where $\tilde{W}_t=\int_0^t R(u(X_r^{x,u})dr+W_t$ is a Wiener process on $[0,T]$ under a suitable probability $\hat{\mathbb{P}}^{u,T}$.
Therefore $Y_t=v(X_t^{x,u})$, $Z_t=\zeta(X_t^{x,u})$ satisfy: $$-dY_t=[\psi(X_t^{x,u},u(X_t^{x,u}))-\lambda]dt-Z_t
R(u(X_t^{x,u}))]dt-Z_tdW_t.$$ Integrating in $[0,\tau_r^T]$ we get $$v(x)=\mathbb E(v(X_{\tau_r^T}^{x,u}))+\mathbb E\int_0^{\tau_r^T}
[\psi(X_s^{u,x},u(X_s^{x,u}))-\lambda-Z_s R(X_s^{x,u})]ds.$$ Thus, $$\label{eq:x}
v(x)\le\mathbb E(v(X_{\tau_r^T}^{x,u}))+\mathbb E\int_0^{\tau_r^T}
[L(X_s^{u,x},u(X_s^{x,u}))-\lambda]ds.$$ Now $$\begin{aligned}
|\mathbb E(v(X_{\tau_r^T}^{x,u}))|\le c\mathbb
E|X^{x,u}_{\tau_r^T}|&\le& c r+ (\mathbb
E(|X_T^{x,u}|^2))^{1/2}(\mathbb P(\tau_r^T=r))^{1/2}\\ &\le & c r+
c (\mathbb P(\tau_r^T=r))^{1/2}\end{aligned}$$ Notice that $\mathbb P(\tau_r^T=r)=\tilde{\mathbb P}(\inf_{t\in
[0,T]}|\tilde{X}_t|\geq r),$ where $\tilde{X}$ is the Markov process on the whole $[0,+\infty)$ corresponding to the equation (\[eq:u\]) with $g=F(\cdot)+GR(u(\cdot))$.
$ $
Since $\tilde{X}$ is recurrent, for all $ r>0$ it holds $\tilde{\mathbb P}(\inf_{t\in [0,T]}|\tilde{X}_t|>r)\rightarrow 0$ as $T\rightarrow \infty.$ Thus $$\limsup_{r\rightarrow 0}\limsup_{T\rightarrow \infty}|
\mathbb E(v(X_{\tau_r^T}^{x,u}))|\rightarrow 0.$$ Hence, $$v(x)\le \limsup_{r\rightarrow 0}\limsup_{T\rightarrow \infty}
\mathbb E\int_0^{\tau_r^T}
[l(X_s^{x,u},u(X_s^{x,u}))-\lambda]ds.$$ The proof is completed noticing that if $u$ is chosen as ${u}(x)=\gamma(x,\zeta(x))$ then the above inequality becomes an equality.
This result combines with Theorems \[th-uniq-lambda\] and \[th-EHJB\] to give the following
\[HJB-uniqueness\] Suppose that all the assumptions of Theorems \[th-uniq-lambda\], \[th-EHJB\] and \[th-uniqueness\] hold. Then $(\bar v, \bar\lambda)$ is the unique mild solution of the Hamilton-Jacobi-Bellman equation (\[hjb\]) satisfying $|\bar v (x)|\le c|x|$.
Application to ergodic control of a semilinear heat equation {#section-heat-eq}
============================================================
In this section we show how our results can be applied to perform the synthesis of the ergodic optimal control when the state equation is a semilinear heat equation with additive noise. More precisely, we treat a stochastic heat equation in space dimension one, with a dissipative nonlinear term and with control and noise acting on a subinterval. We consider homogeneous Dirichlet boundary conditions.
In $\left( \Omega,\mathcal{F},\mathbb{P}\right) $ with a filtration $\left( \mathcal{F}_{t}\right) _{t\geq0}$ satisfying the usual conditions, we consider, for $t \in\left[ 0,T\right] $ and $\xi\in\left[ 0,1\right] $, the following equation $$\left\{
\begin{array}
[c]{l} d_{t }X^{u}\left( t ,\xi\right) =\left[
\frac{\partial^{2}}{\partial \xi^{2}}X^{u}\left( t ,\xi\right)
+f\left( \xi,X^{u}\left( t ,\xi\right) \right)
+\chi_{[a,b]}(\xi) u\left( t ,\xi\right) \right] dt
+\chi_{[a,b]}(\xi) \dot{W}\left(
t ,\xi\right) dt ,\\
X^{u}\left( t ,0\right) =X^{u}\left( t ,1\right) =0,\\
X^{u}\left( t,\xi\right) =x_{0}\left( \xi\right) ,
\end{array}
\right. \label{heat equation}$$ where $\chi_{[a,b]}$ is the indicator function of $[a,b]$ with $0\leq a\leq b\leq 1$; $\dot{W}\left( t ,\xi\right) $ is a space-time white noise on $\left[ 0,T\right] \times\left[
0,1\right] $.
We introduce the cost functional $$J\left( x,u\right) = \limsup_{T\rightarrow\infty}\dfrac{1}{T}
\mathbb{E}\int_{0}^{T}\int_{0}^{1}l\left( \xi ,X^{u}_s\left(
\xi\right) ,u_s(\xi)\right) \mu\left( d\xi\right) \, ds,
\label{heat costo diri}$$ where $\mu$ is a finite Borel measure on $\left[ 0,1\right] $. An admissible control $u\left( t ,\xi\right) $ is a predictable process such that for all $t \geq 0$, and $\mathbb{P}$-a.s. $u\left( t ,\cdot\right)
\in U:=\{v\in C\left( \left[ 0,1\right] \right) :\left\vert
v\left( \xi\right) \right\vert \leq\delta\}$. We denote by $\mathcal{U}$ the set of such admissible controls. We wish to minimize the cost over $\mathcal{U}$, adopting the formulation of Section \[optcontr\], i.e. by a change of probability in the form of (\[def-ergodic-cost\]). The cost introduced in (\[heat costo diri\]) is well defined on the space of continuous functions on the interval $\left[ 0,1\right] $, but for an arbitrary $\mu$ it is not well defined on the Hilbert space of square integrable functions.
We suppose the following:
\[heatipotesi\]
1. $f:\left[ 0,1\right] \times\mathbb{R} \to\mathbb{R}$ is continuous and for every $\xi\in\left[0,1\right] $, $ f(\xi,\,\cdot\,)$ is decreasing. Moreover there exist $C>0$ and $m>0$ such that for every $\xi\in\left[0,1\right] ,$ $x\in\mathbb{R}$, $$|f\left(
\xi,x\right)|\leq C(1+|x|)^m, \qquad f\left( 0,x\right)= f\left(
1,x\right)=0.$$
2. $l:\left[ 0,1\right] \times\mathbb{R} \times
[-\delta,\delta]\rightarrow\mathbb{R}$ is continuous and bounded, and $l(\xi,\cdot,u)$ is Lipschitz continuous uniformly with respect to $\xi \in\left[ 0,1\right]$, $u\in [-\delta,\delta]$.
3. $x_{0}\in C\left( \left[ 0,1\right] \right) $, $x_{0}(0)=x_{0}(1)=0$.
To rewrite the problem in an abstract way we set $H=\Xi=L^{2}\left( 0,1 \right) $ and $E=C_0\left(\left[ 0,1\right] \right)
=\{y\in C\left(\left[ 0,1\right] \right)\,:\, y(0)=y(1)=0\} $. We define an operator $A$ in $E$ by $$D\left( A\right) =\{y\in C^{2}\left( \left[ 0,1\right]
\right)\,:\, y,y''\in C_{0}\left( \left[ 0,1\right] \right)\}
,\text{ \ \ \ \ }\left( Ay\right) \left( \xi\right)
=\frac{\partial^{2}}{\partial\xi^{2}}y\left( \xi\right) \text{
for }y\in D\left( A\right).$$ We notice that $A$ is the generator of a $C_0$ semigroup in $E$, admitting and extension to $H$, and $\left| e^{tA}\right|
_{L\left( E,E\right) }\leq e^{- t}$ see, for instance, Theorem 11.3.1 in [@DP2]. As a consequence, $A+ F+I$ is dissipative in $E$.
We set, for $x\in E$, $\xi \in [0,1]$, $z\in \Xi$, $u\in U$, $$F\left( x\right) \left( \xi\right) =f\left( \xi,x\left(
\xi\right) \right) ,\ \ \left( Gz\right) \left( \xi\right)
=\chi_{[a,b]}\left( \xi\right) z\left(
\xi\right) ,\ \ L\left( x,u\right)
=\displaystyle\int_{0}^{1}l\left( \xi,x\left( \xi\right) ,u\left(
\xi\right) \right) \mu\left( d\xi\right) ,
\label{heatnotazioni}$$ and let $R$ denote the canonical imbedding of $C( \left[
0,1\right])$ in $L^2( 0,1)$.
Finally $\left\{ W_{t },t \geq0\right\} $ is a cylindrical Wiener process in $H$ with respect to the filtration $\left( \mathcal{F}_{t }\right) _{t \geq0}$
$ $
It is easy to verify that Hypotheses \[general\_hyp\_forward\] and \[hyp\_W\_A F(W\_A)\] are satisfied (for the proof of point $4$ in Hypothesis \[general\_hyp\_forward\] and of Hypothesis \[hyp\_W\_A F(W\_A)\] see again [@DP2] Theorem 11.3.1.).
Moreover, see for instance [@C], for some $C>0$, $$\left| e^{tA}\right| _{L\left( H,E\right) }\leq Ct^{-1/4}, \qquad
t\in(0,1] ,$$ thus Hypothesis \[hyp-convol-determ\] holds.
Also Hypothesis \[Hyp-masiero\] is satisfied by taking $\Xi _{0}=\left\lbrace f\in C_0\left( \left[ 0,1\right]
\right):f(a)=f(b)=0 \right\rbrace $.
$ $ Clearly the controlled heat equation (\[heat equation\]) can now be written in abstract way in the Banach space $E$ as $$\left\{
\begin{array}
[c]{l}
dX_{t }^{x_0,u}=\left[ AX_{t }^{x_0,u}+F\left( X_{t
}^{x_0,u}\right) \right] dt +GRu_{t }dt +GdW_{t }\text{\ \ \ }t
\in\left[
t,T\right] \\
X^{x_0,u}_0=x_{0},
\end{array}
\right. \label{heat eq abstract}$$ and the results of the previous sections can be applied to the ergodic cost (\[heat costo diri\]) (reformulated by a change of probability in the form of (\[def-ergodic-cost\])).
In particular if we define, for all $x\in C_0([0,1])$, $z\in L^2(0,1)$, $u\in U$ (identifying $L^2(0,1)$ with its dual) $$\psi(x,z)=\inf _{u\in U}\left\{\int_0^1 l (\xi,x(\xi),u(\xi))
\mu (d\xi)+ \int_a^b z(\xi) u(\xi) d\xi\right\}$$ then there exist $\overline v: E \rightarrow \mathbb{R}$ Lipschitz continuous and with $\overline v(0)=0$, $\overline \zeta : E
\rightarrow \Xi^*$ measurable and $\overline \lambda \in
\mathbb{R}$ such that if $X^{x_0}=X^{x_0,0}$ is the solution of equation (\[heat eq abstract\]) then $(\overline v(X^{x_0}),
\overline \zeta(X^{x_0}),\overline \lambda)$ is a solution of the EBSDE (\[EBSDE\]) and the characterization of the optimal ergodic control stated in Theorem \[Th-main-control\] holds (and $\overline \lambda$ is unique in the sense of Theorem \[th-uniq-lambda\]).
$ $
Moreover if $ f$ is of class $C^1(\mathbb{R})$ (consequently $F$ will be of class ${\cal G}^1(E,E)$) and $\psi$ is of class ${\cal G}^1(E\times \Xi^*,E)$ then by Theorem \[th-diff\] $ \overline v$ is of class ${\cal G}^1(E,E)$ and, by Theorem \[th-EHJB\], it is a mild solution of the ergodic HJB equation (\[hjb\]) and it holds $\overline \zeta=\nabla \overline v G$.
$ $
Let us then consider the particular case in which $[a,b]
=[0,1]$, $f(x,\xi)=f(x)$ is of class $C^1$ with derivative having polynomial growth, and satisfies $f(0)=0$, $[f(x+h)-f(x)]h\leq - c |h|^{2+\epsilon}$ for suitable $c,\epsilon
>0$ and all $x,h\in \mathbb{R}$ (for instance, $f(x)=-x^3$). In that case the Kolmogorov semigroup corresponding to the process $X^{x_0}$ is strongly Feller, see [@C] and [@masiero2], and it is easy to verify that $F$ is genuinely dissipative (see Definition \[gen-diss\]). Moreover we can choose $\Xi_0=C_0([0,1])$ and it turns out that $\psi$ is bounded on each set $E\times B$, where $B$ is any ball of $\Xi_0^*$. Thus the claims of Corollaries \[characterization of lambda\] and \[boundedness of v\] hold true, and in particular $\overline v$ is bounded.
$ $
Finally if we assume that $\mu$ is Lebesgue measure and $f$ is bounded and Lipschitz we can choose $E=\Xi=\Xi_0=H=L^2(0,1)$. Then the assumptions of Theorem \[th-rec-seidler\] are satisfied and we can apply Theorem \[th-uniqueness\] to characterize the function $\overline v$. In particular if $f$ is of class $C^1(\mathbb{R})$ and $\psi$ is of class ${\cal G}^1(H\times \Xi^*,H)$ then $\overline v$ is the unique mild solution of the ergodic HJB equation (\[hjb\]).
[99]{}
, [On ergodic stochastic control]{}, [*Comm. Partial Differential Equations*]{} [**23**]{} (1998), 2187–2217.
, [Équations paraboliques intervenant en contrôle optimal ergodique]{}, [*Mat. Apl. Comput.*]{} [**6**]{} (1987), 211–255.
. [On [B]{}ellman equations of ergodic control in [$\mathbb{ R}\sp n$]{}]{}, [*J. Reine Angew. Math.*]{} [**429**]{} (1992), 125–160.
, Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, [*J. Funct. Anal.*]{} [**155**]{} (1998), 455–494.
S. Cerrai, *Second order PDE’s in finite and infinite dimension, A probabilistic approach.* Lecture Notes in Mathematics, 1762. Springer-Verlag, Berlin, 2001.
G. Da Prato and J. Zabczyk,* Stochastic equations in infinite dimensions.* Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, [1992.]{}
G. Da Prato and J. Zabczyk, *Ergodicity for infinite-dimensional systems.* London Mathematical Society Note Series, 229, Cambridge University Press, Cambridge, 1996.
T.E. Duncan, B. Maslowski and B. Pasik-Duncan, Adaptive boundary and point control of linear stochastic distributed parameter systems, [*SIAM J.Control Optim.*]{}, **32** (1994), 648-672
N. El Karoui, L. Mazliak, *Backward stochastic differential equations.* Pitman Research Notes in Mathematics Series 364. 1997.
S. Kwapień, W.A. Woyczyński, *Random series and stochastic integrals: single and multiple*. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1992.
M. Fuhrman, A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes, [*Stochastic Process. Appl.*]{} **108** (2003), 263–298.
M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, [*Ann. Probab.*]{} **30** (2002), 1397–1465.
M. Fuhrman and G. Tessitore, The Bismut-Elworthy formula for backward SDE’s and applications to nonlinear Kolmogorov equations and control, [*Stochastics Stochastic Rep.*]{} [**74**]{} (2002), 429–464.
, Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces, [*Ann. Probab.*]{} [**32**]{} (2004), 607-660.
, [Ergodic control of semilinear stochastic equations and the [H]{}amilton-[J]{}acobi equation]{}, [*J. Math. Anal. Appl.*]{}, [**234**]{} (1999), 592–631.
Y. Hu, G. Tessitore, BSDE on an infinite horizon and elliptic PDEs in infinite dimension. To appear on *NoDEA*.
F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces. To appear on *Appl. Math. Optim.*.
F. Masiero, Regularizing properties for transition semigroups and semilinear parabolic equations in Banach spaces. *Electronic Journal of Probability* Vol. 12 (2007), Paper no. 13, pages 387-419.
E.J. McShane, R.B. Warfield, [On [F]{}ilippov’s implicit functions lemma]{}, *Proc. Amer. Math. Soc.*, [**18**]{} (1967), [41–47]{}.
E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. *Systems and Control Lett.* [**14**]{} (1990) 55–61.
S. Peng, Backward stochastic differential equations and applications to optimal control. *Appl. Math. Optim.* [**27**]{} (1993) 125-144.
, BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. [*Stochastics Stochastic Rep.*]{} [**76**]{} (2004), 281–307.
, [ Ergodic behaviour of stochastic parabolic equations]{}, [*Czechoslovak Math. J.*]{} [**47(122)**]{} (1997), [277–316]{}.
R. E. Showalter, *Monotone operators in Banach space and nonlinear partial differential equations.* Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997.
J. Yong, X. Y. Zhou, *Stochastic Controls. Hamiltonian Systems and HJB Equations*. Springer, New York, 1999.
|
---
abstract: 'Synchronized oscillations are of critical functional importance in many biological systems. We show that such oscillations can arise without centralized coordination in a disordered system of electrically coupled excitable and passive cells. Increasing the coupling strength results in waves that lead to coherent periodic activity, exhibiting cluster, local and global synchronization under different conditions. Our results may explain the self-organized transition in a pregnant uterus from transient, localized activity initially to system-wide coherent excitations just before delivery.'
author:
- 'Rajeev Singh$^1$, Jinshan Xu$^{2,3}$, Nicolas Garnier$^2$, Alain Pumir$^2$ and Sitabhra Sinha$^1$'
title: ' Self-organized transition to coherent activity in disordered media '
---
Rhythmic behavior is central to the normal functioning of many biological processes [@Glass01] and the periods of such oscillators span a wide range of time scales controlling almost every aspect of life [@Gillette05; @Golubitsky99; @Winfree00; @Hakim09]. Synchronization of spatially distributed oscillators is of crucial importance for many biological systems [@Pikovsky03]. For example, disruption of coherent collective activity in the heart can result in life-threatening arrhythmia [@Keener98]. In several cases, the rhythmic behavior of the entire system is centrally organized by a specialized group of oscillators (often referred to as [*pacemakers*]{}) [@Chigwada06] as in the heart, where this function is performed in the sino-atrial node [@Tsien79]. However, no such special coordinating agency has been identified for many biological processes. A promising mechanism for the self-organized emergence of coherence is through coupling among neighboring elements. Indeed, local interactions can lead to order without an organizing center in a broad class of complex systems [@Gregoire04].
The present work is inspired by studies of the pregnant uterus whose principal function is critically dependent on coherent rhythmic contractions that, unlike the heart, do not appear to be centrally coordinated from a localized group of pacemaker cells [@Blackburn07]. In fact, the uterus remains quiescent almost throughout pregnancy until at the very late stage when large sustained periodic activity is observed immediately preceding the expulsion of the fetus [@Garfield07]. In the USA, in more than $10~\%$ of all pregnancies, rhythmic contractions are initiated significantly earlier, causing preterm births [@Martin09], which are responsible for more than a third of all infant deaths [@MacDorman07]. The causes of premature rhythmic activity are not well understood and at present there is no effective treatment for preterm labor [@Garfield07]. In this paper we have investigated the emergence of coherence using a modeling approach that stresses the role of coupling in a system of heterogeneous entities. Importantly, recent studies have not revealed the presence of pacemaker cells in the uterus [@Shmygol07]. The uterine tissue has a heterogeneous composition, comprising electrically excitable smooth muscle cells (uterine myocytes), as well as electrically passive cells (fibroblasts and interstitial Cajal-like cells \[ICLCs\]) [@Duquette05; @Popescu07]. Cells are coupled in tissue by gap junctions that serve as electrical conductors. In the uterine tissue, the gap junctional couplings have been seen to markedly increase during late pregnancy and labor, both in terms of the number of such junctions and their conductances (by an order of magnitude [@Miller89]), which is the most striking of all electrophysiological changes the cells undergo during this period. The observation that isolated uterine cells do not spontaneously oscillate [@Shmygol07], whereas the organ rhythmically contracts when the number of gap junctions increases, strongly suggests a prominent role of the coupling. The above observations have motivated our model for the onset of spontaneous oscillatory activity and its synchronization through increased coupling in a mixed population of excitable and passive elements. While it has been shown earlier that an excitable cell connected to passive cells can oscillate [@Jacquemet06], we demonstrate that coupling such oscillators with different frequencies (because of varying numbers of passive cells) can result in the system having a frequency [*higher*]{} than its constituent elements. We have also performed a systematic characterization for the first time of the dynamical transitions occurring in the heterogeneous medium comprising active and passive cells as the coupling is increased, revealing a rich variety of synchronized activity in the absence of any pacemaker. Finally, we show that the system has multiple coexisting attractors characterized by distinct mean oscillation periods, with the nature of variation of the frequency with coupling depending on the choice of initial state as the coupling strength is varied. Our results provide a physical understanding of the transition from transient excitations to sustained rhythmic activity through physiological changes such as increased gap junction expression [@Garfield98]. The dynamics of excitable myocytes can be described by a model having the form $C_m \dot{V_e} = -I_{ion}(V_e, g_i)$ where $V_e$(mV) is the potential difference across a cellular membrane, $C_m$ (= 1 $\mu$F cm$^{-2}$) is the membrane capacitance, $I_{ion}$ ($\mu$A cm$^{-2}$) is the total current density through ion channels on the cellular membrane and $g_i$ are the gating variables, describing the different ion channels. The specific functional form for $I_{ion}$ varies in different models. To investigate the actual biological system we have first considered a detailed, realistic description of the uterine myocyte given by Tong [*et al.*]{} [@Tong11]. However, during the systematic dynamical characterization of the spatially extended system, for ease of computation we have used the phenomenological FitzHugh-Nagumo (FHN) system [@Keener98] which exhibits behavior qualitatively similar to the uterine myocyte model in the excitable regime. In the FHN model, the ionic current is given by $I_{ion} = F_e (V_e,g) = A V_e (V_e-\alpha) (1-V_e) - g$, where $g$ is an effective membrane conductance evolving with time as $\dot{g} = \epsilon
(V_e - g)$, $\alpha (=0.2)$ is the excitation threshold, $A
(=3)$ specifies the fast activation kinetics and $\epsilon (=0.08)$ characterizes the recovery rate of the medium (the parameter values are chosen such that the system is in the excitable regime and small variations do not affect the results qualitatively). The state of the electrically passive cell is described by the time-evolution of the single variable $V_p$ [@Kohl94]: $\dot{V_p} = F_p(V_p)=K(V^R_p-V_p)$, where the resting state for the cell, $V^R_p$ is set to 1.5 and $K
(=0.25)$ characterizes the time-scale over which perturbations away from $V^R_p$ decay back to it. We model the interaction between a myocyte and one or more passive cells by:
$$\begin{aligned}
\dot{V_e} &= F_e(V_e,g)+n_p\ C_r (V_p-V_e),
\label{eqn3a} \\
\dot{V_p} &= F_p(V_p)-C_r (V_p-V_e),
\label{eqn3}\end{aligned}$$
where $n_p (= 1, 2, \ldots)$ passive elements are coupled to an excitable element via the activation variable $V_{e,p}$ with strength $C_r$. Here, we have assumed for simplicity that all passive cells are identical having the same parameters $V^R_p$ and $K$, as well as, starting from the same initial state. We observe that the coupled system comprising a realistic model of uterine myocyte and one or more passive cells exhibits oscillations (Fig. \[fig1\] (a)) qualitatively similar to the generic FHN model (Fig. \[fig1\] (b)), although the individual elements are incapable of spontaneous periodic activity in both cases. In Fig. \[fig1\] (a-b), the range of $n_p$ and excitable-passive cell couplings for which limit cycle oscillations of the coupled system are observed is indicated with a pseudocolor representation of the period ($\tau$). We also look at how a system obtained by diffusively coupling two such “oscillators" with distinct frequencies (by virtue of having different $n_p$) behaves upon increasing the coupling constant $D$ between $V_e$ (Fig. \[fig1\] (c)). A surprising result here is that the combined system may oscillate [*faster*]{} than the individual oscillators comprising it.
![(Color online) Oscillations through interaction between excitable and passive elements. A single excitable element described by (a) a detailed ionic model of an uterine myocyte and (b) a generic FHN model, coupled to $n_p$ passive elements exhibits oscillatory activity (inset) with period $\tau$ for a specific range of gap junctional conductances $G_m$ in (a) and coupling strengths $C_r$ in (b). The triangles (upright and inverted) enclosing the region of periodic activity in (b) are obtained analytically by linear stability analysis of the fixed point solution of Eq. (\[eqn3a\]). (c) Frequency of oscillation for a system of two “oscillators" $A$ and $B$ (each comprising an excitable cell and $n_p$ passive cells with $n_p^A = 1$ and $n_p^B = 2$) coupled with strength $D$. Curves corresponding to different values of $C_r$ show that the system synchronizes on increasing $D$, having a frequency that can be [*higher*]{} than either of the component oscillators. (d) Uterine tissue model as a 2-dimensional square lattice, every site occupied by an excitable cell coupled to a variable number of passive cells. []{data-label="fig1"}](fig1.eps){width="0.99\linewidth"}
To investigate the onset of spatial organization of periodic activity in the system we have considered a 2-dimensional medium of locally coupled excitable cells, where each excitable cell is connected to $n_p$ passive cells \[Fig. \[fig1\] (d)\], $n_p$ having a Poisson distribution with mean $f$. Thus, $f$ is a measure of the density of passive cells relative to the myocytes. Our results reported here are for $f=0.7$; we have verified for various values of $f~\geq~0.5$ that qualitatively similar behavior is seen. The dynamics of the resulting medium is described by: $$\frac{\partial V_e}{\partial t} = F_e(V_e,g) + n_p\ C_r (V_p-V_e) +
D \nabla^2 V_e,$$ where $D$ represents the strength of coupling between excitable elements (passive cells are not coupled to each other). Note that, in the limit of large $D$ the behavior of the spatially extended medium can be reduced by a mean-field approximation to a single excitable element coupled to $f$ passive cells. As $f$ can be non-integer, $n_p$ in the mean-field limit can take fractional values \[as in Fig. \[fig1\] (a-b)\].
We discretize the system on a square spatial grid of size $L \times L$ with the lattice spacing set to $1$. For most results reported here $L=64$, although we have used $L$ up to $1024$ to verify that the qualitative nature of the transition to global synchronization with increasing coupling is independent of system size. The dynamical equations are solved using a fourth-order Runge Kutta scheme with time-step $dt \leq 0.1$ and a standard 5-point stencil for the spatial coupling between the excitable elements. We have used periodic boundary conditions in the results reported here and verified that no-flux boundary conditions do not produce qualitatively different phenomena. Frequencies of individual elements are calculated using FFT of time-series for a duration $2^{15}$ time units. The behavior of the model for a specific set of values of $f$, $C_r$ and $D$ is analyzed over many ($\sim 100$) realizations of the $n_p$ distribution with random initial conditions.
![(Color online) Emergence of synchronization with increased coupling. Snapshots (first row) of the activity $V_e$ in a two-dimensional simulation domain ($f=0.7, C_r=1, L=64$) for increasing values of coupling $D$ (with a given distribution of $n_p$). The corresponding time-averaged spatial correlation functions C([**r**]{}) are shown in the middle row. The size of the region around [**r**]{} $= 0$ (at center) where C([**r**]{}) is high provides a measure of the correlation length scale which is seen to increase with $D$. The last row shows pseudocolor plots indicating the frequencies of individual oscillators in the medium (white corresponding to absence of oscillation). Increasing $D$ results in decreasing the number of clusters with distinct oscillation frequencies, eventually leading to global synchronization characterized by spatially coherent, wavelike excitation patterns where all elements in the domain oscillate with same frequency. []{data-label="fig2"}](fig2.eps){width="0.99\linewidth"}
Fig. \[fig2\] (first row) shows spatial activity in the system at different values of $D$ after long durations ($\sim 2^{15}$ time units) starting from random initial conditions. As the coupling $D$ between the excitable elements is increased, we observe a transition from highly localized, asynchronous excitations to spatially organized coherent activity that manifests as propagating waves. Similar traveling waves of excitation have indeed been experimentally observed [*in vitro*]{} in myometrial tissue from the pregnant uterus [@Lammers08]. The different dynamical regimes observed during the transition are accompanied by an increase in spatial correlation length scale (Fig. \[fig2\], middle row) and can be characterized by the spatial variation of frequencies of the constituent elements (Fig. \[fig2\], last row). For low coupling ($D=0.1$), multiple clusters each with a distinct oscillation frequency $\nu$ coexist in the medium. As all elements belonging to one cluster have the same period, we refer to this behavior as [*cluster synchronization*]{} (CS). Note that there are also quiescent regions of non-oscillating elements indicated in white. With increased coupling the clusters merge, reducing the variance of the distribution of oscillation frequencies eventually resulting in a single frequency for all oscillating elements (seen for $D=0.3$). As there are still a few local regions of inactivity, we term this behavior as [*local synchronization*]{} (LS). Further increasing $D$ (=0.4), a single wave traverses the entire system resulting in [*global synchronization*]{} (GS) characterized by [*all*]{} elements in the medium oscillating at the same frequency. Our results thus help in causally connecting two well-known observations about electrical activity in the pregnant uterus: (a) there is a remarkable increase in cellular coupling through gap junctions close to onset of labor [@Miller89] and (b) excitations are initially infrequent and irregular, but gradually become sustained and coherent towards the end of labor [@Blackburn07].
![ (Color online) (a) Different dynamical regimes of the uterine tissue model (for $f=0.7$) in $D-C_r$ parameter plane indicating the regions having (i) complete absence of oscillation (NO), (ii) cluster synchronization (CS), (iii) local synchronization (LS), (iv) global synchronization (GS) and (v) coherence (COH). (b-c) Variation of (b) width of frequency distribution $\langle
\sigma_{\nu} \rangle$ and (c) fraction of oscillating cells $\langle f_{osc} \rangle$ with coupling strength $D$ for $C_r = 1$ \[i.e., along the broken line shown in (a)\]. The regimes in (a) are distinguished by thresholds applied on order parameters $\langle \sigma_{\nu} \rangle$, $\langle f_{osc} \rangle$ and $\langle F \rangle$, viz., NO: $\langle
f_{osc} \rangle< 10^{-3}$, CS: $\langle \sigma_{\nu} \rangle> 10^{-4}$, LS: $\langle \sigma_{\nu} \rangle< 10^{-4}$ and $\langle
f_{osc} \rangle< 0.99$; GS: $\langle f_{osc} \rangle> 0.99$ and COH: $\langle
F \rangle> 0.995$. Results shown are averaged over many realizations. []{data-label="fig3"}](fig3.eps){width="0.99\linewidth"}
The above observations motivate the following order parameters that allow us to quantitatively segregate the different synchronization regimes in the space of model parameters \[Fig. \[fig3\] (a)\]. The CS state is characterized by a finite width of the frequency distribution as measured by the standard deviation, $\sigma_{\nu}$, and the fraction of oscillating elements in the medium, $0<f_{osc}<1$. Both LS and GS states have $\sigma_{\nu} \rightarrow 0$, but differ in terms of $f_{osc}$ ($<1$ in LS, $\simeq 1$ in GS). Fig. \[fig3\] (b-c) shows the variation of the two order parameters $\langle \sigma_{\nu}
\rangle$ and $\langle f_{osc} \rangle$ with the coupling $D$, $\langle~\rangle$ indicating ensemble average over many realizations. Varying the excitable cell-passive cell coupling $C_r$ together with $D$ allows us to explore the rich variety of spatio-temporal behavior that the system is capable of \[Fig. \[fig3\] (a)\]. In addition to the different synchronized states (CS, LS and GS), we also observe a region where there is no oscillation (NO) characterized by $f_{osc}
\rightarrow 0$, and a state where all elements oscillate with the same frequency and phase which we term coherence (COH). COH is identified by the condition that the order parameter $F \equiv {\rm max}_t [f_{act} (t)] \rightarrow 1$ where $f_{act} (t)$ is the fraction of elements that are active ($V_e > \alpha$) at time $t$. In practice, the different states are characterized by thresholds whose specific values do not affect the qualitative nature of the results.
![(Color online) (a) Variation of mean oscillation frequency $\bar{\nu}$ with coupling strength $D$ in the uterine tissue model ($f = 0.7$) for 400 different initial conditions at $C_r = 1$. Continuous curves correspond to gradually increasing $D$ starting from a random initial state at low $D$, while broken curves (overlapping) correspond to random initial conditions chosen at each value of $D$. (b-c) Snapshots of activity in the medium at $D=1.5$ for a random initial condition seen at intervals of $\delta T =5$ time units. (d) Variation of the cumulative fractional volumes $\phi$ of the basins for different attractors corresponding to activation patterns shown in (b-c) and (e), as a function of the coupling strength $D$. (e) Snapshots of topologically distinct patterns of activity corresponding to the five attractors at $D = 1.5$ (broken line in (d)) when $D$ is increased. []{data-label="fig4"}](fig4.eps){width="0.99\linewidth"}
To further characterize the state of the system, we determined the mean frequency $\bar{\nu}$ by averaging over all oscillating cells for any given realization of the system. Fig. \[fig4\] (a) reveals that several values of the mean frequency are possible at a given coupling strength. When the initial conditions are chosen randomly for each value of the coupling (broken curve in Fig. \[fig4\] (a)), the mean frequency decreases with increasing $D$. On the other hand, $\bar{\nu}$ is observed to [*increase*]{} with $D$ when the system is allowed to evolve starting from a random initial state at low $D$, and then adiabatically increasing the value of $D$. The abrupt jumps correspond to drastic changes in the size of the basin of an attractor at certain values of the coupling strength, which can be investigated in detail in future studies. This suggests a multistable attractor landscape of the system dynamics, with the basins of the multiple attractors shown in Fig. \[fig4\] (d) \[each corresponding to a characteristic spatiotemporal pattern of activity shown in Fig. \[fig4\] (e)\] having differing sizes. They represent one or more plane waves propagating in the medium and are quite distinct from the disordered patterns of spreading activity (Fig. \[fig4\] (b-c)) seen when random initial conditions are used at each value of $D$. We note that the period of recurrent activity in the uterus decreases with time as it comes closer to term [@Garfield98] in conjunction with the increase in number of gap junctions. This is consistent with our result in Fig. \[fig4\](a) when considering a gradual increase of the coupling $D$.
Our results explain several important features known about the emergence of contractions in uterine tissue. Previous experimental results have demonstrated that the coupling between cells in the myometrium increases with progress of pregnancy [@Miller89]. This suggests that the changes in the system with time amounts to simultaneous increase of $D$ and $C_r$, eventually leading to synchronization as shown in Fig. \[fig3\] (a). Such a scenario is supported by experimental evidence that disruption of gap-junctional communication is associated with acute inhibition of spontaneous uterine contractions [@Tsai98]. The mechanism of synchronization discussed here is based on a very generic model, suggesting that our results apply to a broad class of systems comprising coupled excitable and passive cells [@Bub02; @Pum05]. A possible extension will be to investigate the effect of long-range connections [@Falcke94].
To conclude, we have shown that coherent periodic activity can emerge in a system of heterogeneous cells in a self-organized manner and does not require the presence of a centralized coordinating group of pacemaker cells. A rich variety of collective behavior is observed in the system under different conditions; in particular, for intermediate cellular coupling, groups of cells spontaneously form clusters that oscillate at different frequencies. With increased coupling, clusters merge and eventually give rise to a globally synchronized state marked by the genesis of propagating waves of excitation in the medium. Our model predicts that a similar set of changes occur in the uterus during late stages of pregnancy.
This research was supported in part by IFCPAR (Project 3404-4). We thank HPC facility at IMSc for providing computational resources.
[2]{} L. Glass, Nature (London) [**410**]{}, 277 (2001). M. U. Gillette and T. J. Sejnowski, Science [**309**]{}, 1196 (2005). M. Golubitsky, I. Stewart, P.-L. Buono and J. J. Collins, Nature (London) [**401**]{}, 693 (1999). A. T. Winfree, [*The Geometry of Biological Time*]{}, Springer, New York, 2000. V. Hakim and N. Brunel, Neural Comput. [**11**]{}, 1621 (1999).
A. Pikovsky, M. Rosenblum and J. Kurths [*Synchronization*]{}, Cambridge Univ. Press, Cambridge, 2003.
J. Keener and J. Sneyd, [*Mathematical Physiology*]{}, Springer, New York, 1998. T. R. Chigwada, P. Parmananda and K. Showalter, Phys. Rev. Lett. [**96**]{}, 244101 (2006). R. W. Tsien, R. S. Kass and R. Weingart, J. Exp. Biol. [**81**]{}, 205 (1979). G. Grégoire and H. Chaté, Phys. Rev. Lett. [**92**]{}, 025702 (2004); T. Vicsek [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 1226 (1995).
S. T. Blackburn, [*Maternal, Fetal and Neonatal Physiology: A Clinical Perspective* ]{}, Saunders-Elsevier, St Louis, Miss., 2007.
R. E. Garfield and W L Maner, Sem. Cell Develop. Biol. [**18**]{} 289 (2007).
J. A. Martin [*et al.*]{}, [*National Vital Statistics Reports*]{} [**57**]{}(7), U.S. Dept. of Health & Human Services, Atlanta, 2009.
M. F. MacDorman [et al.]{}, [*Trends in preterm-related infant mortality by race and ethnicity: United States 1999–2004*]{}, National Center for Health Statistics, Hyattsville, MD, 2007.
A. Shmygol [*et al.*]{}, Ann. N.Y. Acad. Sci. [**1101**]{}, 97 (2007).
R. A. Duquette [*et al.*]{}, Biol. Reproduction [**72**]{}, 276 (2005).
L. M. Popescu, S. M. Ciontea and D. Cretoiu, Ann. N.Y. Acad. Sci. [**1101**]{}, 139 (2007).
S. M. Miller, R. E. Garfield and E. E. Daniel, Am. J. Physiol. [**256**]{}, C130 (1989); H. Miyoshi [*et al.*]{}, Biophys. J. [**71**]{}, 1324 (1996).
V. Jacquemet, Phys. Rev. E [**74**]{}, 011908 (2006); A. K. Kryukov [*et al.*]{}, Chaos [**18**]{}, 037129 (2008); W. Chen [*et al.*]{}, EPL [**86**]{}, 18001 (2009).
R. E. Garfield [*et al.*]{}, Human Reproduction Update [**4**]{}, 673 (1998).
W.-C. Tong [*et al.*]{}, PLoS One, [**6**]{}, 18685 (2011).
P. Kohl [*et al.*]{}, Exp. Physiol. [**79**]{}, 943 (1994).
W. J. E. P. Lammers [*et al.*]{}, Am. J. Physiol. Regul. Integr. Comp. Physiol. [**294**]{}, R919 (2008).
M.-L. Tsai [*et al.*]{}, Toxicol. Appl. Pharmacol. [**152**]{}, 18 (1998).
G. Bub, A. Shrier and L. Glass, Phys. Rev. Lett. [**88**]{}, 058101 (2002).
A. Pumir [*et al.*]{}, Biophys. J. [**89**]{}, 2332 (2005).
M. Falcke and H. Engel, Phys. Rev. E [**50**]{}, 1353 (1994); S. Sinha, J. Saramäki and K. Kaski, Phys. Rev. E [**76**]{}, 015101 (2007).
|
---
abstract: 'Oscillations from high energy photons into light pseudoscalar particles in an external magnetic field are expected to occur in some extensions of the standard model. It is usually assumed that those axionlike particles (ALPs) could produce a drop in the energy spectra of gamma ray sources and possibly decrease the opacity of the Universe for TeV gamma rays. We show here that these assumptions are in fact based on an average behavior that cannot happen in real observations of single sources. We propose a new method to search for photon-ALP oscillations, taking advantage of the fact that a single observation would deviate from the average expectation. Our method is based on the search for irregularities in the energy spectra of gamma ray sources. We predict features that are unlikely to be produced by known astrophysical processes and a new signature of ALPs that is easily falsifiable.'
author:
- Denis Wouters
- Pierre Brun
bibliography:
- 'wb.bib'
title: Irregularity in gamma ray source spectra as a signature of axionlike particles
---
Introduction
============
Light pseudoscalar particles appear in many extensions of the standard model. The most typical example is the axion, which was introduced as a consequence of the Peccei-Quinn mechanism to solve the puzzle of the absence of CP violation in quantum chromodynamics [@Peccei:1977hh; @Peccei:1977ur]. The axion is a hypothetical light particle that has a two-photon vertex described by the interaction term $$\mathcal{L}_{a\gamma}\;\;=\;\;-\frac{1}{4}gF_{\mu\nu}\tilde{F}^{\mu\nu}a\;\; = \;\;g\;\vec{E}\cdot\vec{B}\;a\;\;,$$ where $g$ is the axion-photon coupling constant, $F$ is the electromagnetic tensor, $\tilde{F}$ its dual, $\vec{E}$ the electric field, $\vec{B}$ the magnetic field and $a$ the axion field. This term implies the possibility of photon-axion oscillations in an external magnetic field [@1983PhRvL..51.1415S; @1988PhRvD..37.1237R]. This coupling is used experimentally to search for axions that would be thermally produced in the Sun [@2011PhRvL.107z1302A], or axion dark matter [@2010PhRvL.105q1801W]. In the case of the Peccei-Quinn axion, the photon-axion coupling is predicted to scale with the axion mass; however, other models predict light pseudoscalar particles with the same coupling to the electromagnetic field but [*a priori*]{} unrelated to their mass [@1987PhR...150....1K]. Those are called axionlike particles (ALPs), the phenomenology of which is similar to standard axions. Astrophysical environments can offer ideal conditions for photon-ALP oscillations, with the possibility of long baseline experiments involving magnetic fields [@1996slfp.book.....R]. Progress over the last decade in $\gamma$-ray astronomy allowed one to consider searching for the imprints of $\gamma$-ALP oscillations in the energy spectra of high energy $\gamma$-ray sources [@2007PhRvD..76b3001M]. The effect of $\gamma$-ALP oscillation is usually assumed to be twofold: it is expected to induce a dimming of the fluxes above a given threshold [@2007PhRvL..99w1102H; @2007PhRvD..76l3011H], and possibly decrease the gamma-ray pair production related opacity at high energy. The opacity can be that of the intergalactic medium [@2007PhRvD..76l1301D; @2009PhRvD..79l3511S] or within the sources themselves [@2012arXiv1202.6529T]. A crucial point is the turbulent nature of the magnetic fields the photon beam travels through. It implies a consequential randomness in the prediction of the observable effects. This has been pointed out in [@2009JCAP...12..004M] in the case of the change of opacity due to $\gamma$-ALP oscillations. The authors of [@2009JCAP...12..004M; @2009PhRvL.102t1101B] showed that because of the random nature of the intergalactic magnetic fields, the effect of $\gamma$-ALP mixing should be very different from one source to another. Such an observable is then useless to perform ALP searches through the observation of a single source, leaving only the possibility of a population study in order to average the effect over many sources. This type of study has been conducted in [*e.g.*]{} [@Horns:2012fx], showing a hint for an anomaly in the transparency of the Universe. Though rapidly increasing with the advent of the last generation Cherenkov telescope arrays such as HESS, MAGIC and VERITAS, there are only a handful of high energy sources that are effectively concerned with extragalactic absorption. It is thus interesting to point out some effect of the $\gamma$-ALP mixing that does not rely either on stacking or averaging, in order to exploit observations of single sources. Here for the first time an effect is pointed out that potentially applies to single observations. This article is organized as follows. First, we briefly recall the formalism of $\gamma$-ALP mixing and apply the results to a single coherent magnetic domain. As a second step we show the results of a simulation of photons traveling through a set of magnetic domains. In particular we show that contrary to what is stated in the literature, a sharp drop in the energy spectrum of high-energy $\gamma$-ray sources is not a robust observable and is not what should be searched for. Actually the $\gamma$-ALP mixing would produce an anomalous dispersion of the spectra, which would no longer appear as smooth in a limited energy range. We then give an explicit example of how the effect could appear in the data, in the case of a specific situation, namely an extragalactic TeV emitter whose photons travel through the intergalactic magnetic field (IGMF), and we discuss the robustness of the method.
The photon/axion system in a magnetic field
===========================================
The $\gamma$-ALP system is described following the approach of [@1988PhRvD..37.1237R]. A three-state wave function is used with two states of polarization for the photon and one for the ALP. Let $\theta$ be the angle between the magnetic field direction and the photon momentum. Since only the $\vec{B}$ component transverse to the propagation couples photons and ALPs, the strength of the magnetic field involved in the coupling is $B\sin\theta$. Moreover, for parity issues, only one polarization state parallel to the field is involved in the interaction. This is accounted for by introducing the angle $\phi$ between the transverse component of the field and the direction of the polarization sate $A_1$. The $\gamma$-ALP system is then propagated using the following linearized equations of motion assuming relativistic axions: $$\left(E - i\partial_z + \mathcal{M} \right )
\left(\begin{array}{c} A_1 \\ A_2 \\ a \end{array} \right) = 0 \;\;,$$ with the mixing matrix $$\mathcal{M}\;\;=\;\;\left(\begin{array}{ccc} \Delta_{11}-i\Delta_{\mathrm{abs}} & \Delta_{12} & \Delta_\mathrm{B}\cos\phi \\ \Delta_{21} & \Delta_{22}-i\Delta_{\mathrm{abs}} & \Delta_\mathrm{B}\sin\phi \\ \Delta_\mathrm{B}\cos\phi & \Delta_\mathrm{B}\sin\phi\ & \Delta_\mathrm{a} \end{array}\right)\;\;,$$ where $\Delta_\mathrm{B} = gB\sin\theta/2$ is the coupling term, and $\Delta_\mathrm{a} = -m_\mathrm{a}^2/2E$ is the ALP mass term. Here we neglect the Faraday effect and the vacuum Cotton-Mouton term, as the low magnetic field strength considered in the following makes the corresponding contribution irrelevant for this study. This implies $\Delta_{12} = \Delta_{21} = 0$ and that the other diagonal terms are $\Delta_{11} = \Delta_{22} = -\omega^2_\mathrm{pl}/2E$, $\omega_{\rm pl}$ being the plasma frequency accounting for the effective photon mass. As in [@2003JCAP...05..005C], absorption of photons on their way is introduced with the $\Delta_{\mathrm{abs}} = \tau/2s$ term where $\tau$ is the optical depth assuming a propagation over a domain of size $s$ within which the opacity is homogeneous. Because of that term, the matrix is no longer Hermitian and unitarity is lost. In the following, this term will be used to model the absorption of photons on the extragalactic background light (EBL) while propagating in IGMFs. After diagonalization of the mixing matrix, the equations of motion can be analytically solved and the transfer matrix of the system is obtained. The probability of $\gamma-a$ conversion after crossing one coherent magnetic field domain of size $s$ in the simplest case, without absorption and neglecting the plasma term, yields $$\label{prob}
P_{\gamma\rightarrow a} \;=\;
\frac{2 \Delta_\mathrm{B}^2}{\Delta_{\mathrm{osc}}^2}\sin^2\frac{\Delta_{\mathrm{osc}}s}{2}\;\;,$$ with $\Delta_{\mathrm{osc}} = \sqrt{\Delta_\mathrm{a}^2+4\Delta_\mathrm{B}^2}$. The energy dependence of the mass terms in $\Delta_{\mathrm{osc}}$ implies an energy threshold above which the conversion becomes efficient, $$\label{thresh}
E_\mathrm{thr} = \frac{m_{\mathrm{eff}}^2}{2gB\sin\theta} \;\; ,$$ $m_\mathrm{eff}$ being the effective ALP mass in the presence of charges ([*e.g.*]{} in a plasma). For $E \ll E_\mathrm{thr}$, $\Delta_{\mathrm{osc}} \gg \Delta_\mathrm{B}$ and then no conversion occurs. For $E \sim E_{\mathrm{thr}}$ spectral oscillations happen due to the energy dependent $\sin^2\Delta_{\mathrm{osc}}s/2$ term. For $E \gg E_\mathrm{thr}$, $\Delta_{\mathrm{osc}} \sim \Delta_\mathrm{B}$ and the conversion probability is no longer energy dependent. The conversion probability of Eq. \[prob\] can be parameterized in terms of $E_\mathrm{thr}$ and $$\label{delta}
\delta = gBs\sin\theta/2$$ instead of $B\sin\theta$ and $s$. $\sin^2\delta/2$ is then the conversion probability at very high energy (VHE, $E \gg E_\mathrm{thr}$). The condition required for a significant conversion to occur, $\delta \gtrsim 1$, is similar to the Hillas criterion for the acceleration of ultra high energy cosmic rays, as pointed out in [@2007PhRvL..99w1102H]. Figure \[fig:1domain\] shows the evolution of the photon survival probability as function of the energy for three different values of $\delta$. For allowed large IGMF values of order 1 nG, an ALP mass of 2 neV and a coupling $g=8\times10^{-11}\;\rm GeV^{-1}$ at the limit of current experimental constraints [@2011PhRvL.107z1302A], $E_\mathrm{thr}$ lies at about 1 TeV.
![Survival probability of an unpolarized photon as a function of the energy for three values of $\delta$.[]{data-label="fig:1domain"}](Fig1){width="\columnwidth"}
The asymptotical value of $1-P_{\gamma \rightarrow a}$ gives the level of dimming of the photon flux (independently of eventual additional EBL absorption). One can see that this attenuation is hardly predictable given the uncertainties on the environmental parameters, $\vec{B}$ and $s$, as it depends sinusoidally on the value of $\delta$.
In astrophysical environments, magnetic fields are usually not coherent. In the case of a propagation through a turbulent magnetic field, the beam path can be divided into coherent domains of size of the coherence length of the field (the validity of this simple model is discussed in Sec. \[discussion\]). For each domain, a transfer matrix is generated with a random orientation of the magnetic field yielding a specific value of $\delta$. The total transfer matrix associated with this realization of the turbulent magnetic field is the product of all individual transfer matrices. The spectral shape of the global conversion probability for one single realization is the result of the interference of all oscillation patterns such as those displayed in Fig. \[fig:1domain\]. As the pseudo-period is different in each domain, the photon survival probability has a very complex energy dependence. As an illustration, the survival probability of a photon from a source at redshift $z=0.1$ traveling through a single realization of a 1 nG IGMF with coherent domains of size $s_0=1\;\rm Mpc$ is displayed in Fig. \[fig:turbulent\]. A plasma density of $n_e = 10^{-7} \mathrm{cm}^{-3}$ typical of the intergalactic medium is assumed. In this condition and for ALP masses of order neV, $m_{\rm eff}=m_{\rm a}$. For illustration, the upper panel shows the survival probability without absorption on the EBL, whereas the lower panel results include this effect. Conservatively, the EBL density model used here is the lower limit model from . To account for redshifting, a flat $\rm \Lambda CDM$ Universe with $(\Omega_{\rm \Lambda},\,\Omega_{\rm m})=(0.73,\,0.27)$ and $H_0=71\,\rm km/s/Mpc$ is assumed. Here the dashed red line is the prediction without ALPs, so that the dimming is only due to EBL. From Fig. \[fig:turbulent\] one can see that the prediction of the model including ALPs is the presence of a significant level of noise in the energy spectrum over one decade or so around $E_{\rm thr}$. Because of the unknown nature of the orientation of the magnetic field within the domains, the exact shape of the spectrum in this region is unpredictable. However, as we shall see in the following, the noise level is a prediction of the model. This prediction significantly differs from what usually appears in the literature, namely a smooth transition between no dimming below $E_{\rm thr}$ and a fixed level of attenuation above it. It has been shown in [@2002PhLB..543...23G] that the averaging over a large number of realizations of $N$ domains in each of which the conversion probability is $P_0$ yields an overall conversion probability $$\label{paveraged}
P_{\gamma\rightarrow a} = \frac{1}{3}\left (1-{\rm e}^{-3NP_0} \right )\;\; .$$ This means that the effect as it has been studied so far is valid for an average over a collection of sources. In the case of the observation of one source only, if $N$ is very large and the energy spectrum is binned, then the smooth behavior can be retrieved in principle. In practice $N$ is not large enough, as we shall see in the following. The results presented in Fig. \[fig:turbulent\] are obtained with a single realization. By averaging the results of Fig. \[fig:turbulent\] over a large number of realizations, the value given by Eq. \[paveraged\] is retrieved. From one realization to another, only the orientations of the magnetic fields vary; the number of domain and their sizes are kept fixed.
![Photon survival probability as function of the energy for a realization of a source at $z = 0.1$ using B = 1 nG, $s_0$ = 1 Mpc, $g=8\times10^{-11}$ GeV$^{-1}$ and $m_{\rm a}$ = 2 neV without absorption (upper panel) and with EBL absorption (lower panel). []{data-label="fig:turbulent"}](Fig2){width="\columnwidth"}
Note that above 5 TeV the survival probability for this specific realization is higher with ALPs than with EBL only. This is the so-called opacity effect, because photons are untouched by the EBL as they travel disguised as axions, the Universe appears to be more transparent. This result should be taken with care, however because, as shown in [@2009JCAP...12..004M], there exist realizations of the IGMF where the opposite effect is obtained, basically when most ALPs do not convert back to photons before detection.
Observational effects {#observational}
=====================
The experimental relevance of the proposed signature is now studied in the particular case of a source at redshift $z = 0.1$ for the same parameters as above. The intrinsic spectrum of the source is simulated following a log-parabola shape with an integrated flux in the TeV band at the Crab level. A 50 h observation time is assumed with an energy resolution of 15 $\%$ and assuming a constant effective area of $10^5 \;\rm m^2$, these values being typical of current generation Cherenkov observatories. The intrinsic spectrum is convolved by one randomly generated photon survival probability and eventually binned to obtain the spectrum that would be observed in this model. The result of this simulation is displayed on the left panel of Fig. \[fig:simu\]. A fit of the simulated experimental data by a log-parabola shape convolved with EBL absorption is also shown, as it would be performed by observers. In the right panel of Fig. \[fig:simu\] are displayed the residuals of that fit. It appears that in the case without ALPs the residuals would evenly spread around 0 whereas these residuals would show anomalously strong and chaotic deviations from 0 in the case of $\gamma$-ALP mixing. This is the expected signature of ALPs in the spectrum, induced by the noisy spectral shape of the photon survival probability.
{width=".75\columnwidth"} {width=".75\columnwidth"}
The approach considered here corresponds to what an observer would do. First one would fit a smooth shape (be it a log-parabola, a broken power law or an emission model inspired one) and then pick up the shape providing the best $\chi^2$ and a decent residual distribution ([*e.g.*]{} centered on zero, without obvious biases, etc.). The crucial point is that the observer would fit the data with a smooth function. This is motivated by the fact that no TeV source emission model predicts spectra with local extrema or a noisy shape. So this result holds for any smooth spectral shape provided it gives the best possible fit. For a given observation, a given emission model and a given fit function, the noisy energy range and the variance of the residuals is a prediction of the ALP model. This is illustrated in Tab. \[tab1\], which displays the variance of the fit residuals under different hypotheses: no ALP and two values of $g$, with an ALP mass still yielding $E_{\rm thr}=1\;\rm TeV$. The exact value of the variance of the fit residuals depends on the analysis that would be performed, in particular the energy range chosen by the observer. The use of the variance of the fit residuals is only an example, as observers might choose to use a more sophisticated estimator of the noise in theirs data sets. Additionally, because of the random nature of the predicted effect, it is important here to verify that the scatter in the prediction among realizations is smaller than the effect itself. The predicted uncertainty on the variance of the fit residuals due to the random nature of the prediction is also shown in Tab. \[tab1\] for each considered scenario. These results are obtained by averaging over 5000 realizations. It appears that for the considered parameters, the effect is significant, as one can see that the variance anomaly in the presence of ALP is predicted to be significantly above the conventional value. The observation of a Crab-level source for 50 h was chosen here as an illustration. Actually one finds that for the same redshift and energy range, the effect would still be visible but with less significance by observing only 5 h.
Model Variance of the fit residuals
---------------------------- -------------------------------
No ALP $0.04\pm 0.01$
$g=10^{-11}$, $m=0.7$ $0.11\pm 0.04$
$g=8\times10^{-11}$, $m=2$ $0.20\pm 0.05$
: Values of the RMS of the fit residuals to mock data with different assumptions for $g$ and $m$ (in units of $\rm GeV^{-1}$ and neV resp.), for constant size magnetic field domains.\[tab1\]
The observational signature that is discussed here occurs for energies around $E_{\rm thr}$ given in Eq. \[thresh\]. Therefore the range of accessible ALP parameters with this method depends on the value of the magnetic field and the energy range of the experiment. For instance considering TeV $\gamma$-rays and nG IGMF, the above results show that a typical IACT would be sensitive to ALPs with $g\sim10^{-11}\; \rm GeV^{-1}$ in a range from 0.1 neV to 10 neV. In that range of mass the most stringent constraint currently comes from the CAST helioscope with an upper limit on $g$ of order $10^{-10}\;\rm GeV^{-1}$ [@2011PhRvL.107z1302A]. So in principle this method should allow improving current constraints in this range of mass. To go to larger masses, one has to consider larger magnetic fields (in principle the method discussed in this article is valid for any $\gamma$-ray source behind a turbulent magnetic field) and/or higher energies, as the relevant mass for a given $g$ goes as $\sqrt{E\times B}$.
Discussion
==========
One important point is that should anomalous dispersion be observed some day, one would know how to falsify the interpretation in terms of new physics. This can be done for instance by observing the same object with more exposure. If the ALP interpretation is wrong, local extrema would not hold and all the residual points would be redistributed around zero. If the interpretation is correct though, two effects are predicted due to the increased statistics: [*i*]{}) the significance of the deviant bins would strengthen, and [*ii*]{}) irregularity would disappear at VHE as expected from ALP models. The first point is justified by the fact that a magnetic field that is coherent over a scale $s$ should remain coherent over times of oder $s/c$. For scales of order 1 Mpc as relevant here, this time scale is of order $3\times 10^6$ yrs. Concerning possible effects that could produce similar irregular spectra, one could imagine a complex landscape of background UV-IR photons that produces non-trivial absorption features in the energy spectra and mimics the effect. In the event of a positive detection, this would therefore require studying the effect over more sources and how it depends on $z$ for instance. For observers interested in putting constraints on ALP models, though, this is not an issue since such an effect would add up to the irregularity of the spectrum and by no way it could cancel it.
The modeling of the IGMF as it is done here with domains of same sizes is the simplest model one could think of. It has been used here as it is widely used in the literature. To describe more precisely the magnetic field turbulence, it is possible to account for the power distribution of the modes. The turbulent field can be modeled as a Gaussian random field with each Fourier mode proportional to some power of the wave number $k^{-\alpha}$. In the generic case of isotropic and homogeneous Kolmogorov-like turbulence, $\alpha=5/3$. As shown in [@2007PhRvD..76b3001M], this leads to a variation of the rms intensity of the magnetic field $B$ as a function of the scale $s$ such that $B\propto s^{1/3}$.
Before discussing the effect of such a magnetic field on $\gamma$-ray source spectra, let us remark that magnetic fields that are coherent on small scales should have negligible effects on the spectrum in comparison with the larger scales. For small values of $\delta$, $P_0$, the conversion probability over a scale $s$, is expected to be of order $\delta^2/2\sim g^2B^2s^2/8$ (see Eq. \[delta\]). In that case, the averaged formula of Eq. \[paveraged\] reduces to $P_{\gamma\rightarrow a}\simeq N\delta^2/2$. All in all, for a given $g$, this probability is proportional to $N B^2 s^2$. If $P_s$ is the probability of photon conversion for modes of size $s$, given the above mentioned law for the magnetic field strength, one gets $P_{s/10}\sim 2.5\%\times P_s$ for the conversion probability in a magnetic field mode corresponding to a scale $s/10$. This means that the small scales rapidly become irrelevant for this study and one can safely consider that the largest scales contribute the most in the power distribution of modes. Concerning larger scales, the effect on the noise level is limited by the ratio between the considered scale and the distance to the source. Speaking in terms of domains, if there are only a few equivalent domains, little interference will happen and then the noise in the energy spectra will have wider fluctuations.
Model Variance of the fit residuals
---------------------------- -------------------------------
$g=10^{-11}$, $m=0.35$ $0.18\pm 0.05$
$g=8\times10^{-11}$, $m=1$ $0.42\pm 0.14$
: Values of the RMS of the fit residuals to mock data with different assumptions for $g$ and $m$ (in units of $\rm GeV^{-1}$ and neV resp.), in the case of a Kolmogorov-like turbulent magnetic field.\[tab2\]
To be more quantitative, the study of Sec. \[observational\] has been repeated using a Kolmogorov-like turbulent magnetic field inspired by the modeling used in [@1999ApJ...520..204G]. As for the previous study, 5000 realizations of turbulent magnetic field are performed, with wave numbers ranging from 0.1 Mpc to 100 Mpc, and a rms intensity of $B$ of 1 nG at 100 Mpc. The exact same kind of noise in the $\gamma$-ray spectra is obtained. To illustrate this, the results of these simulations are shown in Table \[tab2\]; in particular, the variance of the fit residuals is still larger than in the no-ALP situation, in a statistically significant way. It has been checked that this results is stable when larger scales are used for the lowest wave number.
Conclusion
==========
In this study we showed a new possible signature of $\gamma$-ALP mixing in the form of an anomalous dispersion in the energy spectra of $\gamma$-ray sources. The smooth-noisy-smooth alternation behavior in the energy spectrum is a peculiar prediction of ALP models that could hardly be mimicked by known astrophysical processes. It has been shown that this effect can be used to constrain ALP models from the observation of single sources. An explicit example has been given in the case of oscillations in IGMF; however, such a signature can be searched in any source for which a turbulent magnetic field is present along the line of sight.
We would like to thank Gilles Henri and Mathieu Langer for interesting discussions about the project, and Pasquale Serpico, Jean-François Glicenstein, Aion Viana, Emmanuel Moulin and Fabian Schüssler for reading and improving the manuscript.
|
---
abstract: 'The inclusive $^{12}\mbox{C}(p,p'')$ and exclusive $^{12}\mbox{C}(p,p''X)$ reactions have been studied with a beam energy of 156 MeV and for $X = p \mbox{ and } \alpha$. The study focuses on the $(p,p''X)$ reaction mechanism and on the structure of $^{12}\mbox{C}$ just above the particle-emission threshold, $14 \leq E_x \leq 28$ MeV. Cross sections were simultaneously measured for all three reactions. The exclusive data were analyzed by making multiple-peak fits of the spectra and by Legendre-polynomial fits of the angular correlations. Multiple-peak fits were also made of the inclusive spectra. The resultant cross sections were compared to theoretical calculations. An analysis of the results shows that this region of $E_x$ consists predominantly of resonant excitations, in contradiction to the findings of previous analyses.'
address:
- 'Indiana University Cyclotron Facility, Bloomington, Indiana 47408, USA'
- 'Brookhaven National Laboratory, Upton, New York 11973, USA'
author:
- 'J. A. Templon'
- 'B. A. Raue'
- 'L. C. Bland'
- 'K. Murphy'
- 'D. S. Carman'
- 'G. M. Huber'
- 'B. C. Markham'
- 'D. W. Miller'
- 'P. Schwandt'
- 'D. J. Millener'
title: 'Study of continuum nuclear structure of via $(p,p''X)$ at intermediate energies'
---
,
,
,
,
,
,
,
,
$(p,p')$, decay, Angular Correlations, Giant Resonances. 21.60.Cs, 23.20.En, 24.30.Cz, 25.40.Ep
Studies of structure in the nuclear continuum suffer from difficulties in interpreting inelastic-scattering data. The biggest problems are that the resonance peaks in the cross-section spectra are often broad and poorly separated, and the amount of nonresonant “background” cross section underlying the resonance peaks is difficult to determine. Observation of the continuum decay products in coincidence with the inelastically-scattered probe can reduce these difficulties in two ways. First, since specific decay modes are accessible to only a subset of the excited states, the density of states seen in a particular decay channel is lower than that seen in inclusive measurements. Second, the angular distribution of the decay products contains information about the reaction dynamics and can be used to judge the relative importance of non-resonant vs. resonant continuum excitation; for the latter, information about the angular momentum of the resonance excitations can be deduced.
These features have been beautifully realized in $(e,e'X)$ [@knop86] and $(\alpha, \alpha' X)$ [@zwar85] experiments, but only a few $(p,p'X)$ experiments [@dera81; @raue95] of this type have been reported. Coincidence studies on $^{12}$C - $(\alpha,\alpha'X)$ [@ried78], $(p,p'\alpha)$ [@dera81], $(e,e'p_0)$ [@cala84; @cala94] and $(e,e'\alpha)$ [@dean95] - have concentrated on searches for isoscalar giant resonance strength. Since intermediate-energy proton inelastic scattering is an important source of our knowledge of nuclear single-particle excitations, it is useful to examine the $(p,p'X)$ reaction to gain a better understanding of the continuum excited by the $(p,p')$ reaction. This letter describes the first such intermediate-energy experiment on a complex nucleus, $^{12}\mbox{C}$. For states, or groups of states, up to 20.6 MeV, these results may be compared with those from the reactions $^{11}$B$(d,nX)$, where $X=p$ or $\alpha$ [@neus85].
Data were accumulated for the reactions $^{12}\mbox{C} (p,p'X)$, where $X = p$ or $\alpha$, at the Indiana University Cyclotron Facility. Inclusive $(p,p')$ data were measured simultaneously. A beam of 156 MeV protons with a DC current of 25–70 nA was focussed on a 2.0 mg/$\mbox{cm}^2$ natural-carbon target foil. Scattered protons were detected in the IUCF K600 spectrometer [@berg86] at central scattering-angle settings of 14.3, 19, and 24 deg (momentum transfers $q_{\rm cm}$ of 0.71, 0.93, and 1.16 $\mbox{fm}^{-1}$.) An energy resolution of 140 keV FWHM was obtained, which is dominated by the energy spread of the incident beam. The proton scattering angle was reconstructed for each event to an accuracy of 0.1$^\circ$. Coincident low-energy charged particles were detected in an array of eight silicon detector telescopes arranged to provide maximal coverage of the in-plane angular range. The residual-nucleus mass resolution was better than 250 keV FWHM. For $^{12}\mbox{C}$ excitation energies $14 \leq E_x \leq 28$ MeV, $\alpha$-particle emission to both the ground and first-excited states of $^{8}\mbox{Be}$ were observed, and proton emission was observed to the ground state and several excited states of $^{11}\mbox{B}$. Only the two most prominent decay channels — $\alpha_1$ (to the $J^\pi = 2^+$ first excited state of $^{8}\mbox{Be}$) and $p_0$ (to the $J^\pi = 3/2^-$ $^{11}\mbox{B}$ ground state) — are discussed below.
Fig. \[fig:channels\] shows representative spectra for the inclusive and exclusive reactions at $q = 0.7\mbox{ fm}^{-1}$. Two observations are apparent from these spectra. The first is that the separation of the continuum spectrum into specific decay channels, by requiring the detection of specific coincident decay products, gives the expected reduction of level density. The $p_0$ and $\alpha_1$ gated spectra are both less complex than the inclusive spectrum and have few common features. The second observation is that the coincidence spectra do not display the large nonresonant backgrounds which are normally associated with inclusive spectra. There is little evidence in the coincidence data for [*any*]{} background for $E_x \leq 24$ MeV. A similar observation has been made in the study of the $^{12}\mbox{C}(e,e'p)$ reaction [@wojt93]. These observations led us to two conjectures: 1) the $^{12}\mbox{C}$ continuum at low excitation energy is less complex than commonly believed and can be analyzed in terms of individual excitations, and 2) the common assumption of large nonresonant contributions to the $^{12}\mbox{C}(p,p')$ spectrum in the low-energy (15–25 MeV) continuum is incorrect.
The level-density reduction is best demonstrated in the $(p,p'\alpha_1)$ spectrum. One can clearly see four peaks below 24 MeV: the well-known states [@ajze90] at $E_x$ = 15.4 MeV ($J^\pi = 2^+, T=0$), 16.1 MeV ($J^\pi = 2^+, T=1$), 18.3 MeV ($J^\pi = 2^-, T=0$), and 21.6 MeV ($J^\pi = 2^+, T=0$). The 16.1 MeV $T=1$ state is known to be isospin-mixed and to decay mainly into the $\alpha_1$ channel [@ajze90]. The lack of background strength is illustrated by the $(p,p'p_0)$ spectrum in the inset of Fig. \[fig:channels\]. The region shown (17–21 MeV) has been well studied [@morr79; @neus83; @buen77; @comf82; @jone83; @hick84] and is dominated at this momentum transfer by states at $E_x$=18.3 MeV (the same mentioned above for the $\alpha_1$ channel), 19.4 MeV ($J^\pi=2^-$,$T=1$) and 20.6 MeV (unknown spin and parity). The curve shows the result of a least-squares fit which assumed the presence of only these three peaks, without inclusion of any background. The inclusive spectrum, on the other hand, suggests the presence of significant nonresonant backgrounds. Large backgrounds have been included in earlier analyses of $(p,p')$ spectra in this region of excitation [@buen77; @comf82; @jone83] (see especially [@jone84]).
The $(p,p'\alpha_1)$ spectrum suggests a possible resolution of this background disparity. The long lorentzian tails of the broad states at 15.4 and 21.6 MeV are clearly visible, and contribute significantly to the yield under the peak at 18.3 MeV. Failure to properly represent these broad resonances when fitting the inclusive spectrum would lead to a false identification of a large nonresonant background. This conclusion might even be drawn if these resonances were included but had incorrect widths. The studies [@comf82; @jone83; @comf80], in which gaussian shapes have been used, provide examples of this effect and result in underestimation of the cross section.
One of the motivations for this experiment was to measure the angular correlation functions (ACFs) and to determine the extent of their sensitivity to the angular momentum of the resonance excitations. The ACF is simply the cross section ${\rm d}^5\sigma/{\rm d}\Omega_{p'} {\rm d}\Omega_X {\rm dE}_X$ plotted as a function of $\theta_{X}^{\rm c.m.}$ (at fixed $\theta_{p'}$), defined in the center-of-mass frame of the recoiling $A=12$ system. This is the natural frame for a multipole decomposition of the ACF. The angle $\theta_{X}^{\rm c.m.} = 0$ corresponds to the $(p,p')$ momentum-transfer direction $\hat{q}$. Under the assumptions that the reaction proceeds by sequential resonance excitation and decay, and that the ACFs are independent of the azimuthal angle $\phi_{X}^{\rm c.m.}$, the ACFs can be described by a Legendre-polynomial ($P_\ell$) series [@temp93]. In regions dominated by a single resonance, this model predicts that only even-$\ell$ terms with $\ell_{\rm max} \leq 2J$, where $J$ is the total angular momentum of the resonance, are needed in the fit.
Fig. \[fig:legfits\] displays several ACFs, for the $p_0$ and $\alpha_1$ channels, for $E_x(^{12}\mbox{C})$ values centered on three of the observed resonances. The ACFs are different for the three regions, indicating a sensitivity to nuclear structure. A single resonance appears to dominate the region around 21.6 MeV in the $(p,p'\alpha_1)$ channel. Several experiments ([@ried78; @dean95] and references therein) assign $J^\pi = 2^+,\ T=0$ to this level. The Legendre polynomial fit shown in the figure has large reduced coefficients ($b_\ell$) for $P_2$ and $P_4$, and those for $P_1$ and $P_3$ are an order of magnitude smaller, consistent with the $2^+$ assignment. The other prominent peak in the $(p,p'\alpha_1)$ spectrum at E$_x$=24.4 MeV has an ACF similar in shape to the 21.6 MeV resonance, suggesting that it too is a $J^\pi = 2^+,\ T=0$ level.
The experiments of [@zwar85; @suko87] have demonstrated that significant quasifree knockout strength leads to large fore-aft asymmetries in the ACFs. No such asymmetry is seen in the two ACFs at lower $E_x$. This further supports the conjecture that little nonresonant background exists in this region.
The channel cross section at a given $E_x$ is given by the integral over $\d \Omega_X$ of the ACF. If the assumption of $\phi_{X}^{\rm c.m.}$-independence for the ACFs holds, this cross section is given by $4\pi a_0$, where $a_0$ is the coefficient of $P_0(\theta)$ in the Legendre-polynomial fit to the data. These cross sections in turn indicate the channel composition of the continuum seen by inclusive $^{12}\mbox{C}(p,p')$. The deduced angle-integrated cross sections for $\alpha_1$ and for $p_0$ account for respectively 24% and 36% of the inclusive cross section for $17 \leq E_x \leq 24$ MeV. The corresponding $n_0$ contribution over this region cannot be deduced from the data; only $s$-wave emission will be important close to the threshold at 18.72 MeV, with $d$-wave emission rising to 50% of $p_0$ at about 22.7 MeV. However, estimates can be made for specific cases. For the peak at 19.4 MeV, $p_0$ and $\alpha_1$ emission account for 48% and 20% of the inclusive cross section at low $q$, where $2^{-}$ strength is known to dominate. The shell-model $n_0$ width for this $2^{-}$ state should be enhanced by the isospin mixing observed in pion scattering [@morr79]. A 5% isospin mixing by intensity, chosen to fit the ratio of $(e,e')$ cross sections [@hick84], of the second $2^-$ shell-model states with $T=0$ and $T=1$ raises the $n_0$ contribution from 16% to 40%, which means that all the inclusive cross section is accounted for. The measured width for this state (see Table \[tab:compare\]) is well reproduced by the calculated nucleon decay width of 468 keV ($\Gamma_{p_0}(d) = 86$ keV, $\Gamma_{p_0}(s) = 168$ keV and $\Gamma_{n_0}(s) = 214$ keV) and the measured $\alpha_1$ width ($\sim 100$ keV.)
The conjectures about the $(p,p')$ continuum, reinforced by this quantitative analysis of the $(p,p'X)$ coincidence data, necessitate a reanalysis of the inclusive data. This analysis was performed on the present $(p,p')$ data sorted into twelve spectra, each corresponding to a one-degree bin of scattering angle $\theta_{p'}$. The twelve spectra were fit simultaneously using identical peak centroids and widths. The background contribution was represented in the fit by a lorentzian function whose centroid, width, and area varied freely at each angle. It is likely that this background represents the low-energy tails of higher-excitation resonances. Broader resonance structures are clearly evident in the $(p,p'\alpha_1)$ spectrum in Fig. \[fig:channels\]. A more quantative treatment of this excitation region would require response functions from a continuum shell model calculation since the broad structure near 25 MeV cannot be represented by a single lorentzian lineshape. Fig. \[fig-incl-peakfit\] shows the results of this analysis for one of the twelve spectra. Comparable representations were obtained for all spectra using a fit that includes thirteen resonances and one background function. The peak parameters for the most prominent of these resonances are tabulated in Table \[tab:compare\]. Also listed are the corresponding data of [@buen77] for comparison. The errors on the peak positions and widths are small since a weighted mean has been made of the results for each of the twelve spectra for most cases. The correlations between the parameters in the fit are only partially included in these errors.
A comparison of the current inclusive analysis with previous data provides information about the effect of background overestimation on the extracted cross sections. Only the pioneering work of Buenerd [*et al.*]{} [@buen77] reports cross sections over the range 14–25 MeV, although the data of Comfort [*et al.*]{} [@comf82; @comf81] extend to the lower edge of the current experiment’s energy acceptance.
Fig. \[fig:bigfig\] displays cross section results for several states along with results from the previous experiments. The data of [@comf82] for the states at 18.3, 19.4, and 19.7 MeV are from a 200 MeV experiment; Comfort has shown previously [@comf81] that $(p,p')$ cross sections over the range 120-200 MeV have a weak beam-energy dependence. The three datasets are in good agreement for the sharp states at 15.1 ($J^\pi = 1^+,\ T = 1$) and 16.1 ($J^\pi = 2^+,\ T = 1$) MeV (not shown), and with half the cross sections for the corresponding states in $(p,n)$ and $(n,p)$ reactions ([@ande96] and references therein). For both states, the current data are identical within errors to that of [@comf81], while those of [@buen77] agree in shape but are lower by a factor 1.6. For the remaining states, the data of [@buen77] agree well in shape with the current results, but usually not in magnitude. Since Comfort [*et al.*]{} [@comf82] made no attempt to decompose the strong peak at 19.7 MeV into $2^-$ and $4^-$ components, comparisons should be made for the summed strength in the 19.4-MeV and 19.7-MeV panels in Fig. \[fig:bigfig\]. The figure illustrates nicely how resonance cross-section results depend strongly on the assumptions of the peak-fitting procedure, and how previous experiments have tended to underestimate the cross sections. With two exceptions, our cross sections are everywhere larger or equal to those of the other two experiments.
A comparison with the results of nucleon charge-exchange reactions corroborates the conclusions reached above. Simple isospin-symmetry arguments predict that the $(p,p')$ cross sections for a given peak should be at least half that of the corresponding peak in $(p,n)$ or $(n,p)$ reactions. For the region around 19.5 MeV at the lowest $q$ measured, near the maximum for the $2^-$ T=1 state which dominates, the $(p,n)$ cross section of [@ande96], also extracted using Lorentzian lineshapes, is equal to twice the summed cross section for the 19.4-MeV and 19.7-MeV peaks in Fig. \[fig:bigfig\]. The cross sections of Comfort at 200 MeV [@comf82] and Jones [@jone83] at higher energies fall well short of expectations based on the charge-exchange data, the more so of other recent data [@yang93] and several older data sets (see references in [@ande96]). Given the similar backgrounds subtracted for the 19.4-MeV and 18.3-MeV peaks [@jone84], the discrepancies seen in Fig. \[fig:bigfig\] for the 18.3-MeV peak are not surprising.
DWIA calculations were compared with the new inclusive cross sections in order to associate the resonances with levels from shell-model calculations [@hick84; @brad91; @ande96] using interactions from [@cohe65; @mill75]. Calculations were performed using the code DW86 [@scha70], the NN effective interaction of [@fran85], the shell-model one-body density-matrix elements with harmonic oscillator wave functions ($b_0 = 1.669$ fm or $b_{rel} = 1.743$ fm [@brad91; @ande96]) and the $^{12}\mbox{C}$ optical potential of [@comp74]. The results of the calculations are displayed in Fig. \[fig:bigfig\]; any normalization factors by which the curves are scaled are shown in parentheses. The normalization factors are also listed in Table \[tab:compare\]. When a specific transition density is identified with a peak, the subscript indicates which of the several states for each $(J^\pi , T)$ was used. The agreement in most cases is good, and the normalization factors have reasonable sizes in the sense that quenching of the shell-model transition densities is expected for all the cases listed in Table \[tab:compare\] [@brad91; @ande96].
Two curves for $J^\pi = 4^-$ excitation are shown with the data for the state at 19.7 MeV. Individually, the normalization factors for these curves are twice as large as those characteristic of the other states but the normalization factor of 0.55 for the sum of the T=0 and T=1 $4^-$ states is consistent with that required for the T=1 state in the $(p,n)$ reaction [@ande96]. A strongly isospin-mixed $J^\pi = 4^-$ doublet, with peaks at 19.25 and 19.65 MeV, was observed in a $\pi^+/\pi^-$ inelastic-scattering experiment [@morr79]. No peak at 19.25 MeV was observed in the current experiment. Its absence suggests that the isospin mixing results in all of the $(p,p')$ strength going to the 19.7 MeV state. However, DWIA calculations for a more proton-like lower state do not support this hypothesis. The fact that the $4^-_2$ and $4^-_3$ T=0 shell-model states are closely spaced and share the $L=3$ $S=1$ excitation strength may complicate the isospin mixing calculation.
The level at 20.6 MeV has been one of the most difficult states to explain in (see [@neus83; @comf82; @hick84; @ande96] and references therein.) States with $J^\pi = 3^+$ and $3^-$, both with $T=1$, are certainly present [@ajze90] but the $(p,n)$ results of [@ande96] suggest that T=1 states account for only 25% of the cross section shown in Fig. \[fig:bigfig\]. Neither of these states can account for the large $^{11}$B$(d,n)$ cross section for the 20.6-MeV peak [@neus83]. However, the $J^\pi = 3^-_4, T=0$ state included in Fig. \[fig:bigfig\] is predicted to lie in the energy region and has a large ground-state spectroscopic factor of 0.56 (mainly $d_{3/2}$). None of these calculations reproduce the data alone, although the sum of the three has at least the correct order of magnitude. Recent $(\pol{d},\pol{d}')$ experiments [@john95] indicate an isoscalar $J^\pi = 1^+$ resonance at 20.5 MeV, but none of the unassigned $1^+,T=0$ wavefunctions of [@cohe65] have significant $(p,p')$ strength. The nature of the states in this region remains unclear.
Evidence was presented above which was consistent with a $J^\pi = 2^+ \
T=0$ assignment for the 21.6 MeV state. For this state, as well as for the 15.4-MeV $2^+$ T=0 state, neither the large inclusive cross section nor the large postive analysing power (not shown) over the $q$ range measured can be reproduced by a $0\hbar\omega$ [@cohe65] shell-model wavefunction. Essentially all the $0\hbar\omega$ $E2$ strength is contained in one state, which corresponds to the 4.44-MeV level. A modest fraction of the $2\hbar\omega$ giant-quadrupole strength built on the $0\hbar\omega$ ground state is required to explain the cross sections and analysing powers - roughly 16% in the case of the of the 21.6 MeV state.
In summary, a $^{12}\mbox{C}(p,p'X)$ experiment has been performed at a beam energy of 156 MeV. Analysis of the coincidence data indicates that the nonresonant component of the low-energy ($14 \leq E_x < 24$ MeV) continuum is much smaller than commonly accepted. The measured angular correlation functions follow the pattern expected for a resonance excitation-decay process, which also points to relatively small nonresonant contributions. The ACFs display sensitivity to nuclear structure.
The inclusive $(p,p')$ data, measured simultaneously in this experiment, were analyzed using more physically motivated peak shapes than used in earlier analyses. This reanalysis produces cross sections that are, in general, much larger than deduced from previous $^{12}\mbox{C}(p,p')$ experiments. Theoretical calculations agree quite well with the present data and require renormalization factors which are qualitatively in line with the quenching expected and to those needed for the corresponding T=1 states excited in recent charge-exchange reactions. At the low momentum transfers probed in the present experiment, we have deduced that the low-energy ($21 < E_x < 24$ MeV) continuum of $^{12}$C is dominated by 1$^-$, T=1 resonances which decay primarily by single-nucleon emission, and 2$^+$, T=0 resonances which decay primarily by alpha emission. There is little, if any, non-resonant background in the spectrum. This is a qualitatively different view of the continuum than has been presented in earlier works. Instead of viewing the continuum as giant resonances sitting atop a sizable non-resonant background, the present analysis suggests that [*most*]{} of the continuum cross section is due to resonance excitation, with clear peak structures melting into the familiar smooth continuum as the width of the overlapping resonances increases with increasing excitation energy.
We thank Dr. J. Comfort for providing a tabulation of his data. This work was financially supported by the National Science Foundation and by the US Department of Energy under Contract No. DE-AC02-76CH00016. One of the authors (J.A.T.) was funded by a National Science Foundation Graduate Fellowship during a portion of this project.
[99]{}
K. T. Knöpfle, in [*Nuclear Structure at High Spin, Excitation, and Momentum Transfer*]{}, edited by H. Nann, AIP Conf. Proc. 142 (AIP, New York, 1986), p. 68; Th. Kihm [*et al.*]{}, Phys. Rev. Lett. 56 (1986) 2789.
F. Zwarts, A. G. Drentje, M. N. Harakeh and A. van der Woude, Nucl. Phys. A 439 (1985) 117.
G. D’Erasmo, I. Iori, S. Micheletti and A. Pantaleo, Z. Phys. A 299 (1981) 41.
B. A. Raue [*et al.*]{}, Phys. Rev. C 52 (1995) R445.
H. Riedsel [*et al.*]{}, Phys. Rev. Lett. 41 (1978) 377.
J. R. Calarco [*et al.*]{}, Phys. Lett. B 146 (1984) 179.
J. R. Calarco, Nucl. Phys. A 569 (1994) 363c.
D. J. DeAngelis [*et al.*]{}, Phys. Rev. C 52 (1995) 61.
G. H. Neuschafer and S. L. Tabor, Phys. Rev. C 31 (1985) 334.
G. A. P. Berg [*et al.*]{}, IUCF Scientific and Technical Report, 1986.
B. B. Wojtsekhowski, Physics on Internal Targets at NEP Ring, Supplement No. 1 to Research Report of Nuclear Science, Tohoku University, Vol. 26, 1993.
F. Ajzenberg-Selove, Nucl. Phys. A 506 (1990) 1.
C. L. Morris [*et al.*]{}, Phys. Lett. B 86 (1979) 31; W. B. Cottingame [*et al.*]{}, Phys. Rev. C 36 (1987) 230.
G. H. Neuschafer, M. N. Stephens, S. L. Tabor, and K. W. Kemper, Phys. Rev. C 28 (1983) 1594.
M. Buenerd, P. Martin, P. de Saintignon and J. M. Loiseaux, Nucl. Phys. A 286 (1977) 377; M. Buenerd, Phys. Rev. C 13 (1976) 444.
J. R. Comfort [*et al.*]{}, Phys. Rev. C 26 (1982) 1800.
K. W. Jones [*et al.*]{}, Phys. Lett. B 128 (1983) 281; Phys. Rev. C 50 (1994) 1982.
R. S. Hicks [*et al.*]{}, Phys. Rev. C 30 (1984) 1.
K. W. Jones, Ph.D. thesis, Rutgers University, 1984 (LANL publication LA-10064-T).
J. R. Comfort [*et al.*]{}, Phys. Rev. C 21 (1980) 2147.
J. A. Templon, Ph.D. Thesis, Indiana University, 1993.
C. Sükösd [*et al.*]{}, Nucl. Phys. A 467 (1987) 365.
J. R. Comfort [*et al.*]{}, Phys. Rev. C 23 (1981) 1858.
F. P. Brady [*et al.*]{}, Phys. Rev. C 43 (1991) 2284.
B. D. Anderson [*et al.*]{}, Phys. Rev. C 54 (1996) 237.
X. Yang [*et al.*]{}, Phys. Rev. C 48 (1993) 1158.
S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1.
D. J. Millener and D. Kurath, Nucl. Phys. A 255 (1975) 315.
R. Schaeffer and J. Raynal, Technical Report No. CEA-R4000, Saclay, (unpublished); IUCF version modified by J.R. Comfort, W. Bauhoff and C. Olmer.
M. A. Franey and W. G. Love, Phys. Rev. C 31 (1985) 488.
V. Comparat [*et al.*]{}, Nucl. Phys. A 221 (1974) 403.
B. N. Johnson [*et al.*]{}, Phys. Rev. C 51 (1995) 1726.
----------------- ---------------- ----------------- ---------- ----------------- ----------------
$E_x$ $\Gamma$ $J^\pi \ \ (T)$ [DWIA]{} $E_x$ $\Gamma$
(MeV $\pm$ keV) (keV) (MeV $\pm$ keV) (keV)
$15.38 \pm 30$ $2800 \pm 170$ $2^+\ \ (0)$ — $15.3 \pm 200$ $2000 \pm 200$
$16.62 \pm 10$ $280 \pm 30$ $2^-\ \ (1)$ — — —
$18.292 \pm 4$ $486 \pm 10$ $2^-_2\ \ (0)$ 0.80 $18.35 \pm 50$ $400 \pm 100$
$19.394 \pm 10$ $520 \pm 30$ $2^-_2\ \ (1)$ 0.38 $19.4 \pm 50$ $530 \pm 100$
$19.671 \pm 6$ $490 \pm 20$ $4^-\ \ (0+1)$ 0.58 $19.6 \pm 50$ $500 \pm 100$
$20.584 \pm 5$ $440 \pm 11$ — — $20.6 \pm 80$ $450 \pm 150$
$21.61 \pm 20$ $1450 \pm 90$ $2^+\ \ (0)$ — $21.3 \pm 250$ $950 \pm 300$
$21.99 \pm 20$ $550 \pm 90$ $1^-_3\ \ (1)$ 0.78 $21.95 \pm 150$ $800 \pm 100$
$22.72 \pm 30$ $1200 \pm 130$ $1^-_4\ \ (1)$ 0.60 $22.6 \pm 150$ $900 \pm 100$
$23.57 \pm 20$ $238 \pm 34$ $1^-\ \ (1)$ $23.50$ $230$
$24.04 \pm 18$ $659 \pm 48$ $1^-\ \ (1)$ — $23.92$ $400$
$24.38 \pm 10$ $671 \pm 49$ $2^+\ \ (0)$ — — —
----------------- ---------------- ----------------- ---------- ----------------- ----------------
: Comparison of current and previous continuum $^{12}\mbox{C}(p,p')$ results (see text.) The subscripts on the spin $J$ indicate the specific wavefunction of [@cohe65; @mill75] which gave the best fit to the data in the DWIA calculation. Quantum numbers without a subscripted $J$ are suggested by systematics or determined in other experiments [@ajze90]. “DWIA Scaling” refers to the best-fit multiplicative factor applied to DWIA calculations.[]{data-label="tab:compare"}
|
---
abstract: 'New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ‘modern’ treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. A conceptual proof is presented of a result of Rohn concerning curves in $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$. The string models of Series XIII (of some ruled quartic surfaces) are based on Rohn’s result.'
address:
- 'Department Matesco, University of Cantabria, Avda. Castros s/n, 39005 Santander, Spain.'
- 'IWI-RuG, University of Groningen, Nijenborgh 9, 9747 AG Groningen, the Netherlands.'
author:
- 'Irene Polo-Blanco, Marius van der Put and Jaap Top'
title: 'Ruled quartic surfaces, models and classification'
---
\[section\]\[definition\][Theorem]{} \[definition\][Lemma]{} \[definition\][Proposition]{} \[definition\][Examples]{} \[definition\][Corollary]{} \[definition\][Remark]{} \[definition\][Remarks]{} \[definition\][Exercise]{} \[definition\][Example]{} \[definition\][Observation]{} \[definition\][Observations]{} \[definition\][Algorithm]{} \[definition\][Criterion]{} \[definition\][Algorithm and criterion]{}
Motivation and History {#motivation-and-history .unnumbered}
======================
The collection of string models of ruled quartic surfaces, present at some mathematical institutes (for instance at the department of mathematics in Groningen) is the direct motivation for this paper. This Series XIII, produced by Martin Schilling in 1886, is based upon a paper of K. Rohn [@Rohn2] containing a classification of ruled quartic surfaces over $\mathbb{C}$ and $\mathbb{R}$. Some authors before Rohn (e.g., M. Chasles [@Chas], A. Cayley [@Cay], L. Cremona [@Cre], R. Sturm [@Sturm], G. Salmon [@Sal]) and many after his time (e.g., B.C. Wong [@Wong], H. Mohrmann [@Mohr], W.Fr. Meyer [@Mey], W.L. Edge [@Ed], O. Bottema [@Bot], T. Urabe [@Ura2]) have contributed to this beautiful topic of 19th century geometry.\
Cremona classified the ruled quartic surfaces in 12 types. He states in [@Cre] that Cayley produced 8 of these without revealing his method. However, Cayley’s third memoir on this subject [@Cay] was written earlier the same year 1868 and contains 10 types. In an addition to this memoir (May 18, 1869), Cayley gives the comparison between his own classification and the one by Cremona and makes it clear what the two types he missed are. The method of Cayley consists of taking three curves in $\mathbb{P}^3$ and to consider the ruled surface $S$ which is the union of the lines meeting all three curves. Using a formula for the degree of $S$, he now computes possibilities of ruled quartic surfaces. The expression ‘the six coordinates of a line’ in Cayley’s work indicates that the Grassmann variety $Gr(2,4)$ of the lines in $\mathbb{P}^3$ plays a role. The work of Cayley contains also explicit calculations for reciprocal surfaces (see below).\
The results of Cremona can be explained as follows. Let $S\subset \mathbb{P}^3$ be a ruled quartic surface (reduced, irreducible and defined over $\mathbb{C}$). The fact that through a general point of $S$ there is only one line of $S$ is tacitly assumed (compare Lemma \[1.2.2\]). The locus $D$ of the points on $S$ through which there are at least two lines of $S$ (in the 1-parameter family) is called the ‘double curve’. Cremona states that $D$ is indeed a curve (hereby excluding cones) and has ‘in general’ degree 3.
(We note that $D$ need not coincide with the singular locus of $S$, that $D$ can also have degree 2 (see [*Number*]{} $15$ in Subsection \[s1.6\]) and that $D=\emptyset$ is possible; compare, for example, Corollary \[1.2.7\](2).)
Two intersecting lines on $S$ determine a plane. The collection $\check{D}$ of all these planes is called the ‘bitangent developable’. This 1-dimensional family (assuming $D\neq \emptyset$) can be seen as a curve in the dual projective space. The genus of $S$ is defined as the genus of the (irreducible, singular) curve $H\cap S$ of degree 4, where $H$ is a general plane. Cremona states that the genus can only be $0$ or $1$. Missing is the nontrivial argument showing that genus 2 is impossible (see Observation \[1.2.obs\] and [@Ura2 Proposition 2.6]). Cremona classifies $S$ according to the nature (degrees and multiplicities of the irreducible components) of the curves $D$ and $\check{D}$ (and in one case a relation between $D$ and $\check{D}$). He obtains his list of possibilities via the following construction:\
Consider a tuple $(C_1,C_2,f)$ consisting of two conics $C_1, C_2\subset \mathbb{P}^3$, not in the same plane, and an isomorphism $f:C_1\rightarrow C_2$. This defines a ruled surface $S$ which is the union of the lines through the pairs of points $\{c_1,f(c_1)\}$ with $c_1\in C_1$. In the general case, the line $H_1\cap H_2$, where $C_i$ lies in the plane $H_i$ for $i=1,2$, intersects $C_1$ in two points $p_1\neq q_1$ and intersects $C_2$ in two points $p_2\neq q_2$. Now $H_2\cap S$ is the union of the conic $C_2$ and the two lines through the pairs of points $(p_1,f(p_1))$ and $(q_1,f(q_1))$. Thus $S$ is an irreducible ruled surface of degree 4. Moreover, the two lines intersect in a point of the ‘double curve’ and $H_2$ is a ‘bitangent plane’, i.e., a point on the ‘bitangent developable.’ The same holds of course for $H_1$. Cremona’s examples are obtained by varying and degenerating $C_1,C_2,f$. His assertion to have found all types in this way is not correct since some ruled quartic surfaces are only obtained from a line and a curve of degree 3. However by including ‘reciprocal surfaces’ and maybe stretching the meaning of ‘degeneration’ some of the latter surfaces can be obtained.
The approach of Cayley (and of Rohn) has the classical name “analytic geometry”, indicating the use of coordinates and algebraic operations with formulas. In contrast, Cremona’s (and Sturm’s) approach is purely “synthetic”. As a consequence, Cremona’s paper is difficult to read and it is hard to verify the results.
Bottema [@Bot p. 349] remarks that Rohn claims to have discovered a type overlooked by his predecessors. Indeed, on p. 147 of Rohn’s paper [@Rohn1], there is an explicit equation and the remark in a footnote: “this ruled surface is not mentioned by Cremona in his treatise”. However, it is easy to verify that Rohn’s equation (in the homogeneous coordinates $x,y,z,w$) $$wx^2(x+3Nz)+F_4(x,y)=0, F_4\mbox{ a binary quartic}, N\mbox{ a constant,}$$ does *not* define a ruled surface, since a general point $(a:b:c:1)$ on it is not contained in any line of the surface. Actually, Rohn’s geometric construction is valid but his formula happens to be mistaken. The construction gives indeed a case which is not explicitly mentioned by Cremona. However it can be interpreted as hiding in Cremona’s species 10. Pascal’s well written Repertorium reviews the classification of Cayley and Cremona, [@Pas], XII, §10. Here Rohn’s extra case reappears on p. 338-339 with the same correct geometric construction and another mistake in the formula. In the classification of the present paper Rohn’s example is [*Number*]{} 5.
There are also critical comments by R. Sturm to the list of Cremona. Moreover, some of the 12 species of Cremona contain surfaces of a rather different nature, as we will see in Section \[section3\].
The classification of ruled quartic surfaces in the book of W.L. Edge [@Ed] is identical with the one of Cremona. Two methods are developed there. The first one classifies the curves (irreducible and of degree 4), corresponding to ruled quartic surfaces, in the Grassmann variety $Gr(2,4)$ (parametrizing the 2-dimensional subspaces of a 4-dimensional vector, or, equivalently, the lines in $\mathbb{P}^3$). The second method obtains the ruled quartic surfaces in $\mathbb{P}^3$ as projections of certain ruled quartic surfaces in $\mathbb{P}^5$ or $\mathbb{P}^4$. This is related to a paper of C. Segre [@Seg] and to a paper by Swinnerton–Dyer[@Swin].\
In the thesis of Wong [@Wong], a rational morphism $\mathbb{P}^3 \cdots\to Gr(2,4)$, associated to the classical ‘tetrahedral complex’, is considered. Certain plane curves in $\mathbb{P}^3$ of degree 2 and 3 have as images in $Gr(2,4)$ curves of degree 4 and correspond therefore to ruled quartic surfaces. It is claimed in this thesis that every ‘species’ in Cremona’s list can be obtained in this way.\
For other details on the early history of the subject we refer to the contribution of W.Fr. Meyer in [@Mey].\
A ruled surface in modern terminology (see [@Har Section V.2.]), is a morphism $Z\rightarrow C$ of a smooth projective surface $Z$ to a smooth curve $C$ such that all fibres are isomorphic to $\mathbb{P}^1$. A classical ruled surface $S\subset \mathbb{P}^3$ is obtained as the image of a suitable morphism $Z\rightarrow\mathbb{P}^3$. This method and the papers of T. Urabe [@Ura1], [@Ura2], [@Ura3], [@Ura4] may lead to a modern classification of ruled quartic surfaces (including moduli). We note, in passing, that Urabe’s important work concerns the discovery of new aspects in the classification of the singularities of quartic surfaces in $\mathbb{P}^3$ (which, generally, are not ruled) and their relation to Dynkin diagrams.\
One aim of the present paper is to give a modern treatment of Rohn’s paper [@Rohn2], namely the‘symmetrization’ of curves in $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$ and the classification of some ruled quartic surfaces over $\mathbb{R}$. The latter is used to obtain explicit equations explaining the visual features of the models of Series XIII.\
The possibilities for the 1-dimensional part of the singular locus of a ruled surface $S$ can be read off from the intersection of $S$ with a general plane. This leads to the elegant elementary treatment of ruled cubic surfaces in Dolgachev’s book [@Dol]. As a didactical step towards ruled quartic surfaces, we present here another method valid over any base field and obtain the three types of ruled cubic surfaces over $\mathbb{R}$.
The other aim of this paper is to present a classification of the quartic ruled surfaces, such that each class is determined by discrete data and the surfaces belonging to a given class give rise to a connected moduli space. This leads to $29$ cases. A combination of the following methods leads to this classification.\
(1). Deriving some properties of the curves $C$ of degree 4 (corresponding to ruled quartic surfaces) lying on the Grassmann variety $Gr(2,4)\subset \mathbb{P}^5$.\
(2). Determining the possibilities for the singular locus of a quartic ruled surface.\
(3). The normalization $C^{norm}$ of $C$ carries a vector bundle of rank two. In case the genus of $C^{norm}$ is 0, there are two possibilities for this vector bundle. Two ‘generating’ meromorphic sections of this vector bundle are brought in some standard form, by some linear base changes. This leads to explicit equations for the corresponding quartic ruled surfaces.\
(4). Classification of the position of $C$ w.r.t. the tangent spaces of $Gr(2,4)$.\
Part (4) is in fact one of the two methods of [@Ed] in deriving Cremona’s list. Although we could not verify this in detail because of a certain vagueness in Edge’s arguments, the results agree with our computations.\
Before giving Cremona’s list we need to explain to notion of ‘[*reciprocal surface*]{}’ or ‘[*dual surface*]{}’ in more modern terms, of a surface $S\subset \mathbb{P}^3$. It is obtained by considering all tangent planes at the nonsingular points of $S$. Each tangent plane is a point in the dual projective space $\check{\mathbb{P}^3}$. The Zariski closure of all these points is the dual surface $\check{S}\subset \check{\mathbb{P}^3}$. In the case that $S$ is ruled, also $\check{S}$ is ruled and has the same degree as $S$. Moreover, the ‘double line’ of $\check{S}$ can be seen to be the ‘bitangent developable’. Cremona’s list shows that the species 3 and 4 are dual, as well as the species 7 and 8. The other species are ‘selfdual’. In the table one has to give ‘double curve’ the interpretation ‘singular locus’. The genus $g$ of the surface is 0, except for Cremona 11, 12 where it is 1.
We adopt a [*notation of Cayley*]{}, namely the expression $d^m$ stands for an irreducible component of the singular locus of degree $d$ and with multiplicity $m$. The difference between Cremona 6 and Cremona 11 is that the two lines intersect in the first case and are skew in the second one. The difference between Cremona 9 and Cremona 10 is somewhat subtle. In case 10 the bitangent planes are the planes containing the singular line, denoted by $1$. In case 9, the bitangent planes are the planes containing another line, denoted by $1'$.
$ \begin{array}{| c | c | l|| c |c | l|} \hline
\mbox{Double} & \mbox{Double curve} & \mbox{Cremona} &
\mbox{Double} & \mbox{Double curve} & \mbox{Cremona}\\
\mbox{curve} & \mbox{recip. surface} & \mbox{type} & \mbox{curve} &
\mbox{recip. surface} & \mbox{type} \\
\hline 3^2&3^2&1&3^2&1^3 &7\\ \hline
2^2,1^2&2^2,1^2&2&1^3&3^2 &8\\ \hline
1^3&2^2,1^2&3&1^3&{1'}^3&9\\ \hline
2^2,1^2& 1^3&4& 1^3& 1^3&10\\ \hline
1^2,1^2,1^2&1^2,1^2,1^2&5&1^2,1^2&1^2,1^2&11,\ g=1\\ \hline
1^2,1^2\mbox{ int} &1^2,1^2\mbox{ int}&6 &1^2&1^2&12,\ g=1\\ \hline
\end{array} $
\
Some models from Series XIII: nr. 1, 4, and 5.
Curves on the Grassmann variety $Gr(2,4)$
=========================================
Properties of the Grassmann variety
-----------------------------------
Let $V$ be a vector space of dimension 4 over the (algebraically closed) field $K$. The lines in theprojective space $\mathbb{P}(V)\cong \mathbb{P}^3$ are points of the Grassmann variety $Gr:=Gr(2,V)=Gr(2,4)$ and the natural way to study a ruled surface $S\subset \mathbb{P}(V)$ is to consider the set of the lines on $S$ as subset of $Gr$. We briefly define $Gr$ and summarize its main properties.
For notational convenience we fix a basis $e_1,e_2,e_3,e_4$ of $V$ and we identify the exterior power$\Lambda ^4 V$ with $K$ by $e_1\wedge \cdots \wedge e_4 \mapsto 1$. The obvious symmetric bilinear map $\Lambda ^2V\times \Lambda ^2V\rightarrow \Lambda ^4V=K$ is nondegenerate. A line in $\mathbb{P}(V)$ correspond to a plane $W\subset V$, a line $\Lambda ^2W\subset \Lambda ^2V$ and to a point in $\mathbb{P}(\Lambda ^2V)\cong \mathbb{P}^5$. If $W$ has basis $v_1,v_2$, then $w=v_1\wedge v_2$ is a basis vector for $\Lambda ^2W$ and $\overline{w}:=Kw$ is this point of $\mathbb{P}(\Lambda ^2V)$. By definition $Gr=Gr(2,V)\subset \mathbb{P}(\Lambda ^2V)$ consists of all these points. Now $\overline{w}$ (with $w\in \Lambda ^2V,\ w\neq 0$) belongs to $Gr$ if and only if $w$ is decomposable, i.e., has the form $v_1\wedge v_2$. The latter is equivalent to $w\wedge w=0$. We use the six elements $e_{ij}:=e_i\wedge e_j,\ i<j$ as basis for $\Lambda ^2V$ and write an element of this vector spaceas $\sum _{i<j} p_{ij}e_{ij}$. The $p_{ij}$ are called the Plücker coordinates. They also serve as homogeneous coordinates for $\mathbb{P}(\Lambda ^2V)$. One finds that $Gr$ is the nondegenerate quadric given by the equation $p_{12}p_{34}-p_{13}p_{24}+p_{14}p_{23}=0$. For notational purposes and for convenience of the reader we recall the following.\
[**List of properties of $Gr$**]{} (of importance for our purposes).
1. $p_0,\ \ell _0, h_0$ are a point, a line and a plane of $\mathbb{P}(V)=\mathbb{P}^3$. One identifies $p_0$ with a $\overline{v}_0,\ v_0\in V,\ v_0\neq 0$ and $\ell _0$ with a $\overline{w}_0,\ w_0\in \Lambda ^2V,\ w_0\wedge w_0=0$. [*We note that $\overline{w}\in \mathbb{P}(\Lambda ^2V)$ with $w\wedge w=0$ is both seen as a point of $Gr$ and as a line in $\mathbb{P}(V)$*]{}.
2. Two lines $\overline{w}_1,\overline{w}_2$ of $\mathbb{P}(V)$ intersect if and only if $w_1\wedge w_2=0$.
3. Every hyperplane of $\mathbb{P}(\Lambda ^2V)$ has the form $\{\overline{z}| \ w\wedge z=0\}$ with $w \in \Lambda ^2V, \ w\neq 0$ and unique $\overline{w}$. If $w$ is indecomposable, then the intersection of the hyperplane with $Gr$ is a nondegenerate quadric. If $w$ is decomposable, i.e., $\overline{w}=\overline{w}_0\in Gr$ and correspond to the line $\ell _0$, then the hyperplane is the tangent plane $T_{Gr,\overline{w}_0}$ of $Gr$ at $\overline{w}_0$. The intersection $T_{Gr,\overline{w}_0}\cap Gr$ is singular and can be identified with the cone in $\mathbb{P}^4$ over a nonsingular quadric in $\mathbb{P}^3$. This intersection identifies with $\sigma _1(\ell _0):=$ the collection of all lines $\ell$ with $\ell \cap \ell _0\neq \emptyset$. Consider for example $w_0=e_1\wedge e_2$. This intersection is now $\{\{p_{ij}\}|\ p_{34}=0,\ -p_{13}p_{24}+p_{14}p_{23}=0\}$. The vertex $\overline{w}_0$ of this cone is its only singular point.
4. $\sigma _2(p_0):=$ the collection of all lines through $p_0$; this is a 2-plane in $Gr$. Indeed, take $p_0=\overline{e}_1$. Then $$\sigma _2(p_0)=
\left\{\overline{\sum _{1<j\leq 4}p_{1j}e_1\wedge e_j}| \mbox{ no relations}\right\}$$ (called a $\omega$-plane in [@Ed]).
5. $\sigma _{1,1}(h_0):=$ the collection of all lines in the plane $h_0$. This is a 2-plane in $Gr$. Indeed, take $h_0=\overline{<e_1,e_2,e_3>}$. Then this collection identifies with $\{ \overline{\sum _{1\leq i<j\leq 3}p_{ij}e_i\wedge e_j}|\mbox{ no relations }\}$ (called a $\rho$-plane in [@Ed]).
6. $\sigma _{2,1}(p_0,h_0):=$ the collection of all lines in $h_0$ through $p_0$. This is a line on $Gr$. Indeed, take $h_0=\overline{<e_1,e_2,e_3>},\ p_0=\overline{e}_1$. Then this collection identifies with $\{\overline{\sum _{j=2,3}p_{1j}e_1\wedge e_j}|\mbox{ no relations}\}$.
7. Every plane in $Gr$ has the form $\sigma _2 (p_0)$ or $\sigma _{1,1}(h_0)$. Every line in $Gr$ has the form $\sigma _{2,1}(p_0,h_0)$ and is thus the intersection of a (uniquely determined) pair of 2-planes in $Gr$ of different type.
8. The are three types of projective subspaces $P\subset \mathbb{P}(\Lambda ^2V)$ of dimension 3 with respect to their relation with $Gr$, namely:\
(a) $Gr\cap P$ is a nondegenerate quartic surface. The equations of $P$ are $p_{12}=p_{34}=0$ for a suitable basis of $V$. Moreover $P$ lies in precisely two tangent space, namely $T_{Gr,\ \overline{e_{12}}}$ and $T_{Gr,\ \overline{e_{34}}}$.\
(b) $Gr\cap P$ is an irreducible degenerate quartic surface. The equations of $P$ are $p_{34}=p_{13}+p_{24}=0$ for a suitable basis of $V$. Now $P$ lies on only one tangent space, namely $T_{Gr,\ \overline{e_{12}}}$ and $Gr\cap P$ is the cone $p_{13}^2+p_{14}p_{23}=p_{34}=p_{13}+p_{24}=0$ over the quadric curve $p_{13}^2+p_{14}p_{23}=0$.\
(c) $Gr\cap P$ is reducible. The equations for $P$ are $p_{12}=p_{13}=0$ for a suitable basis of $V$. Further, $Gr\cap P$ is the union of the planes $p_{14}=0$ and $p_{23}=0$. $\square$
Ruled surfaces and curves on $Gr$
---------------------------------
\[1.2.1\] [(1)]{}. Let $C\subset Gr$ be an irreducible curve of degree $d\geq 2$, not lying in some 2-plane $\sigma _2 (p_0)$. Then $\tilde{S}:=\{(\overline{w},\overline{v})\in C\times \mathbb{P}(V)|\ w\wedge v=0\}$ is an irreducible variety of dimension 2. Its image $S$ under the projection map $pr_2 : \tilde{S}\rightarrow \mathbb{P}(V)$ is an irreducible surface of degree $e$. Suppose that through a general point of $S$ there are $f$ lines $\overline{w}\in C$. Then $d=e\cdot f$.\
[(2)]{}. Let $P(C)$ denote the smallest projective subspace of $\mathbb{P}(\Lambda ^2V)$, containing $C$. If $d\geq 3$ and $S$ is not a cone, a plane or a quadric, then $\dim P(C)\geq 3$.
(1). We note that $C\subset \sigma _2(p_0)$ is not interesting since then $S$ is a cone. The fibers of $pr_1:\tilde{S}\rightarrow C$ are lines in $\mathbb{P}(V)$ and the fibers of $pr_2:\tilde{S}\rightarrow S$ are finite. Thus $S$ is an irreducible ruled surface of some degree $e$. A general line $\overline{w}_0$ in $\mathbb{P}(V)$ intersects $S$ in $e$ points. Through each of these $e$ points there are $f$ lines $\overline{w}\in C$. Thus the intersection of $C$ with the general hyperplane $\{\overline{w}\in \mathbb{P}(\Lambda ^2V)|\ w\wedge w_0=0\}$ consists of $e\cdot f$ points and therefore $d=e\cdot f$.\
(2). Since $d>1$, one has $\dim P(C)>1$. Suppose that $\dim P(C) =2$. If $P(C)\subset Gr$, then either $P(C)$ is a $\sigma _2(p_0)$ and $S$ is a cone, or $P(C)$ is a $\sigma _{1,1}(h_0)$ and $S$ is the plane $h_0$. If $P(C)\not \subset Gr$, then $C\subset P(C)\cap Gr$ is a curve of degree at most 2 and $S$ is a plane or a quadric. Hence $\dim P(C)\geq 3$.
In the sequel we consider ruled surfaces (reduced, irreducible) $S\subset \mathbb{P}(V)$ of some degree $d\geq 3$ which are not cones. One associates to $S$ the subset $\tilde{C}$ of $Gr$ corresponding to the lines on $S$.
\[1.2.2\] $\tilde{C}$ is the union of an irreducible curve $C$ (not lying in some 2-plane $\sigma _2(p_0)$) of degree $d$ and a finite, possibly empty, set. Moreover, through a general point of $S$ there is one line of the surface.
Consider the affine open part of $Gr$ given by $p_{12}\neq 0$. The points of this affine part, actually $\cong \mathbb{A}^4$, can uniquely be written as planes in $V$ with basis $e_1+ae_3+be_4,\ e_2+ce_3+de_4$ and correspond to the vectors $$e_{12}+ce_{13}+de_{14}+ae_{23}+be_{24} + (ad-bc)e_{34}\ .$$ Let $F(t_1,\dots ,t_4)=0$ be the homogeneous equation of $S$. The intersection of $\tilde{C}$ with this affine part consists of the tuples $(a,b,c,d)$ such that $F(s,t,as+ct,bs+dt)=0$ for all $(s,t)\neq (0,0)$. Write this expression as a homogeneous form in $s,t$ and coefficients polynomials in $a,b,c,d$. Then the ideal generated by these polynomials in $a,b,c,d$ defines the intersection of $\tilde{C}$ with this affine part of $Gr$. Thus $\tilde{C}$ is Zariski closed.
Clearly $\tilde{C}$ has dimension 1 and can be written as the union of irreducible curves $C_i,\ i=1, \dots, r$ and a finite set. The image of the projection $\{ (\overline{w},\overline{v})\in C_1\times \mathbb{P}(V) | w\wedge v=0\}
\rightarrow \mathbb{P}(V)$ is a ruled surface contained in $S$. Since $S$ is irreducible, the image is $S$. If $r\geq 2$, then, for through a point $\overline{v}$ of a line $\overline{w}_2\in C_2,\ \overline{w}_2\not \in C_1$ passes a line $\overline{w}_1\in C_1$. Hence $w_1\wedge w_2=0$ for all $\overline{w_1}\in C_1$ and thus $w\wedge w_2=0$ for all $\overline{w}\in P(C_1)$. By symmetry $w_1\wedge w_2=0$ for all $\overline{w}_1\in P(C_1),\ \overline{w}_2\in P(C_2)$. Since the symmetric bilinear form $(w_1,w_2)\mapsto w_1\wedge w_2$ on $\Lambda ^2V$ is not degenerate, one obtains a contradiction by compairing dimensions: $\dim P(C_1)\geq 3,\ \dim P(C_2)\geq 3,\ \dim \mathbb{P}(\Lambda ^2V)=5$. We conclude that the $f$ of Lemma \[1.2.1\] is 1 and that the degree of $C$ is $d$.
\[1.2.3\] Let $\overline{w}_0\in \tilde{C}\setminus C$, then $C$ lies in the tangent space of $Gr$ at $\overline{w}_0$. In other words, the line $\overline{w}_0$ intersects every line on $S$, belonging to $C$.
If the tangent space at $\overline{w}_0$ does not contain $C$, then the intersection $C\cap T_{Gr,\ \overline{w}_0}$ consists of $d$ points, counted with multiplicity. Thus the line $\overline{w}_0$ on $S$ intersects $d$ lines of $S$, corresponding to points of $C$. Let $H\subset \mathbb{P}(V)$ be a plane through $\overline{w}_0$. The intersection $H\cap S$ consists of $\overline{w}_0$ and a curve $\Gamma$ of degree $d-1$. Therefore $\Gamma \cap \overline{w}_0$ consists of $d-1$ points (counted with multiplicity), instead of the $d$ points that we expect. This contradiction proves the lemma.
. [The lines on $S$ corresponding to the points of $\tilde{C}\setminus C$ will be called here [*isolated lines*]{}. A line $\overline{w}_1$ on $S$ is, classically, called a [*directrix*]{} if $\overline{w}_1$ meets every line $\overline{w}_2$ with $\overline{w}_2\in C$. Thus an isolated line is a directrix. It is also possible that a line $\overline{w}_1\in C$ is a directrix. The classical concept of ‘[*double curve*]{}’ on $S$ is, according to [@Ed], p. 8, (the Zariski closure of) the set of points on $S$ lying on at least two, not isolated, lines of $S$. ]{}
\[1.2.obs\] Let $C\subset P=\mathbb{P}^r, r\geq 3$ be an irreducible curve of degree 4 and such that $C$ does not lie in a proper subspace of $P$. Let $g\leq 2$ be the genus of the normalization $n:C^{norm}\rightarrow C$. Then one of the following holds:\
(1) $g=0,\ r=4$, $C$ is the, nonsingular, rational normal quartic.\
(2) $g=0,\ r=3$, $C$ is nonsingular or has one singular point which is a node or ordinary cusp.\
(3) $g=1,\ r=3$ and $C$ is nonsingular.\
Moreover, if $C$ lies on a quadratic cone in $\mathbb{P}^3$, then $g=1$ or $g=0$ and $C$ has a singular point.
Let $I$ be the sheaf of ideals of $C$. The exact sequence $0\rightarrow I\otimes O_P(1)\rightarrow O_P(1)\rightarrow \mathcal{L}\rightarrow 0$ with $\mathcal{L}=O_P(1)\otimes O_C$ and the minimality of $r$ implies that $H^0(O_P(1))\rightarrow H^0(\mathcal{L})$ is injective and thus $1+r\leq \dim H^0(\mathcal{L})$. Define the skyscraper sheaf $\mathcal{Q}$ on $C$ by the exact sequence of sheaves on $C$, $0\rightarrow \mathcal{L}\rightarrow n_*n^*\mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$. Denoting $\dim H^i$ by $h^i$, one finds $$4\leq 1+r\leq h^0(C,\mathcal{L})\leq h^0(C^{norm},
n^*\mathcal{L})=1-g+4+\dim H^1(C^{norm}, n^*\mathcal{L}) .$$ Now $H^1(C^{norm},n^*\mathcal{L})=0$, since the degree of $n^*\mathcal{L}$ is 4 and $g\leq 2$. Thus $g=2$ is not possible. For $g=1$, one has $H^0(C,\mathcal{L})= H^0(C^{norm},n^*\mathcal{L})$ and $C=C^{norm}$ since $n^*\mathcal{L}$ is very ample on $C^{norm}$. Let $E$ be an elliptic curve with neutral element $e$, then $H^0(E, 4[e])$ has basis $t_1=1,t_2=x,t_3=y,t_4=x^2$ (in the standard notation) and $E$ lies on the quadratic cone $t_2^2-t_1t_4=0$.
For $g=0$, the curves $C\subset P$ are parametrized by polynomials of degree $\leq 4$ in a variable $t$. Hence $r\leq 4$. For $r=4$, the only possibility is $t\mapsto (1,t,t^2,t^3,t^4)$. For $r=3$, one has the examples:\
$t\mapsto (1,t,t^3,t^4)$ and $C$ is nonsingular,\
$t\mapsto (1,t^2,t^3,t^4)$ and $C$ has an ordinary cusp,\
$t\mapsto (t,t^2,t^3,t^4-1)$ and $C$ has a node.\
In general, by intersecting $C$ with planes $H\subset \mathbb{P}^3$, through one singular point (or more), one can verify that $C$ has at most one singular point and that such a point can only be a node or an ordinary cusp. Finally, if $g=0$ and $C$ is contained in a quadratic cone in $\mathbb{P}^3$, then $C$ is singular (see [@Har], exercise IV, 6.1). According to the examples, this singular point can be either a node or a cusp. We note that the exercises IV, 3.4, 3.6 and II, Example 7.8.6 of [@Har] are closely related to the above reasoning.
\[1.2.6\] A ruled surface of degree $d\geq 3$ can have at most two isolated lines. If $S$ has two isolated lines $\overline{w}_1,\ \overline{w}_2$, then $\overline{w}_1\cap \overline{w}_2=\emptyset$.
The first statement follows from $\dim P(C)\geq 3$. If $\overline{w}_1\cap \overline{w}_2\neq \emptyset$, then $C$ lies in $Gr\cap T_{Gr,\ \overline{w}_1}\cap T_{Gr,\overline{w}_2}$. According to the list of properties of $Gr$, (viii) part (c), the latter is the union of two planes. One of them contains $C$ and this contradicts $\dim P(C)\geq 3$.
\[1.2.7\] [(1)]{} A general line of a ‘general’ ruled surface $S$ of degree $d\geq 3$ meets $d-2$ other lines of $S$, corresponding to points of $C$. In particular, the ‘double curve’ is not empty. [However:]{}\
[(2)]{} Let $TC\subset \mathbb{P}^3$ be the twisted cubic curve. The equation of the surface $S$ consisting of the tangents of $TC$ is $(t_1t_4-t_2t_3)^2-4(t_1t_3-t_2^2)(t_2t_4-t_3^2)=0$. The singular locus of $S$ is $TC$ and no two distinct lines of the surface intersect.
\(1) For a general point $\overline{w}_0\in C$, the intersection $C\cap T_{Gr,\ \overline{w}_0}$is a positive divisor on $C$ of degree $d$, with support in the nonsingular locus of $C$ and $\geq 2[\overline{w}_0]$. For a ‘general’ $S$ the divisor will be $2[\overline{w}_0]+\sum _{i=1}^{d-2} [\overline{w}_i]$ with distinct points $\overline{w}_i\in C, \ i=0,\dots , d-2$. Thus $\overline{w}_0$ meets precisely $d-2$ other lines corresponding to points of $C$.\
(2) Let $t\mapsto (1,t,t^2,t^3)\in \mathbb{P}^3$ be $TC$ in parametrized form. The tangent line $\overline{w}_t$ contains the point $(0,1,2t,3t^2)$ and has Plücker coordinates $$p_{12}=1,\ p_{13}=2t,\ p_{14}=3t^2,\ p_{23}=t^2,\ p_{24}=2t^3,\ p_{34}=t^4 .$$ This defines the nonsingular curve $C\subset Gr$ corresponding to $S$. From $\overline{w}_t\wedge \overline{w}_s=(t-s)^4$ it follows that the tangent lines do not intersect for $t\neq s$. In other terms $T_{Gr,\overline{w}_0}\cap C =4[\overline{w}_0]$ for every $\overline{w}_0\in C$.
\[1.2.8\] $\ $\
[(1)]{} $pr_2: \tilde{S}:=\{(\overline{w},\overline{v})\in C\times \mathbb{P}(V)|\ w\wedge v=0\} \rightarrow S$ is a birational morphism. Let $C^{norm}\rightarrow C$ denote the normalization of $C$ and let $\tilde{\tilde{S}}=\tilde{S}\times _{C}C^{norm}$ be the pullback of $\tilde{S}\rightarrow C$. Then $\tilde{\tilde{S}}\rightarrow C^{norm}$ is a ruled surface (in the modern sense) and $\tilde{\tilde{S}}\rightarrow S$ is the normalization of $S$.\
[(2)]{} The singular locus of $S$ is purely 1-dimensional or empty.\
[(3)]{} Suppose that the line $\overline{w}$ belongs to the singular locus of $S$ and does not correspond to a singular point of $C$. Then $C$ lies in the tangent space of $Gr$ at the point $\overline{w}$.
\(1) The morphism is finite since $pr^{-1}_2(\overline{v})$ is the finite set of lines of $S$ through $\overline{v}\in S$. For a general $\overline{v}\in S$, this set has one element and therefore the degree of $pr_2$ is 1 and so $pr_2$ is birational. The fibres of $pr_1:\tilde{S}\rightarrow C$ are isomorphic to $\mathbb{P}^1$ and the same holds for the fibres of $\tilde{\tilde{S}}\rightarrow C^{norm}$. Therefore the latter is a ruled surface in the modern terminology. Moreover the morphism $\tilde{\tilde{S}}\rightarrow \tilde{S}$ is birational and so $\tilde{\tilde{S}}\rightarrow S$ is the normalization.\
(2) The local ring of an isolated singular point of $S$ is normal and will remain a singular point of the normalization of $S$. Since $\tilde{\tilde{S}}$ is smooth, $S$ has no isolated singularities.\
(3) The assumption that $\overline{w}$ does not correspond to a singular point of $C$ implies that through any point of $\overline{w}$ there are at least two lines corresponding to points of $C$ (one of them could be $\overline{w}$ itself). Hence $\overline{w}$ meets every line corresponding to a point of $C$ and thus $C\subset T_{Gr,\overline{w}}$.
[The ‘double curve’, as defined above, is seen, by Proposition \[1.2.8\], to be part of the singular locus of $S$. The [*genus of $S$*]{} is defined as the genus of $\tilde{\tilde{S}}$ and thus equal to the genus of $C^{norm}$.]{}$\square$
\[1.2.10\] Suppose that $\dim P(C)=3$ and that $P(C)$ is the intersection of two tangent spaces of $Gr$ at points $\overline{w}_1\neq \overline{w}_2$. Then the lines $\overline{w}_1,\ \overline{w}_2$ do not intersect. For a suitable choice of the homogeneous coordinates $t_1,t_2,t_3,t_4$ of $\mathbb{P}(V)$, the equation $F$ of $S$ is bi–homogeneous of degree $(a_1,a_2)$, with $a_1+a_2=d$, in the pairs $t_1,t_2$ and $t_3,t_4$. Further $\tilde{C}\setminus C= \{\overline{w}_1,\overline{w}_2\}$.
The lines $\overline{w}_1, \overline{w}_2$ are ‘directrices’. The singular locus of $S$ consists of the lines $\overline{w}_i$ with $a_i>1$ and for each singular point $\overline{w}\in C$, the line $\overline{w}\subset S$.
The assumption that the lines $\overline{w}_1,\ \overline{w}_2$ intersect, yields, according to (viii) part (c), the contradiction that $C$ lies in a plane. Take $w_1=e_{12}$ and $w_2=e_{34}$, then $P(C)= T_{Gr,\ \overline{e_{12}}}\cap T_{Gr,\ \overline{e_{34}}}$ is the projective space with coordinates $p_{13},p_{14},p_{23},p_{24}$ and $C$ lies on the quadric surface $Gr\cap P(C)$ given by $-p_{13}p_{24}+p_{14}p_{23}=0$. Identifying $Gr\cap P(C)$ with $\mathbb{P}^1\times \mathbb{P}^1$ leads to $C\subset \mathbb{P}^1\times \mathbb{P}^1$ of bi–degree $(a_1,a_2)$ with $a_1+a_2=d$.
Consider the rational map $f:\mathbb{P}(V)\cdots\rightarrow \mathbb{P}^1\times \mathbb{P}^1$, given by $$(t_1,t_2,t_3,t_4)\mapsto ((t_1,t_2),(t_3,t_4)),$$ which is defined outside the two lines $\overline{w}_1,\overline{w}_2$. The surface $S$ is the Zariski closure of $f^{-1}(C)$ and so the equation $F$ of $S$ is bi-homogeneous and coincides with the equation for $C\subset \mathbb{P}^1\times \mathbb{P}^1$. The other statements of the lemma are easily verified.
\[1.2.11\] $\dim P(C)=3$ and $P(C)$ in a single tangent space of $Gr$.\
For a suitable basis of $V$ the projective subspace $P(C)\subset T_{Gr,\ \overline{e_{12}}}$ is given by the equations $p_{34}=0, p_{13}+p_{24}=0$ and $p_{12},p_{13},p_{14},p_{23}$ are the homogeneous coordinates of $P(C)$. Further $Gr\cap P(C)$ is the cone with equation $p_{13}^2+p_{14}p_{23}=0$ with vertex $\overline{e_{12}}$. Since $C$ lies on this cone we have a rational map $f: C\cdots \rightarrow E:=\{p_{13}^2+p_{14}p_{23}=0\}$. This map can be identified with the rational map $C\cdots\rightarrow \overline{e_{12}}$, given by $\overline{w}\mapsto \overline{w}\cap \overline{e_{12}}$. The rational map $f$ is a morphism if $\overline{e_{12}}\not \in C$ or if $\overline{e_{12}}\in C$ and this is a regular point of $C$.
In case $\overline{e_{12}}\not \in C$ the morphism $f$ has degree $e$. Take two unramified points $e_1,e_2\in E$ and the plane through the corresponding two lines through $\overline{e_{12}}$. This plane meets $C$ in $2e$ points. Hence $d=2e$. In case $\overline{e_{12}}\in C$ and is not a singular point, the same reasoning yields $d-1=2e$.
It seems difficult to investigate the possibilities for general $d$. The cases $d=3$ and $d=4$ will be presented later on.
The vector bundle $B$ on $C^{norm}$
-----------------------------------
Let again $d\geq 3$ denote the degree of the ruled surface $S$ and let $C\subset Gr$ be the corresponding curve. Put $$B:=\{(p,v)\in C^{norm} \times V|\ p\mapsto \overline{w}\in C, w\wedge v=0\}
\subset C^{norm} \times V.$$ This is a (geometric) vector bundle of rank two on $C^{norm}$. We will identify $B$ with its sheaf of sections. We note that $Proj(B)=\tilde{\tilde{S}}$. The line bundle $\Lambda ^2B$ on $C^{norm}$ is the pullback of the restriction of $O_{\mathbb{P}(\Lambda ^2V)}(-1)$ to $C$ and has therefore degree $-d$. The vector space $H^0(C^{norm},B)=0$, otherwise all the lines of $C$ pass through one point and $S$ is a cone.\
[*The vector bundle $B$ is an important tool in case $C^{norm}$ has genus 0*]{}.\
[*For the case $d=3$*]{} it is easily seen that $C^{norm}$ has genus 0. Let $t$ parametrize $C^{norm}$. Then $B$ is isomorphic to $O_{C^{norm}}(-1)\oplus O_{C^{norm}}(-2)$. In particular, $\tilde{\tilde{S}}$ is isomorphic to $\mathbb{P}^2$ with one point blown up (see [@Har], V, Example 2.11.5). The sections of $B$ with a pole of order 1 at $t=\infty$ are $\mathbb{C}a$ and those with a pole of order $\leq 2$ at $t=\infty$ are $\mathbb{C}a+\mathbb{C}b$. By choosing a suitable basis of $V$ one can normalize to the following two cases:\
$a=(1,t,0,0),\ b=(0,0,1,t^2)$ and $S$ has the equation $t_1^2t_4-t_2^2t_3=0$;\
$a=(1,t,0,0),\ b=(0,1,t,t^2)$ and $S$ has the equation $t_3^3+t_4(t_1t_4-t_2t_3)=0$.\
This gives the classification of the ruled cubic surfaces over, say, $\mathbb{C}$. In Section \[section2\] we will follow another method to obtain the classification of ruled cubic surfaces over any field and compare this with Dolgachev’s method.\
[*For $d=4$ and assuming that $C^{norm}$ has genus 0*]{}, there are two possibilities for the vector bundle $B$, namely:\
$B\cong O_{C^{norm}}(-1)\oplus O_{C^{norm}}(-3)$ and $\tilde{\tilde{S}}$ is the Hirzebruch surface $\Sigma _2$,\
$B\cong O_{C^{norm}}(-2)\oplus O_{C^{norm}}(-2)$ and $\tilde{\tilde{S}}$ is $\mathbb{P}^1\times \mathbb{P}^1$.\
We note in passing that the first possibility was overlooked by Cremona. The method of Cayley can be interpreted as taking three sections of the vector bundle $B(d)$ for a certain values of $d\geq 1$. Normalizing sections of $B$ with poles of order 1,2,3 at $t=\infty$, by a choice of the basis of $V$ and possibly changing $t$, we will arrive in Subsection \[3.1\] at a classification of the corresponding ruled quartic surfaces.
If the genus of $C^{norm}$ is 1, the vector bundle $B$ is not helpful for the computation. However, $B$ and also $\tilde{\tilde{S}}=Proj(B)$ will be identified.
The possibilities for the singular locus {#1.4}
----------------------------------------
It is helpful for the classification of the ruled surfaces to consider $Q:=S\cap H$ with $H\subset \mathbb{P}(V)$ a general plane. By Bertini’s theorem, $Q$ is an irreducible reduced curve of degree $d$. The morphism $C\rightarrow Q$, given by $\overline{w}\in C\mapsto \overline{w}\cap H\in Q$, is birational. Thus $C^{norm}$ is the normalization of $Q$. The singular locus of $S$ is written as a union of its irreducible components $C_i,\ i=1,\dots ,s$ of degree $d_i$ and generic multiplicity $m_i\geq 2$. The curve $Q$ meets every $C_i$ with multiplicity in $d_i$ points with multiplicity $m_i$. For every singular point $q\in Q$ one defines a number $\delta _q$ which is the sum of the integers $\frac{k(k-1)}{2}$ taken over the multiplicities $k$ of $q$ itself and of all the singular points that occur in the successive blow ups of $q$. The Plücker formula states that the genus of the normalization $C^{norm}$ of $Q$ is $\frac{(d-1)(d-2)}{2}-\sum \delta _q$.
For $d=3$, there is a single singular point $q$ and $\delta _q=1$ (and $q$ is a node or cusp). The singular locus is described by $s=1, d_1=1, m_1=2$.
For $d=4$, there are more possibilities. The singularities of a simple plane curve (i.e., reduced, multiplicity $\leq 3$ and in the blow ups there are only singularities of multiplicity $\leq 3$) are classified, see [@Bar], p. 62, by formal standard equations $F\in K[[x,y]]$. The condition that $Q$ is irreducible, has degree 4 and the genus of its normalization $C^{norm}$ is 0 or 1, leads to the list of possibilities (with their symbols or names):\
for $m=2$: $$A_2:\ x^2-y^2,\delta =1;\ \ A_3:\ x^2-y^3, \delta =1;\ \
A_4:\ x^2-y^4,\delta =2;\ \ \ \ \ \ \ \ \ \ $$ for $m=3$: $$D_4:\ y(x^2-y^2), \delta =3;\ \ D_5:\ y(x^2-y^3),\delta =3; \ \
E_6:\ x^3-y^4,\delta =3$$ $$\mbox{ and the last case }E_7:\ x(x^2-y^3), \mbox{ which is ruled out by } \delta =4.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ The inequalities $\sum _{i=1}^sd_i\frac{m_i(m_i-1)}{2}\leq \sum \delta _q\leq \frac{(4-1)(4-2)}{2}$ lead to a list of possibilities for the singular locus, again with Cayley’s convention that $d^m$ stands for an irreducible curve of degree $d$ and with multiplicity $m$ and ‘int’ meaning intersecting lines: $ 1^2$; $\ 1^3$; $\ 2^2$; $\ 3^2$; $\ 1^2,1^2\ \mbox{int}$; $\ 1^2,1^2$; $\ 1^2,2^2$; $\ 1^2,1^2,1^2$.
The reciprocal of a ruled surface {#recip}
---------------------------------
As before, $V$ is a vector space of dimension 4 over a field $K$. One identifies $\Lambda ^4V$ with $K$. The nondegenerate symmetric bilinear form on $\Lambda ^2V$, given by $(w_1,w_2)=w_1\wedge w_2\in \Lambda ^4V= K$, yields an isomorphism $f:\Lambda ^2V \rightarrow \Lambda ^2V^*=(\Lambda ^2V)^*$ by $f(w_1)(w_2)=w_1\wedge w_2\in K$. This isomorphism maps decomposable vectors of $\Lambda ^2V$ to decomposable vectors of $\Lambda ^2V^*$.
Indeed, consider $f(v_1\wedge v_2)$. Let $v_1,v_2,v_3,v_4$ be a basis of $V$. The kernel of $f(v_1\wedge v_2)$ has basis $v_1\wedge v_2,\ v_1\wedge v_3,\ v_1\wedge v_4,\ v_2\wedge v_3,\ v_2\wedge v_4$. Let $\ell _1,\ell _2$ be a basis of $(V/Kv_1+Kv_2)^*\subset V^*$. Then $\ell _1\wedge \ell _2$ has the same basis vectors in the kernel. Hence $f(v_1\wedge v_2)$ is a multiple of $\ell _1\wedge \ell _2$.
Thus $f$ induces an isomorphism $\tilde{f}:Gr(2,V)\rightarrow Gr(2,V^*)$. Let $P\subset V$ be a 2-dimensional subspace. Then $\tilde{f}(P)$ is the 2-dimensional subspace $(V/P)^*$ of $V^*$. For 2-dimensional subspaces $P_1,P_2\subset V$ with $P_1\cap P_2\neq 0$ one has $(V/P_1)^*\cap (V/P_2)^*\neq 0$. This also follows from the formula $f(w_1)\wedge f(w_2)=w_1\wedge w_2$ for any $w_1,w_2\in \Lambda ^2V$ (for a suitable identification of $\Lambda ^4V^*$ with $K$).
Any 1-dimensional subspace $L\subset V$ determines the plane in $Gr(2,V^*)$ consisting of all 2-dimensional $P\subset V,\ P\supset L$ (an $\omega$-plane in [@Ed]). The image of this plane under $\tilde{f}$ is the plane in $Gr(2,V^*)$ consisting of all 2-dimensional $Q\subset (V/L)^*\subset V^*$. Since the latter is a plane of the ‘opposite type’ (a $\rho$-plane in [@Ed]), there is no isomorphism $V\rightarrow V^*$ inducing $\tilde{f}$.\
Let $e_1,\dots ,e_4$ denote a basis of $V$ and $e_1^*,\dots ,e_4^*$ the dual basis of $V^*$. Then $e_{ij}:=e_i\wedge e_j,\ i<j$ is a basis of $\Lambda ^2V$ and $e^*_{ij}=e_i^*\wedge e_j^*,\ i<j$ is a basis of $\Lambda ^2V^*$. From the Plücker coordinates $\{p_{ij}\}$ of a line $\overline{w}\in Gr(2,V)$ one easily finds the Plücker coordinates of $f(\overline{w})\in Gr(2,V^*)$ by the identities
$$f(e_{12})=e^*_{34}; f(e_{13})=-e^*_{24}; f(e_{14})=e^*_{23}; f(e_{23})=e^*_{14};$$ $$f(e_{24})=-e^*_{13};
f(e_{34})=e^*_{12} .$$
Let an irreducible ruled surface $S\subset \mathbb{P}(V)$ (of degree $d$) be given by an irreducible curve $C\subset Gr(2,V)$ of degree $d$. Consider a nonsingular point $s\in S$ lying on a single line $\ell $ of the surface. The tangent plane $T_{S,s}$ contains the line $\ell$. The same holds for the tangent planes $T_{S,s'}$ for almost all points $ s'\in \ell$. Since $T_{S,s'}$ cannot be all equal, the reciprocal (or dual) surface contains all planes $H\supset \ell$. It now follows that the reciprocal surface $\check{S}\subset \mathbb{P}(V^*)$ is ruled. The corresponding curve in $Gr(2,V^*)$ is $f(C)$. It has also degree $d$ since the degree of the curve $f(C)$ is $d$.
Using Plücker coordinates one easily finds $\check{S}$. Another useful [*computation of the reciprocal surface*]{} is the following. Consider $\tilde{S}=\{(\overline{w},\overline{v})\in C\times \mathbb{P}(V)|\ v\wedge v=0\}\rightarrow
S\subset \mathbb{P}(V)$ and a nonsingular point $\overline{v}_0\in S$ and the nonsingular point $\overline{w}_0\in C$ with $w_0\wedge v_0=0$. (We note that the tangent plane of $\tilde{S}$ at the point $(\overline{w}_0,\overline{v}_0)$ is mapped isomorphically to the tangent plane of $S$ at the point $\overline{v}_0$. The first tangent plane is the product of the tangent line of $C$ at the point $\overline{w}_0$ and the line $\overline{w}_0$).
Let a (local) parametrization $t\mapsto w(t)$ for $C$ be given, such that $w_0=w(t_0)$. Choose a decomposition $w(t)=a(t)\wedge b(t)$, locally at $t_0$. Then $v_0=s_0a(t_0)+(1-s_0)b(t_0)$ and $S$ has the local parametrization $(t,s)\mapsto sa(t)+(1-s)b(t)$. The linearization of this, i.e., $$v_0+(s-s_0)(a(t_0)-b(t_0))+(t-t_0)(s_0a'(t_0)+(1-s_0)b'(t_0)),$$ is a parametrization of the tangent plane $T_{S,\overline{v}_0}$. This corresponds with the 3-dimensional subspace of $V$ with basis $$v_0, v_0+a(t_0)-b(t_0),v_0+s_0a'(t_0)+(1-s_0)b'(t_0).$$ The exterior product $a(t_0)\wedge b(t_0)\wedge (s_0a'(t_0)+(1-s_0)b'(t_0))$ of these vectors is an element of $\Lambda ^3V=V^*$. This defines a point in $\mathbb{P}(V^*)$ corresponding to the tangent plane $T_{S,\overline{v}_0}$. The reciprocal surface $\check{S}$ consists of all these points. In varying $s_0$ one finds a line on $\check{S}$, through the points $a(t_0)\wedge b(t_0)\wedge a'(t_0)$ and $a(t_0)\wedge b(t_0)\wedge b'(t_0)$. By varying $t_0$ one obtains $\check{S}$.
For the two cases of ruled quartic surfaces $S$ with genus 1, it is easily seen that $\check{S}\cong S$. For the ruled quartic surfaces cases of genus 0 there are explicit global expressions $w(t)=a(t)\wedge b(t)$ (with $t\in \mathbb{P}^1$) and the above can be used for the computation of $\check{S}$.\
The classification of quartic ruled surfaces {#s1.6}
--------------------------------------------
The [*Number*]{} appearing in the tables are introduced for notational reasons in the computation of Subsection \[3.1\]. It has no intrinsic meaning. The cases with $C^{norm}$ of genus 0 and $B$ of type $-1,-3$ are:\
$ \begin{array}{|c|c|c|c|c|c|c|} \hline
\mbox{Number}& \mbox{singular}& \dim & \mbox{singularities }& \mbox{tangent}&
\mbox{Cremona}& {\rm XIII}\\
& \mbox{locus}& P(C) & \mbox{of }C& \mbox{spaces}& & \\ \hline
1\ a,b,c & 1^3&3& - & 2 & 9&\\ \hline
2\ a,b,c & 3^2&4&-&1&7&\\ \hline
3\ a,b & 1^2,2^2&4&-&1&4& \\ \hline
4& 1^3&3& \mbox{node}& 1& 10&7 \\ \hline
5&1^3&3&\mbox{cusp}&1&10&\\ \hline
\end{array} $\
The cases with $C^{norm}$ of genus 0 and $B$ of type $-2,-2$ are:\
$ \begin{array}{|c|c|c|c|c|c|c|} \hline
\mbox{Number}& \mbox{singular}& \dim & \mbox{singularities}& \mbox{tangent }&
\mbox{Cremona}& {\rm XIII}\\
& \mbox{locus}& P(C) & \mbox{of }C& \mbox{spaces}& &\\ \hline
6&1^2,1^2,1^2& 3&\mbox{node} &2&5&\\ \hline
7&1^2,1^2,1^2& 3&\mbox{cusp}&2&5&\\ \hline
8& 1^3& 4& - & 1& 3&\\ \hline
9& 1^3&4&-&1&3&\\ \hline
10\ a & 1^2,1^2, int & 3 &\mbox{node}& 1&6&\\ \hline
10\ b & 1^2,1^2, int & 3 &\mbox{ cusp}& 1&6&\\ \hline
11 & 1^2,2^2 & 4 & - & 1& 2&\\ \hline
12 & 1^2,2^2 & 4 & - & 1& 2&8 \\ \hline
13 \ a,b,c & 1^3&4&-& 1& 8&6 \\ \hline
14 \ ... & 3^2& 4 & - & - & 1&9,10\\ \hline
\end{array} $\
In § \[3.2.4\] it is shown that [*Number*]{} 14 consists of six distinct cases. In the text it is explained that the cases 11 and 12 are different. The reciprocals of $2\ a,b,c$ are $13\ a,b,c$ and the reciprocals of $3\ a,b$ are $8, 9$. The other examples are ‘selfdual’.
The cases with $C^{norm}$ of genus 1 are:\
$ \begin{array}{|c|c|c|c|c|c|} \hline
\mbox{Number}& \mbox{ singular locus} & \dim P(C)& \mbox{tangent spaces}&\mbox{Cremona}&
{\rm XIII}\\ \hline
15& 1^2& 3 & 1&12& 5\\ \hline
16 & 1^2,1^2 & 3 & 2 & 11& 1,2,3,4 \\
\hline \end{array} $
Ruled surfaces of degree 3 {#section2}
==========================
Here we give the classification over an arbitrary field $K$. The singular locus of $S$ is a line, $C^{norm}$ has genus 0 and $\dim P(C)=3$. This implies that $C=C^{norm}$ is the twisted cubic curve in $P(C)$.\
[*In the first case $P(C)$ lies in two tangent spaces*]{} at the points $\overline{w}_1,\overline{w}_2\in Gr$. From Lemma \[1.2.10\] we conclude that $S$ is given by a bi-homogeneous equation $F$ in the pairs of variables $t_1,t_2$ and $t_3,t_4$ of bi-degree $(2,1)$, corresponding to a morphism $f:\overline{w}_1\rightarrow \overline{w}_2$ of degree 2. The line $\overline{w}_1$ is nonsingular and a ‘directrix’. The line $\overline{w}_2$ is the singular locus. Further $\tilde{C}\setminus C=\{\overline{w}_1,\overline{w}_2\}$.
If the field $K$ has characteristic $\neq 2$, then $C,P(C),\overline{w}_1,\overline{w}_2$ are all defined over $K$ and can be put in a standard form. The morphism $f$ is defined over $K$. The ramification points of $f$ are either both in $K$ or belong to a quadratic extension of $K$ and are conjugated.
If the field $K$ has characteristic 2, then $f$ is either separable and has one point of ramification, or $f$ is inseparable. In both cases $f$ can be put into a standard form.\
[*In the second case $P(C)$ lies in only one tangent space*]{}, namely at the point $\overline{w}_0$ which is the singular line of $S$. Then $C$ lies on the quadratic cone in $P(C)$ and $\overline{w}_0\in C$. In this case $\tilde{C}=C$. Now $C$ and $S$ can be put into a standard form. We arrive at the following result.
\[cubic\]\[2.0.1\] The standard equations for ruled cubic surfaces $S/K$, which are not cones, are the following:\
[(1)]{} $t_3t_1^2+t_4t_2^2=0$. If $char K=2$, then there are no twist. For $char K\neq 2$ the twists are $t_3(t_1t_2)+t_4(at_1^2+t_2^2)=0$ with $a\in K^*$ not a square.\
[(2)]{} $t_3t_1t_2+t_4t_1^2+t_2^3=0$ (there are no twists).\
[(3)]{} $t_3t_1^2+t_4(t_2^2+t_1t_2)=0$ if $char K=2$ (there are no twists).
The curves $C$ for (1) and (2) are in parameter form $$p_{12}=0,\ p_{13}=-t^2,\ p_{14}=1,\ p_{23}=-t^3,\ p_{24}=t,\ p_{34}=0
\mbox{ and }$$ $$p_{12}=0,\ p_{13}=t^3,\ p_{14}=t^2,\ p_{23}=-t^2,\ p_{24}=-t,\ p_{34}=-1.$$ The above equations for $S$ are derived in an elegant way by I. Dolgachev [@Dol], using only the information that the singular locus of $S$ is a line with multiplicity 2.
For $K=\mathbb{R}$, there are three types of cubic ruled surfaces (omitting cones). Case (1) of Proposition \[cubic\], without twist, is represented by the plaster model VII, nr. 21 and by the string models XVIII, nr. 2 and 3. The twisted case ($a=-1$) is represented by VII, nr. 20 and XVIII, nr. 1. Finally, case (2) carries the name Cayley’s ruled cubic surface. It is represented by VII, nr. 22 and 23 and XVIII, nr. 4.
Ruled surfaces of degree 4 {#section3}
==========================
The base field $K$ is supposed to be algebraically closed. The only role that the characteristic of $K$ plays is in the classification of the morphisms $\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree 2 and 3. For convenience we suppose that $K$ has characteristic 0 or $>3$. We need both the vector bundle $B$ and the possibilities for the singular locus in order to find all cases and to verify the computations by comparison.
Classification of $S$, using the vector bundle $B$ {#3.1}
--------------------------------------------------
### $C^{norm}$ of genus 0 and $B$ of type $-1,-3$ {#3.1.1}
Choose a parameter $t$ for $C^{norm}\cong \mathbb{P}^1$ and let $p$ be the pole of $t$. Then $H^0(C^{norm}, B([p])$ has basis $a$ and $H^0(C^{norm},B(3\cdot [p]))$ has basis $a,ta,t^2a,b$. Now $a,b\in K[t]\otimes _KV$ have degrees 1 and 3. We note that $b$ is not unique and can be replaced by $\mu b+\lambda _0a+\lambda _1ta+\lambda _2t^2a$ with $\mu \neq 0$. We will derive [*normal forms*]{} for the possibilities of the pair $a,b$. These will depend on the choice of $t$. There is a unique subspace $W\subset V$ of dimension 2 with $a\in K[t]\otimes W$ and the image $b'$ of $b$ in $K[t]\otimes V/W$ is unique up to multiplication by some $\mu \in K^*$ and has degree $\geq 1$.
The above normal form is obtained by choosing $t$ and a basis $e_1,e_2,e_3,e_4$ of $V$ such that $\{e_1,e_2\}$ is a basis of $W$ and such that $a=(a_1,a_2,0,0)$ and a suitable $b=(b_1,b_2,b_3,b_4)$ w.r.t. this basis are as simple as possible.
The Plücker coordinates of the line through $a(t)$ and $b(t)$ are easily computed and this yields $C\subset Gr$ in parametrized form. From this one deduces $\dim P(C)$, possible singularities of $C$ and the relation of $C$ w.r.t. the tangent spaces of $Gr$. The reciprocal surface (needed for the comparison with Cremona’s list) is computed by the method of Subsection \[recip\], again using $a(t),b(t)$. For some cases (especially when the singular locus is $3^2$), the equation of the surface $S$ is rather long and requires a MAPLE computation. We avoid this and compute the singular locus by other means. We start by classifying the pairs $(b_3,b_4)$ which are uniquely determined by $S$, up to taking linear combinations.\
$\max (\deg b_3, \deg b_4)=3$ and $\gcd(b_3,b_4)=1$.\
The morphism $C^{norm}\rightarrow \mathbb{P}^1, t\mapsto (b_3:b_4)$ has degree 3. The possibilities for the ramification indices are: (a) $3,3$, (b) $3,2,2$ or (c) $2,2,2,2$. A change of $t$ and a linear change of $e_3,e_4$ brings the pair $(b_3,b_4)$ into a standard form $$(1,t^3),\ (1,t^2(t+1)), \mbox{ or } ( t-\mu , (2\mu -1)t^3+(2-3\mu )t^2 ) \mbox{ with } \mu \neq 0,1,1/2.$$ (In the last case the 4 ramification points are $0,1,\infty ,\frac{\mu}{2\mu -1}$). One normalizes $a=(1,t,0,0)$, $b=(b_1,b_2,b_3,b_4)$ and $\max (\deg b_1,\deg b_2)\leq 2$. Then $b-b_1\cdot a$ produces a new $b=(0,b_2,b_3,b_4)$. There are now again two cases:
#### 1.
(a,b,c). $b_2=0$ and the data are: $C$ is nonsingular, $P(C)=T_{Gr,\overline{e_{12}}}\cap T_{Gr,\overline{e_{34}}}$ and $\overline{e_{12}},\overline{e_{34}}\not \in C$; equation $t_1^3b_3(\frac{t_2}{t_1})t_4-t_1^3b_4(\frac{t_2}{t_1})t_3=0$; the singular locus of $S$ is the line $\overline{e_{34}}$ with multiplicity 3. Then $1^3$ and Cremona 9.
#### 2.
(a,b,c). $b_2\neq 0$ and the data are: $C$ nonsingular, $P(C)=T_{Gr,\overline{e_{12}}}$ and $\overline{e_{12}}\not \in C$. A direct computation of the equation seems difficult. The points of the line through $a(t)$ and $b(t)$ can be written as $(1,t+\lambda \frac{b_2}{b_3}(t),\lambda ,\lambda \frac{b_4}{b_3}(t))$. Computing with the equality $$(1,t+\lambda \frac{b_2}{b_3}(t),\lambda ,\lambda \frac{b_4}{b_3}(t))=
(1,s+\lambda \frac{b_2}{b_3}(s),\lambda ,\lambda \frac{b_4}{b_3}(s))\mbox{, with } s\neq t$$ leads to the result that the ‘double curve’ and thus also the singular locus is a twisted cubic curve. Then $3^2$ and Cremona 7.
#### 3.
(a,b) $\max (\deg b_3,\deg b_4)= 2$ and $\gcd(b_3,b_4)=1$.\
A normalization is $a=(t+\beta ,1,0,0)$ and $b=(0,t^3+\alpha t,t^2,1)$. Equation $$t_3t_4(t_2-\beta (t_3+\alpha t_4))^2-(t_3(t_3+\alpha t_4)-\beta t_2t_4+t_1t_4)^2=0.$$ One has $P(C)=T_{Gr,\overline{e_{12}}}$ and $\overline{e_{12}}\in C$. The singular locus is the union of the line $\overline{e_{12}}$ and the conic $t_2-\beta (t_3+\alpha t_4)=0, \ t_3(t_3+\alpha t_4)-\beta t_2t_4+t_1t_4=0$. Then $1^2,2^2$ and Cremona 4. The degree morphism $C\rightarrow D$ has two points of ramification. The point $L\cap D$ is a ramification point on $D$ if and only if $\beta =0$. We consider this as two cases.
#### 4.
$\max (\deg b_3,\deg b_4)= 2$ and $\gcd(b_3,b_4)$ has degree 1.\
A normalization is $a=(t,1,0,0),\ b=(0,t^3+\alpha ,t(t+\beta),t+\beta )$ with $\gcd(t^3+\alpha ,t+\beta )=1$. The equation is $$t_1t_4^2(t_3+\beta t_4) -t_2t_3t_4(t_3+\beta t_4)+\alpha t_3t_4^3+t_3^4=0 .$$ Further $\overline{e_{12}}\in C$ is a node (for $t=\infty , t=-\beta$), $\dim P(C)=3$ and $P(C)$ lies in only one tangent space $T_{Gr,\overline{e_{12}}}$. Then $1^3$ and Cremona 10.
#### 5.
$\max (\deg b_3,\deg b_4)= 1$.\
A normalization is $a=(t,1,0,0),\ b=(0,t^3+\alpha t^2,t,1)$. The equation is $$t_1t_4^3-t_2t_3t_4^2+\alpha t_3^3t_4+t_3^4=0.$$ Further $\overline{e_{12}}\in C$ is a cusp (for $t=\infty$), $\dim P(C)=3$ and $P(C)$ lies in only one tangent space, namely $T_{Gr,\overline{e_{12}}}$. Then $1^3$ and Cremona 10.\
Finally we show that the omitted cases can be reduced to the above.\
$\max (\deg b_3,\deg b_4)=3$ and $\gcd(b_3,b_4)$ has degree 1.\
A normalization is $a=(1,t,0,0),\ b=(b_1,b_2,t, t(t+\mu )^2)$. Replacing $t$ by $s^{-1}$ and multiplying by a suitable power of $s$ yields $a=(s,1,0,0),\ b=(s^3b_1(s^{-1}),s^3b_2(s^{-1}), s^2, (1+\mu s)^2)$. Thus reduction to $\max (\deg b_3,\deg b_4)= 2$.\
$\max (\deg b_3,\deg b_4)=3$ and $\gcd(b_3,b_4)$ has degree 2.\
A normalization is $a=(1,t,0,0),\ b=(b_1,b_2,t(t+\mu),t(t+\mu)(t+\lambda ))$. Replacing $t$ by $s^{-1}$ and multiplying by a suitable power of $s$ gives a reduction to $\max (\deg b_3,\deg b_4)= 2$.
### $C^{norm}$ of genus 0 and $B$ of type $-2,-2$ {#3.1.2}
$V,t,p$ have the same meaning as in § \[3.1.1\]. The vector space $H^0(C^{norm},B(2[p]))$ has dimension 2 and consists of elements in $K[t]\otimes V$ of degree $\leq 2$ and the only element of degree $\leq 1$ is 0. We are interested in lines $Ka\subset H^0(C^{norm},B(2[p]))$ such that $a\in K[t]\otimes W$ with $\dim W=2$.\
[*Suppose that there are two such lines $Ka$ and $Kb$*]{}.\
One can normalize by $a=(a_1,a_2,0,0),\ b=(0,0,b_3,b_4)$. The two morphisms $\overline{a}, \overline{b}:C^{norm}\rightarrow \mathbb{P}^1$, $t\mapsto (a_1:a_2)$ and $t\mapsto (b_3,b_4)$ of degree 2 are distinct and their sets of ramification points can be either disjoint or have one point of intersection. Choosing $t$ leads to the following normalizations.
#### 6.
$a=(1,t^2,0,0), \ b=(0,0,(t-1)^2,(t-\lambda ^2 )^2)$. The singular locus consists of the lines $\overline{e_{12}},\overline{e_{34}}$ and a third line corresponding to $t=\pm \lambda$. The morphism $C^{norm}\stackrel{(\overline{a},\overline{b})}{\rightarrow} \mathbb{P}^1\times \mathbb{P}^1$ maps $t=\pm \lambda$ to the same point of $C$. Thus $C$ has a node, $P(C)=T_{Gr,\overline{e_{12}}}\cap T_{Gr,\overline{e_{34}}}$, $1^2,1^2,1^2$ and Cremona 5.
#### 7.
$a=(1,t^2,0,0),\ b=(0,0,1,(t-1)^2)$. The image $C$ of $C^{norm}\stackrel{(\overline{a},\overline{b})}{\rightarrow} \mathbb{P}^1\times \mathbb{P}^1$ has a cusp corresponding to $t=\infty$. The singular locus consists of three lines $\overline{e_{12}},\overline{e_{34}}$ and the line corresponding to $t=\infty$. Thus $P(C)=T_{Gr,\overline{e_{12}}}\cap T_{Gr,\overline{e_{34}}}$, $1^2,1^2,1^2$ and Cremona 5.\
[*Suppose that there exists only one such line $Ka$*]{}.\
Normalize by $a=(1,t^2,0,0),\ b=(b_1,b_2,b_3,b_4)$ with $\deg b_2<2$. The pair $(b_3,b_4)$ is, up to taking linear combinations, uniquely determined by the surface. The morphism $m: C^{norm}\rightarrow \mathbb{P}^1,\ t\mapsto (b_3:b_4)$ cannot be constant and has degree 1 or 2. There are the following cases.
#### 8.
$(b_3,b_4)=(1,t)$. Then $P(C)=T_{Gr,\overline{e_{12}}}$ and $\overline{e_{12}}\not \in C$. The equation is $$t_1t_3t_4^2-t_2t_3^3-t_3^2t_4^2b_1(\frac{t_4}{t_3})+t_3^4b_2(\frac{t_4}{t_3})=0.$$ Thus $1^3$ and Cremona 3.
#### 9.
$(b_3,b_4)=(t-\alpha ,t(t-\alpha ))$. Then $P(C)=T_{Gr,\overline{e_{12}}}$ and $\overline{e_{12}} \in C$. The equation is $$t_1(t_4-\alpha t_3)t_4^2-t_2t_3^2(t_4-\alpha t_3)-t_3^2t_4^2b_1(\frac{t_4}{t_3})+
t_3^4b_2(\frac{t_4}{t_3})=0.$$ Thus $1^3$ and Cremona 3.
#### 10.
(a,b) Now the morphism $m$ has degree two. If $t=0,\infty$ are the ramification points of $m$, then one normalizes to $(b_3,b_4)=(1,t^2)$. Then $\dim P(C)=3$ and $P(C)$ lies in only one tangent space, namely $T_{Gr,\overline{e_{12}}}$, and $\overline{e_{12}}\not \in C$. Write $b_1=b_{12}t^2+b_{11}t+b_{10}$ and $b_2=b_{21}t+b_{20}$. One can normalize further to $b_1=b_{11}t,\ b_2=b_{21}t$. Then $C$ has a node (case (a)) if $b_{21}b_{11}\neq 0$ and has a cusp otherwise (case (b)). Then $1^2,1^2,\ int$ and Cremona 6.
#### 11.
If $m:C^{norm}\rightarrow \mathbb{P}^1$ is ramified for, say, $t=1,\infty$, then one can normalize $a=(1,t^2,0,0),\ b=(b_1(t-1),b_2(t-1),1,(t-1)^2))$ with $b_1,b_2\in K$. The equation is $$t_3t_4(2t_1+(b_2-b_1)t_3-b_1t_4)^2-(t_2t_3-t_1t_3-t_1t_4+2b_1t_3t_4)^2=0.$$ The singular locus is the union of the line $L=\overline{e_{12}}$ and the conic $D$ given by $2t_1+(b_2-b_1)t_3-b_1t_4=0, \ t_2t_3-t_1t_3-t_1t_4+2b_1t_3t_4=0$. Now $P(C)=T_{Gr,\overline{e_{12}}}$, $\overline{e_{12}}\not \in C$, the image of $C\rightarrow L\times D,\ \overline{w}\mapsto (\overline{w}\cap L,\overline{w}\cap D)$ is a rational curve having a cusp. Then $1^2,2^2$ and Cremona 2.
#### 12.
If $m$ is ramified for, say, $t=1, \mu$ with $\mu \neq 0,1,\infty $, then one can normalize to $a=(1,t^2,0,0)\ b=(b_1,b_2,(t-1)^2,(t-\mu)^2)$ with $b_1,b_2\in K$.\
One can replace $b$ by $b-b_1\cdot a$ and normalize further to $a=(1,t^2,0,0),\ b=(0,1,(t-1)^2,(t-\mu )^2)$. A somewhat long computation yields the equation $$4t_3t_4( (\mu-1)^2(t_2-\mu t_1)-2t_3-2t_4)^2-$$ $$((\mu -1)^2(-\mu ^2 t_1t_3-t_1t_4+t_2t_3+t_2t_4)-t_3^2-6t_3t_4-t_4^2)^2=0.$$ The singular locus is the union of the line $L=\overline{e_{12}}$ and the conic $D$ given by the equations $$(\mu-1)^2(t_2-\mu t_1)-2(t_3+t_4)=0, \ (\mu-1)^3t_1(t_3-\mu t_4)-(t_3-t_4)^2=0.$$ Further $P(C)=T_{Gr,\overline{e_{12}}}$, $\overline{e_{12}}\not \in C$, the image of the morphism $C\rightarrow L\times D$ is a rational curve having a node. Then $1^2,2^2$ and Cremona 2.\
[*Suppose that there is no such line and that $P(C)$ lies in a tangent space*]{}.\
The inclusion $P(C)\subset T_{Gr,\overline{e_{12}}}$ yields a morphism $f:C^{norm}\rightarrow \overline{e_{12}}$ induced by $\overline{e_{12}}\neq \overline{w}\in C \mapsto \overline{w}\cap \overline{e_{12}}$. If the degree of $f$ is $1$, then we may assume that $(1,t,0,0)$ lies on $S$. Combining with a nonzero element $a\in H^0(C^{norm},B(2[p]))$, one finds a surface of degree 3 instead of 4.\
The possibility that [*the degree of $f$ is $2$*]{} is excluded by the following reasoning. Let $t$ be a parameter for $C^{norm}$ and write $f=((\alpha t+\beta )^2,(\gamma t+\delta )^2,0,0)$. Let $a,b$ be a basis of $H^0(C^{norm},B(2[p]))$. Then $\lambda _0(t)f=\lambda _1(t)a+\lambda _2(t)b$ holds for some $\lambda _0(t),\lambda _1(t),\lambda _2(t)\in K[t]$ with $\gcd(\lambda _1(t),\lambda _2(t))=1$. The Plücker coordinates of $a\wedge b$ are polynomials in $t$ with greatest common divisor 1 and maximal degree 4, since these parametrize $C$. The same holds for the Plücker coordinates of $f\wedge a$ and $f\wedge b$. The equality $\lambda _0(t)f\wedge a=-\lambda _2(t)\cdot a\wedge b$ implies that $\lambda _0(t)$ is a constant multiple of $\lambda _2(t)$. Similarly, $\lambda _0(t)$ is a constant multiple of $\lambda _1(t)$. We conclude that the $\lambda _i(t)$ are constant. Then$f\in H^0(C^{norm},B(2[p])$ and this contradicts the assumption.\
#### 13.
(a,b,c). [*If the degree of $f$ is $3$*]{}, then $\overline{e_{12}}$ has multiplicity 3 and thus $1^3$. As in case 2, there are three different possibilities for the ramification of $f$. One writes $f(t)=(c_1,c_2,0,0)$ where $c_1,c_2$ are relatively prime polynomials in $t$ and, say, $\deg c_1 <\deg c_2=3$. Let $a(t)=(a_1,a_2,a_3,a_4)$ be a nonzero section of $B(2[p])$. An inspection of the Plücker coordinates of $f\wedge a$ implies that $\max (\deg a_3, \deg a_4)\leq 1$. Moreover $a_3,a_4$ are linearly independent. Thus we may normalize to $(a_3,a_4)=(1,t)$. Because $\overline{e_{12}}$ has multiplicity 3, the equation for $S$ has the form $t_1A_1+t_2A_2+A_3=0$, where $A_1,A_2,A_3$ are homogeneous polynomials in $t_3,t_4$ of degrees $3,3,4$. Substitution of $(\lambda c_1+a_1,\lambda c_2+a_2,1,t)$ in this equation yields $c_1(t)A_1(1,t)+c_2(t)A_2(1,t)=0$ and we can normalize to $A_1(1,t)=c_2(t),\ A_2(1,t)=-c_1(t)$. In particular, $\gcd(A_1,A_2)=1$. Further $A_3(1,t)=-a_1(t)c_2(t)+a_2(t)c_1(t)$. The term $A_3$ cannot be made 0 by a transformation of the form $t_1\mapsto t_1+*t_3+*t_4, \ t_2\mapsto t_2+*t_3+*t_4$, since $P(C)$ does not lie in another tangent space. Therefore, $\max (\deg a_1,\deg a_2)=2$. Further $\dim P(C)=4$ and $\overline{e_{12}}\not \in C$. One verifies that the equations belong to the case that $B$ has type $-2,-2$ by comparing with the cases $1^3$ where $B$ has type $-1,-3$. Further Cremona 8.\
[*Suppose that there is no such line and $P(C)$ does not lie in a tangent space*]{}.
#### 14.
[*We claim that the singular locus is $3^2$ and is of species Cremona* ]{}1.\
The conditions imply that $\dim P(C)=4$ and $C$ is nonsingular. Suppose that the singular locus of $S$ contains a line. This line belongs to $C$ (because of Lemma \[1.2.3\]) and is, say, $\overline{w}(0)\in C$. Take a plane $H$ containing $\overline{w}(0)$. The intersection $H\cap S$ consists of $\overline{w}(0)$ with multiplicity $\geq 2$ and a remaining curve $R$ which is a conic or two lines or one line. For $t\neq 0$ the intersection $\overline{w}(t)\cap H$ lies on $R$. The possibility that $R$ is one or two lines contradicts that $P(C)$ does not lie in a tangent space. Thus $R$ can only be a conic. For $t\neq 0$, the positive divisor $\overline{w}(t)\cap R$ has degree 1 and has degree 2 for $t=0$. This is a contradiction.
We conclude that the singular locus of $S$ does not contain a line. Then, because of Lemma \[3.2.1\] and Subsection \[1.4\], the singular locus of $S$ is the twisted cubic curve. In § \[3.2.4\] it is shown that this [*Number*]{} 14 consists of six subclasses.
### The vector bundle $B$ for a genus 1 curve $C^{norm}$ {#3.1.3}
Here we use the information from § \[3.2.5\] and § \[3.2.7\] below, and deduce the structure of the vector bundle $B$ on the genus 1 curve $C=C^{norm}$.
#### 15.
[*Case $1^2$*]{}. The equation is $(t_1t_4-t_2t_3)^2+H(t_3,t_4)$, where $H$ is homogeneous of degree 4 and defines 4 distinct points on $\mathbb{P}(Ke_3+Ke_4)=\mathbb{P}^1$. We may suppose that these points are $0,1,\lambda ,\infty$. The lines $\overline{w}(t)\in C$ on $S$ are computed to be the lines passing through the points $(1,t,0,0)$ and $(0,y,1,t)$, with $y^2=H(1,t)$. The genus one curve $C$ is made into an elliptic curve by the choice of the neutral element $e$ to correspond to $t=y=\infty$. We note that $\overline{e_{12}}\not \in C$. $(1,t,0,0)$ is a section of $B(2[e])$ and $(0,y,1,t)$ is a section of $B(3[e])$. Further $w(t)=(1,t,0,0)\wedge (0,y,t,1)=ye_{12}+e_{13}+te_{14}+te_{23}+t^2e_{24}+0e_{34}$ is a section of $\Lambda ^2B([4]e)$. Consider the exact sequence $$0\rightarrow O_C(1,t,0,0) \rightarrow B(2[e])\rightarrow O_C(0,0,1,t)\rightarrow 0.$$ From $O_C(1,t,0,0)\cong O_C(0,0,1,t)\cong O_C$ and $H^0(C,B(2[e]))=K(1,t,0,0)$ one concludes that the sequence does not split. Therefore the ruled surface (in the modern sense) $\tilde{S}\rightarrow C$ corresponds to the unique indecomposable vector bundle on $C$ which is an extension of $O_C$ by $O_C$. (see [@Har]).
#### 16.
[*Case $1^2,1^2$*]{}. The equation $F$ for $S$ is bi-homogeneous of degree$(2,2)$ in the pairs of variables $t_1,t_2$ and $t_3,t_4$. The equation $F$ also defines a genus one curve $E\subset \mathbb{P}(Ke_1+Ke_2)\times \mathbb{P}(Ke_3+Ke_4)$. Further $E \rightarrow C\subset Gr$ is the isomorphism which sends $p\in E$ to the line through the points $(pr_1(p),0,0)$ and $(0,0,pr_2(p))$. The vector bundle $B$ is the direct sum of the line bundles $\mathcal{L}_1:=\{ (\overline{w},v)|\overline{w}\in C,\ v\in Ke_1+Ke_2,\ w\wedge v=0\}$ and $\mathcal{L}_2:=\{ (\overline{w},v)|\overline{w}\in C,\ v\in Ke_3+Ke_4,\ w\wedge v=0\}$ of degree $-2$.
A line bundle $\mathcal{L}$ on $E$ of degree $-2$ induces a degree $2$ morphism $E\rightarrow \mathbb{P}(H^0(E,\mathcal{L}^*))$. This yields a bijection between the isomorphy classes of line bundles of degree $-2$ and the equivalence classes of nonconstant morphisms $E\rightarrow \mathbb{P}^1$ of degree $2$. Then $\mathcal{L}_1$ is not isomorphic to $\mathcal{L}_2$, since the two morphisms are not equivalent. The ruled surface $\tilde{S}\rightarrow E$ is equal to $Proj(O_E\oplus \mathcal{L})$, where $\mathcal{L}=\mathcal{L}_1\otimes \mathcal{L}_2^{-1}$ is any line bundle of degree 0, not isomorphic to $O_E$. In particular, $\tilde{S}\not \cong \mathbb{P}^1\times E$.
The classification, using the singular locus {#3.2}
--------------------------------------------
### $2^2$ does not occur as singular locus
\[3.2.1\] The singular locus of a quartic ruled surface cannot be a conic.
Suppose that the conic $D$, lying in a plane $H\subset \mathbb{P}(V)$, is the singular locus of some ruled quartic surface $S$, corresponding to a curve $C\subset \mathbb{P}(\Lambda ^2V)$. If $C$ has genus 1, then $P(C)$ is contained in a tangent space of $Gr$ at some point $\overline{w}_0$. The morphism $\overline{w}\in C\mapsto \overline{w}\cap \overline{w}_0\in \overline{w}_0$ has degree at least 2 and thus $\overline{w}_0$ belongs to the singular locus. Hence $C^{norm}$ has genus 0. The morphism $f: C^{norm}\rightarrow D$, given by $\overline{w}\in C^{norm}\mapsto \overline{w}\cap H\in D$, has degree at most 2, since the multiplicity of $D$ is 2.\
Suppose that the degree of $f$ is 1. One can parametrize $C^{norm}$ with a parameter $t$ and choose coordinates for $\mathbb{P}(V)$ such that the line $\overline{w}(t)\in C^{norm}$ intersects the conic $D$ in the point $(0,1,t,t^2)$. Let $(1,0,a,b)$ with $a,b\in K(t)$ be another point of this line $\overline{w}(t)$. The Plücker coordinates of $\overline{w}(t)$ are $$p_{12}=1,\ p_{13}=t,\ p_{14}=t^2,\ p_{23}=-a,\ p_{24}=-b,\ p_{34}=t^2a-tb .$$ Let $d$ be the common denominator of $a$ and $b$. Then $\{dp_{ij}\}$ are polynomials of degree $\leq 4$ and with $\gcd=1$. If $\alpha$ is a zero of $d$, then the line $\overline{w}(\alpha )$ lies in the plane $H$. Since this is not possible, $d=1$ and $a,b\in K[t]$. One obtains the contradiction that the line $\overline{w}(\infty )$ lies in the plane $H$.\
Suppose that the degree of $f$ is 2. One can parametrize $C^{norm}$ with parameter $t$, and choose coordinates for $\mathbb{P}(V)$ such that $\overline{w}(t)\mapsto (0,1,t^2,t^4)\in D$. The line $\overline{w}(t)$ goes through a point $(1,0,a,b)$ where $a,b\in K(t)$. The Plücker coordinates of $\overline{w}(t)$ are $$p_{12}=1,\ p_{13}=t^2,\ p_{14}=t^4,\ p_{23}=-a,\ p_{24}=-b,\ p_{34}=t^4a-t^2b .$$ Let $d$ be the common denominator of $a$ and $b$. After multiplying the Plücker coordinates with $d$, the degrees are bounded by 4. Hence $d=1$ and $a,b\in K[t]$. Further the degree of $a$ is $\leq 2$ and the degree of $c:=b-t^2a$ is $\leq 2$. The symmetric polynomial $w(s)\wedge w(t)$ in $s,t$ can only have the factors $s+t$ and $s-t$. Indeed, $t\neq s$ and $w(s)\wedge w(t)=0$ implies that $\overline{w}(s)\cap \overline{w}(t)\in D$ and thus $s=-t$. It follows that $a=a_0+a_2t^2,\ c=c_0+c_2t^2$ and this contradicts that $C^{norm}$ is parametrized by $t$.
### $1^2,2^2$ {#3.2.2}
The curve $C^{norm}$, corresponding to a ruled quartic surface $S$ with this type of singular locus, has genus 0 by Observation \[1.2.11\]. The singular locus is the union of a conic $D$ and a line $L$. The plane $H\supset D$ satisfies $S\cap H=C$. Thus $L$ does not lie in $H$ and the intersection $L\cap H$ is a point of $D$. As in the proof of Lemma \[3.2.1\], one shows that the morphism $C^{norm}\rightarrow D$, given by $\overline{w}\in C\mapsto \overline{w}\cap H\in D$, has degree 2. Let $D=\{(0,1,\mu^2 ,\mu )|\ \mu \in \mathbb{P}^1\}$ and $L=\{(1 ,\lambda ,0,0)| \ \lambda \in \mathbb{P}^1\}$. The equations for $D$ and $L$ are $t_1=t_2t_3-t_4^2=0$ and $t_3=t_4=0$. The equation $F$ for $S$ lies in the ideal $(t_1,t_2t_3-t_4^2)^2\cap (t_3,t_4)^2$. Thus $F=t_1^2A_2+t_1(t_2t_3-t_4^2)A_1+(t_2t_3-t_4^2)^2$ where $A_2$ and $A_1$ are homogeneous of degrees 2 and 1. One may suppose that $A_1$ does not contain $t_1$. If $A_1$ contains $t_2$, then $F$ contains the monomial $t_1t_2^2t_3$ which is not possible. Hence $A_1$ is linear in $t_3,t_4$ and it follows that $A_2$ is homogeneous of degree 2 in $t_3,t_4$. Thus $$F=t_1^2(c_1 t_3^2+c_2 t_3t_4+c_3 t_4^2) +t_1(t_2t_3-t_4^2)(c_4 t_3+c_5 t_4) +(t_2t_3-t_4^2)^2 .$$ We will show that an irreducible equation $F$ as above, defines a ruled surface. Consider a point $(0,1,\mu ^2,\mu)\in D,\ \mu \neq 0,\infty$, then there is a $(1,\lambda ,0,0)\in L$ such that the line $\{(s,s\lambda +1,\mu ^2, \mu )|\ s\in \mathbb{P}^1\}$ lies on the surface $F=0$. Indeed, substitution in $F$ yields the equation $$s^2(c_1\mu ^4+c_2\mu ^3+c_3\mu ^2)+s^2\lambda \mu ^2(c_4\mu ^2+c_5\mu )+
s^2\lambda ^2\mu ^4=0.$$ For general constants $c_i$ and general $\mu \neq 0,\infty$, this equations has two solutions for $\lambda$. If the equation has for every $\mu$ only one solution for $\lambda$, then one easily verifies that $F$ is reducible (in fact a square).
Suppose now that $(1,\lambda,0,0)\in L$ is given. The $\mu \neq 0,\infty$ such that the line $\{(s,s\lambda +1,\mu ^2, \mu )|\ s\in \mathbb{P}^1\}$ lies on $F=0$ are solutions of the equation $$\mu ^2(\lambda ^2+c_4\lambda +c_1)+\mu (c_2+\lambda c_5)+c_3=0 .$$ (a) [*Suppose $c_3\neq 0$*]{}. If the equation has only one solution for $\mu \neq 0,\infty$, then $F$ is easily verified to be a square. The assumption that $F$ is irreducible, implies that there are for general $\lambda$ two solutions $\mu$.\
We conclude that the maps $C\rightarrow D$ and $C\rightarrow L$, given by $\overline{w}\in C\mapsto \overline{w}\cap H\in D$ and $\overline{w}\in C\mapsto\overline{w}\cap L\in L=\overline{e_{12}}$ have both degree 2. A further calculation shows that $P(C)=T_{Gr,\overline{e_{12}}}$, $\overline{e_{12}}\not \in C$ and the vector bundle $B$ has type $-2,-2$. There are still two cases, [*Number*]{} 11 and 12.\
(b). If $c_3=0$, then $c_2=c_5=0$ is excluded by $F$ is irreducible. Thus there is only one solution $\mu$. The maps $C\rightarrow D$ and $C-\rightarrow L=\overline{e_{12}}$ have degrees 2 and 1. Further calculation shows that $P(C)=T_{Gr,\overline{e_{12}}}$, $\overline{e_{12}}\in C$ and the vector bundle $B$ has type $-1,-3$. This is [*Number*]{} 3.\
In Rohn’s paper only case (a) is considered and this is treated as follows. The image $E$ of the morphism $C^{norm} \rightarrow D\times L$ is given by an irreducible bi-homogeneous form of bi-degree $(2,2)$. Since $C^{norm}$ has genus 0, the curve $E$ has a singular point which is a node or a cusp. The embedding $E\subset D\times L\cong \mathbb{P}^1\times \mathbb{P}^1$ can be chosen to be symmetric if the field $K$ is algebraically closed. For $K=\mathbb{R}$ one can have a symmetric or an anti-symmetric embedding.\
[*If $E$ has a node*]{}, then the equation $A$, symmetric in $\lambda ,\mu$, for the embedding is written as $a_1\lambda ^2\mu ^2+a_2(\mu ^2\pm \lambda ^2)+2a_3\lambda \mu$, where $\lambda$ and $\mu$ are inhomogeneous coordinates for the rational curves $L$ and $D$. The $\pm$ sign takes care of the real case where one also has to consider an anti-symmetric embedding. The singular point of $E$ corresponds to $\lambda =\mu =0$, which is the point $(0,1,0,0)$. The surface $S$ containing the family of the lines through the the pairs of points $\{(\lambda ,1,0,0), (0,1,\mu ^2,\mu )\}$ satisfying $A(\lambda ,\mu )=0,\ \lambda , \mu \neq 0$ has the equation $$a_1t_1^2t_3^2+a_2((t_2t_3-t_4^2)^2\pm t_1^2t_4^2)+2a_3t_1t_4(t_2t_3-t_4^2)=0.$$ There are various possibilities over $\mathbb{R}$ of the ‘pinch points’ on $L$ and $D$, i.e., the ramification points for the two projections $pr_1:E\rightarrow D,\ pr_2:E\rightarrow L$.
1. All four are real if $\pm =+$ and $\frac{a_3^2-a_2^2}{a_1a_2}>0$.\
Series XIII, no 8, corresponds to this case with additionally $a_1,a_2>0$.
2. No real ones, if $\pm =+$ and $\frac{a_3^2-a_2^2}{a_1a_2}<0$.
3. Real on $L$ and not real on $D$ if $\pm =-$ and $a_1>0,\ a_2<0$.
4. Not real on $D$ and real on $L$ if $\pm =-$ and $a_1>0,a_2>0$.
[*If $E$ has a cusp*]{}, then the equation $A$, symmetric in $\lambda ,\mu$, for the embedding $E\subset \mathbb{P}^1\times \mathbb{P}^1$ can be normalized (following Rohn) to $$(\lambda -\mu )^2-2\lambda \mu (\lambda +\mu )+\lambda ^2\mu^2=0.
\mbox{ This leads to the equation}$$ $$t_1^2t_3^2-2t_1t_3(t_1t_4+t_2t_3-t_4^2)+(t_1t_4-t_2t_3+t_4^2)^2=0\mbox{ for } S.$$
### $1^3$ {#3.2.3}
The line with multiplicity 3 is chosen to be $t_3=t_4=0$. Then the equations have the form $t_1A_1+t_2A_2+A_3=0$ with $A_1,A_2,A_3$ homogeneous in $t_3,t_4$ of degree 3,3,4; $\gcd(A_1,A_2,A_3)=1$ and $A_1,A_2$ are linearly independent. Conversely, one easily verifies that the above equation defines a ruled surface of degree 4.
The pair $(A_1,A_2)$ is unique up to taking linear combinations (and linear changes of $t_3,t_4$). In other words the morphism $f:\mathbb{P}^1\rightarrow \mathbb{P}^1$, given by $(t_3:t_4)\mapsto (A_1:A_2)$, is unique and can have degree 3, 2 or 1. In the first case there are many possibilities for $f$. In the second case one can normalize $(A_1,A_2)=d(t_3,t_4)\cdot (t_3^2,t_4^2)$ and in the third case $(A_1,A_2)=d(t_3,t_4)\cdot (t_3,t_4)$. The term $A_3$ can be changed into $A_3+\ell_1A_1+\ell _2A_2$ with $\ell _1,\ell _2$ homogeneous in $t_3,t_4$ of degrees 1, by replacing $t_1,\ t_2$ by $t_1+\ell _1,\ t_2+\ell _2$.
1. $\gcd(A_1,A_2)=1$ and $A_3=0$. [*Number*]{} $1\ a,b,c$.
2. $\gcd(A_1,A_2)=1$ and $A_3\not \in \{\ell _1A_1+\ell _2A_2\}$. [*Number*]{} $13\ a,b,c$, XIII 6.
3. $\gcd(A_1,A_2)$ has degree 1. [*Number*]{} $8,9$.
4. $\gcd(A_1,A_2)$ has degree 2, not a square. $\overline{e_{12}}$ is a node. [*Number*]{} $4$, XIII 7.
5. $\gcd(A_1,A_2)$ has degree 2 and is a square. $\overline{e_{12}}$ is a cusp. [*Number* ]{}$5$.
### $3^2$ {#3.2.4}
The twisted cubic curve $TC:=\{(1,\lambda ,\lambda ^2,\lambda ^3)|\ \lambda \in \mathbb{P}^1\}$ is the singular locus of $S$. The homogeneous ideal of $TC$ is generated by the three homogeneous forms $X=t_1t_3-t_2^2,\ Y=t_2t_3-t_1t_4,\ Z=t_2t_4-t_3^2$. There are two relations $t_3X+t_2Y+t_1Z=t_4X+t_3Y+t_2Z= 0$. The equation $F$ of $S$ is homogeneous of degree 4 and lies in the ideal $(X,Y,Z)^2\subset K[t_1,t_2,t_3,t_4]$. A computation in the ring $R:=K[\frac{t_2}{t_1},\frac{t_3}{t_1},\frac{t_4}{t_1}] $ shows that the element $G:=F(1,\frac{t_2}{t_1},\frac{t_3}{t_1},\frac{t_4}{t_1})$ of total degree $\leq 4$, lying in the ideal $(R\frac{X}{t_1^2}+R\frac{Y}{t_1^2})^2\subset R$, is a homogeneous polynomial in the terms $\frac{X}{t_1^2},\frac{Y}{t_1^2},\frac{Z}{t_1^2}$ of degree $2$. It follows that $F(t_1,t_2,t_3,t_4)=H(X,Y,Z)$, where $H$ is a homogeneous form of degree 2.\
Consider the morphism $f:\mathbb{P}(V)\setminus TC\rightarrow \mathbb{P}^2$, given by $(t_1,t_2,t_3,t_4)\mapsto (X,Y,Z)$. The fibres of $f$ are the lines of $\mathbb{P}(V)$ intersecting $TC$ with multiplicity 2. Thus a fibre is a corde of $TC$ or a tangent line of $TC$. Let $H(X,Y,Z)$ be homogeneous of degree 2. Then the closure of the preimage under $f$ of the subscheme $H=0$ of $\mathbb{P}^2$ is the ruled surface $S$ given by the equation $F(t_1,t_2,t_3,t_4):=H(X,Y,Z)$. Further $F$ is irreducible and reduced if and only if $H=0$ is a conic. In the sequel we suppose that $\{H=0 \}$ is a conic and we classify the possibilities. The surface with $H=T:=Y^2-4XZ$ is rather special. It consists of [*all tangent lines of $TC$*]{} (see Corollary \[1.2.6\]). For any other conic $H=0$, the intersection with $T=0$ has multiplicity 4. [*In the general case, the intersection of the two conics consists of 4 points*]{}. XIII 9, 10.
Suppose that the intersection has at least one point with multiplicity $>1$. The projective space $\mathbb{P}^3$ admits an automorphism which preserves the curve $T=Y^2-4XZ=0$ and brings this point to $(X,Y,Z)=(0,0,1)$. Then $H$ has the form $XZ+aX^2+bXY+cY^2$. One has the following cases for the intersection.
- $aX^2+bXY+(c+1/4)Y^2=0$ has two distinct solutions (i.e., $b^2-a(4c+1)\neq 0$) and $(c+1/4)\neq 0$. Then the intersection consists of one point with multiplicity 2 and two points with multiplicity 1.
- $aX^2+bXY+(c+1/4)Y^2=0$ has two distinct solutions and $(c+1/4)= 0$. Then the intersection consists of one point with multiplicity 3 and one point with multiplicity 1.
- $aX^2+bXY+(c+1/4)Y^2=0$ has one solution (i.e. $b^2-a(4c+1)=0$) and $(c+1/4)\neq 0$. Then the intersection consists of two points with multiplicity 2.
- $aX^2+bXY+(c+1/4)Y^2=0$ has one solution (i.e. $b^2-a(4c+1)=0$), $(c+1/4)=0$ and $a\neq 0$. Then the intersection consists of one point with multiplicity 4.
[*Thus we found in total six distinct cases for $3^2$*]{} (compare [@Bot]). As we will show below, there is a further natural subdivision of these classes.\
Let $C\subset Gr$ be the curve associated to the surface $S_H$ associated to the irreducible $H=H(X,Y,Z)$ of degree 2. The morphism $C\rightarrow \{H=0\}\subset \mathbb{P}^3$ is clearly an isomorphism. Thus $C$ is a nonsingular rational curve. It is clear that $P(C)$ does not lie in two tangent spaces of $Gr$. Moreover, since $C$ is not singular, one must have $\dim P(C)=4$. For the surfaces $S_T$ and $S_H$ with $H=XZ+aX^2-\frac{1}{4}Y^2$ with $a\neq 0$ (case (iv) above), $P(C)$ is not a tangent space. For the remaining 4 classes there are, a priori, now two possibilities:
\(a) If $P(C)$ is not a tangent space, then $B$ has type $-2,-2$. There are in total six cases and they fill up [*Number* ]{} 14.\
(b) If $P(C)$ is a tangent space $T_{Gr,\overline{w}_0}$. Then $B$ has type $-1,-3$. [*Number*]{} $2\ a,b,c$. These cases are explained as follows.\
The line $\overline{w}_0$ coincides with $\overline{e_{12}}$ of the cases $2\ a,b,c$. The image $f(\overline{w}_0)$ is the conic given by $H=0$. The possibilities for intersection of $H=0$ with $T=0$ reflects the possibilities for the ramification of the degree 3 morphism in $2\ a,b,c$. Case $2 \ a$ corresponds to (iii) above; case $2 \ b$ to (i) above; case $2\ c$ to the case where the intersection consists of 4 points.
### $1^2$ {#3.2.5}
From the Observations \[1.2.obs\] and Subsection \[1.4\], one obtains that the genus of $C$ is 1. Further $P(C)$ lies in only one tangent space of $Gr$, say at the point $\overline{e_{12}}$, since otherwise the surface $S$ has two skew singular lines. The morphism $C\rightarrow \overline{e_{12}}$, given by $\overline{w}\in C\mapsto \overline{w}\cap \overline{e_{12}}$, has degree 2 since $m=2$. This map has 4 ramification points and we obtain for, say, $t\neq 0,1,\lambda ,\infty$ two lines of $C$ through the point $(1,t,0,0)\in \overline{e_{12}}$. The map $t\neq 0,1,\lambda ,\infty \mapsto P(t)$, where $P(t)\supset \overline{e_{12}}$ denotes the plane through these two lines, has degree 1. We may suppose that $P(t)\cap \{(0,0,*,*)\}=(0,0,1,t)$. The equation for $S$ is $$t_1^2A+t_2^2B+t_1t_2C+t_1D+t_2E+F;\ A,B,C,D,E,F \mbox{ homogeneous in } t_3,t_4 .$$ For any point $(a_1,a_2,0,0)\in \overline{e_{12}}$, the plane $a_2t_3-a_1t_4=0$ meets $S$ in $\overline{e_{12}}$ and two lines (or one with multiplicity 2) through $(a_1,a_2,0,0)$. This implies that $t_1^2A(t_3,t_4)+t_2^2B(t_3,t_4)+t_1t_2C(t_3,t_4)$ is a multiple of $(t_4t_1-t_3t_2)^2$ and that $t_1D(t_3,t_4)+t_2E(t_3,t_4)$ is divisible by $(t_4t_1-t_3t_2)$. After changing the variables $t_1,t_2$ we are reduced to two possible equations for $S$: $$\ (t_4t_1-t_3t_2)^2+H(t_3,t_4)=0 \mbox{ and } \ G(t_3,t_4)(t_4t_1-t_3t_2)+H(t_3,t_4)=0 .$$ The line $\overline{e_{12}}$ has multiplicity 3 for the second equation. Thus only the first equation is possible with $H$ not a square since $S$ is irreducible. Moreover, the ruled surface defined by this equation has $\overline{e_{12}}$ as singular locus if and only if $H$ has no multiple factor. Rohn found an equation of this form, namely $$a(t_3^2\pm t_4^2)+2bt_3^2t_4^2+c(t_4t_2-t_4t_1)^2 =0.$$ The sign $\pm$ distinguishes two classes of real cases. For $\pm =+$ and $\frac{b}{a}<-1$, the four ramification points of $C\rightarrow \overline{e_{12}}$ are real. This is [*Number*]{} 15 and Series XIII, no 5.\
[*Remark*]{}. The equation $(t_4t_1-t_3t_2)^2+H(t_3,t_4)=0$ where $H$ has no multiple factors, is valid for any field $K$. If $K$ is algebraically closed, then $H$ is determined by the $j$-invariant of the four zeros of $H$ in $\mathbb{P}^1$. For a general field $K$ there are forms for $H$.\
### $1^2,1^2,\ int $, intersecting lines {#3.2.6}
The two intersecting lines $L_1,L_2$, making up the singular locus of the ruled quartic surface $S$, lie in a plane $H$. For $\overline{w}\in C$ and $\overline{w}\neq L_1,L_2$ the intersection $\overline{w}\cap H$ is a point of $L_1\cup L_2$. The induced morphism $C^{norm}\rightarrow L_1\cup L_2$ has, say, the line $L_1$ as image. Thus we find a nonconstant morphism $f:C^{norm}\rightarrow L_1$ and $P(C)$ lies in the tangent space of $Gr$ at the point $L_1$. For $q\in L_2$ and $q\not \in L_1$, there is no $\overline{w}\in C, \overline{w}\neq L_1,L_2$ with $q\in \overline{w}$. One concludes that $L_2\in C$. Moreover $L_2$ is a singular point $s$ of $C$ since $L_2$ belongs to the singular locus. In particular, $C$ is a rational curve and $\dim P(C)=3$. If $P(C)$ lies in the tangent space of $Gr$ at another point $M\in Gr$, then one obtains a morphism $C\rightarrow M$ by $\overline{w}\mapsto \overline{w}\cap M$. Since $C$ has a singular point, this morphism has degree $>1$ and one finds the contradiction that $M$ belongs to the singular locus. Thus $P(C)$ lies in a single tangent space. The rational map $C-\rightarrow L_1$, given by $\overline{w}\mapsto \overline{w}\cap L_1$, is well defined at the singular point $s\in C$. Then $f$ has degree $>1$ and its degree is 2, since $L_1$ has multiplicity 2. Further $L_1\not \in C$, otherwise the multiplicity of $L_1$ would be 3.\
For a suitable basis of $V$ and parametrization of $C^{norm}$, the morphism $C^{norm}\rightarrow L_1$ has the form $\overline{w}(t)\mapsto (1,t^2,0,0)$. Let $b:=(b_1,b_2,b_3,b_4)$, with all $b_i\in K[t]$ and $\gcd(b_1,\dots ,b_4)=1$, be another point of the line $\overline{w}(t)$. By subtracting a multiple of $(1,t^2,0,0)$ one arrives at $\deg b_2\leq 1$. The Plücker coordinates of $\overline{w}(t)$ are $(b_2-t^2b_1,b_3,b_4,t^2b_3,t^2b_4,0)$ and thus $\deg b_1, \deg b_3,\deg b_4\leq 2$. The morphism $C\rightarrow \mathbb{P}^1$, by $\overline{w}(t)\mapsto (b_3(t):b_4(t))$, is well defined and not constant. Since $C$ is singular, this morphism has degree 2. The corresponding degree $2$ morphism $g: C^{norm}\rightarrow \mathbb{P}^1$ factors over $C$. If the singular point of $C$ is a cusp for $t=\infty$, then $t=\infty$ is a ramification point and $g$ has the form $g(t)=(1:(t+\alpha )^2)$. If the singular point of $C$ is a node, corresponding to $t=\pm 1$, then $g(t)=(1:(\frac{at+b}{ct+d})^2)$ also satisfies $g(1)=g(-1)$. Hence $g(t)=(1:(\frac{t+\beta}{\beta t+1})^2)$ with $\beta ^2\neq 1$.
Suppose that $C$ has a cusp, then $(b_3(t),b_4(t))=(1, (t+\alpha )^2)$ and $b_1,b_2$ can be normalized to constant multiples of $t$. The condition that $t=\infty$ is a cusp for $C$ implies $b_1=0$ and so we arrive at $b=(0,t,1,(t+\alpha )^2)$. The equation reads $$(t_2t_3-t_1t_4-\alpha ^2t_1t_3+\alpha ^2 t_3^2)^2-t_1t_3t_4(t_3-2\alpha t_1)^2=0.
\mbox{ {\it Number} } 10\ b .$$
Suppose that $C$ has a node, then $(b_3,b_4)=((\beta t+1)^2,(t+\beta )^2)$ with $\beta ^2\neq 1$. For $\beta =0$, one can normalize $b_1,b_2$ to constant multiples of $t$. The condition $\overline{w}(1)=\overline{w}(-1)$ implies that $b_1=b_2=ct\neq 0$. The equation reads $$c^2t_3t_4(t_3-t_4)^2-(t_1t_4-t_2t_3)^2=0. \mbox{ {\it Number} } 10 \ a .$$ For $\beta ^2\neq 0,1$, one can normalize $b_1,b_2$ to constants and the condition $\overline{w}(1)=\overline{w}(-1)$ implies $b_1=b_2=c\neq 0$. The equation reads $$t_3t_4\{2\beta (t_1-t_2)+c(1-\beta ^2)(t_3-t_4)\}^2 -\{ -t_1(\beta ^2t_3+t_4)+t_2(t_3+\beta ^2t_4\}^2=0 .$$ Again [*Number* ]{} $10\ a$. Rohn found the two similar equations $$at_3t_4^3+(t_1t_4-t_2t_3)^2=0, \mbox{ and } at_4^4+2bt_3^2t_4^2+(t_1t_4-t_2t_3)^2=0 .$$
### $1^2,1^2$, skew lines {#3.2.7}
The skew lines can be supposed to be $\overline{e_{12}},\overline{e_{34}}$. Every monomial in the equation $F$ of $S$ is divisible by one of the terms $t_1^2,t_1t_2,t_2^2$ and by one of the terms $t_3^2,t_3t_4,t_4^2$. Therefore $F$ is bi–homogeneous of degree $(2,2)$ and $F$ defines a Zariski closed subset of $\overline{e_{12}}\times \overline{e_{34}}\cong \mathbb{P}^1\times \mathbb{P}^1$, which is an irreducible curve $E$. One considers the morphism $f:\mathbb{P}(V)\setminus \overline{e_{12}}\cup \overline{e_{34}}\rightarrow
\mathbb{P}^1\times \mathbb{P}^1$, given by $(a_1,a_2,a_3,a_4)\mapsto ((a_1,a_2),(a_3,a_4))$. Then $S$ is the Zariski closure of $f^{-1}(E)$. The curve $E$ has no singularities since otherwise the singular locus of $S$ would contain another line. Thus $E$ is a curve of genus 1. One easily sees that $C$ identifies with $E$ and that $P(C)$ lies in the two tangent space of $Gr$ at the points $\overline{e_{12}}$ and $\overline{e_{34}}$. [*Number*]{} $16$.
In the above the bases of the two vector spaces $Ke_1+Ke_2$ and $Ke_3+Ke_4$ (or equivalently the parametrization of $\overline{e_{12}}$ and $\overline{e_{34}}$) can be chosen in a suitable way. Rohn (see Section 4) shows that for $K=\mathbb{C}$ these bases can be chosen such that the equation $F$ becomes symmetric, i.e., $F(t_1,t_2,t_3,t_4)=F(t_3,t_4,t_1,t_2)$. For $K=\mathbb{R}$ the results of Rohn are more complicated. These results are essential for the understanding of the models in Series XIII, 1,2,3,4 of quartic ruled surfaces with two skew lines of singularities.
### $1^2,1^2,1^2$ {#3.2.8}
Let $L_1,L_2,L_3$ denote the singular lines with multiplicity 2.
[*Suppose that the lines $L_1,L_2$ are skew*]{}. Then we may suppose $L_1=\overline{e_{12}},\ L_2=\overline{e_{34}}$. From Lemma \[1.2.10\] it follows that the equation $F$ of the surface $S$ is bihomogeneous of degree $(2,2)$ in the pairs of variables $t_1,t_2$ and $t_3,t_4$. The curve $E\subset \overline{e_{12}}\times \overline{e_{34}}\cong\mathbb{P}^1\times \mathbb{P}^1$, defined by $F$, has one singular point corresponding to the line $L_3$. This point is a node or a cusp. [*Number*]{} $6,7$. Not in Series XIII. The parametrization of $\overline{e_{12}}$ and $\overline{e_{34}}$ can be chosen (see Section \[section4\]) in order to obtain the standard equations of Rohn $$a_1\lambda ^2\mu ^2+a_2(\lambda ^2\pm \mu ^2)+2a_3\lambda \mu =0 \mbox{ and }
\lambda ^2\mu ^2+(\lambda -\mu )^2-2\lambda \mu (\lambda +\mu)=0,$$ where $\lambda = \frac{t_2}{t_1},\ \mu =\frac{t_4}{t_3}.$\
The next case to consider is $L_1\cap L_2,\ L_1\cap L_3, L_2\cap L_3 \neq \emptyset$. The three lines cannotlie in a plane $H$ since otherwise the curve $H\cap S$ has degree 6. It follows that $L_1\cap L_2\cap L_3$ is one point. We may suppose that $L_1$ is given by $t_1=t_2=0$, $L_2$ by $t_1=t_3=0$ and $L_3$ by $t_2=t_3=0$. Every monomial of the equation $F$ is divisible by $t_1^{a_0}t_2^{a_1}$ with $a_0+a_1=2$, by $t_1^{b_0}t_3^{b_1}$ with $b_0+b_1=2$ and by $t_2^{c_0}t_3^{c_1}$ with $c_0+c_1=2$. The $t_4$-part of $F$ can only be $c\cdot t_1t_2t_3t_4$. If $c=0$, then $F$ defines a cone. Otherwise one can reduce to the equation $(t_2^2t_3^2+t_1^2t_3^2+t_1^2t_2^2)+t_4t_1t_1t_3=0$ (or equivalently $(t_2t_3+t_1t_3+t_1t_2)^2+t_4t_1t_2t_3=0$). This equation defines the Steiner’s Roman surface and the three singular lines are in fact the only lines on this surface.\
Rohn’s symmetric form for bi–degree $(2,2)$ {#section4}
============================================
K. Rohn proves that over the field $K=\mathbb{C}$, there is an identification of $\overline{e_{12}}\times \overline{e_{34}}$ with $\mathbb{P}^1\times \mathbb{P}^1$ such that the equation $F$ of bi–degree $(2,2)$ is symmetric in the pairs of variables $t_1,t_2$ and $t_3,t_4$, i.e., $F(t_3,t_4,t_1,t_2)=F(t_1,t_2,t_3,t_4)$. This leads to only a few standard forms for $F$. Over the field $\mathbb{R}$, there are more possibilities. First of all, $\overline{e_{12}},\overline{e_{34}}$ can be a pair of conjugated lines over $\mathbb{C}$. Secondly, even if $\overline{e_{12}},\ \overline{e_{34}}$ are real lines, then the above identification need not be defined over $\mathbb{R}$. Thirdly, there are various possibilities over $\mathbb{R}$ for the ramification points of the two projections $C\rightarrow \mathbb{P}^1$. The models Series XIII, nr. 1,2,3,4 represent some of these cases. A ‘modern version’ of this work of Rohn is as follows.\
Consider the closed subset $E$ of $\mathbb{P}^1\times \mathbb{P}^1$, defined by a bi–homogeneous form $F$ of bi–degree $(2,2)$. To start we consider the case that $F$ is irreducible and $E$ is nonsingular and thus $E$ has genus 1. We call the embedding $E\subset \mathbb{P}^1\times \mathbb{P}^1$ [*symmetric*]{} if $(p,q)\in E\Rightarrow (q,p)\in E$.
\[4.0.2\] For a given embedding $E\subset \mathbb{P}^1\times \mathbb{P}^1$ as above, there exists an automorphism $f$ of the first factor, such that the new embedding $E\subset \mathbb{P}^1\times \mathbb{P}^1
\stackrel{f\times 1}{\rightarrow}\mathbb{P}^1\times \mathbb{P}^1$ is symmetric.
The required automorphism $f$ of $\mathbb{P}^1$ has the property $(p,q)\in E\Rightarrow (f^{-1}q,fp)\in E$. In particular, the morphism $C: (p,q)\mapsto (f^{-1}q,fp)$ is an automorphism of $E$ of order 2. We assume that $f$ exists, try to find its explicit form and then use this form to produce an $f$ with the required property. Some explicit information concerning the automorphisms of order 2 of $E$ is needed. For this purpose, we choose a point $e_0\in E$. This makes $E$ into an elliptic curve (and the addition of two points $a,b$ is written as $a+b$). Consider the automorphisms $\sigma$ and $\tau _a$ (any $a\in E$), given by $\sigma (p)=-p$ and $\tau _a(p)=p+a$. One verifies that the automorphisms of order 2 of $E$ are:\
(a) $\sigma \tau _a$ for any point $a$ on $E$,\
(b) $\tau_a$ where $a\neq 0$ is a point of order two on $E$.\
Division of $E$ by the action of an element in the first class yields $\mathbb{P}^1$ and division by the action of an element in the second class yields an elliptic curve. Thus the two projections $pr_i:E \rightarrow \mathbb{P}^1$ correspond to distinct elements $\sigma \tau _{a_1}$ and $\sigma \tau _{a_2}$ of order 2 with the property $pr_i\circ \sigma \tau _{a_i}= pr _i$ for $i=1,2$.
The assumption on $f$ and the definition of $C$ are equivalent to $pr_2(Ce )=f (pr _1(e ))$ for any $e \in E$. Replacing $e$ by $\sigma \tau _{a_1}e$ does not change the right hand side. Thus $C\sigma \tau _{a_1}e$ is either $Ce$ or $\sigma \tau _{a_2}Ce$. The first equality can only hold for four elements $e \in E$. Hence the second equality holds for almost all $e$ and thus holds for all $e$. We conclude that $C\sigma \tau _{a_1}=\sigma \tau _{a_2} C$.
Suppose that $C=\sigma \tau _c$. The equality $\sigma \tau _c\sigma \tau _{a_1}=\sigma \tau _{a_2}\sigma \tau _c$ is equivalent to $2c=a_1+a_2$. There are 4 solutions $c$ of this equation.
Suppose that $C=\tau _c$ with $c$ an element of order 2. Then one finds the contradiction $a_1=a_2$.
Take $C=\sigma \tau _c$ for some $c$ with $2c=a_1+a_2$. Define $f$ by the formula $f(pr_1(e )):=pr_2(Ce )$. This is well defined because of $C\sigma \tau _{a_1}=\sigma \tau _{a_2}C$. It is easily verified that $f$ is an isomorphism and has the required property.
Let $E\subset \mathbb{P}^1\times \mathbb{P}^1$ be a symmetric embedding and the homogeneous coordinates of the two projective lines are denoted by $x_0,x_1$ and $y_0,y_1$. Let $\{p_1,p_2,p_3,p_4\}\subset \mathbb{P}^1$ denote the four ramification points of the projection $pr_1:E\rightarrow \mathbb{P}^1$. There is an automorphism $s$ of order two which permutes each of the pairs $\{p_1,p_2\}$ and $\{p_3,p_4\}$. The two fixed points of $s$ can be supposed to be $0,\infty $ and thus $s$ has the form $s(x_0,x_1)=(x_0,-x_1)$. The four ramification points are then $\{(1,\pm d)\}$ and $\{(1,\pm e)\}$. By scaling $(x_0,x_1)\mapsto (x_0,\lambda x_1)$ with $\lambda ^2ed=\pm 1$ we arrive at four ramification points $\{(1,\pm d^{\pm 1}\}$ (with of course $d^4\neq 1$). Write $F=Ay_0^2+By_0y_1+Cy_1^2$. Then the four ramification points of $pr_1$ are the zeros of the discriminant $B^2-4AC$ and thus $B^2-4AC=x_1^4+bx_1^2x_0^2+x_0^4$ with $b=-(d^2+d^{-2})$. Then we obtain [*the normal form of K. Rohn for $F$*]{}: $$a_1(x_0^2y_0^2+x_1^2y_1^2)+a_2(x_0^2y_1^2+x_1^2y_0^2)+2a_3x_0x_1y_0y_1\ \ \
\mbox{ with } a_1a_2\neq 0$$ $$\mbox{or in Rohn's notation } a_1(\lambda ^2\mu ^2 +1)+a_2(\lambda ^2+\mu ^2)+
2a_3\lambda \mu \mbox{ with } \lambda =\frac{x_1}{x_0},\ \mu =\frac{y_1}{y_0} ,$$ $$\mbox{and with discriminant } x_1^4+bx_0^2x_1^2+x_0^4 \mbox{ and }
b=\frac{a_{1}^2+a_{2}^2-a_{3}^2}{a_{1}a_{2}}\neq \pm 2\ . \ \ \ \ \ \ $$
The above calculations are valid over any algebraically closed field of characteristic $\neq 2$. Now we analyze the more complicated situation over the field $\mathbb{R}$. Assume that the two lines $\overline{e_{12}},\overline{e_{34}}$ and $E$ are defined over $\mathbb{R}$. Assume moreover that $E(\mathbb{ R})$ is not empty (indeed otherwise the real model for the corresponding surface has no points). Fix a real point $e_0$ as the neutral element of $E$. The group $E(\mathbb{R})$ is either isomorphic to the circle $\mathbb{R}/\mathbb{Z}$ ([*the connected case*]{}) or to $\mathbb{R}/\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ ([*the disconnected case*]{}). In the connected case $E(\mathbb{R})$ has two elements of order dividing 2 and in the disconnected case there are 4 such elements. The collection of the real automorphisms of order two of $E$ consists of the $ \sigma \tau _a$ with $a\in E(\mathbb{R})$ and $\tau _v$ where $v$ is a real point of order 2.
[*Now we revisit the proof of the theorem for the case $K=\mathbb{R}$*]{}.\
[*The connected case*]{}. The fixed points $b$ of $\sigma \tau _{a_1}$ (note that$a_1\in E(\mathbb{R})$) are the solutions of $2b =-a_1$. Two of the $b$’s are real. The other two are complex conjugated. Hence two of the ramification points for $pr_1:E \rightarrow \mathbb{P}^1$ are real, the other two are complex conjugated. The same holds for the ramification points of $pr_2:E \rightarrow \mathbb{P}^1$. For the element $c$ with $2c=a_1+a_2$ there are two real choices. Thus the real version of the theorem remains valid in this case. Two of the four ramification points are real and the other two are complex conjugated. One can normalize such that the ramification points are $\pm d,\ \pm id^{-1}$ and this leads to Rohn’s normal equation $$a_{1}(-x_0^2y_0^2+x_1^2y_1^2)+a_{2}(x_0^2y_1^2+x_1^2y_0^2)+2a_{3}x_0x_1y_0y_1
\mbox{ with real }a_{1},a_{2},a_{3}.$$
[*The disconnected case*]{}. There are 4 real fixed points of $\sigma \tau _{a_1}$ if $a_1$ lies in the component of the identity of $E(\mathbb{R})$. In the opposite case there are no real solutions of $2b=a_1$. The same holds for $\sigma \tau _{a_2}$ and for the solutions of the equation $2c=a_1+a_2$. Hence there are cases where no real automorphism$f$ exist. All cases can be listed by:\
(a) 4 real ramification points for $pr_1$ and for $pr_2$ and 4 real solutions for $c$,\
(b) no real ramification points for $pr_1$ and $pr_2$ and 4 real solutions for $c$,\
(c) 4 real ramification points for $pr_1$, none for $pr_2$ and no real solution for $c$,\
(d) 4 real ramification points for $pr_2$, none for $pr_1$ and no real solution for $c$.\
Suppose that $c$ can be chosen to be real. For Rohn’s normal form one needs an automorphism $s$ permuting each pair $\{p_1,p_2\}$ and $\{p_3,p_4\}$. One may suppose that each pair is invariant under complex conjugation. Then the resulting $s$ is also real. For the cases (a) and (b) the standard equation is indeed $$a_{1}(x_0^2y_0^2+x_1^2y_1^2)+a_{2}(x_0^2y_1^2+x_1^2y_0^2)+2a_{3}x_0x_1y_0y_1$$ $$\mbox{ and discriminant }
x_1^4+\frac{a_{1}^2+a_{2}^2-a_{3}^2}{a_{1}a_{2}}x_1^2x_0^2+x_0^4\ ,$$ with $a_{1},a_{2},a_{3}\in \mathbb{R}$. One easily calculates that $\frac{ (a_{1}+a_{2})^2-a_{3}^2}{a_{1}a_{2}}<0$ corresponds to (a) and $\frac{ (a_{1}+a_{2})^2-a_{3}^2}{a_{1}a_{2}}>0$ corresponds to (b).\
For the cases (c) and (d) there is no real symmetric normal form for $F$. In case (c) (case (d) is similar), Rohn’s real normal form could be called [*half-symmetric*]{}, because of its form $$a_{1}(x_0^2y_0^2-x_1^2y_1^2)-a_{2}(x_0^2y_1^2-x_1^2y_0^2)+2a_{3}x_0x_1y_0y_1 \ .$$ The models 1, 2 and 3 of Series XIII deal with a pair of real skew double lines. In the terminology of Rohn, a [*pinch point*]{} is a ramification point for one of the two projections $pr_1,pr_2$ and situated on $L_1$ and $L_2$ with the obvious identification of these lines with the two $\mathbb{P}^1$’s. Series XIII nr. 1 corresponds to (a), Series XIII nr 2 corresponds to (b) and Series XIII no 3 to (c). Rohn also considers the situation where the ruled surface has a pair of complex conjugated lines as double lines and produces a standard form and an example, namely model 4 of series XIII.
[*Rohn’s normal form for other curves $E\subset \mathbb{P}^1\times \mathbb{P}^1$ of type $(2,2)$*]{}.\
These normal forms are useful for §§ \[3.2.2\], \[3.2.5\], \[3.2.6\], and \[3.2.8\]. There are three cases:\
(a) $E$ is irreducible and has a node,\
(b) $E$ is irreducible and has a cusp and\
(c) $E$ is reducible or is not reduced.\
In the following we use the notation and the ideas of the proof of the theorem.\
(a). The nonsingular locus of $E$ is, after a choice of a point $e_0$, the group $\mathbb{G}_m$. Let $\sigma$ denote the automorphism $x\mapsto -x$ and define $\tau _a(x)=ax$. The automorphisms of order 2 of $E$ are $\sigma \tau _a$ (any $a\in \mathbb{G}_m$) and $\tau _{-1}$. Dividing $E$ by the action of $\sigma \tau_a$ yields the quotient $\mathbb{P}^1$ and dividing by the action of $\tau _{-1}$ yields a rational curve with a double point. Thus the two projections $pr_i:E\rightarrow \mathbb{P}^1$ correspond to order two elements $\sigma \tau _{a_i}$ for $i=1,2$ with $a_1\neq a_2$. The required automorphism $C$ of order two should satisfy $C\sigma \tau _{a_1}=\sigma \tau _{a_2}C$. There are two possibilities for $C$, namely $C=\sigma \tau _c$ with $c^2=a_1a_2$. Thus we find a symmetric embedding $E\subset \mathbb{P}^1\times \mathbb{P}^1$ for any algebraically closed field of characteristic $\neq 2$.\
For $\mathbb{R}$ as base field, the situation is more complicated. Suppose that both lines, i.e., the two factors $\mathbb{P}^1$, and $E$ are defined over $\mathbb{R}$. We assume that the nonsingular locus $E^*$ has a real point $e_0$. There are two possibilities for $E^*(\mathbb{R})$, namely: (i) $\mathbb{G}_m(\mathbb{R})=\mathbb{R}^*$ and (ii) $\mathbb{R}/\mathbb{Z}$.\
In case (i), one has to solve the equation $c^2=a_1a_2$ with $c\in \mathbb{R}^*$. If there is a solution, then one has a symmetric embedding $E\rightarrow \mathbb{P}^1\times \mathbb{P}^1$, defined over $\mathbb{R}$. In the opposite case, one makes an anti-symmetric embedding (by adding some minus signs). The [*two standard equations*]{} are $$a_1\lambda ^2\mu ^2+a_2(\lambda ^2\pm \mu ^2)+2a_3\lambda \mu =0,
\mbox{ with }\lambda =\frac{x_1}{x_0},\ \mu =\frac{y_1}{y_0} .$$
In case (ii), the automorphisms of order two are the maps $f_a: x\mapsto -x+a$ (any $a\in \mathbb{R}/\mathbb{Z}$) and $x\mapsto x+1/2$. The last automorphism is ruled out because it does not give a $\mathbb{P}^1$ as quotient. Now we have to solve $Cf_{a_1}=f_{a_2}C$ for some order two automorphism $C$. The two solutions for $C$ are $f_c$ with $2c=a_1+a_2$. There are two solutions for $c\in \mathbb{R}/\mathbb{Z}$ and therefore there is a symmetric embedding. The standard equation is $$a_1\lambda ^2\mu ^2+a_2(\lambda ^2 + \mu ^2)+2a_3\lambda \mu =0,\mbox{ with }
\lambda =\frac{x_1}{x_0},\ \mu =\frac{y_1}{y_0} .$$
Finally, there is the possibility that the two lines form a conjugate pair over $\mathbb{R}$. \[We do not work out the details here.\]\
\
(b). The nonsingular locus $E^*$ of $E$ is isomorphic to the additive group $\mathbb{G}_a$. The automorphism of order two are $f_a: x\mapsto -x+a$ (any $a\in \mathbb{G}_a$). The equation $Cf_{a_1}=f_{a_2}C$ (with $a_1\neq a_2$) has a unique solution $C=f_c$ with $2c=a_1+a_2$. Thus there exists a symmetric embedding $E\subset \mathbb{P}^1\times \mathbb{P}^1$ and this embedding is unique. The above is valid for any field of characteristic $\neq 2$, because the group $\mathbb{G}_a$ has no forms. The standard equation is $$\lambda ^2\mu ^2+(\lambda -\mu )^2-2\lambda \mu (\lambda +\mu )=0.$$
\(c) For a reducible or nonreduced $E$, Rohn obtains the following standard equations $$(\lambda +\mu )^2+2a\lambda \mu =0,\ \lambda ^2\mu ^2\pm (\lambda -\mu )^2=0,
\ (\lambda -\mu )^2=0.$$
[ccc]{} Barth, W. Peters, C., van der Ven, A., *Compact complex surfaces*, Springer-Verlag, Berlin and New York, 1984. Bottema, O., A Classification of Rational Quartic Ruled Surfaces, *Geometriae Dedicata* [**1**]{}, no. 3, (1973), 349-355. Cayley, A., A Third Memoir on Skew Surfaces, Otherwise Scrolls, *Philosophical Transactions of the Royal Society of London* [**159**]{} (1869), 111-126. Chasles, M., Sur les six droites qui peuvent être les directions de six forces en équilibre. Propriétés de l’hyperboloïde à une nappe et d’une certaine surface du quatrième ordre. *Comptes Rendus des Séances de l’Académie des Sciences. Paris.* [ bf 52]{} (1861), 1094-1104. Cremona, L., Sulle Superficie Gobbe di Quarto Grado. *Memorie dell’ Accademia delle Science dell’ Istituto di Bologna*, serie II, tomo VIII (1868), 235-250. *Opere*, II, 420. Dolgachev, I. V., Topics in Classical Algebraic Geometry, *www.math.lsa.umich.edu/idolga/topics1.pdf* Edge, W. L., *The theory of ruled surfaces*, Cambridge, 1931. Hartshorne, R., *Algebraic Geometry*, Springer-Verlag, Berlin etc., 1977. Meyer, W. Fr., Flächen vierter und höherer Ordnung. *Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen*, IIIC, 10b (1930), 1744-1759. Mohrmann, H., Die Flächen vierter Ordnung mit gewundener Doppelkurve, *Mathematische Annalen* [**89**]{} (1923), 1-31. Pascal, E., *Repertorium der Höheren Mathematik (Definitionen, Formeln, Theoreme, Literatur). [II.]{} Theil: die Geometrie*. Teubner, 1902. Rohn, K., Ueber die Flächen vierter Ordnung mit dreifachem Punkte, *Mathematische Annalen* [**24**]{}, No. 1 (1884), 55-151. Rohn, K., Die verschiedenen Arten der Regelflächen vierter Ordnung, (1886). *Mathematische Abhandl ungen aus dem Verlage Mathematischer Modelle von Martin Schilling*. Halle a. S., 1904. *Mathematische Annalen* [**28**]{}, No. 2 (1886), 284-308. Salmon, G., *A Treatise on the Analytic Geometry of Three Dimensions*, Dublin, 1882. Schilling, M., *Catalog mathematischer Modelle für den Höheren mathematischen Unterricht*, Leipzig, 1911. Segre, C., Etude des différentes surfaces du 4e ordre à conique double ou cuspidale (générale ou décomposée) considérées comme des projections de l’intersection de deux variétés quadratiques del’espace à quatre dimensions. *Mathematische Annalen* [**24**]{}, 3 (1884), 313-444. Sturm, R., *Die Gebilde ersten und zweiten Grades der Liniengeometrie in synthetischer Behandlung*, I, B.G. Teubner (1892). Swinnerton-Dyer, H. P. F., An Enumeration of All Varieties of Degree $4$. *American Journal of Mathematics* [**95**]{}, No. 2 (1973), 403-418. Urabe, T., Dynkin Graphs and Combinations of Singularities on Quartic Surfaces, *Proc. Japan. Acad.*, Ser. A [**61**]{} (1985), 266-269. Urabe, T., Classification of Non-normal Quartic Surfaces. *Tokyo Journal of Mathematics* [**9**]{}, No. 2 (1986), 265-295. Urabe, T., Elementary transformations of Dynkin graphs and singularities on quartic surfaces, *Inventiones mathematicae* [**87**]{} (1987) 549-572. Urabe, T., The transformations of Dynkin graphs and singularities on quartic surfaces. *Inventiones mathematicae* [**100**]{} (1990) 207-230. Wong, B. C., A study and Classifications of Ruled Quartic Surfaces by Means of a Point-to-Line Transformation. *University of California Publications of Mathematics* [**1**]{}, No. 17 (1923), 371-387.
|
---
author:
- 'Klemens Esterle$^{1}$, Luis Gressenbuch$^{1}$ and Alois Knoll$^{2}$ [^1] [^2]'
title: '**Formalizing Traffic Rules for Machine Interpretability**'
---
@topnum0 @botnum0
[^1]: $^{1}$fortiss GmbH, Research Institute of the Free State of Bavaria, Munich, Germany
[^2]: $^{2}$Alois Knoll is with Robotics, Artificial Intelligence and Real-time Systems, Technische Universität München, Munich, Germany
|
---
abstract: |
A fundamental assumption in the theory of brane world is that all matter and radiation are confined on the four-dimensional brane and only gravitons can propagate in the five-dimensional bulk spacetime. The brane world theory did not provide an explanation for the existence of electromagnetic fields and the origin of the electromagnetic field equation. In this paper, we propose a model for explaining the existence of electromagnetic fields on a brane and deriving the electromagnetic field equation. Similar to the case in Kaluza-Klein theory, we find that electromagnetic fields and the electromagnetic field equation can be derived from the five-dimensional Einstein field equation. However, the derived electromagnetic field equation differs from the Maxwell equation by containing a term with the electromagnetic potential vector coupled to the spacetime curvature tensor. So it can be considered as generalization of the Maxwell equation in a curved spacetime. The gravitational field equation on the brane is also derived with the stress-energy tensor for electromagnetic fields explicitly included and the Weyl tensor term explicitly expressed with matter fields and their derivatives in the direction of the extra-dimension. The model proposed in the paper can be regarded as unification of electromagnetic and gravitational interactions in the framework of brane world theory.
Classical general relativity, higher-dimensional gravity, brane world theory
author:
- 'Li-Xin Li'
title: Electromagnetic Force on a Brane
---
Introduction
============
The theory of brane world was proposed to address the hierarchy problem in theoretical physics [@ran99a; @ran99b]. In the theory of brane world, the four-dimensional spacetime in which we live is assumed to be a hypersurface (or, a brane) embedded in a five-dimensional bulk spacetime. The gravitational field equation on the four-dimensional brane is derived from the Einstein field equation in the five-dimensional bulk space by the approach of projection [@shi00]. Standard model particles, including electromagnetic fields, strong and weak particles, are assumed to be confined on the four-dimensional brane. The assumption is motivated by D-branes in string theory, on which open strings representing the non-gravitational sector can end [@hor96; @luk99; @luk99b; @luk00]. For a recent and comprehensive review on the theory of brane world and its application in physics and cosmology, please refer to [@maa10].
Kaluza-Klein (KK) theory also attempts to interpret the physics in a four-dimensional spacetime as arising from the gravity in a five-dimensional bulk spacetime [@kal21; @kle26a; @kle26b; @bai87; @ove97]. In the KK theory, both the Maxwell equation and the four-dimensional Einstein field equation are derived from the five-dimensional Einstein field equation. In the KK theory the extra-dimension is assumed to be compactified to a very small scale so that the extra-dimension cannot be seen in normal physical experiments and hence gravity appears four-dimensional. However, in the brane theory the extra-dimension can be noncompact. It is the curvature of the bulk space that keeps gravity to be four-dimensional on scales larger than the curvature radius of the bulk space [@ran99b].
Although both attempt to interpret the four-dimensional physics as arising from the five-dimensional physics, KK and brane world theories are distinctly different in physics: they are defined on two different hypersurfaces in a five-dimensional spacetime and are not related by diffeomorphisms, as explained in detail in [@li15a]. As can be seen from the above description, electromagnetism has different origins in the two theories. In the KK theory, the Maxwell equation is derived from the five-dimensional Einstein field equation, hence electromagnetism and gravity have the same origin. In the brane world theory, electromagnetism is assumed to arise from open strings ending on D-branes hence has no relation to gravity arising from closed strings. In addition, as mentioned above, in the KK theory the extra spatial dimension must be compactified, but in the brane theory the extra spatial dimension can be noncompact.
It would be interesting to adapt the idea of deriving the Maxwell equation from the five-dimensional Einstein field equation in the KK theory to the brane world theory. However, this is not an easy task, since the 4+1 decomposition of the five-dimensional metric tensor adopted in the KK theory is different from that used for obtaining a metric tensor on a brane hypersurface, as explained in [@li15a]. In this paper, we propose a brane world model in which electromagnetic fields and the electromagnetic field equation on the brane are derived from the five-dimensional Einstein field equation. We will see that, this can be realized only if an appropriate boundary condition on the brane is adopted, and the derived electromagnetic field equation differs from the Maxwell equation by a curvature-coupled term. The boundary condition differs from the $Z_2$-symmetry boundary condition used in the standard brane world theory, as will be explained in the paper.
4+1 Decomposition of the Five-dimensional Einstein Field Equation
=================================================================
Assuming a five-dimensional spacetime in which gravity is described by the Einstein field equation $$\begin{aligned}
\t{R}_{ab}-\frac{1}{2}\t{R}\t{g}_{ab}=\t{\kappa}\t{T}_{ab} \;, \label{ein_eq5}\end{aligned}$$ where $\t{g}_{ab}$ is the metric tensor of the spacetime, $\t{R}_{ab}$ is the Ricci curvature tensor, $\t{R}\equiv \t{g}^{ab}\t{R}_{ab}$ is the Ricci scalar, $\t{T}_{ab}$ is the stress-energy tensor of matter, and $\t{\kappa}$ is gravitational coupling constant. The five-dimensional spacetime is sliced by a set of timelike hypersurfaces, and one of the hypersurfaces is just the four-dimensional spacetime in which we live. The hypersurface has a unit normal $n^a$, a metric tensor $g_{ab}\equiv \t{g}_{ab}-n_an_b$, and an extrinsic curvature tensor $K_{ab}\equiv\t{\pounds}_ng_{ab}$ where $\t{\pounds}_n$ denotes the Lie derivative with respect to $n^a$.
It is well known that the Einstein field equation (\[ein\_eq5\]) is equivalent to the following three equations expressed in terms of geometric quantities on the hypersurface [@mis73; @wal84; @shi00; @li15a]: $$\begin{aligned}
R +K_{ab}K^{ab}-K^2 = -2\t{\kappa}\t{T}_{ab}n^an^b \;, \label{scalar_eq}\end{aligned}$$ $$\begin{aligned}
\nabla_aK^{ab}-\nabla^bK = \t{\kappa}g^{ab}\t{T}_{ac}n^c \;, \label{vector_eq}\end{aligned}$$ and $$\begin{aligned}
G_{ab} = \t{\kappa}g_a^{\;\;c}g_b^{\;\;d}\t{T}_{cd} +g_{ab}{}{}^{cd}\t{\pounds}_nK_{cd} -\left(2K_a^{\;\;c}K_{cb}-KK_{ab}\right)+\frac{1}{2}\left(3K_{cd}K^{cd}-K^2\right)g_{ab} \;. \label{tensor_eq}\end{aligned}$$ Here, $g_{abcd}\equiv g_{ac}g_{bd}-g_{ab}g_{cd}$, $\nabla_a$ is the derivative operator associated with the metric $g_{ab}$, $R_{ab}$ is the Ricci curvature tensor of the brane hypersurface, $R\equiv R_a{}^a$ is the Ricci scalar, $G_{ab}\equiv R_{ab}-(1/2)Rg_{ab}$ is the Einstein tensor, and $K\equiv K_a{}^a$.
The scalar equation (\[scalar\_eq\]) is obtained by contraction of equation (\[ein\_eq5\]) with $n^an^b$. The vector equation (\[vector\_eq\]) is obtained by contraction of equation (\[ein\_eq5\]) with $n^b$, then projection onto the hypersurface by the projection operator $g_{ab}$. It is also called the Gauss-Codacci relation [@mis73; @wal84]. The tensor equation (\[tensor\_eq\]) is obtained by full projection of the equation (\[ein\_eq5\]) onto the hypersurface. We use normal letters to denote quantities on the brane hypersurface, and tilded letters to denote quantities defined in the bulk spacetime (except the normal vector $n^a$). The index of a tensor in the bulk spacetime is lowered (raised) by $\t{g}_{ab}$ ($\t{g}^{ab}$). The index of a tensor on the brane can be lowered (raised) by both $g_{ab}$ ($g^{ab}$) and $\t{g}_{ab}$ ($\t{g}^{ab}$), with the same result.
In the brane world theory with the $Z_2$-symmetry boundary condition, the tensor field equation (\[tensor\_eq\]) is interpreted as the Einstein field equation on the bane [@shi00]. Equation (\[tensor\_eq\]) agrees with the eq. 8 of [@shi00], if the traceless tensor $E_{ab}$ in the eq. 8 in [@shi00], which is defined by the Weyl tensor in the bulk space, is expressed in terms of the brane extrinsic curvature $K_{ab}$ and its derivative in the direction orthogonal to the brane. Note that, in derivation of equation (\[tensor\_eq\]), following [@shi00] we have assumed that $n^a$ is tangent to a geodesic so that the acceleration vector $a^b\equiv n^a\t{\nabla}_an^b=0$.
Electromagnetic Field Equations on a Brane
==========================================
Let us consider a discontinuous hypersurface (a brane) in a five-dimensional spacetime, which contains a surface stress-energy tensor as assumed in the theory of brane world. To have a well-defined four-dimensional geometry on the brane, the induced metric $g_{ab}$ must be continuous across it, hence the derivative operator $\nabla_a$ and the Riemann curvature defined by it. Then, by definition, the extrinsic curvature $K_{ab}$ does not contain a Dirac $\delta$-function, although it can be discontinuous across the brane. In fact, by the five-dimensional Einstein field equation (\[ein\_eq5\]), $K_{ab}$ must be discontinuous across the brane.
The bulk stress-energy tensor $\t{T}_{ab}$ must contain a $\delta$-function at the position of the brane. So, we can write $$\begin{aligned}
\t{T}_{ab}=-\frac{\t{\Lambda}}{\t{\kappa}}\t{g}_{ab}+\t{\cal T}_{ab}+\t{S}_{ab}\delta(n) \;, \label{tT_tSab}\end{aligned}$$ where $\t{\Lambda}$ is the cosmological constant in the bulk space, $\t{\cal T}_{ab}$ and $\t{S}_{ab}$ are regular tensors (i.e., they contain no $\delta$-function). Here we have written $n^a=(\partial/\partial n)^a$ and use $n=0$ to denote the position of the brane.
Let $[Q]$ denote the difference in the value of any quantity $Q$ on the two sides of the brane, i.e., $[Q]\equiv Q^+-Q^-$, $Q^+=Q(n=0^+)$, and $Q^-=Q(n=0^-)$. Integration of equation (\[tensor\_eq\]) across the brane hypersurface leads to the Israel junction condition [@isr66; @mis73] $$\begin{aligned}
[K_{ab}]=-\t{\kappa}\left(S_{ab}-\frac{1}{3}Sg_{ab}\right) \;, \label{join1}\end{aligned}$$ where $S_{ab}\equiv g_a^{\;\;c}g_b^{\;\;d}\t{S}_{cd}$ and $S=S_a{}^a$. Similarly, integration of equations (\[scalar\_eq\]) and (\[vector\_eq\]) across the brane leads to $$\begin{aligned}
n^cn^d\t{S}_{cd}=0 \hspace{0.5cm} \mbox{and} \hspace{1cm} g_b^{\;\;d}n^c\t{S}_{cd}=0 \;,\end{aligned}$$ which simply tells that the momentum flow represented by $\t{S}_{ab}$ is entirely in the brane [@mis73].
To introduce electromagnetic fields on the brane, in the neighborhood of the brane hypersurface we choose a general coordinate system $\{x^0,x^1,x^2,x^3,x^4\equiv w\}$ so that the brane is located at $w=0$. The coordinate vector $w^a=(\partial/\partial w)^a$ can be decomposed as $w^a=Nn^a+N^a$, where $N$ is the lapse function, and $N^a=g^a{}_bw^b$ is the shift vector [@wal84]. It can be verified that the acceleration vector $a^a=-\nabla^a\ln N$. Hence, the geodesic condition $a^a=0$ is identical to the condition $\nabla_aN=0$, i.e., $N$ can only be a function of $w$. Define $$\begin{aligned}
w^\prime=2\int Ndw \;, \hspace{0.5cm}
A^a=(2N)^{-1}N^a \;, \hspace{0.5cm}
H_{ab}=\nabla_aA_b+\nabla_bA_a=H_{ba} \;,\end{aligned}$$ the extrinsic curvature $K_{ab}$ can be expressed as $$\begin{aligned}
K_{ab} = \dot{g}_{ab}-H_{ab} \;, \label{Kab_H}\end{aligned}$$ where $\dot{g}_{ab}\equiv\partial g_{ab}/\partial w^\prime\equiv g_a^{\;\;c}g_b^{\;\;d}\t{\pounds}_{w^\prime}g_{cd}$.
Substituting equation (\[Kab\_H\]) into the vector equation (\[vector\_eq\]), we get $$\begin{aligned}
\nabla_aF^{ab}+2R^b{}_aA^a=-4\pi J^b \;, \label{meq1}\end{aligned}$$ where $F_{ab}\equiv\nabla_aA_b-\nabla_bA_a$, $$\begin{aligned}
J^a\equiv\frac{1}{4\pi}\left(\nabla_b\Phi^{ab}+\t{\kappa}g^{ab}\t{T}_{bc}n^c\right) \;, \label{Ja}\end{aligned}$$ and $$\begin{aligned}
\Phi_{ab}\equiv -g_{ab}{}{}^{cd}\dot{g}_{cd}=\Phi_{ba} \;. \label{Phi_ab}\end{aligned}$$ Equation (\[meq1\]) differs from the Maxwell equation only by a curvature-coupled term $2R^b{}_aA^a$, if $A^a$ is interpreted as the electromagnetic potential vector, and $J^a$ interpreted as the electric current density vector. When $R_{ab}=0$, it is exactly the Maxwell equation. Therefore, we can interpret equation (\[meq1\]) as generalization of the Maxwell equation in the brane world theory.
Einstein Field Equations on a Brane
===================================
Substituting equation (\[Kab\_H\]) in to the tensor equation (\[tensor\_eq\]), we get a four-dimensional Einstein field equation on the brane: $$\begin{aligned}
G_{ab}=\kappa T_{ab} \;, \label{ein_eq40}\end{aligned}$$ with the stress-energy tensor $$\begin{aligned}
T_{ab} = T_{\emm,ab}+T_{m,ab}+T_{\intt,ab}+\frac{\t{\kappa}}{\kappa}g_a{}^cg_b{}^d\t{T}_{cd} \;, \label{Tab}\end{aligned}$$ where $$\begin{aligned}
\kappa T_{\emm,ab} = 2\Psi_{ac}\Psi_b^{\;\;c}-\frac{2}{3}\Psi\Psi_{ab} -\frac{1}{2}\left(\Psi_{cd}\Psi^{cd}-\frac{1}{3}\Psi^2\right)g_{ab} -2\nabla^c\left(2A_{(a}\Psi_{b)c}-A_c\Psi_{ab}\right) \;, \label{Tab_em_M}\end{aligned}$$ $$\begin{aligned}
\kappa T_{m,ab} = -2\Phi_{ac}\Phi_b^{\;\;c} +\frac{1}{3}\Phi\Phi_{ab}-\frac{1}{2}\left(\Phi_{cd}\Phi^{cd}-\frac{1}{3}\Phi^2\right)g_{ab} -2\dot{\Phi}_{ab} \;, \label{Tab_m_M}\end{aligned}$$ and $$\begin{aligned}
\kappa T_{\intt,ab} = \frac{1}{3}\Phi\Psi_{ab}-\frac{2}{3}\Psi\Phi_{ab}-\left(\Phi_{cd}\Psi^{cd}-\frac{1}{3}\Phi\Psi\right)g_{ab} -2\nabla^c\left(2A_{(a}\Phi_{b)c}-A_c\Phi_{ab}\right)-2\dot{\Psi}_{ab} \;. \label{Tab_int_M}\end{aligned}$$ Here, $\kappa$ is the gravitational coupling constant in the four-dimensional spacetime, $$\begin{aligned}
\Psi_{ab}\equiv H_{ab}-Hg_{ab}=\Psi_{ba} \;, \label{Psi_ab}\end{aligned}$$ $H\equiv H_c{}^c$, $\Psi\equiv \Psi_c{}^c$, and $\Phi\equiv \Phi_c{}^c$. The braces in tensor indexes denote symmetrization of indexes.
According to [@li15a], $T_{\emm,ab}$ is interpreted as the stress-energy tensor of electromagnetic fields, $T_{m,ab}$ interpreted as the stress-energy tensor of the matter field associated with $\dot{g}_{ab}$, and $T_{\intt,ab}$ interpreted as the stress-energy tensor arising from the interaction between the electromagnetic field and the matter field. The $T_{\emm,ab}$ is related to the ordinary electromagnetic stress-energy tensor $$\begin{aligned}
\!\!~^{(0)}T_{\emm,ab}=\frac{2}{\kappa}\left(F_{ac}F_b{}^c-\frac{1}{4}g_{ab}F_{cd}F^{cd}\right) \label{T_em_0}\end{aligned}$$ by $$\begin{aligned}
T_{\emm,ab} =\, ^{(0)}\!T_{\emm,ab} + ^{(1)}\!T_{\emm,ab} \;,\end{aligned}$$ where $$\begin{aligned}
^{(1)}T_{\emm,ab} = -\frac{2}{\kappa}\left\{\nabla^c\nabla_c(A_aA_b)-2\nabla^c\nabla_{(a}(A_{b)}A_c)+4A^cR_{c(a}A_{b)}+g_{ab}\left[\nabla_c\nabla_d(A^cA^d)-R_{cd}A^cA^d\right]\right\}, \hspace{0.3cm} \label{T_em_1}\end{aligned}$$ which arises from the curvature-coupled term in the field equation (\[meq1\]).
Unlike $\!\!~^{(0)}T_{\emm,ab}$, the $\!\!~^{(1)}T_{\emm,ab}$ does not interact with electric charge and current, since $$\begin{aligned}
\nabla^a\!\!~^{(1)}T_{\emm,ab}=\frac{1}{2\pi}\left[F_b{}^cR_{cd}A^d-A_b\nabla^c\left(R_{cd}A^d\right)\right] . \hspace{0.2cm}\end{aligned}$$ In contrast, for the $\!\!~^{(0)}T_{\emm,ab}$, we have $$\begin{aligned}
\nabla^a\!\!~^{(0)}T_{\emm,ab}=-F_{ba}J^a-\frac{1}{2\pi}F_b{}^cR_{cd}A^d \;.\end{aligned}$$ When the electric charge is conserved, by equation (\[meq1\]) we have $\nabla^c\left(R_{cd}A^d\right)=-2\pi\nabla_aJ^a=0$. Then, for the total $T_{\emm,ab}$ we have just the Lorentz force law: $$\begin{aligned}
\nabla^aT_{\emm,ab}=-F_{ba}J^a \;.\end{aligned}$$ Note that, although the effect of $\!\!~^{(1)}T_{\emm,ab}$ cannot be measured by electromagnetic experiments, $\!\!~^{(1)}T_{\emm,ab}$ affects the spacetime geometry according to the Einstein field equation. Hence, $\!\!~^{(1)}T_{\emm,ab}$ represents a kind of dark electromagnetic energy and momentum [@li15b].
Boundary Conditions
===================
To have a well-defined electromagnetic field on the brane, $A^a$ and $\nabla_aA_b$ must be continuous across the brane. That is, we must have $[A^a]=0$, $[F_{ab}]=[H_{ab}]=[\Psi_{ab}]=0$. Then, by equation (\[Kab\_H\]), we have $[K_{ab}]=[\dot{g}_{ab}]$, and $[\Phi_{ab}]=-g_{ab}{}{}^{cd}[\dot{g}_{cd}]=-g_{ab}{}{}^{cd}[K_{cd}]$. By equation (\[join1\]), we then get $$\begin{aligned}
[\Phi_{ab}]=\t{\kappa}S_{ab} \;. \label{joinx}\end{aligned}$$ In the theory of brane world, it is usually assumed that $K_{ab}^+=-K_{ab}^-$, i.e., $K_{ab}$ is antisymmetric about the brane [@shi00; @maa10]. This $Z_2$-symmetry does not apply when the electromagnetic field is present, by equation (\[Kab\_H\]) and the condition $[H_{ab}]=0$. However, since $[K_{ab}]=[\dot{g}_{ab}]$, we can assume that $$\begin{aligned}
\dot{g}_{ab}^+=-\dot{g}_{ab}^- \;,\end{aligned}$$ i.e., $\dot{g}_{ab}$ is antisymmetric about the brane. This condition is equivalent to $\Phi_{ab}^+=-\Phi_{ab}^-$. With this assumption, from equation (\[joinx\]) we get the boundary condition $$\begin{aligned}
\Phi_{ab}^+=-\Phi_{ab}^-=\frac{1}{2}[\Phi_{ab}]=\frac{\t{\kappa}}{2}S_{ab} \;. \label{join2}\end{aligned}$$
By the symmetry properties of $\Psi_{ab}$ and $\Phi_{ab}$, we have $\dot{\Psi}_{ab}^+=-\dot{\Psi}_{ab}^-$, and $\dot{\Phi}_{ab}^+=\dot{\Phi}_{ab}^-$. Then, by equations (\[Tab\_em\_M\])–(\[Tab\_int\_M\]), we have $[T_{\emm,ab}]=[T_{m,ab}]=0$, and $$\begin{aligned}
\kappa T_{\intt,ab}^+ = -\kappa T_{\intt,ab}^-=-2\dot{\Psi}_{ab}^+-\t{\kappa}\nabla^c\left(2A_{(a}S_{b)c}-A_cS_{ab}\right) +\t{\kappa}\left[\frac{1}{6}S\Psi_{ab}-\frac{1}{3}\Psi S_{ab}-\frac{1}{2}\left(S_{cd}\Psi^{cd}-\frac{1}{3}S\Psi\right)g_{ab}\right] \;, \label{T_int_join}\end{aligned}$$ where we have omitted the indexes “$+$” and “$-$” for $\Psi_{ab}$ and $A_a$ since $\Psi_{ab}^+=\Psi_{ab}^-$ and $A_a^+=A_a^-$. To have a well-defined Einstein field equation on the brane, each term on the right-hand side of equation (\[Tab\]) must be symmetric about the brane, since $G_{ab}^+=G_{ab}^-$. So, we must have $\left[g_a{}^cg_b{}^d\t{T}_{cd}\right]=0$ and $\left[T_{\intt,ab}\right]=0$. Then, by equation (\[T\_int\_join\]), we must have $T_{\intt,ab}^+ = T_{\intt,ab}^-=0$, i.e., $$\begin{aligned}
\dot{\Psi}_{ab}^+=\t{\kappa}\left[-\frac{1}{2}\nabla^c\left(2A_{(a}S_{b)c}-A_cS_{ab}\right) +\frac{1}{12}S\Psi_{ab}-\frac{1}{6}\Psi S_{ab}-\frac{1}{4}\left(S_{cd}\Psi^{cd}-\frac{1}{3}S\Psi\right)g_{ab}\right]\;. \label{T_int_join2}\end{aligned}$$
Let us consider a simple case: $S_{ab}=-\lambda g_{ab}$, where $\lambda$ is constant on the brane but can be a function of $w^\prime$, and $\lambda(-w^\prime)=\lambda(w^\prime)$. That is, the brane has a positive tension (represented by $\lambda$) and is vacuum otherwise. By equation (\[T\_int\_join\]) and the above discussions, we get $\kappa T_{\intt,ab}^+ =-2\dot{\Psi}_{ab}^++(\t{\kappa}/3)\lambda\Psi_{ab}=0$, so we must have $\dot{\Psi}^+_{ab}=-\dot{\Psi}^-_{ab}=(\t{\kappa}/6)\lambda\Psi_{ab}$. Substituting $S_{ab}=-\lambda g_{ab}$ into equation (\[Tab\_m\_M\]), we get $\kappa T_{m,ab}=-2\dot\Phi_{ab}$. If we assume that the relation in equation (\[join2\]) holds in a small neighborhood of the brane, we get $2\dot{\Phi}_{ab}=-\t{\kappa}\left(\dot{\lambda}g_{ab}+\lambda\dot{g}_{ab}\right)=-\t{\kappa}\left(\dot{\lambda}-\t{\kappa}\lambda^2/6\right)g_{ab}$, and $$\begin{aligned}
\kappa T_{m,ab}=\t{\kappa}\left(\dot{\lambda}-\frac{1}{6}\t{\kappa}\lambda^2\right)g_{ab} \;,\end{aligned}$$ corresponding to a cosmological constant term. Then, we get the four-dimensional gravitational field equation on the brane $$\begin{aligned}
G_{ab}+\Lambda_\eff\, g_{ab}=\kappa T_{\emm,ab} \;, \label{ein_eq4a}\end{aligned}$$ where we have assumed $\t{\cal T}_{ab}=0$ (eq. \[tT\_tSab\]) and $$\begin{aligned}
\Lambda_\eff\equiv\t{\Lambda}-\t{\kappa}\left(\dot{\lambda}-\frac{\t{\kappa}}{6}\lambda^2\right) \;. \label{lam_eff}\end{aligned}$$ It is just the four-dimensional Einstein field equation with a cosmological constant and the stress-energy of electromagnetic fields as the source.
To cancel the five-dimensional cosmological constant $\t{\Lambda}$ and have $\Lambda_\eff=0$, we must have $\t{\kappa}\left(\dot{\lambda}-\t{\kappa}\lambda^2/6\right)=\t{\Lambda}$. This result agrees with that discussed in [@shi00] when $\dot{\lambda}=0$. If on the brane $\dot{\lambda}\neq 0$ but $\t{\kappa}^2\lambda^2=-6\t{\Lambda}$, we get a residual cosmological constant $\Lambda_\eff=-\t{\kappa}\dot{\lambda}$.
To find out the relation between the coupling constants $\kappa$ and $\t{\kappa}$, let us include a stress-energy tensor of normal matter in $S_{ab}$ and denote it by $\tau_{ab}$: $S_{ab}=-\lambda g_{ab}+\tau_{ab}$. Then we get $$\begin{aligned}
\kappa T_{m,ab}=\t{\kappa}\left(\dot{\lambda}-\frac{\t{\kappa}}{6}\lambda^2\right)g_{ab}+\frac{\t{\kappa}^2}{6}\lambda\tau_{ab}-\t{\kappa}\dot{\tau}_{ab}+\t{\kappa}^2\pi_{ab} ,\hspace{0cm} \nonumber\\ \label{T_m_ab}\end{aligned}$$ where $$\begin{aligned}
\pi_{ab}=-\frac{1}{2}\tau_{ac}\tau_b{}^c+\frac{1}{12}\tau\tau_{ab}-\frac{1}{8}\left(\tau_{cd}\tau^{cd}-\frac{1}{3}\tau^2\right)g_{ab} \hspace{0.4cm}\label{pi_ab}\end{aligned}$$ contains quadratic terms of $\tau_{ab}$. The requirement that the linear term of $\tau_{ab}$ in (\[T\_m\_ab\]) is identical to that contained in the standard four-dimensional Einstein field equation leads to $$\begin{aligned}
\kappa=\frac{1}{6}\t{\kappa}^2\lambda \;,\end{aligned}$$ in agreement with the result in [@shi00]. Hence, when $\Lambda_\eff=0$, we get the general four-dimensional gravitational field equation on the brane $$\begin{aligned}
G_{ab}=\kappa \left(T_{\emm,ab}+\tau_{ab}\right)+\t{\kappa}\left({\cal T}_{ab}-\dot{\tau}_{ab}\right)+\t{\kappa}^2\pi_{ab} \;, \hspace{0.2cm}\label{ein_eq4}\end{aligned}$$ where ${\cal T}_{ab}=g_a{}^cg_b{}^d\t{\cal T}_{cd}$ is the projection of the bulk stress-energy tensor.
The difference between the $\pi_{ab}$ in equation (\[pi\_ab\]) and the $\pi_{\mu\nu}$ in [@shi00] is caused by the fact that in our treatment the tensor $E_{ab}$ in [@shi00] has been expressed in terms of $K_{ab}$ and its derivative. In fact, the $\dot{\tau}_{ab}$ term in equation (\[ein\_eq4\]) arises from the expression for $E_{ab}$. (Details for the relation between $E_{ab}$ and $K_{ab}$ can be found in [@li15a].)
In equation (\[ein\_eq4\]), the term linear in $\tau_{ab}$ is the stress-energy tensor of normal matter (other than electromagnetic fields) on the brane, which has the same form as in the standard Einstein field equation. It guarantees that the Newtonian gravitational law can be obtained in linear perturbations. The $\pi_{ab}$, which contains quadratic terms of $\tau_{ab}$, is important only in high energy states [@shi00; @maa10]. The $T_{\emm,ab}$, defined by equation (\[Tab\_em\_M\]), represents the stress-energy tensor of electromagnetic fields, which differs from the standard electromagnetic stress-energy tensor by a term $\!\!~^{(1)} T_{\emm,ab}$ (eq. \[T\_em\_1\]). The $\dot{\tau}_{ab}$ is the gradient of $\tau_{ab}$ with respect to the extra dimension $w$, which should be small since its effect has never been detected in normal experiments of gravity. The ${\cal T}_{ab}$ arises from the stress-energy tensor of matter in the bulk space, whose effect on the brane may look like some kind of dark matter.
Relation to Other Work in the Literature
========================================
To understand the boundary condition better, we express $K_{ab}$ in terms of $\Phi_{ab}$ and $\Psi_{ab}$ $$\begin{aligned}
K_{ab}=-\hat{\Phi}_{ab}-\hat{\Psi}_{ab} \;, \label{Kab_Phi_Psi}\end{aligned}$$ where $$\begin{aligned}
\hat{\Phi}_{ab}\equiv\Phi_{ab}-\frac{1}{3}\Phi g_{ab} \;, \hspace{1cm}
\hat{\Psi}_{ab}\equiv\Psi_{ab}-\frac{1}{3}\Psi g_{ab} \;. \label{hat_Phi_Psi}\end{aligned}$$ In our model, we choose the boundary condition so that $\hat{\Psi}_{ab}$ is symmetric about the brane, but $\hat{\Phi}_{ab}$ is antisymmetric about the brane. The $\hat{\Psi}_{ab}$ is interpreted as representing electromagnetic fields on the brane. The $\hat{\Phi}_{ab}$ is related to the stress-energy tensor of matter on the brane through the Israel junction relation (eq. \[join2\]).
The boundary condition adopted in this paper is essentially a non-$Z_2$ symmetric boundary condition, which has been studied in the framework of an asymmetric brane world in the literature ([@bat01; @yam07], and references therein). In [@bat01] and [@yam07], the extrinsic curvature tensor $K_{ab}$ of the brane is separated into two parts, an antisymmetric part and a symmetric part about the brane. As usual, the antisymmetric part is related to the stress-energy tensor of matter confined in the brane by the Israel junction relation. The symmetric part is solved from a constraint equation derived from the requirement that $\left[R_{ab}\right]=0$, with a formal solution determined by the stress-energy tensor of matter confined in the brane and the antisymmetric part about the brane of the geometric quantity in the bulk space (eqs. 31 and 32 in [@bat01], eq. 3.17 in [@yam07]). We call it “a formal solution” because the $\left[{\cal F}_{ab}\right]$ in [@bat01; @yam07] contains the Weyl term $\left[{\cal E}_{ab}\right]$, which itself is a function of the $\langle K_{ab}\rangle$ and its derivative in the direction of the extra-dimension according to equation (B35) of [@li15a]. Electromagnetic fields are not discussed in [@bat01; @yam07].
Mathematically, the result in this paper agrees with that in [@bat01] and [@yam07]. The four-dimensional Einstein field equation (\[ein\_eq4\]) mathematically agrees with the four-dimensional Einstein field equation derived in [@bat01; @yam07]. The electromagnetic field equation (\[meq1\]) is mathematically equivalent to the Gauss-Codacci relation. This is not surprising, since in both models (the model in this paper and the model in [@bat01; @yam07]) the effective field equations on the brane hypersurface are derived from the five-dimensional Einstein field equation by projection, and the boundary conditions on the brane are mathematically equivalent. In particular, the electromagnetic field equation (\[meq1\]) is derived from the Gauss-Codacci relation.
However, the physics represented by our model is different from that represented by the model in [@bat01; @yam07]. In our model, aside from the part resulted from projection of the bulk stress-energy tensor, the stress-energy tensor in the Einstein field equation on the brane contains two parts. One part, represented by $\tau_{ab}$, $\dot{\tau}_{ab}$, and $\pi_{ab}$ in equation (\[ein\_eq4\]), is interpreted as the stress-energy tensor of normal matter confined in the brane and related to the antisymmetric part of $K_{ab}$ (i.e., $-\hat{\Phi}_{ab}$) by the Israel junction relation (eqs. \[Tab\_m\_M\] and \[T\_m\_ab\]). The other part, represented by $T_{\emm,ab}$ in equation (\[ein\_eq4\]), is interpreted as the stress-energy tensor of electromagnetic fields and related to the symmetric part of $K_{ab}$ (i.e., $-\Psi_{ab}$; see eq. \[Tab\_em\_M\]). The $T_{\emm,ab}$ differs from the standard electromagnetic stress-energy tensor $^{(0)}T_{\emm,ab}$ by a $^{(1)}T_{\emm,ab}$, see equations (\[T\_em\_0\])–(\[T\_em\_1\]). As explained in [@li15a; @li15b], the $^{(1)}T_{\emm,ab}$ arises from the curvature-coupled term in the electromagnetic field equation (\[meq1\]).
Our interpretation of $T_{\emm,ab}$ as the stress-energy tensor of electromagnetic fields is motivated by the fact that the vector field equation, i.e., the Gauss-Codacci relation (\[vector\_eq\]), can be expressed in a form very similar to the Maxwell equation (eq. \[meq1\]). By identities $$\begin{aligned}
\nabla_aK^{ab}-\nabla^bK=-\nabla_a\Psi^{ab}-\nabla_a\Phi^{ab}\end{aligned}$$ and $$\begin{aligned}
\nabla_a\Psi^{ab}=\nabla_aF^{ab}+2R^b{}_aA^a \;, \label{Psi_F}\end{aligned}$$ equation (\[meq1\]) is easily derived from equation (\[vector\_eq\]). The electromagnetic field $F_{ab}$ is related to the $\Psi_{ab}$ by equation (\[Psi\_F\]). Equation (\[meq1\]) differs from the Maxwell equation by a curvature-coupled term, $2R^b{}_aA^a$. In a flat spacetime with a vanishing spacetime curvature, equation (\[meq1\]) is exactly the Maxwell equation. Hence, it is natural to consider equation (\[meq1\]) as generalization of the Maxwell equation in a curved spacetime [@li15a; @li15b].
The Gauss-Codacci relation was mentioned and briefly discussed in [@bat01; @yam07], but its relation to the Maxwell equation was not noticed. The authors of [@bat01; @yam07] did not find the electromagnetic field contained in the field equations on the brane. Derivation of the electromagnetic field equation and identification of the electromagnetic part in the total effective stress-energy tensor in the four-dimensional Einstein field equation are the major new contribution of the present work.
Summary and Discussion
======================
We have shown that, similar to the case in the KK theory, electromagnetic fields on a four-dimensional brane can be derived from the gravity in the five-dimensional bulk spacetime. The electromagnetic field is contained in the extrinsic curvature tensor of the brane hypersurface and obeys the field equation (\[meq1\]), which differs from the Maxwell equation by a curvature-coupled term. Since by definition $N^a$ and $A^a$ are vectors tangent to the brane hypersurface, the electromagnetic field can only be seen on the brane and hence its effect is naturally confined on the brane. When $R_{ab}\neq 0$ the field equation is not gauge invariant. However, in a Ricci-flat spacetime with $R_{ab}=0$, the field equation (\[meq1\]) becomes the Maxwell equation and gauge symmetry is restored. Hence, the electromagnetic field equation (\[meq1\]) can be considered as generalization of the Maxwell equation to a curved spacetime, as an alternative to the Einstein-Maxwell equation. The curvature-coupled term $2R^b{}_aA^a$ can be regarded as a pseudo-charge current vector, whose effect is testable in an environment with high mass and energy density [@li15a; @li15b].
With appropriate boundary conditions (eq. \[join2\]), the four-dimensional Einstein field equation is derived, which is given by equation (\[ein\_eq4\]). The right-hand-side of the Einstein field equation explicitly contains the stress-energy tensor of electromagnetic fields defined by equation (\[Tab\_em\_M\]). By the relation $$\begin{aligned}
\t{R}=R-K_{ab}K^{ab}+K^2-2\t{\nabla}_av^a \;,\end{aligned}$$ where $v^a=Kn^a-a^a$, the five-dimensional Einstein-Hilbert action can be written as $$\begin{aligned}
S_\EH=\int \sqrt{-g}\left(L_G+L_\emm+L_m+L_\intt\right)d\t{V} \;, \end{aligned}$$ where $d\t{V}=2^{-1}dx^0dx^1dx^2dx^3dw^\prime$, $L_G=R$, $L_m=-g^{abcd}\dot{g}_{ab}\dot{g}_{cd}$, $L_\intt=-4g^{abcd}A_a\nabla_b\dot{g}_{cd}$, and $$\begin{aligned}
L_\emm = -4\sqrt{-g}\,\left(\frac{1}{4}F_{ab}F^{ab}-R_{ab}A^aA^b\right) \;.\end{aligned}$$ It can be verified that the variation of $S_\EH$ with respect to $A_a$ leads to the electromagnetic field equation (\[meq1\]). The variation of $S_\EH$ with respect to $g^{ab}$ leads to the four-dimensional Einstein field equation with the stress-energy tensor given by equations (\[Tab\_em\_M\])–(\[Tab\_int\_M\]) (see [@li15a] for detail).
The number of degrees of freedom (d.o.f) of gravity determined by the four-dimensional Einstein field equation is two. The number of d.o.f of the electromagnetic field determined by equation (\[meq1\]) is three, since the presence of the curvature-coupled term causes violation of gauge symmetry. This fact means that the curvature-coupled term in the electromagnetic field equation causes an effective mass to photons. Hence, the total number of d.o.f is five, which is equal to the number of d.o.f of the five-dimensional gravity.
So far we have not discussed the scalar constraint equation (\[scalar\_eq\]) yet. In fact, it can be replaced by another scalar equation obtained from the identity $E_a{}^a=0$, or equivalently, from substitution of equation (\[scalar\_eq\]) into the trace of equation (\[tensor\_eq\]). So, the scalar constraint equation gives essentially a constraint on the trace of $\t{\pounds}_nK_{ab}$ [@li15a]. It can be derived that equation (\[scalar\_eq\]) leads to $$\begin{aligned}
S_{ab}\Psi^{ab}-\frac{1}{3}S\Psi=-\left[\t{T}_{ab}n^an^b\right] \;,\end{aligned}$$ where the right-hand side is the difference in the bulk pressure acting on the two sides of the brane.
Similarly, from equation (\[vector\_eq\]) (i.e., eq. \[meq1\]) we can derive that $\nabla_aS^{ab}=0=\nabla_a\tau^{ab}$, if $\left[g^{ab}\t{T}_{ac}n^c\right]=0$. This is just the conservation equation for $S_{ab}$ and $\tau_{ab}$. Then $\nabla^a\Phi^+_{ab}=0$, and by equation (\[Ja\]) we have $J^a=(\t{\kappa}/4\pi)g^{ab}\t{T}_{bc}n^c$. When $J^a=0$, we have $\nabla^a{\cal T}_{ab}=0$ and $\nabla^aT_{\emm,ab}=0$, then by equation (\[ein\_eq4\]) we get $\nabla^a\dot{\tau}_{ab}=\t{\kappa}\nabla^a\pi_{ab}$.
This work was supported by the National Basic Research Program (973 Program) of China (Grant No. 2014CB845800) and the NSFC grant (No. 11373012).
[99]{}
L. Randall & R. Sundrum, Phys. Rev. Lett. [**83**]{}, 3370 (1999a) L. Randall & R. Sundrum, Phys. Rev. Lett. [**83**]{}, 4690 (1999b) T. Shiromizu, K. Maeda, & M. Sasaki, Phys. Rev. D [**62**]{}, 024012 (2000) P. Horava & E. Witten, Nucl. Phys. B [**460**]{}, 506 (1996) A. Lukas, B. A. Ovrut, K. S. Stelle, & D. Waldram, Phys. Rev. D [**59**]{}, 086001 (1999) A. Lukas, B. A. Ovrut, & D. Waldram, Phys. Rev. D [**60**]{}, 086001 (1999) A. Lukas, B. A. Ovrut, & D. Waldram, Phys. Rev. D [**61**]{}, 023506 (2000) R. Maartens & K. Koyama, Living Rev. Relativity [**13**]{}, 5 (2010) T. Kaluza, Sitzungsber. Preuss. Akad. Wiss, 966 (1921) O. Klein, Nature [**118**]{}, 516 (1926a) O. Klein, Zeit. Phys. [**37**]{}, 895 (1926b) D. Bailin & A. Love, Rep. Prog. Phys. [**50**]{}, 1087 (1987) J. M. Overduin & P. S. Wesson, Phys. Rep. [**283**]{}, 303 (1997) L.-X. Li, Front. Phys. [**11**]{}, 110402 (2016) (arXiv:1511.01260) C. W. Misner, K.S. Thorne, & J. A. Wheeler, Gravitation. W. H. Freeman, New York (1973) R. M. Wald, General Relativity. The University of Chicago Press, Chicago (1984) W. Israel, Il Nuovo Cimento B [**44**]{}, 1 (1966) L.-X. Li, Gen. Rel. Grav. [**48**]{}, 28 (2016) (arXiv:1508.06910) R. A. Battye, B. Carter, A. Mennim, & J.-P. Uzan, Phys.Rev. D [**64**]{}, 124007 (2001) D. Yamauchi & M. Sasaki, Prog.Theor.Phys. [**118**]{}, 245 (2007)
|
---
abstract: 'We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups $HF^+(M, {{\mathfrak s}})$ for fibered 3-manifolds $M$ whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus $g\geq 2$, where ${{\mathfrak s}}$ is a non-torsion structure on $M$.'
author:
- 'Stanislav Jabuka, Thomas Mark'
title: Heegaard Floer Homology of Mapping Tori II
---
Introduction
============
After their introduction by Ozsváth and Szabó in a remarkable series of papers [@OS1; @OS2; @OS3; @OS4], Heegaard Floer homology groups and related invariants of 3- and 4-dimensional manifolds have rapidly become central tools in low-dimensional topology. This article is concerned with the calculation of the Heegaard Floer groups associated to certain fibered 3-manifolds, and can be seen as a continuation of, or supplement to, the work in [@us]. In the latter paper, the authors considered 3-manifolds $Y$ fibered over $S^1$ with fiber a surface $\Sigma_g$ of genus $g>1$ and monodromy given as certain combinations of Dehn twists along circles in $\Sigma_g$. Specifically, let $\gamma$ and $\delta$ denote a pair of nonseparating circles on $\Sigma_g$ that intersect transversely in a single point, and let $\sigma$ denote a circle that separates $\Sigma_g$ into components of genus $1$ and $g-1$. Letting $t_\gamma$, $t_\delta$, and $t_\sigma$ denote the right-handed Dehn twists about $\gamma$, $\delta$, and $\sigma$, the monodromies considered in [@us] were those of the form $t_\gamma^nt_\delta^m$ for $n,m\in{\mbox{$\mathbb Z$}}$ and $t^{\pm 1}_\sigma$. For technical reasons the calculations of that paper did not apply to the case of other powers of the separating twist; part of the purpose of this paper is to make the necessary extensions in the arguments so as to include those cases.
Seen in a broader context, the present work extends to more general circumstances a result initially used in [@OSknot] and reformulated and applied in [@us]. Namely, given a nullhomologous knot $K$ in a closed oriented 3-manifold $Y$ we determine (under some hypotheses) the Heegaard Floer homology groups $HF^+(Y_0, {{\mathfrak s}})$ of the result $Y_0$ of 0-framed surgery along $K$ in a structure ${{\mathfrak s}}$ whose first Chern class is not torsion, in terms of the “knot complex” $CFK^{\infty}(Y, K)$. This result is stated formally in Theorem \[0surgthm\] below, and the calculation of the Heegaard Floer groups for the fibered 3-manifolds mentioned above is given as an application.
For an orientation-preserving diffeomorphism $\phi$ of a surface $\Sigma_g$, we denote by $M(\phi)$ the mapping torus $\Sigma_g\times
[0,1]/(x, 1)\sim (\phi(x),0)$. For $\sigma$ a genus-1 separating circle on $\Sigma_g$ as above, note that $M(t_\sigma^n)$ has the homology of $\Sigma_g\times S^1$ for any $n\in{\mbox{$\mathbb Z$}}$. It follows that the following two conditions uniquely determine a structure ${{\mathfrak s}}_k$ on $\Sigma_g$:
1. $\langle c_1({{\mathfrak s}}_k), [c]\times S^1\rangle = 0$ for any class $[c]\in H_1(\Sigma_g; {\mbox{$\mathbb Z$}})$.
2. $\langle c_1({{\mathfrak s}}_k), [\Sigma_g]\rangle = 2k$
According to the adjunction inequality for Heegaard Floer homology [@OS2], $HF^+(M(t_\sigma^n), {{\mathfrak s}}) = 0$ unless ${{\mathfrak s}}$ satisfies (i) above, and also satisfies (ii) with $|k|\leq g-1$. We calculate here the Heegaard Floer groups in all remaining cases except $k = 0$.
To state the result, let $X(g,d)$ denote the graded group whose summand in degree $j$ is $H^{g-j}({{\mbox{Sym}}}^d \Sigma_g; {\mbox{$\mathbb Z$}})$. For a group $G$, we let $G_{(j)}$ denote the graded group isomorphic to $G$ with grading concentrated in degree $j$, and if $H$ is a graded group let $H[j]$ denote $H$ with the grading shifted by $j$.
The following is proved in sections \[proofsec\] and \[lefttwist\], and incorporates Theorem 1.3 of [@us] as the cases $n
= \pm 1$.
\[mainthm\] For $n\neq 0$, let $M(t_\sigma^n)$ denote the mapping torus of the $n$-th power of the right-handed Dehn twist around a genus 1 separating curve on a surface $\Sigma_{g}$ ($g\geq
2$). Let ${{\mathfrak s}}$ be a nontorsion structure on $M(t_\sigma^n)$. Then the Heegaard Floer homology $HF^+(M(t_\sigma^n), {{\mathfrak s}})$ is trivial unless ${{\mathfrak s}}$ satisfies (i) and (ii) above. If these conditions hold we have an isomorphism of relatively graded groups $$\begin{aligned}
HF^+(M(t_\sigma^n), {{\mathfrak s}}) &=& X(g-1,d-1)\otimes H^*(S^1\sqcup
S^1)[\varepsilon(n)]\oplus
\Lambda^{2g-2-d}_{(g-d)}H^1(\Sigma_{g-1})\nonumber\\
&& \oplus \bigoplus_{p = 1}^{d} \left[ \Lambda^{2g-2-d+p}_{(g-d-p+1)}
H^1(\Sigma_{g-1})[\varepsilon(n)] \otimes H_*(\coprod_{|n|-1}
S^{2p-1})\right],\label{answer}\end{aligned}$$ where $d = g-1-|k|$, and where $\varepsilon(n) = 0$ if $n>0$ and $\varepsilon(n) = -1$ if $n<0$.
Note that while the Heegaard Floer groups admit further algebraic structure, in particular an action by the polynomial ring ${\mbox{$\mathbb Z$}}[U]$, the methods in this paper do not give information about that action: the isomorphism above is of ${\mbox{$\mathbb Z$}}$-modules only.
The following should be compared with results of Seidel [@seidel] and Eftekhary [@eaman] on the symplectic Floer homology of surface diffeomorphisms.
There is an isomorphism of relatively graded groups $$HF^+(M(t_\sigma^n), {{\mathfrak s}}_{g-2}) = \left\{ \begin{array}{ll} H^*(\Sigma_g,
C) & n>0 \\ H^*(\Sigma_g \setminus C) & n<0\end{array}\right.$$ where $C$ denotes the union of $|n|$ pairwise disjoint pushoffs of the separating circle $\sigma\subset\Sigma_g$, and the right-hand side above denotes singular cohomology with coefficients in ${\mbox{$\mathbb Z$}}$.
We consider the case $n>0$. In the statement we take $k
= g-2$, so that $d = 1$. Since $X(g-1,0) = {\mbox{$\mathbb Z$}}_{(g-1)}$, the first line of becomes ${\mbox{$\mathbb Z$}}_{(g-1)}\otimes({\mbox{$\mathbb Z$}}_{(0)}^2\oplus{\mbox{$\mathbb Z$}}_{(1)}^2)\oplus
\Lambda^{2g-3}H^1(\Sigma_{g-1})_{(g-1)} =
{\mbox{$\mathbb Z$}}_{(g-1)}^2\oplus{\mbox{$\mathbb Z$}}_{(g)}^2\oplus {\mbox{$\mathbb Z$}}^{2g-2}_{(g-1)}$. The second line of collapses to ${\mbox{$\mathbb Z$}}_{(g-1)}\otimes({\mbox{$\mathbb Z$}}_{(0)}^{n-1}\oplus {\mbox{$\mathbb Z$}}^{n-1}_{(1)}) =
{\mbox{$\mathbb Z$}}_{(g-1)}^{n-1}\oplus{\mbox{$\mathbb Z$}}_{(g)}^{n-1}$. Thus $$HF^+(M(t_\sigma^n), {{\mathfrak s}}_{g-2}) \cong {\mbox{$\mathbb Z$}}^{n+1}_{(g)} \oplus
{\mbox{$\mathbb Z$}}^{2g + n -1}_{(g-1)}.$$ On the other hand, since $\Sigma_g\setminus C$ is the disjoint union of a punctured torus, a punctured surface of genus $g-1$, and $n-1$ annuli, we have $$H^*(\Sigma_g, C) \cong {\mbox{$\mathbb Z$}}^{n+1}_{(2)}\oplus {\mbox{$\mathbb Z$}}^{2g+n -1}_{(1)}$$ which agrees with the above modulo a shift in grading. The case $n<0$ follows similarly.
In the next section we state and prove Theorem \[0surgthm\] on the Floer homology of a manifold obtained by 0-framed surgery on a nullhomologous knot; in Section \[calcsec\] we prove Theorem \[mainthm\].
Floer Homology of 0-Surgeries
=============================
The method we will use to make our calculation is an adaptation to slightly more general circumstances of the one used by [[Ozsváth and Szabó ]{}]{}in [@OSknot] to determine the Heegaard Floer homology of the mapping torus of a single Dehn twist about a nonseparating curve on a surface of genus $g\geq 2$, in nontorsion structures.
Let $K\subset Y$ be a nullhomologous knot in a closed oriented 3-manifold $Y$, and let $Y_n(K)$ denote the result of $n$-framed surgery along $K$. Fix a structure ${{\mathfrak s}}$ on $Y$, and for integers $n>0$ and $k\in
\{0,\ldots,n-1\}$ let $HF^+(Y_n(K), k)$ denote the Heegaard Floer homology of the surgered manifold in the structure ${{\mathfrak s}}_k$ defined as follows. Integer surgery corresponds to a cobordism $W_n$ from $Y$ to $Y_n(K)$, comprising a single 2-handle addition. Fixing a Seifert surface $F$ for $K$ we obtain a closed surface $\hat{F}$ in $W_n$ by capping off $F$ using the core of the 2-handle . Now let ${{\mathfrak s}}_k\in {\mbox{{\rm Spin}${}^c$}}(Y_n(K))$ be the structure cobordant to ${{\mathfrak s}}$ by a structure ${{\mathfrak r}}$ on $W_n$ having $$\langle c_1({{\mathfrak r}}), [\hat{F}]\rangle = n - 2k.$$
Note that if, as will always be the case here, the structure on $Y$ is torsion and $H_2(Y;{\mbox{$\mathbb Z$}})$ is torsion-free, then ${{\mathfrak s}}_k$ is independent of the choice of $F$.
According to [@OS2], there is an integer surgeries long exact sequence $$\cdots\to HF^+(Y_0(K),[k])\to HF^+(Y_n(K),k)\to HF^+(Y)\to
\cdots
\label{les}$$ Here the first term denotes the sum of Floer homology groups over structures in the fiber of ${{\mathfrak s}}_k$ under a certain surjective map ${\mbox{{\rm Spin}${}^c$}}(Y_0(K))\to {\mbox{{\rm Spin}${}^c$}}(Y_n(K))$ (see [@OS2]). Our object is to use knot Floer homology to understand each term in this sequence, and therefore we quickly review the relevant facts about knot Floer homology (for details, see [@OSknot], [@OSknot2]).
Given $K\subset Y$, let $E$ denote the torus boundary of a regular neighborhood of $K$. One can then find a Heegaard surface for $Y$ of the form $E\# \Sigma_{g-1}$, with attaching circles ${\mbox{{\boldmath}$\alpha$}}=
\alpha_1,\ldots,\alpha_g$ and ${\mbox{{\boldmath}$\beta$}}= \beta_1,\ldots,\beta_g$ where $\beta_1\subset E$ is a meridian for $K$ and $(\Sigma,
{\mbox{{\boldmath}$\alpha$}}, \beta_2,\ldots,\beta_g)$ is a Heegaard diagram for the knot complement $Y\setminus K$. Let $w$ and $z$ denote a pair of basepoints, one on each side of the meridian $\beta_1$. The data $(E\#\Sigma, {\mbox{{\boldmath}$\alpha$}},{\mbox{{\boldmath}$\beta$}}, w)$ together with a choice of structure ${{\mathfrak s}}$ on $Y$ can be used to define the Heegaard Floer chain groups $CF^\infty(Y, {{\mathfrak s}})$. The additional basepoint $z$, along with a choice of “relative structure” $\underline{{{\mathfrak s}}}\in Spin^c(Y_0(K))$ lifting ${{\mathfrak s}}$ gives rise to a filtration ${\mathcal}F$ on $CF^\infty(Y,{{\mathfrak s}})$. The “knot chain complex” $CFK^\infty(Y,K,{\underline{{{\mathfrak s}}}})$ is this filtered complex.
More concretely, we fix a Seifert surface $F$ for $K$: then $F$ specifies the zero-framing on $K$, and can be capped off to a closed surface $\hat{F}$ in the zero-surgery $Y_0(K)$. The generators of $CFK^\infty(Y,K,F)$ are triples $[{{\mbox{\bf x}}},i,j]$, where ${{\mbox{\bf x}}}$ denotes an intersection point between the $g$-dimensional tori $T_\alpha =
\alpha_1\times\cdots\times \alpha_g$ and $T_\beta = \beta_1\times
\cdots\times\beta_g$ in the symmetric power ${{\mbox{Sym}}}^g(E\#\Sigma)$, and $i$ and $j$ are integers. The point ${{\mbox{\bf x}}}$ along with the basepoint $w$ determine a structure ${{\mathfrak s}}_w({{\mbox{\bf x}}})$ on $Y$ as well as a relative structure $\underline{{{\mathfrak s}}}_w({{\mbox{\bf x}}})$; we require that ${{{\mathfrak s}}}_w({{\mbox{\bf x}}}) =
{{{\mathfrak s}}}$. Furthermore, $i$ and $j$ are required to satisfy the equation $$\langle c_1({\underline{{{\mathfrak s}}}}_w({{\mbox{\bf x}}})), [\hat{F}]\rangle = 2(j-i).$$ In this notation, the filtration ${\mathcal}F$ is simply ${{\mathcal}F}([{{\mbox{\bf x}}},i,j]) = j$; changing the Seifert surface $F$ shifts ${\mathcal}F$ by a constant.
The boundary map $\partial^\infty$ in $CFK^\infty$ is defined by counting holomorphic disks in ${{\mbox{Sym}}}^g(E\#\Sigma)$ and can only decrease the integers $i$ and $j$. Thus, for example, the subgroup $C\{i<0\}$ of $CFK^\infty$ generated by those $[{{\mbox{\bf x}}},i,j]$ having $i<0$ is a subcomplex, and indeed is simply $CF^-(Y,{{\mathfrak s}})$ with an additional filtration. The quotient of $CFK^\infty$ by $C\{i<0\}$ is written $C\{i\geq 0\}$, and is a filtered version of $CF^+(Y,{{\mathfrak s}})$. We will use other similar notations to indicate other sub- or quotient complexes of $CFK^\infty$. In particular, $CFK^{0,*}$ is by definition the quotient complex $C\{i = 0\}$, and $\widehat{HFK}$ is the homology of the graded object associated to the filtration ${{\mathcal}F}$ of $CFK^{0,*}$. We denote by $\widehat{HFK}(Y,K;j)$ the summand of this group supported in filtration level $j$ (typically suppressing the structure from the notation).
As an additional piece of structure, we have a natural chain endomorphism $U$ on $CFK^\infty$ given by $U:[{{\mbox{\bf x}}},i,j]\mapsto [{{\mbox{\bf x}}},i-1,j-1]$.
[[Ozsváth and Szabó ]{}]{}prove (Theorem 4.4 of [@OSknot]):
\[hello\] For all sufficiently large positive $n$, there exists a $U$-equivariant isomorphism of chain complexes $${^b\Psi^+} : CF^+ (Y_n , k) \rightarrow C\{ i \ge 0 \mbox{ or } j \ge k \}$$
In particular, $HF^+(Y_n(K), k)$ is given by the homology of the portion of the knot complex indicated on the right-hand side. It is important to note that the proof of the above theorem shows that the stated identification is induced by a chain map coming from a particular structure on the cobordism $-W_n$ connecting $Y_n(K)$ to $Y$, namely the structure ${{\mathfrak r}}_0\in{\mbox{{\rm Spin}${}^c$}}(-W_n)$ satisfying $$\label{specialstr}
\langle c_1({{\mathfrak r}}_0),[\hat{F}]\rangle = n
- 2k.$$ By contrast, the homomorphism $HF^+(Y_n(K),k)\to HF^+(Y)$ in the long exact sequence (\[les\]) is given by the sum of the homomorphisms induced by all structures on $-W_n$ that restrict to the given structures on $Y_n(K)$ and $Y$. We will return to this point shortly.
Examining the sequence (\[les\]) again, we can now understand two of the groups appearing in terms of the knot chain complex: the theorem above identifies $HF^+(Y_n,k)$, while $HF^+(Y)$ is simply $H_*(C\{i\geq 0\})$, the homology of the complex obtained by forgetting the filtration. Note that under these identifications, the map $HF^+(Y_n,k)\to HF^+(Y)$ induced by ${{\mathfrak r}}_0$ as above corresponds to the map on homology induced by the natural projection $C\{i\geq 0\mbox{ or } j\geq k\} \to C\{i\geq 0\}$. Supposing the connecting homomorphism $HF^+(Y)\to
HF^+(Y_0(K), [k])$ to be trivial, and ignoring the issue raised in the previous paragraph, we anticipate an isomorphism $$HF^+(Y_0(K), [k]) \cong H_*(C\{i<0\mbox{ and } j\geq k\}).$$ Indeed, it is implicit in [@OSknot] and described concretely in [@us] that if $HF^+_{red}(Y) = 0$ (and certain other, less important, hypotheses hold) then the connecting homomorphism is in fact trivial. We will show that the isomorphism holds even in certain cases when the connecting homomorphism is nontrivial.
\[0surgthm\] Fix a torsion structure ${{\mathfrak s}}$ on $Y$, and let $HF^+(Y, {{\mathfrak s}}; A)$, $HF^+(Y_n(K), k; A)$, and $HF^+(Y_0(K), [k];
A)$ denote Floer homology groups in structures as above, with coefficients in a ring $A$. Assume that $k$ is nonzero, and make the following additional assumptions:
1. $HF^+(Y, {{\mathfrak s}}, A)$ is a free $A$-module.
2. The filtration on $HF_{red}(Y)$ is proportional to the degree: specifically, there is a constant $c$ such that the part of $HF_{red}(Y)$ that lies in filtration level $j$ is supported in absolute grading $j + c$.
3. For $F: HF^+(Y_n(K),k;A) \to HF^+(Y;A)$ the homomorphism in the long exact sequence , we have $${\mbox{\rm Ext}}^1_A(\ker(F), {\mbox{\rm coker}}(F)) = 0.$$
4. The reduced homology $HF_{red}(Y_n(K), k,A)$ is supported in degrees at most $k + c - d$, where $d$ is the degree shift induced by the structure ${{\mathfrak r}}_0$ as in .
Then there is an identification of $A$-modules $$\label{homologyident}
HF^+(Y_0(K), [k]; A) \cong H_*(C\{i<0\mbox{ and } j\geq k\}; A).$$
In particular hypotheses 1 and 3 of the theorem hold if $A$ is a field. If $A =
{\mbox{$\mathbb Z$}}$, these hypotheses hold if, for example, $HF^+(Y)$ and ${\mbox{\rm coker}}(F)$ are torsion-free groups.
For the proof we need to analyze the surgery exact sequence: $$\label{surgseq}
\cdots\rTo^G HF^+(Y_0(K),[k])\rTo^H HF^+(Y_n,k)\rTo^F HF^+(Y)\rTo^G
\cdots$$ In particular, as mentioned above, we study the homomorphism $F$. Recall that since the structures ${{\mathfrak s}}$ and ${{\mathfrak s}}_k$ are torsion, the corresponding Heegaard Floer homology groups admit a (rational-valued) grading that lifts the natural relative ${\mbox{$\mathbb Z$}}$ grading. A structure ${{\mathfrak r}}$ on $-W_n$ extending ${{\mathfrak s}}$ restricts to ${{\mathfrak s}}_k$ on $Y_n(K)$ if and only if it satisfies $$\langle c_1({{\mathfrak r}}), [\hat{F}]\rangle = n - 2k + 2xn$$ for some integer $x\in{\mbox{$\mathbb Z$}}$, where $\hat{F}$ denotes the capped-off Seifert surface in $-W_n$ as before. Recall that we assume $k\in\{1,\ldots, n-1\}$. Now, $F$ is the sum of the maps $F_{-W_n, {{\mathfrak r}}}$ for ${{\mathfrak r}}$ corresponding to all values of $x$. According to [@OS3], the map $F_{{{\mathfrak r}}}: HF^+(Y_n(K), k)\to HF^+(Y)$ induced by ${{\mathfrak r}}$ shifts degree by the quantity $$\deg(F_{{{\mathfrak r}}}) = -nx^2 - (n-2k)x + \frac{n-(n-2k)^2}{4n}.$$ The maximum value of this degree occurs when $x$ is the closest integer to $-\frac{1}{2} + \frac{k}{n}$, which, given our assumption on $k$, is $x = 0$. Comparing with we see that the structure ${{\mathfrak r}}_0$ corresponding to $x = 0$ is both the “leading order term” (i.e., the homogeneous part with maximal degree) in the homomorphism $F$ and also the structure inducing the identification of $HF^+(Y_n(K),k)$ with $H_*(C\{i\geq 0\mbox{ or }
j\geq k\})$.
With the above in mind, we write (as in [@OSknot]) $F = f_1 + f_2$ where $f_1$ is the homogeneous part with highest degree (that is, $f_1
= F_{-W_n, {{\mathfrak r}}_0}$) and $f_2$ is the sum of all lower-degree parts of $F$. Note that the highest-degree part of $f_2$ has degree equal to $\deg(f_1) - \min\{2k, 2(n-k)\}$.
\[rinvlemma\] Suppose there exists a homomorphism $R:
{\mbox{Im}}(f_2)\to HF^+(Y_n(K),k)$ satisfying $f_1\circ R = id$ on ${\mbox{Im}}(f_2)$ (that is, $R$ is a partially-defined right inverse for $f_1$). Then there is an isomorphism $g: HF^+(Y_n(K),k)\to HF^+(Y_n(K),k)$ such that $$f_1 = F\circ g.$$ In particular, ${\mbox{\rm coker}}(F) = {\mbox{\rm coker}}(f_1)$, and $\ker(F) \cong
\ker(f_1)$ (via the map $g$).
Since $F = f_1 + f_2$, we have $$F = f_1( 1 + Rf_2).$$ From the remarks on the degree shift above, the composition $Rf_2$ is strictly decreasing in degree. Since $HF^+(Y_n(K),k)$ is trivial in sufficiently low degrees, the sum $$g = (1 + Rf_2)^{-1} = \sum_{n\geq 0} (-Rf_2)^n$$ is finite, and hence $g$ is the desired isomorphism.
If we can construct the right inverse $R$ as in the lemma, Theorem \[0surgthm\] will follow easily. Indeed, let us identify $HF^+(Y_n(K),k)$ with $H_*(C\{i\geq 0 \mbox{ or } j\geq k\})$ as in Theorem \[hello\]: then we’ve seen that $f_1$ is identified with the natural projection
$$\pi_*: H_*(C\{i\geq 0 \mbox{ or } j\geq k\})\to H_*(C\{i\geq 0\}) = HF^+(Y).$$ Hence we have a commutative diagram[$$\begin{diagram}
\cdots & \rTo & HF^+(Y_0(K)) & \rTo & HF^+(Y_n(K)) & \rTo^F & HF^+(Y) &
\rTo & \cdots \\
& & \dDashto & & \dTo >1+ Rf_2 & & \dTo>id &&\\
\cdots & \rTo & H_*(C\{i<0 \mbox{ and } j\geq k\}) & \rTo &
H_*(C\{i\geq 0 \mbox{ or } j\geq k\}) & \rTo^{\pi_*} & H_*(C\{i\geq
0\}) & \rTo & \cdots
\end{diagram}$$]{} where the solid vertical arrows are isomorphisms and the top row is the surgery long exact sequence . We have not constructed a map corresponding to the dashed arrow, but the above diagram allows us to write another: $$\label{commdiag}
\begin{diagram}
0 & \rTo & {\mbox{\rm coker}}(F) &\rTo& HF^+(Y_0(K)) &\rTo & \ker(F)& \rTo &0\\
& & \dTo>{id} & & & & \dTo<{1 + Rf_2} & \\
0 & \rTo & {\mbox{\rm coker}}(\pi_*) & \rTo & H_*(C\{i<0 \mbox{ and } j\geq k\}) &
\rTo & \ker(\pi_*) & \rTo & 0
\end{diagram}$$ Again the vertical arrows are isomorphisms, so that $HF^+(Y_0(K))$ and $ H_*(C\{i<0 \mbox{ and } j\geq k\})$ are both extensions of ${\mbox{\rm coker}}(F)$ by $\ker(F)$. Under hypothesis 3 of the theorem such an extension is unique, which completes the proof of Theorem \[0surgthm\].
It remains to construct the right inverse $R$.
\[rinvlemma2\] Under hypotheses (1), (2), and (4) of Theorem \[0surgthm\] there exists a right inverse $R$ for $f_1$ that is defined on ${\mbox{Im}}(f_2)$.
Note that the only place that hypothesis 3 of the theorem is required is at the last step of its proof (see above).
Since ${\mbox{Im}}(f_1)\subset HF^+(Y)$ is free by hypothesis, we can find a right inverse $R: {\mbox{Im}}(f_1)\to HF^+(Y_n(K), k)$. We need to check that $R$ is defined on ${\mbox{Im}}(f_2)$, i.e., that ${\mbox{Im}}(f_2)\subset
{\mbox{Im}}(f_1)$. We identify $HF^+(Y)$ with $H_*(C\{i\geq 0\})$ and $HF^+(Y_n(K), k)$ with $H_*(C\{i\geq 0\mbox{ or } j\geq k\})$, so that $f_1$ corresponds to the projection $\pi_*$ as above.
Recall that because of the structure of the chain complex $CF^{\infty}(Y)$ the image of the action of $U^r$ on $HF^+(Y)$ is independent of $r$ for sufficiently large $r$. By definition the reduced Floer homology group $HF_{red}(Y)$ is the quotient of $HF^+(Y)$ by ${\mbox{Im}}(U^r)$ for any such large $r$: in other words there is an exact sequence $$0 \rTo {\mbox{Im}}(U^r) \rTo HF^+(Y) \rTo HF_{red}(Y)\rTo 0.$$ The knot filtration induces a filtration on $HF_{red}$.
Now, since the chain complexes $C\{i\geq 0\}$ and $C\{i\geq 0\mbox{ or } j\geq k\}$ are identical for sufficiently large degrees, it is clear that $f_1 =
\pi_*$ maps onto ${\mbox{Im}}(U^r)$ for $r>>0$. We claim that $f_1$ also maps onto $H_{red}^{<k}(C\{i\geq 0\})$, where $H_{red}^{<k}$ denotes that portion of the reduced homology that lies in filtration level $j=k-1$ or below. To see this, it suffices to show that the connecting homomorphism $\delta: H_*(C\{i\geq 0\})\to H_*(C\{i<0\mbox{ and }j\geq
k\})$ is trivial on the indicated group. But for $x\in C\{i\geq 0\}$ a cycle, $\delta x$ is given by the portion of $\partial^{\infty} x$ that lies in $C\{i< 0\mbox{ and } j\geq k\}$, where $\partial^\infty$ is the boundary map in $CFK^\infty$. The statement follows since $\partial^\infty$ is nonincreasing in $j$.
From the discussion above, $f_1$ maps onto all summands of $HF^+(Y)$ that lie in degree less than $k + c$ as well as ${\mbox{Im}}(U^r)$ (in all degrees). By $U$-equivariance $f_2$ maps ${\mbox{Im}}(U^r)$ into ${\mbox{Im}}(U^r)$, so we need only check that $f_2$ maps $HF_{red}(Y_n(K),k)$ into ${\mbox{Im}}(f_1)$. But this follows immediately from the facts that $f_2$ has degree strictly less than that of $f_1$ and that $HF_{red}(Y_n)$ is supported in degrees $\leq k + c - \deg(f_1)$.
Calculation for Separating Twists {#calcsec}
=================================
Preliminaries: Calculation of Knot Homology
-------------------------------------------
A surgery diagram for the mapping torus $M(t_\sigma^n)$ is obtained from one for $\Sigma_g\times S^1$ by adding $n$ parallel copies of the separating curve $\sigma$ with surgery coefficient $-1$ (here and subsequently we assume $n>0$; the case of negative $n$ is entirely parallel and will be described later). In particular, $M(t_\sigma^n)$ can be obtained from 0-surgery along a knot $\tilde K$ in a connected sum $\#^{2g-2}(S^1\times S^2) \# M_n$, where $M_n$ is described below. The situation is illustrated for $g = 3$ and $n = 1$ in Figure \[sepknot2\].
![The knot $\tilde K$ in the case $g=3$ and $n=2$.[]{data-label="sepknot2"}](knot2 "fig:"){width="12cm"} (5,30)[$\tilde{K}$]{} (-60,138)[$0$]{} (-168,138)[$0$]{} (-275,138)[$0$]{} (-18,95)[$0$]{} (-127,95)[$0$]{} (-235,95)[$0$]{} (-290,25)[$-1$]{} (-290,45)[$-1$]{}
We can think of $\tilde K$ as a connected sum of knots: write $B(p,q)$ for the knot given as the third component of the Borromean rings after performing surgery on the other two components with surgery coefficients $p$ and $q$. Then $\tilde K = \#^{g-1}B(0,0)\# K$, where $K$ is the knot indicated in Figure \[sepknot\].
![Two equivalent pictures for the knot $K$ in $M_n$. In picture (a) there are $n$ parallel $-1$ circles. The $2n$ in picture (b) indicates $2n$ positive half-twists. []{data-label="sepknot"}](knot8 "fig:"){width="4cm"} ![Two equivalent pictures for the knot $K$ in $M_n$. In picture (a) there are $n$ parallel $-1$ circles. The $2n$ in picture (b) indicates $2n$ positive half-twists. []{data-label="sepknot"}](knot7 "fig:"){width="4cm"} (-62,62)[$2n$]{} (-50,127)[0]{} (-266,127)[0]{} (-19,80)[0]{} (-231,80)[0]{} (3,18)[K]{} (-209,18)[K]{} (-275,18)[-1]{} (-262,-25)[(a)]{} (-62,-25)[(b)]{}
We denote by $M_n$ the manifold containing $K$, so $M_n$ is obtained by performing 0-framed surgery on two components of the Borromean rings and $-1$ framed surgery on each of $n$ parallel copies of the third component. By blowing down all the $-1$ circles, we can see $M_n$ as the result of $0$-surgery on both components of an $n$-clasped Whitehead link as shown in figure \[sepknot\](b) (the $2n$ in the box denotes $2n$ positive half-twists). As a preliminary to the calculation of the knot homology of $K$, we find the homology $\widehat{HF}(M_n)$.
Let $Z$ denote the 3-manifold that results from changing the surgery coefficient from $0$ to $-1$ on the smaller $0$-framed circle in the diagram for $M_n$. An easy isotopy shows that after blowing down this $-1$ circle, $Z$ is given as 0-framed surgery on the pretzel knot $P(-2n + 1, -1, -1)$ (here we follow the notation conventions of [@OS4], whereby $P(1,1,1)$ is the right-handed trefoil). It follows from the calculations in section 8 of [@OS4] that for the torsion structure ${{\mathfrak s}}_0$, $$HF^+_k(Z; {{\mathfrak s}}_0) = \left\{\begin{array}{ll} {\mbox{$\mathbb Z$}}& k \equiv
1/2 \mbox{ mod ${\mbox{$\mathbb Z$}}$ and $k\geq 3/2$}\\ {\mbox{$\mathbb Z$}}^n & k = 1/2 \\ 0 &
\mbox{else}\end{array}\right.$$ and that the image of $U$ in degree $1/2$ is nontrivial.
For the rest of this section and throughout the next, we take coefficients in the ring $A = {\mbox{$\mathbb Z$}}$. In particular it suffices for the verification of hypothesis (3) of Theorem \[0surgthm\] to show that ${\mbox{\rm coker}}(F)$ is a free abelian group.
With this information, the long exact sequence in Floer homology connecting $S^1\times S^2$, $Z$, and $M_n$ immediately gives $$HF^+_k(M_n; {{\mathfrak s}}_0) = \left\{\begin{array}{ll} {\mbox{$\mathbb Z$}}^2 & k \equiv 1 \mbox{
mod ${\mbox{$\mathbb Z$}}$ and $k\geq 1$}\\ {\mbox{$\mathbb Z$}}^{n+1} & k = 0 \\ 0 &
\mbox{else}\end{array}\right.$$ From this we infer $\widehat{HF}(M_n) = {\mbox{$\mathbb Z$}}^{n+1}_{(1)}\oplus
{\mbox{$\mathbb Z$}}^{n+1}_{(0)}$.
We are interested in the knot Floer homology $\widehat{HFK}(M_n, K)$. The following was proved in [@us]:
\[M1Klemma\] The knot Floer homology groups for $(M_1, K)$ are given by $$\widehat{HFK}(M_1,K; j) = \left\{\begin{array}{ll} {\mbox{$\mathbb Z$}}_{(1)} & j =
1\\ {\mbox{$\mathbb Z$}}^3_{(0)}\oplus {\mbox{$\mathbb Z$}}_{(1)} & j = 0 \\ {\mbox{$\mathbb Z$}}_{(-1)} & j = -1 \\
0 & \mbox{otherwise}\end{array}\right.$$ The spectral sequence that calculates $\widehat{HF}(M_1)$ from $\widehat{HFK}(M_1,K)$ collapses at the $E_2$ level, and the only nontrivial differential is a surjection $d_1: {\mbox{$\mathbb Z$}}^3_{(0)}\to{\mbox{$\mathbb Z$}}_{(-1)}$.
We now calculate the result for the case of general $n$:
\[MnKprop\] The knot Floer homology groups of $K$ in $M_n$ are given by $$\widehat{HFK}(M_n, K; j) = \left\{ \begin{array}{ll} {\mbox{$\mathbb Z$}}_{(1)} & j =
1\\ {\mbox{$\mathbb Z$}}^{n+2}_{(0)} \oplus {\mbox{$\mathbb Z$}}^n_{(1)} & j = 0 \\ {\mbox{$\mathbb Z$}}_{(-1)} &
j = -1 \\ 0 & \mbox{otherwise} \end{array}\right.$$ The only nontrivial differential in the spectral sequence converging to $\widehat{HF}(M_n)$ is a surjection $d_1: {\mbox{$\mathbb Z$}}^{n+2}_{(0)}\to
{\mbox{$\mathbb Z$}}_{(-1)}$.
We proceed inductively: suppose $n\geq 2$. We look at a surgery sequence arising from figure \[sepknot\](a). Choose one of the $-1$ circles in the picture for $M_n$ and let $A$ denote the 3-manifold obtained by changing the $-1$ to 0. Then the surgery sequence appears as $$\cdots \to \widehat{HFK}(M_{n-1}, K; j)\to \widehat{HFK}(M_n, K ; j)
\to \widehat{HFK}(A, K; j)\to \cdots$$ Now $(A,K)$ is unknotted since $K$ can slide over the 0-framed circle. Therefore the filtration induced by $K$ on $\widehat{CF}(A)$ is trivial, and hence $\widehat{HFK}(A, K)$ is supported in level $j = 0$. It follows immediately that the group $\widehat{HFK}(M_n, K; j)$ has the claimed form when $j\neq
0$.
For the case $j = 0$, note that the calculation so far together with the structure of $\widehat{HF}(M_n)$ already imply that $\widehat{HFK}(M_n, K; 0)$ is supported in degrees 0 and 1, by consideration of the spectral sequence for $\widehat{HF}$. It then follows that the only nontrivial differential in the latter spectral sequence is $d_1$, and in fact $d_1: \widehat{HFK}(M_n, K; 0)\to
\widehat{HFK}(M_n, K, -1) \cong {\mbox{$\mathbb Z$}}$ must be surjective. An argument similar to the case $n = 1$ (see [@us]) together with our inductive knowledge of $\widehat{HFK}(M_{n-1},K)$ shows that $d_1:
\widehat{HFK}(M_n, K, 1)\to \widehat{HFK}(M_n, K; 0)$ is trivial. The proposition follows from this and the fact that $\widehat{HF}(M_n) =
{\mbox{$\mathbb Z$}}^{n+1}_{(1)} \oplus {\mbox{$\mathbb Z$}}^{n+1}_{(0)}$.
It will be convenient in what follows to write $\widehat{HFK}(M_n,
K)$ as $$\widehat{HFK}(M_n,K) \cong \Lambda^*H^1(\Sigma_1)\oplus H^*(\coprod_n S^1).$$ In the above, the grading on $\Lambda^*H^1(\Sigma_1)$ is “centered,” meaning that $\Lambda^iH^1(\Sigma_g)$ is considered to have grading $i-g$. The grading on the second factor above is the usual homological grading. The filtration is not evident from the notation, however we see that it is equal to the (centered) grading on the first factor while the second factor lies in filtration level 0.
In this notation, we can express the single nontrivial differential in the spectral sequence for $\widehat{HF}(M_n)$ as the map $\Lambda^1H^1(\Sigma_1)\to \Lambda^0H^1(\Sigma_1)$ given by contraction with a generator $\gamma$ of $H_1(\Sigma_1)$, which we represent as an embedded circle in the torus also denoted $\gamma$.
The connected sum theorem for $\widehat{HFK}$ then gives:
\[HFKprop\] The knot Floer homology of $\tilde{K}\subset Y = M_n \#^{2g-2} (S^1\times S^2)$ is given by $$\widehat{HFK}(Y,\tilde{K}) = \Lambda^*H^1(\Sigma_{g})\oplus \left[
\Lambda^*H^1(\Sigma_{g-1})\otimes H^*(\coprod_n S^1)\right].
\label{septwistHFK}$$ \[septwistHFKprop\] The only nontrivial differential in the spectral sequence for $\widehat{HF}(Y)$ is given by contraction with a generator $\gamma\in
H_1(\Sigma_g)$ in the first summand above.
Indeed, it is shown in [@OSknot] that $\widehat{HFK}(B(0,0))\cong
\Lambda^*H^1(\Sigma_1)$ with centered grading. Formula follows from this and the Künneth formula for $\widehat{HFK}$ under connected sum [@OSknot].
Verification of Hypotheses
--------------------------
Knowledge of the knot Floer homology $\widehat{HFK}(Y,K)$ can often lead to understanding of the full Heegaard Floer groups $HF^+$ for $Y$ and the surgered manifold $Y_n(K)$. Indeed, there is a spectral sequence for $HF^{\infty}(Y)$ associated to the filtration of $CFK^{\infty}(Y,K)$ given by $[{{\mbox{\bf x}}},i,j] \mapsto i+j$, whose $E_1$-term is $\widehat{HFK}(Y,K)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]$. The $d_1$ differential is a sum of homomorphisms that map the group at position $(i,j)$ in $CFK^\infty$ to those at positions $(i-1,j)$ and $(i, j-1)$: the “vertical” and “horizontal” components of $d_1$. These components can in turn be determined from the spectral sequence for $\widehat{HF}(Y)$ coming from $CFK^{0,*}$: indeed, the vertical component is precisely (after a translation by a power of $U$) the first differential in the latter sequence. On the other hand the complex $CFK^{*,0}$ can also be identified with a filtered version of $\widehat{CF}$, so the horizontal component of $d_1$ is also determined by the differential in the spectral sequence calculating $\widehat{HF}$ from $\widehat{HFK}$. We put these ideas to work in understanding the Floer homologies of the particular $Y$ and $Y_n$ relevant to our situation.
We begin by determining the differentials in the spectral sequence for $CF^{\infty}$, in the case of $(Y,\tilde{K})$ as in Proposition \[septwistHFKprop\]. As noted above, the $E_1$ term is given by $\widehat{HFK}(Y,\tilde{K})\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]$. Explicitly, this is $$\label{E1term}
E_1 = \left( \Lambda^*H^1(\Sigma_g) \otimes
{\mbox{$\mathbb Z$}}[U,U^{-1}]\right) \oplus \left( \Lambda^*H^1(\Sigma_{g-1})\otimes
H^*(\coprod_n S^1)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]\right).$$ The $d_1$ differential is nontrivial only on the first summand, where its action is described as follows. Decompose $\Sigma_g
= \Sigma_1\#\Sigma_{g-1}$ where the generator $\gamma\in
H_1(\Sigma_1)$ of Proposition \[septwistHFKprop\] is contained in the first factor. Let $$\begin{aligned}
E_+ &=& (\Lambda^0H^1(\Sigma_1) \oplus \Lambda^2 H^1(\Sigma_1))\otimes
\Lambda^*H^1(\Sigma_{g-1}) \otimes {\mbox{$\mathbb Z$}}[U,U^{-1}] \\
E_- &=& \Lambda^1H^1(\Sigma_1)\otimes\Lambda^*H^1(\Sigma_{g-1})
\otimes {\mbox{$\mathbb Z$}}[U,U^{-1}],\end{aligned}$$ so that $E_+\oplus E_-$ is isomorphic to the first summand of the $E_1$ term above. Then one can check just as in [@OSknot] or [@us] that the $d_1$ differential is trivial on $E_+$ while on $E_-$ it is given by $$\label{d1diff}
d_1(\omega\otimes U^j) =
\iota_\gamma\omega\otimes U^j + PD(\gamma)\wedge\omega\otimes U^{j+1}.$$ It is a straightforward exercise to check that the homology of this differential is given by $$\begin{aligned}
H(E_+\oplus E_-, d_1) &=& \Lambda^0H^1(\Sigma_1)\otimes
\Lambda^*(\Sigma_{g-1})\otimes {\mbox{$\mathbb Z$}}[U,U^{-1}] \\
&&\quad \oplus PD(\gamma)\wedge \Lambda^*H^1(\Sigma_{g-1})
\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]\\
&\cong& \Lambda^*H^1(\Sigma_{g-1})\otimes H^*(S^1)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}].\end{aligned}$$
Therefore the second term in our spectral sequence appears as $$\begin{aligned}
E_2 &=& \left(\Lambda^*H^1(\Sigma_{g-1})\otimes
H^*(S^1)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]\right) \\&&\oplus \left(
\Lambda^*H^1(\Sigma_{g-1})\otimes H^*(\coprod_n
S^1)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]\right)\\
&=& \Lambda^*H^1(\Sigma_{g-1})\otimes H^*(\coprod_{n+1} S^1)\otimes
{\mbox{$\mathbb Z$}}[U,U^{-1}].\end{aligned}$$ Observe that there is an isomorphism of ${\mbox{$\mathbb Z$}}[U]$-modules $H^*(S^1\amalg S^1)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}] \cong
\Lambda^*H^1(\Sigma_1)\otimes {\mbox{$\mathbb Z$}}[U,U^{-1}]$. Therefore we can write the above as $$\label{E2term}
E_2 = \left( \Lambda^*H^1(\Sigma_g)
\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]\right) \oplus \left(
\Lambda^*H^1(\Sigma_{g-1})\otimes H^*(\coprod_{n-1}
S^1)\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]\right),$$ where we have written $\Lambda^*H^1(\Sigma_{g-1})\otimes
\Lambda^*H^1(\Sigma_1) = \Lambda^*H^1(\Sigma_g)$.
We must now determine subsequent differentials in the spectral sequence, if any. Note first that if $n= 1$ then the above reduces to $\Lambda^*H^1(\Sigma_g)\otimes {\mbox{$\mathbb Z$}}[U,U^{-1}]\cong HF^{\infty}(Y)$. Indeed, this identification follows using the connected sum theorem for $HF^{\infty}$ (recall that $Y = M_n\#(2g-2)S^1\times S^2$) and the fact that since $b_1(M_n) = 2$, the Floer homology $HF^\infty(M_n)$ is “standard” (see [@OS2], [@OS4]). Thus when $n = 1$ there are no subsequent differentials, and in fact this case has already been understood in [@us].
In general, the $d_2$ differential is a sum of three terms, mapping $C\{i, j\}$ into $C\{i-2, j\}\oplus C\{i-1,j-1\}\oplus C\{i,j-2\}$. In our case, however, the vertical and horizontal components must be trivial because those (just as in the case of $d_1$) correspond to differentials in the second term of the spectral sequence for $\widehat{HF}$, which we have seen collapses at the second term. Therefore $d_2$ must be given by a map $d_2: E_2\{i,j\}\to
E_2\{i-1,j-1\}$.
\[sslemma\] In terms of the expression , $d_2$ is trivial on the first factor and acts on the second factor by an isomorphism $$\Lambda^*H^1(\Sigma_{g-1})\otimes H^0(\coprod_{n-1} S^1) \otimes U^k
{\longrightarrow}\Lambda^*(\Sigma_{g-1})\otimes H^1(\coprod_{n-1} S^1)\otimes
U^{k+1}.$$ All subsequent differentials in the spectral sequence are trivial.
We can see that further differentials must be trivial by examining the gradings. In , the exterior algebras are equipped with the centered grading, the grading on $H^*(S^1)$ is the natural homological grading, and $U$ is considered to have grading $-2$. The filtration (the “$j$-coordinate”) is equal to the grading on the exterior algebras and on ${\mbox{$\mathbb Z$}}[U,U^{-1}]$, but $H^*(\coprod_{n-1}S^1)$ is considered to lie in filtration level 0. (All of these observations can be deduced from the remarks after Propositions \[MnKprop\] and \[HFKprop\].) Finally, the “$i$-coordinate” is recovered by recalling that in the expression $E_1 = \widehat{HFK}\otimes{\mbox{$\mathbb Z$}}[U,U^{-1}]$, the subgroup $\widehat{HFK}\otimes 1$ lies in the column $i = 0$. It is now straightforward to see that if $(i+j) - (i' + j') > 2$ then there are no elements $a\in C\{i,j\}$ and $b\in C\{i',j'\}$ whose degrees differ by 1, so differentials beyond $d_2$ vanish for dimensional reasons.
It follows that the homology of $d_2$ must yield $HF^{\infty}(Y)\cong
\Lambda^*H^1(\Sigma_g)\otimes {\mbox{$\mathbb Z$}}[U,U^{-1}]$; furthermore the only factors in $E_2\{i,j\}$ and $E_2\{i-1,j-1\}$ that can be connected by this differential (i.e., factors whose degrees differ by 1) are those that are indicated in the statment. Dimensional considerations ensure that the differential must be an isomorphism between those factors.
With this understanding of the differentials in the spectral sequence it is a straightforward matter to determine $HF^+(Y)$ and $HF^+(Y_n)$ to a degree sufficient to verify the hypotheses of Theorem \[0surgthm\]. Indeed, hypotheses 1 and 2 of that theorem follow from:
With $Y = M_n\#(2g-2)S^1\times S^2$ as above, we have an identification of ${\mbox{$\mathbb Z$}}[U]$-modules $$HF^+(Y, {{\mathfrak s}}) = \left(\Lambda^*(\Sigma_{g-1})\otimes {\mbox{$\mathbb Z$}}[U^{-1}]\right)
\oplus (n-1)\Lambda^*H^1(\Sigma_{g-1}),$$ where ${{\mathfrak s}}$ denotes the torsion structure on $Y$, and $(n-1)\Lambda^*H^1(\Sigma_{g-1})$ denotes the direct sum of $n-1$ copies of the exterior algebra. In particular, $HF_{red}(Y,{{\mathfrak s}}) =
(n-1)\Lambda^*H^1(\Sigma_{g-1})$ as graded groups (where the grading on the exterior algebra is centered as before), and the filtration on $HF_{red}(Y,{{\mathfrak s}})$ is equal to the grading.
Note that one can prove this (except for the information about the filtration) without using the spectral sequence by appealing to the connected sum theorem for Heegaard Floer homology.
By restriction, the filtration ${{\mathcal}F}:
[{{\mbox{\bf x}}},i,j]\mapsto i+j$ used to produce the spectral sequence for $HF^\infty$ also gives a filtration on $CF^+$ and thereby a spectral sequence for $HF^+$ whose differentials are just the restrictions of the originals to this quotient complex. In particular the $E_1$ term appears as $\widehat{HFK}(Y,\tilde{K})\otimes{\mbox{$\mathbb Z$}}[U^{-1}]$, with differential given by . One checks that no new cycles are created in $E_1$ by passing to the quotient complex $C\{i\geq
0\}$, so that the $E_2$ term here looks just like with ${\mbox{$\mathbb Z$}}[U,U^{-1}]$ replaced by ${\mbox{$\mathbb Z$}}[U^{-1}]$.
The second differential takes the same form as previously, but in this case there are additional cycles in $E_2$: since $d_2$ maps $E_2\{i,j\}$ into $E_2\{i-1,j-1\}$, it sends those elements lying in the group $\Lambda^*H^1(\Sigma_{g-1})\otimes H^0(\coprod_{n-1}S^1) \otimes
U^0$ (supported in the column $i = 0$) to 0, where it did not do so in the spectral sequence for $HF^\infty$. Hence this group is precisely $HF_{red}(Y,{{\mathfrak s}})$, and can be written $(n-1)\Lambda^*H^1(\Sigma_{g-1})$ as a graded group. The lemma follows immediately, keeping in mind the structure of the filtration as described in the proof of Lemma \[sslemma\].
For the remaining hypotheses in Theorem \[0surgthm\] we must understand $HF^+(Y_n(\tilde{K}), k)$. Recall the isomorphism ${}^b\Psi^+: CF^+(Y_n)\cong C\{i\geq 0 \mbox{ or }
j\geq k\}$ of Theorem \[hello\]. Using ${}^b\Psi^+$, the filtration ${\mathcal}F$ on $CF^\infty(Y)$ restricts to a filtration on $CF^+(Y_n)$, and thereby gives a spectral sequence for $HF^+(Y_n)$ It is a simple matter to see, by examining the domains and ranges of the differentials, that $HF_{red}(Y_n)$ must be supported along the right-angled strip $\mbox{max}\{i, j-k\} = 0$. (Indeed, just as in the case of $Y$ the reduced homology is a result of the “additional” cycles for $d_2$ that arise from passing to the quotient complex $C\{i\geq 0\mbox{ or }j\geq 0\}$, which can only lie in the indicated region of the $(i,j)$ plane.) Since the $j$-coordinate measures the filtration, hypothesis 4 follows immediately.
Finally for hypothesis 3, note that the proof of Theorem \[0surgthm\] shows that we may replace $F$ by $\pi_*$ since the corresponding kernels and cokernels are isomorphic once the other hypotheses hold (c.f. Lemmas \[rinvlemma\] and \[rinvlemma2\]). But it is clear that $\ker \pi_*$ is equal to that portion of $H_*(C\{i\geq 0 \mbox{ or }j\geq k\})$ lying in the region $i<0$, while its image is ${\mbox{Im}}(U^r) \oplus HF_{red}^{\leq k}(Y)\subset
HF^+(Y)$. In particular $${\mbox{\rm coker}}(\pi_*) = HF_{red}^{>k}(Y) \cong \Lambda^{>g-1+k}H^1(\Sigma_{g-1})$$ is a free ${\mbox{$\mathbb Z$}}$-module.
Calculation {#proofsec}
-----------
We turn our attention to determining the Heegaard Floer homology groups $HF^+(M(t_\sigma^n), {{\mathfrak s}})$ where ${{\mathfrak s}}$ is any nontorsion structure, $\sigma\subset \Sigma_g$ is a separating curve such that $\Sigma_g\setminus\sigma$ consists of components of genus $1$ and $g-1$, $t_\sigma$ denotes the right-handed Dehn twist about $\sigma$, and $M(t_\sigma^n)$ is the mapping torus of the diffeomorphism $t_\sigma^n$ for any $n\neq 0$. We focus first on the case of $n>0$.
The result of the preceding section is that the desired Floer homology can be determined from the knot complex for $\tilde{K}\subset Y$ as $H_*(C\{i<0\mbox{ and } j\geq k\})$, where $\tilde{K} =
K\#(g-1)B(0,0)$ as before. We assume here that $k>0$. We make use of the same spectral sequence as in that section to calculate this homology; note that the differentials have already been determined. Now, the $E_1$ term of this spectral sequence is just the portion of $\widehat{HFK}(Y,\tilde{K})\otimes {\mbox{$\mathbb Z$}}[U,U^{-1}]$ that lies in the relevant part of the $(i,j)$ plane. Following [@OSknot] and [@us], we introduce the notation $$\label{Xgddef}
X(g,d) = \bigoplus ^d _{i = 0} \Lambda ^{2g-i} H^1(\Sigma _g) \otimes
_\mathbb{Z} \mathbb{Z}[U] /U^{d+1-i}.$$ Thus $X(g,d) \cong H^*({{\mbox{Sym}}}^d\Sigma_g)$ as ${\mbox{$\mathbb Z$}}[U]$-modules (see [@macdonald]). Then it is easy to see that in the spectral sequence for $HF^+(Y_0(K), {{\mathfrak s}}_k)$, $$E_1 = X(g,d) \oplus \left[ X(g-1,d-1)\otimes H^*(\coprod_n
S^1)\right],$$ where $d = g-1-k$. The $d_1$ differential is nontrivial only on the first factor (see the discussion after equation ), where it acts as in equation . The complex $(X(g,d), d_1)$ was considered in [@us], where it was shown that its homology is $$H_*(X(g,d), d_1) = (X(g-1,d-1)\otimes H^*(S^1))\oplus
\Lambda^{2g-2-d}H^1(\Sigma_{g-1})_{(g-d)}.$$ Hence, $$\begin{aligned}
E_2 &=& [X(g-1,d-1)\otimes H^*(S^1)\oplus
\Lambda^{2g-2-d}H^1(\Sigma_{g-1})_{(g-d)}] \nonumber\\
&&\oplus \left[ X(g-1,d-1)\otimes H^*(\coprod_n S^1)\right]\nonumber\\
&=& \left[ X(g-1,d-1)\otimes H^*(S^1\sqcup S^1)\oplus
\Lambda^{2g-2-d}H^1(\Sigma_{g-1})_{(g-d)}\right] \label{E2term2} \\
&&\oplus \left[ X(g-1,d-1)\otimes H^*(\coprod_{n-1} S^1)\right].\nonumber\end{aligned}$$ The $d_2$ differential acts as was determined in Lemma \[sslemma\], only on the last summand above (and only when $n>1$). Thus in “most” positions $(i,j)$ the last factor is killed in homology, with the exception of those $(i,j)$ with $i = 0$ or $j = k$: since the differential maps $(i,j)$ to $(i-1,j-1)$ there are additional cycles when $j = k$ and fewer boundaries when $i = 0$. Specifically, the homology of is given by the first term in brackets plus the contributions: $$\begin{aligned}
(i =0 ) &\quad& \bigoplus_{p = 1}^{d} \Lambda^{2g-2-d + p}
H^1(\Sigma_{g-1})\otimes H^1(\coprod_{n-1} S^1)\label{vertcont}\\
(j = k) &\quad& \bigoplus_{p = 1}^{d}
\Lambda^{2g-2-d+ p}H^1(\Sigma_{g-1}) \otimes H^0(\coprod_{n-1} S^1)\otimes
U^{p-1}.\label{horizcont}\end{aligned}$$ where $d = g-1-k$.
[*Proof of Theorem \[mainthm\]*]{}. The results of the preceding sections show that for given $k$ with $0<k\leq g-1$ $$HF^+(M(t_\sigma^n), {{\mathfrak s}}) \cong H_*(C\{i<0\mbox{ and } j\geq k\})$$ where the right-hand side refers to the homology of the indicated quotient complex of $CFK^\infty(Y, \tilde{K})$ (in the torsion structure on $Y = M_n\# (2g-2)S^1\times S^2$). The latter has been shown to be isomorphic as a relatively graded ${\mbox{$\mathbb Z$}}$-module to the sum of the first bracketed term in and the two expressions and .
To understand those two expressions, recall that the summand $\Lambda^{2g-q}H^1(\Sigma_g)\otimes U^j$ of $X(g,d)$ is supported in grading $g-q-2j$. From this it follows (recall that the factors in and arise from $X(g-1,d-1)$ as in the second term of ) that $$\begin{aligned}
\mbox{\eqref{vertcont} $\oplus$ \eqref{horizcont}} &\cong& (n-1)
\bigoplus_{p = 1}^d \Lambda^{2g-2-d+p}_{(g-d+p)}H^1(\Sigma_{g-1}) \oplus
\Lambda^{2g-2-d+p}_{(g-d-p+1)}H^1(\Sigma_{g-1}) \\
&\cong& \bigoplus_{p=1}^d
\Lambda^{2g-2-d+p}_{(g-d-p+1)}H^1(\Sigma_{g-1}) \otimes
H_*(\coprod_{n-1} S^{2p-1}).\end{aligned}$$ Adding this to the first line of gives , when $k$ is positive. For negative $k$, the result follows from the conjugation invariance of $HF^+$ (see [@OS1]).
The Case of Left-Handed Twists {#lefttwist}
------------------------------
The procedure for calculation of $HF^+(M(t_\sigma^{-n}))$, $n>0$, is very similar to the positive-twist case. We outline here the main differences.
The surgery diagram for $M(t_\sigma^{-n})$ is identical to that for $M(t_\sigma^n)$ with the exception that $-1$-surgery curves corresponding to the Dehn twists are replaced by $+1$-curves (c.f. Figure \[sepknot2\]). Thus, $M(t_\sigma^{-n})$ is obtained by 0-surgery on the knot $\tilde{K} = \#(g-1)B(0,0)\# K$ as before, where $K\subset M_{-n}$ is as in Figure \[sepknot\](a) with signs changed, or as in Figure \[sepknot\](b) with positive twists replaced by negative ones. Since $M_{-n}$ is just $M_n$ with the orientation reversed, we have (using the behavior of $\widehat{HF}$ under orientation reversal) $\widehat{HF}(M_{-n}; {{\mathfrak s}}_0) =
{\mbox{$\mathbb Z$}}^{n+1}_{(0)} \oplus {\mbox{$\mathbb Z$}}^{n+1}_{(-1)}$.
To obtain the knot Floer homology $\widehat{HFK}(M_{-n}, K)$, we proceed as before: the result corresponding to Lemma \[M1Klemma\] is $$\widehat{HFK}(M_{-1},K; j) = \left\{\begin{array}{ll} {\mbox{$\mathbb Z$}}_{(1)} & j =
1\\ {\mbox{$\mathbb Z$}}_{(-1)} \oplus {\mbox{$\mathbb Z$}}^3_{(0)} & j = 0 \\ {\mbox{$\mathbb Z$}}_{(-1)} & j = -1 \\
0 & \mbox{otherwise.}\end{array}\right.$$ This is proved in just the same way as Lemma \[M1Klemma\] with some surgery coefficients having opposite sign (see [@us]).
In general, the analogue of Proposition \[MnKprop\] is: $$\widehat{HFK}(M_{-n}, K; j) = \left\{\begin{array}{ll}{\mbox{$\mathbb Z$}}_{(1)} & j =
1\\ {\mbox{$\mathbb Z$}}^n_{(-1)} \oplus {\mbox{$\mathbb Z$}}^{n+2}_{(0)} & j = 0 \\ {\mbox{$\mathbb Z$}}_{(-1)} &
j = -1 \\ 0 & \mbox{otherwise,}\end{array} \right.$$ and is proved in an analogous manner. Put another way, $$\widehat{HFK}(M_{-n}, K; j) \cong \Lambda^*H^1(\Sigma_1)\oplus
H^*(\coprod_{n} S^1)[-1],$$ where “$[-1]$” indicates that the grading on the corresponding factor has been shifted down by 1 (the grading on the exterior algebra, as usual, is centered). Note that the differential $d_1$ in the spectral sequence calculating $\widehat{HF}(M_{-n})$ from the above consists of an injection ${\mbox{$\mathbb Z$}}_{(1)}\to {\mbox{$\mathbb Z$}}^{n+2}_{(0)}$, which we may identify with contraction by a generator in $H_1(\Sigma_1)$.
It follows just as in the previous work that $$\widehat{HFK}(Y,K) \cong \Lambda^*H^1(\Sigma_g) \oplus \left(
\Lambda^*H^1(\Sigma_{g-1})\otimes H^*(\coprod_n S^1)[-1]\right),$$ and in the spectral sequence calculating $HF^+(M(t_\sigma^{-n}))$, $$E_1 = X(g,d) \oplus \left( X(g-1,d-1)\otimes H^*(\coprod_n
S^1)[-1]\right).$$ The $d_1$ differential in this case again acts only on the first factor above, and is given by equation . However, under the splitting $E_+\oplus E_-$ of the first factor (c.f. the remarks after equation ), here $d_1$ is trivial on $E_-$ and acts nontrivially only on $E_+$. The homology of $X(g,d)$ with respect to this differential was also calculated in [@us], and that result gives $$\begin{aligned}
E_2 &=& X(g-1,d-1)\otimes H^*(S^1)[-1] \oplus
\Lambda^{2g-2-d}H^1(\Sigma_{g-1})_{(g-d)} \\
&& \oplus \left( X(g-1,d-1)\otimes H^*(\coprod_n S^1)[-1]\right)\\
&\cong& X(g-1,d-1)\otimes H^*(S^1\sqcup S^1)[-1] \oplus
\Lambda^{2g-2-d}H^1(\Sigma_{g-1})_{(g-d)} \\
&& \oplus \left( X(g-1,d-1)\otimes H^*(\coprod_{n-1} S^1)[-1]\right).\end{aligned}$$
An argument as in the positive case identifies the $d_2$ differential as being nontrivial only on the second factor above, mapping $$d_2: \Lambda^kH^1(\Sigma_{g-1})\otimes
H^0(\coprod_{n-1}S^1)[-1]\otimes U^j \longrightarrow \Lambda^kH^1(\Sigma_{g-1})\otimes
H^1(\coprod_{n-1}S^1)[-1]\otimes U^{j+1}$$ isomorphically. The resulting homology is given again by the sum of and , this time with a grading shift of $-1$. This proves Theorem \[mainthm\] in the case $n<0$.
[99]{} E. Eftekhary, [*Floer cohomology of certain pseudo-Anosov maps*]{}, math.SG/0205029. S. Jabuka and T. Mark, [*Heegaard Floer homology of certain mapping tori*]{}, Alg. Geom. Topol., [**6**]{} (2004) 685–719.
I. G. Macdonald, [*Symmetric products of an algebraic curve*]{}, Topology [**1**]{} (1962), 319–343.
P. Ozsváth and Z. Szabó, [*Holomorphic disks and topological invariants for closed 3-manifolds*]{}, to appear in Annals of Math. math.SG/0101206. P. Ozsváth and Z. Szabó, [*Holomorphic disks and three-manifold invariants: properties and applications*]{}, to appear in Annals of Math. math.SG/0105202. P. Ozsváth and Z. Szabó, [*Holomorphic triangles and invariants for smooth four-manifolds*]{}, math.SG/0110169. P. Ozsváth and Z. Szabó, [*Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary*]{}, Adv. Math. [**173**]{} (2003), no. 2, 179–261. P. Ozsváth and Z. Szabó, [*Holomorphic disks and knot invariants*]{}, Adv. Math. [**186**]{} (2004), no. 1, 58 – 116.
P. Ozsváth and Z. Szabó, [*Heegaard Floer homology and alternating knots*]{}, Geom. Topol. [**7**]{} (2003) 225–254. P. Seidel, [*The symplectic Floer homology of a Dehn twist*]{}, Math. Res. Lett. [**3**]{} (1996), no. 6, 829–834.
|
---
abstract: 'We study a model of spin-1 dark matter which interacts with the Standard Model predominantly via exchange of Higgs bosons. We propose an alternative UV completion to the usual Vector Dark Matter Higgs Portal, in which vector-like fermions charged under SU(2)$_W \times$ U(1)$_Y$ and under the dark gauge group, U(1)$^\prime$, generate an effective interaction between the Higgs and the dark matter at one loop. We explore the resulting phenomenology and show that this dark matter candidate is a viable thermal relic and satisfies Higgs invisible width constraints as well as direct detection bounds.'
author:
- Anthony DiFranzo
- 'Patrick J. Fox'
- 'Tim M.P. Tait'
bibliography:
- 'VDM.bib'
title: Vector Dark Matter through a Radiative Higgs Portal
---
Introduction
============
As the only elementary scalar in the Standard Model (SM), the Higgs boson presents a unique opportunity as a window to physics beyond the Standard Model (SM). The operator $H^\dagger H$ is the lowest dimensional operator which is both a gauge and Lorentz singlet. As such, it occurs time and again as the means by which physics uncharged under the SM gauge symmetries communicates with the Standard Model. In particular, it is an effective mechanism by which scalar dark matter (DM) can talk to the ordinary matter [@Burgess:2000yq], as is required if we wish to understand its abundance in the Universe today as the result of thermal processes acting in a standard cosmological history.
In the present work, we focus on the case in which the dark matter is a spin one vector boson. At first glance, it would appear that this case (much like scalar DM) offers a renormalizable connection between the dark matter and the Higgs [@Djouadi:2011aa; @Lebedev:2011iq], $$\mathcal{L} \supset \lambda ~H^\dagger H ~V_\mu V^\mu~,
\label{eq:naive}$$ where $V_\mu$ is a massive vector field which plays the role of dark matter and $\lambda$ is a dimensionless coupling. But this form, while invariant under the SM gauge symmetries, is misleading. Just like the SM $W$ and $Z$ bosons, a well-behaved UV description of $V$ requires that it be associated with a gauge symmetry (the most simple construction of which would be an Abelian U(1)$^\prime$, though one could also consider non-Abelian theories as well), spontaneously broken to give $V$ a mass. The term in Eq. (\[eq:naive\]) violates the U(1)$^\prime$, and must be engineered via its spontaneous breaking.
One tempting avenue would be to charge the Higgs itself under U(1)$^\prime$. In that case the Higgs kinetic term $(D_\mu H)^\dagger (D^\mu H)$ contains Eq. (\[eq:naive\]), and the mass of $V$ will arise as part of the vacuum expectation value (VEV) of $H$, naturally connecting the scale of the $V$ mass to the electroweak scale. However, this construction contains other terms which mix $V$ with the SM $Z$ boson, with the result that $V$ will inevitably end up unstable and contribute unacceptably to precision electroweak measurements unless it is very light (implying that it is very weakly coupled). This regime, though worth pursuing, is not very interesting for particle physics at the weak scale, and not very amenable to exploration through Higgs measurements at the LHC.
The situation is very different when the $V$ mass is the result of a VEV living in a different scalar particle $\Phi$ which is a SM gauge singlet. In that case, there is no dangerous mixing with the SM $Z$ boson, and the gauge coupling can be relatively large, $$\begin{aligned}
{\cal L} &~ \supset ~ & -\frac{1}{4} V_{\mu \nu} V^{\mu \nu} + \left( D_\mu \Phi \right)^\dagger \left( D^\mu \Phi \right) - V (\Phi)
+ \lambda_P ~ |H|^2 |\Phi|^2~,
\label{eq:module1}\end{aligned}$$ where $D_\mu \Phi \equiv \partial_\mu \Phi - g Q_\Phi V_\mu \Phi$ is the usual covariant derivative for a particle of charge $Q_\Phi$ and $V(\Phi)$ is a U(1)$^\prime$-invariant potential designed to induce a VEV $\langle \Phi \rangle = v_\phi$, producing a mass for $V$, $$\begin{aligned}
m_V^2 & = & g^2Q_\Phi^2~ v_\phi^2~.
\label{eq:Vmass}\end{aligned}$$ We have also included a scalar Higgs portal coupling $\lambda_P$, which leads to tree-level mixing between the SM Higgs boson and the Higgs mode of $\Phi$, effectively implementing the Higgs portal. As a construction implementing the Higgs portal, it is well motivated and has been extensively explored in the literature[^1] [@Hambye:2008bq; @Farzan:2012hh; @Baek:2012se; @Baek:2013qwa; @Baek:2014jga; @Baek:2014goa; @Ko:2014gha; @Gross:2015cwa; @DiChiara:2015bua; @Chen:2015dea; @Kim:2015hda].
However, it does not represent the [*only*]{} possible UV completion. In this work, we explore an alternative completion which realizes the Higgs portal as a consequence of additional heavy fermions which are charged under both U(1)$^\prime$ and the SM gauge symmetries. At one loop, these fermions mediate an interaction between the Higgs and the DM somewhat in analogy with the effective Higgs-gluon vertex induced by the top quarks in the SM. This [*radiative*]{} UV completion leads to different phenomenology and singles out different interesting regions of parameter space.
This article is organized as follows. In Sec. \[model\], we discuss a simplified picture to illustrate the most important physics behind this concept, followed by the full matter content of the UV theory. In Sec. \[results\], we examine the phenomenology in light of experimental probes, such as direct detection, the invisible Higgs width, and relic abundance. We first focus on the case where the simplified picture is valid, with and without also considering mixing generated by a Scalar Higgs Portal. We then examine the effect of the full radiative portion of the UV theory. We reserve Sec. \[conclusion\] for conclusions and summary.
Radiative Higgs Portal for Vector Dark Matter {#model}
=============================================
Particle Content and Structure
------------------------------
A radiative model often has multiple paths to the same low energy physics, since the mediating particles are not themselves involved in the initial and final states. Starting with the basic module of Eq. (\[eq:module1\]), we aim for a construction which adds fermions mediating an interaction of the form (\[eq:naive\]) such that:
- the vector particle $V$ remains stable at the radiative level, which in particular requires that it does not kinetically mix with the SM electroweak interaction;
- the full gauge structure SU(3)$_C \times$ SU(2)$_W \times$ U(1)$_Y \times$ U(1)$^\prime$ remains free from gauge anomalies;
- there are no large contributions to the SM Higgs coupling to gluons or photons in contradiction with LHC measurements [@Flechl:2015foa].
The first of these is the most subtle. Generically, communication between the SM Higgs and $V$ requires that the mediator fermions be charged under both U(1)$^\prime$ and the Standard Model, which typically will induce processes involving an odd number of $V$’s, resulting in their decay. The simplest example of such a process is the kinetic mixing between $V$ and hypercharge. Such dangerous processes can be forbidden by a charge-conjugation symmetry, under which $V$ is odd. In analogy with Furry’s theorem of QED [@Furry:1937], this symmetry forbids processes involving an odd number of $V$’s at energies below the masses of the mediator fermions.
Field (SU(2)$_W$, U(1)$_Y$, U(1)$^\prime$) Field (SU(2)$_W$, U(1)$_Y$, U(1)$^\prime$)
------------------ ------------------------------------------------------ ------------------ -----------------------------------------
$\psi_{1\alpha}$ (2, , 1) $\psi_{2\alpha}$ (2, , -1)
$\chi_{1\alpha}$ (2, , -1) $\chi_{2\alpha}$ (2, , 1)
$n_{1\alpha}$ (1, 0, -1) $n_{2\alpha}$ (1, 0, 1)
$\Phi$ (1, 0, $Q_\Phi$)
: Charge assignments for fermions $\psi$, $\chi$, and $n$ and complex scalar $\Phi$.
\[tab:NPtran\]
Cancelling gauge anomalies further suggests that the additional fermions appear in vector-like pairs under both the SM and U(1)$^\prime$ gauge symmetries, whereas renormalizable coupling to the Higgs requires fields in SU(2)$_W$ representations of size $n$ and $n+1$ (and have hypercharges differing by $1/2$). A minimal set of particles satisfying these conditions is shown in Table \[tab:NPtran\], consisting of four SU(2)$_W$ doublets and two singlets. (Different) pairs of the doublets are vector-like under both U(1)$_Y$ and U(1)$^\prime$, cancelling gauge anomalies, and a U(1)$^\prime$ charge conjugation is implemented by $f_1 \! \leftrightarrow \! f_2$ (where $f = \psi, \chi, n$).
We have left the U(1)$^\prime$ charge of $\Phi$ as a free non-zero parameter which controls the dark matter mass as per Eq. (\[eq:Vmass\]). Choosing $Q_\Phi=\pm1$ would allow the $\Phi$ VEV to mix the SM lepton doublets with the new fermions, which would be strongly constrained by precision measurements and ruin the U(1)$^\prime$ charge conjugation symmetry. Choosing $Q_\Phi=\pm2$ would allow for Yukawa interactions of $\Phi$ with pairs of the new fermions, which would complicate the analysis of their mass eigenstates. We will restrict ourselves to other values for $Q_\Phi$, which avoids these features, and serves simply to adjust the mass of $V$. It’s worth pointing out that this implies that the lightest of the fermionic states is also stable, and will be present in the Universe to some degree as a second component of dark matter. However, provided its mass is much larger than $m_V$, fermion anti-fermion pairs will annihilate efficiently into weak bosons and $V$’s, leaving it as a negligible fraction of the dark matter.
In 2-component Weyl notation, the Lagrangian contains mass terms and Yukawa interactions for the new fermions, $$\begin{aligned}
\mathcal L &~ \supset -m ~ \epsilon^{ab} \left( \psi_{1a} \chi_{1b} + \psi_{2a} \chi_{2b} \right) - m_n ~ n_1 n_2\\
&- y_{\psi}~\epsilon^{ab} \left( \psi_{1a} H_b n_1 + \psi_{2a} H_b n_2 \right) - y_{\chi} \left( \chi_{1} H^* n_2 + \chi_{2} H^* n_1 \right) + h.c.
\end{aligned}$$ where $a$ and $b$ are SU(2)$_W$ indices, the SM Higgs $H$ is defined to transform as a (2, $\nicefrac{-1}{2}$, 0), and spin indices have been suppressed. The U(1)$^\prime$ charge conjugation symmetry, $f_1 \! \leftrightarrow \! f_2$ is manifest. After electroweak symmetry-breaking, the mass terms can be written as, $$\begin{aligned}
\label{eq:mass}
\mathcal L_m &= - N^T M_n N' - E^T M_e E' + h.c.\end{aligned}$$ where $$\begin{aligned}
N &=
\begin{bmatrix}
\psi_{1n}\\
\chi_{2n}\\
n_2
\end{bmatrix},
~~~~~N' =
\begin{bmatrix}
\psi_{2n}\\
\chi_{1n}\\
n_1
\end{bmatrix},
~~~~~E =
\begin{bmatrix}
\psi_{1e}\\
\chi_{2e}\\
\end{bmatrix},
~~~~~E' =
\begin{bmatrix}
\chi_{1e}\\
\psi_{2e}\\
\end{bmatrix},\end{aligned}$$ assemble collections of the electrically neutral ($N$ and $N^\prime$) and charged ($E$ and $E^\prime$) components of the fermions, and the mass matrices are given by, $$\begin{aligned}
M_n =
\begin{bmatrix}
0 & -m & -y_{\psi}v/\sqrt{2} \\
-m & 0 & y_{\chi}v/\sqrt{2} \\
-y_{\psi}v/\sqrt{2} & y_{\chi}v/\sqrt{2} & m_n
\end{bmatrix}, ~~~~~
M_e =
\begin{bmatrix}
m & 0 \\
0 & m
\end{bmatrix}.\end{aligned}$$ In the mass basis, there are three electrically neutral and two charged Dirac fermions, all of which interact with the dark matter $V$ diagonally, since the states that mix all carry the same $U(1)'$ charge. Their coupling to the SM Higgs will involve the mixing matrices which transform from the gauge to the mass basis.
Note that by construction the electrically charged fermions receive no contributions from $\langle H \rangle$, implying that they do not interact with the Higgs boson and lead to no one-loop correction to its effective coupling to photons. Our choice to arrange $N$ such that they also receive no contributions from $\Phi$ implies that the fermions do not renormalize the usual Higgs portal coupling $\lambda_P$ of Eq. (\[eq:module1\]) at one-loop (starting at two loops, there are contributions mediated by a mixture of the fermions and $V$ itself). In order to better extract the features of the radiative model, we self-consistently assume that $\lambda_P$ is small enough to be subdominant in the majority of the remainder of this work.
$\sigma_{\rm SI}$ and Higgs Invisible Width
-------------------------------------------
![Representative triangle diagram contributing to the Higgs–dark matter interaction.[]{data-label="fig:loop"}](loop){width="35.00000%"}
Both the direct detection cross-section and the Higgs invisible decay width result from triangle diagrams (see Fig. \[fig:loop\]). Integrating out the fermion $\psi$ running in the loop, the $h-V-V$ interaction can be encoded by two form factors: $$- \left( \frac{1}{4} A(p^2) ~h ~V^{\mu\nu}V_{\mu\nu} + \frac{1}{2} B(p^2)~ h ~V^{\mu}V_{\mu} \right)
\label{eq:eftloop}$$ with coefficients $A$ and $B$ which are (in the on-shell DM limit, $k_1^2 = k_2^2 = m_V^2$) functions of the fermion masses and mixings, $m_V$, and the momentum through the Higgs line, $p^2$. Reasonably compact analytic expressions for $A$ and $B$ are derived in Appendix \[app:tri\]. We observe that $B(p^2)\rightarrow 0$ in the limit $m_V \rightarrow 0$ (i.e. when the $U(1)^\prime$ symmetry is restored), as is required by gauge invariance, see Appendix \[app:tri\].
In terms of $A$ and $B$, the cross section for non-relativistic scattering of $V$ with a nucleon $n$ is given by, $$\begin{aligned}
\sigma_{\rm SI} &= &\frac{1}{4\pi m_h^4} \left(\frac{f_n}{v}\right)^2 \left(\frac{m_n^2}{m_n+m_V}\right)^2 |B(0) - A(0) ~m_V^2|^2 \label{eq:loopsigma}\end{aligned}$$ where the momentum transfer through the Higgs is approximated as $p^2 \approx 0$, $$\begin{aligned}
f_n = \sum_{q=u,d,s} f_{Tq}^{(n)} + \frac{2}{9} f_{TG}^{(n)},\end{aligned}$$ and we use the hadronic matrix elements $f_{Tq}$, from DarkSUSY [@Gondolo:2004sc]. Because of the tiny up and down Yukawa couplings, scattering mediated by a Higgs is to good approximation iso-symmetric.
The same three point vertex function also describes the invisible decay width of the Higgs boson, $$\begin{aligned}
\Gamma(h \rightarrow V V) &=&
\frac{1}{64\pi m_h} \sqrt{1-4\frac{m_V^2}{m_h^2}} \left[ \left| A(m_h^2) \right|^2 m_h^4 \left(1 - 4\frac{m_V^2}{m_h^2} + 6\frac{m_V^4}{m_h^4}\right)
\right. \label{eq:loopwidth}\\
& & \left.
+ 6 \operatorname{Re}\left( A^*(m_h^2) B(m_h^2) \right) m_h^2 \left(1-2\frac{m_V^2}{m_h^2}\right)
+ \frac{1}{2} \left| B(m_h^2) \right|^2 \frac{m_h^4}{m_V^4} \left(1 - 4 \frac{m_V^2}{m_h^2} + 12 \frac{m_V^4}{m_h^4} \right)\right] \nonumber\end{aligned}$$ where the Higgs is on-shell, $p^2 = m_h^2$. Note that because for small $m_V$ the coefficient $B(p^2) \propto m_V^4$, this expression is finite in the limit $m_V \rightarrow 0$, as it should be.
Annihilation Cross Section and Relic Abundance
----------------------------------------------
![Representative box diagrams which contribute to DM annihilation into pairs of Higgs or electroweak bosons.[]{data-label="fig:box"}](boxhh "fig:"){width="35.00000%"} ![Representative box diagrams which contribute to DM annihilation into pairs of Higgs or electroweak bosons.[]{data-label="fig:box"}](boxww "fig:"){width="35.00000%"}
Pairs of dark matter can annihilate through the three point coupling of Fig. \[fig:loop\] through an (off- or on-shell) SM Higgs, leading to final states containing heavy quarks and/or weak bosons. These contributions exhibit a strong resonant behavior when $m_V \simeq m_h / 2$. The gauge and Higgs boson final states also receive contributions at the same order from box diagrams (see Fig. \[fig:box\]), which contribute to processes including $VV \rightarrow hh, ZZ, WW, \gamma\gamma, hZ, Z\gamma$. These box diagrams are sensitive to more of the details of the UV theory, receiving contributions from the charged fermions as well as the neutral ones. As a result, simple analytic forms are not particularly illuminating, and we evaluate them using FeynArts [@Hahn:2000kx], FormCalc, and LoopTools [@Hahn:1998yk]. In the following section, we compute the full annihilation cross section including all of the accessible SM final states.
Experimental Constraints and Parameter Space {#results}
============================================
In this section, we examine the interesting parameter space, finding the regions consistent with the LUX limits on the spin independent DM-nucleon scattering cross-section [@Akerib:2013tjd]; and the invisible decay width of the Higgs produced via vector boson fusion (VBF) as constrained by CMS with 19.7 fb$^{-1}$ at 8 TeV [@Chatrchyan:2014tja]. In the latter, we include the off-shell Higgs contribution following the technique presented in [@Endo:2014cca], simulating VBF Higgs production with HAWKv2.0 [@Denner:2011id]. We also identify the regions leading to the correct thermal relic abundance for a standard cosmology, computing the loop diagrams with FeynArts [@Hahn:2000kx], FormCalc, and LoopTools [@Hahn:1998yk], which is then linked into micrOMEGAsV4.0 [@Belanger:2013oya].
Because of the relatively large number of parameters, we build up insight into the phenomenology gradually by considering three different limits of the full theory. Initially in Sec. \[sec:singleF\], we consider the limit in which one of the neutral fermions is much lighter than both the other two neutral states and both of the charged ones, and the coupling $\lambda_P$ is small enough to be neglected. We follow this in Sec. \[sec:singleHP\] by allowing $\lambda_P$ to be large enough that there is relevant mixing between $h$ and the Higgs mode of $\Phi$. Finally, in Sec. \[sec:fullF\] we switch off $\lambda_P$ once more, but consider the case where all mediator fermions have comparable masses.
Single Fermion Limit {#sec:singleF}
--------------------
We begin with the case where the charged fermions and the two heavier neutral states are much heavier than the lightest neutral state, effectively decoupling from the phenomenology, and $\lambda_P$ can be ignored. As before we assume the physical scalar contained in $\Phi$ is heavy enough to be ignored. In this limit, the relevant parameters are the $U(1)^\prime$ gauge coupling $g$, Yukawa coupling to the light fermion $y$, light fermion mass $m_\psi$, and the vector dark matter mass $m_V$. As we will see below, the correct thermal relic density can only be achieved for annihilation in the Higgs funnel region, for which one can neglect the box diagram contributions. In that case, the gauge and Yukawa couplings always appear in the combination $y g^2$, leaving only three relevant parameter combinations.
![Left: Upper limits on $yg^2$ from VBF Higgs collider and direct detection constraints, with a fermion of mass 400 GeV. Right: The corresponding lower limit on the relic abundance for a standard cosmology.[]{data-label="fig:simplim"}](couplim "fig:"){width="47.50000%"} ![Left: Upper limits on $yg^2$ from VBF Higgs collider and direct detection constraints, with a fermion of mass 400 GeV. Right: The corresponding lower limit on the relic abundance for a standard cosmology.[]{data-label="fig:simplim"}](relicplot "fig:"){width="47.50000%"}
Fig. \[fig:simplim\], shows the collider and direct detection limits, plotted as the upper bound on $yg^2$ as a function of the dark matter mass, and the translation of those upper limits into a lower limit on the relic abundance, assuming a standard cosmology, for the case when the single relevant fermion has a mass of 400 GeV. Despite the fact that the limits on the couplings are relatively weak, the conclusion is nonetheless that aside from a narrow region in the Higgs funnel region, additional interactions would be required to deplete the dark matter relic density enough to saturate the observed relic density.
Single Fermion with Scalar Mixing {#sec:singleHP}
---------------------------------
Building on the single fermion limit, we now allow for substantial $\lambda_P$ such that the radial modes of $H$ and $\Phi$ experience significant mixing, resulting in two CP even scalars we denote by $h$ and $h_2$. Describing this limit requires three additional free parameters, which we take to be the mass of the second scalar $m_{h_2}$, $\langle \Phi \rangle = v_\phi$, and the Higgs-scalar mixing angle $\alpha$.
For small $\alpha$, the form factors of Eqn. (\[eq:eftloop\]) are shifted: $$\begin{aligned}
A(p^2) &\rightarrow \left(1-\frac{\alpha^2}{2} \right) A(p^2) \\
B(p^2) &\rightarrow \left(1-\frac{\alpha^2}{2} \right) B(p^2) - 2 \alpha \frac{m_V^2}{v_\phi}
\end{aligned}$$ where the additional contribution is the tree level contribution to $B(p^2)$ from the induced $\Phi$ component in $h$. In addition to the shift in the effective $h$-$V$-$V$ coupling, the $h_2$ state acquires a coupling to the SM given by the corresponding SM Higgs coupling multiplied by $\alpha$.
![Exclusion regions on $yg^2$ for various parameters in the Higgs-Scalar mixing model. The left (right) two plots are for a scalar lighter (heavier) than the Higgs. The top (bottom) two plots are for a mixing angle of $\alpha = 0.1(0.01)$.[]{data-label="fig:wmixing"}](WMixg1247 "fig:"){width="49.50000%"} ![Exclusion regions on $yg^2$ for various parameters in the Higgs-Scalar mixing model. The left (right) two plots are for a scalar lighter (heavier) than the Higgs. The top (bottom) two plots are for a mixing angle of $\alpha = 0.1(0.01)$.[]{data-label="fig:wmixing"}](WMixg1242 "fig:"){width="49.50000%"}\
![Exclusion regions on $yg^2$ for various parameters in the Higgs-Scalar mixing model. The left (right) two plots are for a scalar lighter (heavier) than the Higgs. The top (bottom) two plots are for a mixing angle of $\alpha = 0.1(0.01)$.[]{data-label="fig:wmixing"}](WMixg01247 "fig:"){width="49.50000%"} ![Exclusion regions on $yg^2$ for various parameters in the Higgs-Scalar mixing model. The left (right) two plots are for a scalar lighter (heavier) than the Higgs. The top (bottom) two plots are for a mixing angle of $\alpha = 0.1(0.01)$.[]{data-label="fig:wmixing"}](WMixg01242 "fig:"){width="49.50000%"}
In Fig. \[fig:wmixing\], we indicate the bounds on $y g^2$ as a function of the vector mass for various benchmark values of the remaining free parameters as indicated, with shaded regions showing points excluded by the CMS invisible Higgs width bounds (green), and the LUX bounds on $\sigma_{\rm SI}$ (yellow). Note the appearance of “blind spots" in the direct detection plane coming from interference between loop- and tree-level contributions to the $h$-$V$-$V$ vertex and/or between $h$ and $h_2$ exchange [@Baek:2014jga]. Blue shading indicates regions where the dark matter is over-abundant in a standard cosmology. Unshaded regions are allowed by current data and do not over-close the Universe, with points close to the boundaries of the blue shading typically predicting a relic density close to the observed value. Such regions consistent with collider and direct searches are again typically in funnel regions for annihilation through $h$ and $h_2$, when it is heavier than $h$ itself. Additional parameter space also opens up for larger DM masses, where annihilation $VV \rightarrow h ~ h_2$ becomes viable.
Full Matter Content {#sec:fullF}
-------------------
As our final limit, we return to $\lambda_P \ll 1$ but allow for all of the fermions to have comparable masses. We consider three benchmark sets of masses and Yukawa interactions summarized in Table \[tab:bench\], which contains the model parameters associated with the fermion sector, $m$, $m_n$, $y_\psi$, and $y_\chi$, as well as the resulting spectrum of neutral state masses $M_N$ and the coefficient of the $h$-$\bar{N}_i$-$N_j$ coupling in the mass basis, $Y_{ij}$, with the mass eigenstates ordered as $M_{N_1}>M_{N_2}>M_{N_3}$. Table \[tab:benchgauge\] and Eqn. \[eq:gauge\], summarize the corresponding interaction of the gauge bosons with the new fermions. With these quantities fixed, we explore the plane of the U(1)$^\prime$ gauge coupling $g$ and the mass of the dark matter $m_V$.
The new electrically charged fermions may be pair produced or produced in association with a new neutral fermion at colliders. For the regime of interest, the charged fermions decay solely to one of the neutral fermions and a W boson. The charged states are sufficiently similar to charginos in the MSSM that chargino searches may be applied. LEP searches require the charged fermion to be heavier than 100 GeV [@Heister:2002mn; @Abdallah:2003gv; @Acciarri:2000wy; @Abbiendi:2002vz]. LHC searches find similar bounds which strengthen as the charged state becomes very long lived [@Aad:2014vma; @Khachatryan:2014mma]. The lightest charged state among our benchmarks is 300 GeV, which is safe from these constraints.
Some couplings are taken to be quite large to help highlight the features of this model in observables. In choosing such large values for the gauge and yukawa couplings, one may be concerned that perturbativity breaks down or that higher order corrections should not be ignored. The latter case may even reduce the relic abundance when properly taken into account, which would open up available parameter space. Alternately, smaller couplings may be chosen which would reduce the range of viable dark matter masses. However, neither case appreciably alter our conclusions.
$m$ $m_n$ $~~~y_\psi~~~$ $~y_\chi~$ $M_N$ (GeV) Y
-------------- ---------- ---------------- ------------------ ---------------------------------------------------- -----------------------------------------------------------------------------------------------------
800 GeV 250 GeV 1 $-0.5$ $\begin{bmatrix} 832 \\ 807 \\ 274 \end{bmatrix}$ $\begin{bmatrix} -0.25 & -0.04 & 0.71 \\ 0.04 & -0.06 & 0.26 \\ -0.71 & 0.26 & -0.19 \end{bmatrix}$
300 GeV 200 GeV 4 $-2$ $\begin{bmatrix} 848 \\ 810 \\ 238 \end{bmatrix}$ $\begin{bmatrix} -3.0 & -0.81 & -0.56 \\ 0.81 & -3.0 & -0.47 \\ 0.56 & -0.47 & -0.02 \end{bmatrix}$
500 GeV 1000 GeV 4 4 $\begin{bmatrix} 1770 \\ 500 \\ 265 \end{bmatrix}$ $\begin{bmatrix} -3.9 & 0 & 0.98 \\ 0~ & ~~~0~~~ & ~0 \\ -0.98 & 0 & -3.9 \end{bmatrix}$
: Benchmark parameter sets, and resulting neutral fermion masses and Higgs couplings.
\[tab:bench\]
$$\begin{aligned}
\label{eq:gauge}
\mathcal L_{gauge} &= e \left(\bar{E_1}\gamma^\mu E_1 - \bar{E_2}\gamma^\mu E_2\right)\left( A_\mu + \frac{(1 - 2s_w^2)}{2 c_w s_w} Z_\mu\right) ~ + ~ \frac{e}{2 c_w s_w} \bar{N_i}\gamma^\mu ~G^Z_{ij}~ N_j ~Z_\mu \nonumber\\ &+ \frac{e}{\sqrt{2}s_w} \left[ \left( \bar{E_1}\gamma^\mu ~G^{W1}_i~ N_i + \bar{N_i}\gamma^\mu ~G^{W2}_i~ E_2 \right) W_\mu^+ + h.c.\right] \nonumber \\ &+ g \left(\bar{E_i}\gamma^\mu E_i + \bar{N_i}\gamma^\mu N_i\right) V_\mu\end{aligned}$$
$m$ $m_n$ $~~y_\psi~~$ $~y_\chi~$ $G^Z$ $G^{W1}$ $G^{W2}~$
----------- ---------- -------------- ---------------- ---------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------
800 GeV 250 GeV 1 $-0.5$ $\begin{bmatrix} 0.01~\gamma^5 & -0.98 & -0.11 \\ -0.98 & 0.03~\gamma^5 & -0.17~\gamma^5 \\ -0.11 & -0.17~\gamma^5 & -0.04~\gamma^5 \end{bmatrix}$ $\begin{bmatrix} 0.70 \\ 0.70-0.01~\gamma^5 \\ 0.08+0.12~\gamma^5 \end{bmatrix}$ $\begin{bmatrix} 0.70 \\ -0.70-0.01~\gamma^5 \\ -0.08+0.12~\gamma^5 \end{bmatrix}$
300 GeV 200 GeV 4 $-2$ $\begin{bmatrix} 0.20~\gamma^5 & -0.36 & 0.69 \\ -0.36 & 0.38~\gamma^5 & -0.35~\gamma^5 \\ 0.69 & -0.35~\gamma^5 & -0.59~\gamma^5 \end{bmatrix}$ $\begin{bmatrix} -0.56+0.09~\gamma^5 \\ -0.26+0.36~\gamma^5 \\ 0.65+0.23~\gamma^5 \end{bmatrix}$ $\begin{bmatrix} -0.56-0.09~\gamma^5 \\ 0.26+0.36~\gamma^5 \\ -0.65+0.23~\gamma^5 \end{bmatrix}$
500 GeV 1000 GeV 4 4 $\begin{bmatrix} 0 & -0.61 & 0 \\ -0.61 & 0 & 0.79~\gamma^5 \\ 0 & 0.79~\gamma^5 & 0 \end{bmatrix}$ $\begin{bmatrix} -0.43 \\ -0.71 \\ 0.56~\gamma^5 \end{bmatrix}$ $\begin{bmatrix} 0.43 \\ -0.71 \\ -0.56~\gamma^5 \end{bmatrix}$
: Gauge couplings matrices defined in Eqn. \[eq:gauge\] represented in fermion mass basis.
\[tab:benchgauge\]
![Upper bound on the gauge coupling, $g$, for the three benchmark parameters. VBF Higgs collider constraints are in solid and direct detection constraints are dashed lines. Note that for the direct detection constraints we assume the local abundance of DM is $0.3$ GeV/cm$^3$ whereas the prediction from the model, for conventional thermal history, is often smaller, see Figure \[fig:fullrelic\].[]{data-label="fig:fulllim"}](Boxes_ddcol){width="90.00000%"}
In Fig. \[fig:fulllim\], we show upper bounds on $g$ as a function of the vector mass. We find that the collider and direct detection constraints are relatively weak, often less constraining than perturbativity. Despite the mass of the lightest neutral state being similar for all three benchmarks, constraints are significantly stronger for the second and third cases, where the Yukawa couplings are stronger. In terms of the dominant contribution to the effective $h$-$V$-$V$ coupling, in the first and third models, the lightest neutral state is the dominant contribution, whereas in the second benchmark model the lightest state has a small Yukawa coupling and is less important than the second lightest state, which has a much larger coupling.
![The vector relic abundance for the three benchmark parameters. The gauge coupling here is chosen to be $g=3.5$.[]{data-label="fig:fullrelic"}](Boxes_benchrelic){width="90.00000%"}
In Fig. \[fig:fullrelic\], we plot the relic abundance for the benchmark parameters with a large, fixed gauge coupling of $g = 3.5$, to make comparisons between the benchmarks more apparent. Note that for our second and third benchmark models, this value is mildly excluded by limits on the invisible width of the Higgs for $m_V \leq 60$ GeV. All benchmarks can be thermal relics when the vector can resonantly annihilate through a Higgs, causing the sharp dip at $m_V\sim m_h/2$. We also find that the second benchmark can attain a thermal relic for vector masses above 100 GeV, and third may be a thermal relic above 80 GeV. The success at larger DM masses is due to annihilation channels with two bosons in the final state. Of the three benchmarks, the second has the lightest charged states. This allows efficient annihilation through loops involving the charged fermions, such as those which result in the $WW$ and $ZZ$ final states. The third benchmark, also benefits from this with slightly heavier charged states. However, this case also has large Yukawas causing a marked drop in the relic abundance when DM is heavy enough to annihilate to two Higgs bosons.
Conclusion
==========
We have explored a simplified model in which the dark matter is a spin one vector particle which interacts with the Standard Model predominantly through Higgs exchange. Unlike the more usually considered Higgs portal based on the quartic interaction $\lambda_P$, we mediate the interaction radiatively, via a loop of heavy fermions charged under both the dark U(1)$^\prime$ as well as the SM electroweak interaction. By construction, the theory is anomaly free, has a heavy vector particle which is effectively stable, and leads to no large deviations in the properties of the SM Higgs. This last feature, together with the possibility to completely decouple the U(1)$^\prime$-breaking Higgs $\Phi$ from the SM are the primary features which distinguish the radiative model from the quartic-induced Higgs portal as far as dark matter phenomenology is concerned.
Of course, the UV structure of the radiative model is also far richer, with a family of electroweakly charged particles whose decays produce gauge bosons and missing momentum, a signature already under study in the context of the neutralinos and charginos of a supersymmetric theory. These states are the true avatars of the radiative Higgs portal. The thermal relic density suggests that their masses are at most around TeV, raising the hope that they could be found at the LHC run II or a future high energy collider.
AD is supported by the Fermilab Graduate Student Research Program in Theoretical Physics and in part by NSF Grant No. PHY-1316792. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. The work of TMPT is supported in part by NSF grant PHY-1316792 and by the University of California, Irvine through a Chancellor’s Fellowship.
$h$-$V$-$V$ Effective Vertex at One Loop {#app:tri}
========================================
Here we outline the details of the triangle loop calculation. The following results are for a single fermion species running in the loop. While the Higgs has off-diagonal couplings with the three neutral fermions in the mass basis, the vector only has diagonal couplings and thus only the diagonal Higgs interactions appear in the triangle diagrams. As a result, the functions $A$ and $B$ of Eq. (\[eq:eftloop\]) are the sum of the contributions from each individual fermion species.
Momenta are defined as in Fig. \[fig:loop\], with $k_1$ and $k_2$ the two (on-shell) vector momenta coming into the diagram, and $p = - (k_1 + k_2)$ the momentum incoming through the Higgs line. In addition to the diagram shown explicitly in Fig. \[fig:loop\], there is a second contribution related to it by $k_1\!\leftrightarrow \! k_2$, $\mu \leftrightarrow \nu$.
The contribution to the matrix element from a single fermion of mass $m$ and Yukawa coupling $y$ is given by: $$\mathcal M = g^2 \frac{y}{\sqrt{2}} \frac{i \pi^2 8m}{(2\pi)^4} \times \mathcal I^{\mu\nu}\left( k_1, k_2 \right) \times \epsilon_{\mu}(k_1) \epsilon_{\nu}(k_2)$$ where, $$\hspace*{-0.25cm}
\mathcal I^{\mu\nu}\left( k_1, k_2 \right) =
\frac{1}{8m} \int \frac{d^d k}{i\pi^2}\frac{{\rm Tr}\left[(\slashed{k}+m)\gamma^{\nu}(\slashed{k}+\slashed{k}_2+m)(\slashed{k}-\slashed{k}_1+m)\gamma^{\mu} \right]}
{(k^2-m^2)((k-k_1)^2-m^2)((k+k_2)^2-m^2)}
+(k_1,\mu\leftrightarrow k_2,\nu).$$ Evaluating the trace in the numerator and making use of the fact that $k_1 \cdot \epsilon(k_1) = k_2 \cdot \epsilon(k_2) = 0$ for on-shell vectors results in, $${\rm Tr}[...] = 4m \left(g^{\mu\nu}(m^2-k_1\!\cdot \! k_2-k^2) + 4 k^{\mu}k^{\nu} + k_1^{\nu} k_2^{\mu}\right)~.$$ After Passarino–Veltman decomposition [@Passarino:1978jh] we find, $$\begin{aligned}
\mathcal I^{\mu\nu}\left( k_1, k_2 \right)
&= \Big\{g^{\mu\nu}\Big[(4-d)C_{00} + m^2 C_0+ k_1\! \cdot \! k_2 (2C_{12}-C_0) - m_V^2(C_{11}+C_{22})\Big] \\ &
~~~~~~~~~~~ + k_1^{\nu}k_2^{\mu}\Big[C_0-4C_{12}\Big] \Big\},
\label{eq:PVform}
\end{aligned}$$ where the arguments of the $C$ functions are (uniformly) $C_0(k_1, k_2; m, m, m)$, etc.
Reducing to scalar functions results in a finite expression of the form, $$\mathcal I^{\mu\nu} = F_1(p^2, m)~ (k_1\! \cdot \! k_2 g^{\mu\nu} - k_1^{\nu}k_2^{\mu}) + F_2(p^2,m)~ g^{\mu\nu}$$ corresponding to an effective three-point vertex described by $$- \left(\frac{g^2ym}{2\sqrt{2}\pi^2}\right) \left( \frac{1}{4} F_1(p^2, m)~ hV^{\mu\nu}V_{\mu\nu} + \frac{1}{2} F_2(p^2, m)~ h V^{\mu}V_{\mu} \right)$$ where the form factors $F_1$ and $F_2$ are given by, $$\begin{aligned}
F_1(p^2, m) &= \frac{1}{2bm^2(b-4a)^2} \Big\{2 m^2 (b-2a)\Big[4a(a-1)+b(1+6a-b)\Big] C_0 \\ & \qquad\qquad\qquad\qquad- 2a(2a+b) \Delta B_0 + (b-2a)(b-4a)\Big\} \\
F_2(p^2, m) &= \frac{4a^2}{b(b-4a)^2}\left\{2(b-a) \Delta B_0 - 2 m^2 \Big[4a(a-1)+b(1-2a+b)\Big] C_0 + 4a-b \right\}
\end{aligned}$$ with, $$a\equiv\frac{m_V^2}{4m^2}, ~~~~~b\equiv\frac{p^2}{4m^2}.$$ The scalar integrals $C_0$ and $\Delta B_0$ can be expressed analytically as, $$\begin{aligned}
C_0&=\frac{1}{4 m^2 b \beta} \sum_{j,k=1}^2 \left[2 {\rm Li}_2\left(\frac{1+(-1)^j \beta}{1+(-1)^k {\rm X} \beta}\right) - {\rm Li}_2\left(\frac{(1+(-1)^j \beta)^2}{1+(-1)^k2 {\rm Y} \beta+\beta^2} \right) \right]~,\\
\Delta B_0 &\equiv B_0(m_V^2; m, m) - B_0(p^2; m, m)\\
&= 2\sqrt{\frac{1-b}{b}} \arctan\left[\sqrt{\frac{b}{1-b}}\right]-2\sqrt{\frac{1-a}{a}} \arctan\left[\sqrt{\frac{a}{1-a}}\right].
\end{aligned}$$ with $$\beta \equiv\sqrt{1-4\frac{a}{b}}, ~~~{\rm X}\equiv\sqrt{1-\frac{1}{a}}, ~~~{\rm Y}\equiv\sqrt{1-\frac{1}{b}}~.$$
As mentioned above, the coefficients $A$ and $B$ in Eq. (\[eq:eftloop\]) are given by the sum over the contributions from all three neutral mediator fermions, $$\begin{aligned}
\quad A(p^2) &= \sum_i \left(\frac{g^2 y_i m_i}{2\sqrt{2}\pi^2}\right) F_1(p^2, m_i), \\
\quad B(p^2) &= \sum_i \left(\frac{g^2 y_i m_i}{2\sqrt{2}\pi^2}\right) F_2(p^2, m_i).
\end{aligned}$$
In the $m_V\rightarrow 0$ limit the two form factors become, F\_1 &=& (1 + ) + (m\_V\^2) ,\
F\_2 &=& ( 4 -5+ )\
&& + (m\_V\^6) .
[^1]: It also provides a mechanism to stabilize the Higgs potential [@Duch:2015jta] and/or generate a first order electroweak phase transition [@Chao:2014ina].
|
---
abstract: 'Elastic scattering amplitudes dominated by the Pomeron singularity which obey the principal unitarity bounds at high energies are constructed and analyzed. Confronting the models of double and triple (at $t=0$) Pomeron pole (supplemented by some terms responsible for the low energy behaviour) with existing experimental data on $pp$ and $\bar pp$ total and differential cross sections at $\sqrt{s}\geq 5$ GeV and $|t|\leq 6$ GeV$^{2}$ we are able to tune the form of the Pomeron singularity. Actually the good agreement with those data is received for both models though the behaviour given by the dipole model is more preferable in some aspects. The predictions made for the LHC energy values display, however, the quite noticeable difference between the predictions of models at $t\approx -0.4$ GeV$^{2}$. Apparently the future results of TOTEM will be more conclusive to make a true choice.'
author:
- 'E. Martynov'
title: |
Elastic $pp$ and $\bar pp$ scattering\
in the models of unitarized pomeron
---
\[sec:level1\]Introduction
==========================
The forthcoming TOTEM experiment at LHC will provide us, in fact, with the first measurements of soft pomeron (strictly speaking pomeron and odderon) as the contributions of secondary reggeons are negligible at such high energies. Then obviously the precise measurement of $pp$ differential cross section makes it possible to discriminate the various pomeron models comparing their particular predictions. Certainly, such an analysis makes sense only if the same data set is used with the model parameters reliably fixed. For the time being there are three model types for elastic hadron scattering amplitudes which reproduce rising cross sections experimentally measured with a high precision.
- Models treating Pomeron (and odderon as well) as a simple pole in a complex momentum plane located righter of unity, $\alpha_{P}(0)=1+\varepsilon\approx 1.1$ [@PVL; @DGP]. In order to describe a dip-bump structure in differential cross section one should take into account the cuts in one or another form. Such a pomeron violates unitarity bound $\sigma_{t}(s)\leq Cln^{2}s$ at $s\to \infty$. However, the argument that unitarity corrections are important only at higher energies justifies this approach.
- Pomeron with $\alpha_{P}(0)>1$ is an input in some scheme of unitarization (for example, eikonal or quasieikonal [@eik], $U$-matrix models [@UM]). Having done the unitarization all such models give $\sigma_{t}(s)\propto ln^2{s}$, whereas other characteristic predictions depend on the concrete model.
- Another way to construct amplitude is just to take into account unitarity and analytical requirements from the beginning as well as experimental information on the cross sections (e.g. growth of total cross sections). Such a model we named in what follows as model of unitarized pomeron. Here most successful examples are tripole pomeron ($\sigma_{t}(s)\propto ln^{2}s$) [@LNGLN; @GLN; @AGN] and dipole pomeron ($\sigma_{t}(s)\propto lns$) [@JMS; @DGMP].
Within the third approach we consider the models of tripole and dipole pomeron. These models are most successful in a description of all data on the forward scattering data [@COMPETE].
As it was shown [@COMPETE; @DGMP2] the total cross sections of meson and nucleon interactions are described with the minimal $\chi^{2}$ in the dipole and tripole models in which forward scattering amplitudes are parameterized in explicit analytic form. This conclusion was confirmed by analysis applying the dispersion relations for real part of amplitudes [@clms-disp].
The elastic scattering at small-$|t|$ ($0.1\leq |t|\leq 0.5$ GeV$^{2}$) ($pp, \bar pp, \pi^{\pm}p$ and $K^{\pm}p$) was analyzed in detail [@CLM]. The particular model was considered as a combination of hard (with $\alpha_{h}(0)\approx 1.4$) and soft $\alpha_{s}(0)\approx 1.1$ pomeron contributions. It was noticed in [@CLMS] that additional hard pomeron essentially improves the description of the meson and nucleon data on parameter $\rho=\Re eA(s,0)/\Im mA(s,0)$ comparing with ordinary soft pomeron model. Extension of the model to higher $|t|$ can be done within some scheme of unitarization (e.g. eikonal, quasieikonal, $U-matrix$) taking into account pomeron rescatterings or cuts.
Here we focus on the dipole and tripole pomeron models. Without entering the details we note here these models describe the small-$|t|$ differential cross sections with the same level of precision ($\chi^{2}/dof\lesssim 1.05$, dof$\equiv$ degrees of freedom) as the model of [@CLM] did. The purpose of the present paper is to demonstrate the description of the data on elastic $pp$ and $\bar pp$ scattering at low and middle $t$ in the dipole and tripole pomeron models.
In Sec. \[sec:Gen.Con\] we remind the general restrictions on hardness of pomeron singularity and form of its trajectory at small $t$, imposed by unitarity bounds on cross sections. In Sec. \[sec:Dpar\] and in Sec. \[sec:Tpar\] parametrizations of $pp$ and $\bar pp$ elastic scattering amplitudes are presented dealing with dipole and tripole pomeron models, correspondingly. Results of least square analysis for both models as well as their comparison are given in the Sec. \[sec:expdata\].
General constraints {#sec:Gen.Con}
===================
Let us reiterate here that the model with $\sigma_{t}(s)\propto
ln^{2}s$ is not compatible with a linear pomeron trajectory having the intercept 1. Indeed, let us assume that $$\alpha_{P}(t)=1+\alpha_{P}'t$$ and the partial wave amplitude develops the form $$\label{eq:j-gen}
\varphi (j,t)=\eta(j)\frac{\beta(j,t)}{\left
[j-1-\alpha_{P}'t\right]^{n}}\approx \frac{i\beta(1,t)}{\left
[j-1-\alpha_{P}'t\right]^{n}},$$ $$\eta(j)=\frac{1+e^{-i\pi
j}}{-\sin\pi j}.$$
In ($s,t$)-representation amplitude $\varphi(j,t)$ is transformed to $$\label{eq:s,t-gen}
a(s,t)=\frac{1}{2\pi i}\int dj \varphi (j,t)e^{\xi (j-1)}, \quad
\xi=ln(s/s_{0}).$$ Then, we have pomeron contribution at large $s$ as $$\label{eq:s,t-gen1}
a(s,t)\approx -g(t)[ln(-is/s_{0})]^{n-1}(-is/s_{0})^{\alpha'_{P}t}$$ where $$g(t)=\beta (t)/\\sin(\pi\alpha_{P}(t)/2).$$ If as usually $g(t)=g\exp(bt)$ then we obtain $$\begin{aligned}
\label{eq:sigma-0}
\sigma_{t}(s)&&\propto ln^{n-1}s,\nonumber\\
\sigma_{el}(s)&&\propto
\frac{1}{s^{2}}\int \limits_{-\infty}^{0}dt|a(s,t)|^{2}\propto
ln^{2n-3}s.\end{aligned}$$ According to the obvious inequality, $$\label{eq:unitbound}
\sigma_{el}(s)\leq \sigma_{t}(s)$$ we have $$\label{eq:nbound}
2n-3\leq n-1 \qquad \Rightarrow \qquad n\leq 2.$$
Thus we come to the conclusion that a model with $\sigma_{t}(s)\propto
ln^{2}s$ is incompatible with a linear pomeron trajectory. In other words the partial amplitude Eq. (\[eq:j-gen\]) with $n=3$ (but used in some papers) in principle is incorrect.
If $n=1$ we have a simple $j$-pole leading to constant total cross section and vanishing elastic cross section. However such a behaviour of the cross sections is not supported by experimental data.
If $n=2$ we have the model of dipole pomeron ($\sigma_{t}(s)\propto
ln(s)$) and would like to emphasize that double $j$-pole is the maximal singularity of partial amplitude settled by unitarity bound (\[eq:unitbound\]) if its trajectory is linear at $t\approx
0$.
Thus, constructing the model leading to cross section which increases faster than $ln(s)$, we need to consider a more complicated case: $$\label{eq:j-gen**2}
\varphi (j,t)=\eta(j)\frac{\beta(j,t)}{\left
[j-1+k(-t)^{1/\mu}\right]^{n}}\approx \frac{i\beta(1,t)}{\left
[j-1+k(-t)^{1/\mu}\right]^{n}}.$$ Making use of the same arguments as above, we obtain $$\label{eq:nmbound}
\sigma_{t}(s)\propto ln^{n-1}s,$$ $$\sigma_{el}(s)\propto
ln^{2n-2-\mu}s \qquad \mbox{and} \qquad \mu\geq n-1. \nonumber$$ However in this case amplitude $a(s,t)$ has a branch point at $t=0$ which is forbidden by analyticity.
A proper form of amplitude leading to $t_{eff}$ [^1] decreasing faster than $1/lns$ (it is necessary for $\sigma_{t}$ rising faster than $lns$) is the following $$\label{eq:j-gencorrect}
\varphi (j,t)=\eta(j)\frac{\beta(j,t)}{\left
[(j-1)^{m}-kt\right]^{n}}.$$ Now we have $m$ branch points colliding at $t=0$ in $j$-plane and creating the pole of order $mn$ at $j=1$ (but there is no branch point in $t$ at $t=0$). At the same time $t_{eff}\propto 1/ln^{m}s$ and from the $\sigma_{el}\propto ln^{2mn-2-m}s\leq \sigma_{t}\propto ln^{mn-1}s\leq
ln^{2}s$ one can obtain $$\label{eq:mnbound}
\left \{
\begin{array}{ll}
mn &\leq m+1, \\
mn &\leq 3.
\end{array}
\right .$$ If $\sigma_{el}\propto \sigma_{t}$ then $n=1+\frac{1}{m}$. Furthermore, if $\sigma_{t}\propto lns$ then $m=1$ and $n=2$ what corresponds just to the dipole pomeron model. In the tripole pomeron model $m=2$ and $n=3/2$ what means $\sigma_{t}\propto ln^{2}s$.
Dipole parametrizations {#sec:Dpar}
=======================
The dominating term at high energy in this model is double pole $$\label{eq:dipole_d}
\varphi_{d}(j,t)\propto \frac{1}{(j-1-\alpha_{d}'t)^{2}}.$$ Apparently in accordance with the inequalities (\[eq:mnbound\]) the double pole obeys the unitarity limit for linear pomeron trajectory ($m=1$). Adding to the partial amplitude less singular term (simple pole with a trajectory having intercept $\alpha(0)=1$ and a different slope $\alpha'$) we obtain dipole pomeron model in the form $$\label{eq:dipole_c}
\varphi(j,t)=\eta(j)\frac{\beta_{d}(t)}
{(j-1-\alpha_{d}'t)^{2}}+\eta(j)\frac{\beta_{s}(t)}
{j-1-\alpha_{s}'t}.$$ It can be rewritten in $(s,t)$ representation as$$\begin{aligned}
\label{eq:dipole_st}
a(s,t)=&&g_{d}ln(-iz)(-iz)^{1+\alpha_{d}'t}\exp(b_{d}t)\nonumber\\&&+
g_{s}(-iz)^{1+\alpha_{s}'t}\exp(b_{s}t),\end{aligned}$$ where variable $z$ is proportional to cosine of scattering angle in $t$-channel $$\label{eq:zvar}
z=(t+2(s-2m_{p}^{2}))/z_{0}, \qquad z_{0}=1 {\rm GeV}^{2}.$$ Generally, the form factors (or residues) $\beta(t)$ may be chosen in various forms (e.g. exponential, factorized powers [*etc.*]{}). However we consider the simplest exponential ones.
Let us consider two effective reggeons: crossing-even, $R_{+}(s,t)$, and crossing-odd, $R_{-}(s,t)$) instead of four contributions - $f, \omega$ and $\rho, a_{2}$ (the latter two reggeons are of less importance at high energy). We take into account their contribution in the standard form. However, we insert additional factor $Z_{R}(t)$ that changes a sign at some $t$ [^2]. $$\label{eq:reggeons}
R(s,t)=\eta_{R}g_{R}Z_{R}(t)(-iz)^{\alpha_{R}(t)}\exp(b_{R}t),$$ where $\eta_{R}=-1/\sin(0.5\pi \alpha_{+}(0))$ for $R_{+}$-reggeon and $\eta_{R}=i/\cos(0.5\pi \alpha_{-}(0))$ for $R_{-}$-reggeon. Obviously these terms are very close to $f$- and $\omega$-reggeons, respectively. There are some arguments [@CLM] to use the factors $Z_{R}(t)$ in the form: $$\label{eq:Rzero}
Z_{R}(t)=\frac{\tanh(1+t/t_{R})}{\tanh(1)}.$$ Going to extend wide regions of $s$ ($\sqrt{s}\geq 5$ GeV) and $t$ ($0.1\leq |t|\leq 6$ GeV$^{2}$) [^3] we certainly need a few extra terms in amplitude to reach a good fit to the data. First of all it concerns the odderon contribution. The existing data on total cross section and parameters $\rho=\Re ea(s,0)/\Im ma(s,0)$, as well known, do not show any visible odderon contribution. However, it appears definitely to provide the difference of $pp$ and $\bar pp$ differential cross sections at $\sqrt{s}=53$ GeV and $t$ around the dip. So, we add the odderon contribution vanishing at $t=0$ $$\begin{aligned}
\label{eq:odd_d}
%\begin{array}
{\cal O}(s,t)=t^{2}zZ_{R_{-}}(t)\biggl\{&& \!\!\!o_{1}ln^{2}(-iz)\exp(b_{o1}t)\nonumber\\
&&\!\!\!+o_{2}ln(-iz)\exp(b_{o2}t)\nonumber\\
&&\!\!\!+o_{3}\exp(b_{o3}t)\biggr\}(-iz)^{1+\alpha_{o}'t}
%\end{array}\end{aligned}$$ The term $\propto ln^{2}(s)$ in Eq. \[eq:odd\_d\] does not violate unitarity restriction $\sigma_{el}(s)\leq \sigma_{t}(s)$ at very large $s$ due to presence of factor $t^{2}$ (in the dipole model $t_{eff}\sim 1/lns $, therefore $\sigma_{el}\propto t^{3}_{eff}ln^{4}s\propto lns$).
At high energy and at $t=0$ two main rescattering terms of dipole pomeron (or cut terms) have the same form as the input amplitude - double pole plus simple pole. It means that comparing the model with experimental data we are not able to distinguish unambiguously input terms and cuts. Then as result, at $t=0$ one may use the input amplitude only. At $t\neq 0$ the situation occurs more complicated because the slopes of trajectories in the cut terms are different from the input one. These terms are important at large $|t|$ but, in fact, they are already taken into account at $t=0$.
Keeping in mind the above arguments and preserving a good description of the data at $t=0$ we take pomeron, pomeron-pomeron and pomeron-reggeons cuts vanishing at $t=0$. Certainly they are not “genuine” rescatterings but mimic them quite efficiently at $t\neq 0$. Thus we write down:
the pomeron contribution $$\label{eq:pom}
P(s,t)=-g_{P}(-iz)^{1+\alpha_{P}'t}\left[\exp(b_{P1}t)-\exp(b_{P2}t)\right],$$ the pomeron-pomeron cut $$\label{eq:Pcut}
C_{P}(s,t)=-\frac{t}{ln(-iz)}g_{PP}(-iz)^{1+\alpha_{P}'t/2}\exp(b_{PP}t),$$ the pomeron-even reggeon cut $$\begin{aligned}
\label{eq:P+cut}
C_{R_{+}}(s,t)=&&-\frac{tZ_{R_{+}}(t)}{ln(-iz)}\eta_{R_{+}}
g_{P+}Z_{R_{+}}(t)\nonumber \\&&\times(-iz)^{\alpha_{+}(0)+\alpha'_{P+}t}\exp(b_{P+}t),\end{aligned}$$ where $$\label{eq:slopeC+}
\alpha'_{P+}=\frac{\alpha_{P}'\alpha_{R_{+}}'}{\alpha_{P}'+\alpha_{R_{+}}'},$$ and the pomeron-odd reggeon cut $$\begin{aligned}
\label{eq:P-cut}
C_{R_{-}}(s,t)=&&-i\frac{tZ_{R_{-}}(t)}{ln(-iz)}\eta_{R_{-}}
g_{P-}Z_{R_{-}}(t)\nonumber \\&&\times(-iz)^{\alpha_{-}(0)+\alpha'_{P-}t}\exp(b_{P-}t),\end{aligned}$$ $$\label{eq:slopeC-}
\alpha'_{P-}=\frac{\alpha_{P}'\alpha_{R_{-}}'}{\alpha_{P}'+\alpha_{R_{-}}'}.$$
Tripole pomeron model {#sec:Tpar}
=====================
As it follows from Eq.(\[eq:mnbound\]) for the dominating contribution in a tripole pomeron model with $\sigma_{t}(s)\propto
ln^{2}(s)$, i.e. $n=2$, $m=3/2$, we should take $$\label{eq:tripole1}
\varphi_{1}(j,t)=\eta(j)\frac{\beta_{1}(j,t)}{\left
[(j-1)^{2}-kt\right]^{3/2}}.$$ It seems to be natural to write the subleading terms as the following $$\label{eq:tripole2}
\varphi_{2}(j,t)=\eta(j)\frac{\beta_{2}(j,t)}{\left
[(j-1)^{2}-kt\right]},$$ $$\label{eq:tripole3}
\varphi_{3}(j,t)=\eta(j)\frac{\beta_{3}(j,t)}{\left
[(j-1)^{2}-kt\right]^{1/2}}.$$ Then the amplitude has a form $$\label{eq:j-tripole}
\varphi
(j,t)=\varphi_{1}(j,t)+\varphi_{2}(j,t)+\varphi_{3}(j,t)+R(j,t),$$ where $R(j,t)$ means the contribution of other reggeons and possible cuts (which are important at low energies).
Taking into account that $$\label{eq:besselgen}
\int \limits_{0}^{\infty} dx x^{\alpha-1}e^{-\omega
x}\textit{J}_{\nu}(\omega_{0})=I_{\nu}^{\alpha}$$ where $$\begin{array}{ll}
{I_{\nu}^{\nu+1}=\frac{(2\omega_{0})^{\nu}}{\sqrt{\pi}}
\frac{\Gamma(\nu+1/2)}
{(\omega^{2}+\omega_{0}^{2})^{\nu+1/2}}},\\ \\
{I_{\nu}^{\nu+2}=2\omega\frac{(2\omega_{0})^{\nu}}{\sqrt{\pi}}
\frac{\Gamma(\nu+3/2)}
{(\omega^{2}+\omega_{0}^{2})^{\nu+3/2}}},
\end{array}$$ one can find $$\label{eq:phi1}
\frac{1}{(\omega^{2}+\omega_{0}^{2})^{3/2}}=\frac{1}{2\omega_{0}}
\int\limits_{0}^{\infty}
dx xe^{-x\omega}J_{1}(\omega_{0}x),$$ $$\label{eq:phi2}
\frac{1}{\omega^{2}+\omega_{0}^{2}}=\frac{1}{\omega_{0}}\int\limits_{0}^{\infty}
dx e^{-x\omega}\sin(x\omega_{0})$$ and $$\label{eq:phi3}
\frac{1}{(\omega^{2}+\omega_{0}^{2})^{1/2}}=\int\limits_{0}^{\infty}
dx e^{-x\omega}J_{0}(\omega_{0}x).$$ Thus tripole amplitude with the subleading terms can be presented as $$\begin{aligned}
\label{eq:trpole-st}
%\begin{array}{ll}
a_{tr}(s,t)=iz\biggl\{
%\dst
&&g_{+1}\exp(b_{+1}t)ln(-iz)\frac{2J_{1}(\xi_{+}\tau_{+})}{\tau_{+}}\nonumber \\
&&+g_{+2}\frac{\sin(\xi_{+}\tau_{+})}{\tau_{+}}\exp(b_{+2}t)\nonumber\\
&&+g_{+3}J_{0}(\xi_{+}\tau_{+})\exp(b_{+3}t)\biggr \}
%\end{array}\end{aligned}$$ where $\xi_{+}=ln(-iz)+\lambda_{+}$, $z$ is defined by Exp.(\[eq:zvar\]), and $\tau_{+}=r_{+}\sqrt{-t/t_{0}}$, $t_{0}=1$ GeV$^{2}$, $r_{+}$ is a constant.
Similar expression for odderon contribution (but introducing the factors $t$ and $Z_{R_{-}}(t)$) is given by $$\begin{aligned}
\label{eq:odd_tr}
%\begin{array}{ll}
{\cal O}(s,t)=ztZ_{R_{-}}(t)\biggl \{
&&g_{-1}ln(-iz)
\frac{2J_{1}(\xi_{-}\tau_{-})}{\tau_{-}}\exp(b_{-1}t)
\nonumber\\&&+g_{-2}
\frac{\sin(\xi_{-}\tau_{-})}{\tau_{-}}\exp(b_{-2}t)\nonumber\\&&+
g_{-3}J_{0}(\xi_{-}\tau_{-})\exp(b_{-3}t)\biggr \}
%\end{array}\end{aligned}$$ where $\xi_{-}=ln(-iz)+\lambda_{-}$ and $\tau_{-}=r_{-}\sqrt{-t/t_{0}}$.
Again, similarly to the dipole model we add the “soft” pomeron $$\label{eq:softpom}
P(s,t)=-g_{P}(-iz)^{1+\alpha_{P}'t}\exp(b_{P}t),$$ the reggeon and cut contributions which are of the same form as in dipole pomeron model Eqs.(\[eq:Pcut\],\[eq:P+cut\],\[eq:P-cut\]).
[**AGLN-model.**]{} Let us give a few comments about another version of tripole pomeron model presented in the papers [@GLN; @AGN].
1\. If $\xi=ln(-is/s_{0}), s_{0}=1 {\rm GeV}^{2}$, then the first pomeron term in [@GLN; @AGN] is identical to the term in Eq. (\[eq:trpole-st\]) while for the second and third terms authors use $$g_{2}(t)\xi J_{0}(\xi\tau)$$ and $$g_{3}(t)[J_{0}(\xi\tau_{+})-\xi_{0}\tau_{+} J_{1}(\xi\tau_{+})]$$ which originated from the partial amplitudes $$\varphi(j,t)=\eta(j)g_{2}(t)\frac{2(j-1)}{\left
[(j-1)^{2}-kt\right]^{3/2}},$$ and $$\varphi(j,t)=\eta(j)g_{2}(t)\frac{(j-1)^{2}+kt}{\left
[(j-1)^{2}-kt\right]^{3/2}},$$ respectively.
2\. They used another form of odderon terms. The maximal odderon contribution in the form $$\begin{aligned}
\label{eq:maxodd}
{\cal
O}_{m}(s,t)&&=g_{-1}ln^{2}(-iz)\frac{\sin(\xi\tau_{-})}{\tau_{-}}\exp(b_{-1}t)
\nonumber\\&&+
g_{-2}ln(-iz)\cos(\xi\tau_{-})\exp(b_{-2}t)\nonumber\\&&+g_{-3}\exp(b_{-3}t)\end{aligned}$$ as well as a simple pole odderon and odderon-pomeron cut are also taken into account.
3\. Omitting the details we note that because of the chosen form of signature factors the AGLN amplitude has pole in physical region at $t=-1/\alpha'=-4$ GeV$^{2}$. This feature of the model restricts its applicability region. AGLN amplitude has similar poles even at lower values of $|t|$ in the reggeon terms. Thus the model requires a modification to describe wider region of $t$ than was considered in [@AGN], namely $|t|\leq 2.6$ GeV$^{2}$.
4\. Clearly this model leads to the unreasonable intercept value for the crossing-odd reggeon, $\alpha_{-}(0)=0.34$. It is in strong contradiction with the values known from meson resonance spectroscopy data. One could expect it close to the intercept of $\omega$-trajectory, $\alpha_{\omega}(0)\approx 0.43 - 0.46$ [@DGMP2].
Nevertheless, in the Section \[sec:expdata\] we demonstrate the curves for differential cross sections obtained in AGLN model in comparison with the results of our dipole and tripole models at energies available and future LHC.
Comparison with experimental data {#sec:expdata}
=================================
Total cross sections
--------------------
Analyzing the $pp$ and $\bar pp$ data we keep in mind a further extension of the models to elastic $\pi^{\pm} p$ and $K\pm p$ scattering, which are quite precisely measured. One important point should be underlined from the beginning. Fitting the $pp$ and $\bar pp$ data on $\sigma_{t}$ and $\rho$ gives the set of parameters which is essentially different from those derived from the fit of all ($p, \bar p, \pi$- and $K$-meson) the data.
Following this procedure at the first stage we determine all parameters which control the amplitudes at $t=0$. We use the standard data set for the $\pi^{\pm} p$ and $K\pm p$ total cross sections and the ratios $\rho$ (at 5 GeV$\leq\sqrt{s}<$2000 GeV) [@PDG] to find intercepts of $C_{\pm}$-reggeons and couplings of the reggeon and pomeron exchanges. There are 542 experimental points in the region under consideration (see Table \[tab:t0-1\]).
An extension of the $pp\rightarrow pp$ and $\bar pp\rightarrow \bar pp$ dipole and tripole amplitudes to $\pi^{\pm} p$ and $K^{\pm} p$ elastic scattering is quite straight forward. All the couplings are various in these amplitudes at $t=0$ but the odderon does not contribute to the $\pi p$ and $Kp$ amplitudes. In the simplest unitarization schemes (eikonal, $U$-matrix) all total cross sections at asymptotically high energies have the universal behaviour, $\sigma_{t}(s)\to \sigma_{0}\log^{2}(s_{ap}/s_{0})$, where $\sigma_{0}$ is independent of the initial particles. Today the data available support this conclusion and advocates putting the same couplings $g_{+1}^{p}=g_{+1}^{\pi}=g_{+1}^{K}$ for the leading pomeron terms in all amplitudes. Besides, in order to avoid uncertainty at $t=0$ (constant contributions to total cross sections come additively from the third term of Eq.(\[eq:trpole-st\]) and from “soft” pomeron, Eq.(\[eq:softpom\])) we substitute couplings $g_{+3}$ for $g_{+3}-g_{P}$ in Eq.(\[eq:trpole-st\]). As a result we have energy independent contribution to the total cross from $g_{+3}$ only.
The following normalization of $ab\rightarrow ab$ amplitude is used $$\label{eq:norm}
\sigma_t=\frac{1}{s_{ab}}\Im mA(s,0), \qquad
\frac{d\sigma}{dt}=\frac{1}{16\pi s_{ab}^{2}}|A(s,t)|^{2}$$ where $$s_{ab}=\sqrt{(s-m_a^2-m_b^2)^2-4m_a^2m_b^2}=2p_a^{lab}\sqrt{s}$$ and $p_a^{lab}$ is the momentum of hadron $a$ in laboratory system of $b$.
The details of the fit at $t=0$ are presented in the Tables \[tab:t0-1\] and \[tab:t0-2\].
-------------------------- ---------------- -------------- ---------------
quantity number of data Dipole model Tripole model
$\sigma_{t}^{pp}$ 104 0.88260E+00 0.87055E+00
$\sigma_{t}^{\bar pp}$ 59 0.95280E+00 0.96273E+00
$\sigma_{t}^{\pi^{+} p}$ 50 0.66216E+00 0.66792E+00
$\sigma_{t}^{\pi^{-}p}$ 95 0.10023E+01 0.99864E+00
$\sigma_{t}^{K^{+}p}$ 40 0.72357E+00 0.72104E+00
$\sigma_{t}^{K^{-}p}$ 63 0.61392E+00 0.60883E+00
$\rho^{pp}$ 64 0.16612E+01 0.16965E+01
$\rho^{\bar pp}$ 11 0.40392E+00 0.40675E+00
$\rho^{\pi^{+} p}$ 8 0.15107E+01 0.15036E+01
$\rho^{\pi^{-}p}$ 30 0.12560E+01 0.12122E+01
$\rho^{K^{+}p}$ 10 0.10869E+01 0.10016E+01
$\rho^{K^{-}p}$ 8 0.12185E+01 0.11611E+01
Total 542 0.99450E+00 0.99345E+00
-------------------------- ---------------- -------------- ---------------
: Quality of the fit to $\sigma_{t}$ and $\rho$[]{data-label="tab:t0-1"}
------------------ -------------- ------------- ------------------ ------------- -------------
[parameter]{} value error [parameter]{} value error
$\alpha_{R+}(0)$ 0.80846E+00 0.36035E-02 $\alpha_{R+}(0)$ 0.71947E+00 0.18496E-02
$\alpha_{R-}(0)$ 0.46505E+00 0.91416E-02 $\alpha_{R-}(0)$ 0.46356E+00 0.73746E-02
$g_{d}^{p}$ 0.89435E+01 0.19499E+00 $g_{+1}^{p}$ 0.15330E+02 0.31619E+00
$g_{s}^{p}$ -0.52159E+02 0.30078E+01 $g_{+2}^{p}$ 0.19153E+01 0.38589E-01
$g_{R+}^{p}$ 0.15857E+03 0.30846E+01 $g_{+3}^{p}$ 0.20672E+00 0.26747E-02
$g_{R-}^{p}$ 0.58961E+02 0.28775E+01 $g_{R+}^{p}$ 0.96906E+02 0.84678E+00
$g_{d}^{\pi}$ 0.70477E+01 0.22719E+00 $g_{R-}^{p}$ 0.59294E+02 0.23228E+01
$g_{s}^{\pi}$ -0.45720E+02 0.29022E+01 $g_{+1}^{\pi}$ 0.69901E+01 0.18396E+00
$g_{R+}^{\pi}$ 0.10691E+03 0.33590E+01 $g_{+2}^{\pi}$ 0.10106E+01 0.27123E-01
$g_{R-}^{\pi}$ 0.10710E+02 0.52507E+00 $g_{R+}^{\pi}$ 0.56296E+02 0.45360E+00
$g_{d}^{K}$ 0.54351E+01 0.29205E+00 $g_{R-}^{\pi}$ 0.10784E+02 0.43687E+00
$g_{s}^{K}$ -0.29703E+02 0.33234E+01 $g_{+1}^{K}$ 0.12186E+02 0.22424E+00
$g_{R+}^{K}$ 0.70785E+02 0.43792E+01 $g_{+2}^{K}$ 0.14683E+00 0.30439E-01
$g_{R-}^{K}$ 0.23674E+02 0.11159E+01 $g_{R+}^{K}$ 0.28730E+02 0.51047E+00
$g_{R-}^{K}$ 0.23831E+02 0.91697E+00
------------------ -------------- ------------- ------------------ ------------- -------------
Differential cross sections
---------------------------
At the second stage of the fitting procedure we fix all the intercept and coupling values obtained at the first stage. The other parameters are determined by fitting the $d\sigma/dt$ data in the region $$\label{eq:t-region}
0.1 {\rm GeV}^{2}\leq |t|\leq 6 {\rm GeV}^{2}, \quad \sqrt{s}\geq 5
{\rm GeV}.$$
The measurements of the differential elastic cross sections were very intensive for the last 40 years. Fortunately, most of them have been collected in the Durham Data Base [@DDB]. However, there are 80 papers, with different conventions, and various units. The complete list of the references is given by [@CLM]. We have uniformly formatted them, found and corrected some errors in the sets and gave a detailed description of a full set which contains about 10000 points. Analyzing each subset of these data we have payed [@CLM] particular attention to the data at small $t$. Some of subsets which are in strong disagreement with the rest of the dataset were excluded from the fit.
A similar work has been done for the data at $|t|>0.7$ GeV$^{2}$. We have found out and corrected some mistakes in the data base. Furtermore, we excluded from the final dataset the subsets [@BRUNETON] at $\sqrt{s}=9.235$ GeV, [@CONETTI] at $\sqrt{s}=19.47$ and $27.43$ GeV from $pp$ data and [@BOGOLYUBSKY] at $\sqrt{s}=7.875$ GeV, [@AKERLOF] at $\sqrt{s}=9.778$ GeV from $\bar pp$ data because they strongly contradict the bulk of data. Thus, the considered models were fitted to 2532 points of $d\sigma/dt$ in the region Eq.(\[eq:t-region\]). The results are given in the Table \[tab:chi2-t\] for a quality of fitting and in the Table \[tab:par-t\] for the fitting parameters.
In the Figs. \[fig:paplow\] - \[fig:lhc\] we show experimental data at some energies and theoretical curves obtained in three models: AGLN [@AGN], Dipole and Tripole. As to the AGLN model, we would like to emphasize that the corresponding curves were calculated at the parameters given in [@AGN]. However, in contrast to Dipole and Tripole models AGLN model was fitted to differential cross sections at $\sqrt{s}>9.7$ GeV and $|t|<2.6$ GeV$^{2}$ but not with a complete dataset. Therefore a disagreement between curves and data behaviours at lowest energies is not surprising in the given model. The AGLN model works well at high energies.
![$\bar pp$ at low energies[]{data-label="fig:paplow"}](fig1.eps)
![$ pp$ at low energies[]{data-label="fig:pplow"}](fig2.eps)
![$\bar pp$ at high energies[]{data-label="fig:paphigh"}](fig3.eps)
![$pp$ at high energies[]{data-label="fig:pphigh"}](fig4.eps)
![Predictions for LHC energy[]{data-label="fig:lhc"}](fig5.eps)
------------------------ ----------------- -------------- ---------------
Number of
points, $N_{p}$ Dipole model Tripole model
$d\sigma^{pp}/dt$ 1857 0.15122E+01 0.18153E+01
$d\sigma^{\bar pp}/dt$ 675 0.14183E+01 0.16697E+01
------------------------ ----------------- -------------- ---------------
: Quality of the fit to $d\sigma/dt$[]{data-label="tab:chi2-t"}
---------------- -------------- ------------- ---------------- -------------- -------------
parameter value error parameter value error
$\alpha_{d}'$ 0.30631E+00 0.16923E-02 $r_{+}$ 0.25417E+00 0.33181E-02
$\alpha_{s}'$ 0.28069E+00 0.19026E-03 $\lambda_{+}$ 0.11575E+01 0.14824E+00
$b_{d}$ 0.38675E+01 0.22767E-01 $b_{+1}$ 0.34583E+01 0.37982E-01
$b_{s}$ 0.55679E+00 0.14694E-02 $b_{+2}$ 0.19091E+01 0.30598E-01
$\alpha_{R+}'$ 0.82000E+00 fixed $b_{+3}$ 0.45970E+00 0.29750E-02
$b_{R+}$ 0.29226E+01 0.30019E-01 $\alpha_{R+}'$ 0.82000E+00 fixed
$t_{R+}$ 0.48852E+00 0.26683E-02 $b_{R+}$ 0.10668E+01 0.30350E-01
$\alpha_{R-}'$ 0.91000E+00 fixed $t_{R+}$ 0.54237E+00 0.13817E-01
$b_{R-}$ 0.15201E+01 0.68671E-01 $\alpha_{R-}'$ 0.91000E+00 fixed
$t_{R-}$ 0.14497E+00 0.24811E-02 $b_{R-}$ 0.61435E-01 0.20044E-01
$o_{1}$ 0.30738E+00 0.31368E-02 $t_{R-}$ 0.15755E+00 0.26068E-02
$o_{2}$ -0.63119E+01 0.55282E-01 $r_{-}$ 0.78807E-01 0.60563E-02
$o_{3}$ 0.13456E+00 0.21551E-02 $\lambda_{-}$ 0.16281E+02 0.19020E+01
$\alpha_{o}'$ 0.21810E-01 0.70324E-03 $o_{1}$ -0.56075E-01 0.51702E-02
$b_{o1}$ 0.39317E+01 0.81697E-02 $o_{2}$ 0.17372E+01 0.17893E+00
$b_{o2}$ 0.45007E+01 0.76861E-02 $o_{3}$ -0.61193E+02 0.36330E+01
$b_{o3}$ 0.12947E+01 0.76773E-02 $b_{o1}$ 0.12038E+01 0.26425E-01
$g_{P}$ 0.58961E+02 0.98576E-01 $b_{o2}$ 0.15152E+01 0.31722E-01
$\alpha_{P}'$ 0.30696E+00 0.20475E-03 $b_{o3}$ 0.26331E+01 0.78349E-01
$b_{P1}$ 0.54894E+00 0.15036E-02 $g_{P}$ 0.16042E+02 13856E-01
$b_{P2}$ 0.59365E+01 0.34863E-01 $\alpha_{P}'$ 0.36060E+00 0.76335E-02
$g_{PP}$ -0.39324E+02 0.36883E+00 $b_{P}$ 0.14662E+01 0.29152E-01
$b_{PP}$ 0.11828E+01 0.44025E-02 $g_{PP}$ 0.91195E+01 0.84044E+00
$g_{P+}$ -0.22656E+03 0.27529E+01 $b_{PP}$ 0.44977E+00 0.46404E-01
$b_{P+}$ 0.17522E+01 0.98459E-02 $g_{P+}$ 0.11772E+02 0.57753E+00
$g_{P-}$ -0.15255E+02 0.28213E+00 $b_{P+}$ 0.81585E-01 0.28185E-01
$b_{P-}$ 0.24068E-01 0.61336E-02 $g_{P-}$ 0.81908E+01 0.88408E+00
$b_{P-}$ -0.79115E-01 0.45833E-01
---------------- -------------- ------------- ---------------- -------------- -------------
Conclusion
==========
In this paper we compare three unitarized models of elastic scattering amplitude fitting the Dipole and Tripole models to all existing data. We emphasize that the amplitude leading to the behaviour of $\sigma_{t}\propto \ln^{2}s$ should be parameterized with a special care of the unitarity and analyticity restrictions on properties of the leading partial wave singularity.
The Figures and Tables demonstrate good description of the data within the considered models. However the obtained $\chi^{2}$ (Table \[tab:chi2-t\]) hints that the Dipole pomeron model looks more preferable.
We believe the most interesting and instructive result for further search of more realistic model is shown in Fig. \[fig:lhc\]. Our predictions of the compared models (together with AGLN model) for $pp$ cross section at LHC energy are crucially different at $|t|$ around 0.3 - 0.5 GeV$^{2}$. Certainly the future TOTEM measurement will allow to distinguish between three considered models.
I would like to thank Prof. B. Nicolescu and Dr. J.R. Cudell for many useful discussions.
The work was supported partially by the Ukrainian Fund of Fundamental Researches.
[99]{}
A. Donnachie and P.V. Landshoff, Phys. Lett. [**B296**]{} 227 (1992) ; P.V. Landshoff, arXiv:hep-ph/0509240; Nucl. Phys. Proc. Suppl. [**A99**]{} 311 (2001), \[arXiv:hep-ph/0010315\] and references therein. M. Giffon, P. Desgrolard and E. Predazzi, Z. Phys. [**C63**]{} 241 (1994). Discussion of an eikonal approximation in the Regge theory as well as references to the original papers on this subject can be found in P.D.B. Collins, An introduction to Regge theory & high energy physics, Cambridge University press (Cambridge) 1977;\
See also: K.A. Ter-Martirosyan, Sov. ZhETF Pisma [**15**]{} 519 (1972);\
V.A. Petrov and A.V. Prokudin, Eur. Phys. J. [**C23**]{} 135 (2002);\
C. Bourrely, J. Soffer and T.T. Wu, Eur. Phys. J. C [**28**]{} 97 (2003) \[arXiv:hep-ph/0210264\];\
P.A.S. Carvalho, A.F. Martini and M.J. Menon, Eur. Phys. J. [**C39**]{} 359 (2005) \[arXiv:hep-ph/0312243\]. S.M. Troshin and N.E. Tyurin, Phys. Part. Nucl. [**35**]{} 555 (2004); Fiz. Elem. Chast. Atom. Yadra [**35**]{} 1029 (2004); \[arXiv:hep-ph/0308027\] and references therein;\
L.L. Jenkovszky, B.V. Struminsky and A.N. Wall, Sov. Journ. Part. Nucl. [**19**]{} 180 (1988). L. Lukaszuk and B. Nicolescu, Nuovo Cimento Letters [**8**]{} 405 1973;\
P. Gauron, E. Leader and B. Nicolescu, Nuclear Physics [**B299**]{} 640 (1988). P. Gauron, E. Leader and B. Nicolescu, Phys. Lett. [**B238**]{} 406 (1990). R. Avila, P. Gauron and B. Nicolescu, Eur. Phys. J. [**C49**]{} 581 (2007) \[arXiv:hep-ph/0607089\]. L.L. Jenkovszky, E.S. Martynov and B.V. Struminsky, Phys. Lett. [**B249**]{} 535 (1990). P. Desgrolard, M. Giffon, E. Martynov and E. Predazzi, Eur. Phys. J. [**C18**]{} 359 (2000). J.R. Cudell [*et al.*]{}, Phys. Rev. [**D65**]{}, 074024 (2002) \[arXiv:hep-ph/0107219\]; J.R. Cudell [*et al.*]{}, Phys. Rev. Lett. [**89**]{} 201801 (2002); \[arXiv:hep-ph/0206172\]. P. Desgrolard, M. Giffon, E. Martynov and E. Predazzi, Eur. Phys. J. [**C18**]{} 555 (2001) \[arXiv:hep-ph/0006244\]. J.R. Cudell, E. Martynov and O. Selyugin, Eur. Phys. J. [**C33**]{} s533 (2004); \[arXiv:hep-ph/0311019\]. J.R. Cudell, E. Martynov, O. Selyugin and A. Lengyel, Phys. Lett. [**B587**]{} 78 (2004) \[arXiv:hep-ph/0310198\]. J.R. Cudell, A. Lengyel and E. Martynov, Phys. Rev. [**D73**]{} 034008 (2006) \[arXiv:hep-ph/0511073\]. Review of Particle Physics, S. Eidelman et al., Phys. Lett. [**B592**]{} (2004) 1. Encoded data files are available at http://pdg.lbl.gov/2005/hadronic-xsections/hadron.html. Durham Database Group (UK), M.R. Whalley et al., http://durpdg.dur.ac.uk/hepdata/reac.html. C. Bruneton [*et al.*]{}, Nucl. Phys. [**B 124**]{} 391 (1977) . S. Conetti [*et al.*]{}, Phys. Rev. Lett. [**41**]{} 924 (1978) . J.C.M. Armitage [*et al.*]{}, Nucl. Phys. [**B132**]{} 365 (1978). M.Y. Bogolyubsky [*et al.*]{}, Yad. Fiz. [**41**]{} 1210 (1985) \[Sov. J. Nucl. Phys. [**41**]{} (1985) 773\]. C.W. Akerlof [*et al.*]{}, Phys. Rev. [**D14**]{} 2864 (1976).
[^1]: $t_{eff}$ can be defined by behaviour of elastic scattering amplitude at $s\to \infty $. If $a(s,t)\approx sf(s)F(t/t_{eff}(s))$ then $\sigma_{el}(s)\propto|f(s)|^{2}\int_{-\infty}^{0}dt|F(t/t_{eff})|^{2}=
t_{eff}|f(s)F(1)|^{2}$.
[^2]: For crossing-odd term of amplitude such a factor is well known and describes crossover effect, i.e. intersection of the $ab$ and $\bar ab$ differential cross sections at $t\approx -0.15$ GeV$^{2}$. Our analysis [@CLM] has shown that similar factor is visible in crossing-even reggeon term.
[^3]: A more sophisticated form for residues should be considered for larger $|t|$.
|
---
abstract: 'A generalization of the Bowen–York initial data to the case with a positive cosmological constant is investigated. We follow the construction presented recently by Bizoń, Pletka and Simon, and solve numerically the Lichnerowicz equation on a compactified domain $\mathbb S^1 \times \mathbb S^2$. In addition to two branches of solutions depending on the polar variable on $\mathbb S^2$ that were already known, we find branches of solutions depending on two variables: the polar variable on $\mathbb S^2$ and the coordinate on $\mathbb S^1$. Using Vanderbauwhede’s results concerning bifurcations from symmetric solutions, we show the existence of the corresponding bifurcation points. By linearizing the Lichnerowicz equation and solving the resulting eigenvalue problem, we collect numerical evidence suggesting the absence of additional branches of solutions.'
address:
- 'Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-438 Kraków, Poland'
- 'Gravitational Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria'
- 'Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-438 Kraków, Poland'
author:
- Patryk Mach
- Jerzy Knopik
title: 'Rotating Bowen–York initial data with a positive cosmological constant'
---
Introduction
============
In a recent paper Bizoń, Pletka and Simon investigated Bowen–York type initial data for the Einstein equations with a positive cosmological constant [@BPS]. Bowen–York initial data are obtained within the framework of the so-called conformal method. From the technical point of view, its most important part amounts to solving a corresponding Lichnerowicz equation. In this paper we find numerically new, symmetry-breaking branches of solutions of the Lichnerowicz equation that describes the ‘rotating’ case investigated in [@BPS]. Their existence was predicted in [@BPS]. Basing on a bifurcation theorem due to Vanderbauwhede [@vanderbauwhede], we formalise the arguments presented in [@BPS] and show the existence of bifurcation points that give rise to these symmetry-breaking branches of solutions. We also collect numerical evidence suggesting the absence of additional solutions.
We are interested in the initial data consisting of a triple $(\tilde {\mathcal M}, \tilde g_{ij}, \tilde K_{ij})$, where $\tilde {\mathcal M}$ is a 3-dimensional compact manifold, $\tilde g_{ij}$ is a smooth Riemannian metric, and $\tilde K_{ij}$ is a trace-free ($\tilde g^{ij}\tilde K_{ij} = 0$) tensor satisfying the Einstein vacuum constraint equations $$\label{einsteinid}
\tilde R - \tilde K_{ij} \tilde K^{ij} - 2 \Lambda = 0, \quad \tilde \nabla_i \tilde K^{ij} = 0.$$ Here $\Lambda$ is a positive cosmological constant, $\tilde R$ is the scalar curvature of $(\tilde {\mathcal M}, \tilde g_{ij})$, and $\tilde \nabla_i$ is the corresponding covariant derivative with respect to the metric $\tilde g_{ij}$.
The conformal method of finding such initial data can be summarized as follows. Let $(\mathcal M, g_{ij})$ be a compact 3-dimensional Riemannian manifold with a smooth metric $g_{ij}$ in the positive Yamabe class. In addition, let $K_{ij}$ be a smooth trace-free ($g^{ij}K_{ij} = 0$) and divergence-free ($\nabla_i K^{ij} = 0$) tensor on $\mathcal M$. The initial data $(\tilde {\mathcal M}, \tilde g_{ij}, \tilde K_{ij})$ can be found as $$\tilde g_{ij} = \phi^4 g_{ij}, \quad \tilde K^{ij} = \phi^{-10} K^{ij},$$ where the conformal factor $\phi$ is a positive, smooth solution of the Lichnerowicz equation $$\label{lich1}
- \Delta_g \phi + \frac{1}{8}R \phi - \frac{1}{4} \Lambda \phi^5 - \frac{\Omega^2}{8 \phi^7} = 0.$$ In the above formulas $\Delta_g$, $\nabla_i$, and $R$ denote the Laplacian, the covariant derivative and the scalar curvature with respect to metric $g_{ij}$. Note that setting $\tilde K^{ij} = \phi^{-10} K^{ij}$ is equivalent to $\tilde K_{ij} = \phi^{-2} K_{ij}$. Following [@BPS] we also denote $\Omega^2 = K^{ij}K_{ij}$.
Clearly, apart from the technical difficulty of solving Eq. (\[lich1\]) for the conformal factor, the key point of the above conformal method is to find an appropriate ‘seed’ metric $g_{ij}$ together with the trace- and divergence-free tensor $K_{ij}$. A simple, but physically important choice was introduced by Bowen and York in 1980 [@bowen_york], for the case with vanishing cosmological constant. Their initial data are obtained by choosing as a ‘seed’ manifold $\mathcal M = \mathbb R^3 \setminus \{ 0 \}$ (which is noncompact) endowed with the flat Euclidean metric. The original forms of $K_{ij}$ given by Bowen and York were later generalized by Beig [@beig]. In this latter version they read, in Cartesian coordinates $(x^1,x^2,x^3)$, $$\begin{aligned}
K_{ij}(x) & = & \frac{3}{2r^2} \left[ P_i n_j + P_j n_i - (\delta_{ij} - n_i n_j) P^k n_k \right], \\
K_{ij}(x) & = & \frac{3}{r^3} \left( \epsilon_{kli} J^k n^l n_j + \epsilon_{klj} J^k n^l n_i \right), \label{byrot} \\
K_{ij}(x) & = & \frac{C}{r^3} (3 n_i n_j - \delta_{ij}), \\
K_{ij}(x) & = & \frac{3}{2r^4} \left[ -Q_i n_j - Q_j n_i - (\delta_{ij} - 5n_i n_j) Q^k n_k \right].\end{aligned}$$ Here $r = \sqrt{x_i x^i}$, $n_i = x_i/r$; $P_i$, $S_i$ and $Q_i$ are constant; $\delta_{ij}$ and $\epsilon_{klj}$ denote the Kronecker delta and the usual 3-dimensional permutation symbol, respectively. Vectors $P_i$ and $S_i$ can be interpreted respectively as the Arnowitt–Deser–Misner (ADM) linear and angular momenta at $r \to \infty$. In this paper we are only interested in the rotational case (\[byrot\]).
A transformation of the Bowen–York data (\[byrot\]) to the case with a positive cosmological constant can be done along the lines of the derivation described in [@BPS]. The flat metric $g = dr^2 + r^2 (d\theta^2 + \sin^2 \theta d \varphi^2)$ is first transformed conformally into $$\hat g = \hat \phi^4 g = \hat \phi^4 \left[ dr^2 + r^2 (d\theta^2 + \sin^2 \theta d \varphi^2) \right]$$ with $\hat \phi^4 = 1/(\Lambda r^2)$. This conformal factor satisfies the Lichnerowicz equation $$- \Delta_g \hat \phi - \frac{1}{4} \Lambda \hat \phi^5 = 0,$$ with the vanishing extrinsic curvature. Consequently, the metric $\hat g$ and the extrinsic curvature $\hat K_{ij} = 0$ already satisfy the constraint equations with the cosmological constant $\Lambda$. A coordinate transformation $r = \exp(\alpha)$ yields $$\label{torus}
\hat g = \frac{1}{\Lambda} \left( d\alpha^2 + d \theta^2 + \sin^2 \theta d \varphi^2 \right).$$ If we choose to identify $\alpha$ periodically with a period $T$, we get a round metric on $\mathbb S^1(T) \times \mathbb S^2$. The scalar curvature $\hat R$ of $\hat g$ is constant; $\hat R = 2 \Lambda$.
The aim at this point is to construct an equivalent to the original Bowen–York expression (\[byrot\]) on $\mathbb S^1(T) \times \mathbb S^2$ with the metric $\hat g$. This can also be done by exploiting the relations of the conformal method described above. Setting $\hat K^{ij} = \hat \phi^{-10} K^{ij}$, we get $\hat \nabla_i \hat K^{ij} = 0$, where $\hat \nabla_i$ denotes the covariant derivative with respect to the metric $\hat g_{ij}$. In spherical coordinates $(r,\theta,\phi)$ with the axis parallel to the angular momentum $J^i$ the only nonvanishing components of (\[byrot\]) are $K_{r \varphi} = -3 J \sin^2 \theta/r^2$, where $J^2 = J_i J^i$. This leads to $\hat K_{\alpha \varphi} = \hat \phi^{-2} \frac{\partial r}{\partial \alpha} K_{r \varphi} = -3 J \sqrt{\Lambda} \sin^2 \theta$. A direct calculation yields, for the original Bowen–York extrinsic curvature (\[byrot\]), $$K_{ij}K^{ij} = \frac{18}{r^6} \left[ J_i J^i - (J_i n^i)^2 \right],$$ or, in spherical coordinates defined above, $K_{ij} K^{i,j} = 18 \sin^2 \theta J^2/r^6$. Accordingly, $\hat K_{ij} \hat K^{ij} = 18 J^2 \Lambda^3 \sin^2 \theta = 8 b^2 \Lambda \sin^2 \theta$, where $b = 3 J \Lambda / 2$. The initial data satisfying constraint equations (\[einsteinid\]) can be obtained as $\tilde g_{ij} = \phi^4 \hat g_{ij}$, $\tilde K_{ij} = \phi^{-2} \hat K_{ij}$, provided that the conformal factor $\phi$ satisfies the equation $$- \Delta_{\hat g} \phi + \frac{1}{8} \hat R \phi - \frac{1}{4} \Lambda \phi^5 - \frac{\hat K_{ij} \hat K^{ij}}{8 \phi^7} = 0,$$ where $\Delta_{\hat g}$ denotes the Laplacian with respect to metric $\hat g_{ij}$. In explicit terms we get $$\label{lichnerowicz}
- \partial_{\alpha \alpha} \phi - \frac{1}{\sin \theta} \partial_\theta \left( \sin \theta \partial_\theta \phi \right) - \frac{1}{\sin^2 \theta} \partial_{\varphi \varphi} \phi + \frac{1}{4} \phi - \frac{1}{4} \phi^5 - \frac{b^2 \sin^2 \theta}{\phi^7} = 0.$$ Here $\alpha \in \mathbb S^1(T)$, $(\theta, \varphi) \in \mathbb S^2$; by $\alpha \in \mathbb S^1(T)$ we mean that the solution of Eq. (\[lichnerowicz\]) should be periodic in $\alpha$ with the period $T$. This article is devoted to the analysis of solutions of Eq. (\[lichnerowicz\]), their dependence on the parameter $b$, and bifurcations.
The remaining sections are organized as follows. Section \[sec\_sols\] is devoted to the analysis of solutions of Eq. (\[lichnerowicz\]). In subsection \[sec\_bifurcation\] we discuss the bifurcation pattern of solutions, and collect basic existence results. A detailed account of these and related facts can be found in [@BPS]. In subsection \[bzero\] we deal with the exactly solvable case of Eq. (\[lichnerowicz\]) with $b = 0$. In subsection \[general\_method\] we describe the general numerical spectral method used to obtain solutions for $b \neq 0$.
Section \[linear\] is dedicated to the analysis of the linearized Eq. (\[lichnerowicz\]). The main reason of presenting this analysis here is that it is directly related to the bifurcation structure of the solutions. Best known theorems formalizing this relation were proved by Crandall and Rabinowitz [@crandall_rabinowitz1; @crandall_rabinowitz2]. Roughly speaking, they connect the bifurcation of solutions with an occurrence of a zero mode in the linearized equation. In the context of Eq. (\[lichnerowicz\]), the analysis of the corresponding linearized equation clarifies the bifurcation structure described in subsection \[sec\_bifurcation\], but it can also be used as an evidence suggesting that no additional branches of solutions bifurcate from the already known ones. Strictly speaking, the theorems of Crandall and Rabinowitz [@crandall_rabinowitz1; @crandall_rabinowitz2] apply to bifurcations from simple zero eigenvalues. We will see in Sec. \[linear\] that in our case bifurcations occur for degenerate zero eigenvalues. A generalization of theorems of Crandall and Rabinowitz that works in the case investigated in this paper was given by Vanderbauwhede [@vanderbauwhede]. Vanderbauwhede’s theorem requires that a background solution (a solution from which a new branch bifurcates) shares a continuous symmetry — the $O(2)$ symmetry in our case. We summarize the results on existence of the symmetry-breaking bifurcations for the solutions of Eq. \[lichnerowicz\] in Corollary \[cor1\]. Section \[linear\] is also divided into subsections. In subsection \[separation\] we discuss the separation of variables in the linearized equation. The case with $b = 0$ is then treated in subsection \[linb0\]. The general spectral numerical method used to solve the linear eigenvalue problem is described in subsection \[lin\_general\_method\]. The numerical results are discussed in subsection \[lin\_results\]. Section \[conclusions\] contains a few concluding remarks.
Solutions of the Lichnerowicz equation {#sec_sols}
======================================
Bifurcation diagrams {#sec_bifurcation}
--------------------
![\[fig2\]Energies of different solutions for $T = 5\pi$.](bifurcation_diagram.pdf){width="80.00000%"}
![\[bifdiag\]Same as in Fig. \[fig2\], but for $T = 7\pi$.](Energy1.pdf){width="80.00000%"}
We begin our discussion with an example of a family of solutions of Eq. (\[lichnerowicz\]) obtained numerically for $T = 5 \pi$. For conciseness we will restrict most of our numerical examples to this sample period. Our numerical solutions do not depend on $\varphi$, i.e., they all admit the $O(2)$ symmetry group acting on $\mathbb S^2$. Figure \[fig2\] shows a bifurcation diagram created by computing the value of the ‘energy’ functional $$E = \int_0^T d\alpha \int_0^\pi \sin \theta d \theta \left[ \frac{1}{2} (\partial_\alpha \phi)^2 + \frac{1}{2} (\partial_\theta \phi)^2 + \frac{1}{8} \phi^2 - \frac{1}{24} \phi^6 + \frac{1}{6} \frac{b^2 \sin^2 \theta}{\phi^6} \right]$$ associated with each of the solutions depending on $\alpha$ and $\theta$. Note that the ‘full’ energy associated with solutions that would potentially depend also on $\varphi$, and that corresponds directly to Eq. (\[lichnerowicz\]), reads $$\begin{aligned}
I & = & \int_0^T d\alpha \int_0^\pi \sin \theta d \theta \int_0^{2 \pi} d \varphi \left[ \frac{1}{2} (\partial_\alpha \phi)^2 + \frac{1}{2} (\partial_\theta \phi)^2 + \frac{1}{2} \frac{1}{\sin^2 \theta} (\partial_\varphi \phi)^2 \right. \\
& & \left. + \frac{1}{8} \phi^2 - \frac{1}{24} \phi^6 + \frac{1}{6} \frac{b^2 \sin^2 \theta}{\phi^6} \right].\end{aligned}$$ For the solutions that do not depend on $\varphi$ one has $I = 2 \pi E$. The abscissa in Fig. \[fig2\] shows the bifurcation parameter $b$.
There are four branches of solutions for $T = 5\pi$: two branches consisting of solutions that only depend on $\theta$ \[admitting the $O(2) \times O(2)$ symmetry group acting on $\mathbb S \times \mathbb S^2$; they are denoted as branch (a) and (b)\], and two branches of solutions that do depend on both $\alpha$ and $\theta$ \[branch (c) and (d)\]. The solutions exist only for sufficiently small values of the parameter $b < b_\mathrm{max}$. This stays in agreement with a result obtained in [@BPS] that is based on a general theorem by Premoselli [@premoselli], and which we also quote at the end of this section. There are four solutions for $b = 0$, each belonging to one of the branches (a–d). Two of these solutions are constant: $\phi \equiv 1$ ($E = T/6$) and $\phi \equiv 0$ ($E = 0$). The remaining two are non trivial, but they only depend on $\alpha$. They are depicted in Fig. \[fig1\]. We derive these solutions in Sec. \[bzero\].
The $O(2) \times O(2)$-symmetric branches (a) and (b) were obtained numerically already in [@BPS]. They originate at $b = 0$ from the two elementary solutions $\phi \equiv 0$ \[branch (a)\] and $\phi \equiv 1$ \[branch (b)\], and they join each other as $b \to b_\mathrm{max}$. The existence of symmetry-breaking of branches (c) and (d) was suggested in [@BPS]. They bifurcate from branch (b). As $b \to 0$, these branches join the two nontrivial solutions that depend on $\alpha$ only, and that were plotted in Fig. \[fig1\]. We refer to the branch that bifurcates at $b = b_1 = 0.235$ as branch (c) and to the one bifurcating at $b = b_2 = 0.188$ as branch (d). Sample solutions belonging to these two branches are shown in Figs. \[fig3\] and \[fig4\].
An example of a bifurcation diagram analogous to that of Fig. \[fig2\], but obtained for $T = 7\pi$ is shown in Fig. \[bifdiag\]. There are five branches of solutions: two $O(2) \times O(2)$ symmetric branches (a) and (b), and three symmetry-breaking branches (c), (d) and (e). In general for a given period $T$, we expect $k$ symmetry-breaking branches, where $k$ is the largest integer satisfying $k < T/(2\pi)$. We will return to this point in Corollary \[cor1\].
Except for the case with $b = 0$, all solutions discussed here are found numerically. There are, however, partial analytic existence, symmetry and stability results. They are described in detail in [@BPS]; here we only review those that are relevant for our discussion.
Denote the left-hand side of Eq. (\[lich1\]) by $$F(\phi) = - \Delta_g \phi + \frac{1}{8}R \phi - \frac{1}{4} \Lambda \phi^5 - \frac{\Omega^2}{8 \phi^7}.$$ Stability of solutions of Eq. (\[lich1\]) is understood in terms of the derivative $$F_\phi w = - \Delta_g w + \frac{1}{8} R w - \frac{5}{4} \Lambda \phi^4 w + \frac{7 \Omega^2}{8 \phi^8} w.$$ A solution $\phi$ of Eq. (\[lich1\]) is called strictly stable, stable, marginally stable, unstable, or strictly unstable, if the lowest eigenvalue $\lambda$ of $F_\phi$ satisfies $\lambda > 0$, $\lambda \ge 0$, $\lambda = 0$, $\lambda \le 0$, or $\lambda < 0$, respectively. A motivation of this definition comes from the bifurcation theory, where the bifurcation points are identified by zero eigenvalues of $F_\phi$. In particular, it is not connected with the dynamical stability of solutions to the Einstein equations with the initial data implied by a given conformal factor $\phi$.
The following important facts are proved/listed in [@BPS].
Let $(\mathcal{M},g_{ij},K_{ij})$ be a seed manifold such that $g_{ij}$ and $\Omega^2 = K_{ij}K^{ij}$ admit a continuous symmetry $\xi$, i.e., $$\mathcal{L}_\xi g_{ij} = 0, \quad \mathcal{L}_\xi \Omega = 0,$$ where $\mathcal{L}$ is the Lie derivative. Then all stable solutions of (\[lich1\]) are also symmetric, that is $$\mathcal{L}_\xi \phi = 0.$$
The next result by Bizoń, Pletka and Simon is based on the work of Hebey, Pacard and Pollack [@hebey_pacard_pollack]. The proof can be found in [@pletka; @thesis].
Consider a seed manifold $(\mathcal{M}, g_{ij}, K_{ij})$ as defined in the Introduction, but with a constant scalar curvature $R$.
1. Let $$\int_\mathcal{M} \Omega^2 dV \le \frac{Y^6}{256\Lambda^2 R^3 V^3},$$ then Eq. (\[lich1\]) has a smooth positive solution.
2. Assume that $$\int_\mathcal{M} \Omega^{5/6} dV > \frac{R^{5/4}V}{3^{5/4} \Lambda^{5/6}},$$ then Eq. (\[lich1\]) has no smooth positive solution.
Here $V = \int_\mathcal{M}dV$ is the volume of $\mathcal{M}$, $dV$ is the volume element associated with the metric $g_{ij}$, and $$Y = \inf_{\gamma \in C^\infty(\mathcal{M}), \, \gamma \not\equiv 0} \frac{\int_\mathcal{M} (8 |\nabla \gamma|^2 + R \gamma^2) dV}{\left( \int_\mathcal{M} \gamma^6 dV \right)^{1/3}}.$$ is the Yamabe constant.
Another important result was obtained by Premoselli [@premoselli].
\[thm\_prem\] Let us decompose $\Omega= \tilde b \Omega_0$ in (\[lich1\]) in terms of a constant $\tilde b$ and a fixed function $\Omega_0$. There exists $0 < b_\mathrm{max} < \infty$ such that Eq. (\[lich1\]) has
1. At least two positive solutions for $\tilde b < b_\mathrm{max}$, at least one of which is strictly stable. In addition, one of the strictly stable solutions, called $\phi(\tilde b)$, is ‘minimal’ — for any positive solution $\phi \not\equiv \phi(\tilde b)$ one has $\phi > \phi(\tilde b)$.
2. A unique, marginally stable, positive solution for $\tilde b = b_\mathrm{max}$.
3. No solution for $\tilde b > b_\mathrm{max}$.
The decomposition $\Omega= \tilde b \Omega_0$ is, of course, non-unique. A natural choice in case of Eq. (\[lichnerowicz\]) is to assume $\tilde b = b = 3 J \Lambda / 2$.
The branch of stable solutions whose existence is asserted by the above theorem can be identified with the one that starts from $\phi \equiv 0$ at $b = 0$, which we consequently denote as branch (a) in Figs. \[fig2\], \[bifdiag\].
$O(3)$-symmetric solutions ($b = 0$) {#bzero}
------------------------------------
![\[fig1\]Periodic solutions with period $T = 5\pi$ and $b = 0$.](periodic_lichnerowicz_wykres.pdf){width="80.00000%"}
For $b = 0$ Eq. (\[lichnerowicz\]) admits solutions that do not depend on $\theta$ and $\varphi$. In this case it can be written as an ordinary differential equation $$\label{lane_emden_a}
\frac{d^2 \phi}{d\alpha^2} = \frac{1}{4} \phi (1 - \phi^4),$$ where $\phi$ should be a periodic function of $\alpha$.
Two obvious solutions are $\phi \equiv 0$ and $\phi \equiv 1$. The existence of the solution $\phi \equiv 1$ is a consequence of the fact that metric (\[torus\]) satisfies the constraint equations (\[einsteinid\]) itself. It represents a time-symmetric slice through the Nariai spacetime [@nariai] $$g = \frac{1}{\Lambda}(-dt^2 + \cosh^2 t d\alpha + d\theta^2 + \sin^2 \theta d\varphi^2).$$
Remarkably, Eq. (\[lane\_emden\_a\]) corresponds to the famous Lane–Emden equation with index $n = 5$, $$\label{lane_emden_b}
\frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d \Phi}{dr} \right) + \Phi^5 = 0,$$ where $r$ is a spherical radius. The relation between the solutions of Eqs. (\[lane\_emden\_a\]) and (\[lane\_emden\_b\]) is given by $\Phi = \phi/\sqrt{2r}$, $\alpha = - \ln r$. All real solutions of Eqs. (\[lane\_emden\_a\]) and (\[lane\_emden\_b\]) are derived in [@mach]. Here we are interested only in those that are periodic and everywhere positive. Multiplying Eq. (\[lane\_emden\_a\]) by $d\phi/d\alpha$ and integrating with respect to $\alpha$ one obtains $$\left( \frac{d\phi}{d\alpha} \right)^2 = \frac{1}{12} \left( -\phi^6 + 3 \phi^2 + C \right),$$ where $C$ is an integration constant. Positive and periodic solutions exist for $C \in (-2,0)$. In this case the polynomial $w(\phi) = -\phi^6 + 3 \phi^2 + C$ has 4 real roots. It can be factorized as $$w(\phi) = (\phi^2 - a)(b - \phi^2)(\phi^2 + c),$$ where $$\begin{aligned}
a & = & 2 \sin \left[ \frac{1}{3} \mathrm{arc\, sin} \left( \frac{|C|}{2} \right) \right], \quad b = 2 \cos \left[ \frac{1}{3} \mathrm{arc\, cos} \left( - \frac{|C|}{2} \right) \right], \\
c & = & 2 \cos \left[ \frac{1}{3} \mathrm{arc\, cos} \left( \frac{|C|}{2} \right) \right].\end{aligned}$$ It is elementary to show that $$0 < a < 1 < b < \sqrt{3} < c < 2.$$ Solutions of Eq. (\[lane\_emden\_a\]) can be then written as $$\label{sol1}
\phi = \sqrt{\frac{a b y^2}{b y^2 - (b - a)}}, \quad y = \mathrm{dc} \left[ \frac{1}{2} \sqrt{\frac{(a+c)b}{3}} (\alpha - \alpha_0) , \sqrt{\frac{(b - a)c}{(a + c)b}} \right],$$ where $\mathrm{dc}$ is a subsidiary Jacobian elliptic function (in the standard Glaisher notation) [@mach]. Note that $|\mathrm{dc}(x,k)| \ge 1$ for $k \in [0,1]$. It follows that the solution (\[sol1\]) is strictly positive.
The function $\mathrm{dc}(x,k)$ is periodic with the period $4K(k)$, where $$K(k) = \int_0^\frac{\pi}{2} \frac{d \theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$ is the complete elliptic integral of the first kind. The period of $y$ with respect to $\alpha$ is $$2 T_\phi \equiv 8 \sqrt{\frac{3}{(a+c)b}} K\left( \sqrt{\frac{(b - a)c}{(a + c)b}} \right),$$ and because $\phi$ depends on $y^2$, it is periodic with the period $T_\phi$. The period $T_\phi$ given by the above formula is a strictly increasing function of $C \in [-2,0)$. It is equal to $2 \pi$ for $C = -2$ and tends to $+ \infty$, as $C \to 0$. Finding the value of $C$ corresponding to a given period $T_\phi$ is a simple numerical task. In Figure \[fig1\] we plot two sample solutions $\phi(\alpha)$ corresponding to the period $T = 5\pi$. They are given by Eq. (\[sol1\]) for $T_\phi = 5\pi$ (in this case $C = -0.0161381$) and $T_\phi = 5 \pi/2$ ($C = -0.848422$).
In general, for a given period $T$, there are $k$ periodic solutions, where $k$ is the largest integer satisfying $k < T/(2 \pi)$. The periods of these solutions read $T_\phi = T/j$, where $j \le k$. It is easy to show that they correspond to time-symmetric initial data obtained by gluing together $j$ copies of the $t = \mathrm{const}$ slices through the Kottler (Schwarzschild–de Sitter) metric $$g = - \left(1 - \frac{2M}{r} - \frac{\Lambda}{3}r^2 \right) dt^2 + \frac{dr^2}{1 - \frac{2M}{r} - \frac{\Lambda}{3}r^2} + r^2 (d \theta^2 + \sin^2 \theta d\phi^2).$$ Such initial data are periodic in the radial coordinate and contain $j$ pairs of horizons — each pair consists of a ‘cosmological’ and a ‘black hole’ horizon. The constant $C$ is related with the Kottler mass by $C = - 6 M \sqrt{\Lambda}$. Showing this amounts to a simple modification of a calculation done in [@mach_niall]. It is expected that each of these periodic solutions belongs to one of symmetry-breaking branches discussed in Sec. \[sec\_bifurcation\].
For $C = -2$ one gets a solution of the form $\phi \equiv 1$. The other limiting case, with $C = 0$, yields the well known Schuster solution of the Lane–Emden equation $$\Phi(r) = \frac{1}{\sqrt{1 + \frac{1}{3}r^2}}.$$ In terms of $\phi$ and $\alpha$ it is given by $$\label{sol2}
\phi = \left( \frac{12}{4} \right)^\frac{1}{4} \sqrt{\mathrm{sech} (\alpha - \alpha_0)},$$ which may be interpreted as a case with infinite period.
$O(2)$-symmetric solutions {#general_method}
--------------------------
![\[fig4\]Contour plots of sample solutions depending on $\alpha$ and $\theta$. Here $T = 5\pi$. The above solutions belong to branch (c) plotted in Fig. \[fig2\].](lichnerowicz_wykresy_2.pdf){width="80.00000%"}
![\[fig3\]Contour plots of sample solutions depending on $\alpha$ and $\theta$. Here $T = 5\pi$. The above solutions belong to branch (d) in Fig. \[fig2\].](lichnerowicz_wykresy_1.pdf){width="80.00000%"}
In the more general case of solutions depending both on $\alpha$ and $\theta$, but not on $\varphi$, we resort to numerical methods. In this case Eq. (\[lichnerowicz\]) can be written as $$\label{lichnerowicz2}
- \partial_{\alpha \alpha} \phi - \frac{1}{\sin \theta} \partial_\theta \left( \sin \theta \partial_\theta \phi \right) + \frac{1}{4} \phi - \frac{1}{4} \phi^5 - \frac{b^2 \sin^2 \theta}{\phi^7} = 0.$$ We solve this equation numerically using a spectral scheme, which we now describe briefly.
The solution is approximated as $$\label{expansion}
\tilde \phi (\alpha, \theta) = \sum_{k = 0}^M \sum_{l=0}^N a_{kl} y_{kl}(\alpha, \theta),$$ where the expansion functions $y_{kl}$, $k = 0, \dots, M$, $l = 0, \dots, N$, are eigenfunctions of the operator $L$ defined as $$L \phi \equiv - \partial_{\alpha \alpha} \phi - \frac{1}{\sin \theta} \partial_\theta \left( \sin \theta \partial_\theta \phi \right) + \frac{1}{4} \phi$$ \[$L \phi$ is the linear part of the left-hand side of Eq. (\[lichnerowicz2\])\]. We choose specifically $$\begin{aligned}
\label{ya}
y_{0l}(\alpha, \theta) & = & \frac{1}{2} P_l(\cos \theta), \quad l = 0, \dots, N, \\
y_{kl}(\alpha, \theta) & = & \cos \left( k \frac{2 \pi}{T} \alpha \right) P_l (\cos \theta), \quad k = 1, \dots, M, \quad l = 0, \dots, N,
\label{yb}\end{aligned}$$ where $P_l$ denotes the $l$-th Legendre polynomial. The corresponding eigenvalues of $L$ read $$L y_{kl} = \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l(l+1) + \frac{1}{4} \right] y_{kl}.$$ Since the eigenfunctions $u_{kl}$ satisfy the periodic boundary conditions, so does the solution $\tilde \phi$. This approach is sometimes referred to as the Galerkin method.
The nonlinear part in Eq. (\[lichnerowicz2\]) is more troublesome. The task is to expand the expression $$-\frac{1}{4} \tilde \phi^5 - \frac{b^2 \sin^2 \theta}{\tilde \phi^7}$$ in $y_{kl}$. The coefficients of this expansion are given by the integrals $$\begin{aligned}
\nonumber
c_{kl} & = & \frac{2l+1}{T} \int_0^\pi d\theta \sin \theta \int_0^T d\alpha \left( - \frac{1}{4} \tilde \phi^5 - \frac{b^2 \sin^2 \theta}{\tilde \phi^7} \right) \\
& & \times \cos\left( k \frac{2\pi}{T} \alpha \right) P_l(\cos \theta).
\label{bkl}\end{aligned}$$ The values of $c_{kl}$ are computed using Gauss–Legendre–Fourier quadratures. We describe this procedure in the Appendix.
Equation (\[lichnerowicz2\]) yields the following set of equations for the coefficients $a_{kl}$ $$\begin{aligned}
& & r_{kl} \equiv \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l(l+1) + \frac{1}{4} \right] a_{kl} + c_{kl} = 0, \\
& & \quad k = 0, \dots, M, \quad l = 0, \dots, N.\end{aligned}$$ Note that the coefficients $c_{kl}$ depend on $a_{kl}$ via Eqs. (\[expansion\]) and (\[bkl\]). We solve the above set of equations iteratively, using the Newton–Raphson scheme. In each iteration corrections $\Delta_{kl}$ to the coefficients $a_{kl}$ are computed as a solution of the set of linear equations $$\sum_{m=0}^M \sum_{n=0}^N \frac{\partial r_{kl}}{\partial a_{mn}} \Delta_{mn} = - r_{kl}, \quad k = 0, \dots, M, \quad l = 0, \dots, N,$$ where the Jacobians $\partial r_{kl}/\partial a_{mn}$ are computed numerically. A new set of coefficients $a^\prime_{kl}$ is then given by $a^\prime_{kl} = a_{kl} + \Delta_{kl}$. It turns out that the above method converges relatively fast. Around 10 to 15 iterations suffice to converge up to machine precision. The numerical error is thus controlled by $N$, $M$, and the accuracy with which the Jacobians $\partial r_{kl}/\partial a_{mn}$ are computed.
Sample solutions obtained with the above method are shown in Figs. \[fig4\] and \[fig3\]. They belong to branches (c) and (d) illustrated in Fig. \[fig2\] and discussed in previous sections. It is worth noticing that both sequences of solutions join smoothly with the solutions depending only on $\alpha$ for $b = 0$ and those depending only on $\theta$ for $b = b_1$ and $b_2$, respectively.
Linearized eigenvalue problem {#linear}
=============================
In this section we prove existence of bifurcation points giving rise to the symmetry-breaking branches of solutions depending on $\alpha$ and $\theta$. We are also concerned with the question of existence or nonexistence of additional solutions of Eq. (\[lichnerowicz\]). This general problem can be naturally divided into following specific cases or tasks:
1. Symmetry of solutions with respect to $\varphi$. Is it possible to prove that there are no solutions depending on $\varphi$?
2. A weaker version of the above question: Is it possible to prove that no solution that depends on $\varphi$ bifurcate from the known solutions, e.g. those depicted in Fig. \[fig2\]?
3. Are there any other solutions depending on $\alpha$ and $\theta$ only that do not belong to the already known branches of solutions?
4. Analogously: can one prove that no branch of solutions depending only on $\alpha$ and $\theta$ bifurcates from the known branches?
We believe that it might be possible to answer the first of the above questions by means of techniques that stem from the method of moving planes [@ni_nirenberg], including in particular a version that was called the method of moving spheres [@jin]. In this manner Chruściel and Gicqaud were able to show that solutions of an equation similar to Eq. (\[lichnerowicz\]) are symmetric with respect to $\varphi$ [@chrusciel].
In this paper we collect numerical evidence concerning questions (ii) and (iv) — that no additional branches of solutions bifurcate from the already known ones. The reasoning is based on the linearization of Eq. (\[lichnerowicz\]), which is also a main tool allowing one to understand the bifurcation structure of its solutions. We exploit the fact that new branches of solutions bifurcate from solutions for which zero modes occur in the linearized equation.
It is well known that an existence of a zero eigenvalue of the linearized equation is not a sufficient condition for the occurrence of bifurcation. The best known rigorous result assuring the occurrence of bifurcation from simple zero eigenvalues is due to Crandall and Rabinowitz [@crandall_rabinowitz1; @crandall_rabinowitz2]. We will see, however, that in our case bifurcations do occur from degenerate zero modes. A suitable generalization of the original theorem of Crandall and Rabinowitz was given by Vanderbauwhede [@vanderbauwhede Thm. 6.2.6]; it applies to potentially degenerate zero eigenvalues for which the eigenspace is invariant under the action of the orthogonal group $O(n)$. Below we quote this theorem after [@smoller_wasserman].
\[thm4\] Let $X$, $B$ and $Z$ be real Banach spaces, and let $Y \subset X \times B$ be a neighborhood of the origin $(0,0) \in Y$. Consider a mapping $F \colon Y \to Z$ of class $C^2$ such that $F(0,0) = 0$. Assume further that the following hyphoteses are satisfied.
1. There exist representations of the orthogonal group $O(n)$ on $X$ and $Z$ denoted by $\Gamma \colon O(n) \to GL(X)$ and $\tilde \Gamma \colon O(n) \to GL(Z)$, where $GL(X)$ and $GL(Z)$ denote the general linear groups on $X$ and $Z$, respectively. Moreover, for each $s \in O(n)$ and all $(x,b) \in Y$, the following conditions hold:
1. $(\Gamma(s) x, b) \in Y$, and
2. $F(\Gamma(s)x,b) = \tilde \Gamma(s) F(x,b)$.
2. The partial derivative $F_x(0,0)$ is a Fredholm operator of index zero, and the representation $\Gamma_0$ of $O(n)$ induced by $\Gamma$ on the kernel $\mathrm{ker} \, F_x(0,0)$ is irreducible.
3. There is a non-zero vector $u_0$ in $\mathrm{ker} \, F_x(0,0)$ such that $F_{xb}(0,0)(u_0,0) \notin \mathrm{Range}\,(F_x(0,0))$.
Then $(0,0)$ is a bifurcation point of the equation $F(x,b) = 0$.
The above theorem applies to our case as follows. We define $$F(\phi, b^2) = - \Delta_{\hat g} \phi + \frac{1}{4} \phi - \frac{1}{4} \phi^5 - \frac{b^2 \sin^2 \theta}{\phi^7},$$ so that Eq. (\[lichnerowicz\]) can be written simply as $F(\phi,b^2) = 0$. Instead of the origin, we consider a point $(\phi_0,b_0^2)$ (which, of course, can be trivially shifted to the origin). The derivative $F_\phi$ at $(\phi_0,b_0^2)$ reads $$F_\phi (\phi_0,b_0^2) w = - \Delta_{\hat g} w + \frac{1}{4} w - \frac{5}{4} \phi_0^4 w + \frac{7 b^2 \sin^2 \theta}{\phi_0^8} w.$$ Clearly, $F_\phi$ is a self-adjoint operator with respect to the standard $L^2$ inner product on $\mathbb S^1(T) \times \mathbb S^2$, and thus it has a zero Fredholm index (i.e., the dimension of its kernel is the same, as the co-dimension of its range). Suppose now that $F_\phi (\phi_0,b_0^2)$ admits a zero eigenvalue. To see that the condition (iii) is satisfied, note that the mixed derivative $F_{\phi b^2}(\phi_0,b_0^2)$ reads $$F_{\phi b^2}(\phi_0,b_0^2) w = \frac{7 \sin^2 \theta}{\phi_0^8} w.$$ Let $w_0$ be a vector from $\mathrm{ker} \, F_\phi (\phi_0,b_0^2)$. We need to show that $$\frac{7 \sin^2 \theta}{\phi_0^8} w_0 \neq - \Delta_{\hat g} w + \frac{1}{4} w - \frac{5}{4} \phi_0^4 w + \frac{7 b^2 \sin^2 \theta}{\phi_0^8} w,$$ where $w$ is any vector in the domain of $F_\phi (\phi_0,b_0^2)$. To see that this condition indeed holds, assume the contrary, and multiply both sides of the obtained equation by $w_0$. We get $$\begin{aligned}
\frac{7 \sin^2 \theta}{\phi_0^8} w_0^2 & = & - w_0 \Delta w + \frac{1}{4} w_0 w - \frac{5}{4} \phi_0^4 w_0 w + \frac{7 b^2 \sin^2 \theta}{\phi_0^8} w_0 w \\
& = & - w_0 \Delta w + w \Delta w_0,\end{aligned}$$ where we have used the fact that $w_0 \in \mathrm{ker} \, F_\phi (\phi_0,b_0^2)$. Integrating over $\mathbb S(T) \times \mathbb S^2$ we get $$\int_0^T d\alpha \int_0^\pi d\theta \sin \theta \int_0^{2\pi} d \varphi \frac{7 \sin^2 \theta}{\phi_0^8} w_0^2 = 0,$$ which cannot hold, as the left-hand side of the above equation is strictly positive. In our case, bifurcations occur for the branch of solutions depending only on $\theta$ \[denoted as branch (b) in Fig. \[fig2\]\], which are clearly symmetric with respect to translations in $\alpha$, i.e., they are $O(2)$ symmetric. This symmetry is broken by the bifurcating solutions \[branches (c) and (d)\]. We will see in the subsequent section that in our case $\mathrm{ker} \, F_\phi (\phi_0,b_0^2)$ is in fact two dimensional and that the representation of $O(2)$ induced on $\mathrm{ker} \, F_\phi (\phi_0,b_0^2)$ is irreducible.
In fact, Theorem \[thm4\] and the above discussion imply the following result.
\[cor1\] Consider Eq. (\[lichnerowicz\]) on $\mathbb S^1(T) \times \mathbb S^2$. Let the branch of solutions of Eq. (\[lichnerowicz\]) which originates as $\phi \equiv 1$ for $b = 0$, and ends at $b = b_\mathrm{max}$, be denoted by (b), as in Figs. \[fig2\] and \[bifdiag\]. Let $k$ be the largest integer satisfying $k < T/(2\pi)$. Then there are $k$ bifurcation points on branch (b).
The missing step in the proof of the above corollary, which we fill in below, is to show that the equation $$\label{linf}
F_\phi (\phi,b^2) w = \lambda w$$ has a zero eigenvalue at $k$ isolated points on branch (b).
In the following sections we will discuss particular cases in which the eigenvalue problem (\[linf\]) can be solved exactly, or where the separation of variables can be carried out to some degree. We will then describe the numerical method which was used to compute the eigenvalues of Eq. (\[linf\]) in more general cases.
Linearization around solutions depending on $\alpha$ and $\theta$: separation of variables {#separation}
------------------------------------------------------------------------------------------
Consider a solution $\phi$ of Eq. (\[lichnerowicz\]) that does not depend on $\varphi$. The linearization of Eq. (\[lichnerowicz\]) around $\phi$ leads to the following eigenvalue problem: $$\label{schr}
- \partial_{\alpha \alpha} w - \frac{1}{\sin \theta} \partial_\theta (\sin \theta \partial_\theta w) - \frac{1}{\sin^2 \theta} \partial_{\varphi \varphi}w + V(\alpha,\theta) w = \lambda w,$$ where $$\label{schr_potential}
V(\alpha,\theta) = \frac{1}{4} - \frac{5}{4} \phi^4(\alpha,\theta) + \frac{7 b^2 \sin^2 \theta}{\phi^8(\alpha,\theta)}.$$ We require that the eigenfunctions $w \colon \mathbb S^1(T) \times \mathbb S^2 \to \mathbb R$ should be smooth on $\mathbb S^1(T) \times \mathbb S^2$.
The first immediate observation is that it is possible to factor out the dependence on $\varphi$. Assuming $w(\alpha, \theta, \varphi) = v(\alpha, \theta) z(\varphi)$ we obtain $$\label{schr2}
- \partial_{\alpha \alpha} v - \frac{1}{\sin \theta} \partial_\theta (\sin \theta \partial_\theta v) + \frac{m^2}{\sin^2 \theta}v + V(\alpha,\theta) v = \lambda v,$$ where $V$ is given by Eq. (\[schr\_potential\]) and $m = 0, \pm 1, \pm 2, \dots$. Of course, one can also introduce a new potential $V_m(\alpha, \theta) = V(\alpha,\theta) + m^2/\sin^2 \theta$, and write $$\label{schr3}
- \partial_{\alpha \alpha} v - \frac{1}{\sin \theta} \partial_\theta (\sin \theta \partial_\theta v) + V_m(\alpha,\theta) v = \lambda v.$$
Let $V_{min} = \min_{0 \le \alpha \le T, \; 0 \le \theta \le \pi} V(\alpha, \theta)$. Then $\lambda \ge V_{min}$.
This is a well known fact. To obtain this estimate, it is enough to multiply both sides of Eq. (\[schr\]) by $w$ and integrate with respect to $\int_0^T d \alpha \int_0^\pi d \theta \sin \theta \int_0^{2 \pi} d \varphi$. Integrating by parts one obtains $$\begin{aligned}
\lefteqn{\lambda \int_0^T d \alpha \int_0^\pi d \theta \sin \theta \int_0^{2 \pi} d \varphi w^2 = \int_0^T d \alpha \int_0^\pi d \theta \sin \theta \int_0^{2 \pi} d \varphi \left[ (\partial_\alpha w)^2 \right.} \\
& & \left. + (\partial_\theta w)^2 + \frac{1}{\sin^2 \theta} (\partial_\varphi w)^2 \right] + \int_0^T d \alpha \int_0^\pi d \theta \sin \theta \int_0^{2 \pi} d \varphi w V w,\end{aligned}$$ from which the inequality follows immediately.
Applying an analogous reasoning to Eq. (\[schr2\]) one gets an improved estimate $\lambda \ge V_{min} + m^2$ (note that $\min_{0 \le \theta \le \pi} m^2/\sin^2 \theta = m^2$).
If $\phi$ depends on $\alpha$, the function $w = \partial_\alpha \phi$ is an eigenfunction of Eq. (\[schr\]) corresponding to the eigenvalue $\lambda = 0$.
To see this, it is enough to differentiate Eq. (\[lichnerowicz\]) with respect to $\alpha$.
The following remark applies to Eq. (\[schr2\]) or Eq. (\[schr3\]).
\[remark\_separation\] If $\phi(\alpha, \theta) = \phi(\theta)$, then it makes sense to separate the variables as $v(\alpha,\theta) = v_1(\alpha) v_2(\theta)$.
Inserting this ansatz into Eq. (\[schr2\]) one obtains $$\label{separation_theta}
\lambda = \lambda_\theta + \left( \frac{2\pi j}{T} \right)^2,$$ where $j = 0, 1, \dots$, and $\lambda_\theta$ is an eigenvalue of the equation $$\label{separation_eq}
- \frac{1}{\sin \theta} \frac{d}{d\theta} \left( \sin \theta \frac{d v_2}{d \theta} \right) + \frac{m^2}{\sin^2 \theta} v_2 + V(\theta) v_2 = \lambda_\theta v_2.$$ The space $W$ of solutions $v_1$ is spanned by $$v_1 = A_j \sin(2 \pi j \alpha/T) + B_j \cos(2 \pi j \alpha/T), \quad j = 0, 1, \dots,$$ satisfying standard harmonic oscillator equation on $\mathbb S^1(T)$: $$\partial_{\alpha \alpha} v_1 + \left( \frac{2\pi j}{T} \right)^2 v_1 = 0.$$ Clearly the representation of the $O(2)$ group induced on $W$ is irreducible. Eigenvalues $2 \pi j/T$ are of course two-fold degenerate. Consequently, eigenvalues $\lambda$ are also degenerate.
Solutions for $b = 0$ {#linb0}
---------------------
### Linearization around $\phi \equiv 0$ and $\phi \equiv 1$
Equation (\[separation\_eq\]) can be solved in two limiting cases corresponding to $b = 0$ and $\phi \equiv 0$ or $\phi \equiv 1$, respectively. In both cases the potential $V$ is constant. We have $V \equiv -1$ for $\phi \equiv 1$ and $V \equiv 1/4$ for $\phi \equiv 0$. Defining $\mu = \cos \theta$ we can write Eq. (\[separation\_eq\]) as $$\frac{d}{d\mu}\left[ (1 - \mu^2) \frac{d}{d \mu} v_2 \right] + \left( \lambda_\theta - V - \frac{m^2}{1 - \mu^2} \right) v_2 = 0,$$ i.e., in the form of the general Legendre equation. It has regular solutions (associated Legendre polynomials $P^m_n(\mu)$), if and only if $\lambda_\theta - V = n(n+1)$, $n = 0, 1, \dots$, and $0 \le m \le n$ (or with equivalent negative values).
In summary, we get $$\label{eigen_phi_1}
\lambda = - 1 + \left( \frac{2\pi j}{T} \right)^2 + n(n+1), \quad j,n = 0, 1, \dots,$$ for $\phi \equiv 1$ and $$\lambda = \frac{1}{4} + \left( \frac{2\pi j}{T} \right)^2 + n(n+1), \quad j,n = 0, 1, \dots,$$ for $\phi \equiv 0$.
Clearly, for a given period $T$, there are $k + 1$ negative eigenvalues corresponding to $\phi \equiv 1$, where $k$ is the largest integer satisfying $k < T/(2\pi)$. The lowest eigenvalue is simply $\lambda_0 = -1$. The branch of solutions that originates at $\phi \equiv 1$, denoted as branch (b), ends at $b = b_\mathrm{max}$. On the other hand, we know from Theorem \[thm\_prem\] that the unique solution for $b = b_\mathrm{max}$ is marginally stable, i.e., the lowest eigenvalue corresponding to that solution is zero. It follows from continuity that there exist $k$ points in the interval $b \in (0, b_\mathrm{max})$, for which Eq. (\[schr2\]) admits zero eigenvalues. Another way of seeing this follows directly from Eq. (\[separation\_theta\]), where the lowest eigenvalue $\lambda_\theta$ reads $\lambda_\theta = -1$ for $b = 0$ and $\lambda_\theta = 0$ for $b = b_\mathrm{max}$. Consequently, it spans the whole range $(-1,0)$ for $b \in (0, b_\mathrm{max})$. The $k$ bifurcation points correspond to the values of $j$ in Eq. (\[separation\_theta\]) equal $j = 1, \dots, k$, each yielding $\lambda = 0$. This concludes the proof of Corollary \[cor1\]. Our numerical results obtained for $T = 5\pi$ and $T = 7\pi$ suggest that the lowest eigenvalue $\lambda_\theta$ is in fact strictly increasing with $b$; consequently, there are exactly $k$ bifurcation points on branch (b).
It is also not surprising (but of course highly nontrivial) that the number $k$ coincides with the number of periodic solutions obtained for $b = 0$ in Sec. \[bzero\]. This suggests that each of the bifurcating branches of solutions ends at a separate periodic solution at $b = 0$. We do not have any strict proof of this fact, but it is confirmed by the numerical results presented in Sec. \[sec\_bifurcation\].
Finally, note that there are no negative eigenvalues for $\phi \equiv 0$. This agrees with the fact that the branch (a) that originates at $\phi \equiv 0$ at $b = 0$ is stable.
### Linearization around $\phi = \phi(\alpha)$
The case with $b = 0$ and $\phi = \phi(\alpha)$ is also relatively simple, as the potential $V(\alpha)$ is now known explicitly \[here $\phi$ is given by Eq. (\[sol1\])\]. Separation of variables leads to a one-dimensional Schrödinger equation. Assuming $v(\alpha,\theta) = v_1(\alpha) v_2(\theta)$ we get, from Eq. (\[schr2\]), $$\label{schr_one_dim}
-\partial_{\alpha \alpha} v_1 + V(\alpha) v_1 = \lambda_\alpha v_1$$ and $$\frac{d}{d\mu}\left[ (1 - \mu^2) \frac{d}{d \mu} v_2 \right] + \left( \lambda - \lambda_\alpha - \frac{m^2}{1 - \mu^2} \right) v_2 = 0,$$ where again $\mu = \cos \theta$. Consequently, $\lambda = \lambda_\alpha + n(n+1)$, $n = 0, 1, \dots$
Equation (\[schr\_one\_dim\]) has the form $$-\partial_{\alpha \alpha} v + \frac{1}{4}v - \frac{5}{4} \phi^4 v = \lambda_\alpha v$$ or $$\label{eq_schr}
-\partial_{\alpha \alpha} v - \frac{5}{4} \phi^4 v = \left( \lambda_\alpha - \frac{1}{4} \right) v,$$ which is a one-dimensional Schrödinger equation with the potential $\tilde V = -\frac{5}{4}\phi^4$. Surprisingly, for the limiting solution (\[sol2\]) one gets $\tilde V = - \frac{15}{4} \mathrm{sech}^2 (\alpha - \alpha_0)$, which is a Pöschl–Teller potential, for which the one-dimensional Schrödinger equation is solvable [@pt]. In general, for the potential $\tilde V = - l(l+1)\mathrm{sech}^2 \alpha$, negative eigenvalues of the corresponding Schrödinger equation are given by $-(l - j)^2$, $j = 0, \dots, N$, where $N$ is the largest integer less than $l$. For $\tilde V = - \frac{15}{4} \mathrm{sech}^2 (\alpha)$, Eq. (\[eq\_schr\]) has a unique negative eigenvalue $(\lambda_\alpha - 1/4) = -9/4$ or, equivalently, $\lambda_\alpha = - 2$.
General numerical method {#lin_general_method}
------------------------
In the general case, we compute solutions of the eigenvalue problem Eq. (\[schr3\]) numerically, using the following variant of the Riesz method. We search for the solution by expanding it in $y_{kl}$, given by Eqs. (\[ya\]) and (\[yb\]), and $$z_{kl} = \sin \left( k \frac{2 \pi}{T} \alpha \right) P_l (\cos \theta), \quad k = 1, \dots, M, \quad l = 0, \dots, N,$$ i.e., we assume $v(\alpha, \theta)$ in the form $$\label{expansion2}
v(\alpha, \theta) = \sum_{k=0}^M \sum_{l=0}^N a_{kl} y_{kl} (\alpha, \theta) + \sum_{k=1}^M \sum_{l=0}^N b_{kl} z_{kl} (\alpha, \theta).$$ The functions $y_{kl}$ and $z_{kl}$ are the eigenfunctions of the operator $$\tilde L v \equiv - \partial_{\alpha \alpha} v - \frac{1}{\sin \theta} \partial_\theta \left( \sin \theta \partial_\theta v \right)$$ satisfying $$\tilde L y_{kl} = \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l (l+1) \right] y_{kl},$$ $$\tilde L z_{kl} = \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l (l+1) \right] z_{kl}.$$ Inserting expansion (\[expansion2\]) into Eq. (\[schr\]) we obtain the relation $$\begin{aligned}
\sum_{k=0}^M \sum_{l=0}^N \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l (l+1) + V_m(\alpha,\theta) - \lambda \right] a_{kl} y_{kl} + && \\
\sum_{k=1}^M \sum_{l=0}^N \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l (l+1) + V_m(\alpha,\theta) - \lambda \right] b_{kl} z_{kl} & = & 0.\end{aligned}$$ It is convenient to project the above equation on the functions $$\cos \left( k^\prime \frac{2 \pi}{T} \alpha \right) P_{l^\prime} (\cos \theta), \quad k^\prime = 0, \dots, M, \quad l^\prime = 0, \dots, N,$$ and $$\sin \left( k^\prime \frac{2 \pi}{T} \alpha \right) P_{l^\prime} (\cos \theta), \quad k^\prime = 1, \dots, M, \quad l^\prime = 0, \dots, N,$$ exploiting the fact that $V_m(\alpha, \theta)$ can be chosen as a symmetric function in $\alpha$. Then $$\int_0^T d \alpha \sin \left( k^\prime \frac{2 \pi}{T} \alpha \right) V_m(\alpha, \theta) y_{kl}(\alpha,\theta) = 0,$$ and $$\int_0^T d \alpha \cos \left( k^\prime \frac{2 \pi}{T} \alpha \right) V_m(\alpha, \theta) z_{kl}(\alpha,\theta) = 0.$$ This leads to the following algebraic eigenvalue problems: $$\label{lina}
\sum_{k=0}^M \sum_{l=0}^N \left\{ \tilde r_{k^\prime l^\prime k l} + \delta_{kk^\prime} \delta_{l l^\prime} \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l(l+1) - \lambda \right] \right\} a_{kl} = 0,$$ $k^\prime = 0, \dots, M$, $l^\prime = 0, \dots, N$, where $$\tilde r_{k^\prime l^\prime k l} = \frac{2 l + 1}{T} \int_0^\pi d \theta \sin \theta \int_0^T d \alpha P_{l^\prime}(\cos \theta) \cos\left( k^\prime \frac{2 \pi}{T} \alpha \right) V_m(\alpha,\theta) y_{kl},$$ and $$\label{linb}
\sum_{k=1}^M \sum_{l=0}^N \left\{ \tilde s_{k^\prime l^\prime k l} + \delta_{kk^\prime} \delta_{l l^\prime} \left[ \left( \frac{2 \pi}{T} \right)^2 k^2 + l(l+1) - \lambda \right] \right\} b_{kl} = 0,$$ $k^\prime = 1, \dots, M$, $l^\prime = 0, \dots, N$, where $$\tilde s_{k^\prime l^\prime k l} = \frac{2 l + 1}{T} \int_0^\pi d \theta \sin \theta \int_0^T d \alpha P_{l^\prime}(\cos \theta) \sin\left( k^\prime \frac{2 \pi}{T} \alpha \right) V_m(\alpha,\theta) y_{kl}.$$ Both eigenvalue equations (\[lina\]) and (\[linb\]) can be solved easily. In practice, it is convenient to introduce ‘one dimensional’ indices numbering the expansion functions. This allows one to rewrite Eqs. (\[lina\]) and (\[linb\]) in the standard matrix notation. For the indices $k = 0, \dots, M$ and $l = 0, \dots, N$ we introduce the index $J = k(N+1) + l + 1$, with the inverse relation given by $$k = \lfloor \frac{J - 1}{N + 1} \rfloor, \quad l = J - 1 - k(N+1),$$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. For the indices $k = 1, \dots, M$ and $l = 0, \dots, N$ we define $J = (k - 1)(N + 1) + l + 1$, with the inverse $$k = \lfloor \frac{J - 1}{N + 1} \rfloor + 1, \quad l = J - 1 - (k - 1)(N+1).$$
Numerical results {#lin_results}
-----------------
![\[widmoa\]Lowest eigenvalues of the linearized Eq. (\[schr2\]) corresponding to branches (b) and (c). Here $T = 5\pi$, $m = 0$.](wwlasne25.pdf){width="80.00000%"}
![\[widmob\]Same as in Fig. \[widmoa\], but for branches (b) and (d).](wwlasne45.pdf){width="80.00000%"}
![\[widmoc\]Lowest eigenvalues of the linearized Eq. (\[schr2\]). Here $T = 7 \pi$, $m = 0$.](Spectrum.pdf){width="80.00000%"}
![\[m1\]Lowest eigenvalues of the linearized Eq. (\[schr2\]). Here $T = 5 \pi$, $m = 1$.](m1.pdf){width="80.00000%"}
We will now discuss numerical solutions of the linearized equations introduced in Sec. \[linear\]. The purpose of this analysis is twofold. Firstly, we would like to illustrate the origin of the already known bifurcations in the system — they should correspond to the occurrence of zero modes of the linearized equation. Secondly, we would like to collect numerical evidence suggesting that no additional bifurcation points exist. As before we restrict ourselves to sample periods $T = 5 \pi$ and $T = 7 \pi$, and solutions of the Lichnerowicz equation that were already discussed in Sec. \[sec\_sols\].
In practice, we solve Eqs. (\[schr2\]) or (\[schr3\]) with the potential $V$ determined by a given numerical solution $\phi$. Figures \[widmoa\] and \[widmob\] show examples of the eigenvalues corresponding to branches (b), (c) and (d), discussed in Sec. \[sec\_bifurcation\]. We assumed $T = 5 \pi$ and $m = 0$. In both graphs the abscissa corresponds to the parameter $b$; the ordinate gives appropriate eigenvalues. Since we are interested in the occurrence of zero modes, we only plotted eigenvalues satisfying $\lambda \lessapprox 0.8$. We deliberately omit plotting strictly positive eigenvalues corresponding to the stable branch (a).
In both Figs. \[widmoa\] and \[widmob\] we plot eigenvalues corresponding to branch (b). They are denoted with dotted lines. Solid lines in Figs. \[widmoa\] and \[widmob\] denote, respectively, eigenvalues corresponding to bifurcating branches (c) and (d).
Branch (b) originates at $b = 0$ with a limiting solution $\phi \equiv 1$, and ends at the maximum value of the parameter $b = b_\mathrm{max}$. It consists of solutions $\phi$ that depend only on $\theta$. Consequently, one expects to find a spectrum predicted by Eq. (\[separation\_theta\]). The branch of lowest eigenvalues $\lambda_0 = \lambda_\theta$ starts at $b = 0$ with the value $\lambda_0 = \lambda_\theta = -1$, as given by Eq. (\[eigen\_phi\_1\]) with $j = n = 0$, and increases monotonically to $\lambda_0 = 0$ at $b = b_\mathrm{max}$. For $T = 5\pi$ higher eigenvalues are given by $\lambda_1 = \lambda_\theta + 4/25$, $\lambda_2 = \lambda_\theta + 16/25$, $\lambda_3 = \lambda_\theta + 36/25$, etc. The branch $\lambda_3$ is already strictly positive, but branches $\lambda_1$ and $\lambda_2$ have zeros at $b = b_1$ and $b = b_2$ respectively. These zeros give rise to two bifurcating branches of solutions: (c) and (d). Note that $\lambda_0 = -4/25$ at $b = b_1$. At $b = b_2$ we have $\lambda_0 = -16/25$ and $\lambda_1 = -12/25$.
Eigenvalues corresponding to branch (c) are shown in Fig. \[widmoa\]. For each solution $\phi$ belonging to this branch there is a nontrivial eigenfunction $\partial_\alpha \phi$ and an eigenvalue $\lambda = 0$. The only negative eigenvalue reads $\lambda_0 = -1.994$ at $b = 0$ and grows monotonically up to $\lambda_0 = -4/25$ at $b = b_1$ \[it bifurcates from the $\lambda_0$ eigenvalue that corresponds to branch (b)\]. The remaining eigenvalues are already nonnegative with possible zeros at $b = 0$ or $b = b_1$. Consequently they do not give rise to any bifurcating branches of solutions $\phi$.
The spectrum corresponding to branch (d) is more complex. There are 3 negative eigenvalues $\lambda_0$, $\lambda_1$, and $\lambda_2$. The lowest eigenvalue $\lambda_0$ reads $\lambda_0 = -1.659$ at $b = 0$. It grows monotonically up to $b = b_2$, where $\lambda_0 = - 16/25$ \[it bifurcates from the $\lambda_0$ eigenvalue that corresponds to branch (b)\]. The two eigenvalues $\lambda_1$ and $\lambda_2$ bifurcate from the eigenvalue $\lambda_1$ corresponding to branch (b) at $b = b_2$. We have, at $b = b_2$, $\lambda_1 = \lambda_2 = - 12/25$. At $b = 0$, $\lambda_1$ and $\lambda_2$ read $\lambda_1 = -1.658$ and $\lambda_2 = -0.127$. Note that at $b = 0$ the eigenvalues $\lambda_0$ and $\lambda_1$ are almost degenerate. This is a characteristic feature of solutions of the Schrödinger equation with a double-minimum potential[^1]. The eigenvalue $\lambda_1$ is monotonically growing with $b$, but $\lambda_2$ is not monotonic. Similarly to the spectrum of branch (c), there is also a zero eigenvalue corresponding to the eigenfunction $\partial_\alpha \phi$. All other eigenvalues are nonnegative.
We believe that this picture is generic. We give another example in Fig. \[widmoc\], which shows spectra analogous to those shown in Figs. \[widmoa\] and \[widmob\], but obtained for the case with $T = 7 \pi$.
In principle, one can also search for possible zero eigenvalues for $m \neq 0$. Figure \[m1\] shows a sample plot of the eigenvalues obtained for $m = 1$ and $T = 5\pi$. Except at $b = 0$, all eigenvalues are strictly positive; consequently, there is no need to investigate the spectra corresponding to higher values of $m$. For $m = 1$ and $b = 0$ the lowest eigenvalue $\lambda_0 = 0$. There is a numerical subtlety connected with this result. Our numerical computations actually yield a slightly negative eigenvalue $\lambda_0$ that tends to zero with an increase of the numerical accuracy. All these results support the conjecture that no additional branches of solutions depending on $\varphi$ bifurcate from branches (b), (c) and (d).
Concluding remarks {#conclusions}
==================
The so-called conformal method is probably the best known way of solving the Einstein constraint equations. In some form it was present already in the work of Lichnerowicz [@lichnerowicz]. A more recent version of the conformal method can be found in [@york; @niall_york]. We believe that the investigation of the Bowen-York initial data can provide new insights into the nature of the initial value problem in the case with a positive cosmological constant.
Here, we only dealt with the ‘rotating’ case, for which Bizoń, Pletka and Simon introduced a particularly elegant and simple generalization. In this approach the resulting Lichnerowicz equation depends explicitly only on one variable — the polar coordinate $\theta$. The compactification of the ‘radial’ variable $\alpha$ is another restriction. It is a remarkable property that there exist solutions periodic with respect to $\alpha$, which, in a sense, justify calling such a compactification a ‘natural’ one. In any case, the properties and the classification of solutions depend strongly on the fact that we are dealing with a Lichnerowicz equation on a compact domain.
Except for the case with $b = 0$ and the proof of the existence of symmetry-breaking bifurcations, the results presented in this work are mostly numerical. We focused on new branches of solutions, whose existence was predicted by Bizoń, Pletka and Simon in [@BPS]. It should be also noted that proving the completeness of the pattern of solutions presented in this paper remains an open problem. We hope that the numerical evidence given in the second part of this paper can motivate further research in this direction.
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank Piotr Bizoń and Walter Simon for many stimulating discussions concerning this project. PM acknowledges the financial support of the Narodowe Centrum Nauki Grant No. DEC-2012/06/A/ST2/00397. JK was supported by the Polish Ministry of Science and Higher Education within the ‘Diamentowy Grant’ program (No. 0153/DIA/2016/45).
Appendix: Gauss–Legendre–Fourier quadratures {#appendix-gausslegendrefourier-quadratures .unnumbered}
============================================
Gauss–Legendre and Gauss–Fourier quadratures are frequently used in the implementation of spectral methods. Here we recall appropriate formulae for completeness.
Let us define $x = \cos \theta$. The task is to compute integrals of the form $$\begin{aligned}
I_{kl} & \equiv & \frac{2l+1}{T} \int_0^\pi d\theta \sin \theta \int_0^T d\alpha f(\alpha, \theta) \cos\left( k \frac{2\pi}{T} \alpha \right) P_l(\cos \theta) \\
& = & \frac{2l+1}{T} \int_{-1}^1 dx \int_0^T d\alpha f(\alpha, x) \cos\left( k \frac{2\pi}{T} \alpha \right) P_l(x),\end{aligned}$$ where for simplicity no separate symbol is reserved for the function $f$ in the new coordinate system. Integration with respect to $x$ can be performed by introducing the $N$ zeros of the polynomial $P_N$ (they are denoted by $x_i$, $i = 1, \dots, N$) and $N$ weights $$w_i = \frac{2(1 - x_i^2)}{(N+1)^2 P_{N+1}^2(x_i)}, \quad i = 1, \dots, N.$$ Integration with respect to $\alpha$ requires $\tilde M = 2M + 1$ collocation points $\alpha_j = T (j - 1)/\tilde M$, $j = 1, \dots, \tilde M$. The integral $I_{kl}$ is then computed as $$I_{kl} = \frac{2l + 1}{\tilde M} \sum_{j = 1}^{\tilde M} \sum_{i = 1}^N f(\alpha_j,x_i) \cos\left( k \frac{2\pi}{T} \alpha_j \right) w_i P_l(x_i).$$ The above formula is exact for all functions $f$ that can be spanned by $y_{kl}$, $k = 0, \dots, M$, $l = 0, \dots, N$. Similarly, the integral $$\begin{aligned}
J_{kl} & \equiv & \frac{2l+1}{T} \int_0^\pi d\theta \sin \theta \int_0^T d\alpha f(\alpha, \theta) \sin\left( k \frac{2\pi}{T} \alpha \right) P_l(\cos \theta) \\
& = & \frac{2l+1}{T} \int_{-1}^1 dx \int_0^T d\alpha f(\alpha, x) \sin\left( k \frac{2\pi}{T} \alpha \right) P_l(x),\end{aligned}$$ is computed as $$J_{kl} = \frac{2l + 1}{\tilde M} \sum_{j = 1}^{\tilde M} \sum_{i = 1}^N f(\alpha_j,x_i) \sin\left( k \frac{2\pi}{T} \alpha_j \right) w_i P_l(x_i).$$
[99]{} P. Bizoń, S. Pletka, W. Simon, Initial data for rotating cosmologies, Class. Quantum Grav. 32, 175015 (2015) A. Vanderbauwhede, Local bifurcation and symmetry, Research Notes in Mathematics, 75, Pitman, Boston MA (1982) J.M. Bowen, J.W. York, Time-asymmetric initial data for black holes and black-hole collisions, Phys. Rev. D21, 2047 (1980) R. Beig, Generalized Bowen-York Initial Data, in S. Cotsakis and G.W. Gibbons (Eds.) Mathematical and Quantum Aspects of Relativity and Cosmology Lecture Notes in Physics 537, 55 (2000) M.G. Crandall, P.H. Rabinowitz, Bifurcation from Simple Eigenvalues, Journal of Functional Analysis 8, 321 (1971) M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Ration. Mech. and Anal. 52, 161 (1973) B. Premoselli, Effective multiplicity for the Einstein-scalar field Lichnerowicz equation, Calc. Var. 53, 29 (2015) E. Hebey, F. Pacard, D. Pollack, A Variational Analysis of Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds, Commun. Math. Phys. 278, 117 (2008) S. Pletka, Initial data for rotating cosmologies, Master thesis, University of Vienna (2015) H. Nariai, On a new cosmological solution of Einstein’s field equations of gravitation, Gen. Relativity Gravitation 31, 963 (1999), reprinted from Reports of Tohoku University (1951) P. Mach, All solutions of the $n = 5$ Lane–Emden equation, Journal of Mathematical Physics 53, 062503 (2012) P. Mach, N. Ó Murchadha, Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations, Class. Quantum Grav. 31, 135001 (2014) B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68, 209 (1979) Q. Jin, Y. Li, H. Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Diff. Equ. 13, 601 (2008) P.T. Chruściel, R. Gicquaud, Bifurcating Solutions of the Lichnerowicz Equation, Ann. Henri Poincaré 18, 643 (2017) J. Smoller, A. Wasserman, Symmetry-Breaking for Positive Solutions of Semilinear Elliptic Equations, Archive for Rational Mechanics and Analysis 95, 217 (1986) G. Pöschl, E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Zeitschrift für Physik 83, 143 (1933) F. Hund, Zur Deutung der Molekelspektren. III, Zeitschrift für Physik 43, 805 (1927) R.J.W. Hodgson, Y.P. Varshni, Splitting in a double-minimum potential with almost twofold degenerate lower levels, J. Phys. A 22, 61 (1989) A. Lichnerowicz, L’intégration des équations de la gravitation relativiste et le problème des n corps, Journal de Mathématiques Pures et Appliquées. Neuvième Série 23, 37 (1944) J.W. York, Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity, Journal of Mathematical Physics 14, 456 (1973) N. Ó Murchadha, J.W. York, Initial-value problem of general relativity. I. General formulation and physical interpretation, Physical Review D 10, 428 (1974)
[^1]: It is a well-known effect, observed already in 1927 by Hund [@hund]; see also the discussion and references in [@hodgson].
|
---
abstract: 'In the tight-binding approximation the Harper like equation describing an electron in 3D crystal subject to a uniform magnetic field is obtained. It is supposed that the vector ${\bf H}$ can be oriented along several directions in the lattice. The Fermi surfaces relevant to a magnetic flux $p/q=1/2$ in a simple cubic lattice are built. The quantization rules in magnetic fields slightly distinguished from $p/q=1/2$ are investigated.'
address: |
Nizhny Novgorod State University\
RUSSIA, 603600 Nizhny Novgorod, Gagarin Ave., 23
author:
- 'V.Ya.Demikhovskii[^1], A.A.Perov, D.V.Khomitsky'
title: Formation of New Fermi Surfaces in 3D Crystals at Ultra High Magnetic Field with Different Orientations
---
=0.5cm
number(s): 71.18.+y, 71.20.-b
[*Keywords:*]{} Fermi surfaces, energy band structure, ultra high magnetic field
The effects of a high magnetic field on Bloch electrons are fascinating problem with very rich physics. This is not surprising since there are two fundamental periods in the problem – period of the potential and the period of the phase of the wave function – and the interplay of these two periods gives very interesting spectrum (Hofstadter butterfly) and eigenstate structure. Starting from classical works of Azbel \[1\], Hofstadter \[2\], Wannier \[3\] (see also \[4-7\]) this problem attracts the increasing attention. During the last decade the one electron quantum states in lateral superlattices in the presence of perpendicular magnetic field have been studied in a series of theoretical \[8-10\] and experimental \[11,12\] works. But unfortunately, all works mentioned above deals only with 2D crystals subject to a perpendicular magnetic field. Nowadays when in VNIIEF (Sarov) the magnetic field up to 28 MGs is reached \[13\] the idea of observation of details forecasted by the theory in the crystals with the lattice spacing of $a=0.6~nm$ and more does not seem to be unaccessible.
In this paper we shall consider electronic states in 3D crystals subject to ultrahigh magnetic field. We consider that the vector H is parallel to arbitrary translation vector of a lattice. In a tight-binding approximation the wave function satisfying the generalized Bloch-Peierls conditions is obtained and Harper like equation for different orientations of a magnetic field is derived. As an example, the Fermi surfaces for magnetic subbands relevant to rational value of a magnetic flux $p/q=1/2$ are built. The quantization rules for magnetic flux near this rational value of flux quanta are studied.
First of all in the tight-binding approximation let us derive Harper’s equation for the simple cubic lattice subject to uniform magnetic field. To be specific let the basis vectors on the $x,y$ plane be ${\bf a}_1$ and ${\bf a}_2$, and the magnetic field is applied along lattice basis vector ${\bf a}_3\parallel 0z$ ($|{\bf a}_1|=
|{\bf a}_2|=|{\bf a}_3|=a$). Let vector potential is be chosen in Landau gauge ${\bf A}=(0,H\,x,0)$. Then electron wave function must satisfy the generalized Bloch conditions (Peierls conditions) \[14\] $$\psi_{{\bf k}}({\bf r})=\psi_{{\bf k}}(x+qa,y+a,z+a)\exp(-ik_xqa)
\exp(-ik_ya)\exp(-ik_za)\exp(-2\pi ipy/a),\eqno(1)$$ due to the fact that the vector potential is not a periodic function of the coordinates. Here $p$ and $q$ are mutually prime integers which define the number of magnetic flux quanta per two-dimensional square elementary cell $$\frac{p}{q}=\frac{\Phi}{\Phi_0}=\frac{|e|H|[{\bf a}_1\times{\bf a}_2]|}
{2\pi\hbar c},\eqno(2)$$ where $\Phi_0=hc/|e|$ is the magnetic flux quanta. In a presence of a magnetic field the electron quantum states must be classified in accordance with irreducible representations of magnetic translation group $${\bf a}_{mag}=nq{\bf a}_1+m{\bf a}_2+l{\bf a}_3, \quad (n,m,l{\rm ~are~
integer~numbers}).$$ The quasimomentum ${\bf k}$ takes values in the magnetic Brillouin zone $$-\frac{\pi}{qa}\le k_x\le \frac{\pi}{qa},\quad
-\frac{\pi}{a}\le k_y\le \frac{\pi}{a},\quad
-\frac{\pi}{a}\le k_z\le \frac{\pi}{a}.\eqno(3)$$
In the tight-binding approximation the electron wave function which satisfies the conditions (1) can be written in the form \[15\] $$\psi_{{\bf k}}({\bf r})=\sum_{n,m,l}g_n({\bf k})\exp(i{\bf ka_n})
\exp\Bigg(-2\pi i\frac{p}{q}\frac{(y-ma)}{a}n\Bigg)\psi_0(x-na,y-ma,z-la)
\eqno(4)$$ where $\psi_0({\bf r}-{\bf a_n})$ is the atomic function. The path-dependent geometric phase $2\pi\frac{p}{q}\frac{(y-ma)}{a}n$ of the wave function is known to play a fundamental role in the problem.
By substituting of (4) into Schrödinger equation and evaluating the transfer integrals between neighbour sights we shall obtain a system of difference equations for coefficients $g_n$. The magnetic field has a non-trivial influence on the transfer integrals between neighbour lattice sights in the $x$ direction $$A=\int\exp\Bigg(\pm2\pi i\frac{p}{q}\frac{y-ma}{a}\Bigg)\psi_0({\bf r}-
{\bf a_n^{\prime}})
(V({\bf r})-U({\bf r-a_n}))\psi_0({\bf r}-{\bf a_n})\, d\tau,\eqno(5)$$ where ${\bf a_n^{\prime}}=\Big((n\pm 1)a,ma,la\Big)$, $V({\bf r})$ is the crystal scalar potential with periodicity $a$ in three dimensions and $U$ is the atomic potential. Calculating the transfer integrals in the mean-value approximation we substitute $y=ma$ and obtain $A=E_0$ where $E_0$ is the transfer integral (5) in the absence of a magnetic field. In the $y$ direction the transfer integral is $$B=\exp\Bigg(\pm2\pi i\frac{p}{q}n\Bigg)\int\psi_0({\bf r}-{\bf a_n^{\prime}})
(V({\bf r})-U({\bf r-a_n}))\psi_0({\bf r}-{\bf a_n})\, d\tau,\eqno(6)$$ where ${\bf a_n^{\prime}}=\Big(na,(m\pm 1)a,la\Big)$ and in the $z$ direction the transfer integral is equal to $E_0$. As a result we obtain the well-known Harper’s equation \[4\] $$\exp(ik_xa)g_{n+1}+\exp(-ik_xa)g_{n-1}+2g_n\cos\Big(2\pi n\frac{p}{q}+k_ya
\Big)=\varepsilon(k_x,k_y,k_z)g_n,\eqno(7)$$ where $$\varepsilon(k_x,k_y,k_z)=\varepsilon_{\perp}(k_x,k_y)-2\cos k_za\eqno(7a)$$ is the dimensionless energy measured in terms of $E_0$. The spectrum $\varepsilon_{\perp}(k_x,k_y)$ in Eq.(7) consists of $q$ bands and depend on one single parameter $p/q$ counting the number of magnetic flux quanta per unit cell (Fig.1). If flux is irrational the spectrum is a singular-continuum – uncountable but measure zero set of points (Cantor set). Originally the Eq.(7) was derived by Harper \[4\] using Peierls substitution \[14\].
For some values of $p/q=1/2,\,1/3,\,1/4$ it is possible to solve the Eq.(7) analytically. As a result one can obtain the following expressions for dispersion laws for $q$ magnetic subbands: for $p/q=1/2$ (two subbands) $$\displaylines{\varepsilon^{(1,2)}_{\perp}(k_x,k_y)=
\pm 2\sqrt{\cos^2k_xa+\cos^2k_ya},\hfill\llap{(8)}\cr}$$ for $p/q=1/3$ (three subbands) $$\displaylines{\varepsilon^{(1)}_{\perp}(k_x,k_y)=-2\sqrt{2}\cos\Big(\frac{2\pi}{3}
+(\arctan\beta)/3\Big),\hfill\cr
\varepsilon^{(2)}_{\perp}(k_x,k_y)=2\sqrt{2}\cos\Big(\frac{\pi}{3}
+(\arctan\beta)/3\Big),\hfill\cr
\varepsilon^{(3)}_{\perp}(k_x,k_y)=-2\sqrt{2}\cos
\Big((\arctan\beta)/3\Big),\hfill\llap{(9)}\cr}$$ where $\beta=\sqrt{8-\alpha^2}/\alpha,\,\alpha=\cos 3k_xa+\cos 3k_ya$; and for $p/q=1/4$ (four subbands) $$\displaylines{\varepsilon^{(1,2,3,4)}_{\perp}(k_x,k_y)=
\pm\sqrt{4\pm 2\sqrt{4-\sin^22k_xa-\sin^22k_ya}}.\hfill\cr}$$
Fig.2 shows the Fermi surfaces in the lowest subband for $p/q=1/2$ obtained using (7a) and (8) in the first Brillouin zone $-\pi/qa\le k_x\le
\pi/qa,\;-\pi/qa\le k_y\le\pi/qa,\;-\pi/a\le k_z\le\pi/a$. Three surfaces for different representative energies are plotted. For an ellipsoid-type surface such as (1) ($|{\bf k}|\to 0$) the simple analytical expression for an energy spectrum $\varepsilon(\bf k)$ can be easily obtained with the help of (7a) and (8). The relevant effective masses are determined as $$m^{*}_{x,y}=\frac{\hbar^2}{\sqrt{2}E_0a^2},\; m^{*}_z=\frac{\hbar^2}
{2E_0a^2}.\eqno(11)$$ The asymptotic spectrum near the point $k_{x,y}\to\pi/2a$ ((2)-type surface) is $\varepsilon(k_x,k_y,k_z)\sim\sqrt{k_x^2+k_y^2}-2\cos k_z$.
Now we shall discuss the magnetic quantization rules at values of a flux slightly distinguished from $p/q=1/2$: $$\frac{p}{q}=\frac{1}{2}+\frac{1}{q^{\prime}},\quad q^{\prime}\gg 1.\eqno(12)$$ This corresponds to the magnetic field $H=H(1/2)+\Delta H(1/q^{\prime})$. As one can see from Fig.1 near value of $p/q=1/2$ the spectrum represents a system of narrow subbands (practically discrete levels). Here it is possible to see an equidistant spectrum, points of level accumulation and ranges where $\varepsilon_{\perp}\sim\sqrt{N}$, ($N$ is the level number).
The spectrum $\varepsilon_{\perp}(k_x,k_y)$ of Harper’s equation near one half of flux quanta can be studied analytically due to the self-similarity of its structure. In the case $p/q=1/2$ two subbands and two subsystems in the eigenvector distribution can be observed. It may be derived analytically from Harper’s equation that every eigenvector function from a subsystem has an argument step $\Delta n=2$.
The magnetic flux (12) gives us the diagonal term in Harper’s equation of the form $2\cos(\pi n+2\pi n/q^{\prime})g_n=2\cos\pi n\cos(2\pi n/q^{\prime})
g_n$. Now $q^{\prime}$ is a new period of the equation. Since $\cos\pi n=(-1)^n$ our system of equations (7) consists of two subsystems where only the sign varies. It is easy to construct an equation for each subsystem. After simple algebra we obtain $(k_x=k_y=0)$: $$g_{n+2}+g_{n-2}+2\cos\frac{4\pi}{q^{\prime}}n\,g_n=
(\varepsilon^2_{\perp}-4)g_n.\eqno(13)$$ Now $\varepsilon^2_{\perp}-4=\varepsilon_{\perp}^{\prime}$ is the energy in Harper’s equation (7) and $\varepsilon_{\perp}=
\pm\sqrt{\varepsilon_{\perp}^{\prime}+4}$ describes the structure of the spectrum near one half of flux quanta and $\cos k_za=0$. Two signs at the square root correspond to two groups of energy subbands. The energy $\varepsilon_{\perp}$ describes all parts of the spectrum near $p/q=1/2$ (Fig.1). We can obtain from (13) the “square root” dependence at $\varepsilon_{\perp}=0$, the clustering points $\varepsilon_{\perp}=\pm 2$ and Landau levels near $\varepsilon_{\perp}=\pm 2\sqrt{2}$. These peculiarities of a spectrum can be obtained with the help of Onsager-Lifshitz quasiclassical quantization rules on the Fermi surfaces (Fig.2a,b). It is easy to see that magnetic quasiclassical quantization on the (1)-type surface results to the eqidistant energy spectrum at the bottom of the lowest magnetic subband. Accordingly, the magnetic Onsager-Lifshitz quantization near $k_z\to 0$ on (2)-type surface yields the “square root” spectrum. At last, it is clear that in magnetic field ${\bf H\parallel 0z}$ there are self-crossing orbits on (3)-type surface. It gives the accumulation point in the spectrum $\varepsilon_{\perp}$ near $p/q=1/2$ (see Fig.1). It can be stressed that level spacing obtained from (13) has the same value that was derived using quasiclassical quantization.
In accordance with Harper’s equation (7) and (7a) the partial overlapping between $q$ energy subbands is observed. When $p/q=1/2$ the overlapping takes place in the energy interval $(-2,\,2)$. It is clear that when the electron concentrations leading to $\varepsilon_{F}\in (-2,\,2)$ in the magnetic field $H=H(1/2)+\Delta H(1/q^{\prime})$ the de Haas - van Alfen oscillations with different periods corresponding to the different energy level series (Fig.1) can be observed.
Further it is necessary to make the following relevant note. The wave function (4) describes non-homogeneous probability distribution on the lattice sights. This is due to the amplitudes $g_n({\bf k})$ which give us non-uniform shapes. However, in real crystals due to Coloumb interaction the static electron density distribution must maintain its symmetry even in the presence of magnetic field.
Fortunately, the spectrum of Harper’s equation is $q$-fold degenerate since $\varepsilon(k_x,k_y+2\pi j/qa,k_z)=\varepsilon(k_x,k_y,k_z),\;
j=0,1,2,\dots,q-1$ and a wave function with homogenious electron density distribution can be written as a combination of (4). Let us introduce $$\psi^{\prime}_{{\bf k}}({\bf r})=\sum_{j=0}^{q-1}C_j({\bf k})\hat K_j
\hat T_{ja}\psi_{{\bf k}}({\bf r}),\eqno(14)$$ where $\psi_{{\bf k}}({\bf r})$ is from (4), $C_j({\bf k})=\exp(-ik_xja)$, $\hat T_{ja}\psi=\psi(x+ja,y,z)\exp(ief/\hbar c)$ ($f=-Hyja$), the operator $\hat K_j$ transforms $\psi_{{\bf k}}$ with ${\bf k}=(k_x,k_y,k_z)$ into $\psi_{{\bf k}^{\prime}}$ with ${\bf k}^{\prime}=(k_x,k_y-2\pi pj/qa,k_z)$. After this procedure we can write (14) in the form $$\psi_{{\bf k}}^{\prime}({\bf r})=D({\bf k})\sum_{{\bf n}}\exp\Bigg(-2\pi i
\frac{p}{q}\frac{(y-ma)}{a}n\Bigg)\exp(i{\bf ka_n})\psi_0({\bf r-a_n}),
\eqno(15)$$ where $D({\bf k})=\sum_{j=0}^{q-1}g_{n+j}({\bf k})$ does not depend on $n$ due to the periodicity of the system. Now (15) describes homogeneous electron density distribution on the lattice sights and corresponds to the same energy as $\psi_{{\bf k}}$.
Let us consider now the case of a simple cubic lattice subjected to a uniform magnetic field oriented along the diagonal of a square in the plane $(x,y)$. The magnetic field with amplitude $H$ has now the following Cartesian coordinates: $${\bf H}=\frac{H}{\sqrt{2}}(1,1,0).\eqno(16)$$ It is convenient to choose a new coordinate system making a rotation about the old $z$ axis by $\pi/4$ and defining the magnetic field orientation as $x_3$. This transformation is written as following: $$\displaylines{\hfill x=\frac{x_3-x_1}{\sqrt{2}},\hfill\cr
\hfill y=\frac{x_3+x_1}{\sqrt{2}},\hfill\cr
\hfill z=x_2.\hfill\cr}$$
We choose a new elementary cell: it is a rectangular parallelepiped based on the new coordinate system vectors: a square with the side $\sqrt{2}a$ at the base in the plane $(x_1,x_3)$ and the height $a$. This cell is non-primitive: it consists of two atoms (additional sights are located at the centers of top and bottom faces). In this coordinate system the magnetic field is oriented along one of the elementary cell basic vectors (namely, ${\bf a}_3$). According to the general principles in this case the classification of electron states is possible \[16\]. Now the first Brillouin zone has an area two times less with respect to the initial value. The magnetic field in this coordinate system is written: $${\bf H}=H(0,0,1)\eqno(17)$$ and it is convenient to choose a vector potential using Landau gauge: $${\bf A}=(-H\, x_2,0,0).\eqno(18)$$
The wavefunction in the tight-binding approximation can be written as following: $$\displaylines{\hfill\psi_{{\bf k}}({\bf r})=\sum_{{\bf n}}g_n({\bf k})
\Bigg\{\exp\Big(2\pi i\frac{p}{q}\frac{x_1-m\sqrt{2}a}
{a/\sqrt{2}}n\Big)\exp(i{\bf ka_n})\psi_0({\bf r-a_n})+\hfill\cr
\hfill +\exp\Big(2\pi i\frac{p}{q}\frac{x_1-(m+1/2)\sqrt{2}a}
{a/\sqrt{2}}n\Big)\exp(i{\bf k(a_n+d)})\psi_0({\bf r-(a_n+d)})\Bigg\}.
\hfill\llap{(19)}\cr}$$ Here ${\bf ka_n}=k_1\sqrt{2}am+k_2an+k_3\sqrt{2}al$, $\psi_0$ is an isolated atomic function in the presence of the magnetic field, the amplitudes $g_n({\bf k})$ are the subject of further research and $g_{n+q}=g_n$. The indexes $(m,n,l)$ are taking all integer values independently, $n$ numerates sights along $x_2$ axis, perpendicular to the magnetic field. The sum is taken over all lattice sights and two addendums in (22) correspond to the two existing crystalline subsystems. Each of them is a simple tetragonal lattice with the elementary cell being a rectangular parallelepiped with square of the side $\sqrt{2}a$ in its base and the height $a$. These subsystems are displaced on the vector ${\bf d}$ with new coordinates $(a/\sqrt{2},0,a/\sqrt{2})$. The magnetic translation group is constructed by the $q$ - time multiplication of the translation period along $x_2$ axis: $x_2\to x_2+qa$. The other translations are not changed. Their periods in the new coordinate system are $\sqrt{2}a$ which is the diagonal of a square with the side $a$. The classification of wavefunctions is possible when the number of flux quanta $\Phi/\Phi_0$ through the area $a^2/\sqrt{2}$ (being the minimal inter-sight area in the transversal to magnetic field projection) is a rational number: $\Phi/\Phi_0=p/q$ where $p$ and $q$ are mutually prime integers.
The substitution of (19) into Schrödinger equation is performed and the transfer integrals are calculated. We will have the equation for the energy $\varepsilon$ and the amplitudes $g_n({\bf k})$: $$e^{ik_2a}g_{n+1}+e^{-ik_2a}g_{n-1}+4\cos\Big(\frac{k_3a}{\sqrt{2}}\Big)
\cos\Big(2\pi\frac{p}{q}n+\frac{k_1a}{\sqrt{2}}\Big)g_n=\varepsilon g_n.
\eqno(20)$$ It should be mentioned that this equation may be obtained using Peierls substitution ${\bf k}\to \hat k-\frac{e{\bf A}}{\hbar c}$ with the initial spectrum at null magnetic field. In old coordinates it was written as $\varepsilon=2(\cos k_xa+\cos k_ya+\cos k_za)$, and in new coordinates $(x_1,x_2,x_3)$ it is $$\varepsilon=2\cos k_2a+4\cos\frac{k_3a}{\sqrt{2}}\cos\frac{k_1a}{\sqrt{2}}.$$
The equation (20) is a generalization of Harper’s equation for the case of magnetic field parallel to a diagonal of a square which is a face of the elementary cell cube. Keeping a three-diagonal structure (20) has some remarkable features with respect to standart Harper’s equation. First, (20) describes total (not “transversal” $\varepsilon_{\perp}(k_1,k_3)$) energy as a function of quasimomentum ${\bf k}$: $\varepsilon=\varepsilon(k_1,k_2,k_3)$. Now $k_3=const$ corresponds to the plane perpendicular to ${\bf H}$ and $k_1=const$ describes the plane parallel to magnetic field. Second, (20) corresponds to anisotropic Harper’s equation with anisotropy ratio $$4\cos\Big(\frac{k_3a}{\sqrt{2}}\Big)=\lambda\eqno(21)$$ depending on the cross-section transversal to the magnetic field which is fixed by choosing the value of $k_3$. It is well-known that Harper’s equation in the case $\lambda\ne 2$ describes 2D square lattice with anisotropic transfer integrals \[5,6\]. According to (21) the persistent interval for $\lambda$ is $-4\le\lambda\le 4$. In the quasiclassical limit it describes the situation where open-type trajectories are available. The contribution of open trajectories to the whole topological structure varies with the respect to a cross-section $k_3=const$ being chosen. For example, the cross-section $k_3=\pi\sqrt{2}/3a$ corresponds to $\lambda=2$ which is the case of isotropic Harper’s equation and only close-type orbits are avaliable. This is illustrated in Fig.3 where Fermi surfaces (Fig.3a,b) and spectrum (Fig.3c) are shown for $p/q=31/60$. Then, the cross-section $k_3=\pi/\sqrt{2}a$ gives us $\lambda=0$ which is a “full anisotropic” limit for Harper’s equation. Here as one can see in Fig.4a,b only opened-type orbits are avaliable. It gives a continious energy spectrum (Fig.4c). For remaining values of $\lambda$ both opened and closed orbits are present. In the last case the spectrum $\varepsilon_{\perp}$ consists of a continious and quasi discrete (narrow subbands) parts.
As a conclusion, we suggested the expression for electron wave function which satisfies to Peierls conditions and obtained Harper’s equation from initial principles of the tight-binding model. The Fermi surfaces are built for different orientations of the magnetic field. In the following paper we plan to study the electron states in 3D crystals in the case of the magnetic field orientation parallel to the arbitrary translation vector of the lattice. The properties of de Haas - van Alfen effect in this problem will be studied.
One of us (V.Ya.Demikhovskii) wants to thank V.D.Selemir for constant support and Prof. Michael von Ortenberg for numerous discussions of this problem. This research was made possible due to financial support from the Russian Foundation for Basic Research (Grant No. 98-02-16412).
1. M.Ya.Azbel’, Zh. Exper. i Teor. Fiz. [**46**]{}, 929 (1964) \[[*in Russian*]{}\].
2. D.R.Hofstadter, Phys.Rev.B [**14**]{}, 2239 (1976).
3. F.H.Claro, G.H.Wannier, Phys.Rev.B [**19**]{}, 6068 (1979).
4. P.G.Harper, Proc.Phys.Soc.(London) [**A 68**]{}, 874 (1955).
5. H.Hiramoto, M.Kohmoto, Int.Journ.of Modern Physics [**6**]{}, Nos. 3&4, 281 (1992).
6. M.Kohmoto, Y.Hatsugai, Phys.Rev.B [**41**]{}, 9527 (1990).
7. A.Barelli, R.Fleckinger, Phys.Rev.B [**46**]{}, 11559 (1992).
8. H.Silberbauer, J.Phys.: Condens. Matter [**4**]{}, 7355 (1992).
9. V.Ya.Demikhovskii, A.A.Perov, JETP [**87**]{}, 973 (1998)
10. V.Ya.Demikhovskii, A.A.Perov, Phys. Low-Dim. Structures [**7/8**]{}, 135 (1998).
11. D.Weiss, M.L.Roukes, A.Menschig, et al., Phys.Rev.Lett. [**66**]{}, 27 (1991).
12. T.Schlösser, K.Ensslin, J.P.Kotthaus, et al., Semicond. Sci. Technol. [**11**]{}, 1582 (1996).
13. B.A.Boyko, A.I.Bykov, M.I.Dolotenko et al, book of abstracts, VIIIth Int. Conference on Megagauss Magnetic Field Generation and Related Topics, p.149, Tallahassee, USA.
14. R.E.Peierls, Z.Phys. [**80**]{}, 763 (1933).
15. V.Ya.Demikhovskii, A.A.Perov, D.V.Khomitsky, Proc. of the VIIIth International Conference on Megagauss Magnetic Field Generation and Related Topics (MEGAGAUSS-VIII), Tallahassee, USA (1998) (in press).
16. J.Zak, Phys.Rev.A [**134**]{}, 1602 (1964); Phys.Rev.A [**134**]{}, 1607 (1964).
[^1]: Corresponding author. Mail: Russia, 603006, Nizhny Novgorod, M.Gorky St., 156-3; E-mail: demi@phys.unn.runnet.ru; FAX: 007 8 312 658592
|
---
abstract: |
Scintillation properties of pure CsI crystals used in the shower calorimeter being built for precise determination of the $\pi^+$$\rightarrow$$\pi^0e^+\nu_e$ decay rate are reported. Seventy-four individual crystals, polished and wrapped in Teflon foil, were examined in a multiwire drift chamber system specially designed for transmission cosmic muon tomography. Critical elements of the apparatus and reconstruction algorithms enabling measurement of spatial detector optical nonuniformities are described. Results are compared with a Monte Carlo simulation of the light response of an ideal detector. The deduced optical nonuniformity contributions to the FWHM energy resolution of the PIBETA CsI calorimeter for the $\pi^+$$\rightarrow$$e^+\nu$ 69.8$\,$MeV positrons and the monoenergetic 70.8$\,$MeV photons were 2.7$\,$% and 3.7$\,$%, respectively. The upper limit of optical nonuniformity correction to the 69.8$\,$MeV positron low-energy tail between 5$\,$MeV and 55$\,$MeV was $+\,$0.2$\,$%, as opposed to the $+\,$0.3$\,$% tail contribution for the photon of the equivalent total energy. Imposing the 5 MeV calorimeter veto cut to suppress the electromagnetic losses, [GEANT]{}-evaluated positron and photon lineshape tail fractions summed over all above-threshold ADCs were found to be 2.36$\pm\,$0.05(stat)$\pm\,$0.20(sys)$\,$% and 4.68$\pm\,$0.07(stat)$\pm\,$0.20(sys)$\,$%, respectively.
[PACS Numbers: 87.59.F; 29.40.Mc; 24.10.Lx]{}
[*Keywords:*]{} Computed tomography; Scintillation detectors; Monte Carlo simulations
address:
- 'Department of Physics, University of Virginia, Charlottesville, VA 22901, USA'
- 'Institute Rudjer Bošković, Bijenička 46, HR-10000 Zagreb, Croatia'
- 'Physics Department, Hampton University, Hampton, VA 23668, USA'
- 'Paul Scherrer Institut, Villigen PSI, CH-5232, Switzerland'
- 'Institut für Teilchenphysik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland'
- 'Deutsches Elektronen Synchrotron, D-22603 Hamburg, Germany'
- 'Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85281, USA'
- 'Department of Physics, University of Massachusetts, Amherst, MA 01003, USA'
- 'Institute for High Energy Physics, Tbilisi State University, 380086 Tbilisi, Georgia'
- 'Jet Propulsion Laboratory, Pasadena, CA 91109, USA'
author:
- 'E. Frlež'
- 'I. Supek'
- 'K. A. Assamagan'
- 'Ch. Brönnimann'
- 'Th. Flügel'
- 'B. Krause'
- 'D. W. Lawrence'
- 'D. Mzavia'
- 'D. Počanić'
- 'D. Renker'
- 'S. Ritt'
- 'P. L. Slocum'
- 'N. Soić'
title: |
Cosmic muon tomography of pure\
cesium iodide calorimeter crystals
---
psfig.sty
,
,
,
,
,
,
,
,
,
,
,
,
Introduction
============
The PIBETA collaboration has proposed an experimental program [@Poc88] with the aim of making a precise determination of the $\pi^+$$\rightarrow$$\pi^0e^+\nu_e$ decay rate at the Paul Scherrer Institute (PSI). The proposed technique is designed to achieve an overall level of accuracy in the range of $\sim\,$0.5$\,$%, improving thus on the present branching ratio uncertainty of 4$\,$% [@McF85]. The (1.025$\pm\,$0.034)$\times$$10^{-8}$ pion beta decay ($\pi\beta$) branching ratio will be remeasured relative to the $10^{4}$ times more probable $\pi^+$$\rightarrow$$e^+\nu_e$ decay rate, that is known with the combined statistical and systematic uncertainty of $\sim\,$0.40$\,$% [@Cza93; @Bri94].
The Standard Model description of the $\pi\beta$ decay and its radiative corrections [@Sir78] enables a stringent test of the conserved vector current (CVC) hypothesis and the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. The most accurate extraction of the CKM matrix element $V_{ud}$, based on the superallowed Fermi transitions in nuclei, involves the theoretical nuclear overlap corrections that dominate the 0.3$\,$% total uncertainty [@Sir89]. Recent measurements of nine different nuclear transition rates confirm the CVC hypothesis at the level of 4$\times$$10^{-4}$, but violate the three-generation CKM unitarity condition by more than twice the estimated error [@Tow95b]. Complementary neutron decay experiments yield a result that also differs from unitarity by over two standard deviations, but in the opposite sense [@Tow95b]. It is of fundamental interest to measure the branching ratio of the $\pi^+$$\rightarrow$$\pi^0e^+\nu_e$ decay at the 0.5$\,$% level or better to check the consistency with the nuclear and neutron $\beta$ decay values and the unitary prediction of the Minimal Standard Model (MSM). Moreover, the precisely measured $\pi\beta$ branching ratio could be used to constrain masses and couplings of additional neutral gauge bosons in grand unified theories. If such couplings are directly discovered at future collider experiments, precision low-energy data will be an essential ingredient in extending the MSM framework [@Mar87].
The central part of the PIBETA detector system is a high-resolution, highly segmented fast shower calorimeter surrounding the active stopping target in a near-spherical geometry. The requirements imposed on the calorimeter design were:
- high energy resolution for effective suppression of background processes;
- high segmentation and fast timing response to handle the high event rates;
- acceptable radiation, mechanical and chemical resistance;
- compact geometry to simplify the operation and reduce the cost.
These design constraints are met in a spherical detector with individual calorimeter modules made from undoped cesium iodide (CsI) crystals. The nonactivated alkali iodides have been known for more than thirty years to exhibit a near-ultraviolet emission component when excited by ionizing radiation [@Mur65; @Bat77]. The pure CsI material was reintroduced as a fast, rugged, and relatively inexpensive scintillator material by Kobayashi et al. [@Kob87] and Kubota et at. [@Kub88a; @Kub88b]. The scintillation characteristics of pure CsI crystals have been reported subsequently by a number of experimental groups. These investigations have covered light readout techniques, scintillation decay times, origins of different emission components, crystal light yields, and energy and timing resolutions [@Kes91; @Woo90; @Sch90; @Utt90; @Gek90; @Ham95], radiation resistances [@Woo92; @Wei93], wrapping and tuning methods and the uniformity of light responses [@Bro95; @Dah96]. The design and performance of pure CsI electromagnetic calorimeters have been recently described in Refs. [@Mor89] (NMS at LAMPF/BNL), [@Ray94; @Kess96] (KTeV at Fermilab), and [@Dav94] (PHENIX at RHIC).
The PIBETA calorimeter geometry shown in Fig. \[fig:ball\] is obtained by the class II geodesic triangulation of an icosahedron [@Ken76]. Selected geodesic breakdown results in 220 truncated hexagonal, pentagonal and trapezial pyramids covering the total solid angle of 0.77$\times$$4\pi$ sr. Additional 20 crystals cover two detector hemispheres open to the beam and act as shower leakage vetoes. The inner radius of the crystal ball is 26$\,$cm, and the axial module length is 22$\,$cm, corresponding to 12 CsI radiation lengths ($X_0$=1.85$\,$cm [@PGP]). There are nine different module shapes: four irregular hexagonal truncated pyramids (we label them HEX–A, HEX–B, HEX–C, and HEX–D), one regular pentagonal (PENT) and two irregular half-hexagonal truncated pyramids (HEX–D1 and HEX–D2), and two trapezohedrons which function as vetoes (VET–1, VET–2). The volumes of our CsI crystal detector modules vary from 797$\,$cm$^3$ (HEX–D1/2) to 1718$\,$cm$^3$ (HEX–C). Dimensioned drawings of the HEX–A and HEX–D1 crystal shapes are shown as examples in two panels of Fig. \[fig:shape\].
All key components of the complete PIBETA detector have been prototyped and built and have met the required specifications [@Ass95]. All the detector components were delivered to PSI by mid-1998. The final assembly of the apparatus was followed by the in-beam commissioning and calibration. “Production” measurements are scheduled for 1999, and will extend over two more years to complete the first phase of the project, a $\sim\,$0.5$\,$% measurement of the $\pi^+$$\rightarrow$$\pi^0e^+\nu$ decay rate.
The quality of the delivered CsI calorimeter crystals was controlled in the following program of measurements:
1. check precisely the crystal physical dimensions against the specifications;
2. study the effects of different crystal surface treatments and wrapping configurations on the energy and timing resolution and on the uniformity of light response;
3. determine the percentage of the fast scintillation component in the light output;
4. measure the contribution of detector photoelectron statistics to energy resolution;
5. find the temperature dependence of a detector’s ADC readings;
6. determine axial and transverse optical nonuniformities for each crystal.
The ultimate goal of the program was to provide the input parameters for a Monte Carlo simulation of the calorimeter response and to deduce the effects of CsI crystal light yields and optical nonuniformities on the energy lineshapes of detected photons and positrons ranging in energy between 10$\,$MeV and 120$\,$MeV.
The spectra generated by cosmic muons have been used to examine the uniformity of light response of two large (24$\times$36$\,$cm) cylindrical NaI(Tl) calorimeter crystals in the earlier work of Dowell et al. [@Dow90]. Their apparatus relied on plastic scintillator hodoscopes with 3.8$\times$3.8$\,$cm$^2$ cross section for the charged particle tracking. The precision of the trajectory reconstruction was therefore limited to $\sim\,$4$\,$cm. Such positional resolution was deemed inadequate for our application because the PIBETA CsI crystals have an axial length of 22$\,$cm with front (back) surface side lengths of less than 4$\,$cm (7$\,$cm).
Most of our measurements were done with the tomography apparatus built around three pairs of multiwire drift chambers (MWDC) using cosmic muons as the probe. One dozen PIBETA CsI detectors were examined in a 350$\,$MeV/c minimum ionizing muon beam of the PSI $\pi$E1 area, as well as in the 405$\,$MeV/c penetrating pion and stopping proton beams in the $\pi$M1 channel. Selected CsI crystals were also scanned with a 662 keV $^{137}$Cs $\gamma$-source. The light yield nonuniformities measured in the beam and with a radioactive-source scan method were compared with more precise cosmic muon tomography data. The experimental apparatus used for a $\gamma$-source crystal scans and the subsequent data reduction were simple enough to be easily adopted by the crystal manufacturers, thus accelerating the subsequent quality control cycle.
Crystal Production, Mechanical Quality Control and Surface Treatment
====================================================================
Pure CsI crystals produced for the PIBETA calorimeter came from two different sources. Twenty-five crystals were grown and cut to specified shapes in the Bicron Corporation facility in Newbury, Ohio. The rest of the scintillators were grown in the Institute for Single Crystals in Kharkov (AMCRYS), Ukraine. Preliminary examination of fifty-five AMCRYS CsI crystals, including the mechanical and physical tests, was done in the High Energy Institute of Tbilisi University, Georgia.
Manufacturing tolerances of the crystals were specified for the linear dimensions ($+$150 $\mu$m/$-$50 $\mu$m) and for the angular deviations ($+$0.040$^\circ$/$-$0.013$^\circ$). Geometrical dimensions of the machined crystals were measured upon delivery at PSI using the computer-controlled distance-measuring device [*WENZEL Precision*]{}. The machine was programmed to automatically probe the surfaces of a subject crystal with a predefined shape. Each crystal surface plane was scanned with a touch head at six points and the equations of planes were found through these surveyed points. Body vertices, measured with an absolute precision of 2 $\mu$m and reproducible within 20 $\mu$m, were then compared with the expected theoretical values. Those crystals that failed the imposed geometrical tolerances were returned to the manufacturer for reuse as raw crystal growing material.
After physical measurements crystal surfaces were polished with a mixture of 0.2 $\mu$m aluminum oxide powder and etylenglycol. Next came a measurement of the light output of the fast CsI scintillation component (F) that is completely decaying in the first 100 ns, relative to the total CsI signal (T), integrated in a 1 $\mu$s ADC gate. These measurements were made with unwrapped crystals using air-gap coupled photomultiplier tubes (PMTs) and a Tektronix TDS 744 digital oscilloscope. Only the crystals with a fast-to-total component ratio (F/T) better than 0.7 were accepted. The mean value of fast-to-total ratio for all accepted CsI crystals was 0.788, as shown in Fig. \[fig:ft\].
EMI photomultiplier tubes 9821QKB [@EMI] with 75$\,$mm diameter cathode were glued to the back faces of hexagonal and pentagonal CsI crystals using a 300 $\mu$m layer of silicone Sylgard 184 elastomer (Dow Corning RTV silicon rubber plus catalyst). Smaller half-hexagonal and trapezial detector modules were equipped with two inch EMI 9211QKA phototubes [@EMI]. The resulting crystal–photomultiplier tube couplings were strong and permanent, but could be broken by application of a substantial tangential force. The PMT quartz window transparency, peaking at $\sim\,$380 nm [@EMI], is approximately matched to the spectral excitation of a pure CsI fast scintillation light component with a maximum room temperature emission at $\sim\,$310 nm [@Woo90]. The PMT high voltage dividers were modified EMI-recommended bases designed and built at the University of Virginia. These dividers minimized the so-called “super”-linearity exhibited by many PMTs well below the onset of saturation. The maximum PMT nonlinearity measured with a pair of light-emitting diodes was less than 2$\,$% over the full dynamic range expected in the $\pi\beta$ decay rate measurement [@Col95].
Four different wrapping materials for lateral crystal surfaces were investigated in wrapping and tuning studies with 405$\,$MeV/c pion and proton beams in PSI $\mu$M1 area:
1. one to five layers of a 38$\,$$\mu$m PTFE Teflon sheet (CF$_2$ monomer, $X_0$=16.0$\,$cm);
2. one to five layers of 30$\,$$\mu$m Mylar sheet (C$_5$H$_4$O$_2$, $X_0$=28.7$\,$cm);
3. a 250 $\mu$m thick Tyvek fleece (polyethylene CH$_2$, $X_0$=131.3$\,$cm);
4. a 110$\,$$\mu$m thick Millipore filter (polyvinylidene fluoride PVDP, $X_0$=19.0$\,$cm) with 0.22$\,$$\mu$m pores;
5. a wavelength shifter lacquer treatment of crystal surfaces.
In all cases a primary diffuse reflector was protected by an additional 20 $\mu$m thick aluminized Mylar cover. These preliminary tests showed more uniform axial light collection from the crystal front section when a polished front crystal surface was covered with a black paper sheet. A similar tuning method, using black paper strips at the front section of 5$\times$6$\times$36$\,$cm$^3$ CsI(Tl) crystals in order to improve the uniformity of light collection, was recently described in Ref. [@Dah96].
The maximum fast-to-total ratio was always achieved with unwrapped crystals because all applied wrapping materials absorb more ultraviolet light which dominates the fast component, than visible light. A two-layered Teflon cover was found to be superior in delivering $\ge$20$\,$% more fast scintillation light component than the other reflectors. It was also comparatively better in not degrading the F/T ratio by more than $\sim\,$2$\,$%. These differences in measured light output were reproducible in the beam tests with $\sim\,$1$\,$% event statistics and estimated 2$\,$% systematic uncertainty. They are subsequently confirmed in more controlled cosmic muon tomography measurements.
On the basis of these findings the adopted standard wrapping configuration for all studied CsI crystals consisted of: (a) the lateral surfaces being wrapped with two layers of Teflon foil plus one layer of aluminized Mylar sheet, and (b) the front crystal surface covered with the black paper template. The back crystal surface with the glued phototube was left uncovered.
Light yield measurements were repeated for a subset of crystals after a six month period to confirm that no appreciable changes occurred as a result of degradation of the surface reflectivities and wrapper material qualities. The measured light output was typically within 5$\,$% of the originally measured value.
The most successful surface treatment of CsI crystals, however, involved painting the lateral crystal surfaces with a waveshift lacquer. We recently studied this method on a large sample of CsI detectors. The light response and uniformity properties were noticeably improved, resulting in $\sim\,$20$\,$% better energy resolutions for the 70$\,$MeV monoenergetic $e^\pm$ and 50–82$\,$MeV tagged photons. The possible degradation and change in the crystal surface reflectivities that could be difficult to account for in the multi-year long precision experiment like the PIBETA is also expected to be arrested by such a treatment. These measurements will be covered in detail in a forthcoming paper [@Frl98].
Tomographic Apparatus
=====================
A simplified sketch of the tomographic apparatus layout is illustrated in a [GEANT]{} rendering showing a few simulated cosmic muon trajectories on Fig. \[fig:drift\]. Three identical delay-line readout drift chambers were used to define the cosmic muon tracks intersecting CsI crystals. The chambers were built at the Los Alamos Meson Physics Facility (LAMPF) by the M. Sadler group from the Abilene Christian University. This type of drift chambers has been used reliably for years with several LAMPF spectrometers [@Ate81; @Mor82; @Ran81; @Ran82]. Each chamber consists of two signal and three ground planes with the nominal active area of 60$\times$60$\,$cm$^2$. The horizontally oriented “$x$” and “$y$” signal planes are two orthogonal sets of alternating cathode and anode wires evenly spaced at 0.4064$\,$cm.
The light-tight aluminum box had two drift chamber pairs mounted on its top plate, and one pair fixed below the bottom side. Distances between the centers of chamber pairs 1–3 and 2–3 were 24$\,$cm and 27$\,$cm, respectively. The dark box could accommodate up to six CsI calorimeter modules at one time. It had feedthrough connectors for twelve signal and twelve high voltage cables as well as six temperature sensor lines and six LED calibration signal cables. A pair of 1$\,$cm thick plastic scintillators were placed directly below the apparatus and separated from the chamber system by a 5$\,$cm layer of lead bricks shielding out the soft cosmic ray component. The Monte Carlo simulation (see Sec. \[mcd\])showed that hard cosmic muons penetrating the frames of the apparatus, CsI crystals and shielding material and triggering two scintillators had a smooth energy threshold starting at $\sim\,$120$\,$MeV. The zenithal angle range of accepted cosmic muon trajectories intersecting all three MWDCs and at least one CsI detector was $\pm\,$45$^\circ$.
The chambers were operated with anode wires held at positive voltage between 2400–2600 V. The chamber cathode wires were grounded. The gas mixture was 65$\,$% argon, 35$\,$% isobutane plus 0.1$\,$% isopropyl alcohol. Detection efficiencies of individual chambers for penetrating cosmic muons exceeded 90$\,$%, with the combined six-chamber efficiency routinely surpassing 50$\,$%.
Anode wires were attached directly to a fast 2.5 ns/cm delay-line. Signals from both ends of the delay line were amplified twenty-fold, discriminated at a threshold of 10 mV and connected to two channels of a time-to-digital converter (LCR 2229A TDC). The time difference in the two TDC readings identified the fired anode wire. Cathode lines defining the alternate field [@Wal78; @Ers82] were connected to one “odd” (O) and one “even” (E) line, and the signals on these lines were processed by a custom-made electronics unit which added and subtracted the analog pulses [@Bro79]. The electronic sum of cathode pulses (O$+$E) was discriminated and the resulting delayed signal determined the drift position timing, and was used to stop another TDC channel. The difference of the cathode signals (O$-$E) was digitized with an analog-to-digital converter (LCR 2249A ADC). That information was used to discriminate between the events that produced the ionization tracks left and right of the given anode wire. The 100 ns integration gate timing was defined by the (O+E) logic pulse timing. Four measurements were therefore required to find an intersection between a cosmic muon track and one drift chamber plane, namely three TDC values and one ADC value. The schematic diagram of the electronic logic is shown in Fig. \[fig:tmelec1\].
Advantages of the system were good charged particle track resolution ($\sim\,$0.5$\,$mm root-mean-square in the horizontal plane, Fig. \[fig:deviation\]), stability to fluctuations in the outside environmental parameters (humidity, temperature and pressure) and low cost of the associated electronics logic and readout. Limited counting rate and restriction to single hit events in the chambers did not represent a drawback in this tomographic application. The apparatus was operated in an air-conditioned room with a controlled humidity level kept below 30$\,$%. The temperature at six points inside the dark box as well as absolute time were recorded for every triggered event. The temperature range recorded inside the dark box during three years of data taking was (22$^\circ$$\pm\,$3$^\circ\,$)C. The typical temperature gradient was 0.4$^\circ\,$C/day, leading to average absolute temperature variation of 0.2$^\circ\,$C and average CsI ADC gain drift of $\sim\,$0.4$\,$% in a single data taking run (fixed at 250000 triggers, $\sim\,$6 hours). Measured light output variations of the CsI crystals and the PMT gain drifts caused by daily temperature cycles were compensated for in the replay analysis (see Sec. \[raw\]).
Data Acquisition System
=======================
The computer code used for data acquisition was HIX (Heterogeneous Information Exchange), a data acquisition system developed by S. Ritt at PSI, originally intended for use in small and medium-size nuclear physics experiments [@Rit95].
The “frontend” 486 personal computer was connected to a CAMAC crate via a HYTEC 1331 interface that read ADC, TDC, scalar, and temperature sensor units. The C program running under MS-DOS accessed data and sent it over the network to a VAX 3100 server. A simple communication program on a VAXstation received data from the frontend computer and stored it in a global section buffer. Buffered data were passed by the Logger application to a user-written Analyzer program.Analyzed events were subject to predefined cuts, filled in predefined histograms, and stored in a raw data stream directed simultaneously to a hard disk drive and an 8-mm tape system. The experiment was controlled from a PC computer by a Microsoft Windows control program. A windows-based graphical user interface to the program allowed starting, pausing and stopping data acquisition, as well as online inspection of individual raw data words, calculated data words, scalers, assorted efficiencies, one and two-dimensional histograms, and gate, box and Boolean tests. Different PC computers on the Ethernet could make a connection to the VAX server called Link and access the same information remotely.
During data acquisition only those events for which at least two $x$ and two $y$ drift chambers had nonzero ADC and TDC values were written to an disk-resident ASCII Data Summary File (DSF). One good event contained 55 raw data words specifying the response of the MWDCs and CsI detectors, instantaneous temperatures and absolute times. That was a full set of observables for which a cosmic muon trajectory could be unambiguously reconstructed. The raw trigger rate was $\sim\,$13 Hz with six CsI crystals in the dark box. The average DSF event rate was $\sim\,$4 Hz. Individual runs were stopped and restarted automatically after 250,000 collected triggers. A total of about 10$^5$ DSF events per crystal were typically collected in one week of data acquisition. Tomography data for one set of six crystals were usually collected over a two week period.
Drift Chamber Calibration
=========================
Two different trigger configurations were used interchangeably during the data collection. Drift chamber calibration was done with a trigger requiring a two-scintillator coincidence and good ADC and TDC data for at least one drift chamber. In the tomography data acquisition mode, the triple coincidence between two tag scintillators and one CsI detector was required, accompanied by at least one good pair of $x$ and $y$ chamber hits. The rare accidental coincidences involving signals in more than one CsI detector were eliminated from the DSF records in the offline analysis.
The hit anode wire number $n_A$ was calculated from $$n_A={{(T_1-T_2+N\cdot D)}\over {2D}},$$ where $T_1$ and $T_2$ were the TDC values for two ends of the anode delay line, $N$ was the number of wires in a chamber (between 71 and 73), and $D$ was the $\sim\,$2.05 ns time delay between the adjacent wires. Truncated wire position $x_T$ was defined as the nearest integer multiple $n_A$ of the 0.8128$\,$cm wire separation $g$: $$x_T={\tt NINT}(n_A\cdot g).$$
The anode delay time difference $(T_1-T_2)$ depended nonlinearly on the anode position. Linearized anode wire positions for each chamber were approximated by a polynomial expansion $$x_A=\sum_{i=0}^m c_i\cdot (T_1-T_2)^i,$$ where the order $m$ of the fit with $\chi^2$ per degree of freedom $\chi^2/(m-1)$$\approx\,$1 depended on the particular delay line. For our chambers the orders of the fits were 3, 4, or 5. The coefficients $c_i$ were optimized with the program [MINUIT]{} [@Jam89] that minimized the sum of squared deviations $(x_T-x_A)^2$ for each chamber in every run.
The drift distance $d_x$ was determined from the drift timing $T_x$ using the calibration lookup table $f(T_x)$ $$d_x=f(T_x)\cdot T_x+\Delta T_{x},$$ where $\Delta T_{x}$ was the drift time offset. The lookup table was calculated assuming that incoming cosmic muons are distributed uniformly over the equidistantly spaced wires.
The final hit position $x$ was given by expression: $$x=x_T+(-1)^n\cdot sign({\rm ADC_{O-E}+ADC_0})\cdot d_x+x_{\rm OFF},$$ where $x_{\rm OFF}$ was the chamber $x$ coordinate offset, and ${\rm ADC_{O-E}}$ and ${\rm ADC_{0}}$ were the (O$-$E) ADC signal and associated offset, respectively.
The individual chamber phases $n$ in the Eq. (5) and absolute horizontal coordinate offsets $x_{\rm OFF}$ and $y_{\rm OFF}$ between six chambers were adjusted with help of two-dimensional histograms $(x_1-x_3)$ vs $(x_2-x_3)$ and $(y_1-y_3)$ vs $(y_2-y_3)$ [@McN87; @Sup89].
Raw Data Reduction {#raw}
==================
Data summary files were analyzed offline by compressing the DSF event data into [PAW]{} Ntuples [@Bru93]. Drift time-to-distance lookup tables and assorted chamber calibration constants, as well as CsI detector ADC pedestals and ADC temperature corrections, were determined for each run separately. The high voltages of the CsI detector phototubes were selected to give $\sim\,$8 ADC channels/MeV scale. The FWHMs of the pedestal peaks were typically 2 ADC channels. Therefore, the pedestal widths corresponded to an energy deposition of 0.25$\,$MeV, 24 $\mu$m pathlength or 0.5$\,$% of the most probable energy deposition by a minimum ionizing charged particle. Drifts in the pedestal position over a one month period amounted to less than 2 ADC channels.
The precise horizontal coordinate offsets of six CsI crystals inside the dark box were determined in the next stage of analysis. Using the preliminary offsets read from the plastic template on which the crystals were laying, between 5 and 10 percent of the reconstructed cosmic muon trajectories were found not to intersect any of the predefined detector volumes. After adjusting the $x$ and $y$ translation software offset parameters of the CsI modules by maximizing the number of tracks intersecting individual crystal volumes with nonzero ADCs, the real number of events undergoing scattering in the apparatus or having the improperly reconstructed tracks was shown to be below 1$\,$%. That percentage was in agreement with the Monte Carlo simulation of cosmic muons interacting with the experimental apparatus, showing the most probable muon scattering angle of 0.25$^\circ$ and the 2.5$^\circ$ mean scattering angle. The Monte Carlo root-mean-square of the scattering angle for the accepted events was 0.66$^\circ$, Fig. \[fig:ths\]. That value translates into an average pathlength uncertainty of 1$\,$mm. The estimated error in finding the correct crystal position inside the dark box was smaller than $\sim\,$0.5$\,$mm, Fig. \[fig:position22\].
The energy deposited in CsI crystal by cosmic muons along the fixed pathlength is a broadened distribution due to the statistical nature of energy transfers. Maximal measured pathlengths were up to 12$\,$cm, while the average pathlength was $\sim\,$6$\,$cm. A Gaussian model for the energy loss distribution of the minimum ionizing particles in CsI thicknesses of less than 12$\,$cm is not a good approximation. The distribution of energy losses is asymmetric, with a long high energy tail; the most probable energy loss is smaller than mean energy loss [@Leo87].
Light yield temperature coefficients for the individual CsI detectors were determined by the “robust” estimation [@Hub81] of the ADC values per unit pathlength as a linear function of temperature recorded inside the dark box. The least-squares condition assumes normally distributed measurements with constant standard deviations and is therefore not appropriate in this application, as pointed out above. Typically, more than $10^5$ cosmic muon events, collected over at least one week and spanning the temperature range of $\sim\,$2$^\circ$C, were used as an input experimental data set. The [FORTRAN]{} subroutine [MEDFIT]{}, documented in the Ref. [@Pre86], was adopted as the fitting procedure by imposing the requirement of minimum absolute deviation between the measured and calculated ADC values per unit path that were dependent linearly on the temperature variable. The average light output temperature coefficient for 74 different CsI detectors extracted by that method was $-1.4\pm\,1.4$$\,$%/$^\circ$C, both for fast and total scintillation light components. These numbers are in good agreement with the previously reported value of $-$1.5$\,$%/$^\circ$C [@Kob87], but the spread of extracted coefficients for different CsI detectors was large: $\pm\,$4$\,$%/$^\circ$C (Fig. \[fig:tf\]). We point out that the method did not allow us to separate the temperature dependence of the crystal light outputs from the temperature instabilities of the phototubes and high voltage dividers as well as the temperature instabilities of the ADC modules themselves. The LeCroy catalogue specification [@LCR95] lists the typical temperature coefficient of a LCR 2249A ADC unit as zero, and the maximum coefficient up to $\pm\,$3$\,$% for an ADC gate longer than 100 ns and an average ADC reading of 75 pC (about a quarter of a full 256 pC scale).
The raw ADC values were corrected for the temporal temperature variations and written into the revised DSF files used in subsequent analysis.
Detector Photoelectron Statistics
=================================
The contribution of the photoelectron statistics to the energy resolution of CsI crystals was determined by a photodiode-based system. Six individual detectors (CsI crystals with photomultiplier tubes and high voltage dividers) having identical light emitting diodes (LEDs) coupled to their back sides were placed inside the cosmic muon tomography apparatus. The LEDs were pulsed at a 10 Hz rate using a multichannel diode driver with continuously adjustable output voltage. One split output signal from each channel of the driver generated a 100 ns wide ADC gate. The LED light in a CsI detector produced fast ($\sim\,$20 ns FWHM) PMT anode pulses whose integrated values depended on the driving voltage and were equivalent to the fixed cosmic muon energy depositions between 10$\,$MeV and 100$\,$MeV.
A total of five different LED amplitudes were used in measurements of each CsI detector. Examples of the LED spectra are shown in Fig. \[fig:set2\]. The ADC pedestals were recorded simultaneously during the run. The intensity of LED light was cross-calibrated against the cosmic muon spectra peaks in CsI crystals. These muons were tracked in tomography drift chambers and both their pathlengths and energy depositions in the crystals could be easily calculated. That calculation enabled the establishment of the absolute energy scale in MeV.
The variance $\sigma_E^2$ of a photodiode peak depended upon the mean number of photoelectrons $\bar N_{pe}$ on the photocathode created per unit energy deposition in the crystal: $$\sigma_E^2=\sum_i\sigma_i^2+\bar E/\bar N_{pe},
\label{eq:led}$$ where $\bar E$ was the LED spectrum peak position and $\sigma_i^2$’s were assorted variances, such as instabilities of the LED driving voltage and temporal pedestal variations. Five measured points ($\sigma_E^2$,$\bar E$) were fitted with a linear function (\[eq:led\]) and the mean number of photoelectrons per MeV $\bar N_{\rm pe}$ for each CsI detector was determined. The statistical error of the least-squares fits was typically less than one photoelecton/MeV. Using the previously extracted light output temperature coefficients all $\bar N_{\rm pe}$ values were scaled to the 18$^\circ$C point. That value was the designed operating temperature of the PIBETA calorimeter.
The number of extracted photoelectrons per MeV for the fast scintillation component fell into the 20–130 range as shown in Fig. \[fig:npf\]. The hexagonal and pentagonal crystals equipped with three inch phototubes averaged 73 photoelectrons/MeV, while the half-hexagonal and veto detectors with two inch phototubes had a mean of 33 photoelectrons/MeV. The 73/33 ratio is very well explained by the (3/2)$^2$ ratio of photosensitive areas for two different photocathode sizes. The measured photoelectron statistics were in agreement with the 100 ns ADC gate values of $\bar N_{\rm pe}$=(20–260) for large ($\sim\,$10$\,$cm) pure CsI crystals equipped with two or three inch PMTs reported in the past in Refs. [@Woo90; @Sch90; @Utt90; @Mor89].
The quantum efficiency of our bialkali photocathodes for pure CsI scintillation light was 23$\,$% [@EMI], the average light collection probability for our detector shapes 23$\,$% (Sec. \[tks\]) and the fraction of deposited energy converted into the scintillation light was about 12$\,$%. Therefore, starting from the mean number of 73 photoelectrons/MeV, we calculated that 1$\,$MeV energy deposited in the CsI crystal produced on average about 10$^4$ scintillation photons.
Monte Carlo Description
=======================
Tomography System {#mcd}
-----------------
Geometrical layout of the experimental apparatus was defined using the [GEANT]{} detector description and simulation tools [@Bru94]. The active elements of the simulated apparatus were an aluminum box housing six CsI modules, six multiwire drift chambers, two scintillator planes and a layer of lead brick shielding, all shown in Fig. \[fig:drift\]. Generated events were muons with the energy, angular and charge distribution of a hard cosmic ray component at sea level. The zenithal angle $\theta_z$ distribution of muons at the ground was assumed to be proportional to $\cos^2\theta_z$; the momentum spectrum between 0.1 and 10$^3$ GeV/c and the energy-dependent ratio of number of positive to number of negative cosmic muons was parameterized from the data given in Refs. [@PGP; @Ros48].
The simulation trigger was defined by requiring a minimum energy deposition of more than 0.2$\,$MeV in each scintillator plane. This threshold was about one-tenth of the minimum-ionizing peak in the triggered scintillator and corresponded to the discriminator level used in the data acquisition electronics.
Penetrating cosmic muons and generated secondary particles were tracked through the apparatus, and energy depositions, pathlengths in CsI crystals and hits in the drift chambers were digitized. All physical processes were turned on in the [GEANT]{} programs with the cutoff energies of 0.2$\,$MeV for charged particles and photons. Inspection of simulated energy deposition spectra showed deviations from theoretical Vavilov distributions [@Leo87] expected in the planar detector geometries. The differences were caused by multiple scattering in the irregularly shaped crystals, and were particularly prominent for the events with shorter pathlengths close to the crystal edges. The simulated ADC histograms revealed that triggering cosmic muons deposit on average 10.33$\,$MeV/cm in a CsI detector, with the most probable energy loss 5.92$\,$MeV/cm.
Cosmic muon spectra in CsI detectors were described in a satisfactory way ($\chi^2/(m-1)$$\approx\,$1.2) by the combination of a Gaussian distribution and a falling exponential tail: $${A}(\epsilon)=\theta(\epsilon_0-\epsilon)e^{ -{1\over 2}[{{(
\epsilon-\bar\epsilon})/{\sigma_\epsilon}}]^2 }+\theta(\epsilon-
\epsilon_0)e^{\alpha-\beta\epsilon},$$ where $\epsilon$ is energy (in MeV) deposited in one full CsI module. Parameters $\bar\epsilon$, $\sigma_\epsilon$, $\epsilon_0$, $\alpha$, and $\beta$ are extracted from the [GEANT]{} spectra by imposing the least squares constraint to the fit and leaving the pathlength $d$ (cm) as free parameter: $$\begin{aligned}
\bar\epsilon(d)&=&5.079+0.1876d-9.9390\cdot 10^{-3}d^2, \\
\sigma_\epsilon(d)&=&0.3545+7.2329\cdot 10^{-3}d-3.4148\cdot 10^{-4}d^2,
\\
\epsilon_0(d)&=&5.1800+0.1989d-1.0297\cdot 10^{-2}d^2, \\
\alpha(d)&=&4.0581+0.4465d-1.5760\cdot 10^{-2}d^2, \\
\beta(d)&=&0.7952+5.0313\cdot 10^{-1}d-1.5240\cdot 10^{-3}d^2, \
\label{eq:mc}\end{aligned}$$
and
$$\begin{aligned}
\theta(\epsilon_0-\epsilon)=\cases{ 0,& if $\epsilon_0<0$,\cr
1,& otherwise.\cr }\end{aligned}$$
These Monte Carlo spectra, broadened with photoelectron statistics, were used to describe our cosmic muon lineshapes produced by the optically uniform CsI detectors.
Light Collection Simulation: [TkOptics]{} Code {#tks}
----------------------------------------------
Propagation of scintillating photons through a uniform detector with ideal reflecting dielectric surfaces and different wrapping materials was studied using the [TkOptics]{} simulation program [@Wri92; @Wri94]. The code is a library of [FORTRAN]{} subroutines with an X-Windows-based user interface written with [Tck/Tk]{} toolkit [@Ous94]. The program can simulate the light output response of an arbitrarily shaped scintillation detector with given bulk and surface optical properties.
The polygonal detector shape is defined by the coordinates of its vertices. The detector attributes are reflector types of lateral and front crystal surfaces and wrapper material, crystal surface-wrapper gap distances and characteristics of the photomultiplier tube and a phototube-crystal joint coupling. The program handles normal dielectric, specular, diffuse and partially absorbent reflector surfaces with arbitrary diffuse fraction, roughness, and specular $r_s$ and diffuse reflectivity $r_d$. Predefined bulk properties of a detector are the medium scattering and attenuation length as well as refractive index. A photomultiplier tube is specified by the diameter, position and quantum efficiency of the photocathode.
Different choices of initial scintillating photon distribution are possible, the default being a uniform starting distribution throughout the scintillating volume. A working volume is a box divided into elementary cubic cells of fixed size. Output menu options include the initial and endpoint photon coordinates, and direction vectors and timing distributions of the detected scintillation photons organized in a [PAW]{} Ntuple. Every elementary cell is flagged as a bulk or edge crystal cell. Center coordinates of the cells and a fraction of photons generated in every cell, and subsequently detected on the PMT sensitive surface, are always recorded. Results of high statistics runs with $10^7$ photons generated uniformly through the detector volume were plotted to show the number of photoelectrons as a function of scintillation source position inside the crystal. The relative statistical errors of calculated light collection probabilities for the bulk crystal cells were $\le\,$2$\,$%.
The bulk attenuation length and the light scattering length of the near-ultraviolet emission component inside the CsI crystals were set to 150$\,$cm and 200$\,$cm, respectively [@Frl89]. The index of refraction for the CsI medium increases from 1.82 for the blue-green light to about 2.08 for ultraviolet light [@Frl89].
The simulated light collection probabilities $P(x,z)$ that depend on axial $z$ and transverse $x$ positions are shown integrated in the vertical coordinate ($y$) in Figs. \[fig:hexa\_9\] and \[fig:hh1d\_9\]. in the form of “lego” plots for one wrapping configuration and two different crystal shapes. The coordinate system is defined with the $z$=0$\,$cm origin at the front face of the crystal and a photocathode window at the $z$=22$\,$cm plane.
We find that simulated light response of the ideal hexagonal or pentagonal PIBETA CsI detector with specular lateral surfaces and a diffuse wrapping material can be described with the following parameters:
1. The average photon collection probability $P$ for a three inch photocathode is about 23$\,$% for ideal $r_s$$=$1.0 crystal surfaces and a $r_d$$=$0.9 diffuse wrapper, decreasing by half, to 12$\,$% for $r_s$$=0.8$.
2. The axial light collection probability variation through the first 10 centimeters is in the $\pm\,5\,$% range, with a positive slope $dP/dz\ge 0$ for higher specular and diffuse reflectivities, namely for $r_{s,d}$$\ge$0.9.
3. The axial detected light variation in the back crystal half ($z$$\ge$10$\,$cm) is up to $-30\,$%.
4. The transverse light response referenced to the light yield at the crystal axis typically increases towards the crystal surfaces by up to $+\,$5$\,$% for $z$$\le$10$\,$cm, but is generally declining away from the central axis by $-\,$30$\,$% at the $z$=18$\,$cm plane.
5. The root-mean-square of a three-dimensional light nonuniformity is between 2.5$\,$% and 3.5$\,$% for $z$$\le$10$\,$cm, compared to a $\sim\,$20$\,$% root-mean-square for a $z$$\ge10$$\,$cm volume, where a large spread is caused by the inefficient light collection from crystal back corners.
For lateral surfaces painted with a $r_d$$\approx\,$0.9 diffuse substance without an air gap we find that the average photon collection probability is lower, $\sim\,$17$\,$%, and the axial light collection nonuniformity as a function of axial position is always positive and is usually larger than $+\,$10$\,$% in a 22$\,$cm long detector, making the root-mean-square of the 3D nonuniformity function $\ge\,$20$\,$%.
The Monte Carlo results for the optically uniform half-hexagonal and trapezial CsI crystals with the same range of optical properties predict smaller light yields and somewhat higher light collection nonuniformities:
1. The average photon collection probability with a two inch photocathode is about 12$\,$%.
2. The typical axial nonuniformity is positive in front, $+\,$5$\,$%/10 cm, and negative in the back crystal half, with a $-30\,$%/10$\,$cm variation,
3. The simulated transverse light output from the front crystal half is very similar to one for the full crystal shapes, increasing up to $+\,$3$\,$% away from the detector central axis, but decreasing by almost a third in the back corners of the crystal.
4. The root-mean-square of detected light output varying between 3.0$\,$% and 4.5$\,$%, in the front ($z$$\le$10$\,$cm) and back ($z$$\ge$10$\,$cm) crystal halves, respectively.
These light collection probability distributions calculated for optically homogeneous crystals explain the major features of the measured optical nonuniformities presented below.
Measured Nonuniformities of CsI Detector Light Responses
========================================================
The three-dimensional (3D) spatial distribution of the light output of a scintillation detector can be specified by giving the number of photoelectrons $N_{\rm pe}(x,y,z)$ produced by 1$\,$MeV energy deposition at point ($x,y,z$) (“3D light nonuniformity function”). In the following discussion we limit ourselves to the linear one-dimensional variations of the detected light output separable in the axial and transverse directions: $$\begin{aligned}
N_{\rm pe}(z,x)\propto\cases{
N_1+a_{z1}\cdot z+a_t(z)\cdot x,& $z\le10\ {\rm cm}$, and $x=\pm15\
{\rm cm}$ \cr
N_2+a_{z2}\cdot z+a_t(z)\cdot x,& $z\ge10\ {\rm cm}$, and $x=\pm15\
{\rm cm}$\cr
}\end{aligned}$$ where $a_z$ ($a_t$) is the linear optical nonuniformity coefficient in the $z$ ($x$) coordinate, and the coordinate system origin is at the center of the detector front face, as explained in Sec. \[tks\].
A simple and straightforward parameterization of the detector light response nonuniformity can be made on the basis of scatter plots showing the light output per unit pathlength as a function of longitudinal or transverse cosmic muon-CsI crystal intersection coordinates. In this analysis we take only cosmic muon events with almost perpendicular trajectories ($\theta_z$$\ge$85$^\circ$), so that the measured two-dimensional light outputs are averaged over the pathlengths and over $\sim\,1\,$cm$^2$ vertical cross sections. The cosmic muons deposit the energy along the well-defined ionization track of the length $d$ in the crystal. This description will somewhat underestimate the real light collection probabilities (see Sec. \[137s\]) due to the integration of a three-dimensional light nonuniformity function along the charged particle track. Our work on the fully 3D reconstruction of the scintillator light response will be presented in a forthcoming publication [@Frl96].
Fig. \[fig:expected6\] shows the axial variation of normalized ADC values as a function of distance of energy deposition from the front crystal face for the six representative crystals. The axial positions were calculated by averaging two $z$ values of the cosmic muon track intersection with the crystal surfaces.
Panels on Figs. \[fig:expected7a\] and \[fig:expected7c\] show the transverse dependence of measured light output per unit pathlength, where the independent variable is the distance from the crystal axis measured in the horizontal plane. The selected scatter plots show the data points for the axial slices at $z_0$=6$\pm$1$\,$cm and $z_0$=18$\pm$1$\,$cm.
We have chosen to parameterize the axial light output nonuniformity with two piecewise linear functions: the light output nonuniformity coefficient $a_{z1}$($z$$\le$10$\,$cm) for the front half of the crystal, and the light nonuniformity coefficient $a_{z2}$($z$$\ge$10$\,$cm) for the back half of the crystal, both values expressed in %/cm. The transverse light output variation at a fixed axial distance $z_0$, $a_t(z_0)$, is described by the average value of a change in luminosity left and right of the crystal axis.
Several general features are readily noticeable: the spread of measured points due to energy deposition straggling, the gradual variation in the collected light along and perpendicular to the detector axis, and the decrease in detected light when the photon generation occurs close to lateral detector surfaces and, in particular, near the back corners of the detector volumes.
The calculated ADC/pathlength data points shown on the panels of Figs. \[fig:expected6\]–\[fig:expected7c\] have been fitted with linear functions imposing the “robust” condition of the minimum absolute deviation between the measured and calculated values, Eqs. 14. The normalized light output scatter plots in the axial and transverse coordinates, as well as two-dimensional ADC/$d$ distributions in the $x$-$z$ bins and associated linear light output nonuniformity coefficients have been documented for all studied CsI crystals and are available for inspection at the PIBETA WWW site [@pb]. The average scintillation properties of all the measured CsI crystals are listed in Tables \[tab1\] and \[tab2\].
In summary, for a set of fifty-nine full hexagonal and pentagonal crystals we find:
1. The average axial light nonuniformities are $a_{z1}$($z\le10$$\,$cm)=$-$0.1$\,$%/cm and $a_{z2}$($z\ge10$$\,$cm)=$-$1.3$\,$%/cm, respectively (see Fig. \[fig:a1a2\]).
2. Distribution of nonuniformity coefficients could be described by a Gaussian with $\sigma_{a_{z1}}$$\approx\,$1.3$\,$%/cm.
3. The average transverse light output is flat for $z$$\le$10$\,$cm, and the typical light variation is $-15\,$% at $z$=18$\,$cm.
4. The Kharkov-grown crystals on average have twice the axial optical nonuniformity of the Bicron-grown crystals.
For fifteen half-hexagonal and trapezial crystal shapes we find:
1. The average axial nonuniformities are $a_{z1}$($z$$\le$10 cm)=$-\,$0.3$\,$%/cm and $a_{z2}$($z$$\ge$10$\,$cm)=$-$1.0$\,$%/cm.
2. The transverse light output variation is limited in the front crystal half, but increases up to $-\,$30$\,$% in the back of the crystal.
With the insights gained in the simulation of the light collection probabilities (Sec. \[tks\]) we conclude that the $\pm\,$2$\,$%/cm spread of the nonuniformity coefficients corresponds to an equivalent range of 0.8–1.0 in crystal surface reflectivities, and/or 100–250$\,$cm range in CsI attenuation lengths.
Light output uniformity scans of CsI crystals with $^{137}$Cs gamma source {#137s}
==========================================================================
A light-tight plywood box was made to house a single CsI detector and associated measurement apparatus described below. A 0.662$\,$MeV $^{137}$Cs gamma source was placed on a moveable support next to the detector and collimated by a 20$\times$10$\times$5$\,$cm$^3$ lead brick with a 6$\,$mm collimator hole. The photomultiplier analog signal from the detector was processed with an ORTEC 454 timing amplifier with a gain of 30 and a 50 ns integration time constant. The amplifier output was digitized with a peak sensing ORTEC AD811 ADC unit. The same signal was discriminated and produced the trigger rate of about 5 kHz with a $^{137}$Cs source present. The background rate, without the source present, was 50–100 Hz.
Pedestal runs were taken with a clock trigger and used to properly correct offsets of the ADC spectra. Temperature variation during one run was less than 0.3$^\circ$C, typically causing the overall light output variation of less than 0.5$\,$% in a ADC spectrum that was gated with a 100 ns window. Several runs were taken with the source removed to find the shape of the background spectrum. It was confirmed that the background spectrum does not depend on the position of the lead collimator, so the same background lineshape was used in the analysis for all source positions.
The collimated $^{137}$Cs source was placed by remote control in turn at five points along the axis of each crystal at 2, 6, 10, 15 and 20$\,$cm from the front face of the crystal.
Following background subtraction, recorded spectra were fitted with a sum of a Gaussian and exponential function, Fig. \[fig:cs137\]. The peak position and the FWHM for each spectrum were extracted with the statistical uncertainty lower than 0.3$\,$%.
The dependence of the peak position on the placement of the source is illustrated in Fig. \[fig:cs137f\] and follows the trend of the tomography results. The [GEANT]{} simulations revealed that the 662 keV gamma rays could probe all the volume of CsI crystal, but the shower energy deposited at the central axis is only about one-sixth of the energy deposition near the crystal surface. The cosmic muons transfer the energy uniformly along their tracks in CsI material: the scintillation volume over which the measured light output is integrated is therefore larger, and inherent averaging leads to smaller extracted optical nonuniformity coefficients. This feature is borne out in the panels of Fig. \[fig:cs137\] where the tomography data for three different crystals yield consistently smaller axial light nonuniformities when compared with the radioactive source measurements.
The simple $^{137}$Cs scans with the described apparatus were used to evaluate the light collection nonuniformities for up to six crystals per day.
Radioactive source scan measurements similar to ours are described in Refs. [@Bro95; @Dow90].
A [GEANT]{} simulation of the PIBETA calorimeter response to 10–120$\,$MeV positrons and photons
================================================================================================
We have studied the simulated PIBETA calorimeter response to the 69.8$\,$MeV $\pi^+$$\rightarrow$$e^+\nu$ positrons and to the (69.8+2$m_{e^+}$) MeV photons, where $m_{e^+}$ is the positron rest mass. The goal was to find the intrinsic difference between the responses to monoenergetic positrons and photons with identical total incident energies, emphasizing the correct modeling of the low-energy tail below the edge of the Michel $\mu$$\rightarrow$$e^+\nu\nu$ spectrum at 52.8$\,$MeV.
The 0.46$\,$% accuracy of the new TRIUMF measurement [@Bri94] of the $\pi^+$$\rightarrow$$e^+\nu$ branching ratio was limited by event statistics and systematic uncertainties in evaluating the low-energy tail of the positron peak. The $\pi^+$$\rightarrow$$e^+\nu\gamma$ positrons and the forward-peaked bremsstrahlung gammas were detected in a 46-cm-diameter$\times$51-cm-long NaI(Tl) crystal “TINA”. The tail correction of 1.44$\pm\,$0.24$\,$% for the energy region from 0$\,$MeV to 52.3$\,$MeV was determined by subtracting the measured Michel positron spectrum from the detector positron response functions measured with the 20–85$\,$MeV $e^+$ monoenergetic beams. An additional tail component of $\sim\,$0.4$\,$% due to radiative processes was estimated by Monte Carlo method. The TRIUMF group apparently made no attempt to account for the potential difference in the scintillator light response of positrons and photons and neglected potential nonlinearities of the energy scale. Both effects could arise from the light collection nonuniformities of the NaI(Tl) detector.
In the recent PSI $\pi^+$$\rightarrow$$e^+\nu$ experiment [@Cza93] the quoted 0.29$\,$% systematic uncertainty of the extracted branching ratio was also dominated by the 1.64$\pm\,$0.09$\,$% electromagnetic loss fraction as well as by the 0.95$\pm\,$0.19$\,$% contribution from the photonuclear reactions. The $\sim\,$4$\pi$ calorimeter used in that measurement was made of 132 hexagonally shaped BGO crystals, with the light yield claimed to be homogeneous to within 1.5$\,$% over the whole 20$\,$cm length of every crystal.
The lineshapes of positrons and photons in our PIBETA calorimeter and their dependence on the optical properties of 240 constituent CsI crystals were predicted using the [GEANT]{} detector description and simulation tools. The code defined the geometry and tracking media for the complete PIBETA detector [@Pib95]. The inner detector region was occupied by two cylindrical wire chambers and a segmented plastic veto detector. The incident particles, monoenergetic positrons and photons, were generated in the center of the crystal sphere.
Photonuclear reactions in the active detectors as well as in the surrounding passive material were modeled in a user subroutine [@Frl95] that was added to the default [GEANT]{} version [3.21]{}. The probabilities of photonuclear reactions were calculated using the published cross sections from the $\gamma N$ reaction thresholds up to the energy of 120 MeV [@Ahr75; @Lep81; @Heb76; @Ber69; @Bra66; @Jon68].
The 240 individual CsI module shapes were specified, taking into account the mechanical tolerances of the physical CsI crystals, with Teflon and aluminized Mylar wrappings filling the 200 $\mu$m intermodule gaps. The irregular CsI modules had to be constructed from up to six [GEANT]{} generalized trapezoidal wedges. Every CsI crystal volume was considered a sensitive detector with the associated luminosity $\bar N_{\rm pe}$ expressed as the number of photoelectrons per MeV, and two axial and one transverse light collection nonuniformity coefficients, $a_{z1}$, $a_{z2}$, and $a_t$, respectively. A set of seventy-four detector luminosity and nonuniformity values, extracted in the tomography analysis, were initialized in the [GEANT]{}-accessible database. The optical parameters of the remaining 163 modules were drawn randomly from the distributions in Figs. \[fig:npf\] and \[fig:a1a2\]. The assumption was that the crystals produced in the future will be of the same optical quality as the ones that are already delivered.
We have always simulated at least 10$^5$ events for every fixed set of the calorimeter parameters ($\bar N_{\rm pe}$,$a_{z1}$,$a_{z2}$,$a_t$). The low-threshold trigger was defined by requiring a sum of the ADC readings exceed 5$\,$MeV of the light-equivalent energy for one calorimeter supercluster containing 32 crystals. The integrated detector acceptances with a low-level trigger were 85.8$\,$% for $\sim\,$70$\,$MeV positrons and 85.5$\,$% for $\sim\,$70$\,$MeV photons, so all extracted quantities had the relative statistical uncertainties of $\sim\,$0.2$\,$%. The response of the calorimeter was parameterized by the FWHM of the simulated ADC spectrum and the tail contribution being between the adjustable low level (default [LT]{} being 5$\,$MeV) and high level thresholds (two default values [HT$_1$]{} and [HT$_2$]{} being 54$\,$MeV and 55$\,$MeV, respectively).
For the purposes of comparison and gain normalization we first studied the response of an ideal, homogeneous calorimeter, with realistic CsI crystal light outputs. The “software” gain factor of every individual CsI detector was determined from the fitted peak positions of the simulated ADC spectrum sum over the crystal with maximum energy deposition and its nearest neighbors. The peak value positions were found after smoothing the histograms with a multiquadric function, to improve on the limited simulation statistics. The ratios of peak positions in the ideal, homogeneous calorimeter and the corresponding values for the physical, nonuniform calorimeter, were, by definition, the individual detector gain corrections. The values of these ratios were refined in three steps of the iterative procedure. The extracted energy resolution of the nonuniform detector in principle depends on the convergence of the gain-matching process and the resulting uncertainties of the individual detector gains. That gain-matching process was equivalent to the PMT high voltage adjustments in the real experiment that are effected to obtain the best energy resolution with the modular detector. We estimate that our simulation procedure fixes the software detector gains with the accuracy of $\sim\,$1$\,$%.
The set of gain constants depended on the optical properties of our CsI crystals as defined in our [GEANT]{} database, but also upon the incident particle chosen for the Monte Carlo calibration runs and its energy because of differences in shower developments of photons and positrons. The gain normalizations of the calorimeter detectors were calculated by aligning the simulated $\pi^+$$\rightarrow$$e^+\nu$ positron peak positions. The same procedure for matching the CsI detector gains was used in the real data-taking runs. Due to the gain renormalization, the influence of transverse light nonuniformities smaller than 10$\,$% on the ADC spectrum peak positions and the ADC lineshape could be neglected in comparison with the axial light nonuniformity effects. The simulated ADC values were summed over all detectors with the energy deposition above the 1$\,$MeV TDC threshold. We also examined the calculated ADC sums for the clusters which contained the crystal with the maximum energy deposition and its nearest neighbors. The full-widths at half maximum for these spectra are labeled in the following figures and tables as FWHM$_{(220)}$ and FWHM$_{\rm (NN)}$, respectively. No event-to-event uncertainties of the ADC pedestal values were assumed in the simulation. Stability of the PIBETA electronics tested under real experimental conditions in 1996 calibration runs and the quality of the algorithms used for the first and second pedestal correction reduces the pedestal peak root-mean-square to $\sim\,$8 channels, equivalent to $\sim\,$0.3$\,$MeV.
The [GEANT]{}-calculated resolution of the PIBETA calorimeter consisting of 220 CsI detector crystals and 20 veto crystals was parameterized by the fractional full width at half maximum:
$$\begin{aligned}
{ {\rm FWHM_{(220)}}\over {E_{e^+}} }(\%)&=&
{ {2.36\cdot\sigma_{E_{e^+}} }\over {E_{e^+}} }= \\
&=&\cases{ {{(54.69+1.197 E_{e^+}+0.4656\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.8030}} }\cr
{ {(37.19+0.060 E_{e^+}+0.2734\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.5256}} }\cr} \\end{aligned}$$
for the positrons with the most probable peak energy $E_{e^+}$ MeV and $$\begin{aligned}
{ {\rm FWHM_{(220)}}\over {E_\gamma} }(\%)=\cases{
{ {(37.07+0.5507 E_\gamma+0.3233\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.6549}} }\cr
{ {(40.52+0.9437\cdot E_\gamma+0.4931\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.6942}} }\cr}\end{aligned}$$ for the monoenergetic photons with the detected peak energy $E_\gamma$ MeV (Fig. \[fig:fit\_fwhm\_220\]). The top lines refer to the cases of an optically uniform detector ($a_z$=0) while the bottom equations describe the predicted response of the nonuniform calorimeter ($a_z$=[TOMOGRAPHY]{}).
The average fractional energy resolution FWHM$_{(220)}$ of 5.3$\,$% (6.0$\,$%) was achieved for the $\sim\,$70$\,$MeV kinetic energy positrons in the optically uniform (nonuniform) calorimeter, as compared to 5.7$\,$% (6.8$\,$%) resolution for the equivalent energy photons. That was the resolution found with no cuts applied on the light-equivalent energy generated in the calorimeter vetoes. Requiring less than 5$\,$MeV detected in the veto shield decreased the statistics by about 10$\,$% and suppressed the low-energy tail, but did not improve the energy resolution at the peak position.
Limiting the ADC sums to the nearest-neighbor crystals (six or seven crystal clusters) the fractional FWHM$_{\rm (NN)}$ energy resolution could be parameterized by: $$\begin{aligned}
{ {\rm FWHM_{\rm (NN)}}\over {E_{e^+}} }(\%)=
\cases{ {{(69.19+3.057 E_{e^+}+0.4198\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.8560}} }\cr
{ {(68.13+2.479 E_{e^+}+0.3368\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.8018}} }\cr} \end{aligned}$$ for the positrons with the detected peak energy $E_{e^+}$ MeV and $$\begin{aligned}
{ {\rm FWHM_{\rm (NN)}}\over {E_\gamma} }(\%)=\cases{
{ {(43.16+2.318 E_\gamma+0.4148\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.7723}} }\cr
{ {(39.57+2.302 E_\gamma+0.1329\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.7199}} }\cr}\end{aligned}$$ for the monoenergetic photons with the detected peak energy $E_\gamma$ MeV (Fig. \[fig:fit\_fwhm\_nn\]).
The average FWHM$_{\rm (NN)}$ for the 70$\,$MeV positrons and gammas in nearest-neighbor nonuniform crystal clusters was 8.7$\,$% (6.1$\,$MeV) and 9.9$\,$% (6.8$\,$MeV), respectively. Imposing the 5$\,$MeV cut on the detector veto signals improves these resolutions only marginally. These numbers should be compared with the response width of the ideal uniform nearest-neighbor clusters: 8.2$\,$% (5.7$\,$MeV) and 8.7$\,$% (6.2$\,$MeV).
The percentage of the events in the tail between the preset low and high energy threshold was tracked in the same [GEANT]{} simulation.
The listing of the low-energy tail contributions for 69.8$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$ in Table \[tab3\] shows that in the optically uniform calorimeter they differ by 1.8$\,$%. The optical nonuniformity does not change the positron tail contribution, but increases the photon tail by $\sim\,$0.2$\,$%. The imposition of the 5$\,$MeV veto-shield cut decreases the electromagnetic leakage to 2.4$\,$% for positrons and 4.7$\,$% for photons, a $e^+$-$\gamma$ tail difference of $+\,$2.3$\,$%.
Table \[tab4\] shows the low-energy tail components for the simulated nearest-neighbor cluster ADC sums: with no applied cuts the contributions were 4.7$\,$% and 6.7$\,$% for $e^+$’s and $\gamma$’s, respectively. The veto shielding cuts decreased both fractions by about 0.3$\,$%.
If the positron originates from a radiative $\pi^+$$\rightarrow$$e^+\nu\gamma$ decay, its average Monte Carlo peak position is essentially unchanged. Its tail contribution is unaffected if simulated ADC sums extend over full calorimeter but are increased by $\sim\,$1.4$\,$% if only the nearest-neighbor clusters are summed. The radiative decay matrix elements used in the calculation were taken from Ref. [@Bro64]. All quoted numbers have an approximate systematic uncertainty of $\sim\,$0.2$\,$%.
The detector response to positrons and photons in the energy range between 10$\,$MeV and 120$\,$MeV is also nonlinear because the volume distribution of the shower energy deposition depends on the incident particle type and the energy. Our [GEANT]{} studies showed that the magnitude and energy dependence of these nonlinearities does not change significantly because of CsI crystal light collection nonuniformity. Results of the calculations, both for the case of a homogeneous and optically nonhomogeneous calorimeter, are displayed in Figs. \[fig:fit\_nonlin\_220\] and \[fig:fit\_nonlin\_nn\]. Nonlinearities of the detector response in the covered energy range for positrons and photons are very close in magnitude and shape of energy variation and amounted to $\sim\,$1.5$\,$% for the ADC sums over 220 crystals and up to $\sim\,$2.8$\,$% if the simulated ADC sums were restricted to the over-the-threshold ADCs of nearest neighbors.
Conclusion
==========
We have measured the optical properties of seventy-four pure cesium iodide crystals that were polished and wrapped in the diffuse Teflon reflector. The results are summarized separately for the full-sized and half-sized crystals in Tables \[tab1\] and \[tab2\].
The deduced light yields parameterized by two axial and one transverse light collection nonuniformity coefficients constitute a minimum set of parameters necessary for a realistic Monte Carlo simulation of the modular CsI calorimeter.
The predicted energy resolutions FWHM$_{(220)}$ for $\sim\,$70$\,$MeV positrons and photons in the full PIBETA calorimeter with the ideal, optically uniform CsI modules with specified luminosities were shown to be close, 3.7$\,$MeV and 4.0$\,$MeV, respectively. The upper limit of the low energy tail contributions in the region between 5$\,$MeV and 55$\,$MeV were calculated to be 6.9$\,$% and 8.7$\,$% for the positrons and gammas in an optically homogeneous detector, respectively. After applying the 5$\,$MeV calorimeter veto cut, these tail corrections decrease to 2.2$\,$% and 4.5$\,$%.
The average deduced axial light nonuniformities of real CsI crystals wrapped in a Teflon sheet had negative slopes, $-\,$0.18$\,$%/cm and $-\,$1.6$\,$%/cm, for the front and back half of the crystal volume, respectively. The corresponding [GEANT]{} simulation of the nonuniform PIBETA apparatus for 68.9$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$’s shows that both energy responses will be broadened to the average FWHM$_{(220)}$ of 5.9$\,$% and 6.9$\,$% while the associated tail contributions will change only for photon spectra, increasing the low-energy tail by $\sim\,$0.3$\,$%. The simulated calorimeter ADC spectra are shown in Figs. \[fig:tails1\] and \[fig:tails2\].
The nonlinearity of the measured energy scale caused by the optical nonuniformity is $\le\,$2.8$\,$% throughout the relevant $e^+$/$\gamma$ energy range. This spread is consistent with the precision of energy calibration required to extract the tail corrections with the systematic uncertainty of $\sim\,$0.2$\,$%.
The predicted ADC spectra of the monoenergetic positrons, electrons and tagged photons in the energy range 10–70$\,$MeV will be compared with the measured responses of the partial CsI calorimeter arrays in a forthcoming publication [@Frl97].
Acknowledgements
================
The authors wish to thank Micheal Sadler of the Abilene Christian University (ACU) for lending us the drift chamber tomography apparatus. Derek Wise, also of ACU, has helped with the cosmic muon tomography measurements. Roger Schnyder of Paul Scherrer Institut has maintained and repaired the faulty electronics modules. Their help is gratefully acknowledged.
This work is supported and made possible by grants from the US National Science Foundation and the Paul Scherrer Institute.
[9]{} D. Počanić, D. Day, E. Frlež, R. M. Marshall, J. S. McCarthy, R. C. Minehart, K. O. H. Ziock, M. Daum, R. Frosch and D. Renker, PSI R-89.01 Experiment Proposal (Paul Scherrer Institute, Villigen, 1988).
W. K. McFarlane, L. B. Auerbach, F. C. Gaille, V. L. Highland, E. Jastrzembski, R. J. Macek, F. H. Cverna, C. M. Hoffman, G. E. Hogan, R. E. Morgado and R. D. Werbeck, Phys. Rev. D32 (1985) 547.
G. Czapek, A. Federspiel, A. Fluckiger, D. Frei, B. Hahn, C. Hug, E. Hugentobler, W. Krebs, U. Moser, D. Muster, E. Ramseyer, H. Scheidiger, P. Schlatter, G. Stucki, R. Abela, D. Renker and E. Steiner, Phys. Rev. Lett. 70 (1993) 17.
D. I. Britton, S. Ahmad, D. A. Bryman, R. A. Burnham, E. T. H. Clifford, P. Kitching, Y. Kuno, J. A. Macdonald, T. Numao, A. Olin, J. M. Poutissou and M. S. Dixit, Phys. Rev. D49 (1994) 28.
A. Sirlin, Rev. Mod. Phys. 50 (1978) 573.
A. Sirlin, private communication (1989).
I. S. Towner and J. C. Hardy, Currents and Their Couplings in the Weak Sector of the Standard Model, NUCL-TH/9504015 Preprint (1995).
W. J. Marciano and A. Sirlin, Phys. Rev. D35 (1987) 1672.
R. B. Murray and F. J. Keller, Phys. Rev. 137A (1965) A942.
C. W. Bates, A. Salau, and D. Leniart, Phys. Rev. B15 (1977) 5963.
H. Kobayashi, A. Konaka, K. Miyake, T. T. Nakamura, T. Nomura, N. Sasao, T. Yamashita, S. Sakuragi, and S. Hashimoto, Kyoto University Preprint KUNS-900 (1987).
S. Kubota, H. Murakami, J. Z. Ruan, N. Iwasa, S. Sakuragi, and S. Hashimoto, Nucl. Inst. and. Meth. A273 (1988) 645.
S. Kubota, S. Sakuragi, S. Hashimoto and J. Z. Ruan, Nucl. Inst. and Meth. A268 (1988) 275.
S. Keszthelyi-Lándori, I. Földvári, R. Voszka, Z. Fodor, and Z. Seres, Nucl. Inst. and Meth. A303 (1991) 374.
C. L. Woody, P. W. Levy, J. A. Kierstead, T. Skwarnicki, Z. Sobolewski, M. Goldberg, N. Horwitz, P. Souder and D. F. Anderson, IEEE Trans. Nucl. Sci. 37 (1990) 492.
P. Schotanus, R. Kamermans and P. Dorenbos, IEEE Trans. Nucl. Sci. 37 (1990) 177.
B. K. Utts and S. E. Spagno, IEEE Trans. Nucl. Sci. 37 (1990) 134.
A. V. Gektin, A. I. Gorelov, V. I. Rykalin, V. I. Selivanov, N. V. Shiran, and V. G. Vasilchenko, Nucl. Inst. and Meth. A294 (1990) 591.
M. M. Hamada, Y. Nunoya, S. Kubota, and S. Sakuragi, Nucl. Inst. and. Meth. A365 (1995) 98.
C. Woody, BNL-47776-MC (1992).
Z. Y. Wei and R. Y. Zhu, Nucl. Inst. and Meth. A326 (1993) 326.
J. Brose, G. Dahlinger, P. Eckstein, K. R. Schubert and R. Seitz, SLAC BaBar Note \#175, Stanford (1995).
G. Dahlinger, SLAC BaBar Note \#241, Stanford (1996).
C. Morris and R. L. Boudrie, A Proposal for a High Resolution Spectrometer for Neutral Mesons (LAMPF, Los Alamos, 1989).
R. E. Ray, FERMILAB-CONF-94-418 (1994).
R. S. Kessler, A. Roodman, P. Shawhan, N. Solomey, B. Winstein, S. Hansen, H. Nguyen, R. Ray, R. Tschirhart, J. Whitmore, T. Nakaya, and M. Lindgren, Nucl. Inst. and Meth. A368 (1996) 653.
G. David, A. Hershcovitch, S. Stoll, C. Woody, P. B. Munzinger, R. Hutter, J. Stachel, C. M. Zou, N. Horwitz, and Z. Sobolewski, Nucl Inst. and Meth. A348 (1994) 87.
H. Kenner, Geodesic Math and How to Use It (University of California Press, Berkeley, 1976).
Review of Particle Physics, Particle Data Group, Phys. Rev. D54 (1996) 72.
K. A. Assamagan, Ph. D. Thesis (University of Virginia, Charlottesville, 1995) unpublished.
D. H. Dowell, A. M. Sandorfi, A. Q. R. Baron, B. J. Fineman, O. C. Kistner, G. Matone, C. E. Thorn, and R. M. Sealock, Nucl. Instr. and Meth. A286 (1990) 183.
THORN-EMI Electron Tubes Catalog: Photomultipliers and Accessories, Rockaway, NJ (1993).
C. L. Smith, PMT Evaluation PIBETA Note, accessible at URL [http://helena.]{}[ -0.3em phys.virginia.edu/ -0.3em pibeta/]{} or [http://psw340.psi.ch/ -0.3em pibeta/]{}, (1995).
E. Frlež, Ch. Brönnimann, T. Flügel, J. E. Koglin, B. Krause, D. Počanić, D. Renker, S. Ritt, P. L. Slocum, [*The Light Response of Pure CsI Calorimeter Crystals Painted with Waveshift Lacquer*]{}, to be submitted to Nucl. Phys. and Meth.
L. G. Atencio, J. F. Amann, R. L. Boudrie and C. L. Morris, Nucl. Inst. and Meth. 187 (1981) 381.
C. L. Morris, Nucl. Inst. and Meth. 196 (1982) 263.
R. D. Ransome, Ph. D. Thesis, LA-8919-T (LAMPF, Los Alamos, 1981).
R. D. Ransome, S. J. Greene, C. L. Hollas, B. E. Bonner, M. W. McNaughton, C. L. Morris and H. A. Thiessen, Nucl. Inst. and Meth. 201 (1982) 309.
A. H. Walenta, Nucl. Inst. and Meth. 151 (1978) 461.
G. A. Erskine, Nucl. Inst. and Meth. 198 (1982) 325.
D. Brown and E. Sandler, Drift Chamber Electronics, LAMPF E-Division Semiannual Report LA-8069-PR (1979) 63.
S. Ritt, Data Acquisition PIBETA Note, accessible at URL [http://helena.]{}[ -0.3em phys.virginia.edu/ -0.3em pibeta/]{} or [http://psw340.psi.ch/ -0.3em pibeta/]{}, (1995).
F. James and M. Roos, MINUIT—Function Minimization and Error Analysis, CERNLIB D506 (CERN, Geneva, 1989).
M. W. McNaughton, Introduction to the LAMPF Recoil Polarimeter (Janus) MP-13/MWM/X87-01 (LAMPF, Los Alamos, 1987).
I. Supek, Ph. D. Thesis, (LAMPF, Los Alamos, 1989) unpublished.
R. Brun, O. Couet, C. Vandoni, P. Zanarini and M. Goossens, PAW 2.03—Physics Analysis Workstation CN/Q121 (CERN, Geneva, 1993).
W. R. Leo, Techniques for Nuclear and Particle Physics Experiments, (Springer-Verlag, New York, 1987).
P. J. Huber, Robust Statistics (Wiley, New York, 1981).
W. H. Press, B. P. Fannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes—Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).
LeCroy 1995 Research Instrumentation Catalogue (Chestnut Ridge, NY, 1995).
R. Brun, F. Bruyant, M. Maire, A. C. McPherson and P. Zanarini, [GEANT ]{}3.21 DD/EE/94-1 (CERN, Geneva, 1994).
B. Rossi, Rev. Mod. Phys. 20 (1948) 537.
B. K. Wright, Program [OPTICS]{} (University of Virginia, Charlottesville, 1992), unpublished.
B. K. Wright, Program [TkOptics]{} (University of Virginia, Charlottesville, 1994), unpublished.
J. K. Ousterhout, Tck and Tk Toolkit (Addison-Wesley, Melno Park, 1994).
E. Frlež, Measurement of Bulk CsI Index of Refraction and Attenuation Length (University of Virginia, Charlottesville, 1989), unpublished.
E. Frlež, S. Ritt, K. A. Assamagan, Ch. Brönnimann, T. Flügel, B. Krause, D. Počanić, D. Renker, P. L. Slocum, N. Soić, I. Supek, [*Three-Dimensional Reconstruction of Light Collection Probability Function for Pure CsI Scintillators*]{}, to be submitted to Nucl. Phys. and Meth.
The PIBETA Decay Experiment Home Page, accessible at URL [http://helena.phys.virginia.edu/ -0.3em pibeta/]{} or [http://psw340.psi.ch/ -0.3em pibeta/]{}, (1997).
The PIBETA Collaboration, Status Update of PSI Experiment R-89.01.1 (Paul Scherrer Institute, Villigen, 1995) unpublished.
E. Frlež, Photonuclear Reactions in [GEANT]{} Code, accessible at URL [http://helena.]{}[ -0.3em phys.virginia.edu/ -0.3em pibeta/]{} or [http://psw340.psi.ch/ -0.3em pibeta/]{}, (1995).
J. Ahrens, H. Borchert, K. H. Czock, H. B. Eppler, H. Gimm, H. Gundrun, M. Kröning, P. Riehn, G. Sita Ram, A. Zieger and B. Ziegler, Nucl. Phys. A251 (1975) 478.
A. Lepretre, H. Beil, R. Bergère, P. Carlos, J. Fagot, A De Miniac and A. Veyssière, Nucl. Phys. A367 (1981) 237.
H. Hebach, A. Wortberg, and M. Gari, Nucl. Phys. A267 (1976) 425.
B. L. Berman, R. L. Bramblett, J. T. Caldwell, H. S. Davis, M. A. Kelly and S. C. Fultz, Phys. Rev. 177 (1969) 1745.
R. L. Bramblett, J. T. Caldwell, B. L. Berman, R. R. Harvet and S. C. Fultz, Phys. Rev. 148 (1966) 1198.
G. G. Jonsson and B. Forkman, Nucl. Phys. A107 (1968) 52.
S. G. Brown and S. A. Bludman, Phys. Rev. 136 (1964) 1160.
E. Frlež, K. A. Assamagan, Ch. Brönnimann, T. Flügel, J. E. Koglin, B. Krause, D. W. Lawrence, R. C. Minehart, D. Počanić, D. Renker, B. G. Ritchie, S. Ritt, P. L. Slocum, and I. Supek, [*The Response of the PIBETA CsI Calorimeter Array to 10–70 MeV Positrons, Electrons and Tagged Photons*]{}, to be submitted to Nucl. Phys. and Meth.
--------------------------------------------- ---------- ----------
Fast-to-Total Ratio (100 ns/1 $\mu$s gate) 0.835 0.739
\# Photoelectrons/MeV (100 ns ADC gate) 83.4 65.1
\# Photoelectrons/MeV (1 $\mu$s ADC gate) 99.9 88.0
Fast Light Temp. Coefficient (%/$^\circ$C) $-1.20$ $-1.61$
Total Light Temp. Coefficient (%/$^\circ$C) $-1.25$ $-1.40$
Axial Nonuniformity Coefficient (%/cm),
$z$$\le$10$\,$cm, 100 ns ADC gate $-0.200$ $-0.124$
Axial Nonuniformity Coefficient(%/cm),
$z$$\ge$10$\,$cm, 100 ns ADC gate $-1.62$ $-1.61$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$6$\,$cm, 100 ns ADC gate $-0.400$ $-0.500$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$18$\,$cm, 100 ns ADC gate $-1.10$ $-1.20$
--------------------------------------------- ---------- ----------
: Average scintillation properties of hexagonal and pentagonal PIBETA CsI calorimeter shapes PENTAs, HEX–As, HEX–Bs, HEX–Cs and HEX–Ds (59 crystals). All crystals were polished and wrapped in two layers of a Teflon membrane and one layer of aluminized Mylar film. The light yields are normalized to the temperature of 18$^\circ$C. All other parameters were measured at the average laboratory room temperature of 22$^\circ$C.[]{data-label="tab1"}
--------------------------------------------- ---------- ----------
Fast-to-Total Ratio (100 ns/1 $\mu$s gate) 0.806 0.728
\# Photoelectrons/MeV (100 ns ADC gate) 34.5 31.5
\# Photoelectrons/MeV (1 $\mu$s ADC gate) 42.8 43.3
Fast Light Temp. Coefficient (%/$^\circ$C) $-2.08$ $-0.63$
Total Light Temp. Coefficient (%/$^\circ$C) $-1.78$ $-0.69$
Axial Nonuniformity Coefficient (%/cm),
$z$$\le$10$\,$cm, 100 ns ADC gate $-0.326$ $-0.500$
Axial Nonuniformity Coefficient(%/cm),
$z$$\ge$10$\,$cm, 100 ns ADC gate $-1.00$ $-1.83$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$6$\,$cm, 100 ns ADC gate $-0.755$ $-0.827$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$18$\,$cm, 100 ns ADC gate $-4.10$ $-5.14$
--------------------------------------------- ---------- ----------
: Average scintillation properties of half-hexagonal and trapezial PIBETA CsI calorimeter shapes HEX–D1/2s and VETO–1/2s (15 crystals). All crystals were polished and wrapped in two layers of Teflon foil and one layer of aluminized Mylar.[]{data-label="tab2"}
-------------------------------- ---------------- ---------------- ---------------- ----------------
Peak Position (MeV) 68.87$\pm$0.03 68.90$\pm$0.03 70.10$\pm$0.03 69.97$\pm$0.03
FWHM$_{(220)}$ (MeV) 3.66$\pm$0.03 4.10$\pm$0.03 3.98$\pm$0.03 4.76$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 6.44$\pm$0.09 6.46$\pm$0.09 8.20$\pm$0.10 8.47$\pm$0.10
5$\le$Events$\le$55$\,$MeV (%) 6.89$\pm$0.09 6.98$\pm$0.09 8.86$\pm$0.10 9.11$\pm$0.11
Peak Position (MeV) 68.88$\pm$0.03 68.92$\pm$0.03 70.07$\pm$0.03 69.99$\pm$0.03
FWHM$_{(220)}$ (MeV) 3.67$\pm$0.03 4.09$\pm$0.03 3.99$\pm$0.03 4.76$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 1.91$\pm$0.05 2.02$\pm$0.05 3.92$\pm$0.07 4.15$\pm$0.07
5$\le$Events$\le$55$\,$MeV (%) 2.24$\pm$0.05 2.36$\pm$0.05 4.46$\pm$0.07 4.68$\pm$0.07
-------------------------------- ---------------- ---------------- ---------------- ----------------
: The predicted energy resolutions and tail contributions for 69.8$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$’s events in the full PIBETA calorimeter. The light output in photoelectrons/MeV and the linear axial light collection nonuniformities measured for individual CsI crystals ($a_{z1,z2}$=[TOMOGRAPHY]{}) were used in a [GEANT]{} simulation. The values for the perfect optically homogeneous crystals ($a_{z1,z2}$=0) were shown for comparison.[]{data-label="tab3"}
-------------------------------- ---------------- ---------------- ---------------- ----------------
Peak Position (MeV) 67.74$\pm$0.03 67.85$\pm$0.03 68.80$\pm$0.03 68.65$\pm$0.03
FWHM$_{\rm (NN)}$ (MeV) 5.22$\pm$0.03 5.90$\pm$0.03 6.01$\pm$0.03 6.80$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 3.83$\pm$0.07 4.01$\pm$0.07 5.47$\pm$0.08 5.78$\pm$0.08
5$\le$Events$\le$55$\,$MeV (%) 4.54$\pm$0.07 4.73$\pm$0.08 6.33$\pm$0.09 6.67$\pm$0.09
Peak Position (MeV) 67.75$\pm$0.03 67.83$\pm$0.03 68.80$\pm$0.04 68.64$\pm$0.03
FWHM$_{\rm (NN)}$ (MeV) 5.52$\pm$0.03 5.83$\pm$0.03 5.92$\pm$0.03 6.76$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 3.53$\pm$0.07 3.73$\pm$0.07 5.28$\pm$0.08 5.59$\pm$0.08
5$\le$Events$\le$55$\,$MeV (%) 4.19$\pm$0.07 4.39$\pm$0.07 6.10$\pm$0.09 6.42$\pm$0.09
-------------------------------- ---------------- ---------------- ---------------- ----------------
: The predicted energy resolutions and tail contributions for 69.8$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$’s events in the PIBETA clusters containing the crystal with maximum energy deposition and its nearest neighbors.[]{data-label="tab4"}
FIGURE 1
FIGURE 2
-9cm
FIGURE 3
-1cm
FIGURE 4
FIGURE 5
-8cm
FIGURE 6
-10cm
FIGURE 7
-9cm
FIGURE 8
-9cm
FIGURE 9
-9cm
FIGURE 10
-9cm
FIGURE 11
-4.5cm
FIGURE 12
-4.5cm
FIGURE 13
-1cm
0.5cm
FIGURE 14
-1cm
0.5cm
FIGURE 15
-1cm
0.5cm
FIGURE 16
-9cm
FIGURE 17
-9cm
FIGURE 18
-1cm
0.5cm
FIGURE 19
-1cm
0.5cm
FIGURE 20
-1cm
0.5cm
FIGURE 21
-1cm
0.5cm
FIGURE 22
-1cm
0.5cm
FIGURE 23
-8.5cm
FIGURE 24
-8.5cm
FIGURE 25
|
---
abstract: 'We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boundary value problems are also considered.'
---
\[firstpage\]
Introduction
=============
In the the proof of the reiterated homogenization results obtained in [@llpw1] (see also [@llpw2]) the following two facts were used (the exact defintions and properties are given later in Section 2 and 3):
- If $(v_h)$ is a sequence of $Y$-periodic functions in $L_{\text{loc}}^p(R^n),$ $p>1,$ such that $v_h\rightarrow v$ weakly in $L^p(Y)$ and if $w_h
$ is defined as $w_h(x)=v_h(hx)$, then $w_h\rightarrow (1/\left| Y\right|
)\int_Yv(x)\,dx.$
- If $$-\hbox{div}(a(x,hx,\xi +Du_h^\xi ))=0\text{ }$$ on $Y=[0,y]^n,\ u_h^\xi \in W_{\text{per}}^{1,p}(Y),$ then $u_h^\xi
\rightarrow u_0^\xi $ weakly in $W_{\text{per}}^{1,p}(Y),$ where $u_0^\xi $ is the solution of the corresponding limit-equation. Here $a$ is monotone, continuous and satisfies suitable coerciveness and growth conditions in the third variable and periodic in the second.
We have not found proofs of these facts in the literature. The aim of this paper is to present such proofs. Moreover, we show that the first statement also holds for the case $p=1$.
The two facts described above are used in the proof of the reiterated homogenization result for monotone operators, see [@llpw1] and [@llpw2]. The solution $u_h^\xi $ is used to define a sequence of functions similar to the ones in Tartar’s celebrated method of oscillated test functions (see e.g. the book [@cioranescu99]). The first fact described above in combination with compensated compactness is used to analyze the asymptotic behavior of this sequence of functions.
For information concerning reiterated homogenization we recommend the papers [@llpw1] and [@llpw2] and the references given there. Concerning explicit engineering applications see e.g. [@bylund2001].
A weak convergence result
=========================
Let us first recall the following lemma (for the proof see e.g. [@dellacherie1975]).
\[lemdunford\]\[lemcompensated\]Let $\left\{ u_h\right\} $ be a sequence in $L^1(\Omega ).$ The following statements are equivalent:
1. every subsequence of $\left\{ u_h\right\} $ contains a subsequence which converges weakly in $L^1(\Omega ).$
2. for all $\varepsilon >0$ there exists $t_\varepsilon >0$ such that for all $h\in \left\{ 1,2,...\right\} $ it holds that $$\int_{\left\{ u_h\geq t_\varepsilon \right\} }\left| u_h\right| dx\leq
\varepsilon ,$$ where $\left\{ u_h\geq t_\varepsilon \right\} $ denotes the set $\left\{
x\in \Omega :u_h(x)\geq t_\varepsilon \right\} $.
The following Proposition is a generalization of the well-known fact that a periodic function converges weakly to its mean value as the oscillation increases.
\[lemfundamental\]Let $1\leq p\leq \infty $ and let $u_h\in L_{\text{loc}}^p(R^n)$ be $Y$ periodic for $h\in N$. Moreover, suppose that $u_h\rightarrow u~$weakly in $L^p(Y)$ (weak-$*$ if $p=\infty $) as $h\rightarrow \infty $. Let $w_h$ be defined by $w_h(x)=u_h(hx\dot{)}$. Then as $h\rightarrow \infty $ it holds that $$w_h\rightarrow \frac 1{\left| Y\right| }\int_Yu(x)dx$$ weakly in $L^p(\Omega )$ (weak-$*$ if $p=\infty $).
We first consider the case $1<p\leq \infty $. For simplicity we put $Y=(0,1)^{n}$, i.e. the unit cube in $R^{n}$, since the general case is principally the same. Let $Y_{h}^{k}=(1/h)\left( k+Y\right) $, where $k\in
Z^{n}$, i.e. the translated image of $1/hY$ by the vector $k/h$. We note that $\left| Y_{h}^{k}\right| =\left( 1/h\right) ^{n}$. Let $\phi \in
C_{0}(\Omega )$ and $\phi ^{h}$ the function which takes a constant value equal to the value $\phi (k/h)$ in each cell $Y_{h}^{k}$. Due to the uniform continuity of $\phi $ on the compact set $\overline{\Omega }$, we obtain that $\phi ^{h}\rightarrow \phi $ uniformly on $\Omega $. Thus, $$\int_{\Omega }\left| \phi ^{h}-\phi \right| ^{q}dx\leq \int_{\Omega }\left(
\max_{x\in \Omega }\left| \phi ^{h}(x)-\phi (x)\right| \right)
^{q}dx\rightarrow 0, \label{lqconverg}$$ as $h\rightarrow \infty $, i.e. $\phi ^{h}\rightarrow \phi $ in $L^{q}(\Omega )$. Since $\phi $ has compact support in $\Omega $ we have that each cell $Y_{h}^{k}$, for which $\phi (k/h)\neq 0$, is contained $\Omega $ for sufficiently large values of $h$. This and the $Y$-periodicity of $u_{h}$ implies that
$$\begin{aligned}
\int_{\Omega }u_{h}\left( hx\right) \phi ^{h}(x)dx &=&\int_{\mathbf{R}^{n}}u_{h}\left( hx\right) \phi ^{h}(x)dx \nonumber \\
&=&\sum_{k\in \mathbf{Z}^{n}}\int_{Y_{h}^{k}}u_{h}\left( hx\right) \phi
(k/h)dx \nonumber \\
&=&\sum_{k\in \mathbf{Z}^{n}}\left( \frac{1}{\left| Y_{h}^{k}\right| }\int_{Y_{h}^{k}}u_{h}\left( hx\right) dx\right) \left( \int_{Y_{h}^{k}}\phi
(k/h)dx\right) \nonumber \\
&=&\frac{1}{\left| Y\right| }\int_{Y}u_{h}\left( x\right) dx\sum_{k\in
\mathbf{Z}^{n}}\left( \int_{Y_{h}^{k}}\phi (k/h)dx\right) \label{large} \\
&=&\frac{1}{\left| Y\right| }\int_{Y}u_{h}\left( x\right) dx\int_{\Omega
}\phi ^{h}(x)dx, \nonumber\end{aligned}$$
for sufficiently large values of $h$. Moreover, we have that $u_{h}\left(
h\cdot \right) $ is bounded in $L^{p}(\Omega )$. This fact is shown as follows: Define the index set $I_{h}$ as $$I_{h}=\left\{ k\in Z^{n}:Y_{h}^{k}\cap \Omega \neq \emptyset \right\} .$$ Since $\Omega $ is bounded there exists a constant $K$ such that the number of elements in $I_{h}$ is less than $Kh^{n}$. We obtain that $$\begin{aligned}
\int_{\Omega }\left| u_{h}\left( hx\right) \right| ^{p}dx &\leq &\sum_{k\in
I_{h}}\int_{Y_{h}^{k}}\left| u_{h}\left( hx\right) \right| ^{p}dx \nonumber
\\
&=&\sum_{k\in I_{h}}\left( \frac{1}{h}\right) ^{n}\int_{k+Y}\left|
u_{h}\left( x\right) \right| ^{p}dx \label{ulikhet} \\
&\leq &K\int_{Y}\left| u_{h}\left( x\right) \right| ^{p}dx. \nonumber\end{aligned}$$ Now it follows that $u_{h}\left( h\cdot \right) $ is bounded in $L^{p}(\Omega )$ by taking into account that any weakly convergent sequence is bounded. By Hölder’s inequality we have that $$\begin{aligned}
&&\left| \int_{\Omega }\left( u_{h}\left( hx\right) -\frac{1}{\left|
Y\right| }\int_{Y}u\left( x\right) dx\right) \phi (x)dx\right| \\
&\leq &\left( \int_{\Omega }\left| u_{h}\left( hx\right) -\frac{1}{\left|
Y\right| }\int_{Y}u\left( x\right) dx\right| ^{p}dx\right) ^{\tfrac{1}{p}}\left( \int_{\Omega }\left| \phi (x)-\phi ^{h}(x)\right| ^{q}dx\right) ^{\tfrac{1}{q}} \\
&&+\left| \int_{\Omega }\left( u_{h}\left( hx\right) -\frac{1}{\left|
Y\right| }\int_{Y}u_{h}(x)dx\right) \phi ^{h}(x)dx\right| \\
&&+\left| \frac{1}{\left| Y\right| }\int_{Y}\left( u_{h}(x)-u(x)\right) \phi
^{h}(x)\,dx\right| \left| \Omega \right| .\end{aligned}$$ This together with (\[lqconverg\]), (\[large\]) and (\[ulikhet\]) implies that $$\int_{\Omega }\left( u_{h}\left( hx\right) -\frac{1}{\left| Y\right| }\int_{Y}u\left( x\right) dx\right) \phi (x)dx\rightarrow 0,$$ as $h\rightarrow \infty $ for every $\phi \in C_{0}(\Omega )$. By using a density argument it also holds for every $\phi \in L^{q}(\Omega )$ and we are done.
Let us now turn to the case $p=1$. Let $u_{h}^{i}$ be defined as follows $$u_{h}^{i}=\left\{
\begin{array}{l}
u_{h}\text{ \ \ if }u_{h}(x)<t_{1/i} \\
\\
0\;\;\;\text{ if }u_{h}(x)\geq t_{1/i}
\end{array}
\right. \text{\thinspace }.$$ According to Lemma \[lemdunford\] there exists a constant $t_{1/i}>0$ for each positive integer $i$ such that $$\int_{\Omega }\left| u_{h}^{i}-u_{h}\right| dx=\int_{\left\{ u_{h}\geq
t_{1/i}\right\} }\left| u_{h}\right| dx\leq \frac{1}{i}, \label{dag1}$$ for all $h,i\in N$. By a diagonalization argument each subsequence of $(h)$ has a subsequence, denoted by $(h^{^{\prime }})$, such that $u_{h^{^{\prime
}}}^{i}$ converges weak\* in $L^{\infty }(Y)$ to some function $u^{i}$ for every $i$. It is easy to see that the proof for the case ($1<p\leq \infty $) also holds if $(h)$ is replaced with $(h^{\prime })$, which implies that $$u_{h^{\prime }}^{i}(h^{\prime }\cdot )\rightarrow \frac{1}{\left| Y\right| }\int_{Y}u^{i}(x)dx \label{h-merked}$$ weak\* in $L^{\infty }(\Omega )$ for every $i$. Let $v\in L^{\infty }(\Omega
) $. Then $$\begin{aligned}
&&\limsup_{h^{\prime }\rightarrow \infty }\left| \int_{\Omega }v(x)\left(
u_{h^{\prime }}\left( h^{\prime }x\right) dx-\left( \frac{1}{\left| Y\right|
}\int_{Y}u(x)dx\right) \right) dx\right| \label{limsup} \\
&=&\limsup_{i\rightarrow \infty }\limsup_{h^{\prime }\rightarrow \infty
}\left| \int_{\Omega }v(x)\left( u_{h^{\prime }}\left( h^{\prime }x\right)
dx-\left( \frac{1}{\left| Y\right| }\int_{Y}u(x)dx\right) \right) dx\right|
\nonumber \\
&\leq &\limsup_{i\rightarrow \infty }\limsup_{h^{\prime }\rightarrow \infty
}\left| \int_{\Omega }v(x)\left( u_{h^{\prime }}\left( h^{\prime }x\right)
-u_{h^{\prime }}^{i}\left( h^{\prime }x\right) \right) dx\right| \nonumber
\\
&&+\limsup_{i\rightarrow \infty }\limsup_{h^{\prime }\rightarrow \infty
}\left| \int_{\Omega }v(x)\left( u_{h^{\prime }}^{i}\left( h^{\prime
}x\right) dx-\left( \frac{1}{\left| Y\right| }\int_{Y}u(x)dx\right) \right)
dx\right| . \nonumber\end{aligned}$$ Both of the last terms are zero. For the first term this is seen by replacing $u_{h}$ with $u_{h^{\prime }}^{i}-u_{h^{\prime }}$ in (\[ulikhet\]) and using (\[h-merked\]) and (\[dag1\]) to obtain that $$\begin{aligned}
\left| \int_\Omega v(x)\left( u_{h^{\prime }}\left( h^{\prime }x\right)
-u_{h^{\prime }}^i\left( h^{\prime }x\right) \right) dx\right| &\leq
&\left\| v(x)\right\| _\infty \int_\Omega \left| u_{h^{\prime }}^i\left(
h^{\prime }x\right) -u_{h^{\prime }}\left( h^{\prime }x\right) \right| dx \\
&\leq &\left\| v(x)\right\| _\infty K\int_Y\left| u_{h^{\prime }}^i\left(
x\right) -u_{h^{\prime }}\left( x\right) \right| dx \\
&\leq &\left\| v(x)\right\| _\infty \frac Ki.\end{aligned}$$ From this it is clear that $$\limsup_{i\rightarrow \infty }\limsup_{h^{\prime }\rightarrow \infty }\left|
\int_{\Omega }v(x)\left( u_{h^{\prime }}\left( h^{\prime }x\right)
-u_{h^{\prime }}^{i}\left( h^{\prime }x\right) \right) dx\right| =0.$$ For the second term we use (\[h-merked\]), the weak lower semicontinuity of the $L^{1}(\Omega )$ norm and (\[dag1\]) in order to obtain that $$\begin{aligned}
&&\limsup_{h^{\prime }\rightarrow \infty }\left| \int_{\Omega }v(x)\left(
u_{h^{\prime }}^{i}\left( h^{\prime }x\right) dx-\left( \frac{1}{\left|
Y\right| }\int_{Y}u(x)dx\right) \right) dx\right| \\
&\leq &\frac{1}{\left| Y\right| }\int_{Y}\left| u^{i}(x)-u(x)\right|
dx\left| \int_{\Omega }v(x)dx\right| \\
&\leq &\frac{1}{\left| Y\right| }\liminf_{h^{\prime }\rightarrow \infty
}\int_{Y}\left| u_{h^{\prime }}^{i}\left( x\right) -u_{h^{\prime }}\left(
x\right) \right| dx\left| \int_{\Omega }v(x)dx\right| \\
&\leq &\frac{1}{\left| Y\right| i}\left| \int_{\Omega }v(x)dx\right|
\rightarrow 0\end{aligned}$$ as $i\rightarrow \infty $. Summing up from (\[limsup\]) we have that any subsequence of $\left( u_{h}(h\cdot )\right) $ contains a subsequence $\left( u_{h^{\prime }}(h^{^{\prime }}\cdot )\right) $ which converges weakly to $\left| Y\right| ^{-1}\int_{Y}u(x)dx$ in $L^{1}(\Omega ).$ Thus this is also true for the whole sequence $\left\{ u_{h}(h\cdot )\right\} $.
Homogenization of some periodic boundary value problems
=======================================================
Before we state the result of this section we introduce some definitions and notations. Let $Y$ and $Z$ be a open bounded rectangles in $R^{n}$, $\left|
E\right| $ denotes the Lebesgue measure of the set $E\subset R^{n}$ and $(\cdot ,\cdot )$ is the Euclidean scalar product on $R^{n}$. Moreover, $c$ will be a constant that may differ from one place to an other and $h\in N$. The function $\widetilde{\omega }:R\rightarrow R$ is an arbitrary function which is continuous, increasing and $\widetilde{\omega }(0)=0$. By $W_{per}^{1,p}(Y)$ we denote the set of all functions $u\in W^{1,p}(Y)$ with mean value zero which have the same trace on opposite faces of $Y$, $W_{per}^{1,p}(Z)$ is defined in the corresponding way. Every function $u\in
W_{per}^{1,p}(Y)$ can be extended by periodicity to a function in $W_{loc}^{1,p}(R^{n})$ (in this paper we will not make any distinction between the original function and its extension). Let us fix a function $a:Y\times R^{n}\times R^{n}\rightarrow R^{n}$ which fulfills the conditions:
1. $a(y,\cdot ,\xi )$ is $Z$-periodic and Lebesgue measurable for every $\xi \in R^n$ and every $y\in R^n,$
2. There exists two constants $c_1,c_2>0$ and two constants $\alpha $ and $\beta $, with $0\leq \alpha \leq \min \left\{ 1,p-1\right\} $ and $\max
\left\{ p,2\right\} \leq \beta <\infty $ such that $a$ satisfies the following boundedness, continuity and monotonicity assumptions: $$a(y,z,0)=0\;\;\text{for a.e. }y,z\in R^n, \label{ayz0}$$ $$\left| a(y,z,\xi _1)-a(y,z,\xi _2)\right| \leq c_1(1+\left| \xi _1\right|
+\left| \xi _2\right| )^{p-1-\alpha }\left| \xi _1-\xi _2\right| ^\alpha ,
\label{acont}$$ $$(a(y,z,\xi _1)-a(y,z,\xi _2),\xi _1-\xi _2)\geq c_2(1+\left| \xi _1\right|
+\left| \xi _2\right| )^{p-\beta }\left| \xi _1-\xi _2\right| ^\beta
\label{amon}$$
3. $a$ is on the form $a(y,z,\xi )=\sum_{i=1}^N\chi _{\Omega
_i}(y)a_i(y,z,\xi )$ and satisfies a continuity condition of the form $$\left| a(y_1,z,\xi )-a(y_2,z,\xi )\right| ^q\leq \omega (\left|
y_1-y_2\right| )\left( 1+\left| \xi \right| ^p\right) ,\text{ \ \ }
\label{a1cont}$$ for $y_1,y_2\in \Omega _i$, $i=1,\ldots ,N$,$\;$a.e. $z\in R^n$ and every $\xi \in R^n$, and where $\omega :R\rightarrow R$ is continuous, increasing and $\omega (0)=0$.
By (\[ayz0\]), (\[acont\]), and (\[amon\]) it follows that $$\left| a(y,z,\xi )\right| \leq c\left( 1+\left| \xi \right| ^{p-1}\right) ,
\label{a1}$$ $$\left| \xi \right| ^{p}\leq c\left( 1+(a(y,z,\xi ),\xi )\right) , \label{a2}$$ hold for $y\in R^{n},\;$a.e. $z\in R^{n}$ and every $\xi \in R^{n}$.
We are now in the position to state the result in this section.
\[lemaux2\]Let $a$ satisfy (\[ayz0\]), (\[acont\]), (\[amon\]) and (\[a1cont\]). Moreover, let $(u_h^\xi )$ be the solutions of $$\left\{
\begin{array}{l}
\int_Y(a(x,hx,\xi +Du_h^\xi ),D\phi )\,dx=0\text{ for every }\phi \in
W_{per}^{1,p}(Y), \\
\\
u_h^\xi \in W_{per}^{1,p}(Y).
\end{array}
\right. \label{kalle}$$ Then $$\begin{aligned}
u_h^\xi &\rightarrow &u^\xi \text{ weakly in }W_{per}^{1,p}(Y), \\
a(x,hx,\xi +Du_h^\xi ) &\rightarrow &b(x,\xi +Du^\xi )\text{ weakly in }L^q(Y,R^n),\end{aligned}$$ as $h\rightarrow \infty $, where $u^\xi $ is the unique solution of $$\left\{
\begin{array}{l}
\int_Y(b(x,\xi +Du^\xi ),D\phi )\,dx=0\text{ for every }\phi \in
W_{per}^{1,p}(Y), \\
\\
u^\xi \in W_{per}^{1,p}(Y).
\end{array}
\right.$$ The operator $b:Y\times R^n\rightarrow R^n$ is defined as $$b(y,\tau )=\frac 1{\left| Z\right| }\int_Za(y,z,\tau +Dv^{\tau ,y}(z))\,dz,$$ where $v^{\tau ,y}$ is the unique solution of the cell-problem $$\left\{
\begin{array}{l}
\int_Z(a(y,z,\tau +Dv^{\tau ,y}(z)),D\phi )\,dz=0\text{ for every }\phi \in
W_{per}^{1,p}(Z), \\
\\
v^{\tau ,y}\in W_{per}^{1,p}(Z).
\end{array}
\right. \label{localeq1}$$
We divide the proof into several steps.
**Step 1.** Let $\left\{ \Omega _i^k\subset \Omega :i\in I_k\right\} $ denote a family of disjoint open sets with diameter less than $\frac 1k$ such that $\left| \Omega \backslash \cup _{i\in I_k}\Omega _i^k\right| =0$ and $\left| \partial \Omega _i^k\right| =0$. We define the function $a^k$ as $$a^k(y,z,\xi )=\sum_{i\in I_k}\chi _{\Omega _i^k}(y)a(y_i^k,z,\xi ),$$ where $y_i^k\in \Omega _i^k$. Consider the auxillary periodic boundary value problems (transmission problems) $$\left\{
\begin{array}{l}
\int_Y(a^k(x,hx,\xi +Du_h^{k,\xi }),D\phi )\,dx=0\text{\thinspace \thinspace
\thinspace \thinspace \thinspace \thinspace for every }\phi \in
W_{per}^{1,p}(Y), \\
\\
u_h^{k,\xi }\in W_{per}^{1,p}(Y).
\end{array}
\right. \label{uhk}$$ Then we have that $$u_h^{k,\xi }\rightarrow u^{k,\xi }\text{ weakly in }W_{per}^{1,p}(Y),$$ $$a^k(x,hx,\xi +Du_h^{k,\xi })\rightarrow b^k(x,\xi +Du^{k,\xi })\text{ weakly
in }L^q(Y,R^n),$$ where $u^{k,\xi }$ is the unique solution of the homogenized problem $$\left\{
\begin{array}{l}
\int_Y(b^k(x,\xi +Du^{k,\xi }),D\phi )\,dx=0\text{ for every }\phi \in
W_{per}^{1,p}(Y), \\
\\
u^{k,\xi }\in W_{per}^{1,p}(Y).
\end{array}
\right. \label{homprob}$$ The operator $b^k:Y\times R^n\rightarrow R^n$ is defined a.e. as $$b^k(y,\tau )=\sum_{i=I_k}\chi _{\Omega _i^k}(y)\int_Za(y_i^k,z,\tau
+Dv^{\tau ,y_i^k}(z))\,dz=\sum_{i=I_k}\chi _{\Omega _i^k}(y)b(y_i^k,\tau ),$$ where $v^{\tau ,y_i^k}$ is the unique solution of the cell problem $$\left\{
\begin{array}{l}
\int_Z(a(y_i^k,z,\tau +Dv^{\tau ,y_i^k}(z)),D\phi (z))\,dz=0\text{ for every
}\phi \in W_{per}^{1,p}(Z), \\
\\
v^{\tau ,y_i^k}\in W_{per}^{1,p}(Z).
\end{array}
\right. \label{cellproblem3}$$ The proof of these convergence results follows by suitable modifications of well-known homogenization techniques. Indeed, let $\phi =u_h^{k,\xi }$ in (\[kalle\]) then it follows by (\[a2\]), (\[ayz0\]), (\[acont\]) and Hölder’s inequality that $$\begin{aligned}
\int_Y\left| \xi +Du_h^{k,\xi }\right| ^p\,dx &\leq &c\int_Y1+(a^k(x,hx,\xi
+Du_h^{k,\xi }),\xi +Du_h^{k,\xi })\,dx \nonumber \\
&=&c\int_Y1+(a^k(x,hx,\xi +Du_h^{k,\xi }),\xi )\,dx \nonumber \\
&\leq &c\left( 1+\int_Y\left( 1+\left| \xi +Du_h^{k,\xi }\right| \right)
^{p-1}\,dx\right) \label{uhxibound1} \\
&\leq &c\left( 1+\left( \int_Y\left( \left| \xi +Du_h^{k,\xi }\right|
\right) ^p\,dx\right) ^{^{\tfrac 1q}}\right) . \nonumber\end{aligned}$$ If $\left( \int_Y\left( \left| \xi +Du_h^{k,\xi }\right| \right)
^p\,dx\right) ^{1/q}\leq 1$ it is clear that the sequence of solutions $(u_h^{k,\xi })$ is bounded in $L^p(Y,R^n)$, so let us assume that $\left(
\int_Y\left( \left| \xi +Du_h^{k,\xi }\right| \right) ^p\,dx\right)
^{1/q}\geq 1$, then (\[uhxibound1\]) implies that $$\int_Y\left| \xi +Du_h^{k,\xi }\right| ^p\,dx\leq c$$ which means that $(u_h^{k,\xi })$ is bounded in . Since $\left\| D\cdot
\right\| _{L^p(Y,R^n)}$ is an equivalent norm on $W_{per}^{1,p}(Y)$ it follows that there exists a constant $c>0$ independent of $h$ such that $$\left\| u_h^{k,\xi }\right\| _{W_{per}^{1,p}(Y)}\leq c.$$ From the reflexivity of $W_{per}^{1,p}(Y)$ there exists a subsequence, still denoted by $(u_h^{k,\xi })$ such that $$u_h^{k,\xi }\rightarrow u_{*}^{k,\xi }\text{ weakly in }W_{per}^{1,p}(Y).$$ Let us now define $$\eta _h^{i,k,\xi }=a(x_i^k,hx,\xi +Du_h^{k,\xi }),\;\;\;i\in I_k$$ By (\[acont\]), (\[ayz0\]), Hölder’s inequality and (\[uhxibound1\]) we have that $\eta _h^{i,k,\xi }$ is bounded in $L^q(\Omega _i^k,R^n)$. Indeed $$\begin{aligned}
\int_{\Omega _i^k}\left| \eta _h^{i,k,\xi }\right| ^q\,dx &=&\int_{\Omega
_i^k}\left| a(x_i^k,hx,\xi +Du_h^{k,\xi })\right| ^q\,dx \\
&\leq &c\int_{\Omega _i^k}\left( 1+\left| \xi +Du_h^{k,\xi })\right| \right)
^{q(p-1-\alpha )}\left| \xi +Du_h^{k,\xi })\right| ^{q\alpha }\,dx \\
&\leq &c\int_{\Omega _i^k}1+\left| \xi +Du_h^{k,\xi })\right| ^p\,dx\leq c\end{aligned}$$ where $c$ is a constant independent of $h$. This means that there exists a subsequence, still denoted by $(\eta _h^{i,k,\xi }),$ and a $\eta
_{*}^{i,k,\xi }\in L^q(\Omega _i^k,R^n)$ such that $$\eta _h^{i,k,\xi }\rightarrow \eta _{*}^{i,k,\xi }\text{ weakly in }L^q(\Omega _i^k,R^n).$$ From our original problem (\[kalle\]) we have that $$\left\{
\begin{array}{l}
\sum_{i\in I_k}\int_{\Omega _i^k}(a(x_i^k,hx,\xi +Du_h^{k,\xi }),D\phi
)\,dx=0\text{ for every }\phi \in W_{per}^{1,p}(Y), \\
\\
u_h^{k,\xi }\in W_{per}^{1,p}(Y).
\end{array}
\right.$$ In the limit we get $$\sum_{i\in I_k}\int_{\Omega _i^k}(\eta _{*}^{i,k,\xi },D\phi )\,dx=0\text{
for every }\phi \in W_{per}^{1,p}(Y).$$ Especially this means that $$\int_{\Omega _i^k}(\eta _{*}^{i,k,\xi },D\phi )\,dx=0\text{ for every }\phi
\in C_0^\infty (\Omega _i^k)\text{, \thinspace \thinspace \thinspace
\thinspace \thinspace \thinspace \thinspace }i\in I_k.$$ If we now could show that $$\eta _{*}^{i,k,\xi }=b(x_i^k,\xi +Du_{*}^{k,\xi })\text{ for a.e }x\in
\Omega _i^k, \label{eq15}$$ then it follows by the uniqueness of the homogenized problem (\[homprob\]) that $u_{*}^{k,\xi }=u^{k,\xi }$. To this aim we define the function $$w_h^{\tau ,x_i^k}(x)=(\tau ,x)+\frac 1hv^{\tau ,x_i^k}(hx),$$ where $v^{\tau ,x_i^k}$ is defined as in (\[cellproblem3\]). By periodicity we have that $$\begin{aligned}
w_h^{\tau ,x_i^k} &\rightarrow &(\tau ,\cdot )\text{ weakly in }W^{1,p}(\Omega _i^k), \\
Dw_h^{\tau ,x_i^k} &\rightarrow &\tau \text{ weakly in }L^p(\Omega _i^k,R^n),
\\
a(x_i^k,hx,Dw_h^{\tau ,x_i^k}) &\rightarrow &b(x_i^k,\tau )\text{ weakly in }L^q(\Omega _i,R^n).\end{aligned}$$ By the monotonicity of $a_i$ we have for a fix $\tau $ that $$\int_{\Omega _i}(a(x_i^k,hx,\xi +Du_h^{k,\xi })-a(x_i^k,hx,Dw_h^{\tau
,x_i^k}),\xi +Du_h^{k,\xi }-Dw_h^{\tau ,x_i^k})\phi \,dx\geq 0,$$ for every $\phi \in C_0^\infty (\Omega _i),\phi \geq 0.$ By density we obtain that $$(\eta _{*}^{i,k,\xi }(x)-b(x_i^k,\tau ),\xi +Du_{*}^{k,\xi }(x)-\tau )\geq 0\text{ for a.e. }x\in \Omega _i^k\text{ and for every }\tau \in R^n.$$ Since $b^k$ is monotone and continuous, see Proposition \[lemmab1prop\], we have that $b^k$ is maximal monotone and the crucial relation (\[eq15\]) follows. We have now proved step 1 up to a subsequence of $(u_h^{k,\xi })$. By the uniqueness of the solution of the homogenized equation (\[homprob\]) it follows that it is true for the whole sequence.
**Step 2.** Let us now prove that $u_{h}^{\xi }\rightarrow u^{\xi }$ weakly in $W_{per}^{1,p}(Y)$. Let $g\in \left( W_{per}^{1,p}(Y)\right) ^{*},$ then $$\begin{aligned}
\lim_{h\rightarrow \infty }\left\langle g,u_{h}^{\xi }-u^{\xi }\right\rangle
&=&\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\langle
g,u_{h}^{\xi }-u^{\xi }\right\rangle \\
&\leq &\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\|
g\right\| _{\left( W_{per}^{1,p}(Y)\right) ^{*}}\left\| u_{h}^{\xi
}-u_{h}^{k,\xi }\right\| _{W_{per}^{1,p}(Y)} \\
&&+\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\langle
g,u_{h}^{k,\xi }-u^{k,\xi }\right\rangle \\
&&+\lim_{k\rightarrow \infty }\left\| g\right\| _{\left(
W_{per}^{1,p}(Y)\right) ^{*}}\left\| u^{k,\xi }-u^{\xi }\right\|
_{W_{per}^{1,p}(Y)}.\end{aligned}$$ It is enough to prove that all three terms on the right hand side are zero.
**Term 1.** Let us prove that $$\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\| u_h^\xi
-u_h^{k,\xi }\right\| _{W_{per}^{1,p}(Y)}=0. \label{step1a}$$ By definition $$\begin{aligned}
\int_Y(a^k(x,hx,\xi +Du_h^{k,\xi }),D\phi )\,dx &=&0\text{ for every }\phi
\in W_{per}^{1,p}(Y), \\
\int_Y(a(x,hx,\xi +Du_h^\xi ),D\phi )\,dx &=&0\text{ for every }\phi \in
W_{per}^{1,p}(Y).\end{aligned}$$ This implies that we for $\phi =u_h^{k,\xi }-u_h^\xi $ have $$\begin{aligned}
&&\int_Z(a^k(x,hx,\xi +Du_h^{k,\xi })-a^k(x,hx,\xi +Du_h^\xi ),Du_h^{k,\xi
}-Du_h^\xi )\,dx \\
&=&\int_Z(a(x,hx,\xi +Du_h^\xi )-a^k(x,hx,\xi +Du_h^\xi ),Du_h^{k,\xi
}-Du_h^\xi )\,dx.\end{aligned}$$ By using (\[amon\]), and Hölder’s reversed inequality on the left hand side and Hölder inequality (\[a1cont\]) and the fact that $(u_h^\xi )$ and $(u_h^{k,\xi })$ is bounded in $W_{per}^{1,p}(Y)$ on the right hand side we obtain that $$\begin{aligned}
&&c_2\left( \int_Y\left| Du_h^{k,\xi }-Du_h^\xi \right| ^p\,dx\right)
^{\tfrac \beta p} \\
&&\times \left( \int_Y\left( 1+\left| \xi +Du_h^{k,\xi }\right| +\left| \xi
+Du_h^\xi \right| \right) ^p\,dx\right) ^{\tfrac p{p-\beta }} \\
&\leq &c_2\int_Y\left( 1+\left| \xi +Du_h^{k,\xi }\right| +\left| \xi
+Du_h^\xi \right| \right) ^{p-\beta }\left| Du_h^{k,\xi }-Du_h^\xi \right|
^\beta \,dx \\
&\leq &\left( \int_Y\left| (a(x,\frac x{\varepsilon _h},\xi +Du_h^\xi
)-a^k(x,\frac x{\varepsilon _h},\xi +Du_h^\xi )\right| ^q\,dx\right)
^{\tfrac 1q} \\
&&\times \left( \int_Y\left| Du_h^{k,\xi }-Du_h^\xi \right| ^p\,dx\right)
^{\tfrac 1p} \\
&\leq &\widetilde{\omega }(\frac 1k)\left( \int_Y1+\left| \xi +Du_h^\xi
)\right| ^p\,dx\right) ^{\tfrac 1q}\left( \int_Y\left| Du_h^{k,\xi
}-Du_h^\xi \right| ^p\,dx\right) ^{\tfrac 1p} \\
&\leq &\widetilde{\omega }(\frac 1k)\left( \int_Y\left| Du_h^{k,\xi
}-Du_h^\xi \right| ^p\,dx\right) ^{\tfrac 1p}.\end{aligned}$$ Since $\left\| D\cdot \right\| _{L^p(Y,R^n)}$ is an equivalent norm on $W_{per}^{1,p}(Y)$ this implies that $$\left\| u_h^{k,\xi }-u_h^\xi \right\| _{W_{per}^{1,p}(Y)}\leq \widetilde{\omega }(\frac 1k)\rightarrow 0 \label{bound}$$ as $k\rightarrow \infty $ uniformly in $h$. This means that we can change the order in the limit process in (\[step1a\]) and (\[step1a\]) follows by taking (\[bound\]) into account.
**Term 2.** We observe that $$\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\langle
g,u_h^{k,\xi }-u^{k,\xi }\right\rangle =0,$$ as a direct consequence of Step 1.
**Term 3.** Let us prove that $$\lim_{k\rightarrow \infty }\left\| u^{k,\xi }-u^{\xi }\right\|
_{W_{per}^{1,p}(Y)}=0. \label{step3}$$ By definition we have that $$\begin{aligned}
\int_{Y}(b^{k}(x,\xi +Du^{k,\xi }),D\phi )\,dx &=&0\text{ for every }\phi
\in W_{per}^{1,p}(Y), \\
\int_{Y}(b(x,\xi +Du^{\xi }),D\phi )\,dx &=&0\text{ for every }\phi \in
W_{per}^{1,p}(Y).\end{aligned}$$ Thus $$\begin{aligned}
&&\int_{Y}(b^{k}(x,\xi +Du^{k,\xi })-b^{k}(x,\xi +Du^{\xi }),D\phi )\,dx \\
&=&\int_{Y}(b(x,\xi +Du^{\xi })-b^{k}(x,\xi +Du^{\xi }),D\phi )\,dx,\end{aligned}$$ for every $\phi \in W_{per}^{1,p}(Y)$. Choose $\phi =u^{k,\xi }-u^{\xi }$ and take the strict monotonicity of $b^{k}$, see (\[bmon2\]), into account on the left hand side and apply the Hölder inequality and (\[bcont2\]) on the right hand side to obtain $$\begin{aligned}
&&c\left( \int_{Y}\left| Du^{k,\xi }-Du^{\xi }\right| ^{p}\,dx\right) ^{\tfrac{\beta }{p}}\times \\
&&\left( \int_{Y}\left( 1+\left| \xi +Du^{k,\xi }\right| +\left| \xi
+Du^{\xi }\right| \right) ^{p}\,dx\right) ^{\tfrac{p-\beta }{p}} \\
&\leq &c\int_{Y}\left( 1+\left| \xi +Du^{k,\xi }\right| +\left| \xi +Du^{\xi
}\right| \right) ^{p-\beta }\left| Du^{k,\xi }-Du^{\xi }\right| ^{\beta }\,dx
\\
&\leq &\left( \int_{Y}\left| b(x,\xi +Du^{\xi })-b^{k}(x,\xi +Du^{\xi
})\right| ^{q}\,dx\right) ^{\tfrac{1}{q}}\times \\
&&\left( \int_{Y}\left| Du^{k,\xi }-Du^{\xi }\right| ^{p}\,dx\right) ^{\tfrac{1}{p}} \\
&\leq &\widetilde{\omega }(\frac{1}{k})\left( \int_{Y}1+\left| \xi +Du^{\xi
}\right| ^{p}\,dx\right) ^{\tfrac{1}{q}}\left( \int_{Y}\left| Du^{k,\xi
}-Du^{\xi })\right| ^{p}\,dx\right) ^{\tfrac{1}{p}}.\end{aligned}$$ By using the fact that $u^{\xi }$ and $u^{k,\xi }$ are bounded in $W_{per}^{1,p}(Y)$ it follows that $$\left\| Du^{k,\xi }-Du^{\xi }\right\| _{L^{p}(Y,R^{n})}\leq \widetilde{\omega }(\frac{1}{k}), \label{bound2}$$ and the result follows by noting that $\left\| D\cdot \right\|
_{L^{p}(Y,R^{n})}$ is an equivalent norm on $W_{per}^{1,p}(Y)$.
**Step 3.** Next we prove that $a(x,hx,\xi +Du_{h}^{\xi })\rightarrow
b(x,\xi +Du^{\xi })$ weakly in $L^{q}(Y,R^{n})$. In fact if $g\in
(L^{q}(Y,R^{n}))^{*},$ then $$\begin{aligned}
&&\lim_{h\rightarrow \infty }\left\langle g,a(x,hx,\xi +Du_{h}^{\xi
})-b(x,\xi +Du^{\xi })\right\rangle \\
&=&\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\langle
g,a(x,hx,\xi +Du_{h}^{\xi })-b(x,\xi +Du^{\xi })\right\rangle \\
&\leq &\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\|
g\right\| \left\| a(x,hx,\xi +Du_{h}^{\xi })-a^{k}(x,hx,\xi +Du_{h}^{k,\xi
})\right\| _{L^{q}(Y,R^{n})} \\
&&+\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\langle
g,a^{k}(x,hx,\xi +Du_{h}^{k,\xi })-b^{k}(x,\xi +Du^{k,\xi }))\right\rangle \\
&&+\lim_{k\rightarrow \infty }\left\| g\right\| \left\| b^{k}(x,\xi
+Du^{k,\xi })-b(x,\xi +Du^{\xi })\right\| _{L^{q}(Y,R^{n})}.\end{aligned}$$ It is sufficient to prove that all three terms on the right hand side are zero.
**Term 1.** Let us show that $$\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\| a(x,hx,\xi
+Du_{h}^{\xi })-a^{k}(x,hx,\xi +Du_{h}^{k,\xi })\right\| _{L^{q}(Y,R^{n})}=0.
\label{step12}$$ By using elementary estimates we find that $$\begin{aligned}
&&\int_{Y}\left| a^{k}(x,hx,\xi +Du_{h}^{k,\xi })-a(x,hx,\xi +Du_{h}^{\xi
})\right| ^{q}\,dx \\
&\leq &c\int_{Y}\left| a^{k}(x,hx,\xi +Du_{h}^{k,\xi })-a^{k}(x,hx,\xi
+Du_{h}^{\xi })\right| ^{q}\,dx \\
&&+c\int_{Y}\left| a^{k}(x,hx,\xi +Du_{h}^{\xi })-a(x,hx,\xi +Du_{h}^{\xi
})\right| ^{q}\,dx.\end{aligned}$$ Hence, by applying the continuity conditions (\[acont\]) and Hölder inequality to the first term and (\[a1cont\]) to the second term we obtain that $$\begin{aligned}
&&\int_{Y}\left| a^{k}(x,hx,\xi +Du_{h}^{k,\xi })-a(x,hx,\xi +Du_{h}^{\xi
})\right| ^{q}\,dx \\
&\leq &c\left( \int_{Y}\left( 1+\left| \xi +Du_{h}^{k,\xi }\right| +\left|
\xi +Du_{h}^{\xi }\right| \right) ^{p}\,dx\right) ^{\tfrac{p-1-\alpha }{p-1}}
\\
&&\times \left( \int_{Y}\left| Du_{h}^{k,\xi }-Du_{h}^{\xi }\right|
^{p}\,dx\right) ^{\tfrac{\alpha }{p-1}}+\widetilde{\omega }(\frac{1}{k})\int_{Y}1+\left| \xi +Du_{h}^{\xi }\right| ^{p}\,dx.\end{aligned}$$ By using the fact that $u_{h}^{k,\xi }$ and $u_{h}^{\xi }$ are bounded in $W_{per}^{1,p}(Y)$ and (\[bound\]) it follows that $$\left\| a(x,hx,\xi +Du_{h}^{\xi })-a^{k}(x,hx,\xi +Du_{h}^{k,\xi })\right\|
_{L^{q}(Y,R^{n})}\leq \widetilde{\omega }(\frac{1}{k})\rightarrow 0.
\label{step121}$$ as $k\rightarrow \infty $ uniformly in $h.$ This implies that we may change the order in the limit process in (\[step12\]) and we obtain (\[step12\]) by taking (\[step121\]) into account.
**Term 2.** We observe that $$\lim_{k\rightarrow \infty }\lim_{h\rightarrow \infty }\left\langle
g,a^k(x,hx,\xi +Du_h^{k,\xi })-b^k(x,\xi +Du^{k,\xi })\right\rangle =0,$$ as a direct consequence of Step 1.
**Term 3**. Let us show that $$\lim_{k\rightarrow \infty }\left\| b^{k}(x,\xi +Du^{k,\xi })-b(x,\xi
+Du^{\xi })\right\| _{L^{q}(Y,R^{n})}=0. \label{step13}$$ We have that $$\begin{aligned}
&&\int_{Y}\left| b^{k}(x,\xi +Du_{*}^{k,\xi })-b_{1}(x,\xi +Du^{\xi
})\right| ^{q}\,dx \\
&\leq &c\int_{Y}\left| b^{k}(x,\xi +Du_{*}^{k,\xi })-b^{k}(x,\xi +Du^{\xi
})\right| ^{q}\,dx \\
&&+c\int_{Y}\left| b^{k}(x,\xi +Du^{\xi })-b_{1}(x,\xi +Du^{\xi })\right|
^{q}\,dx.\end{aligned}$$ By applying the continuity condition (\[bcont12\]) and Hölders’s inequality to the first term and the continuity condition (\[bcont2\]) to the second term we see that $$\begin{aligned}
&&\int_{Y}\left| b^{k}(x,\xi +Du^{k,\xi })-b(x,\xi +Du^{\xi })\right|
^{q}\,dx \\
&\leq &c\left( \int_{Y}\left( 1+\left| \xi +Du^{k,\xi }\right| +\left| \xi
+Du^{\xi }\right| \right) ^{p}\,dx\right) ^{\tfrac{p-1-\gamma }{p-1}} \\
&&\times \left( \int_{Y}\left| Du^{k,\xi }-Du^{\xi }\right| ^{p}\,dx\right)
^{\tfrac{\gamma }{p-1}}+\widetilde{\omega }(\frac{1}{k})\int_{Y}\left|
Du^{\xi }\right| ^{p}\,dx.\end{aligned}$$ By using the fact that $u^{k,\xi }$ and $u^{\xi }$ are bounded in $W_{per}^{1,p}(Y)$ and (\[bound2\]) it follows that $$\left\| b^{k}(x,\xi +Du^{k,\xi })-b(x,\xi +Du^{\xi })\right\|
_{L^{q}(Y,R^{n})}\leq \widetilde{\omega }(\frac{1}{k})\rightarrow 0$$ and we are done.
$\square $
We remark that we have only considered the case when $a$ satisfies (\[a1cont\]) over the whole $Y$ the piecewise case follows by using the technique used in step 1.
\[lemmab1prop\]Let $b$ be the homogenized operator defined in Theorem \[lemaux2\]. Then
- $b(\cdot ,\xi )$ satisfies the continuity condition $$\left| b(y_1,\xi )-b(y_2,\xi )\right| ^q\leq \widetilde{\omega }(\left|
y_1-y_2\right| )\left( 1+\left| \xi \right| ^p\right) . \label{bcont2}$$
- $b(x,\cdot )$ is strictly monotone, more precisely $$(b_1(y,\xi _1)-b_1(y,\xi _2),\xi _1-\xi _2)\geq c\left( 1+\left| \xi
_1\right| +\left| \xi _2\right| \right) ^{p-\beta }\left| \ \xi _1-\xi
_2\right| ^\beta , \label{bmon2}$$ $\xi _1,\xi _2\in R^n.$
- $b(x,\cdot )$ is Lipschitz continuous, more precisely $$\left| b(x,\xi _1)-b(x,\xi _2)\right| \leq c\left( 1+\left| \xi _1\right|
+\left| \xi _2\right| \right) ^{p-1-\gamma }\left| \ \xi _1-\xi _2\right|
^\gamma , \label{bcont12}$$ for every $\xi _1,\xi _2\in R^n$, where $\gamma =\alpha /(\beta -\alpha )$.
- $$b(x,0)=0\text{ for }x\in Z. \label{b2x0}$$
(i): By the definition of $b$ and Jensen’s inequality we have that $$\begin{aligned}
&&\left| b(y_1,\tau )-b(y_2,\tau )\right| ^q \\
&=&\left| \int_Za(y_1,z,\tau +Dv^{\tau ,y_1}(z))\,-a(y_2,z,\tau +Dv^{\tau
,y_2}(z))\,dz\right| ^q \\
&\leq &c\int_Z\left| a(y_1,z,\tau +Dv^{\tau ,y_1}(z))\,-a(y_2,z,\tau
+Dv^{\tau ,y_1}(z))\right| ^q\,dz \\
&&+c\int_Z\left| a(y_2,z,\tau +Dv^{\tau ,y_1}(z))-a(y_2,z,\tau +Dv^{\tau
,y_2}(z))\right| ^q\,dz.\end{aligned}$$ By applying (\[a1cont\]) to the first term and (\[acont\]) in combination with Hölder’s inequality to the second term we obtain that $$\begin{aligned}
&&\left| b(y_1,\tau )-b(y_2,\tau )\right| ^q\leq \widetilde{\omega }(\left|
y_1-y_2\right| )\int_Z1+\left| \tau +Dv^{\tau ,y_1}\right| ^p\,dz \nonumber
\\
+ &&c\left( \int_Z\left( 1+\left| \tau +Dv^{\tau ,y_1}\right| +\left| \tau
+Dv^{\tau ,y_2}\right| \right) ^p\,dz\right) ^{\tfrac{p-1-\alpha }{p-1}}
\nonumber \\
&&\times \left( \int_Z\left| Dv^{\tau ,y_1}-Dv^{\tau ,y_2}\right|
^p\,dz\right) ^{\tfrac \alpha {p-1}} \label{bcont1}\end{aligned}$$ Let us now study the two terms in (\[bcont1\]) separately. The first term: (\[a2\]), (\[localeq1\]) and (\[a1\]) yields
$$\begin{aligned}
\int_Z\left| \tau +Dv^{\tau ,y_1}\right| ^p\,dz &\leq &c\int_Z1+(a(y,z,\tau
+Dv^{\tau ,y_1}),\tau +Dv^{\tau ,y_1})\,dz \\
&=&c\int_Z1+(a(y,z,\tau +Dv^{\tau ,y_1}),\tau )\,dz \\
&\leq &c\int_Z1+c\left( 1+\left| \tau +Dv^{\tau ,y_1}\right| ^{p-1}\right)
\left| \tau \right| \,dz.\end{aligned}$$
By using the Young inequality we obtain that $$\int_Z\left| \tau +Dv^{\tau ,y_1}\right| ^p\,dz\leq c\left( 1+\left| \tau
\right| ^p\right) . \label{bconta}$$ Let us now study the second term in (\[bcont1\]): By definition we have that $$\begin{aligned}
\int_Z(a(y_1,z,\tau +Dv^{\tau ,y_1}),D\phi )\,dz &=&0\text{ for every }\phi
\in W_{per}^{1,p}(Z), \\
\int_Z(a(y_2,z,\tau +Dv^{\tau ,y_2}),D\phi )\,dz &=&0\text{ for every }\phi
\in W_{per}^{1,p}(Z).\end{aligned}$$ This implies that $$\begin{aligned}
&&\int_Z(a(y_1,z,\tau +Dv^{\tau ,y_1})-a(y_1,z,\tau +Dv^{\tau ,y_2}),D\phi
)\,dz \\
&=&\int_Z(a(y_2,z,\tau +Dv^{\tau ,y_2})-a(y_1,z,\tau +Dv^{\tau ,y_2}),D\phi
)\,dz,\end{aligned}$$ for every $\phi \in W_{per}^{1,p}(Z)$. In particular, for $\phi =v^{\tau
,y_1}-v^{\tau ,y_2}$, we have that $$\begin{aligned}
&&\int_Z(a(y_1,z,\tau +Dv^{\tau ,y_1})-a(y_1,z,\tau +Dv^{\tau
,y_2}),Dv^{\tau ,y_1}-Dv^{\tau ,y_2})\,dz \\
&=&\int_Z(a(y_2,z,\tau +Dv^{\tau ,y_2})-a(y_1,z,\tau +Dv^{\tau
,y_2}),Dv^{\tau ,y_1}-Dv^{\tau ,y_2})\,dz.\end{aligned}$$ By applying the reversed Hölder inequality and (\[amon\]) on the left hand side and Schwarz’s and Hölder’s inequalities on the right hand side it follows that $$\begin{aligned}
&&c\left( \int_Z\left| Dv^{\tau ,y_1}-Dv^{\tau ,y_2}\right| ^p\;dz\right)
^{\tfrac \beta p}\times \\
&&\left( \int_Z\left( 1+\left| \tau +Dv^{\tau ,y_1}\right| +\left| \tau
+Dv^{\tau ,y_2}\right| \right) ^p\,dz\right) ^{\tfrac{p-\beta }p} \\
&\leq &c\int_Z\left( 1+\left| \tau +Dv^{\tau ,y_1}\right| +\left| \tau
+Dv^{\tau ,y_2}\right| \right) ^{p-\beta }\left| Dv^{\tau ,y_1}-Dv^{\tau
,y_2}\right| ^\beta \,dz \\
&\leq &\left( \int_Z\left| a(y_2,z,\tau +Dv^{\tau ,y_2})-a(y_1,z,\tau
+Dv^{\tau ,y_2})\right| ^q\,dz\right) ^{\tfrac 1q} \\
&&\times \left( \int_Z\left| Dv^{\tau ,y_1}-Dv^{\tau ,y_2}\right|
^p\,dz\right) ^{\tfrac 1p},\end{aligned}$$ which means that $$\begin{aligned}
&&\left( \int_Z\left| Dv^{\tau ,y_1}-Dv^{\tau ,y_2}\right| ^p\;dz\right)
^{\tfrac \alpha {p-1}} \nonumber \\
&\leq &c\left( \int_Z\left( 1+\left| \tau +Dv^{\tau ,y_1}\right| +\left|
\tau +Dv^{\tau ,y_2}\right| \right) ^p\,dz\right) ^{\tfrac{\alpha (\beta -p)}{(\beta -1)(p-1)}} \nonumber \\
&&\left( \int_Z\left| a(y_2,z,\tau +Dv^{\tau ,y_2})-a(y_1,z,\tau +Dv^{\tau
,y_2})\right| ^q\,dz\right) ^{\tfrac \alpha {\beta -1}} \nonumber \\
&\leq &c\left( 1+\left| \tau \right| ^p\right) ^{\tfrac{\alpha (\beta -p)}{(\beta -1)(p-1)}}\widetilde{\omega }(\left| y_1-y_2\right| )\left( 1+\left|
\tau \right| ^p\right) ^{\tfrac \alpha {\beta -1}} \nonumber \\
&\leq &\widetilde{\omega }(\left| y_1-y_2\right| )\left( 1+\left| \tau
\right| ^p\right) ^{\tfrac \alpha {p-1}}. \label{bcontc}\end{aligned}$$ The result follows by taking (\[bcont1\]), (\[bconta\]) and (\[bcontc\]) into account.
(ii), (iii) and (iv): The proofs follows by similar arguments as in e.g. [@bystrom00].$\square $
\[rema\]By similar arguments it follows that (ii), (iii), and (iv) hold up to boundaries, for the homogenized operator $b^k$ in Step 1.
[99]{} Bensoussan A, Lions J L, and Papanicolaou G C, , North Holland, Amsterdam, 1978.
Braides A and Defranceschi A, , Oxford Science Publications, Oxford, 1998.
Braides A and Lukkassen A, Reiterated Homogenization of Integral Functionals, *Math. Mod. Meth. Appl. Sci* [**10**]{} (2000), 1–25.
Bylund N, Byström J and Persson L E , Editors David Hui, Tenerife, to appear 2001.
Byström J, Correctors for Some Nonlinear Monotone Operators, *J. Nonlin. Math. Physics* [**8**]{} (2000), 8-30.
Cherkaev A and Kohn R (Editors), , Birkhäuser, Boston, 1997.
Chiado Piat V and Defranceschi A, Homogenization of Monotone Operators, *Nonlinear Anal.* [**14**]{} (1990), 717–732.
Cioranescu D and Donato P , Oxford University Press, Oxford, 1999.
Dal Maso G, , Birkhäuser, Boston, 1993.
Dal Maso G and Defranceschi A, Correctors for the Homogenization of Monotone Operators, *Differential and Integral Equations* [**3**]{} (1990), 1151–1166.
De Giorgi E and Spagnolo S, Sulla Convergenza Degli Intehrali Dell’energia per Operaori Ellittici Del Secondo Ordine, *Boll. Un. Mat. Ital.* [**8**]{} (1973), 391–411.
Dellacherie P-A, Meyer C, , Herman, Paris, 1975.
Hartman P and Stampaccia G, On Some Nonlinear Elliptic Differential Functional Equations, *Acta Math.* [**115**]{} (1966), 153–188.
Lions J L, Lukkassen D, Persson L E and Wall P Reiterated Homogenization of Monotone Operators, *C. R. Acad. Sci. Paris* [**330**]{} (2000) Série I, 675–680.
Lions J L, Lukkassen D, Persson L E and Wall P Reiterated Homogenization of Nonlinear Monotone Operators, *Chin. Ann. of Math.* [**22B**]{} (2001), 1–12.
Lukkassen D, , Ph[D]{} thesis, Dept. of Math., Troms[ö]{} University, Norway, 1996.
Lukkassen D, A New Reiterated Structure with Optimal Macroscopic Behaviour, *SIAM J. Appl. Math.* [**59**]{} (1999), 1825–1842.
Suquet P, . Ph[D]{}. thesis, Univ. Paris VI, 1982.
Wall P, , Ph[D]{} thesis, Dept. of Math., Luleå University of Technology, Luleå, Sweden, 1998.
Wall P, Optimal Bounds on the Effective Shear Moduli for Some Nonlinear and Reiterated Problems, *Acta. Sci. Math.* [**65**]{} (1999), 553-566.
\[lastpage\]
|
---
abstract: 'Interstellar complex organic molecules (COMs) are commonly observed during star formation, and are proposed to form through radical chemistry in icy grain mantles. Reactions between ions and neutral molecules in ices may provide an alternative cold channel to complexity, as ion-neutral reactions are thought to have low or even no energy barriers. Here we present a study of a the kinetics and mechanisms of a potential ion-generating acid-base reaction between NH$_{3}$ and HCOOH to form the salt NH$_{4}^{+}$HCOO$^{-}$. We observe salt growth at temperatures as low as 15K, indicating that this reaction is feasible in cold environments. The kinetics of salt growth are best fit by a two-step model involving a slow “pre-reaction” step followed by a fast reaction step. The reaction energy barrier is determined to be 70 $\pm$ 30K with a pre-exponential factor 1.4 $\pm$ 0.4 x 10$^{-3}$ s$^{-1}$. The pre-reaction rate varies under different experimental conditions and likely represents a combination of diffusion and orientation of reactant molecules. For a diffusion-limited case, the pre-reaction barrier is 770 $\pm$ 110K with a pre-exponential factor of $\sim$7.6 x 10$^{-3}$ s$^{-1}$. Acid-base chemistry of common ice constituents is thus a potential cold pathway to generating ions in interstellar ices.'
author:
- 'Jennifer B. Bergner'
- 'Karin I. Öberg'
- Mahesh Rajappan
- 'Edith C. Fayolle'
title: 'Kinetics and mechanisms of the acid-base reaction between NH$_3$ and HCOOH in interstellar ice analogs'
---
Introduction
============
Complex organic molecules (COMs) have been detected towards a wide range of interstellar environments [@Herbst2009] and are thought to be the precursors to prebiotic molecules [e.g. @Jørgensen2012; @Belloche2013]. It is of particular interest to understand how they are formed and inherited through different stages of evolution in star-forming regions that may ultimately develop into solar systems capable of sustaining life. Current models of COM formation involve electron- or photon-induced dissociation of molecules in the icy mantles coating interstellar grain surfaces, producing radical species which diffuse and recombine to form larger molecules. This formation pathway requires temperatures exceeding $\sim$30K [@Garrod2006; @Herbst2009]. COMs were first detected in the cores of high-mass protostars [e.g. @Blake1987; @Helmich1997], which undergo heating as they collapse and therefore experience temperatures above 30K; thus, these early detections are readily explained by the radical diffusion mechanism.
Recently, COM detections in cold prestellar environments [e.g. @Oberg2010; @Bacmann2012; @Cernicharo2012; @Vastel2014] have challenged the established need for lukewarm ice chemistry, as there must be efficient cold channels to chemical complexity in order to produce COMs in these regions. Several mechanisms, both gas-phase and grain-surface, have been suggested to explain COM production at low temperatures. Based on observations of the B1-b core and the pre-stellar core L1544, respectively, @Cernicharo2012 and @Vastel2014 propose grain-surface formation of methanol and other smaller species, followed by nonthermal desorption and gas-phase reaction to form more complex molecules. @Balucani2015 have developed a gas-grain model to account for such a mechanism. This model relies on efficient non-thermal desorption of methanol, followed by gas-phase radical-neutral and radical-radical reactions, and fairly well reproduces the observations of dimethyl ether and methyl formate towards L1544. Ice chemistry alone could also be a viable pathway, provided that non-thermal diffusion is efficient, since it is the diffusion step that limits surface reactivity in low-temperature regimes. For example, experiments involving UV photoprocessing of CH$_3$OH-rich ices have demonstrated efficient production of COMs at temperatures as low as 20K [@Oberg2009a], and @Oberg2010 suggest this mechanism to explain COM detections in the B1-b core. @Bacmann2012 similarly rely on non-thermal ice processing to explain the observed abundances towards the pre-stellar core L1698B: since it is well-shielded from UV radiation, they suggest that chemistry is likely induced by cosmic ray bombardment, secondary UV radiation from cosmic ray interactions with H$_{2}$, or energy from exothermic chemical reactions.
Radical-radical reactions are not the only reactions without barriers, and reactions between ions and neutral molecules offer an alternative cold route to chemical complexity. However, the importance of this channel is unknown since most previous work on COM formation has focused on radical reactions. In the gas phase, ion-neutral reactions are responsible for many of the observed molecules in cold interstellar regions, and there may be an analogous grain-surface pathway. Notably, recent theoretical work by @Woon2011 demonstrates several such surface reactions to be barrierless. Furthermore, compared to radical chemistry, ion chemistry does not rely on access to dissociative radiation. While diffusion is still an obstacle for both ion chemistry and radical chemistry, it is important to understand their relative contributions to complex molecule formation.
Ions have been observed in ice mantles in a range of different interstellar environments, particularly OCN$^{-}$, and potentially also HCOO$^{-}$, and NH$_{4}^{+}$ [see e.g. @Grim1987; @Grim1989; @Schutte1997; @Schutte1999; @Keane2001; @Knez2005; @Bisschop2007a; @Oberg2011]. Acid-base reactions are one potential source of ion generation in ice mantles. This study will focus on the grain-surface reaction between NH$_{3}$ and HCOOH, which are among the most common constituents of ices in star-forming regions; observations indicate abundances with respect to water of 1-5% for HCOOH [@Bisschop2007a; @Boogert2008; @Boogert2014] and 3-8% for NH$_{3}$ [@Boogert2008; @Bottinelli2010; @Oberg2011; @Boogert2014].
Similar acid-base systems have been studied experimentally in the past. There have been several qualitative studies on the reaction of HNCO with NH$_{3}$, motivated by the identification of solid OCN$^{-}$ towards many different astrophysical objects. @Raunier2003a observe the proton transfer at 10K when HNCO is codeposited with an excess of NH$_{3}$ (1:10); however, HNCO deposited on top of NH$_{3}$ does not react until warmed to 90K. To explain this, the authors perform quantum calculations and determine that the proton transfer is only spontaneous when the HNCO-NH$_3$ pair is stabilized by three or more solvating NH$_3$ molecules. @VanBroekhuizen2004 deposit mixed H$_2$O/NH$_3$/HNCO gases with varying H$_2$O concentrations and confirm that thermal processing is a robust mechanism for OCN$^-$ production. They suggest that the reaction they observe at 10-15K may be due to kinetic energy brought by molecules when they freeze onto the surface, in addition to the solvation-induced reaction described in @Raunier2003a. Subsequent reaction during warm-up is attributed to increasing NH$_3$ mobility.
In a quantitative study on the reaction between HCN and NH$_3$, @Noble2013 co-deposit both species with an excess of NH$_{3}$. The reaction is found to be thermally active, with some reaction occuring during deposition at 10K and further growth during warm-up. They model isothermal growth curves as pseudo-first order with respect to HCN concentration and determine an activation energy of 324K. In another quantitative proton transfer study, @Mispelaer2012 characterize and model the kinetics of the reaction between NH$_3$ and HNCO. Fitting isothermal growth curves with rate equations, they find that it follows a two-step process: an initial, slow orientation step, followed by a fast reaction step, with an activation energy of 48K for the reaction. They also perform fitting using a gamma-distribution of reaction constants to account for the fact that there will be a distribution of energy barriers to the orientation process, depending on the original position of different molecules. Using this second method they determine an activation energy of 73K.
There has been no quantitative study of the NH$_{3}$-HCOOH reaction at cryogenic temperatures; however, there have been several studies demonstrating that this reaction can occur at temperatures as low as 10K. Early work by @Schutte1997 was focused on spectroscopic assignments for the NH$_{4}^{+}$HCOO$^{-}$ ion pair, which was extended in @Schutte1999 to demonstrate that in situ proton transfer occurs after deposition of an H$_2$O/NH$_3$/HCOOH mixture and the conversion increases during sample warm-up. The kinetics are not quantified, but the authors note that the reaction seems to have a very small reaction barrier and that growth is limited by diffusion of reactant molecules. Later theoretical work by @Park2006 suggests that the proton transfer is barrierless as long as at least three water molecules are present per reacting pair to stabilize the system. More recently, experimental work by @Galvez2010 confirms a thermally-induced in situ reaction of co-deposited NH$_{3}$ and HCOOH, with a small fraction (15%) of reaction occurring upon deposition at 14K, and continuing as temperature is increased.
The objective of this paper is to elucidate the kinetics and mechanisms of this process, and thus its feasibility and importance for the evolution of interstellar ices. Experimental details are described in Section \[Experimental\], followed by the data analysis and modeling procedures in Section \[Analysis\]. The experimental results are presented in Section \[Results\], including the kinetic parameters extracted from the experiments as well as mechanistic inferences. In Section \[Discussion\] we compare our results with previous cryogenic acid/base studies and discuss the implications for surface channels to chemical complexity.
Experimental Details {#Experimental}
====================
Experimental Setup
------------------
The experimental setup used for this experiment has been described previously by @Lauck2015. Briefly, it consists of a CsI substrate window capable of being cooled to 11K by a closed-cycle He cryostat, with temperature monitored by a temperature controller (LakeShore 335) with an estimated accuracy of 2K and a relative uncertainty of 0.1K. The substrate is suspended inside an ultra-high vacuum chamber with a base pressure of $\sim$5x10$^{-10}$ Torr. Ices are grown by introduction of gas vapors at a normal incidence through 4.8mm diameter pipes 0.7 inches from the substrate, unless otherwise noted.
A Fourier transform infrared spectrometer (Bruker Vertex 70v) in transmission mode was used to measure the amount of each infrared-active species in the ice. Gas-phase species present in the chamber were continuously monitored by a quadrupole mass spectrometer (Pfeiffer QMG 220M1). The experiments were performed using NH$_{3}$ gas ($\geq$99.99% purity, Sigma), HCOOH (98%, Sigma), and deionized water (Sigma). The HCOOH and water were purified using three freeze-thaw cycles with liquid nitrogen.
Experimental Procedures {#methods}
-----------------------
Table \[tab1\] summarizes all experiments presented in this paper. HCOOH and NH$_3$ were either co-deposited from separate pipes to form a mixed ice or sequentially deposited to form a layered structure. All dosing took place at 14K. Apart from the TPD experiments, the ices were heated at 5K min$^{-1}$ to a target temperature and maintained there for 1-4 hours while monitoring the ice composition, and finally the ices were heated to desorption. IR scans taken at intervals of 10 minutes or less throughout the duration of each experiment.
Temperature programmed desorption (TPD) experiments (1-3) were performed for pure NH$_{3}$, pure HCOOH, and a layered NH$_3$/HCOOH ice. By monitoring the masses of desorbing molecules and the temperatures at which they desorb, TPDs show chemical conversions that have occurred over the course of warm-up. These experiments were performed by depositing $\sim$10 monolayers (ML) of each species and ramping the temperature at 5K/min until desorption (see section \[IRcalc\] for how thickness is determined).
For the co-deposition experiments (4-6), NH$_{3}$ and HCOOH were simultaneously deposited from two separate dosers at a distance of $\sim$1.2 inches and at an angle normal to the substrate. The temperature was ramped at 5K/min to a specified temperature, held for 4 hours, and then ramped again at 5K/min to 240K. The co-deposition at normal incidence resulted in a high degree of ice mixing, as evidenced by the large amount of salt growth. A quantitative analysis of these experiments would require constraining the ice mixture homogeneity, which is beyond the scope of this study.
Most experiments (7-23) were performed using a layered ice configuration to reduce the amount of reaction that occurs during dosing. Figure \[experimental\] (top) shows a comparison of the co-deposited and layered experimental setups. For all layered experiments, NH$_{3}$ was deposited first, followed by HCOOH. The sample was heated at 5K min$^{-1}$ to a target temperature and held for a given amount of time, and then heated to 240K.
We perform two types of layered experiments: “step” and “single-temperature”, shown schematically in Figure \[experimental\] (bottom). Step experiments are each held for 1h at several increasing temperatures within an experiment; thus, each step experiment yields several short isothermal growth curves at different temperatures. Single-temperature experiments are each held for 4h at a single temperature per experiment, yielding a single long isothermal growth curve. Experiments 7-9 are step experiments of different thicknesses: 1ML, 3ML, and 8ML. Experiments 10-13 are single-temperature experiments of thin ices (1ML NH$_{3}$: 1ML HCOOH), and experiments 14-23 are single-temperature experiments of thick ices (9ML NH$_{3}$: 7ML HCOOH). Experiment 24 is a single-temperature experiment of a thick ice, with an 8 hour isothermal hold instead of 4 hours in order to verify that the growth curve is sufficiently sampled within a 4 hour timescale. By varying the conditions of thickness and time, we can assess the salt growth kinetics under different experimental conditions and thereby better constrain the mechanism of reaction.
Analysis and Modeling {#Analysis}
=====================
IR spectra {#IRcalc}
----------
Concentrations of ice species of interest were determined from baseline-subtracted infrared spectra. Each spectrum is averaged over 128 interferograms and takes approximately 2 minutes to complete. Ice thickness was calculated from the formula: $$N_{i} = \frac{2.3\int\tau_{i}(\nu)d\nu}{A_{i}} \label{col_density}$$ where $N_{i}$ is column density (molecule cm$^{-2}$), $\int\tau_{i}(\nu)d\nu$ is the integrated area of the IR band (absorbance units), and $A_{i}$ is the band strength in optical depth units. The standard monolayer coverage of 10$^{15}$ molecules cm$^{-2}$ was assumed.
The band positions and strengths used to determine the thickness of NH$_{3}$ and HCOOH are taken from @Bouilloud2015 and were chosen to be the strongest features that do not overlap with the growing salt features (Table \[Tab2\]). Band strength uncertainties of 20% will result in the same uncertainty for the measured ice thicknesses listed in Table \[tab1\].
IR scans are taken throughout the course of an experiment, enabling in situ monitoring of the concentrations of ice species. These ice concentrations are plotted as a function of time over the course of an experiment to yield growth curves.
Growth curve fitting
--------------------
In order to extract kinetic parameters, isothermal growth curves are fit according to rate equations. To measure the reaction rate ideally, growth would depend only on the barrier to proton transfer; in reality, however, it is likely that an ensemble of diffusion and reorientation processes (“pre-reaction” steps) will inhibit growth. Because it is not clear which of these will contribute under different experimental conditions, we have used several different models to see which provides the best fit to the experimental growth curves. We first give a brief overview of the kinetic modeling, and then describe the four models used in this work.
The general formalism for reaction kinetics is: $$\frac{dX}{dt} = k(T)f(X) \label{gen_kin}$$ where $X$ is the reaction fraction, $k$ is the rate constant, $T$ is temperature (K), and $f(X)$ is the reaction model. The reaction fraction is determined by normalizing each growth curve to the final amount of salt at the completion of the reaction (ie, just prior to desorption). The rate constant can in turn be expanded to $$k(T) = Ae^{-E_{a}/T} \label{arrhenius}$$ where $A$ is the Arrhenius prefactor and $E_{a}$ is the activation energy. For solid-state reactions, the choice of reaction model $f(X)$ is not obvious since local inhomogeneities and diffusion effects preclude the kinetics from being described by concentration-dependent rate-laws used for homogeneous fluid systems. However, order-based methods are still useful as empirical models that allow kinetic parameters to be extracted from experiments [@Vyazovkin1997; @Khawam2006]. Recently, order methods have been successfully applied to a number of thermal reactions of astrochemical relevance [@Bossa2009a; @Bossa2009; @Theule2011; @Mispelaer2012; @Noble2013]. For the reaction studied here: $$\mathrm{NH_{3} + HCOOH \rightarrow NH_{4}^{+} + HCOO^{-}} \label{rxn}$$ we define the change in reaction fraction $$\frac{dX}{dt} = \frac{d(\mathrm{HCOO}^{-})}{dt} = \frac{d(\mathrm{NH}_{4}^{+})}{dt} = \frac{-d(\mathrm{HCOOH})}{dt} \label{fractions}$$ where parentheses denote the fraction of each individual species relative to its maximum value in the course of the experiment. The fractional changes in concentration of HCOO$^-$, NH$_4^+$, and HCOOH can be equated since each have a maximum value equal to the initial value of HCOOH, the limiting reactant. The reaction order model is then: $$\frac{dX}{dt} = k(T)(\mathrm{NH}_{3})^{\alpha}(\mathrm{HCOOH})^{\beta} \label{gen_order}$$ In this work, growth curves were fit with different variations of this general model in order to evaluate different mechanisms by which the reaction may take place.
*Pseudo-first order:* Because NH$_{3}$ in in excess of HCOOH for the majority of experiments, the simplest kinetic model would be pseudo-first order with respect to HCOOH: $$\frac{dX}{dt} = k(T)(\mathrm{HCOOH}) \label{pseudo_first}$$
*Second order:* The next step in complexity is if the reaction is first-order with respect to both HCOOH and NH$_{3}$, for a total order of two: $$\frac{dX}{dt} = k(T)(\mathrm{NH_{3}})(\mathrm{HCOOH}). \label{first_first}$$ Modeling the reaction as second-order with respect to either reactant did not improve the fit, and no higher-order processes were considered.
Both and are single-step processes that consider only the reaction rate. We also tested a single-step model that assumes diffusion-regulated kinetics, and a more complex model that accounts for both mobility and reaction kinetics potentially at play in the solid-state environment.
*Diffusion model:* A Fickian diffusion model was adapted (see @Lauck2015 for details) to describe the layered system of NH$_{3}$ beneath HCOOH. In this model, the NH$_{3}$ is assumed to be the mobile diffusant into a matrix of immobile HCOOH. The mixed fraction is described by: $$N_{mix}(t) = \frac{N_o(h-d)}{h} - \sum_{n = 1}^{\infty}\frac{2N_oh}{n^{2}\pi^{2}d}\sin^{2}(\frac{n\pi d}{h}) \exp[{\frac{-n^{2}\pi^{2}}{h^{2}}D(t + t_0)}]$$ where $h$ and $d$ are, respectively, the total ice height and the height to the interface (both in nm), $D$ is the diffusion constant (cm$^{2}$ s$^{-1}$), $N_o$ accounts for thickness uncertainties, and $t_0$ is a time offset. This model assumes that the kinetics are determined entirely by diffusion, and that reaction occurs immediately; in other words, the rate of mixing entirely determines the rate of salt growth.
*Two-step model:* The final model used was a two-step model involving a slow pre-reaction step followed by a fast reaction step. A similar treatment was used by @Mispelaer2012 in fitting the reaction of NH$_{3}$ with HNCO. Several mechanistic possibilities exist for the pre-reaction step; as we discuss in more detail in the following section, it likely consists of both orientation and diffusion, with different processes dominating the pre-reaction kinetics under different conditions. Because of this mixed nature, the pre-reaction step may depend on the temperature, ice structure, and elapsed time of a given experiment. The two-step model allows us to absorb the pre-reaction processes into the first step and therefore isolate the actual reaction step.
Here, we express the slow pre-reaction step as a unimolecular conversion of HCOOH from an inactive form to an active form. We write this step as: $$\mathrm{HCOOH}^{o} \rightarrow \mathrm{HCOOH}^{*}$$ with a rate $k_{p}$, where $^{o}$ and $^{*}$ represent the inactive and active form, respectively. The fast reaction step that follows is then: $$\mathrm{NH}_{3} + \mathrm{HCOOH}^{*} \rightarrow \mathrm{NH}_{4}^{+} \mathrm{HCOO}^{-}$$ with a rate $k_{r}$. We can express the kinetics of the total process in terms of the fractional conversion of HCOOH: $$\begin{aligned}
\frac{d(\mathrm{HCOOH}^{o})}{dt} = - k_{p}(\mathrm{HCOOH}^{o}) \\
\frac{d(\mathrm{HCOOH}^{*})}{dt} = k_{p}(\mathrm{HCOOH}^{o}) - k_{r}(\mathrm{HCOOH}^{*})\end{aligned}$$ This system of equations is solved for the concentrations of both HCOOH$^{o}$ and HCOOH$^{*}$ as a function of time; since the two are indistinguishable by IR spectroscopy, our observable quantity (the total rate of consumption of HCOOH) is the sum of these two contributions. Recalling equation , we derive an expression for the reaction fraction $\alpha$ as a function of time: $$X(t) = \frac{e^{-t(k_{p} + k_{r})}[e^{k_{p}t}(k_{p} - k_{r} c) - e^{k_{r}t}k_{r}(1 - c)]}{k_{p} - k_{r}}$$ where $c$ is the initial fraction of HCOOH in the active form, or (HCOOH$^{*})_{t=0}$.
Because kinetic fits are performed with respect to the reaction fraction, any uncertainties in the absolute band strength of the salt will not impact the kinetic parameters derived from the growth curves, as we are fitting a ratio. However, because the final amount of salt is measured at a different temperature than the growth curves, uncertainties in the temperature dependence of the band strength may introduce error into the fitting (see Section \[uncert\] and Appendix I).
All modeling is done only for isothermal growth curves. We do not attempt to model salt growth that occurs during deposition, as this may be governed by a different mechanism. For instance, it is possible that a “hot molecule” mechanism by which energy dissipated by gas-phase molecules is in part responsible for the initial salt formation that occurs during deposition.
Uncertainty analysis {#uncert}
--------------------
We consider five sources of error: spectral line fit errors, band strength uncertainties, kinetic fit uncertainties, dispersion between identical experiments, and absolute temperature calibration errors. The spectral line fit uncertainty consists of both gaussian fit uncertainty and uncertainty due to baseline selection. The former is very small, with errors generally at least 4 orders of magnitude smaller than the values of the peak areas. The latter is thickness-dependent and results in uncertainties of 0.5%-1% in the peak areas of measured salt features. This source of uncertainty dominates the error bars for individual measurements of IR peak area.
Uncertainties in the IR band strength of the measured salt feature will not contribute to the uncertainty for an individual fit of a growth curve, as we fit the reaction fraction rather than the total number of monolayers of salt. Thus, any uncertainty in the absolute value of the band strength will cancel out. However, the band strength likely varies with temperature and this must be considered when comparing salt growth in experiments run at different temperatures. We assume a 5% uncertainty in the magnitude of growth curves due to this temperature dependence (see Appendix I for a detailed explanation of band strength uncertainty). This is the most important contribution to the error bars on rate constants for individual experiments.
Statistical errors from fitting the kinetic growth curves are very small and do not contribute significantly to the rate constants uncertainties (less than 1%).
Dispersion measurements are determined using identical experiments to incorporate variations in chamber and ice morphology that cannot be directly controlled for. This is the main source of uncertainty when comparing multiple experiments, with a 15% uncertainty for reaction rate constants and 17% uncertainty for pre-reaction rate constants which are propagated into fitting for the reaction barriers and pre-exponential factors.
The error from calibration of the absolute temperature reading will contribute only to the derived energy barriers and pre-exponential factors. However, this uncertainty is very small compared to the dispersion error.
Results {#Results}
=======
The experimental results are presented in the following order: IR spectra in \[sec\_IRspec\], TPDs in \[sec\_TPD\], co-deposited experiments in \[sec\_codep\], qualitative analysis of layered experiments in \[qual\], growth curve fitting in \[sec\_modeling\], reaction barrier determination in \[sec\_kr\], and pre-reaction barrier determination in \[sec\_kp\]. It should be noted that the ultimate aim is to determine the reaction barrier of the proton transfer, which is necessarily tangled with other processes due to the limits of experimental constructs. We attempt to constrain pre-reaction steps as well as possible in order to isolate the reaction barrier, but the kinetic parameters derived for the pre-reaction are not necessarily relevant to astrophysical conditions since our system consists of simple, well-controlled ice structures rather than a more complicated and water-dominated mixture.
IR spectra {#sec_IRspec}
----------
Figure \[IR\]a shows the IR spectra for pure NH$_{3}$, pure HCOOH, and a layered NH$_3$/HCOOH mixture at increasing temperatures. Salt features, identified as those that grow over the course of the reaction, are consistent with those reported by @Galvez2010 and are listed in Table \[Tab2\]. Small salt features are apparent at 15K, with slow growth until 80K and rapid growth beyond this. Desorption is complete by $\sim$220K.
The four most prominent peaks associated with salt growth occur within the spectral window of 1300cm$^{-1}$ to 1600cm$^{-1}$; while there is some overlap of especially the inner two peaks, a Gaussian fitting of the four peaks produces an excellent fit to the overall shape of the spectrum, as shown in Figure \[IR\]b. The fitting procedure uses the spectral window from $\sim$1320-1600 cm$^{-1}$, excluding the tail from 1600-1750 cm$^{-1}$ which is due to the pure HCOOH feature at 1710 cm$^{-1}$ and the pure NH$_3$ feature at 1625 cm$^{-1}$. Regions outside of this window are assigned a value of zero to allow the model to return to the baseline. The growth curves derived from the areas of the two outermost peaks at 1346cm$^{-1}$ and 1573cm$^{-1}$ are in very good agreement with one another, providing confirmation that these peaks are indeed originating from the growth of the same species. The inner two peaks follow less consistent growth curves, possibly due to their greater degree of overlap. In addition, the NH$_{4}^{+}$ $\nu_{4}$ band is very broad and it is difficult to distinguish an exact peak position. The HCOO$^{-}$ $\nu_5$ band at 1377cm$^{-1}$ may also have a contaminating component from the pure HCOOH band at 1380cm$^{-1}$, although this is weaker by an order of magnitude [@Bouilloud2015]. The $\nu_{2}$ formate band at 1573cm$^{-1}$ is used for all growth-curve fitting since it is the most consistent and prominent salt feature. Band strengths for the salt IR modes were derived for each of the four peaks and are listed in Table \[Tab2\]. Details of their derivation are given in Appendix 1, along with an analysis of structure and temperature dependencies.
TPDs {#sec_TPD}
----
Temperature programmed desorption experiments were performed for pure NH$_{3}$, pure HCOOH, and a layered experiment of NH$_3$ under HCOOH. The ices were heated at 5K/min until desorption while continuously monitoring gas-phase concentrations with the QMS. Figure \[TPDs\] shows the resulting QMS traces for these experiments. Mass 29 traces the dominant HCOOH fragment, and mass 17 traces NH$_3$. The mixture of NH$_{3}$ and HCOOH desorbs at higher temperatures than either pure ice, indicating a chemical conversion has indeed occurred (peak desorption temperatures for NH$_3$, HCOOH, and the salt are 96K, 144K, and 203K, respectively). Furthermore, some NH$_{3}$ desorbs prior to salt desorption, but HCOOH desorbs only as the salt; this indicates that the HCOOH has undergone complete conversion, and in experiments with excess NH$_{3}$, the total salt formation will be determined by the initial dose of HCOOH.
We note that mass 17 also traces a fragment of H$_2$O in addition to the major NH$_3$ channel. We expect that water will be a minor contribution to the total QMS signal since the samples are formed from high purity gases maintained under ultra-high vacuum. However, there may be background deposition of water onto the sample throughout the experiment and thus some degree of this signal could be due to water. Based on an analysis of the dominant water channel at mass 18, around the water desorption temperature of 160K, water contamination contributes up to 20% of the total mass 17 signal; at all other temperatures, the water contamination contributes 10% or less to the measured mass 17 signal.
Salt formation in co-deposited ices {#sec_codep}
-----------------------------------
Growth curves for co-deposited experiments are shown in Figure \[codep\]. For all co-deposited experiments, considerable salt growth occurs immediately following dosing at 14K ($\sim$30% of the total salt conversion), with further growth during subsequent warm-up from 14K to the target temperature. For the target temperature of 85K, growth during the isothermal hold is evident; however, no isothermal growth was seen for the co-deposited experiments with 20K and 65K target temperatures. After the isothermal holds, the temperature was increased until desorption, and growth continued once again during this final warm-up. The rapid growth during the initial warm-up followed by a lack of growth at target temperatures below 85K suggests that the kinetics of salt formation under these conditions are complex: growth is evidently accessible at low temperatures as long as the sample is being warmed, but during isothermal periods the growth is arrested unless the temperature is sufficiently high.
Qualitative analysis of salt growth in layered ices {#qual}
---------------------------------------------------
A comparison of layered and co-deposited experiments, both with 85K target temperatures, is shown in Figure \[colay\]. Following deposition, both experiments exhibit similar patterns of salt growth over the course of the experiment. However, a layered setup offers the advantage of greatly reducing the amount of salt formation that occurs during dosing ($\sim$0.5ML, compared to $\sim$1.8ML in co-deposited ices). This enables a better estimation of the amount of available reactants compared to the co-deposited case. Additionally, it minimizes the amount of salt that has formed during the warm-up to the targeted temperature, enabling a better evaluation of the early kinetics of the reaction. Because of this, we performed our analysis using a layered ice setup, rather than co-deposition as is used by many previous kinetic studies [e.g. @Mispelaer2012; @Noble2013].
We note that, while the layered experiments exhibit a lower absolute growth, it is in fact a higher reactivity per molecule in contact: $\sim$50% salt conversion occurs in the 1ML HCOOH in contact for the layered experiment, compared $\sim$25% salt conversion for the 7ML HCOOH in contact in the co-deposited experiment. It is possible that the salt formation during deposition may be occurring by different mechanisms in the co-deposited and layered experiments. For instance, a hot-molecule mechanism might contribute to growth preferentially in the layered experiment.
We first look qualitatively at the amount of salt that has formed at the end of each isothermal hold for the different categories of experiments described in Section \[methods\]. Figure \[steps\]a shows the growth differences between step experiments of ices with different thicknesses. Above 80K, the 3ML and 8ML experiments produce more salt than the 1ML experiment. In this regime, diffusion must occur since non-interface molecules contribute to salt growth in the multilayer ices. Furthermore, the 3ML and 8ML experiments grow identically until about 120K, at which point the 8ML ice growth continues faster than the 3ML growth. This can be attributed once again to diffusion: while short-range diffusion appears to contribute above 80K, long-range diffusion is important only above 120K.
In Figure \[steps\]b-c, the 8ML and 1ML step experiments are compared to the single-temperature experiments with the same thicknesses. Regardless of the fact that prior reaction has occurred in the step experiments, roughly the same amount of salt is produced by a given temperature as for the single-temperature long-hold experiments. This has several implications, notably that most salt growth occurs in the first hour at a given temperature, and that at each temperature the salt formation kinetics does not depend on how pre-existing salt has formed. We discuss this observation in more detail in Section \[Discussion\].
Growth curves and model fitting {#sec_modeling}
-------------------------------
In fitting the kinetics of salt formation, we do not attempt to model growth that occurs during deposition or temperature ramps. Our modeling of isothermal growth takes into account previous salt formation, however, and so this warm-up growth should not be problematic to our kinetic derivations. Previous works [@Mispelaer2012; @Noble2013] offset the reaction fraction to equal zero at time zero, which models salt growth as if there is no salt present in the ice at the start of the isothermal period. Instead, we fit for a time offset that encompasses any salt growth prior to time zero of the isothermal hold. This allows us to account for prior salt growth regardless of its origin, i.e. whether it formed during deposition, warm-up, or isothermal growth. All resulting reaction constants ($k_r$) and pre-reaction constants ($k_p$) are summarized in Table \[ks\].
*Thin short experiments:* The layered step experiment of $\sim$1ML NH$_3$ under 1ML HCOOH is shown in Figure \[1mlstep\]. The salt grows rapidly during warm-up periods, and isothermal periods exhibit slower growth. Isothermal holds at 85K, 95K, and 105K produce the most salt growth; the kinetics are likely quite slow at 75K, and above 105K the reactants are mostly consumed and of limited availability, thus slowing growth. The growth curves are well-fit by the pseudo-first order model, and more complex models do not improve the fit. Indeed, the diffusion model does not converge on a solution at all, which is not surprising given that diffusion should not contribute much to growth in single-ML ice layers. The two-step model produced negative values for k$_r$, indicating that in the case of short reaction timescales and thin ices, salt growth is determined by only the reaction barrier. In other words, under these conditions the energy barrier of the proton transfer dominates the kinetics, and dynamical processes do not play an important role.
*Thin long experiments:* The growth curves for single-temperature experiments of 1ML ices are shown in Figure \[1mllong\]. Early reaction times were fit with the single-step model as was done for the step experiments of 1ML ices. We expect that if a single kinetic process contributes to growth, then the same model should work at early and late timescales. However, it is clear that different kinetics contribute at early and late times, and a model of early growth cannot reproduce late growth. In the limit of longer experiments, salt growth is instead best modeled by two-step kinetics, dependent on both the reaction barrier and a pre-reaction barrier. This suggests that while the growth seen in the shorter experiments is dominated by reaction between molecules already in an active state, in the limit of longer time when these active molecules are fully consumed, the contribution of the slow pre-reaction step becomes evident. Because each ice layer is $\sim$1ML thick, it is unlikely that diffusion is the important pre-reaction step in this case. Instead, we propose a re-orientation of reactant molecules into a favorable reaction configuration, similar to that described in @Mispelaer2012.
*Thick long experiments:* The 8ML step and single-temperature experiments were initially fit with all four kinetic models; an example growth curve is shown in Figure \[modelfits\]. Again the two-step model produces an excellent fit, while the other models do not capture the shape of the growth curve. In this case, the pre-reaction step is likely a combination between diffusion and orientation since, as discussed previously, diffusion is seen to contribute to salt growth in multilayer ices.
The pre-reaction barrier becomes an important contribution to growth on long timescales; thus, to verify that it is being sufficiently sampled on 4 hour timescales, we performed an experiment with an 8 hour isothermal hold. Comparing experiment 24 with the 4hr experiments 19 and 20 in Table 3, we find that both the reaction and pre-reaction barrier for the 8hr experiment are consistent with the values extracted from the 4hr experiments. The 4 hour growth curves are therefore sufficient for use in fitting, as the inferred kinetics do not change on longer timescales. The two-step model predicts that the growth curve should approach a stable value equal to a reaction fraction of 1 on the timescale of tens of hours.
Reaction barrier determination {#sec_kr}
------------------------------
The reaction rate constants derived from the growth curve fitting were used to generate an Arrhenius plot (Figure \[Arrhenius\]a). Fitting all data points simultaneously, we derive a reaction energy barrier $E_{a}$ = 70 $\pm$ 30 K and pre-exponential factor $A$ = 1.4 $\pm$ 0.4 x 10$^{-3}$ s$^{-1}$. Note that rate constants from below 70K and above 130K are excluded in this fitting, as the reaction is too slow to fit growth at low temperatures and most of the reactants have already been consumed by high temperatures. The error bars for thin ices are much larger than for the thick ices, as it is more difficult to constrain the fit when the magnitude of growth is smaller, and especially so when the timescale is short as for the step experiments.
The reaction rate constants for the thin short, thin long, and thick long experiments at the same temperature are in very good agreement, and fall within each others error bars as seen in Figure \[Arrhenius\]a. Likewise, when each category is fit individually, the derived values for E$_a$ and A are all consistent within the combined uncertainties (Table \[Eas\]). Thus, despite differences in thickness and timescale, we still find consistent values for the reaction barrier using the two-step model, indicating that we are in fact isolating the reaction step from the pre-reaction processes.
Pre-reaction barrier determination {#sec_kp}
----------------------------------
A similar treatment can be performed on the pre-reaction constants as is described above. Unlike for the reaction barrier, each category of experiments is likely to have a different pre-reaction barrier depending on the experimental conditions; in other words, orientation and diffusion are expected to contribute to growth to different extents between setups.
Fitting the thick long (single-temperature) experiments results in a pre-reaction activation barrier of 950K with a pre-factor of order 0.04; however, the uncertainties on these numbers are very large. To better constrain the pre-reaction barrier, we have re-fit all experiments with a fixed reaction constant $k_{r}(T)$ based on the combined $E_{a}$ and $A$ values in Table \[Eas\]. This is especially important for experiments with thin ices or short timescales since these fits are more uncertain. The resulting fits to growth curves using the fixed parameter method are in excellent agreement with the data. The pre-reaction rate constants from the fixed fitting reaction kinetics are shown in Figure \[Arrhenius\]b, and the resulting kinetic parameters are listed in Table \[Eas\].
The pre-reaction rates from thick experiments with short and long isothermal holds appear reasonably consistent, and Arrhenius fitting results in a much steeper slope than for the thin experiments, corresponding to a higher energy barrier. We expect the pre-reaction process to be dominated by diffusion for the thick ices and by orientation for the thin ices, and it is not surprising that orientation would have a lower energy barrier than diffusion. Additionally, the pre-reaction barrier for the thick ices appears to have a tail at the 85K and 95K temperature points. Based on the qualitative analysis of Figure \[steps\]a, diffusion appears to contribute to multilayer growth above 80K; it is likely that the shallower slope of the Arrhenius plot at lower temperatures corresponds to an orientation-dominated pre-reaction step, and the steeper slope at higher temperatures represents a diffusion-dominated pre-reaction step.
Discussion {#Discussion}
==========
NH$_3$ + HCOOH reaction {#rxndisc}
-----------------------
We extract an energy barrier of 70K for the proton transfer between NH$_3$ and HCOOH using the isothermal rate constants from different experiments (Figure \[Arrhenius\]). This barrier is consistent for different experimental conditions, demonstrating that it is indeed isolated from the pre-reaction processes that depend on the experimental construct.
The salt growth observed in this work is consistent with the qualitative description by @Schutte1999, who observed some reaction upon deposition at 10K and further growth during warm-up. The authors speculated that the reaction would have a low barrier and be limited by the diffusion of reactants, both of which we have demonstrated here. Indeed, we find that not only diffusion but also an orientation of reactant molecules contribute to the overall growth rate of the salt.
Comparing our results with previous quantitative proton transfer studies, the reaction barriers are quite similar, despite differences in fitting methodologies: @Noble2013 derive the reaction barrier for NH$_3$ + HCN to be 324K, @Mispelaer2012 find the barrier for HNCO + NH$_3$ to be 48K, and we find the barrier for NH$_3$ + HCOOH to be 70K. The HCN barrier perhaps appears high in comparison, but HCN is also a far weaker acid than HCOOH in the aqueous phase.
We note three differences with these existing studies that may impact the derived kinetic parameters. First, @Mispelaer2012 use a two-step model as in this work, but @Noble2013 use a single-step pseudo-first order model; if the reaction indeed follows two-step kinetics, a single-step fit would likely result in too high of a reaction barrier since the slow pre-reaction step is also being incorporated into the fit. This may contribute to the relatively high barrier found for the HCN + NH$_3$ reaction. Second, both previous studies analyzed reaction in co-deposited ices, whereas our ice is layered. This introduces diffusion as an important pre-reaction process in our salt growth, whereas orientation may be the most important pre-reaction step in previous works. Finally, the previous works perform fitting on growth curves that are normalized such that at time zero the pure ice has a fraction of unity and the product has a fraction of zero. This method assumes that any reaction that has already occurred during the warm-up phase of the experiment does not influence the kinetics during the isothermal period. In this work, we do not offset our growth curves to begin at a reaction fraction of zero because prior reaction could inhibit further reaction and should therefore be taken into account in the isothermal modeling.
Pre-reaction processes
----------------------
Unlike the reaction barrier, the derived pre-reaction barriers differ between experimental setups, reflecting the different dynamical processes at play under different experimental conditions. Re-orientation of reactant molecules into favorable reaction configurations should occur in all ices; this process contributes to growth once molecules that are already oriented to react are consumed. Diffusion of reactant molecules occurs only in thick ices, with molecules not originally at the interface replenishing the reactive stock. We find diffusion-limited (thick) ices to have much higher pre-reaction barriers than orientation-limited (thin) ices, which is consistent with the degree of mobility required for each process.
The pre-reaction mechanism will depend on the position and orientation of the molecules in the ice. This may differ from other experimental setups if a hot-molecule mechanism is at play during deposition. As mentioned in Section \[qual\], it is possible that molecules could be preferentially oriented to react due to energy dissipation following condensation. If molecules were deposited cold instead of at room temperature, it may result in a smaller fraction of activated reactants. However, since the two-step model fits for the initial active fraction of reactants, this will not impact our derived reaction barrier. A hot-molecule mechanism may impact whether a pre-reaction step is required or not, but the barrier should not be impacted since pre-reaction kinetics depend only on molecules that are in the inactive state.
Comparing with previous studies, the pre-reaction step in the case of @Mispelaer2012 is likely only an orientation process, as diffusion is not expected to play a role under their experimental conditions of co-deposition with a great excess of NH$_3$. We cannot make a comparison to their pre-reaction barrier, however, since they do not observe a temperature dependence for their pre-reaction step, and instead claim that re-orientation is induced by IR photons used to monitor their sample.
In our experiments, is likely that NH$_3$ is the main diffusing species due to its greater mobility over HCOOH. Thus, the diffusion-limited pre-reaction barrier of $\sim$770K derived in this work can be considered an upper limit for the diffusion barrier for NH$_3$, since it also incorporates orientation. The energy barrier to NH$_3$ diffusing up through water and desorbing has been measured by @Mispelaer2013 to be around 8000K, with values as low as 2000K allowed within the uncertainties. This range of values is much higher than the value measured in this work. It is possible that the NH$_3$/HCOOH ice is a more favorable environment for diffusion than a water ice. Alternatively, it is possible that the presence of charge in the ice impacts diffusion. For instance, diffusion barriers might be overcome more easily in a charged environment with Coulombic attractions present.
It is likely that each pre-reaction barrier derived in this work is in reality a distribution of barriers rather than a single value. A pre-reaction barrier distribution would explain the rapid salt growth occurring at low temperatures during the initial warm-up, contrasting with the slow growth observed during isothermal holds (see Figures \[codep\], \[colay\], and \[1mlstep\]): the low-energy end of the distribution reacts initially, and at increasing temperatures we are probing the higher-barrier pre-reaction processes. This would also explain the similarity in salt production between the single-temperature experiments and the step experiments (\[steps\]b-c) since the amount of salt that can form at a given temperature would be controlled by the fraction of pre-reaction barriers that are thermally accessible at that temperature.
Physically, a distribution of energies for the pre-reaction step would not be surprising given that it represents both orientation and diffusion. These two processes will contribute to a different extent to growth depending on, ice thickness, temperature, and reaction progress. In particular, the barrier for each process may change over the course of an experiment; for instance, diffusion later in the course of an experiment is likely more difficult as more salt will have built up in between the pure ice layers.
However, it is not obvious why we would observe an Arrhenius-like relationship between pre-reaction rates and temperature in the case of a temperature-dependent distribution of energy barriers. While we see a tail end to the Arrhenius plot at low temperatures in Figure \[Arrhenius\], the experiments above $\sim$90K appear linear and thus consistent with Arrhenius behavior. We performed a toy calculation to evaluate whether a distribution of energy barriers could produce a log-linear relationship between measured rates and temperatures. We assume that the measured energy barrier increases as a function of temperature since the lower barriers would be overcome at lower temperatures. Rate constants are calculated using the Arrhenius equation with a fixed value of A and with E$_{a}$ varying according to three different distributions: a symmetric equally-spaced distribution from 80% to 120%; a symmetric Gaussian distribution from 80% to 120%; and an asymmetric equally-spaced distribution from 90% to 130%.
For both linear distributions we see that the rate constants are still linear on the Arrhenius plot despite being derived from different energy barriers. The Gaussian distribution can appear linear or deviate substantially from linear depending on the choice of standard deviation; we show the results using a moderate value, which could certainly be possible within the scatter of the rate constants we derive for the pre-reaction barrier. In all three models the derived barrier would be lower than the median barrier, and thus the reported pre-reaction barriers should be considered lower limits.
These plots demonstrate that it is possible under a number of conditions to reproduce linear Arrhenius behavior despite each rate constant being defined by a different energy barrier. Given the nature of the pre-reaction step, such a distribution is likely to be at play in this system. It should be noted that, by contrast, the reaction barrier appears to be a single value, independent of temperature and ice environment, as discussed in Section \[rxndisc\].
Astrophysical Implications
--------------------------
We next extend our laboratory results to assess the temperature at which the NH$_4^+$/HCOO$^-$ salt formation reaction should be efficient under interstellar conditions. Because this is an ice process, it should not be impacted by the lower pressure of the interstellar medium compared to the ultra-high vacuum setup. However, the timescales of astrophysical processes are many orders of magnitude longer than laboratory timescales. Following @Pontoppidan2008, we relate the critical temperature $T_{\mathrm{crit}}$ of the reaction under interstellar conditions to that under laboratory conditions: $$\frac{\tau_{\mathrm{astro}}}{\tau_{\mathrm{lab}}} = \mathrm{exp}\bigg[E_a\bigg(\frac{1}{T_{\mathrm{astro}}} - \frac{1}{T_{\mathrm{lab}}}\bigg)\bigg]$$ where $\tau_{\mathrm{astro}}$ and $\tau_{\mathrm{lab}}$ are the e-folding timescales (i.e., a single exponential lifetime) for the reaction in the interstellar medium and the laboratory, respectively, and $T_{\mathrm{astro}}$ and $T_{\mathrm{lab}}$ correspond to the critical temperatures for each environment. For $\tau_{\mathrm{astro}}$ we assume a typical star formation timescale of 10$^6$ years for a cold cloud, and for $E_a$ we use the reaction barrier of 70K derived in this work. Since our reaction model is first-order, $\tau_{\mathrm{lab}}$ is simply the inverse of the rate constant (i.e. 1/s$^{-1}$) measured at a given temperature $T_{\mathrm{lab}}$. Using the rate constants measured in this work we find a critical temperature $T_{\mathrm{astro}}$ of 3K. Such low critical temperature implies that, in the case where reactants are in the appropriate configuration, we expect reaction to proceed under cloud core conditions of $<$10K. Note that this treatment assumes HCOOH and NH$_3$ are already in the proper position and orientation, i.e. we do not consider pre-reaction steps. The availability of reactants may therefore limit the degree to which this reaction occurs.
The impact of pre-reaction barriers on growth kinetics is of particular importance to astrochemical modeling. Most Monte Carlo models of grain-surface chemistry already account for diffusion using a “hopping” barrier; however, we have demonstrated that other mobility-related barriers may be equally important to ice chemistry. Orientation has a lower barrier than diffusion, but is not currently incorporated in models and needs to be considered to create a full picture of the chemistry. The matrix-dependent nature of mobility barriers means that the values of pre-reaction barriers derived in this work are likely not suitable for use in astrochemical models; further work is required to derive appropriate pre-reaction barriers for modeling use.
Additionally, more work is required to assess the role of ions as initiators of surface chemical pathways. As discussed by @Woon2011, surface reactions involving ions offer a promising low-energy pathway to chemical complexity, without the destructive effects of energetic processing. Woon presents theoretical calculations of several protonated species undergoing barrierless reactions with neutral water molecules; analogous to the gas-phase ion-neutral reactions that are responsible for many gas-phase processes in low-energy interstellar environments, these surface ion-neutral reactions are spontaneous and offer an energetic advantage over neutral-neutral reactions mediated by barriers. Woon’s calculations consider ions as depositing onto water ices, but an in-situ method of ion formation such as that considered in this paper could equally drive these ion-neutral reaction pathways. As NH$_{3}$ and HCOOH are fairly abundant components of ice mantles, the reaction described in this work has the potential to be a robust source of ion generation, and could increase chemical complexity via low-barrier ion-neutral reactions.
Conclusion
==========
The kinetics and mechanisms of the reaction between NH$_3$ and HCOOH were examined by continuously monitoring product formation in layered ices held at constant temperatures. Based on our results we conclude:
1\. Under laboratory conditions the reaction between NH$_3$ and HCOOH is accessible as low as 14K and proceeds to completion upon warm-up.
2\. The proton transfer to form the salt has a barrier of 70K if the molecules are positioned to react.
3\. A two-step process (a slow pre-reaction step followed by a rapid reaction step) best describes the mechanism of salt growth. The pre-reaction step is likely a combination between orientation and diffusion of reactant molecules.
4\. Multilayer ices exhibit a pre-reaction barrier of $\sim$770K while that of single layer ices is only $\sim$400K. The former includes both diffusion and orientation barriers, while the latter is only orientation.
5\. The pre-reaction barrier may actually be a distribution of barriers. Pre-reaction barriers at the higher end of the energy distribution are sampled at higher temperatures. As a result, the values of the pre-reaction barriers listed here are likely upper limits to the actual median barriers.
6\. NH$_4^+$HCOO$^-$ salt formation in the interstellar medium is potentially possible below 10K provided that the reactants are correctly positioned at neighboring sites.
J.B.B acknowledges funding from the National Science Foundation Graduate Research Fellowship under Grant DGE1144152. K.I.Ö. acknowledges funding from the Simons Collaboration on the Origins of Life (SCOL) investigator award.
Appendix 1 {#sec_appendix}
==========
Band strengths were derived as follows: a layered ice with NH$_{3}$ beneath HCOOH was heated at 5K/min to 165K. At this point the HCOOH band at 1216 cm$^{-1}$ has disappeared, but HCOOH has not begun to desorb (as monitored by QMS). This indicates that the reactant has been fully converted, and none is lost to desorption. Also at this point the salt has not yet crystallized (as seen by IR spectra) or desorbed. The final peak area for the salt upon reaching 165K is then equated to the inital amount of formic acid. This method assumes that the salt features do not change as a result of the temperature or the structure of the ice. This is likely not entirely accurate, and we have assessed the dependencies of each. Figure \[BSexp\] shows the experiment used to do so.
A structure dependence would arise from the fact that the salt forming at low temperatures may not be configured in the same way as salt at higher temperatures. Salt crystallization occurs above the range of temperatures we are interested in for this study, but more minor bulk rearrangements may occur at lower temperatures. We explore the magnitude of this effect by comparing the consumption of HCOOH with the formation of salt during warm-up of a layered HCOOH/NH$_3$ ice from 14K to 150K (Figure \[BSexp\], left of vertical dotted line). We then assume that there is a one-to-one conversion from HCOOH to HCOO$^-$ and assign a temperature-dependent band strength such that the sum of HCOOH and HCOO$^-$ remains constant throughout the conversion. When there are low abundances of either species the uncertainties of the derived band strengths are large, but we focus on the range 60-130K in which most experiments are performed. Here, the determined band strength varies by less than 5%.
We next assess the temperature dependence of the salt band strength for an ice with a constant structure by comparing the strength of the salt feature at 165K with the strength of the same feature after the salt has been cooled down (Figure \[BSexp\], right of vertical dotted line). After reaching 165K the salt was cooled back to 11K at 5K/min and then heated once again at 5K/min until fully desorbed. The band strength does change by $\sim$5% over this temperature range, and the change in band strength is fully reversible.
We do not attempt to correct for these dependencies in performing our fitting, as this would introduce many more uncertainties but not ultimately change the values substantially. Instead, these uncertainties are incorporated into the error analysis.
natexlab\#1[\#1]{}
Bacmann, A., Taquet, V., Faure, A., Kahane, C., & Ceccarelli, C. 2012, A&A, 541, L12
Balucani, N., Ceccarelli, C., & Taquet, V. 2015, MNRAS, 449, L16
Belloche, a., M[ü]{}ller, H., Menten, K., Schilke, P., & Comito, C. 2013, A&A, 559, A47
Bisschop, S. E., J[ø]{}rgensen, J. K., & [Van Dishoeck]{}, E. F. 2007, A&A, 465, 913
Blake, G., & Sutton, E. 1987, ApJ, 315, 621
Boogert, A. C. A., Gerakines, P. A., & Whittet, D. C. 2014, ARA&A, 53, 541
Boogert, A. C. A., Pontoppidan, K. M., Knez, C., [et al.]{} 2008, ApJ, 678, 985
Bossa, J. B., Duvernay, F., Theul[é]{}, P., [et al.]{} 2009, A&A, 506, 601
Bossa, J. B., Theule, P., Duvernay, F., & Chiavassa, T. 2009, ApJ, 707, 1524
Bottinelli, S., Boogert, A. C. A., Bouwman, J., [et al.]{} 2010, ApJ, 718, 1100
Bouilloud, M., Fray, N., Benilan, Y., [et al.]{} 2015, MNRAS, 451, 2145
Cernicharo, J., Marcelino, N., Roueff, E., [et al.]{} 2012, ApJ, 759, L43
G[á]{}lvez, O., Mat[é]{}, B., & Herrero, V. J. 2010, ApJ, 724, 539
Garrod, R. T., & Herbst, E. 2006, A&A, 457, 927
Grim, R. J. A., & Greenberg, J. M. 1987, ApJ, 321, L91
Grim, R. J. A., Greenberg, J. M., Schutte, W. A., & Schmitt, B. 1989, ApJ, 341, L87
Helmich, F. P., & van Dishoeck, E. F. 1997, A&AS, 124, 205
Herbst, E., & van Dishoeck, E. F. 2009, ARA&A, 47, 427
J[ø]{}rgensen, J. K., Favre, C., Bisschop, S. E., [et al.]{} 2012, ApJ, 757, L4
Karssemeijer, L. J., Ioppolo, S., van Hemert, M. C., [et al.]{} 2014, ApJ, 781, 16
Keane, J. V., Tielens, A. G. G. M., Boogert, A. C. A., Schutte, W. A., & Whittet, D. C. B. 2001, A&A, 376, 254
Khawam, A., & Flanagan, D. R. 2006, JPCB, 110, 17315
Knez, C., Boogert, A. C. A., Pontoppidan, K. M., [et al.]{} 2005, ApJ, 635, L145
Lauck, T., Karssemeijer, L., Shulenberger, K., [et al.]{} 2015, ApJ
Mispelaer, F., Theule, P., Duvernay, F., & Roubin, P. 2012, A&A
Mispelaer, F., Theul[é]{}, P., Aouididi, H., [et al.]{} 2013, A&A, 555, A13
Noble, J. A., Theul[é]{}, P., Borget, F., [et al.]{} 2013, MNRAS, 428, 3262
berg, K. I., Boogert, A. C. A., Pontoppidan, K. M., [et al.]{} 2011, ApJ, 740, 109
berg, K. I., Bottinelli, S., J[ø]{}rgensen, J. K., & van Dishoeck, E. F. 2010, ApJ, 716, 825
berg, K. I., Fayolle, E. C., Cuppen, H. M., van Dishoeck, E. F., & Linnartz, H. 2009, A&A, 505, 183
berg, K. I., Garrod, R. T., van Dishoeck, E. F., & Linnartz, H. 2009, A&A, 504, 891
Park, J. Y., & Woon, D. E. 2006, ApJ, 648, 1285
Pontoppidan, K. M. M., Boogert, A. C. A., Fraser, H. J. J., [et al.]{} 2008, ApJ, 678, 1005
Raunier, S., Chiavassa, T., Marinelli, F., Allouche, A., & Aycard, J. 2003, CPL, 368
Schutte, W. A., Greenberg, J. M., [Van Dishoeck]{}, E. F., [et al.]{} 1998, Ap&SS, 255, 61
Schutte, W. A., & Khanna, R. K. 2002, A&A, 1062, 16
Schutte, W. A., Boogert, A. C. A., Tielens, A. G. G. M., [et al.]{} 1999, A&A, 343, 966
Theule, P., Duvernay, F., Ilmane, A., [et al.]{} 2011, A&A, 530, A96
van Broekhuizen, F. A., Keane, J. V., & Schutte, W. A. 2004, A&A, 415, 425
Vastel, C., Ceccarelli, C., Lefloch, B., & Bachiller, R. 2014, ApJ, 795, L2
Vyazovkin, S., & Wight, C. a. 1997, ARPC, 48, 125
Woon, D. E. 2011, ApJ, 728, 44
|
---
abstract: 'In this paper, we show that high-dimensional sparse wavelet signals of finite levels can be constructed from their partial Fourier measurements on a deterministic sampling set with cardinality about a multiple of signal sparsity.'
author:
- 'Yang Chen, Cheng Cheng and Qiyu Sun[^1]'
title: Reconstruction of sparse wavelet signals from partial Fourier measurements
---
Introduction
============
Sparse representation of signals in a dictionary has been used in signal processing, compression, noise reduction, source separation, and many more fields. Wavelet bases are well localized in time-frequency plane and they provide sparse representations of many signals and images that have transient structures and singularities ([@daubechiesbook; @mallatbook]). In this paper, we consider recovering sparse wavelet signals of finite levels from their partial Fourier measurements. Let ${\bf D}$ be a dilation matrix with integer entries whose eigenvalues have modulus strictly larger than one, and set $M=|\det {\bf D}|\geq 2$. Wavelet vectors ${\Psi}_m=(\psi_{m, 1}, \ldots, \psi_{m, r})^T, 1\le m\le M-1$, used in this paper are generated from a multiresolution analysis $\{V_j\}_{j\in \ZZ}$, a family of closed subspaces of $L^2:=L^2(\RR^n)$, that satisfies the following: (i) $ V_j\subset V_{j+1}$ for all $j\in \ZZ$; (ii) $V_{j+1}=\{f({\bf D}\cdot), \ f\in V_j\}$ for all $j\in \ZZ$; (iii) $\overline{\cup_{j\in \ZZ}V_{j}}=L^{2}$; (iv) $\cap_{j\in \ZZ}V_{j}=\{0\}$; and (v) there exists a scaling vector $\Phi=(\phi_1,\cdots,\phi_r)^{T}\in V_{0}$ such that $\{\phi_l(\cdot -{\bf k}),\ 1\leq l\leq r, {\bf k}\in \ZZ^n\}$ is a Riesz basis for $V_0$ ([@daubechiesbook; @mallatbook; @grochenig92; @cohen93; @geronimo94; @goodmanlee94; @heller95; @han98; @bds99; @sbhbook]). They generate a Riesz basis $\{M^{j/2}{\Psi}_m({\bf D}^j{\bf x}-{\bf k}): 1 \le m \le M-1,\ {\bf k} \in \ZZ^n \}$ for the wavelet space $W_j:=V_{j+1}\ominus V_j$, the orthogonal complement of $V_j$ in $V_{j+1}$, for every $j\in \ZZ$. Therefore any signal $f$ in the scaling space $V_J$ of level $J\ge 0$ has a unique wavelet decomposition, $$\label{sparsesignal.def} f = f_0+ g_0+\cdots+ g_{J-1},$$ where $$\label{sparsesignal.def1}
f_0=\sum_{k\in \ZZ^n} {\bf a}_0^T({\bf k}) {\pmb\Phi}(\cdot-{\bf k})\in V_0$$ and $$\label{sparsesignal.def2}
g_j=\sum_{m=1}^{M-1}\sum_{{\bf k}\in \ZZ^n} {\bf b}_{m,j}^T({\bf k}) M^{j} {\pmb\Psi}_{m}({\bf D}^j\cdot-{\bf k})\in W_j, 0\le j\le J-1.$$ In this paper, we consider wavelet signals $f\in V_J$ with $f_0$ and $g_j, 0\le j\le J-1$, in the above wavelet decomposition having sparse representations.
Define Fourier transform of an integrable function $f$ on $\RR^n$ by $$\hat f({\pmb \xi})=\int_{\RR^n} f({\bf t}) e^{-i{\bf t}\cdot {\pmb \xi}} d{\bf t}.$$ Due to coherence of wavelet bases between different levels, the conventional optimization method does not work well to reconstruct a sparse wavelet signal $f$ of finite level from its partial Fourier measurements $\hat f(\xi), \xi\in \Omega$, on a finite sampling set $\Omega$ ([@donoho03; @candes06; @donoho06; @baraniuk07; @candes08; @sun12; @yang10b; @foucartbook]). Recently, Prony’s method was introduced in [@zhang; @ccssampta15] for the exact reconstruction of one-dimensional sparse wavelet signals.
Denote by $\#E$ the cardinality of a set $E$. We say that a wavelet signal $f\in V_J$ has sparsity ${\mathbf s}= (s_0, \cdots, s_{J-1})$ if it has sparsity $$s_j:= \left\{\begin{array} {l}
\max\{\# K_0, \# K_{1, 0}, \ldots, \# K_{M-1,0}\} \ {\rm if} \ j=0\\
\max\{ \# K_{1, j}, \ldots, \# K_{M-1, j}\}\ {\rm if }\ j=1, \ldots, J-1,
\end{array}\right.$$ at level $j, 0\le j\le J-1$, where $K_0$ and $K_{m, j}$ are supports of coefficient vectors $(a_0({\bf k}))_{{\bf k}\in \ZZ^n}$ and $(b_{m, j}({\bf k}))_{{\bf k}\in \ZZ^n}$ in the wavelet decomposition , and respectively. For the classcial one-dimensional scalar case (i.e. $n=1, r=1$ and ${\bf D}=2$), under the assumption that Fourier transform of the scaling function $\phi$ does not vanish on $(-\pi, \pi)$, $$\label{nonzerophi}
\hat\phi(\xi)\ne 0, \ \ \xi\in (-\pi, \pi),$$ Zhang and Dragotti proved in [@zhang] that a compactly supported sparse wavelet signal of the form can be reconstructed from its Fourier measurements on a sampling set $\Omega$ of size about twice of its sparsity $s_0+\cdots+s_{J-1}$. In this paper, we extend their result to high-dimensional sparse wavelet signals without nonvanishing condition on the scaling vector $\Phi$. Particularly in Theorem \[maintheorem\], we show that any ${\mathbf s}$-sparse wavelet signal $f$ of the form can be reconstructed from its Fourier measurements on a sampling set $\Omega$ with cardinality less than $2 Mr (s_0+\cdots+s_{J-1})$, which is independent on dimension $n$.
Multiresolution analysis and wavelets {#multiwavelets.section}
=====================================
Set ${\bf M}={\bf D}^T$. Then the scaling vector ${\Phi}=(\phi_1, \ldots, \phi_r)^T$ of a multiresolution analysis $\{V_j\}_{j\in \ZZ}$ satisfies a matrix refinement equation, $$\label{refinefourier.def}
\widehat\Phi({\pmb\xi})={\bf G}_0({\bf M}^{-1}{\pmb \xi})\widehat\Phi({\bf M}^{-1}{\pmb \xi}),$$ where the matrix function ${\bf G}_0$ of size $r\times r$ is bounded and $2\pi$-periodic. In this paper, we assume that ${\bf G}_0$ has trigonometric polynomial entries. Hence $\Phi$ is compactly supported, and the Riesz basis property for the scaling vector $\Phi$ can be reformulated as that $(\widehat{\Phi}(\pmb \xi+2\pi{\bf k}))_{{\bf k}\in \ZZ^n}$ has rank $r$ for every $\pmb\xi\in \RR^n$. Therefore for any $\pmb\xi\in \RR^n$ there exist ${\bf k}(\pmb\xi, l)\in \ZZ^n, 1\le l\le r$, such that $$\label{rankcondition3.new}
\big(\widehat \Phi( \pmb\xi+2\pi{\bf k})\big)_{{\bf k}\in \Lambda (\pmb\xi)}\
\text{has full rank} \ r,$$ where $$\begin{aligned}
\label{Upsilon.def}
\Lambda (\pmb\xi) &\hskip-0.05in=\hskip-0.05in&\{{\bf k}(\pmb\xi, l)\in \ZZ^n: \ 1\le l\le r\}.
\end{aligned}$$
Let ${\bf p}_m,0\leq m \leq M-1$, be representatives of $\ZZ^n/{\bf M}\ZZ^n$, and write $$\ZZ^n=\bigcup_{m=0}^{M-1}({\bf p}_m+{\bf M}\ZZ^n).$$ Take matrices ${\bf G}_m$, $1\leq m\leq M-1$, with trigonometric polynomial entries such that $$\label{orthogonalcondition}
\sum_{m'=0}^{M-1}{\bf G}_0({\pmb\xi}+2\pi{\bf M}^{-1}{\bf p}_{m'})\overline{{\bf G}_{m}({\pmb \xi}+2\pi{\bf M}^{-1}{\bf p}_{m'})}^T=0$$ for all $1\le m\le M-1$, and $$\label{waveletrankcondition}
{\bf G}(\pmb \xi) {\rm \ has\ rank} \ Mr {\rm\ for\ all}\ \pmb\xi\in \RR^n,$$ where[$${\bf G}(\pmb\xi)\hskip-0.02in = \hskip-0.05in \left(\hskip-0.05in \begin{array} {ccc}
{\bf G}_0(\pmb\xi+2\pi {\bf M}^{-1}{\bf p}_0) & \hskip-0.05in \cdots & \hskip-0.05in {\bf G}_0(\pmb\xi+2\pi {\bf M}^{-1}{\bf p}_{M-1})\\
{\bf G}_1(\pmb\xi+2\pi{\bf M}^{-1}{\bf p}_0) & \hskip-0.05in \cdots & \hskip-0.05in {\bf G}_1(\pmb\xi+2\pi{\bf M}^{-1}{\bf p}_{M-1})\\
\vdots & \hskip-0.05in \ddots & \hskip-0.05in \vdots\\
{\bf G}_{M-1}(\pmb\xi+2\pi{\bf M}^{-1}{\bf p}_0) &\hskip-0.05in \cdots & \hskip-0.05in {\bf G}_{M-1}(\pmb\xi+2\pi{\bf M}^{-1} {\bf p}_{M-1})\\
\end{array}\hskip-0.05in\right).$$]{} In this paper, wavelet vectors $\Psi_m, 1\leq m\leq M-1$, are defined as follows: $$\label{wavelet.def}
\widehat\Psi_m({\pmb \xi})={\bf G}_m({\bf M}^{-1}{\pmb \xi})\widehat\Phi({\bf M}^{-1}{\pmb \xi}),\ \ 1\leq m\leq M-1.$$ Then $\Psi_m$ are compactly supported and $\{M^{j/2}{\Psi}_m({\bf D}^j{\bf x}-{\bf k}): 1 \le m \le M-1,\ {\bf k} \in \ZZ^n \}$ forms a Riesz basis for the wavelet space $W_j:=V_{j+1}\ominus V_j$ for every $j\in \ZZ$. For the scaling vector $\Phi$ and wavelet vectors $\Psi_m, 1\le m\le M-1$, constructed above, one may verify that any signal in $V_J$ has the unique wavelet decomposition , and .
Reconstruction of sparse wavelet signals {#sparserecovery.section}
========================================
Take ${\bf h}=(h_1, \ldots, h_n)\in \RR^n$ and sparsity vector ${\mathbf s}=(s_0, \ldots, s_{J-1})$, and set $\|{\bf s}\|_\infty=\max_{0\le j\le J-1} s_j$. For $0\le j\le J-1$ and $0\le m\le M-1$, let $$\begin{aligned}
\Gamma_{j}& \hskip-0.1in = & \hskip-0.1in \{ (-s_j+1/2){\bf h}, (-s_j+3/2){\bf h}, \ldots, (s_j-1/2){\bf h}\},\end{aligned}$$ and $$\begin{aligned}
\Omega_{j} &\hskip-0.1in
= &\hskip-0.1in \cup_{\pmb\gamma\in\Gamma_{j}} \cup_{m=0}^{M-1}
\big( \pi \pmb\gamma+2\pi{\bf M}^{j} {\bf p}_m\nonumber\\
& & \ + 2\pi {\bf M}^{j+1}
\Lambda (\pi{\bf M}^{-j-1}\pmb\gamma+ 2\pi{\bf M}^{-1}{\bf p}_m)\big),
$$ where the set $\Lambda$ of cardinality $r$ is defined by . Set $$\label{omega.def}
\Omega=\cup_{j=0}^{J-1}\Omega_{j}.$$ Then $$\Omega\subset
\{ (-\|{\bf s}\|_{\infty}+1/2){\bf h}\pi, \ldots, (\|{\bf s}\|_{\infty}-1/2){\bf h}\pi\}
+2\pi\ZZ^n,$$ and $$\# \Omega\le \sum_{j=0}^{J-1} \#\Omega_{j}=2 M r (s_0+s_1+\cdots+s_{J-1}). $$ The following is the main theorem of this paper.
\[maintheorem\] Let ${\bf D}$ be a dilation matrix, $\Phi$ be a compactly supported scaling vector, $\Psi_m, 1\le m\le M-1$, be wavelet vectors satisfying and , let $\Omega$ be the set in with ${\bf h}=(h_1, \ldots, h_n)$. If $1, h_1, \ldots, h_n$ are linearly independent over the field of rationals, then any ${\mathbf s}$-sparse wavelet signal of the form , and can be reconstructed from its Fourier measurements on $\Omega$.
Let $f$ be an ${\mathbf s}$-sparse signal with wavelet representation , and . Set $$\label{maintheorem.pf.eq2}
\widehat{{\bf a}}_0(\pmb\xi)=\sum_{{\bf k}\in \ZZ^n} {\bf a}_0({\bf k}) e^{-i{\bf k}\cdot\pmb\xi},$$ and $$\label{bmj.eq}
\widehat{ {\bf b}}_{m,j}(\pmb\xi)=\sum_{{\bf k}\in \ZZ^n} {\bf b}_{m,j}({\bf k}) e^{-i{\bf k}\cdot\pmb\xi}$$ for $1\le m\le M-1$ and $0\le j\le J-1$. Then taking Fourier transform on both sides of the equation gives $$\label{maintheorem.pf.eq1}
\hat f(\pmb\xi)= {\widehat{\bf a}}_0^{T}(\pmb\xi) \widehat {\Phi}(\pmb\xi)+\sum_{j=0}^{J-1} \sum_{m=1}^{M-1}\widehat{{\bf b}}_{m, j}^{T}({\bf M}^{-j}\pmb\xi) \widehat {\Psi}_m({\bf M}^{-j}\pmb\xi).$$
Define $f_i, 0\le i\le J-1$, by $$\label{maintheorem.pf.eq4}
\widehat f_i(\pmb\xi)={\widehat{\bf a}}^{T}_0(\pmb\xi) \widehat {\Phi}(\pmb\xi)+\sum_{j=0}^i \sum_{m=1}^{M-1}\widehat{{\bf b}}_{m, j}^{T}({\bf M}^{-j}\pmb\xi) \widehat {\Psi}_m({\bf M}^{-j}\pmb\xi).$$ Then $$\label{level.eq0}
f_{J-1}=f,$$ and $$\begin{aligned}
\label{level.eq}
\widehat f_i({\bf M}^i\pmb\xi) &\hskip-0.05in = &\hskip-0.05in \widehat f_{i-1}({\bf M}^i{\pmb\xi})+
\sum_{m=1}^{M-1}{\widehat{{\bf b}}_{m, i}}^{T}(\pmb\xi) \widehat\Psi_m(\pmb\xi)\nonumber\\
&\hskip-0.05in = &\hskip-0.05in {\widehat{{\bf a}}_i}^{T}(\pmb\xi) \widehat\Phi(\pmb\xi) +
\sum_{m=1}^{M-1}{\widehat{{\bf b}}_{m, i}}^{T}(\pmb\xi)\widehat\Psi_m(\pmb\xi)\nonumber\\
&\hskip-0.05in = &\hskip-0.05in \big({\widehat{{\bf a}}_i}^{T}(\pmb\xi){\bf G}_0({\bf M}^{-1}\pmb\xi)+\sum_{m=1}^{M-1}{\widehat{{\bf b}}_{m, i}}^{T}(\pmb\xi)\nonumber\\
&&\hskip0.1in \times {\bf G}_m({\bf M}^{-1}\pmb\xi)\big)\widehat\Phi({\bf M}^{-1}\pmb\xi)\end{aligned}$$ for some vectors $\widehat{{\bf a}}_i(\pmb\xi)$ with trigonometric polynomial entries, where the last equality follows from and .
For $0\le j\le J-1$, $\pmb\gamma\in \Gamma_{j}$ and $0\le m'\le M-1$, set $$\eta_j(\pmb\gamma,m')=\pi{\bf M}^{-j}\pmb\gamma+2\pi{\bf p}_{m'}.$$ Applying and with $i=J-1$, replacing $\pmb \xi$ in by $ \eta_{J-1}(\pmb\gamma,m')+2\pi{\bf M}{\bf k}, {\bf k}\in \Lambda({\bf M}^{-1} \eta_{J-1}(\pmb\gamma,m'))$, and using periodicity of $\widehat{{\bf a}}_{J-1}$ and $\widehat{{\bf b}}_{m,J-1}$, we obtain $$\begin{aligned}
\label{maintheorem.pf.eq60+}
\hskip-0.1in&\hskip-0.1in&\hskip-0.1in\hat f({\bf M}^{J-1}\eta_{J-1}(\pmb\gamma,m')+2\pi{\bf M}^{J}{\bf k})
\nonumber\\
\hskip-0.1in&\hskip-0.1in = &\hskip-0.1in A(J-1,\pmb\gamma,m')
\widehat\Phi({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m')+2\pi{\bf k})\end{aligned}$$ for all ${\bf k}\in \Lambda({\bf M}^{-1} \eta_{J-1}(\pmb\gamma,m'))$, where $$\begin{aligned}
\label{anonsingular}
\hskip-0.1in{}&\hskip-0.1in&\hskip-0.1in A(J-1,\pmb\gamma,m')={\widehat{\bf a}_{J-1}}^{T}(\pi{\bf M}^{-J+1}\pmb\gamma) {\bf G}_0({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m'))\nonumber\\
& \hskip-0.1in& \hskip-0.1in+\sum_{m=1}^{M-1} {\widehat{{\bf b}}_{m, J-1}}^{T}(\pi{\bf M}^{-J+1}\pmb\gamma){\bf G}_m({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m')).\end{aligned}$$ Recall from that $$\Big(\widehat\Phi({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m')+2\pi{\bf k})\Big)_{{\bf k}\in \Lambda({\bf M}^{-1} \eta_{J-1}(\pmb\gamma,m'))}$$ is nonsingular. Then $A(J-1,\pmb\gamma,m')$ can be solved from the linear system for all $0\le m'\le M-1$ and $\pmb\gamma\in \Gamma_{J-1}$. Recall from and that [$$\label{gnonsingular}
{\bf G}(h\pi{\bf M}^{-J}\pmb\gamma)= \left(\hskip-0.05in \begin{array} {c}
{\bf G}_0({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m')) \\
{\bf G}_1({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m')) \\
\vdots \\
{\bf G}_{M-1}({\bf M}^{-1}\eta_{J-1}(\pmb\gamma,m')) \\
\end{array}\hskip-0.05in\right)_{0\le m'\le M-1}$$ ]{} is nonsingular. Thus, for every $\pmb\gamma \in \Gamma_{J-1}$ and $1\le m\le M-1$, $$\label{abhat}
{\widehat{{\bf a}}_{J-1}}(\pi{\bf M}^{-J+1}\pmb\gamma)\ \ {\rm and} \ \
{\widehat{{\bf b}}_{m, J-1}}(\pi {\bf M}^{-J+1} \pmb\gamma)$$ are uniquely determined from samples of $\hat f$ on $\Omega_{J-1} \subset \Omega$ by and .
For $1\le m\le M-1$, it follows from the linear independence assumption of $1, h_1, \ldots, h_n$ on the field of rationals that $$e^{-i \pi {\bf k}\cdot {\bf M}^{-J+1} {\bf h}}, {\bf k}\in K_{m, J-1}, {\rm \ are\ distinct\ to\ each\ other}.$$ For $\pmb \gamma= n {\bf h}$ with $n\in \{-s_{J-1}+1/2, \ldots, s_{J-1}-1/2\}$, $$\widehat{{\bf b}}_{m, J-1}(\pi {\bf M}^{-J+1} \pmb\gamma)=
\sum_{{\bf k}\in K_{m, J-1} } {\bf b}_{m,J-1}({\bf k}) \big(e^{-i \pi {\bf k}\cdot {\bf M}^{-J+1} {\bf h}}\big)^n$$ by . Therefore applying Prony’s method ([@foucartbook; @zhang; @kuma84; @scharf91; @barbieri92; @vmb02; @osborne06; @plonka13]) recovers trigonometric polynomials $\widehat{{\bf b}}_{m, J-1}, 1\le m\le M-1$, from their measurements on $\pi {\bf M}^{-J+1} \pmb\gamma, \pmb\gamma\in \Gamma_{J-1}$. Hence ${\bf b}_{m,J-1}({\bf k})$, ${\bf k}\in\ZZ^n$, can be recovered from samples of $\hat f$ on $\Omega$ for all $1\le m\le M-1$.
By the above argument, $$\label{fJ-1}
f_{J-1}-f_{J-2}\ {\rm and}\ {\widehat f}_{J-2}(\pmb\xi),\ \pmb\xi \in\Omega,$$ can be obtained from samples of $\hat f$ on $\Omega$, because $$\begin{aligned}
\label{le.eq}
\widehat f_{J-2}(\pmb\xi) &\hskip-0.1in = &\hskip-0.1in \hat f({\pmb\xi}) -
\sum_{m=1}^{M-1}\Big(\sum_{{\bf k}\in \ZZ^n} {\bf b}_{m,J-1}({\bf k}) e^{-i{\bf k}\cdot{\bf M}^{-J+1}\pmb\xi}\Big)\\
& & \qquad\qquad\quad \times \widehat\Psi_m({\bf M}^{-J+1}\pmb\xi)\end{aligned}$$ by and . Inductively we can reconstruct $$\label{finduct}
f_i-f_{i-1}\ {\rm and} \ {\widehat f}_{i-1}(\pmb\xi), \ \pmb\xi\in\Omega,$$ from the samples of $\hat f$ on $\Omega$ for $ i=J-2, \cdots, 1$.
Taking $i=1$ in determines samples of ${\widehat f}_0$ on $\Omega$. Next we recover the function $f_0$ from its Fourier measurements on $\Omega_0\subset \Omega$. By and , $$\widehat{{\bf a}}_0(n\pi{\bf h}) = \sum_{{\bf k}\in {\bf K}_0} {\bf a}_0({\bf k}) \big(e^{-i\pi {\bf k}\cdot {\bf h}}\big)^n$$ and $$\widehat{\bf b}_{m, 0}( n\pi{\bf h})\nonumber\\
=\hskip-0.08in \sum_{{\bf k}\in {\bf K}_{m,0}} {\bf b}_{m,0}({\bf k})\big(e^{-i\pi {\bf k}\cdot {\bf h}}\big)^n,$$ where $n\in \{-s_0+1/2, -s_0+3/2, \ldots, s_0-1/2\}$. Similar to , we can show that $${\widehat{{\bf a}}_0}(n\pi {\bf h})\ \ {\rm and} \ \
{\widehat{{\bf b}}_{m, 0}}(n\pi {\bf h}), \ 1\le m\le M-1,$$ are uniquely determined from samples of $\widehat f_0$ on $\Omega$. Applying Prony’s method again recovers ${\bf a}_0({\bf k})$ for ${\bf k}\in {\bf K}_0$ and ${\bf b}_{m, 0}({\bf k})$ for $1\le m\le M-1$ and ${\bf k} \in {\bf K}_{m,0}$. Therefore $f_0$ could be completely recovered from its Fourier measurements on $\Omega$. This together with , and completes the proof.
The linear independence requirement on ${\bf h}=(h_1, \ldots, h_n)$ in Theorem \[maintheorem\] can be replaced by a quantitative condition if the sparse signal has some additional information on its support, c.f. [@zhang].
Let ${\bf D}$, $\Phi$ and $\Psi_m, 1\le m\le M-1$, be as in Theorem \[maintheorem\], and let $f$ be an ${\mathbf s}$-sparse signal in satisfying $$\label{support.additional}
K_0\subset [a, b)^n \ \ {\rm and} \ \
K_{m,j}\subset {\bf D}^j[a, b)^n,$$ where $1\le m\le M-1$, $0\le j\le J-1$ and $a<b$. Then $f$ can be recovered from its Fourier measurements on $\Omega$ in with ${\bf h}=(h_1, \ldots, h_n)$ satisfying $$\label{hrequire}
0<(b-a)(h_1+h_2+\cdots+h_n)\le 2.$$
Following the argument in Theorem \[maintheorem\], it suffices to prove that $ e^{-i\pi {\bf k}\cdot {\bf h}}, {\bf k} \in K_0$, are distinct, and also that $ e^{-i\pi {\bf k} \cdot {\bf M}^{-j} {\bf h}}, {\bf k}\in K_{m,j}$, are distinct for every $1\le m\le M-1$ and $0\le j\le J-1$. The above distinctive property follows from and immediately.
From the proof of Theorem \[maintheorem\], we have the following result on the reconstruction of an $s$-sparse trigonometric polynomial from its samples on a set of size $2s$.
Let ${\bf h}=(h_1, \ldots, h_n)$ with $1, h_1, \ldots, h_n$ being linearly independent over the field of rationals, and define $$\Theta_s=
\{ (-s+1/2){\bf h}, (-s+3/2){\bf h}, \ldots, (s-1/2){\bf h}\}, \ s\ge 1.$$ Then any $n$-dimensional trigonometric polynomial $$P(\pmb\xi)= \sum_{{\bf k}\in \ZZ^n} p({\bf k})e^{-i{\bf k}\cdot \pmb \xi}$$ with sparsity $s$, $$\#\{{\bf k}:\ p({\bf k})\ne 0\}\le s,$$ can be reconstructed from its samples on $\Theta_s$.
Simulations
===========
The following algorithm for sparse wavelet signal recovery is proposed in the proof of Theorem \[maintheorem\].
[**Algorithm 1:**]{}
- Input sparsity vector ${\bf s}=(s_0,\cdots,s_{J-1})$.
- Input Fourier measurements $\hat f(\pmb\xi)$, $\pmb\xi\in\Omega$ and set $f_{J-1}=f$.
- [**for**]{} $j=J-1$ to $0$ [**do**]{}
- every $\pmb\gamma\in\Gamma_j$ [**do**]{}
- every $m'=0,\cdots,M-1$ [**do**]{}
- 3a) $\eta_j(\pmb\gamma,m')=\pi{\bf M}^{-j}\pmb\gamma+2\pi{\bf p}_{m'}$.
- 3b) Solve the linear system $$\begin{aligned}
\hskip-0.2in&\hskip-0.1in&\hskip-0.1in\big(\widehat f_j\big({\bf M}^{j}\eta_j(\pmb\gamma,m')+2\pi{\bf M}^{j+1}{\bf k}\big)\big)_{{\bf k}\in \Lambda (M^{-1}\eta_j(\pmb\gamma,m'))}\\
\hskip-0.2in&\hskip-0.1in=&\hskip-0.1in A(j,\pmb\gamma,m')\big(\widehat\Phi({\bf M}^{-1}\eta_j(\pmb\gamma,m')+2\pi{\bf k})\big)_{{\bf k}\in \Lambda (M^{-1}\eta_j(\pmb\gamma,m'))}\end{aligned}$$ to get $$\begin{aligned}
&&\hskip-0.1in A(j,\pmb\gamma,m'):={\widehat{\bf a}_j}^{T}(\pi{\bf M}^{-j}\pmb\gamma) {\bf G}_0({\bf M}^{-1}\eta_j(\pmb\gamma,m'))\\
&\hskip0.05in&+\sum_{m=1}^{M-1} {\widehat{{\bf b}}_{m, j}}^{T}(\pi{\bf M}^{-j}\pmb\gamma){\bf G}_m({\bf M}^{-1}\eta_j(\pmb\gamma,m')).\end{aligned}$$
- - 3c) Solve the linear equation [$$\begin{aligned}
&&\big({\widehat{\bf a}_j}^{T}(\pi{\bf M}^{-j}\pmb\gamma), {\widehat{{\bf b}}_{1, j}}^{T}(\pi{\bf M}^{-j}\pmb\gamma),\cdots, {\widehat{{\bf b}}_{M-1, j}}^{T}(\pi{\bf M}^{-j}\pmb\gamma)\big)\\
&&\hskip0.1in\times{\bf G}(h\pi{\bf M}^{-j-1}\pmb\gamma)=\big(A(j,\pmb\gamma,0),\cdots,A(j,\pmb\gamma,M-1)\big).\end{aligned}$$ ]{}
- - 3d) Recover ${\bf b}_{m, j}$ from $\widehat{{\bf b}}_{m, j}(\pi{\bf M}^{-j}\pmb\gamma)$, $\pmb\gamma\in\Gamma_j$ with Prony’s method for every $1\le m\le M-1$.
- 3e) Subtract $\sum_{m=1}^{M-1}\widehat{{\bf b}}^{T}_{m, j}({\bf M}^{-i}\pmb \xi)\widehat{\Psi}_m({\bf M}^{-i}\pmb \xi)$ from $\hat f_j(\pmb\xi)$ to get $\hat f_{j-1}(\pmb\xi)$, $\pmb \xi\in\Omega$.
- [**end for**]{}
- Recover ${\bf a}_0$ from $\widehat{{\bf a}}(\pi\pmb\gamma)$, $\pmb\gamma\in\Gamma_0$ with Prony’s method.
- Reconstruct the sparse wavelet signal $$\begin{aligned}
f({\bf t})&\hskip-0.1in=&\hskip-0.1in\sum_{{\bf k}\in\ZZ^n}{\bf a}_0^T({\bf k})\Phi({\bf t}-{\bf k})\\
&&+\sum_{j=0}^{J-1}\sum_{m=1}^{M-1}\sum_{{\bf k}\in\ZZ^n}{\bf b}_{m,j}^T({\bf k})M^j\Psi_m({\bf D}^{j}{\bf t}-{\bf k}).
\end{aligned}$$
Next we present simulations to demonstrate the above algorithm for perfect reconstruction of sparse wavelet signals of finite levels. Let $\phi_1(t)=\chi_{[0,1)}(t)$ and $\phi_2(t)=2\sqrt{3}(t-1/2)\chi_{[0,1)}(t)$ be scaling functions, and let $$\psi_1(t)=(6t-1)\chi_{[0,1/2)}(t)+(6t-5)\chi_{[1/2,1)}(t),$$ and $$\psi_2(t)= 2\sqrt{3}(2t-1/2)\chi_{[0,1/2)}(t)-2\sqrt{3}(2t-3/2)\chi_{[1/2,1)}(t)$$ be wavelet functions. Consider reconstructing the sparse signal $$\begin{aligned}
\label{signaldefi}
f(t)&\hskip-0.1in=&\hskip-0.1in {\bf a}_{0}^{T}(2)\Phi(t-2)+{\bf a}_0^{T}(4)\Phi(t-4)\nonumber\\
&&\hskip-0.1in+{\bf b}^{T}_{0}(1)\Psi(t-1)+{\bf b}^{T}_{0}(5)\Psi(t-5)\nonumber\\
&&\hskip-0.1in+{\bf b}^{T}_{1}(6)\Psi(2t-6)+{\bf b}^{T}_{1}(12)\Psi(2t-12)\end{aligned}$$ from its Fourier measurements on the sampling set $$\Omega=\Big\{- \frac{\sqrt{2}}{128} n \pi+2k\pi:\ n=\pm 1, \pm 3 \ {\rm and} \ k=0,\pm1,\pm2,4\Big\}$$ in , where $\Phi=(\phi_1,\phi_2)^{T}$, $\Psi=(\psi_1, \psi_2)^T$, and the nonzero components of ${\bf a}_{0}$, ${\bf b}_{0}$ and ${\bf b}_{1}$ are randomly chosen in $[-1,1]\setminus(-0.1,0.1)$, see Figure \[signal.fig\]. Applying the proposed algorithm, our numerical results support the conclusion on perfect recovery of sparse wavelet signals from their Fourier measurements on $\Omega$.
![Plotted on the left is the sparse wavelet signal $f$ in , while on the right is the magnitude of its Fourier transform and the measurements on $\Omega$.[]{data-label="signal.fig"}](signal.jpg "fig:"){width="42mm" height="30mm"} ![Plotted on the left is the sparse wavelet signal $f$ in , while on the right is the magnitude of its Fourier transform and the measurements on $\Omega$.[]{data-label="signal.fig"}](xhatomega.jpg "fig:"){width="42mm" height="30mm"}
The proposed algorithm is tested when the Fourier measurements of the signal $f$ are corrupted by random noises $\epsilon$, $$h(\xi)=\hat f(\xi)+\epsilon(\xi),\ \ \xi\in\Omega.$$ In this case, sparsity locations obtained by Prony’s method in the algorithm are not necessarily integers, but it is observed that they are not far away from the sparsity locations of the signal $f$, when the signal-to-noise-ratio (SNR), $$SNR=-20\log_{10}{\frac{\max_{\xi\in\Omega}|\epsilon(\xi)|}{\max_{\xi\in\Omega}|\hat f(\xi)|}}$$ is above 50 dB. Taking nearest integers of those locations may perfectly recover the sparsity positions $\{2,4\}$ for the scaling component of level $0$, $\{1,5\}$ for the wavelet component of level $0$, and $\{6,12\}$ for the wavelet component of level $1$. Then the signal $f$ can be reconstructed by the proposed algorithm approximately, see Figure \[error.fig\].
![The difference between the original signal and the reconstructed signal. In this figure, the reconstruction is generated by the proposed algorithm with modified Prony’s method, and the noise level on Fourier measurements is SNR=50 dB. []{data-label="error.fig"}](error50.jpg){width="68mm" height="30mm"}
We also tested our proposed algorithm for two-dimensional wavelet signals with dilation ${\bf D}=\left(\hskip-0.05in \begin{array}{cc}
0 & -2\\
1 & 0\\
\end{array}\hskip-0.05in\right)$. Presented on the left of Figure \[2dsignalfourier.fig\] is the amplitude of a sparse wavelet signal $$\begin{aligned}
\label{twodimensionalsignaldefi}
\hskip-0.3in f(t_1,t_2)&\hskip-0.1in=&\hskip-0.1in a_0\phi(t_1-1,t_2)+ a_1\phi(t_1-2,t_2-3)\nonumber\\
&\hskip-0.1in+&\hskip-0.1in b_{0}\psi(t_1-2,t_2-1)+ b_{1}\psi(t_1-3,t_2-5),\end{aligned}$$ where $a_0, a_1, b_0, b_1\in [-1,1]\setminus(-0.1,0.1)$ are selected randomly, the scaling function is $\phi(t_1,t_2)=\chi_{[0,1)}(t_1)\chi_{[0,1)}(t_2)$, and the wavelet function is $\psi(t_1,t_2)=\chi_{[0,1)}(t_1)(\chi_{[0,1/2)}(t_1)-\chi_{[1/2,1)}(t_2))$. Our simulations show that the signal $f$ in can be reconstructed from its Fourier measurements on $$\label{omegatwodimensional}
\Omega=\Big\{\Big(\frac{\sqrt{2}}{64} n+2k, \frac{\sqrt{3}}{64} n+2l\Big)\pi,\ n=\pm 1, \pm 3 \ {\rm and}\ k,l=0, 1\Big\},$$ which is plotted on the right of Figure \[2dsignalfourier.fig\].
![Fourier amplitudes of the signal $f$ in and the sampling set $\Omega$ in for sparse recovery. []{data-label="2dsignalfourier.fig"}](2dsignalfourier.jpg "fig:"){width="42mm" height="30mm"} ![Fourier amplitudes of the signal $f$ in and the sampling set $\Omega$ in for sparse recovery. []{data-label="2dsignalfourier.fig"}](2domega.jpg "fig:"){width="42mm" height="30mm"}
Conclusion
==========
In this paper, we show that sparse wavelet signals of finite level can be reconstructed from their Fourier measurements on a deterministic sampling set, whose cardinality is independent on signal dimension and almost proportional to signal sparsity. A difficult problem on this aspect is exact reconstruction of signals having sparse wavelet-like (e.g. wavelet packet, framelet, curvelet, and shearlet) representations from their partial Fourier information ([@coifman92; @ronshen97; @candes06siam; @chui06; @kutyniokbook]).
[99]{}
I. Daubechies, [*Ten Lectures on Wavelets*]{}, SIAM, 1992.
S. Mallat, [*A Wavelet Tour of Signal Processing*]{}, Academic Press, 1999.
K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of ${\mathbb R}^n$, [*IEEE Trans. Inform. Theory*]{}, [**38**]{}(1992), 556–568.
A. Cohen and I. Daubechies, Non-separable bidimensional wavelet bases, [*Rev. Mat. Iberoamericana*]{}, [**9**]{}(1993), 51–137.
J. S. Geronimo, D. P. Hardin and P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, [*J. Approx. Theory*]{}, [**78**]{}(1994), 373–401.
T. N. T. Goodman and S. L. Lee, Wavelets of multiplicity $r$, [*Trans. Amer. Math. Soc.*]{}, [**342**]{}(1994), 307–342.
P. N. Heller, Rank $M$ wavelets with $N$ vanishing moments, [*SIAM J. Matrix Anal. & Appl.*]{}, [**16**]{}(1995), 502–519.
B. Han and R. Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, [*SIAM J. Math. Anal.*]{}, [**29**]{}(1998), 1177–1199.
N. Bi, X. Dai and Q. Sun, Construction of compactly supported $M$-band wavelets, [*Appl. Comp. Harmonic Anal.*]{}, [**6**]{}(1999), 113–131.
Q. Sun, N. Bi and D. Huang, [*An Introduction to Multiband Wavelets*]{}, Zhejiang University Press, 2001.
D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $\ell^1$ minimization, [*Proc. Natl. Acad. Sci. U.S.A*]{}, [**100**]{}(2003), 2197–2202.
E. J. Candes, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, [*IEEE Trans. Inform. Theory*]{}, [**52**]{}(2006), 489–509.
D. L. Donoho, Compressed sensing, [*IEEE Trans. Inform. Theory*]{}, [**52**]{}(2006), 1289–1306.
R. Baraniuk, Compressive sensing, [*IEEE Signal Process. Mag.*]{}, [**24**]{}(2007), 118–124.
E. J. Candes and M. Wakin, An introduction to compressive sampling, [*IEEE Signal Process. Mag.*]{}, [**25**]{}(2008), 21–30.
Q. Sun, Recovery of sparsest signals via $\ell^q$-minimization, [*Appl. Comput. Harmonic Anal.*]{}, [**32**]{}(2012), 329–341.
J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, [*IEEE J. Sel. Topics Signal Process.*]{}, [**4**]{}(2010), 288–297.
S. Foucart and H. Rauhut, [*A Mathematical Introduction to Compressive Sensing*]{}, Springer, 2013.
Y. Zhang and P. L. Dragotti, On the reconstruction of wavelet-sparse signals from partial Fourier information, [*IEEE Signal Proc. Letter*]{}, [**22**]{}(2015), 1234–1238.
Y. Chen, C. Cheng and Q. Sun, Reconstruction of sparse multiband wavelet signals from Fourier measurements, Proceeding of the 11th International Conference on Sampling Theory and Applications (SAMPTA 2015), accepted.
R. Kumaresan, D. W. Tufts and L. L. Scharf, A Prony method for noisy data: choosing the signal components and selecting the order in exponential signal models, [*Proc. IEEE*]{}, [**72**]{}(1984), 230–233.
L. L. Scharf, [*Statistical Signal Processing: Detection, Estimation, and Time Series*]{}, Prentice Hall, 1991.
M. M. Barbieri and P. Barone, A two-dimensional Prony’s method for spectral estimation, [*IEEE Trans. Signal Process.*]{}, [**40**]{}(1992), 2747–2756.
M. Vetterli, P. Marziliano, and T. Blu, Sampling signals with finite rate of innovation, [*IEEE Trans. Signal Process.*]{}, [**50**]{}(2002), 1417–1428.
M. R. Osborne and G. K. Smyth, A modified Prony algorithm for exponential function fitting, [*SIAM J. Sci. Comput.*]{} [**16**]{}(2006), 119–138.
T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, [*Inverse Problems*]{} [**29**]{}(2013), 025001
R. Coifman and M. V. Wickerhauser, Entropy-based algorithms for best basis selection, [*IEEE Trans. Inform. Theory*]{}, [**38**]{}(1992), 713–718.
A. Ron and Z. Shen, Affine systems in $L^2(\RR^d)$: the analysis of the analysis operator, [*J. Funct. Anal.*]{}, [**148**]{}(1997), 408–447.
E. J. Candes, L. Demanet, D. L. Donoho and L. Ying, Fast discrete curvelet transforms, [*SIAM J. Multiscale Model. Simul.*]{}, [**5**]{}(2006), 861–899.
C. K. Chui and Q. Sun, Affine frame decompositions and shift-invariant spaces, [*Appl. Comput. Harmonic Anal.*]{}, [**20**]{}(2006), 74–107.
G. Kutyniok and D. Labate, [*Shearlets: Multiscale Analysis for Multivariate Data*]{}, Springer, 2012.
[^1]: Chen is with the Department of Mathematics, Hunan Normal University, Changsha 100044, Hunan, China; Cheng and Sun are with the Department of Mathematics, University of Central Florida, Orlando 32816, Florida, USA. Emails: yang\_chenww123@163.com; cheng.cheng@knights.ucf.edu; qiyu.sun@ucf.edu. The project is partially supported by National Science Foundation (DMS-1412413).
|
---
abstract: 'Despite the rapid progresses in the field of quantum spin Hall (QSH) effect, most of the QSH systems studied up to now are based on crystalline materials. Here we propose that the QSH effect can be realized in quasicrystal lattices (QLs). We show that the electronic topology of aperiodic and amorphous insulators can be characterized by a spin Bott index $B_s$. The nontrivial QSH state in a QL is identified by a nonzero spin Bott index $B_s=1$, associated with robust edge states and quantized conductance. We also map out a topological phase diagram in which the QSH state lies in between a normal insulator and a weak metal phase due to the unique wavefunctions of QLs. Our findings not only provide a better understanding of electronic properties of quasicrystals but also extend the search of QSH phase to aperiodic and amorphous materials that are experimentally feasible.'
author:
- Huaqing Huang
- 'Feng Liu[^1]'
title: Quantum Spin Hall Effect and Spin Bott Index in Quasicrystal Lattice
---
#### Introduction.
QSH states, which are characterized by topologically protected metallic edge states with helical spin polarization residing inside an insulating bulk gap, have attracted much interest in the last decade [@PhysRevLett.95.226801; @PhysRevLett.95.146802; @PhysRevLett.96.106802; @BHZ; @konig]. QSH states have been discovered in various two-dimensional (2D) materials [@huanghqCMS; @wang2017computational]. Recently, QSH systems grown on conventional semiconductor substrates have been theoretically proposed and experimentally realized, [@zhou2014epitaxial; @zhou2014formation; @hsu2015nontrivial; @wang2016quantum; @reis2017bismuthene] which bridges topological states with conventional semiconductor platforms offering an attractive route towards future quantum device applications. However, almost all the existing QSH states are based on crystals and identified by the $\mathbb{Z}_2$ topological invariant that is defined only for periodic systems [@PhysRevLett.95.146802; @PhysRevB.74.195312]. In this Letter, we extend the QSH state to quasicrystalline systems and define a new topological invariant, the spin Bott index, to identify nontrivial topology in aperiodic and amorphous systems.
Quasicrystal phases with long-range orientational order but no translational symmetry have attracted considerable attentions since their first discovery in 1982 [@PhysRevLett.53.1951]. Because of the unique structural characteristics, quasicrystals exhibit various unusual physical properties, such as extremely low friction [@dubois1991quasicrystalline], self-similarity [@PhysRevB.34.3904; @PhysRevB.37.2797] and critical (power-law decay) behaviors of wavefunctions [@PhysRevB.43.8879; @PhysRevB.43.8892] [in between extended Bloch states of periodic systems and exponentially localized states of disorder systems due to Andersion localization.]{} Additionally, 2D QLs have been epitaxially grown on quasicrystalline substrates recently [@jenks1998quasicrystals; @mcgrath2002quasicrystal; @sharma2007quasicrystal; @thiel2008quasicrystal; @mcgrath2012memory; @ledieu2014surfaces]. Here we answer an intriguing question: is it possible to realize a QSH state in a quasicrystal to support topologically protected extended edge states along the quasicrystal’s boundary, contrary to the general critical states of quasicrystals? If so, how to define a topological invariant in analogue of the $\mathbb{Z}_2$ index of periodic systems, to identify the QSH state in quasicrystals and more generally in amorphous systems?
We note that so far only very few works have been done on topological states of aperiodic systems, in particular, the Chern insulator states analogous of quantum anomalous Hall (QAH) states [@PhysRevX.6.011016; @mitchell2018amorphous; @PhysRevLett.118.236402]. However, they are based on either artificial systems, such as optical waveguide with an artificial gauge field [@PhysRevX.6.011016] and networks of interacting gyroscopes [@mitchell2018amorphous], or lattice models with artificial hoppings, such as an amorphous lattice model [@PhysRevLett.118.236402]. In contrast, we propose the realization of the QSH state in real quasicrystal materials with atomic orbitals, especially the surface-based QLs, which are readily makable by experiments. We characterize its nontrivial topology using the newly derived spin Bott index. As in crystals, the QSH phase in quasicrystals manifests also with robust metallic edge states and quantized conductance. Moreover, we determine a topological phase diagram for QLs and show that the QSH state lies in between a normal insulator and a gapless phase exhibiting weak metallic behavior due to the critical wavefunctions of QLs. Our proposed atomic model of quasicrystalline QSH states can be possibly grown by deposition of atoms on the surfaces of quasicrystal substrates.
#### Model.
2D QLs are constructed according to the Penrose tiling with fivefold rotational symmetry [@penrose1974role], as shown in Fig. \[fig1\](a). Since the QL possesses long-range orientational order but lacks translational symmetry, we cannot use the Bloch theorem as for the crystal calculations. However, it is still possible to generate a series of periodic lattices with a growing number of atoms that approximate the infinite QL according to the quasicrystal tiling approximants [@tsunetsugu1986eigenstates; @entin1988penrose; @PhysRevB.43.8879; @PhysRevX.6.011016]. In our model, the atoms are located on vertices of rhombuses of the Penrose tiling. The first three nearest neighbors are considered, as shown in the inset of Fig. \[fig1\](a); their separations are $r_0:r_1:r_2=2\cos\frac{2\pi}{5}: 1: 2\sin\frac{\pi}{5}$, respectively.
We consider a general atomic-basis model for QLs with three orbitals ($s,p_x,p_y$) per site. The Hamiltonian is given by $$\begin{aligned}
H&=&\sum_{i\alpha}\epsilon_\alpha c_{i\alpha}^\dag c_{i\alpha}+\sum_{<i\alpha,j\beta>}t_{i\alpha,j\beta}c_{i\alpha}^\dag c_{j\beta}\nonumber\\
&+&i\lambda\sum_i(c_{ip_y}^\dag \sigma_z c_{ip_x}-c_{ip_x}^\dag \sigma_z c_{ip_y}),
\label{eq1}\end{aligned}$$ where $c_{i\alpha}^\dag=(c_{i\alpha\uparrow}^\dag,c_{i\alpha\downarrow}^\dag)$ are electron creation operators on the $\alpha(=s,p_x,p_y)$ orbital at the $i$-th site. $\epsilon_\alpha$ is the on-site energy of the $\alpha$ orbital. The second term is the hopping term where $t_{i\alpha,j\beta}=t_{\alpha,\beta}(\mathbf{d}_{ij})$ is the hopping integral which depends on the orbital type ($\alpha$ and $\beta$) and the vector $\mathbf{d}_{ij}$ between sites $i$ and $j$. $\lambda$ is the spin-orbit coupling (SOC) strength and $\sigma_z$ is the Pauli matrix. In our model, the hopping integral follows the Slater-Koster formula [@SlaterKoster] $$t_{\alpha,\beta}(\mathbf{d}_{ij})=\mathrm{SK}[V_{\alpha\beta}(d_{ij}),\hat{\mathbf{d}}_{ij}],$$ where $\hat{\mathbf{d}}_{ij}$ is the unit direction vector. The distance dependence of the bonding parameters $V_{\alpha\beta}(d_{ij})$ is captured approximately by the Harrison relation [@harrison2012electronic]: $$V_{\alpha\beta}(d_{ij})=V_{\alpha\beta,0}\frac{d_0^2}{d_{ij}^2},$$ where $d_0$ is a scaling factor to uniformly tune the bonding strengths. Since only the band inversion between *s* and *p* states of different parities is important for the realization of topological states, we focus mainly on 2/3 filling of electron states hereafter, unless otherwise specified.
#### Results.
The calculated results of a particular realization of the QSH state in the QL with open boundary condition (OBC) and periodic BC (PBC) are shown in Fig. \[fig2\]. It is found that the PBC system clearly shows an energy gap, as displayed in Fig. \[fig2\](a). This indicates that the system is an insulator. However, there are some eigenvalues within the gap region of the energy spectrum in the presence of OBC, implying that the system becomes metallic. In Fig. \[fig2\](b), we plotted the wavefunction distribution of a typical mid-gap state \[marked as a star in Fig. \[fig2\](a)\]. Interestingly, the mid-gap state is an “edge state” which is localized on the boundary of the finite QL. We also studied other finite QL samples with different boundary geometries, and found that the edge state is robust, which always remains on the boundaries regardless of their detailed shapes [^2]. In contrast, typical bulk states show the well-known localized or critical characters of quasicrystals . Due to the time-reversal symmetry, the eigenvalues always appear in pairs with the same energy for the mid-gap edge states; while the wavefunctions of the two “degenerate” states are mainly contributed from spin up and down components, respectively. This implies that the system should be a topologically nontrivial QSH insulator.
To verify the conductive feature of the edge states, we further investigated the transport properties based on the non-equilibrium Green’s function method [@datta1997electronic; @PhysRevB.38.9375; @huanghqInterface]. In the transport simulation, one finite QL is coupled to two semi-infinite periodic leads and the two-terminal conductance is calculated as shown in Fig. \[fig2\](c). Remarkably, there is a clear quantized plateau at $G =2e^2/h$ for the two-terminal charge conductance, which resembles that of the QSH state in graphene [@PhysRevLett.95.226801]. As shown in Fig. \[fig2\](d), the local density of state of the central quasicrystal at $E=0$ eV \[the blue star marked in Fig. \[fig2\](c)\] mainly distributes on two edges of the quasicrystal, indicating that the conductive channels are mostly contributed by the topological edge states.
The bulk energy gap, robust mid-gap edge states as well as the quantized conductances connote the nontrivial topology of the QL. However, the most critical quantity to identify the electronic topology is the topological invariant which classifies insulators into different topological classes. For example, the Chern number ($C$) [@Haldane; @chang2013experimental; @huanghqSemiDirac] distinguishes the QAH states ($C\neq 0$) with trivial time-reversal-broken insulators ($C=0$); the topological $\mathbb{Z}_2$ invariant [@PhysRevLett.95.146802; @PhysRevB.74.195312] determines the QSH states ($\mathbb{Z}_2=1$) with normal time-reversal-invariant insulators ($\mathbb{Z}_2=0$). However, these topological invariants are only applicable to periodic systems. Recently, the Bott index, which is equivalent to the Chern number [@toniolo2017equivalence], is proposed to determine QAH state in nonperiodic system [@loring2011disordered; @hastings2011topological; @loring2015k; @PhysRevLett.118.236402; @PhysRevX.6.011016]. Here we derive a topological invariant for the QSH state of aperiodic systems.
#### Spin Bott index.
In order to verify the QSH states in the QLs, we define the spin Bott index for QSH states (For a detailed discussion see .) in reference to the definition of the spin Chern number [@PhysRevLett.97.036808; @PhysRevB.75.121403; @PhysRevB.80.125327; @*prodan2010non; @*prodan2011disordered]. We firstly construct the projector operator of the occupied states, $$P=\sum_i^{N_{occ}}|\psi_i\rangle\langle\psi_i|.$$ Then we make a smooth decomposition $P_z=P_+\oplus P_-$ for spin-up and spin-down sectors. At the first sight, this seems an easy job as long as two spin channels are decoupled. However, it becomes more complicated if there are spin-mixing terms in the Hamiltonian. The key idea is to use the projected spin operator, $$P_z=P\hat{s}_zP,$$ where $\hat{s}_z=\frac{\hbar}{2}\sigma_z$ is the spin operator ($\sigma_z$ is the Pauli matrix). For a spin conserving model, $\hat{s}_z$ commutes with the Hamiltonian $H$ and $P_z$, the Hamiltonian as well as eigenvectors can be divided into spin-up and spin-down sectors. Therefore, the eigenvalues of $P_z$ consist of just two nonzero values $\pm\frac{\hbar}{2}$. For systems without spin conservation (for example, the Kane-Mele model with a nonzero Rashba term $\lambda_R$ ), the $\hat{s}_z$ and $H$ no longer commute, and the spectrum of $P_z$ spreads toward zero. However, as long as the spin mixing term is not too strong (e.g., $\lambda_R/\lambda_{SO}<3$ in the Kane-Mele model, where $\lambda_{SO}$ is the intrinsic spin-orbit coupling ), the eigenvalues of $P_z$ remain two isolated groups which are separated by zero. Because the rank of matrix $P_z$ is $N_{occ}$, the number of positive eigenvalues equals to the number of negative eigenvalues, which is $N_{occ}/2$. The corresponding eigenvalue problem can be denoted as $$P_z|\pm \phi_i\rangle=S_\pm|\pm\phi_i\rangle.$$ In this way we can construct new projector operators $$P_\pm=\sum_i^{N_{occ}/2}|\pm\phi_i\rangle\langle\pm\phi_i|,$$ for two spin sectors. Next, we calculate the projected position operators $$\begin{aligned}
U_\pm=P_\pm e^{i2\pi X}P_\pm+(I-P_\pm),\\
V_\pm=P_\pm e^{i2\pi Y}P_\pm+(I-P_\pm),\end{aligned}$$ where $X$ and $Y$ are the rescaled coordinates which are defined in the interval $[0,1)$. To make the numerical algorithm more stable, we perform a singular value decomposition (SVD) $M=Z\Sigma W^\dag$ for $U_\pm$ and $V_\pm$, where $Z$ and $W$ are unitary and $\Sigma$ is real and diagonal, and set $\tilde{M}=ZW^\dag$ as the new unitary operators. The SVD process does not destroy the original formalism, but effectively improves the convergence and stability of the numerical algorithm . The Bott index, which measures the commutativity of the projected position operators [@bellissard1994noncommutative; @hastings2010almost; @exel1991invariants; @katsura2016Z2; @*katsura2018noncommutative], are given by $$B_\pm=\frac{1}{2\pi}\textrm{Im}\{\textrm{tr}[\log(\tilde{V}_\pm \tilde{U}_\pm \tilde{V}_\pm^\dag \tilde{U}_\pm^\dag)]\},$$ for two spin sectors, respectively. Finally, we define the spin Bott index as the half difference between the Bott indices for the two spin sectors $$B_s=\frac{1}{2}(B_+-B_-).$$ We checked the above definition for crystalline and disorder systems and found that the spin Bott index is the same as the $\mathbb{Z}_2$ topological invariant .
Similar to the spin Chern number [@PhysRevLett.97.036808; @PhysRevB.75.121403; @PhysRevB.80.125327; @*prodan2010non; @*prodan2011disordered], the spin Bott index is a well-defined topological invariant. It is applicable to quasiperiodic and amorphous systems, which provides a useful tool to determine the electronic topology of those systems without translational symmetry. For the QL in Fig. \[fig2\], we found that the spin Bott index $B_s = 1$, indicating indeed a QSH state. Due to the bulk-edge correspondence, it is natural to expect the existence of robust boundary states for systems with nontrivial spin Bott index. Hence, the nontrivial spin Bott index is consistent with the above calculations of edge states and electronic conductance, all confirming the nontrivial topological character of the QL.
#### Topological phase diagram.
In order to achieve the QSH state, one prerequisite is the band inversion between conduction and valance states. Generally, one can realize the band inversion by tuning either the on-site energy difference $\Delta=\epsilon_s-\epsilon_p$ or the SOC strength $\lambda$ [@huanghqCMS; @wang2017computational; @wang2016quantum]. To investigate the necessary condition for the realization of topological states in QLs, we calculated the topological phase diagram in the $\Delta$-$\lambda$ plane. As shown in Fig. \[fig3\](a), the normal insulator (NI) and QSH state are divided by an energy gap closing line \[white dashed line in Fig. \[fig3\](a)\]. To achieve the QSH phase, one has to reduce $\Delta$ so as to realize a band inversion between *s* and *p* states and then increase $\lambda$ to open a nontrivial energy gap.
Additionally, interactions between different atomic sites also play an important role in determining the spectrum and localization of electronic states in quasicrystals [@PhysRevB.33.2184; @PhysRevB.34.3904; @PhysRevB.43.1378]. We further investigated the phase diagram with the increasing bonding strengths. As shown in Fig. \[fig3\](b), the QL undergoes a topological phase transition from a NI to a QSH insulator at a critical value of about $d_0^c=0.89$. By further increasing the interaction, the system eventually enters a gapless phase which turns out to be a weak metal, as discussed later. Such a process can be understood by considering the evolution of states as following: starting from the atomic orbital limit, *s* and *p* states are initially separated by a trivial charge gap \[Fig. \[fig3\](c)\]. By increasing the interaction one gradually enlarges the band width, therefore reduces and eventually closes the charge gap, realizing a *s*-*p* band inversion; the SOC effect then reopens an energy gap with nontrivial topology \[Fig. \[fig3\](d)\]. Further increasing the interaction, to overcome the SOC gap, will drive the system into a gapless phase \[Fig. \[fig3\](e)\]. Interestingly, in the QSH region a defect mode induced by periodic approximation of quasicrystal moves downwards into the energy gap gradually with the increasing $d_0$, while the whole gap remains to be topologically nontrivial with $B_s=1$. We also calculated phase diagrams of quasicrystal approximants with different sizes and found similar phase transitions in all approximants.This implies that the topological phase transition as well as the QSH effect should appear in QLs in the thermodynamic limit of infinite lattice size.
Finally, we studied the localization of wavefunctions in the QL model. Whether the gapless state in the phase diagram is metallic or insulating depends on the localization of wavefunction around the Fermi level [@PhysRevB.34.3904; @PhysRevB.43.1378]. For nonperiodic systems, we illustrate the localization of each state by its participation ratio [@PhysRevB.33.2184], $$\gamma_n=\frac{(\sum_i^N|\langle i|\psi_n\rangle|^2)^2}{N\sum_i^N|\langle i|\psi_n\rangle|^4},$$ where $|i\rangle$ is the *i*-th local orbital. The participation ratio takes the value $1/N$ if the wavefunction is localized in a single orbital and unity if the wavefunction is extended uniformly over the whole system. The participation ratio of wavefunctions in QLs (Fig. 10 of Ref. ) is less than 0.25 which is much smaller than that of extended wavefunctions in periodic crystals. However, the localization behavior is also different from disordered Anderson localization, where the state around the mobility gap tend to be more localized [@schreiber1985numerical; @odagaki1986properties; @de2013electronic]. This indicates that critical wavefunctions are induced by local structural topology of the QLs [@PhysRevB.51.15827; @PhysRevB.34.5208]. The low participation ratio also suggests a weak metallic behavior in the electronic transport [@PhysRevB.35.1456; @PhysRevB.38.10109; @PhysRevB.43.8892; @de2013electronic]. Our transport simulation gives a conductance about an order of magnitude smaller than that of pure periodic leads, confirming its weak metallic behavior .
#### Experimental feasibility.
Our proposed atomic model of quasicrystalline QSH states is expected to be realized in surface-based 2D QLs. With the development of growth technique with atomic precision, 2D quasicrystals have been epitaxially grown on quasicrystalline substrates in the last two decades [@jenks1998quasicrystals; @mcgrath2002quasicrystal; @sharma2007quasicrystal; @thiel2008quasicrystal; @mcgrath2012memory; @ledieu2014surfaces]. For example, single-element epitaxial quasicrystalline structures have been successfully grown by deposition of Si [@PhysRevB.73.012204], Pb [@PhysRevB.77.073409; @PhysRevB.82.085417; @PhysRevB.79.165430; @PhysRevB.79.245405; @smerdon2008formation; @sharma2013templated], Sn [@PhysRevB.72.045428], Sb [@PhysRevLett.89.156104], Bi [@PhysRevB.78.075407], Co [@smerdon2006adsorption], and Cu atoms [@PhysRevLett.92.135507; @PhysRevB.72.035420; @PhysRevB.74.035429] on the fivefold (tenfold) surfaces of icosahedral (decagonal) quasicrystals which serve as template substrates. Moreover, aperiodic quasicrystalline phases can also be realized on crystalline surfaces [@forster2013quasicrystalline]. For example, atomically flat epitaxial Ag films with quasiperiodicity was synthesized on GaAs(110) surface [@ScienceAgGaAs]. Recently, Collins *et al.* [@collins2017imaging] realized a synthetic QL by arranging carbon monoxide molecules on the surface of Cu(111) to form a Penrose tiling using scanning tunneling microscopy and atomic manipulation. We, therefore, expect that it is experimentally feasible to realize our theoretically proposed atomic model of QSH states on surface-based QLs.
#### Conclusion
We have proposed the realization of QSH states in Penrose-type QLs. We characterize the topological nature by deriving a newly-defined topological invariant, the spin Bott index, in addition to conventional evidences including robust edge states and quantized conductances. Beyond the Penrose tiling, other QLs of different classes of local isomorphism should also be able to realize the QSH state [@senechal1996quasicrystals]. The essential band inversion is not limited to the $s$ and $p$ orbitals, and other types of band inversion mechanism [@zhang2013topological] such as *p*-*p* [@PhysRevB.93.035135], *p*-*d* [@PhysRevLett.106.156808], *d*-*d* [@si2016large], and *d*-*f* band inversion [@PhysRevLett.104.106408], are all feasible to achieve QSH states in quasicrystals. Our finding therefore extends the territory of topological materials beyond crystalline solids, to surface-based aperiodic systems with a range of choices of structural and element species. Our proposed approach is also applicable to other 2D QLs with different symmetries and to 3D quasicrystal structures [@PhysRevB.34.596; @PhysRevB.34.617], which may open additional exciting possibilities.
This work was supported by DOE-BES (Grant No. DE-FG02-04ER46148). The calculations were done on the CHPC at the University of Utah and DOE-NERSC.
[88]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase 10.1103/PhysRevLett.95.226801) [****, ()](\doibase 10.1103/PhysRevLett.95.146802) [****, ()](\doibase 10.1103/PhysRevLett.96.106802) [****, ()](\doibase 10.1126/science.1133734) @noop [****, ()]{} @noop [**** ()]{} @noop [**** ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevB.74.195312) [****, ()](\doibase 10.1103/PhysRevLett.53.1951) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevB.34.3904) [****, ()](\doibase 10.1103/PhysRevB.37.2797) [****, ()](\doibase 10.1103/PhysRevB.43.8879) [****, ()](\doibase 10.1103/PhysRevB.43.8892) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevX.6.011016) @noop [ ()]{} [****, ()](\doibase 10.1103/PhysRevLett.118.236402) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRev.94.1498) @noop [**]{} (, ) @noop [**]{} (, ) [****, ()](\doibase 10.1103/PhysRevB.38.9375) [****, ()](\doibase 10.1103/PhysRevB.92.075138) [****, ()](\doibase 10.1103/PhysRevLett.61.2015) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevB.92.161115) @noop [ ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevLett.97.036808) [****, ()](\doibase 10.1103/PhysRevB.75.121403) [****, ()](\doibase 10.1103/PhysRevB.80.125327) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevB.33.2184) [****, ()](\doibase 10.1103/PhysRevB.43.1378) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevB.51.15827) [****, ()](\doibase 10.1103/PhysRevB.34.5208) [****, ()](\doibase 10.1103/PhysRevB.35.1456) [****, ()](\doibase 10.1103/PhysRevB.38.10109) [****, ()](\doibase
10.1103/PhysRevB.73.012204) [****, ()](\doibase
10.1103/PhysRevB.77.073409) [****, ()](\doibase 10.1103/PhysRevB.82.085417) [****, ()](\doibase 10.1103/PhysRevB.79.165430) [****, ()](\doibase
10.1103/PhysRevB.79.245405) @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevB.72.045428) [****, ()](\doibase
10.1103/PhysRevLett.89.156104) [****, ()](\doibase
10.1103/PhysRevB.78.075407) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevLett.92.135507) [****, ()](\doibase
10.1103/PhysRevB.72.035420) [****, ()](\doibase 10.1103/PhysRevB.74.035429) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevB.93.035135) [****, ()](\doibase 10.1103/PhysRevLett.106.156808) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevLett.104.106408) [****, ()](\doibase 10.1103/PhysRevB.34.596) [****, ()](\doibase 10.1103/PhysRevB.34.617)
[^1]: Corresponding author: fliu@eng.utah.edu
[^2]: \[fn\] See the joint publication: H. Huang and F. Liu, Phys. Rev. B xx,xxx (2018), for more details.
|
[DIRICHLET BRANES AND A COHOMOLOGICAL\
DEFINITION OF TIME FLOW]{} [**José M. Isidro and P. Fernández de Córdoba**]{}\
Grupo de Modelización Interdisciplinar Intertech,\
Instituto Universitario de Matemática Pura y Aplicada,\
Universidad Politécnica de Valencia, Valencia 46022, Spain\
[joissan@mat.upv.es, pfernandez@mat.upv.es]{} [**Abstract**]{} Dirichlet branes are objects whose transverse coordinates in space are matrix–valued functions. This leads to considering a matrix algebra or, more generally, a Lie algebra, as the classical phase space of a certain dynamics where the multiplication of coordinates, being given by matrix multiplication, is nonabelian. Further quantising this dynamics by means of a $\star$–product introduces noncommutativity (besides nonabelianity) as a quantum $\hbar$–deformation. The algebra of functions on a standard Poisson manifold is replaced with the universal enveloping algebra of the given Lie algebra. We define generalised Poisson brackets on this universal enveloping algebra, examine their properties, and conclude that they provide a natural framework for dynamical setups (such as coincident Dirichlet branes) where coordinates are matrix–valued, rather than number–valued, functions.
Introduction {#labastidachupamelapolla}
============
Classical mechanics can be formulated on a Poisson manifold ${\cal M}$ (classical phase space). This means that the algebra $C^{\infty}({\cal M})$ of smooth functions on ${\cal M}$ supports Poisson brackets, [*i.e.*]{}, an antisymmetric, bilinear map $$\left\{\cdot\,,\cdot\right\}_{\rm Poisson}\colon C^{\infty}({\cal M})\times C^{\infty}({\cal M})\longrightarrow C^{\infty}({\cal M})
\label{labastidaquetepartaunrayo}$$ satisfying the Jacobi identity and the Leibniz derivation rule [@RATIU].
On the other hand, Lie algebras $\mathfrak{g}$ support Lie brackets $$[\cdot\,,\cdot]\colon \mathfrak{g}\times \mathfrak{g}\longrightarrow \mathfrak{g}
\label{ramallocabron}$$ that are antisymmetric, bilinear and satisfy the Jacobi identity. For reasons that will become clear presently, we would like to regard $\mathfrak{g}$ as a classical phase space for a certain dynamics, and its universal enveloping algebra ${\cal U}(\mathfrak{g})$ as its [*algebra of functions*]{}, in a sense to be specified below.
When $\mathfrak{g}$ is simple, we can use the Killing metric to identify $\mathfrak{g}$ with its dual $\mathfrak{g}^*$. Then the Kirillov brackets on $\mathfrak{g}^*$, $$\left\{\cdot\,,\cdot\right\}_{\rm Kirillov}\colon C^{\infty}(\mathfrak{g}^*)\times C^{\infty}(\mathfrak{g}^*)\longrightarrow C^{\infty}(\mathfrak{g}^*)
\label{barbonchupamelapolla}$$ turn $\mathfrak{g}$ into a Poisson manifold [@KIRILLOV]. Although we will make use of them, we are not primarily interested in the Kirillov brackets (\[barbonchupamelapolla\]). Instead we will pass from the Lie algebra $\mathfrak{g}$ to its universal enveloping algebra ${\cal U}(\mathfrak{g})$, where an associative multiplication is defined, and the Leibniz derivation rule can be made to hold. The price to pay is the loss of the abelian property for the associative multiplication: while the algebra $C^{\infty}({\cal M})$ was abelian under the pointwise multiplication of functions, ${\cal U}(\mathfrak{g})$ will not be abelian. This reflects the nonabelian property of $\mathfrak{g}$, which in turn is necessary for $\mathfrak{g}$ to be simple.
Evolution equations of the type $$\dot F = \left[F, H\right],
\label{labastidaputonverbenero}$$ for $F\in\mathfrak{g}$ and for a certain Hamiltonian $H\in\mathfrak{g}$, are ubiquitous in physics. Interesting generalisations of the above equation can be obtained as follows. For definiteness we will consider the Lie algebra $\mathfrak{su}(n)$ throughout, although our results can be easily generalised to any finite–dimensional, simple, compact Lie algebra $\mathfrak{g}$, provided one pays due attention to its corresponding cohomology ring [@KNAPP]. (Since $\mathfrak{g}$ is compact we may alternatively consider de Rham cohomology on the corresponding Lie group [@GOLDBERG]). Let $\omega_{2j+1}$ be a nonzero cocycle of the cohomology of $\mathfrak{g}$, with order $2j+1$. This cocycle defines $2j$–fold Lie brackets on $\mathfrak{g}^{\times (2j)}$, $$[\cdot\,,\cdots,\cdot]_{\omega_{2j+1}}\colon \mathfrak{g}\times{}^{(2j)\atop\ldots}\times \mathfrak{g}\longrightarrow
\mathfrak{g},
\label{pajarescabron}$$ linear in all $2j$ entries, completely antisymmetric, and satisfying a generalised Jacobi identity [@BUENO]. When $j=1$, the $2j$–fold Lie brackets (\[pajarescabron\]) reduce to the Lie brackets (\[ramallocabron\]), where one omits the subindex $\omega_3$. Then an equation of motion for $F\in\mathfrak{g}$ in the dynamics generated by $2j-1$ Hamiltonians $H_2, \ldots H_{2j}\in\mathfrak{g}$ is $$\dot F = \left[F,H_2, \ldots, H_{2j}\right]_{\omega_{2j+1}}.
\label{sanchezguillencabron}$$ We will generalise eqns. (\[labastidaputonverbenero\]) and (\[sanchezguillencabron\]) by allowing $F$ and the Hamiltonians to be elements of ${\cal U}(\mathfrak{g})$. Such evolution equations allow one to regard $\mathfrak{g}$ as a classical phase space, and the universal enveloping algebra ${\cal U}(\mathfrak{g})$ as an algebra of functions. We will define $2j$–fold Poisson brackets on ${\cal U}(\mathfrak{g})$ that will satisfy the Leibniz derivation rule, and that will reduce to the $2j$–fold Lie brackets (\[pajarescabron\]) when acting on elements of $\mathfrak{g}$. With this formal viewpoint, the impossibility of setting ${\cal U}(\mathfrak{g})$ (for $\mathfrak{g}$ compact and simple) equal to $C^{\infty}({\cal M})$ for any smooth Poisson manifold ${\cal M}$ becomes irrelevant. Of course, the restriction to the Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ leads to the abelian ${\cal U}(\mathfrak{h})$, a subalgebra of the full enveloping algebra ${\cal U}(\mathfrak{g})$; if $\mathfrak{g}$ has rank $l$, then ${\cal U}(\mathfrak{h})$ is the subalgebra of polynomials within $C^{\infty}(\mathbb{R}^l)$.
In our terminology, [*abelian*]{} and [*commutative*]{} are not interchangeable, nor are their opposites [*nonabelian, noncomutative*]{}. We reserve the term [*noncommutative*]{} for those multiplications performed using the $\star$–product [@GRONEWALD; @MOYAL; @BAYEN; @KONTSEVICH; @GMS]; [*commutative*]{} are those products that use the pointwise multiplication of functions. On the other hand, matrix multiplication is termed [*nonabelian*]{}, although it remains commutative because it uses the pointwise product. A quantum deformation of matrix multiplication, eqn. (\[labamaricon\]), will yield a multiplication that will be both nonabelian (because we will be dealing with matrices) and [*noncommutative*]{} (because of the $\hbar$–deformation). To illustrate our terminology, the algebra of functions $C^{\infty}({\cal M})$ is abelian and commutative. Replacing the pointwise product with a $\star$–product we obtain $C^{\star}({\cal M})$, which is abelian and noncommutative. The universal enveloping algebra ${\cal U}(\mathfrak{g})$ is nonabelian and commutative; its quantum deformation ${\cal U}^{\star}(\mathfrak{g})$ will be nonabelian and noncommutative. All these algebras are associative.
We will address the deformation quantisation of the previous classical dynamics, by replacing the (pointwise) multiplication on ${\cal U}(\mathfrak{g})$ with a Kontsevich $\star$–product. The latter requires the specification of classical Poisson brackets. Roughly speaking, there is a 1–to–1 correspondence between classical Poisson structures and quantum $\star$–products [@KONTSEVICH], hence the quantisation obtained depends on the classical brackets one starts out with. This quantisation can be carried out for any compact, simple Lie algebra $\mathfrak{g}$, but it is best performed by specifying a faithful representation for $\mathfrak{g}$ and regarding the matrices so obtained as [*nonabelian coordinate functions*]{}, their multiplication being nonabelian and, after quantisation, also noncommutative. One is thus led to the conclusion that $n$ coincident, parallel Dirichlet branes (D–branes for short) [@SZABO], having matrices as their [*transverse*]{} coordinates [@WITTEN], provide a natural realisation of the abstract setup described previously. Moreover, the identification established in ref. [@WITTEN] between transverse components of bulk gauge fields and D–brane coordinates suggests the adjoint representation as the preferred one, but our treatment holds in any faithful representation just as well.
Generalisations of the standard Poisson brackets involving more than 2 entries, in particular the Nambu brackets [@NAMBU], have appeared in connection with branes and integrable systems [@CZ; @HOPPE]. The approach of ref. [@CZ] is based on the observation that the worldvolume element on a membrane, being a Jacobian determinant, can be identified with the Nambu brackets. Although we also quantise by replacing the pointwise product with a $\star$–product, our approach differs from that of ref. [@CZ] in several respects. We do not regard the longitudinal brane coordinates are the basic variables entering the brackets. Instead our starting point is motivated in the consideration of matrix–valued functions as [*transverse*]{} coordinate functions to the brane. Thus our approach is strongly motivated in M–theory [@SZABO], where the Lie algebra in which coordinates take values plays a prominent role: it is the Lie algebra of the gauge group within the stack of coincident D–branes [@WITTEN]. In turn, the gauge symmetry $\mathfrak{su}(n)$ is determined only by the brane content, [*i.e.*]{}, by the number $n$ of coincident branes. (One may eventually add orientifolds in order to obtain an orthogonal/symplectic gauge symmetry within the branes, but we will basically consider the case of $\mathfrak{su}(n)$). In other words, transverse D–brane coordinate functions are determined by Yang–Mills gauge fields within the D–branes themselves. Moreover, being Lie–algebra valued, our transverse coordinate functions exhibit nonabelianity already at the classical level.
Since we are addressing the mechanics (classical and quantum) of [*transverse*]{} coordinates, and time is always [*longitudinal*]{}, or parallel to a brane, we can think of our construction as providing the mechanics (classical and quantum) of [*matrix–valued, spacelike coordinate functions*]{} and their time evolution—in the absence of time! Indeed we will see that time evolution can be defined algebraically, by means of Lie algebra cohomology, without any recourse to a continuously flowing parameter.
This article is organised as follows. Section \[casposoramallo\] presents the mathematical prerequisites concerning (classical and quantum) Lie and Poisson multibrackets [@BUENO]. Section \[labastidaquetefollen\] works out the connection between multibrackets and $n$ parallel D$p$–branes (coincident or separated), where the gauge symmetry is $\mathfrak{u}(n)$ or a simple subalgebra thereof. Section \[mierdaparatodos\] presents our conclusions.
Dynamics on a simple, compact Lie algebra $\mathfrak{g}$ {#casposoramallo}
========================================================
Let $\mathfrak{g}$ be a simple, finite–dimensional, compact Lie algebra over $\mathbb{R}$; as a rule we have $\mathfrak{su}(n)$ in mind.
Classical {#barbonputon}
---------
The universal enveloping algebra ${\cal U}(\mathfrak{g})$ is the associative algebra obtained as the $\mathbb{R}$–linear span of all formal products of powers $X^{p}$, for all $X\in \mathfrak{g}$ and all $p=1,2,\ldots$, subject to the requirement that $$XY-YX=[X,Y] \qquad \forall X, Y \in \mathfrak{g}.
\label{cesargomezmekagoentuputakaramarikondemierda}$$ In eqn. (\[cesargomezmekagoentuputakaramarikondemierda\]), the left–hand side contains the associative product on ${\cal U}(\mathfrak{g})$, while the the right–hand side contains the Lie brackets on $\mathfrak{g}$.
The Lie brackets (\[ramallocabron\]) extend to Poisson brackets $$\{\cdot\,,\cdot\}\,\colon{\cal U}(\mathfrak{g})\times{\cal
U}(\mathfrak{g})\longrightarrow {\cal U}(\mathfrak{g})
\label{ramallocasposo}$$ by setting $$\left\{X,Y\right\}:=\left[X,Y\right]\qquad \forall X,Y\in\mathfrak{g},
\label{barbonmaricondeplaya}$$ by requiring multilinearity, complete antisymmetry, and by imposing the Leibniz derivation rule when applied to products of Lie algebra elements, [*i.e.*]{}, $$\left\{XY, Z\right\}:=\left\{X, Z\right\}\,Y + X\, \left\{Y, Z\right\}\qquad \forall X,Y,Z\in\mathfrak{g}.
\label{labastidacasposo}$$ Higher powers of Lie algebra elements can be reduced to smaller powers by repeated application of eqn. (\[labastidacasposo\]). Picking a Hamiltonian $H\in{\cal U}(\mathfrak{g})$, the evolution equation for any $F\in{\cal U}(\mathfrak{g})$, or classical equation of motion, reads $$\dot F= \left\{F, H\right\}.
\label{ramalloquetepartaunrayo}$$
Choose now a $(2j+1)$–cocycle $\omega_{2j+1}$ in the cohomology ring of $\mathfrak{g}$. Then the $2j$–fold Lie brackets (\[pajarescabron\]) can be extended to $2j$–fold Poisson brackets on ${\cal U}(\mathfrak{g})^{\times(2j)}$, $$\{\cdot\,, \cdots,\cdot \}_{\omega_{2j+1}}\colon {\cal U}(\mathfrak{g})\times{}^{(2j)\atop\ldots}\times {\cal U}(\mathfrak{g})
\longrightarrow {\cal U}(\mathfrak{g}),
\label{pajareshijodeputa}$$ by setting $$\left\{X_1, \ldots, X_{2j}\right\}_{\omega_{2j+1}}:=\left[X_1,\ldots, X_{2j}\right]_{\omega_{2j+1}}
\label{barbonladron}$$ for all $X_1, \ldots, X_{2j}\in \mathfrak{g}$, by demanding $2j$–linearity, complete antisymmetry, and further requiring that the Leibniz derivation rule hold. If we pick $2j-1$ Hamiltonians $H_2, \ldots, H_{2j}\in{\cal U}(\mathfrak{g})$ we can write a classical equation of motion for any $F\in {\cal U}(\mathfrak{g})$: $$\dot F = \left\{F, H_2,\ldots, H_{2j}\right\}_{\omega_{2j+1}}.
\label{ramayoquetepartaunrayo}$$ The $2j$–fold Poisson brackets (\[pajareshijodeputa\]) reduce to the $2j$–fold Lie brackets (\[pajarescabron\]) when restricted to $\mathfrak{g}^{\times (2j)}$, and to the Poisson brackets (\[ramallocasposo\]) when $j=1$. In this latter case one omits the subindex $\omega_3$.
In this way the algebra ${\cal U}(\mathfrak{g})$ supports the Poisson multibrackets (\[pajareshijodeputa\]) even if ${\cal U}(\mathfrak{g})$ cannot be identified with the algebra of smooth functions $C^{\infty}({\cal M})$ for any manifold ${\cal M}$. If $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$, and the latter has rank $l$, then ${\cal U}(\mathfrak{h})$ can be identified with the subalgebra of polynomials within $C^{\infty}(\mathbb{R}^l)$. However, the restriction of the Poisson structure (\[pajareshijodeputa\]) from ${\cal U}(\mathfrak{g})$ to ${\cal U}(\mathfrak{h})$ vanishes identically, because $\mathfrak{h}$ is abelian. We will see next that ${\cal U}(\mathfrak{g})$ can be naturally associated with a certain algebra of functions. This is best done by specifying a representation for $\mathfrak{g}$, which brings us to our next point.
Eqns. (\[cesargomezmekagoentuputakaramarikondemierda\])–(\[ramayoquetepartaunrayo\]) above hold for any abstract Lie algebra $\mathfrak{g}$ (simple and compact). Given now a faithful $d$–dimensional representation for $\mathfrak{g}$, eqns. (\[cesargomezmekagoentuputakaramarikondemierda\])–(\[ramayoquetepartaunrayo\]) above are represented as (antisymmetrised sums of) compositions of elements of ${\rm End}\,(\mathbb{R}^d)$, and we have a (representation–dependent) isomorphism $${\cal U}(\mathfrak{g})\simeq {\rm End}\,(\mathbb{R}^d).
\label{barbonquetepartaunrayo}$$ Further fixing a basis on $\mathbb{R}^d$, endomorphisms are represented by matrices, and the associative multiplication law on ${\cal U}(\mathfrak{g})$ becomes matrix multiplication. This gives isomorphisms $${\rm End}\,(\mathbb{R}^d)\simeq {\rm Mat}_{d\times d}(\mathbb{R})\simeq{\cal U}(\mathfrak{g}),
\label{labastidajodete}$$ and eqns. (\[cesargomezmekagoentuputakaramarikondemierda\])–(\[ramayoquetepartaunrayo\]) above can be written as antisymmetrised sums of powers of $(d\times d)$ matrices. Thus elements $F\in{\cal U}(\mathfrak{g})$ are represented by $(d\times d)$–dimensional matrices with entries $F_{jm}$. The $F_{jm}$ are the coordinate functions of $F\in {\cal U}(\mathfrak{g})$ in the given representation. As such they are polynomials of arbitrary degree in the coordinates $x^1,\ldots,x^{d^2}$ on $\mathbb{R}^{d^2}$; these polynomials are homogeneous of degree 1 when $F\in \mathfrak{g}$. So, in the given representation for $\mathfrak{g}$, the matrix entries $F_{jm}$ specifying $F\in{\cal U}(\mathfrak{g})$ are polynomial functions, $$F_{jm}\colon\mathbb{R}^{d^2}\longrightarrow\mathbb{R}\qquad j,m=1,\ldots, d,
\label{labastidahijoputa}$$ and the nonabelian, pointwise multiplication law on ${\cal U}(\mathfrak{g})$ is matrix multiplication, $$(FG)_{jk}=\sum_{m=1}^{d}F_{jm}G_{mk}, \qquad j,k=1,\ldots, d.
\label{labacabron}$$
Quantum {#barbonmaricon}
-------
On the linear space $\mathbb{R}^{N}$ we have an associative, commutative algebra of functions $C^{\infty}(\mathbb{R}^{N})$ with respect to the pointwise product: if $f,g\in C^{\infty}(\mathbb{R}^{N})$, then their pointwise product is the function $$(f\cdot g)\colon\mathbb{R}^N\longrightarrow \mathbb{R}\qquad (f\cdot g)(x):=f(x)g(x) \;\; \forall x\in\mathbb{R}.
\label{barbonkabronquetepartaunrayo}$$ Let a Poisson structure $\left\{\cdot\,,\cdot\right\}_{\rm Poisson}$ be given on $\mathbb{R}^{N}$. Picking coordinates $x^1, \ldots, x^N$ on $\mathbb{R}^{N}$ we can write $$\left\{f,g\right\}_{\rm Poisson}(x)=\Omega^{jm}(x)\partial_jf(x)\partial_mg(x),
\label{ramallokaka}$$ where $\Omega^{jm}(x)=-\Omega^{mj}(x)$ is the matrix of $\left\{\cdot\,,\cdot\right\}_{\rm Poisson}$ at $x\in\mathbb{R}^N$, and all products involved are pointwise. Associated with $\left\{\cdot\,,\cdot\right\}_{\rm Poisson}$ there is a $\star$–product [@KONTSEVICH] which is an associative, noncommutative deformation of the pointwise product on $C^{\infty}(\mathbb{R}^{N})$, such that $$f\star g=f\cdot g + O(\hbar)
\label{labastidamecagoentuputasombramarikon}$$ and $$\left\{f,g\right\}_{\rm Poisson}=\frac{1}{{\rm i}\hbar}\left(f\star g-g\star f\right)+O(\hbar)
\label{labastidamecagoentuputacaracabron}$$ for all $f,g\in C^{\infty}(\mathbb{R}^N)$. In fact the Kontsevich $\star$–product is uniquely determined (up to gauge equivalence) by the given Poisson structure $\left\{\cdot\,,\cdot\right\}_{\rm Poisson}$ on ${\cal M}$ [@KONTSEVICH]. Replacing the pointwise product on $C^{\infty}(\mathbb{R}^{N})$ with the Kontsevich $\star$–product provides a quantum deformation of this latter algebra, denoted $C^{\star}(\mathbb{R}^{N})$. We should point out that the Kontsevich $\star$–product reduces to Grönewald–Moyal’s [@GRONEWALD; @MOYAL] $$(f\star g)(x)=f(x)\,\exp{\left({\rm i}\hbar\,\stackrel{\leftarrow}{\partial_j}\Omega^{jm}\stackrel{\rightarrow}{\partial_m}\right)}\,g(x)
\label{marikitasjaviermas}$$ in the case when the Poisson structure $\Omega^{jm}$ is constant, [*i.e.*]{}, independent of $x\in\mathbb{R}^N$. However in our setup the Kirillov–Poisson brackets (\[barbonchupamelapolla\]) are the natural choice. The reason is the explicit presence of the structure constants $f^{jm}_k$ of $\mathfrak{g}$ in the Kirillov–Poisson brackets: for the latter we have $\Omega^{jm}(x)=f^{jm}_kx^k$, where the $x^k$ are coordinates on $\mathfrak{g}^*$.
We define the nonabelian, noncommutative algebra ${\cal U}^{\star}(\mathfrak{g})$ as the one resulting from ${\cal U}(\mathfrak{g})$ upon replacing the matrix pointwise multiplication (\[labacabron\]) with the matrix $\star$–product $$(F\star G)_{jk}=\sum_{m=1}^{d}F_{jm}\star G_{mk} \qquad j,k=1,\ldots, d,
\label{labamaricon}$$ where $F_{jm}\star G_{mk}$ is the Kontsevich $\star$–product of functions on $\mathbb{R}^{d^2}$. That is, we are setting $N=d^2$ in eqns. (\[barbonkabronquetepartaunrayo\])–(\[labastidamecagoentuputacaracabron\]) above. However this requires previous Poisson brackets $\left\{\cdot\,,\cdot\right\}_{\rm Poisson}$ on the algebra $C^{\infty}(\mathbb{R}^{d^2})$. In order to define them we observe that $$\mathfrak{g}^*\simeq\mathfrak{g}\subset{\rm End}\,(\mathbb{R}^d)\simeq\mathbb{R}^{d^2},
\label{alvarezgaumecabron}$$ where the first $\simeq$ sign is due to the Killing form, and the inclusion sign reminds us that not all endomorphisms qualify as elements of $\mathfrak{g}$. For example the identity endomorphism, having nonzero trace, cannot belong to any simple $\mathfrak{g}$. It follows that $$C^{\infty}(\mathfrak{g}^*)\subset C^{\infty}(\mathbb{R}^{d^2}).
\label{alvarezgaumemecagoentuputacaramarikondemierda}$$ By eqn. (\[barbonchupamelapolla\]), at least a subalgebra of $C^{\infty}(\mathbb{R}^{d^2})$ supports natural Poisson brackets: the Kirillov brackets on $C^{\infty}(\mathfrak{g}^*)$. One can try and extend the latter to all of $C^{\infty}(\mathbb{R}^{d^2})$, and in fact any extension will do the job. However any such extension will be redundant since the $\star$–product will only enter our equations through antisymmetrised expressions. This is so because the conditions restricting an endomorphism $X\in{\rm End}\,(\mathbb{R}^d)$ to be also an element of the Lie algebra $\mathfrak{g}$ ([*e.g.*]{}, the tracelessness condition) carry over unchanged to the $\hbar$–deformed case. Hence the resulting antisymmetrised $\star$–matrix multiplication on ${\cal U}^{\star}(\mathfrak{g})$ is independent of which extension is picked for the Kirillov brackets. To summarise, eqn. (\[labamaricon\]) correctly defines a nonabelian, noncommutative multiplication on the algebra ${\cal U}^{\star}(\mathfrak{g})$. Moreover, the Poisson structure picked to define the $\star$–product is the natural one, namely, the Kirillov brackets. Next we have the $\star$–isomorphism $${\cal U}^{\star}(\mathfrak{g})\simeq {\rm End}^{\star}(\mathbb{R}^{d});
\label{pajareseresuntontodebaba}$$ the superscript $\star$ reminds us that matrices are to be multiplied according to eqn. (\[labamaricon\]). Picking a basis of vectors in $\mathbb{R}^d$, the algebra ${\rm End}^{\star}(\mathbb{R}^{d})$ is $\star$–isomorphic to the algebra of $(d\times d)$ matrices whose entries are $\star$–polynomial functions of arbitrary degree in the $x^1,\ldots,x^{d^2}$.
Finally the quantum dynamics on ${\cal U}^{\star}(\mathfrak{g})$ is described by $2j$–fold Poisson brackets, $$\{\cdot\,,\cdots,\cdot \}_{\omega_{2j+1}}^{\star}\colon {\cal U}^{\star}(\mathfrak{g})\times{}^{(2j)\atop\ldots}\times
{\cal U}^{\star}(\mathfrak{g})
\longrightarrow {\cal U}^{\star}(\mathfrak{g}),
\label{pajaresmaricon}$$ that can be obtained from the classical $2j$–fold Poisson brackets (\[pajareshijodeputa\]) by just replacing all pointwise matrix products (\[labacabron\]) with $\star$–matrix products, as per eqn. (\[labamaricon\]). The result provides an $\hbar$–deformation of the classical $2j$–fold brackets (\[pajareshijodeputa\]), to which it reduces in the limit $\hbar\to 0$. The time evolution of an observable $F\in {\cal U}^{\star}(\mathfrak{g})$ is governed by the equation $$\dot F = \{F,H_2,\ldots, H_{2j}\}^{\star}_{\omega_{2j+1}},
\label{labakaka}$$ which is the quantum analogue of the classical equation of motion (\[ramayoquetepartaunrayo\]). The latter can be obtained from the above by letting $\hbar\to 0$.
The connection with D–branes {#labastidaquetefollen}
============================
Our previous correspondence between coordinates and matrices is essential in order to understand the latter as a natural generalisation of the former. While standard geometry has number–valued functions as coordinates, matrix–valued functions arise naturally as [*transverse coordinate functions*]{} for D$p$–branes [@WITTEN]. Next we demonstrate that the Poisson multibrackets of section \[casposoramallo\] are appropriate to describe the classical and quantum dynamics of the transverse coordinates to branes.
The superposition of $n$ parallel, identical D$p$–branes produces a $\mathfrak{u}(n)$ gauge theory on their common $(p+1)$–dimensional worldvolume [@WITTEN]. Now $\mathfrak{u}(n)=\mathfrak{u}(1)\times \mathfrak{su}(n)$ is not simple, but separating out the centre–of–mass motion we are left with the simple algebra $\mathfrak{su}(n)$. Let $A_{\mu}$ be an $\mathfrak{su}(n)$–valued gauge field on the D$p$–brane stack, and separate its components into longitudinal and transverse parts to the D$p$–branes, $A_{\mu}=(A_l, A_t)$. Longitudinal components $A_l$ are then adjoint–valued $\mathfrak{su}(n)$ matrices, [*i.e.*]{}, Yang–Mills gauge fields. Transverse components $A_t$ describe D$p$–brane fluctuations that are orthogonal to the D$p$–branes themselves. They are thus identified with transverse coordinates, so they are more properly denoted $X_l$ instead of $A_l$. Modulo numerical factors, the bosonic part of the mechanical action of super Yang–Mills theory dimensionally reduced to $p+1$ dimensions is [@SZABO] $$S_{\rm YM}^{(p+1)}=\int{\rm d}^{p+1}\xi\,{\rm tr}\,({\cal F}_{ll'}^2+2{\cal F}^2_{lt}+{\cal F}^2_{tt'}),
\label{labastidachupamelapollamarikondemierda}$$ where $l,l'$ are longitudinal indices, $t,t'$ are transverse, and the trace is taken is the adjoint representation. Dirichlet boundary conditions remove all derivatives in the $t$ directions, and (again up to numerical factors) eqn. (\[labastidachupamelapollamarikondemierda\]) becomes $$S_{\rm YM}^{(p+1)}=\int{\rm d}^{p+1}\xi\,{\rm tr}\,{\cal F}_{ll'}^2 - \int{\rm d}^{p+1}\xi\,{\rm tr}\,\left(\frac{1}{2} (D_lX^t)^2-\frac{1}{4}[X^t,X^{t'}]^2
\right),
\label{cesargomezquetefollenkabrondemierda}$$ where $D_lX_t=\partial_lX_t+{\rm i}[A_l, X_t]$ is the longitudinal, gauge–covariant derivative of transverse coordinates. The appearance of matrix–valued coordinate functions can be motivated in the relation of D$p$–branes to Chan–Paton factors via T–duality [@SZABO]. For $p=-1$ (the case of instantons), all spacelike directions are transverse; for $p=0$, all but one. The latter is the important case of the M(atrix) model [@BFSS] of M–theory, where the limit $n\to\infty$ is taken. What follows can be regarded as applying to the lagrangian density describing the transverse coordinates $X_t$, which is the integrand of the second summand on the right–hand side of (\[cesargomezquetefollenkabrondemierda\]); the corresponding action will be the volume integral of this lagrangian over [*transverse*]{} space. (Notice that the integral (\[cesargomezquetefollenkabrondemierda\]) extends over [*longitudinal*]{} space instead). Thus our action integral reads, in the 11 dimensions of M–theory [@SZABO], $$S_{\rm transverse}=\int{\rm d}^{10-p}\xi\,{\rm tr}\,\left( (D_lX^t)^2-\frac{1}{2}[X^t,X^{t'}]^2
\right),
\label{ramallomechupalapolla}$$ as always up to overall factors.
We recall that $\mathfrak{su}(n)$ has $n-1$ simple roots [@HUMPHREYS], $$\alpha_1={\bf e}_1-{\bf e}_2,\quad \alpha_2={\bf e}_2-{\bf e}_3,\quad \ldots\quad \alpha_{n-1}={\bf e}_{n-1}-{\bf e}_n,
\label{javiermasmarikondeplaya}$$ the ${\bf e}_j$, $j=1,\ldots n$, being an orthonormal basis in $\mathbb{R}^n$. The simple root $\alpha_j={\bf e}_j-{\bf e}_{j+1}$ can be understood as corresponding to a string connecting the D–branes $j$ and $j+1$ within the stack of $n$ coincident D–branes. Nonsimple, positive roots such as, [*e.g.*]{}, $\beta=\alpha_j+\alpha_{j+2}$, correspond to strings connecting nonadjacent D–branes; negative roots correspond to oppositely oriented strings. (The strings themselves are stretched only when the corresponding D–branes are separated, thus breaking the $\mathfrak{su}(n)$ gauge symmetry to a subalgebra [@WITTEN]). Separating now the $n$–th D–brane from the remaining $n-1$ coincident D–branes reduces the gauge symmetry down to $\mathfrak{su}(n-1)\times\mathfrak{u}(1)$; this corresponds to eliminating the simple root $\alpha_{n-1}$. In this process the $\mathfrak{su}(n)$ generators $e_{\pm\alpha_{n-1}}$ are removed, but not so their diagonal commutator $h_{\alpha_{n-1}}=[e_{\alpha_{n-1}},e_{-\alpha_{n-1}}]$, which remains as the generator of the $\mathfrak{u}(1)$ corresponding to the separated brane. Further separating out more branes from the stack one can reduce this matrix dynamics all the way down to $\mathfrak{u}(1)^{\times n}$.
In the given representation we can arrange to have $e_{\alpha_j}^{\dagger}=e_{-\alpha_j}$ for all $j=1,\ldots, n$. That is, the adjoint of the generator $e_{\alpha_j}$, with $\alpha_j$ a simple root, is the generator $e_{-\alpha_j}$ corresponding to the opposite root. Within ${\cal U}(\mathfrak{su}(n))$ let us consider the $2n-2$ selfadjoint matrices defined as $$H_j^{(\pm)}:=\frac{1}{2}\sum_{l=1}^{j}\left(e_{\alpha_l}e_{-\alpha_l}\pm e_{-\alpha_l}e_{\alpha_l}\right), \qquad j=1, \ldots, n-1.
\label{ramallocasposoquetedenpurculo}$$ We claim that the $H_j^{(\pm)}$ play the role of selfadjoint Hamiltonian operators (matrices) for the dynamics (\[ramallomechupalapolla\]) describing the coordinates transverse to a stack of $n$ coincident D$p$–branes. In order to justify our claim we first recall that the integral (\[ramallomechupalapolla\]) does not extend over the time coordinate, because time is longitudinal. Hence the canonical Hamiltonian that one would naively construct out of (\[ramallomechupalapolla\]), and the corresponding Poisson brackets, are meaningless. We need a geometric, Lie–algebraic prescription to give the dynamics (\[ramallomechupalapolla\]) a meaning. Let us further recall that $\mathfrak{su}(n)$ has the nontrivial cohomology cocycles $\omega_3, \omega_5$, $\ldots$, $\omega_{2n-1}$ [@KNAPP]. The cohomology of the corresponding Lie group, $SU(n)$, is a product of spheres, $S^3\times S^5\times \ldots\times S^{2n-1}$ [@GOLDBERG]. Any such sphere $S^{2j-1}$, for $j=2,\ldots, n$, is the submanifold of $\mathbb{R}^{2j}$ defined by $$\sum_{l=1}^{2j}(x^l)^2=1.
\label{barbonmatrikonquetelametanporkulo}$$ Pairwise grouping the $2j$ real coordinates $x^l$ into $j$ complex ones $z^l$ on $\mathbb{C}^{j}$, (\[barbonmatrikonquetelametanporkulo\]) becomes $$\sum_{l=1}^{j} z^l \bar z^l=1,
\label{labastidamechupalapolla}$$ which we write more suggestively as $$\frac{1}{2}\sum_{l=1}^{j}\left( z^l \bar z^l+\bar z^l z^l\right)=1.
\label{ramalloquetelametanporculo}$$ Identifying the generator $e_{\alpha_l}$ with the complex variable $z^l$ and $e_{-\alpha_l}$ with its complex conjugate $\bar z^l$, the $n-1$ operators $H_j^{(+)}$ have a clear geometric origin. The remaining $n-1$ matrices, given by $H_j^{(-)}$, are linearly independent of the $H_j^{(+)}$. The $H_j^{(-)}$ actually equal a sum of the diagonal Cartan generators for the $\mathfrak{su(2)}$ subalgebras within $\mathfrak{su(n)}$, but this fact is immaterial to what follows. We will presently provide a physical interpretation for the appearance of 2 Hamiltonians, $H_j^{(+)}$ and $H_j^{(-)}$, for each value of $j=1, \ldots, n-1$. Altogether the $2n-2$ operators $H_j^{(\pm)}$ form a linearly independent set of selfadjoint matrices within ${\cal U}(\mathfrak{su}(n))$.
We can write down a classical evolution equation for $\mathfrak{su}(n)$ involving all the Hamiltonians (\[ramallocasposoquetedenpurculo\]). For this we consider the top cocycle $\omega_{2n-1}$, whose Poisson multibrackets involve $2n-2$ entries and set, for any $F\in{\cal U}(\mathfrak{su}(n))$, $$\dot F = \left\{F, H_{1}^{(+)}, H_{1}^{(-)}, H_{2}^{(+)}, H_{2}^{(-)}\ldots, H_{n-1}^{(+)}\right\}_{\omega_{2n-1}}.
\label{cesargomezmekagoentuputakaramarikondeplaya}$$ We observe that the final entry above is $H_{n-1}^{(+)}$, while $H_{n-1}^{(-)}$ is missing. In fact we could just as well write $$\dot F = \left\{F, H_{1}^{(+)}, H_{1}^{(-)}, H_{2}^{(+)}, H_{2}^{(-)}\ldots, H_{n-1}^{(-)}\right\}_{\omega_{2n-1}}.
\label{alvarezgaumekaka}$$ Thus we have two independent evolution equations, (\[cesargomezmekagoentuputakaramarikondeplaya\]) and (\[alvarezgaumekaka\]), that we can regard as corresponding to the two different orientations that the top cohomology cocycle $S^{2n-1}$ can have. This makes sense since, in the absence of a continuously flowing time parameter, there is no canonical choice of an orientation; the latter has to be determined geometrically. All other (lower–dimensional) cocycles are represented within (\[cesargomezmekagoentuputakaramarikondeplaya\]) and (\[alvarezgaumekaka\]) in their two possible orientations. To summarise, we have the classical equations of motion: $$\dot F = \left\{F, H_{1}^{(+)}, H_{1}^{(-)}, H_{2}^{(+)}, H_{2}^{(-)}\ldots, H_{n-1}^{(\pm)}\right\}_{\omega_{2n-1}}.
\label{labastidacasposoquetedenpurculo}$$ Upon quantisation, the above becomes $$\dot F = \left\{F, H_{1}^{(+)}, H_{1}^{(-)}, H_{2}^{(+)}, H_{2}^{(-)}\ldots, H_{n-1}^{(\pm)}\right\}_{\omega_{2n-1}}^{\star}.
\label{cesargomezmekagoentuputakaramarikonazodemierda}$$
Discussion {#mierdaparatodos}
==========
Apparently there is nothing compelling about the stack of $n$ coincident D–branes that forces one to describe its mechanics using Poisson multibrackets of [@BUENO]. The Lie algebra $\mathfrak{su}(n)$ arises naturally when superimposing $n$ D–branes, but the equations of motion we have written down have an algebraic origin in the Lie algebra cohomology, that is apparently independent of any branes whatsoever. After all one could just as well continue to use the standard binary Poisson brackets, with the quadratic Casimir of the Lie algebra as the Hamiltonian. Nothing seems to require more than one Hamiltonian, [*i.e.*]{}, more than one generator of translations along a timelike coordinate.
However, the time coordinate itself is parallel to the brane, so all transverse coordinates are spacelike. In particular there is no transverse time to a D–brane. Transverse coordinates to a brane are all matrix–valued and all spacelike. The goal we set out to achieve was the description of the [*transverse*]{} directions to a brane. So, if the Hamiltonian is the generator of time translations, with time being longitudinal, either there is no Hamiltonian at all, or there is no reason to restrict to just one Hamiltonian. In this article we adopt this latter point of view. This opens up many possibilities for evolution equations, now that time evolution becomes an algebraic property instead of a smooth evolution along a distinguished, continuous parameter.
The lesson we learn is that evolution equations for transverse, Lie–algebra valued coordinate functions such as those considered here are determined by the gauge symmetry present in the branes, rather than by the [*coordinate*]{} aspect of those coordinate functions. In other words, the [*Lie–algebra*]{} aspect prevails over the [*coordinate*]{} aspect. This is in accord with branes as worldvolumes for (supersymmetric) gauge theories, at least at low energies [@SZABO]. The corresponding dynamics must therefore take this fact into account; the Poisson multibrackets considered here do precisely that. Last but not least, matrix–valued coordinate functions provide an interesting example of noncommutative geometry [@CONNES; @LANDI].
[**Acknowledgements**]{} J.M.I. thanks Max–Planck–Institut für Gravitationsphysik, Albert–Einstein–Institut (Golm, Germany) for hospitality during the preparation of this article. This work has been supported by Generalitat Valenciana (Spain).
[99]{}
J. Marsden and T. Ratiu, [*Introduction to Mechanics and Symmetry*]{}, Springer, Berlin (1998).
A. Kirillov, [*Elements of the Theory of Representations*]{}, Springer, Berlin (1976).
A. Knapp, [*Lie groups, Lie Algebras and Cohomology*]{}, Princeton University Press, Princeton (1988).
S. Goldberg, [*Curvature and Homology*]{}, Dover, New York (1982).
J. de Azcárraga, J. Izquierdo and J. Pérez Bueno, [*An Introduction to Some Novel Applications of Lie Algebra Cohomology and Physics*]{}, Rev. R. Acad. Cien. Exactas Fis. Nat. Ser. A Mat. [**95**]{} (2001) 225.
H. Grönewald, [*On the Principles of Elementary Quantum Mechanics*]{}, Physica [**12**]{} (1946) 405.
J. Moyal, [*Quantum Mechanics as a Statistical Theory*]{}, Proc. Camb. Phil. Soc. [**45**]{} (1949) 99.
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, [*Deformation Theory and Quantization, I: Deformations of Symplectic Structures*]{}, Ann. Phys. (NY) [**111**]{} (1978) 61.
M. Kontsevich, [*Deformation Quantization of Poisson Manifolds, I*]{}, Lett. Math. Phys. [**66**]{} (2003) 157.
G. Giachetta, L. Mangiarotti and G. Sardanashvily, [*Geometric and Algebraic Topological Methods in Quantum Mechanics*]{}, World Scientific, Singapore, (2005).
R. Szabo, [*An Introduction to String Theory and D–Brane Dynamics*]{}, Imperial College Press, London (2004).
E. Witten, [*Bound States of Strings and p-Branes*]{}, Nucl. Phys. [**B460**]{} (1996) 335.
Y. Nambu, [*Generalized Hamiltonian Dynamics*]{}, Phys. Rev. [**D7**]{} (1973) 2405.
T. Curtright and C. Zachos, [*Classical and Quantum Nambu Mechanics*]{}, Phys. Rev. [**D68**]{} (2003) 085001.
J. Hoppe, [*On M Algebras, the Quantization of Nambu Mechanics, and Volume Preserving Diffeomorphisms*]{}, Helv. Phys. Acta [**70**]{} (1997) 302.
T. Banks, W. Fischler, S. Shenker and L. Susskind, [*M Theory as a Matrix Model: a Conjecture*]{}, Phys. Rev. [**D55**]{} (1997) 5112.
J. Humphreys, [*Introduction to Lie Algebras and Representation Theory*]{}, Springer, Berlin (1994).
A. Connes, [*Noncommutative Geometry*]{}, Academic Press, London (1994).
G. Landi, [*An Introduction to Noncommutative Spaces and their Geometry*]{}, Springer Lecture Notes in Physics [**51**]{}, Springer, Berlin (1997).
|
---
abstract: 'Multiple analyses from ATLAS and CMS collaborations, including searches for ttH production, supersymmetric particles and vector-like quarks, observed excesses in the same-sign dilepton channel containing $b$-jets and missing transverse energy in the LHC Run 1 data. In the context of little Higgs theories with T parity, we explain these excesses using vector-like T-odd quarks decaying into a top quark, a $W$ boson and the lightest T-odd particle (LTP). For heavy vector-like quarks, decay topologies containing the LTP have not been searched for at the LHC. The bounds on the masses of the T-odd quarks can be estimated in a simplified model approach by adapting the search limits for top/bottom squarks in supersymmetry. Assuming a realistic decay branching fraction, a benchmark with a 750 GeV T-odd $b^\prime$ quark is proposed. We also comment on the possibility to fit excesses in different analyses in a common framework.'
author:
- 'Chuan-Ren Chen$^{\, a}$, Hsin-Chia Cheng$^{\, b}$ and Ian Low$^{\, c,d}$'
title: 'Same-Sign Dilepton Excesses and Vector-like Quarks'
---
Introduction {#sec:intro}
============
It was recently pointed out that both ATLAS and CMS observed in their Run 1 data mild excesses in the same-sign dilepton (SS2l) channel, which contains final states with same-sign dilepton, $b$-jets, and missing transverse energy ($E_{\rm T}^{\rm miss}$) [@Huang:2015fba]. The excesses were observed in five semi-independent analyses, three from the ATLAS [@Aad:2014pda; @Aad:2015gdg; @atlastth] and two from the CMS [@Khachatryan:2014qaa; @Chatrchyan:2013fea], using different background subtraction methods. In particular, the ATLAS exotica search [@Aad:2015gdg] and the CMS ttH Higgs search [@Khachatryan:2014qaa] reported about $2\sigma$ and $2.5\sigma$ significance, respectively, for the excesses. More recently, measurements on the standard model (SM) process ${\rm t\bar{t}W}$ from ATLAS [@Aad:2015eua] and CMS [@Khachatryan:2015sha] have also reported seeing excesses above the SM expectation in the SS2l channel.
A supersymmetric interpretation of the SS2l excess was put forward in Ref. [@Huang:2015fba], which proposed using top squarks (stops) or bottom squarks (sbottoms) decaying into two top quarks, two $W$ bosons, and $E_{\rm T}^{\rm miss}$ to explain the excess. However, it is worth recalling that SS2l, $b$-jets and $E_{\rm T}^{\rm miss}$ are generic signatures of many beyond-the-SM theories and, therefore, not unique to supersymmetry. In fact, it is well-known that many scenarios could “fake" supersymmetry at the LHC, due to the complex environment of $pp$ collision [@Cheng:2002ab]. For example, decays of the stop could be mimicked by that of a fermionic top partner, unless one can measure the spin of the new particles involved in the decay chain, which is a challenging measurement at the LHC [@Wang:2008sw]. Indeed, should the SS2l excess be confirmed at the LHC Run 2, it would be of paramount importance to determine the specific quantum number of the new particles associated with the excess.
In this study we would like to proceed in an exploratory spirit that is appropriate for this nascent subject, and consider alternative possibilities, other than supersymmetry, for the SS2l excess. The simplest possibility is to invoke a new conserved quantum number at the TeV scale, the T parity, under which the SM particles are neutral and the new particles are charged [@Cheng:2003ju]. Collider phenomenology of T parity is very similar to that of R parity in supersymmetry [@Martin:1997ns] and KK parity in Universal Extra Dimensions [@Appelquist:2000nn], in that all T-odd particles can only be pair-produced and subsequently cascade-decay into SM particles plus the lightest T-odd particle (LTP), which carries away extra $E_{\rm T}^{\rm miss}$ in collider detectors.
This work is organized as follows. In Section \[sec:Tparity\] we give a brief overview of T parity and an estimate on the collider bound on the mass of T-odd quarks, followed by Section \[sect:ss2llhc\] which discusses fitting the SS2l excess in the context of SM ttH searches at the LHC Run 1 and beyond. In Section \[sect:broadpic\] we consider whether it is possible to explain using a common benchmark the excesses observed by analyses outside of the SM ttH searches. Then we provide an outlook in Section \[sect:outlook\].
A Simplified Overview of T parity {#sec:Tparity}
=================================
In this section we provide a “simplified" overview of T parity and consider collider bounds on T-odd quarks.
Simplified T Parity
-------------------
The case for a new symmetry at the TeV scale, under which all SM particles are neutral, is motivated by two considerations [@Cheng:2003ju]:
- Precision Electroweak Constraints: If we parameterize flavor-conserving new physics in terms of higher-dimensional operators, precision electroweak constraints indicate that the mass scale suppressing these operators tend to be at around 5 – 10 TeV, assuming all dimensionless coefficients to be order unity [@Barbieri:1999tm]. This is referred to as the little hierarchy problem, since naturalness principle expects new physics at around 1 TeV scale to stabilize the Higgs mass. If there exists a new symmetry at the TeV scale such that all SM particles are neutral while all new particles are charged under the new symmetry, then new particles would enter into precision electroweak observables only through loop-induced effects, thereby introducing a loop factor of about $1/16\pi^2$ in front of the higher dimensional operators. This allows one to lower the scale of new physics down to below 1 TeV.
- Dark Matter: The existence of dark matter calls for (at least) a new particle that is electrically neutral and stable at the cosmological time scale. A simple possibility is to make the dark matter particle absolutely stable. This can be achieved again by postulating a new symmetry such that all SM particles are neutral. Then the lightest particle charged under the new symmetry will be stable, assuming the symmetry is exact, or almost exact so that the particle is stable cosmologically. If such a particle is also electrically neutral, it can be a dark matter candidate.
There are many realizations of such a new TeV symmetry in explicit models. The most popular models use the simplest symmetry: a new parity ($Z_2$) symmetry. R parity in supersymmetry [@Martin:1997ns], KK parity in Universal Extra Dimensions [@Appelquist:2000nn] and warped extra dimension [@Randall:1999ee; @Agashe:2007jb], and T parity in little Higgs models [@ArkaniHamed:2001nc; @Cheng:2003kk] all fall into this simple category. Nevertheless, larger symmetry groups such as the $Z_3$ group have also been employed [@Agashe:2004ci].
It is possible to adopt the “simplified model" approach [@Alves:2011wf], by postulating the existence of a new parity, without committing to a specific model realization. In particular, one could assume there is a “simplified T parity," under which all SM particles are neutral and all new particles are charged. Then at colliders any new particles must be pair-produced, in order to conserve the simplified T parity, and eventually cascade-decay into the LTP. The LTP would serve as the dark matter candidate, assuming it is electrically neutral, and manifest itself as $E_{\rm T}^{\rm miss}$ in colliders. Phenomenology of the simplified T parity at the LHC is very similar to that of supersymmetry with R parity, which is characterized by leptons, jets and $E_{\rm T}^{\rm miss}$.[^1]
In the end, many collider signatures of R parity conserving SUSY can be mimicked by simplified T parity, which allows more freedom in choosing the spin quantum number of the mother and intermediate particles in the decay chain. For example, decays of bottom squarks in SUSY, \[eq:sbottom\] \_1 t + (\^-\_1 W\^- + \_1\^0) , can be impersonated in simplified T parity by a T-odd vector-like quark with the same quantum number as the bottom quark: \[eq:bprime\] b\^t + ( W\_H\^[-]{} W\^- + A\_H ) , where $W_H^\pm$ is a pair of heavy charged vector boson and $A_H$ is a heavy neutral vector boson that is the LTP. For the SS2l excess, the bottom squark decay chain Eq. (\[eq:sbottom\]) was discussed to some extent in Ref. [@Huang:2015fba], which focused on supersymmetric theories. In this work we consider the alternative decay chain in Eq. (\[eq:bprime\]) based on the (simplified) T parity. While both models can produce SS2l events, the kinematic distributions in general are different, and might be used to distinguish different models. Of course, to fully identify the decay chain requires measuring the spin of the mother and intermediate particles [@Wang:2006hk], which would be a top priority should the excess be confirmed at the LHC.
Explicit constructions of the T-odd top and bottom partners in little Higgs models were given in Ref. [@Cheng:2003kk; @Cheng:2005as; @Hubisz:2005tx]. It was shown that in order to cut off the contributions to the standard model four-fermion interactions from the Goldstone boson loop, there should be a vector-like T-odd doublet partner for every standard model fermionic doublet. In these models the branching fraction of the desired decay chain in Eq. (\[eq:bprime\]) is typically not 100%, in contrast to the simplified model approach. Because the dominant contributions to the masses of the T-odd vector bosons are $SU(2)_L\times U(1)_Y$ preserving, the corresponding mixing between the T-odd partners of the neutral gauge bosons due to the electroweak breaking is small. As a result, $A_H$ is mostly the partner of the Standard Model hypercharge gauge boson $B_\mu$ and $Z_H$ is mostly the partner of the $SU(2)_L$ gauge boson $W_\mu^3$. In this case, the decay branching fractions of the T-odd $b^\prime$ and $t^\prime$ are more or less determined by the Goldstone equivalence theorem [@Belyaev:2006jh]: Br(b\^tW\_H) : Br(b\^bZ\_H)& & 2:1\
Br(t\^bW\_H) : Br(t\^tZ\_H) && 2:1 . The other decay channels, $b^\prime \to b A_H$ and $t^\prime \to t A_H$, are subdominant. These observations will be taken into account when we consider model-specific T-odd quark decays.[^2]
Collider Bounds on Third Generation T-odd Quarks
------------------------------------------------
Although there are many searches for stops and sbottoms in SUSY at the LHC Run 1, there hasn’t been many dedicated searches on the closely related decay chains involving T-odd quarks.[^3] In particular, the stop and sbottom searches usually adopt the simplified model approach, by assuming 100% decay branching ratio (BR) into the final states being searched for. Current limits from the LHC Run 1 data on the stop and bottoms are typically around 500 – 700 GeV [@Aad:2015pfx].
Since the decays of T-odd quarks often give the same final state signatures as the squarks, in this subsection we will provide a rough estimate on the bounds on the masses of T-odd quarks, based on the experimental searches for third generation squarks. Recasting the search limits on squarks is not a straightforward task, as the signal selection efficiencies depend non-trivially on the kinematics of the decay products [@Low:2013aza], which in turn is determined by the mass spectrum of the mother and daughter particles. For example, the strong limits on the sbottom mass in the tripleton channel of the decay chain in Eq. (\[eq:sbottom\]) varies significantly when the lightest neutralino \[which is assumed to be the lightest supersymmetric particle (LSP)\] and chargino masses change [@cmstrilepton]. Moreover, the bound disappears completely when the LSP becomes heavier than 170 – 240 GeV, depending on the chargino mass. It is clear that a full-fledge study on the experimental constraints of T-odd quarks requires dedicated efforts and is beyond the scope of this paper.
In this work we will settle for a naïve estimate on the experimental bounds on third generation T-odd quarks by translating the [*strongest*]{} limits on the third generation squark masses into upper bounds on the production cross-sections, and then computing the $b^\prime$ and $t^\prime$ masses that give rise to the same production cross-section as the upper bounds. The outcome based on the simplified model assumption is shown in Fig. \[fig:1\], where the squark and T-odd quark cross sections are quoted from the LHC SUSY Cross-section Working Group [@Kramer:2012bx; @Borschensky:2014cia] and from [HATHOR]{} [@Moch:2008ai; @Langenfeld:2009tc], respectively. We see that the lower bound on the $b^\prime$ and $t^\prime$ quarks are about 800 GeV and 825 GeV, respectively. The bounds in Fig. \[fig:1\] are conservative in the sense that they correspond to the most stringent limits on the $m_{\rm LSP}$ – $m_{\tilde{b}_1/{\tilde{t}_1}}$ plane, which would loosen when the $m_{\rm LSP}$ becomes larger.
Beyond the simplified model approach, the exclusion limit degrades quickly as the decay BR decreases from the assumed 100%, because the signal strength is usually proportional to the square of the BR due to the assumption of pair-production of the mother particles. For example, in the littlest Higgs with T parity model (LHT) [@Cheng:2003kk; @Belyaev:2006jh], the decay BR of T-odd $b^\prime$ into $tW_H$ final state is typically about 55% at around 800 GeV, as shown in Fig. \[fig:2\]. Consequently, the collider bounds on the T-odd quarks are considerably weaker in a full model than in the simplified model approach. This comparison is shown in Fig. \[fig:3\], where we compared the bound assuming 100% BR versus 55% for both T-odd $b^\prime$ and $t^\prime$. The bounds on the $b^\prime$ ($t^\prime$) mass is only about 680 (700) GeV.
![\[fig:2\]*Decay BR of T-odd $b^\prime$ quark in the littlest Higgs theory with T parity. As the $b^\prime$ becomes heavy, the decay BR into $bZ_H$ gets close to 1/2 of the BR into $tW_H$, as predicted by the Goldstone equivalence theorem. The remaining decay BR to the $bA_H$ final state is quite small.*](lhtb_decay.pdf){width="3.35in"}
Same-Sign Dilepton Signals at the LHC {#sect:ss2llhc}
=====================================
Having estimated the LHC bounds on third generation T-odd quarks, in this Section we consider an interpretation of the SS2l excess from the pair-production of T-odd quarks $b^\prime$. In particular, the decay chain we concentrate on is pp b\^|[b]{}\^(tW\^-\_H)(|[t]{}W\^+\_H) (tW\^-A\_H) (|[t]{}W\^+A\_H) , which gives $2b+4W+E_{\rm T}^{\rm miss}$ final states and contributes to the SS2l signal region.
Following the strategy in Ref. [@Huang:2015fba], we base our numerical simulations on the selection cuts implemented in the CMS ttH analysis in Ref. [@Khachatryan:2014qaa] and then normalize the $b^\prime$ signal strength to the SM ttH signal strength. More specifically, we generate both the T-odd $b^\prime$ pair production and SM ttH with [Madgraph/MadEvent]{} [@Alwall:2007st], pass the events through [Pythia]{} [@Sjostrand:2006za] for showering and then to [Delphes]{} [@deFavereau:2013fsa] for detector simulations. The particular benchmark we study has the mass spectrum motivated by the LHT model [@Cheng:2003kk; @Belyaev:2006jh]: m\_[b\^]{} = 750 [ GeV]{} , m\_[W\_H]{} = 320 [ GeV]{} , m\_[A\_H]{} = 66 [ GeV]{} . The production cross section of the 750 GeV $b'$ pair at the 8 TeV LHC is 34.1 fb. The relevant decay branching fraction is BR$(b^\prime \to tWA_H)\approx 55$%. The selected events are required to have exactly two same-sign leptons, at least 4 jets among which two are $b$-jets. Following Ref. [@Khachatryan:2014qaa] we impose further the following kinematic cuts: \[eq:cmscuts\] p\^\_T > 20 [GeV]{} , p\^j\_T > 25 [GeV]{} ,> 30 [GeV]{} , S\_T >100 [GeV]{} , where $S_T=p^{\ell_1}_T+p^{\ell_2}_T+E_{\rm T}^{\rm miss}$ is the scalar sum of transverse momentum of two charged leptons and $E_{\rm T}^{\rm miss}$ and $\rm L_D=0.6\times E_{\rm T}^{\rm miss} + 0.4\times H^{miss}_T$ with $\rm H^{miss}_T$ being the negative vector $p_T$ sum of selected jets and two same-sign leptons. Numbers of after-the-cut events from $b^\prime$ and SM ttH are then used to calculate the ratio of signal strength $\mu_{b^\prime}/\mu_{\rm ttH}$. In the end, the total signal strength is = \_[b\^]{} + \_[ttH]{} = 2.0 , which is to be compared with the ATLAS result $\mu = 2.8_{-1.9}^{+2.1}$ [@atlastth] and the CMS fit $\mu = 5.3_{-1.8}^{+2.1}$ [@Khachatryan:2014qaa]. If we consider the “simplified T parity model," namely the 100% branching fraction of $b^\prime \to t W_H$, a $b^\prime$ with a mass of $850$ GeV and production cross-section 12.85 fb will generate the similar signal strength.
It is interesting to compare with the SUSY benchmark considered in Ref. [@Huang:2015fba]. There the spectrum was taken to be: m\_[\_1]{} = 550 [ GeV]{} , m\_[\^0\_2]{} = 340 [ GeV]{} , m\_[\_1\^]{} m\_ =260 [ GeV]{} . The pair production cross section of 550 GeV $\tilde{t}_1$ is 45.2 fb at 8 TeV and the branching fraction of $\tilde{t}_1\to tW^\pm \tilde{\chi}^\mp_1$ is close to 100%. The total signal strength is $\mu= \mu_{\tilde{t}_1} + \mu_{\rm ttH} = 2.83$. We see that the selection efficiency on the $2b+4W+E_{\rm T}^{\rm miss}$ final states for our $b'$ benchmark model is about 2.4 times the selection efficiency of the SUSY benchmark model of Ref. [@Huang:2015fba]. This is because the final state particles in our benchmark model are much more energetic due to the heavy mother particle mass as well as the large splitting in the spectrum, and hence are easier to pass the cuts.
![\[fig:4-1\]*Ratios of the cross sections for $b^\prime$ and stops/sbottoms at 13 TeV over 8 TeV. The increase in the SM ttH cross section is a factor of 3.9.* ](ratio_13to8.pdf){width="3.35in"}
At the LHC Run 2 with the same cuts imposed in Eq. (\[eq:cmscuts\]), the total signal strength for the $b'$ benchmark model increases to $\mu = 3.2$. The reason of the increase is similar to the situation in the stop interpretation: the production cross-section for the heavy particles grows at a faster pace than that of SM ttH [@Huang:2015fba]. The increase is also more significant than that of the SUSY benchmark model \[which has $\mu(13\mbox{ TeV}) =3.69$\]. In Fig. \[fig:4\] we show the production cross sections for the T-odd $b^\prime$ and the stop/sbottom in SUSY at 8 TeV and 13 TeV LHC, while Fig. \[fig:4-1\] shows the ratios of the production cross sections at 13 TeV LHC over the 8 TeV LHC. In comparison, the increase in the SM ttH cross section is only a factor of 3.9 in going from 8 TeV to 13 TeV at the LHC. Fig. \[fig:4\] also allows for a simple scaling of cross section should one be interested in increasing (decreasing) the signal strength using a lighter (heavier) mass for the $b^\prime$ or stop/sbottoms. For example, assuming the signal acceptance stays roughly the same, a T-odd $b^\prime$ at around 620 GeV and a decay branching of 55% into $tW_H$ could give rise to a total signal strength $\mu\approx 4$ at 8 TeV, in unit of the SM ttH signal strength. The corresponding $b^\prime$ mass for $\mu\approx 4$ in the simplified T parity model is about 720 GeV.
It will also be interesting to contemplate further kinematic cuts at the LHC Run 2 that could help disentangle the $b^\prime$ signal from that coming from the SM ttH or the stop/sbottom SUSY signals. In this regard, we show in Fig. \[fig:kin\] some kinematic distributions of events from the SM ttH, the T-odd $b^\prime$ and the stop in SUSY. One sees that $b^\prime$ has the hardest spectra among the three benchmarks, which is due to the heaviness of the $b^\prime$ in the benchmark, resulting in more energetic decay products. This feature is quite generic, since the fermion has a significantly larger cross section than the scalar at the same mass. Therefore, given a particular signal strength, one can always fit it with a fermion mass that is heavier than that of a scalar. As long as the spectrum is not degenerate, it will result in harder distributions of the decay products.
Based on the above observation, we can consider the following additional cuts on the leading jet $p_T$, $E_{\rm T}^{\rm miss}$ and $S_T$, \[eq:hardcuts\] p\^[leading jet]{}\_T > 100 [GeV]{} , E\_[T]{}\^[miss]{} > 150 [GeV]{} , S\_T >200 [GeV]{}, to enhance the contribution of $b^\prime$ to SS2l signal region in the context of the CMS SM ttH analysis. With these further cuts, the total signal strength, in unit of the SM ttH strength, raises dramatically to 19 . In other words, the signal would come almost exclusively from T-odd $b^\prime$.
A Broader Picture {#sect:broadpic}
=================
So far we have focused on fitting the SS2l excess in the context of the ttH multilepton analyses [@atlastth; @Khachatryan:2014qaa], by normalizing to the SM ttH signal strength. Given that several other analyses [@Aad:2014pda; @Aad:2015gdg; @Chatrchyan:2013fea], outside of the Higgs searches, have also observed excessive events in the SS2l category, it is worth exploring whether the T-odd $b^\prime$, or even the supersymmetric stop/sbottom, can simultaneously explain the excesses observed in these other analyses. Since these semi-independent analyses utilize different background subtraction methods and signal selection cuts, $b^\prime$ and sbottoms/stops models can have varied responses to these analyses. A detailed study on the consistency of these excesses is best carried out by the experimental collaborations. Here we will be content with some crude estimates based on publicly available information provided by the experimental collaborations, as well as our own Monte Carlo simulations. Given that the number of events in the excess of each analysis is small, it is unlikely that one can draw definite conclusions from the comparisons right now. However, it gives us a flavor on what can be done in distinguishing different models from analyses based on different selection criteria if the excesses are confirmed at 13 TeV LHC with a larger luminosity.
CMS SUSY Analysis
-----------------
The SS2l analysis in the CMS SUSY working group in Ref. [@Chatrchyan:2013fea] presented exclusion limits on the sbottom decay chain pp\_1\^\* \_1 (tW\^-\^0\_1)(|[t]{}W\^+\^0\_1) \[eq:sbottom-pair\] that are degraded from the expected limits, implying more events were observed than expected. The most significant excess appeared in the signal region SR24, which requires SS2l and N\_[b-jets]{}2 ,N\_[jets]{}4, 50 [GeV]{} E\_[T]{}\^[miss]{} 120 [GeV]{}, H\_[T]{}400 [GeV]{}. The expected number of events in SR24 is 4.4$\pm 1.7$ (2.8$\pm 1.2$) in the low (high) $p_T$ region and the observed numbers are 11 (7).
We simulated the contribution to SR24 from both the $b^\prime$ and the stop benchmarks. The $b^\prime$ (stop) would give rise to 0.4 (3) and 0.3 (2.3) events in the low and high $p_{\rm T}$ region, respectively. It is clear that T-odd $b^\prime$ has a lot more difficulty fitting the SR24 excess than the stop benchmark. This is due to the fact that SR24 only selects events with a relatively small $E_{\rm T}^{\rm miss}$, while the $E_{\rm T}^{\rm miss}$ distribution from $b^\prime$ benchmark is much harder than that from the stop, as can be seen from Fig. \[fig:kin\].
It is also interesting to note that the $b$-tagging selection efficiency is typically 70%. As a result, events contributing to $N_{\rm b-jets}=2$ region will also contribute to $N_{\rm b-jet}=1$ region. Therefore both the $b^\prime$ and the stop benchmarks should contribute to SR14, the $N_{\rm b-jet}=1$ cousin of SR24. However, SR14 in Ref. [@Chatrchyan:2013fea] observed 6 events, which is less than the expected number of 10$\pm 4$ events. In addition, a model that could generate the excess events in the SR24 region may also produce events in other signal regions. For example, both the $b'$ and the SUSY benchmarks have significant portions of their $E_{\rm T}^{\rm miss}$ distributions beyond the upper limit 120 GeV of the SR24 region. They are expected to show up also in the signal regions SR18 and SR28 which are similar to SR14 and SR24 but requiring $E_{\rm T}^{\rm miss}>120$ GeV. The SR18 region ($N_{\rm b-jets}=1$) does have a small excess ($6.8\pm 2.5$ expected and 11 observed) but no excess ($3.4\pm 1.3$ expected and 3 observed) is seen in SR28 ($N_{\rm b-jets}\geq2$). Clearly, a consistent picture has not emerged from the current CMS SS2l SUSY search data. Of course, we are considering a very low number statistics, of the order of five signal events or less, so we should not attempt to over-fit the current data. More data from LHC Run 2 could certainly help to clarify the situation.
ATLAS SUSY Analysis
-------------------
The ATLAS SS2l search in the SUSY group [@Aad:2014pda] also searched for sbottom decays as in Eq. (\[eq:sbottom-pair\]). The signal region SR1b, which requires SS2l and[^4] N\_[b-jets]{}1, N\_[jets]{}3, E\_[T]{}\^[miss]{}> 150 [GeV]{},m\_[T]{}>100 [GeV]{}, m\_[eff]{}> 700 [GeV]{} , expects 4.7$\pm$2.1 events and observed 10 events with a $p$-value of 0.07, while the SR3b region expects $2.2\pm0.8$ and observed 1 event.
We implemented the above cuts in both the $b^\prime$ and the stop benchmarks, which contributed 3.7 and 3.1 events, respectively, to the SR1b signal region. Therefore, the excess observed by the ATLAS SUSY search could be explained by both the $b^\prime$ and the stop benchmarks. The selection efficiency for the $b^\prime$ benchmark is much better than that from the stop because of the hard cuts on $E_{\rm T}^{\rm miss}$ and $H_{\rm T}$, which can be seen from the distributions shown in Fig. \[fig:kin\].
It is interesting to note that the $E_{\rm T}^{\rm miss}$ cuts in the CMS SR24 and ATLAS SR1b regions of the SUSY analyses are mutually exclusive. On the other hand, the CMS ttH analysis made public the $E_{\rm T}^{\rm miss}$ distributions of the SS2l events in the public twiki page [@twiki], which exhibit broad excesses over a wide range of $E_{\rm T}^{\rm miss}$, including those overlapping with the CMS and ATLAS SUSY analyses. Again, should the excess be confirmed, the $E_{\rm T}^{\rm miss}$ distribution of the events would be a key observable to pay attention to.
ATLAS Exotica Analysis
----------------------
The ATLAS search for heavy vector-like quarks in the SS2l + 2 $b$-jets region reported an excess in the SR4t3 signal region, which is defined by requiring SS2l and N\_[b-jets]{}=2, E\_[T]{}\^[miss]{} 100 [GeV]{}, H\_[T]{}700 [GeV]{} . In this signal region $4.4\pm 1.1\pm 1.1$ events are expected and 12 are observed. There is also a separate excess in SR4t4, the 3 $b$-jets category, which sees 6 events while expecting only $1.1\pm 0.9 \pm 0.4$ events. The $p$-values for both excesses are about 2$\sigma$. However, recall that the ATLAS SUSY search in Ref. [@Aad:2014pda] sees no excess in SR3b, the three $b$-jets region.
Implementing the above selection cuts in our simulation, we find that the $b^\prime$ benchmark contributes about 2.3 events to SR4t3, while the stop benchmark contributes about 1.8 events. Again $b^\prime$ contributes more to the signal region because it has a harder decay spectrum.
Summary and Outlook {#sect:outlook}
===================
In this work we studied the possibility of fitting the SS2l excess observed in the LHC Run 1 data by vector-like quarks that are odd under a new parity at the TeV scale, the T parity. Phenomenology of T-odd quarks is quite different from those heavy quark decay chains that are being searched for at the LHC so far, since the T-odd quarks decays into SM particles plus the LTP, which carries away extra missing transverse energy at collider detectors. The current bounds on the masses of the third generation T-odd quarks can be estimated by recasting the exclusion limits for stops and sbottoms in supersymmetry, which have similar decays chains to the third generation T-odd quarks,.
We proposed a vector-like quark benchmark containing a 750 GeV $b^\prime$ quark, decaying 55% of the time into a SM top quark and a 320 GeV $W_H$. The $W_H$ subsequently decays into the SM $W$ boson and the LTP, whose mass is at 66 GeV. Using the ttH searches as a starting point, we normalized the $b^\prime$ signal strength to that of the SM ttH expectation and obtained a total signal strength of $\mu=2$ in the context of ttH searches.
At the LHC Run 2, kinematic cuts are suggested to further enhance the $b^\prime$ signal over the SM ttH signal. Moreover, we also studied differences in the kinematic distributions between the $b^\prime$ benchmark and the stop benchmark proposed in Ref. [@Huang:2015fba], and observed that the decay spectra of $b^\prime$ are generically more energetic than those from the stop decays. Should the SS2l excess be confirmed at the LHC Run 2, it would be a top priority to determine whether the excess is due to the production of new colored particles, as well as the possible quantum numbers of such new particles.
Looking beyond the excess in the ttH searches, we also considered whether the SS2l excesses in searches for new particles can be explained in a common framework. As it stands there is some difficulty in explaining all the excesses using the $b^\prime$ or the stop benchmarks. However, the number of expected signal events is quite small at the LHC Run 1 and obviously more data from Run 2 is needed to clarify the nature of the excess. In particular, if the excess is due to the pair-production of new colored particles, a small amount of data, of the order of 5 fb$^{-1}$, at the 13 TeV LHC is sufficient to reach the same sensitivity as the entire Run 1 dataset.
Last but not least, even if the SS2l excess disappears as more data is collected, it is clear that searches for vector-like quarks at the LHC need to be extended to the well-motivated scenario of T-odd quarks, whose decay phenomenology requires dedicated analyses that are not covered by current searches for vector-like quarks.
C.-R. C. would like to acknowledge the support of National Center for Theoretical Sciences (NCTS). H.-C. C. would like to thank Academia Sinica in Taiwan for hospitality while part of this work was done. I. L. acknowledges helpful discussions with Aurelio Juste regarding the ATLAS exotica search. We thank the authors of Ref. [@Huang:2015fba] for providing the event file for the stop benchmark used in their study. The work of C.-R. C. is supported in part by the National Science Council of R.O.C. under Grants No. NSC 102-2112-M-003-001-MY3. H.-C. C is supported in part by U.S. Department of Energy grant DE-SC-000999. I. L. is supported in part by the U.S. Department of Energy under Contracts No. DE-AC02-06CH11357 and No. DE-SC0010143.
[nn]{}
P. Huang, A. Ismail, I. Low and C. E. M. Wagner, arXiv:1507.01601 \[hep-ph\].
G. Aad [*et al.*]{} \[ATLAS Collaboration\], JHEP [**1406**]{}, 035 (2014) \[arXiv:1404.2500 \[hep-ex\]\]. G. Aad [*et al.*]{} \[ATLAS Collaboration\], arXiv:1504.04605 \[hep-ex\]. G. Aad [*et al.*]{} \[ATLAS Collaboration\], arXiv:1506.05988 \[hep-ex\]. CMS Collaboration \[CMS Collaboration\], CMS-PAS-HIG-13-020; V. Khachatryan [*et al.*]{} \[CMS Collaboration\], JHEP [**1409**]{}, 087 (2014) \[JHEP [**1410**]{}, 106 (2014)\] \[arXiv:1408.1682 \[hep-ex\]\].
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], JHEP [**1401**]{}, 163 (2014) \[arXiv:1311.6736 \[hep-ex\]\].
G. Aad [*et al.*]{} \[ATLAS Collaboration\], arXiv:1509.05276 \[hep-ex\]. V. Khachatryan [*et al.*]{} \[CMS Collaboration\], arXiv:1510.01131 \[hep-ex\]. See, for example, H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D [**66**]{}, 056006 (2002) \[hep-ph/0205314\]. L. T. Wang and I. Yavin, Int. J. Mod. Phys. A [**23**]{}, 4647 (2008) \[arXiv:0802.2726 \[hep-ph\]\]. H. C. Cheng and I. Low, JHEP [**0309**]{}, 051 (2003) \[hep-ph/0308199\]. See, for example, S. P. Martin, Adv. Ser. Direct. High Energy Phys. [**21**]{}, 1 (2010) \[hep-ph/9709356\]. T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D [**64**]{}, 035002 (2001) \[hep-ph/0012100\]; H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D [**66**]{}, 036005 (2002) \[hep-ph/0204342\]; H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D [**66**]{}, 056006 (2002) \[hep-ph/0205314\].
R. Barbieri and A. Strumia, Phys. Lett. B [**462**]{}, 144 (1999) \[hep-ph/9905281\]; Z. Han and W. Skiba, Phys. Rev. D [**71**]{}, 075009 (2005) \[hep-ph/0412166\]; A. Falkowski and F. Riva, JHEP [**1502**]{}, 039 (2015) \[arXiv:1411.0669 \[hep-ph\]\]. L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{}, 3370 (1999) \[hep-ph/9905221\].
K. Agashe, A. Falkowski, I. Low and G. Servant, JHEP [**0804**]{}, 027 (2008) \[arXiv:0712.2455 \[hep-ph\]\]. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Lett. B [**513**]{}, 232 (2001) \[hep-ph/0105239\]; N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire and J. G. Wacker, JHEP [**0208**]{}, 021 (2002) \[hep-ph/0206020\]; N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP [**0207**]{}, 034 (2002) \[hep-ph/0206021\].
H. C. Cheng and I. Low, JHEP [**0408**]{}, 061 (2004) \[hep-ph/0405243\]; I. Low, JHEP [**0410**]{}, 067 (2004) \[hep-ph/0409025\]. K. Agashe and G. Servant, Phys. Rev. Lett. [**93**]{}, 231805 (2004) \[hep-ph/0403143\]. D. Alves [*et al.*]{} \[LHC New Physics Working Group Collaboration\], J. Phys. G [**39**]{}, 105005 (2012) \[arXiv:1105.2838 \[hep-ph\]\]. A. Anandakrishnan, J. H. Collins, M. Farina, E. Kuflik and M. Perelstein, arXiv:1506.05130 \[hep-ph\]. L. T. Wang and I. Yavin, JHEP [**0704**]{}, 032 (2007) \[hep-ph/0605296\]. H. C. Cheng, I. Low and L. T. Wang, Phys. Rev. D [**74**]{}, 055001 (2006) \[hep-ph/0510225\].
J. Hubisz, P. Meade, A. Noble and M. Perelstein, JHEP [**0601**]{}, 135 (2006) \[hep-ph/0506042\]. A. Belyaev, C. R. Chen, K. Tobe and C.-P. Yuan, Phys. Rev. D [**74**]{}, 115020 (2006) \[hep-ph/0609179\].
M. Carena, J. Hubisz, M. Perelstein and P. Verdier, Phys. Rev. D [**75**]{}, 091701 (2007) \[hep-ph/0610156\]; D. Choudhury and D. K. Ghosh, JHEP [**0708**]{}, 084 (2007) \[hep-ph/0612299\]; G. Cacciapaglia, S. R. Choudhury, A. Deandrea, N. Gaur and M. Klasen, JHEP [**1003**]{}, 059 (2010) \[arXiv:0911.4630 \[hep-ph\]\]; M. Perelstein and J. Shao, Phys. Lett. B [**704**]{}, 510 (2011) \[arXiv:1103.3014 \[hep-ph\]\]. See, for example, G. Aad [*et al.*]{} \[ATLAS Collaboration\], arXiv:1506.08616 \[hep-ex\]. I. Low, Phys. Rev. D [**88**]{}, no. 9, 095018 (2013) \[arXiv:1304.0491 \[hep-ph\]\].
CMS Physics Analysis Summary SUS-13-008.
M. Kramer, A. Kulesza, R. van der Leeuw, M. Mangano, S. Padhi, T. Plehn and X. Portell, arXiv:1206.2892 \[hep-ph\]. C. Borschensky, M. Krämer, A. Kulesza, M. Mangano, S. Padhi, T. Plehn and X. Portell, Eur. Phys. J. C [**74**]{}, no. 12, 3174 (2014) \[arXiv:1407.5066 \[hep-ph\]\]. S. Moch and P. Uwer, Nucl. Phys. Proc. Suppl. [**183**]{}, 75 (2008) \[arXiv:0807.2794 \[hep-ph\]\]. U. Langenfeld, S. Moch and P. Uwer, arXiv:0907.2527 \[hep-ph\]. J. Alwall [*et al.*]{}, JHEP [**0709**]{}, 028 (2007) \[arXiv:0706.2334 \[hep-ph\]\]. T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP [**0605**]{}, 026 (2006) \[hep-ph/0603175\]. J. de Favereau [*et al.*]{} \[DELPHES 3 Collaboration\], JHEP [**1402**]{}, 057 (2014) \[arXiv:1307.6346 \[hep-ex\]\].
<https://twiki.cern.ch/twiki/bin/view/CMSPublic/Hig13020TWiki#Dilepton_final_state>.
[^1]: For a related approach to the simplified T parity, see Ref. [@Anandakrishnan:2015yfa].
[^2]: For collider signatures of the first and second generation T-odd quarks, see Ref. [@Carena:2006jx].
[^3]: There are many collider searches for vector-like fermionic top partners. However, typically T parity was not assumed in these searches and there is no LTP carrying away additional $E_{\rm T}^{\rm miss}$ in the final states.
[^4]: There is an additional veto on SR3b signal region, which requires SS2l or three leptons, $N_{\rm b-jets}\ge 3$, $N_{\rm jets}\ge 5$ and $m_{\rm eff}> 350$ GeV.
|
---
author:
- 'Maarten Reyniers[^1]'
- Jan Cuypers
date: 'Received 30 September 2004 / Accepted 31 October 2004'
title: |
The evolutionary status of the bright\
high-latitude supergiant HD190390[^2]
---
Introduction {#sect:ntrdctn}
============
Since their discovery by William @bidelman51, the evolutionary stage of (b=$+$25$^{\circ}$), ($+$22$^{\circ}$) and ($+$30$^{\circ}$), three high-luminosity stars at moderately high galactic latitude, has been a matter of debate. At least three different possibilities were suggested: (1) young, massive objects that recently escaped from a star forming region, (2) low-mass, evolved objects or (3) a rare product in a binary star scenario. After the high resolution, high signal-to-noise spectroscopic study of @luck90, there was general consensus that these objects are low-mass objects in an evolved, post-AGB evolutionary stage. Although [*no*]{} s-process enhancement was observed in these stars, there were enough indications for a classification as a post-AGB star (slightly metal-poor, overabundant carbon and oxygen and a strong infrared dust excess indicating high mass loss in a previous evolutionary (AGB) stage).
[rrr]{}\
Coordinates & $\alpha_{2000}$ & 20 05 05.41\
& $\delta_{2000}$ &$-$11 35 57.9\
Galactic & $l$ &30.60\
coordinates & $b$ &$-$21.53\
Mean magnitude & B &6.88\
& V &6.39\
Spectral type$^*$ & & F1III\
Parallax &$\pi$(mas) & 2.56$\pm$0.97\
IRAS fluxes &$f_{12}$ & 0.65\
(Jy) &$f_{25}$ & $<$0.32\
&$f_{60}$ & $<$0.40\
&$f_{100}$ & $<$1.08\
[$^*$very differing spectral types have been reported in the literature: gF4, F1III, F6II, F6Ib, F3Ib, F2p(shell) [cit. @fernie86]]{}
There was also a fourth star in the @luck90 sample, (, Table \[tab:smbdgegevens\]). The inclusion of this object in their sample was motivated not only by its similar galactic latitude and spectral type, but also by the variability pointed out by @waelkens85 and @fernie86. This variablity made a candidate member of the UU Her type stars, a heterogeneous class of luminous variables at high galactic latitudes defined by @sasselov84, which also includes and . For , however, @luck90 found a clearly different chemical signature than for the other three stars, with a C and O deficiency and a high Li abundance. The authors proposed the object to be a descendant of a Li-rich S-type star, with the Li attributed to a “hot bottom burning” event.
In this paper, new observational material is presented to clarify the status of this intriguing object. Both the chemical composition and the variability are reanalysed, based on high quality data coming from different instruments and telescopes. In Sect. \[sect:chmclcomposition\] we report on the reduction and analysis of our high resolution, high signal-to-noise NTT+EMMI spectra and discuss the chemical photospheric composition of . In Sect. \[sect:vrbltanalysis\], a detailed variability analysis is presented, based on both photometric data and radial velocities. Some results are presented originating from the new Flemish Mercator telescope located at La Palma, Spain. Our results are discussed in Sect. \[sect:dscssn\]. We give our conclusions in Sect. \[sect:cnclsns\].
Chemical composition {#sect:chmclcomposition}
====================
Previous studies
----------------
@luck90 were the first to carry out a detailed abundance analysis of , based on spectra with medium to high resolution ($R=\lambda/\delta\lambda$$\simeq$15000–47000) and a signal-to-noise ratio between 75 and 100. They confirmed the metal deficiency discovered by @mcdonald76 and quantified it: \[Fe/H\]=$-$1.1. Further, they found a slight C and O deficiency and also claimed a slight s-process enhancement by a factor of 4 above solar. Lithium was also detected with an abundance of log$\epsilon$(Li)=2.4. The main shortcomings of this analysis were the quite low signal-to-noise ratio obtained, and the inhomogeneous collection of spectra with different resolution, leading to quite large line-to-line scatters of the reported abundances (up to 0.92dex). A more recent abundance analysis was presented by @giridhar97. This analysis was based on moderately high resolution spectra ($R\simeq30000$) with a rather narrow wavelength coverage (5260–5600Å), and it is therefore rather limited.
In the analysis we present here, we avoided the shortcomings of the previous studies: a large spectral domain covered with the same resolution and a high signal-to-noise was used. Such spectra were taken with ESO-NTT telescope and the EMMI spectrograph.
Observations and reduction {#subsect:obsrvtns}
--------------------------
In the framework of our ongoing program to study the photospheric chemical composition of stars in their last stages of evolution [e.g. @vanwinckel03; @reyniers04], high resolution, high signal-to-noise spectra were taken with the EMMI Spectrograph mounted on the 3.58m New Technology Telescope (NTT) located in La Silla, Chile. The observations of were made by Dr. H. Van Winckel during two nights in 1998 (see Table \[tab:obsdtls\]). The reduction of the data was performed in the specific echelle package [echelle]{} of the [midas]{} data analysis system. For a more detailed description of the different steps of the reduction procedure, see @reyniers02a. A sample spectrum can be found in Fig. \[fig:figr2004\].
------------------ ----------------- ------ ---------
29/09/1998 02:55 Ech\#14 Grsm\#5 900 398-662
30/09/1998 04:18 Ech\#14 Grsm\#6 1200 597-832
------------------ ----------------- ------ ---------
: Log of the high resolution, high signal-to-noise NTT+EMMI spectra of . The resolving power $\lambda/\delta\lambda$ was $\sim$60000 for both settings.[]{data-label="tab:obsdtls"}
Abundance analysis {#sect:abndc}
------------------
A detailed abundance analysis of was performed with an extended version of the line list described in @vanwinckel00.
### Atmospheric parameters and equivalent widths
Since the atmospheric parameters are usually based on a fine analysis of the iron lines, the @vanwinckel00 list only contains the critical compilation of Lambert [@lambert96], completed with values of @blackwell80 for the singly ionised iron lines. Two different methods were used to measure the equivalent widths: a Gaussian approximation and direct integration. Due to a small asymmetry in the blue wing of the stronger lines, a Gaussian approximation leads to an underestimation of the real equivalent width. This is illustrated in the top panels of Fig. \[fig:abndcplots\]. On the other hand, a Gaussian approximation allows a better continuum placement (especially important for weaker lines) and also allows for blends in the wing of a line. We have chosen a strictly homogeneous way of measuring the equivalent widths of the iron lines: for lines with $W_{\lambda}$$<$10.5mÅ, we applied a Gaussian approximation, for the other lines we used direct integration, except if the line was blended in the wings. 84 Fe[i]{} lines and 24 Fe[ii]{} lines were measured in such way.
The effective temperature T$_{\rm eff}$ of the Kurucz model atmosphere [@kurucz93] was then obtained by forcing the Fe abundance to be independent of the excitation potential of the lines, the gravity $\log g$ by forcing ionisation equilibrium, the microturbulent velocity $\xi_t$ by forcing the Fe abundance to be independent of (reduced) equivalent widths. Abundance calculations were made with the LTE line analysis program MOOG (version April 2002). This iterative process finally yielded a model with (T$_{\rm eff}$,$\log g$,$\xi_t$,\[M/H\]) = (6250K,1.0(cgs),3.0kms$^{-1}$,$-$1.5). E.P.-abundance and $\log(W_{\lambda}/\lambda$)-abundance diagrams for this model can be found in Fig. \[fig:abndcplots\]. The model parameters slightly differ from the model parameters found by @luck90 and @giridhar97. The former authors found (6600K,1.5(cgs),2.3kms$^{-1}$), the latter (6500K,1.25(cgs),3.2kms$^{-1}$). However, due to the superior quality of our spectra and the severe selection of our Fe line list, we feel confident about our slightly differing parameters. Note that the difference in atmospheric parameters can also be a consequence of the variability of the object.
The method of the equivalent width measurement (either Gaussian approximation or direct integration) for the other lines was chosen for each line individually, without a strict criterion. The @vanwinckel00 list was extended for the s-process elements in the following way. We extracted all s-process elements from the VALD database [@kupka99]. These lists were then sorted based on the calculated equivalent widths of the lines using the model obtained above and an ad-hoc abundance value. Each line of these lists was then checked in this order on the spectrum of , until the lines were too small to detect. 136 lines of s-process elements were used in the analysis. The oscillator strengths of La and Eu from VALD were replaced by the more recent values published by @lawler01a and @lawler01b respectively. The total number of spectral lines used in this analysis is 462. The line list can be obtained from the authors upon request.
### Abundance results {#subsubs:abundancerslts}
The resulting abundances are given in Table \[tab:abndctabel\]. For each ion we list the number of lines N, the mean equivalent width $\overline{W_{\lambda}}$, the absolute abundance $\log\epsilon$ (i.e. relative to H: $\log\epsilon = \log X/{\rm H} + 12$), the internal scatter $\sigma$, if more than one line is used, the solar abundances and the abundance ratio relative to iron \[el/Fe\]. The latter values are also graphically presented in the top panel of Fig. \[fig:cnoelfedefic\]. For the solar iron abundance we used the meteoritic iron abundance of 7.51; the references for the solar CNO are C: @biemont93, N: @hibbert91 and O: @biemont91; for Mg and Si the latest Holweger values [@holweger01] were used; for La and Eu we took the recent values derived by @lawler01a and @lawler01b respectively; other solar abundances were taken from the review by @grevesse98. Despite the fact that there are more recent values for some of the solar abundances (especially for the solar CNO), we take these references to ensure as much as possible that the $gf$ values that we have used in the present paper are consistent with the adopted solar abundances. Note the excellent agreement in abundance for elements with two ionisation stages, Mg, Si, Ti, Cr and Ni, indicating a good model parameter choice. Note also the small line-to-line scatter which rarely exceeds 0.15dex, including the scatter for the s-process ions.
[lrrrrrr]{}\
\
ion & N &
------------------------------------------------------------------------
$\overline{W_{\lambda}}$
&$\log\epsilon$&$\sigma$& sun &\[el/Fe\]\
Li[i]{} & 1 & ss & 1.9&0.2& &\
C[i]{} & 9 & 4 & 6.44 & 0.13 & 8.57 &$-$0.57\
N[i]{} & 3 & 5 & 6.94 & 0.16 & 7.99 & 0.51\
O[i]{} & 3 & 5 & 7.76 & 0.05 & 8.86 & 0.46\
Na[i]{} & 2 & 11 & 4.65 & 0.12 & 6.33 &$-$0.12\
Mg[i]{} & 3 & 91 & 6.47 & 0.09 & 7.54 & 0.49\
Mg[ii]{} & 1 & 8 & 6.46 & & 7.54 & 0.48\
Si[i]{} & 17 & 11 & 6.50 & 0.12 & 7.54 & 0.52\
Si[ii]{} & 5 & 46 & 6.47 & 0.17 & 7.54 & 0.49\
S[i]{} & 5 & 9 & 6.06 & 0.06 & 7.33 & 0.29\
Ca[i]{} & 23 & 36 & 5.10 & 0.13 & 6.36 & 0.30\
Sc[ii]{} & 13 & 54 & 1.72 & 0.12 & 3.17 & 0.11\
Ti[i]{} & 2 & 27 & 3.80 & 0.01 & 5.02 & 0.34\
Ti[ii]{} & 42 & 76 & 3.82 & 0.10 & 5.02 & 0.36\
V[ii]{} & 1 & 96 & 2.66 & & 4.00 & 0.22\
Cr[i]{} & 16 & 18 & 4.07 & 0.13 & 5.67 &$-$0.04\
Cr[ii]{} & 23 & 28 & 4.01 & 0.11 & 5.67 &$-$0.10\
Mn[i]{} & 9 & 9 & 3.52 & 0.10 & 5.39 &$-$0.31\
Fe[i]{} & 84 & 30 & 5.96 & 0.10 & 7.51 & 0.01\
Fe[ii]{} & 24 & 41 & 5.95 & 0.12 & 7.51 & 0.00\
Ni[i]{} & 35 & 10 & 4.78 & 0.10 & 6.25 & 0.09\
Ni[ii]{} & 2 & 7 & 4.79 & 0.01 & 6.25 & 0.10\
Cu[i]{} & 1 & 3 & 2.56 & & 4.21 &$-$0.09\
Zn[i]{} & 3 & 19 & 3.25 & 0.03 & 4.60 & 0.21\
Y[ii]{} & 22 & 34 & 0.80 & 0.13 & 2.24 & 0.12\
Zr[ii]{} & 31 & 35 & 1.48 & 0.15 & 2.60 & 0.44\
Ba[ii]{} & 3 & 30 & 0.63 & 0.11 & 2.13 & 0.06\
La[ii]{} & 16 & 19 &$-$0.21& 0.09 & 1.13 & 0.22\
Ce[ii]{} & 32 & 11 & 0.15 & 0.22 & 1.58 & 0.13\
Nd[ii]{} & 14 & 10 & 0.07 & 0.16 & 1.50 & 0.13\
Sm[ii]{} & 15 & 6 &$-$0.41& 0.13 & 1.01 & 0.14\
Eu[ii]{} & 2 & ss &$-$0.71& 0.22 & 0.52 & 0.33\
Hf[ii]{} & 1 & 4 &$-$0.43& & 0.88 & 0.25\
\
\
\
### Lithium synthesis
The lithium line in with equivalent width $W_{\lambda}$=9.5mÅ was first reported by @luck90. To account for the doublet structure of the line, spectrum synthesis is necessary in deriving the Li abundance. The program MOOG was also used for this purpose. The line list is identical to the line list that was used in @reyniers02b, but due to the combined effect of the quite high temperature and the metal deficiency of , no other lines are detectable in the 6707Å region.
For a spectral synthesis, a broadening factor is needed, in addition to the model atmosphere parameters. This broadening factor is a combined effect of the instrumental, rotational and macroturbulent broadening. An instrumental broadening of 5kms$^{-1}$ was adopted (the spectral resolution of the EMMI spectrograph). The rotational broadening is thought not to be very high due to the supergiant character of . The macroturbulent broadening $\xi_m$ was determined in the following way. We selected 27 unblended iron lines (neutral and ionised) with $W_{\lambda}$$<$25mÅ. For each of these lines, we made a synthesis using the abundance calculated via their $W_{\lambda}$. In such way, the macroturbulent broadening is the only free parameter in this synthesis. We found a quite large spread in $\xi_m$ for these lines: $\xi_m$=8.0$\pm$1.2kms$^{-1}$. This is also exactly the value found in the @luck90 analysis. Hence for the lithium synthesis $\xi_m$=8.0kms$^{-1}$ was used. This yielded log$\epsilon$(Li)=1.9.
An error analysis on the Li abundance was made by varying all parameters involved in the synthesis. As expected, a temperature change has far the largest impact on the abundance: a temperature variation of $\pm$250K yields a change in the Li abundance of $\sim$0.2 dex. Other variations are listed in Table \[tab:lthmvaria\].
### Europium synthesis
The effect of hyperfine splitting ([*hfs*]{}) on the europium abundance was investigated by the synthesis of the two Eu[ii]{} lines of our analysis. $\log gf$ data and hyperfine splitting constants were taken from @lawler01b. The influence of hfs was determined by integrating the profile of the synthesized line, with and without hfs decomposition. This calculated equivalent width is then compared with the measured equivalent width. In this way, the exact profile of the line is eliminated, and no broadening factors have to be applied. The effect on abundance of hfs for the resonance line at 4129.725Å (W$_{\lambda}$=43.1mÅ) is significant. A hfs treatment of this line yields $\log\epsilon({\rm Eu})$=$-$0.87dex, while without hfs it is $-$0.79dex. For the weak line at 6645.064Å (W$_{\lambda}$=5.9mÅ), the difference between an hfs and a non-hfs treatment is not significant.
Abundances: summary
-------------------
Referring to Fig. \[fig:cnoelfedefic\], we will now discuss the derived abundances of .\
[**Metallicity**]{} With an iron abundance of log$\epsilon({\rm Fe})$=5.95, is clearly metal deficient by almost a factor of 40. The other iron peak elements follow this deficiency within 0.1dex, except Mn and Zn. The slight manganese deficiency (\[Mn/Fe\]=$-$0.3) is expected given the galactic chemical evolution [e.g. @nissen00; @prochaska00]. The zinc abundance (\[Zn/Fe\]=$+$0.2) is somewhat higher than expected, but it is based on only three lines.\
[**CNO**]{} Although the abundance of the CNO elements is solely based on very small lines, a clear pattern for these three elements is observed: \[C/Fe\]=$-$0.6 and \[N/Fe\]=\[O/Fe\]=$+0.5$. The deficient carbon and the overabundant oxygen lead to a very low C/O number ratio: C/O=0.05. Nitrogen seems to be enhanced at the same level as oxygen.\
[**$\alpha$-elements**]{} The mean of the \[el/Fe\] values of the $\alpha$-elements Mg, Si, S, Ca and Ti yields \[$\alpha$/Fe\]= $+$0.4. This enhancement is what is expected for stars in this metallicity range and hence the $\alpha$-elements are [*not*]{} intrinsically enhanced. The abundances of the individual $\alpha$-elements all agree within 0.1dex with this $+$0.4 number.\
[**S-process elements**]{} A very complete abundance pattern could be obtained for the s-process elements, with 53 lines of the light ([*ls*]{}) and 83 lines of the heavy ([*hs*]{}) s-process elements. The s-process elements are all slightly enhanced, but the abundance pattern reveals that the enhancement is probably [*not*]{} intrinsic. All elements are compatible with \[el/Fe\]=0 considering the line-to-line scatters, except Zr, La, Eu and Hf. For Zr, a large spread is seen in the \[Zr/Fe\] of unevolved disk and halo stars from $-$0.1 to $+$0.6dex [e.g. @travaglio04]. Moreover, the same relatively high Zr abundance is also seen in our analysis of the non-enriched RVTauri star DSAqr [@deroo05]. Eu is an element with a clear r-process origin, and an enhancement of \[Eu/Fe\]=$+$0.4 is certainly expected in this metallicity range [e.g. @travaglio99]. The Hf abundance is based on only one very small line and should therefore be interpreted with care. Moreover, the galactic chemical evolution of this very heavy element is still unknown. Finally, La seems to be a little bit more enhanced than the other s-process elements, but still compatible with [*no*]{} intrinsic enhancement regarding the scatter (although not as pronounced as for Zr) for unevolved halo stars [e.g. @travaglio99]. This result is contrary to the s-process enrichment claimed by @luck90. They found a mild enhancement by a factor of 4 compared to solar. A detailed inspection of their results, however, reveals large line-to-line scatters for all elements ($>$0.25, except for Ba) and a smaller number of lines per element. Also, the abundances of the s-process elements as determined by @luck90 do not display a consistent pattern in the sense that some elements are much more enhanced than others. \[Sm/Fe\]=$+$2.1 together with \[Ce/Fe\]=$+$0.2 is theoretically very difficult to reconcile with an s-process signature. We conclude that the s-process abundances presented here are much more reliable than the ones from the @luck90 paper.\
[**Lithium**]{} The lithium abundance found in this analysis (log$\epsilon$(Li)=1.9) is 0.5dex lower than that found in the @luck90 analysis. The reason is the higher effective temperature in the latter analysis. Indeed, taking a model with a temperature of 6500K in the Li synthesis, our derived Li abundance rises by almost 0.2dex.\
[**Other elements**]{} Sodium is slightly underabundant, but not at a significant level [e.g. @carretta00]. The slight scandium enhancement (\[Sc/Fe\]=$+$0.1) is not unusual in this metallicity range [e.g. @nissen00; @prochaska00]. Finally, the vanadium (V) abundance is quite high, but it is based on only one line.
--------------------- --------------- ------------------ --------------------------
parameter symbol variation variation
$\log\epsilon({\rm Li})$
temperature T$_{\rm eff}$ $\pm$250K $\pm$0.20
gravity $\log(g)$ $\pm$0.5 (cgs) $\pm$0.05
microturb. velocity $\xi_t$ $\pm$2kms$^{-1}$ $\pm$0.04
macroturb. velocity $\xi_m$ $\pm$2kms$^{-1}$ $\pm$0.04
--------------------- --------------- ------------------ --------------------------
: Error analysis of the Li synthesis. A temperature change has the largest effect on the derived Li abundance.[]{data-label="tab:lthmvaria"}
Variability analysis {#sect:vrbltanalysis}
====================
has been known to be variable for quite a long time. According to the SIMBAD database, the variability of this star was first mentioned by @strohmeier65 who reported a photographic range of 0.4mag. However, dedicated observational efforts to study this variability were made more than twenty years later. @fernie86 published an analysis based on 32 observations in the $uvby\beta(RI)$ system and mentions a range in the $y$ magnitude of 0.3mag and a possible period of 28.4 or 11.8 days. At nearly the same time, @waelkens85 presented their frequency analysis of 123 measurements in the Geneva photometric system [@rufener81], obtained with the Swiss photometric telescope at the European Southern Observatory. The data covered 1040 days and revealed a most significant period of 28.49 days, accounting for 80% of the total variance. The residual scatter around the fitted light curve was attributed to irregular cycle-to-cycle variations.
The frequency analysis was made using different algorithms: the Jurkevich-Stellingwerf [pdm]{} method [@stellingwerf78], the [clean]{} algorithm [@roberts87] and a general multifrequency least squares fitting method [@schoenaers04]. In this analysis, we will give a rough estimate of this error by using the formula given by @montgomery99: $$\label{eq:mont}
\sigma_f = \frac{\sqrt{6}\sigma_R}{\pi \sqrt{N}AT}$$ with $\sigma_R$ the observational error, $N$ the number of measurements, $A$ the amplitude of the signal without noise, $T$ the total time span of the observations. Different forms of this formula are given in the literature. @schwarzenbergczerny91 introduced an extra parameter $D$ in the formula in order to take correlations in the noise into account. To avoid an underestimation of the calculated error, we took $T$=$N$. The motivation is that always $N$$\le$$T$ for our different data sets of , because the object is measured at most only once per night. For $A$ we took in each analysis the amplitude of the curve fitted with the found frequency. The obtained error estimate for the proposed frequencies is only an indicative number and should be interpreted with care. It is, however, useful to compare frequencies from two data sets obtained with the same or a comparable instrument.
------ --------- ----------- ------ --------- -----------
Year [*N*]{} Time span Year [*N*]{} Time span
(days) (days)
1978 2 74 1985 52 163
1979 1 - 1986 24 77
1982 29 57 1987 6 13
1983 45 129 1988 21 118
1984 22 128
------ --------- ----------- ------ --------- -----------
: Log of the observations in the Geneva photometric system[]{data-label="tab:genlog"}
[rrrl]{} algorithm & $f$ & P & note\
& (c/d) & (days) &\
\
& 0.03501 & 28.56 & $\Theta$=0.39\
[clean]{} & 0.03503 & 28.55 & Amplitude 0.080\
[least squares fit]{} & 0.03502 & 28.56 & [*Variance reduction*]{} 62%\
\
& 0.03535 & 28.29 & $\Theta$=0.72\
[clean]{} & 0.03535 & 28.29 & Amplitude 0.033\
[least squares fit]{} & 0.03535 & 28.29 & [*Variance reduction*]{} 25%$^*$\
\
& 0.01729 & 57.84 & $\Theta$=0.79\
[clean]{} & 0.03457 & 28.93 & Amplitude 0.027\
[least squares fit]{} & 0.03468 & 28.84 & [*Variance reduction*]{} 22%$^*$\
\
-.5 cm
Geneva photometry {#subsect:geneva}
-----------------
The photometric data consist of 202 measurements in the Geneva photometric system obtained with the P7 photometer on the (now decommissioned) Swiss photometric telescope in La Silla from September 1982 to August 1988. In a first approach we chose not to add the new Mercator measurements to this data set for two reasons: (1) for the sake of the homogeneity of our data set and (2) there would be a time gap of more than ten years in our data. The Mercator measurements are discussed in a separate section (Sect. \[subsect:merc\]). A small log of the observations is presented in Table \[tab:genlog\]. The analysis is performed on the V magnitude. We looked for frequencies between 0.01 and 0.5c/d using a frequency step of 0.00001c/d.
### Frequency analysis
All frequency analysis methods show a highly significant peak near 0.035c/d, indicated by an arrow in the high resolution $\Theta$ statistic ([pdm]{}) or in the [clean]{} spectrum in Fig. \[fig:spectraallfr\] (a). The results for this frequency $f_1$ are summarized in Table \[tab:sumfr\]. In the following, we will use for $f_1$ the value 0.03501($\pm$0.00001)c/d (P$_1$=28.56d). The error estimate is obtained with formula (\[eq:mont\]). $f_1$ accounts for 62% of the total variance. A phase diagram for this frequency is shown in Fig. \[fig:fasen\].
After removing (“prewhitening”) this first frequency from the data, we performed a second frequency analysis on the residuals. We found a new frequency very close to $f_1$: $f'_1$=0.03535($\pm$0.00003)c/d. This frequency is documented in Table \[tab:sumfr\] and Fig. \[fig:spectraallfr\] (b). Together with $f_1$, $f'_1$ accounts for 72% of the total variance (an improvement of 11%). A phase diagram for $f'_1$ is shown in Fig. \[fig:fasen\].
In the periodograms of the once prewhitened data (Fig. \[fig:spectraallfr\], b), we note, apart from the peak at 0.03535c/d ($f'_1$), another peak very close to the main frequency $f_1$, situated at 0.03469c/d. This second peak can easily be seen on Fig. \[fig:spectraallfr\] (c), which is a zoom-in of panel (b). The peak at 0.03469c/d is at the position $f_1-(f'_1-f_1)$ and therefore it could be the third component of a frequency [*triplet*]{}, together with $f_1$ and $f'_1$. A natural question is then if this frequency is also visible in an analysis of the twice prewhitened data. The answer is positive for the [clean]{} spectrum, which has a maximum at $f''_1$=0.03457($\pm$0.00005)c/d. $\Theta$ is minimal for $f$=0.01729c/d, which is $\nicefrac{1}{2}$$f''_1$ (see Fig. \[fig:spectraallfr\], d). $f''_1$ accounts for an additional 6% of the variance (from 72% to 78%). Other details can be found in Table \[tab:sumfr\]. A phase diagram for this frequency is shown in Fig. \[fig:fasen\].
A strong argument for the presence of additional frequencies is given by a simple inspection of Fig. \[fig:genevadat\]. If $f'_1$ and/or $f''_1$ are physical, then a [*beating*]{} should be observed with frequency $f_b$=$f'_1$$-$$f_1$=0.00034c/d or with a period of P$_b$$\simeq$3000d. This beating can be seen in Fig. \[fig:genevadat\]: the peak-to-peak amplitude [*per year*]{} clearly varies, reaching its maximum in 1983 (0.355mag) and its minimum in 1986 (0.099mag). The dashed line is a simple harmonic fit with frequency $f_b$ through the extrema (either minimum or maximum) of each year in which observations have taken place.
In the periodograms of the once prewhitened data, a prominent peak is also seen at 0.09154c/d (Fig. \[fig:spectraallfr\], b and c). The same peak is also present in the twice prewhitened data (Fig. \[fig:spectraallfr\], d). Since this frequency is not linked to the frequencies around 0.035c/d, we can consider this frequency as a genuine second frequency $f_2$=0.09154($\pm$0.00005)c/d. $f_2$ is also the highest peak in the [clean]{} periodogram after removing the three frequencies around 0.035c/d. A phase diagram for this second frequency is shown in Fig. \[fig:fasen\].
### Light curves and colour variations
A light curve was constructed using the four frequencies we found above. The fits were obtained using a simple least-squares procedure and the results can be seen in Fig. \[fig:f1h3f2f3\]. In the upper panel of this figure, we show how the light curve behaves [*globally*]{}. The beating is present in the light curve, even when making use of only two frequencies. However, the [*amplitude*]{} of the beating seems to be underestimated. In this respect, the inclusion of the fourth frequency ($f_2$) improves the global behaviour of the light curve. In the lower panel of Fig. \[fig:f1h3f2f3\], a detail of the light curve between JD2445217 and JD2445288 is shown.
The frequency analysis presented above was based on the V magnitude because this magnitude is mostly used in this kind of analysis. In Fig. \[fig:lghtclrsfolded\] the colour variations and the light variations are folded on the same period. As was noted by @waelkens85, the U-B and B-V variations are respectively in antiphase and in phase with the light variations. This result is important in the discussion of the pulsational nature of (Sect. \[subsect:uhrclassification\] and \[subsect:wvrclassification\]).
HIPPARCOS data {#subsect:hipp}
--------------
53 HIPPARCOS measurements of were extracted from the HIPPARCOS catalogue [@ESA97]. 43 measurements had a non-zero quality flag, but at 10 more epochs the star was 0.10 to 0.15 mag brighter than on average. We have no explanation for this brightening, but since we cannot exclude that these brightenings are instrumental, we used only the remaining 34 points. They cover a total time span of 948.6 days, from April 1990 to November 1992. The precision of these HIPPARCOS magnitudes (which is also given in the catalogue for each measurement individually) is between 0.005 and 0.01mag.
It is remarkable that also in this very limited data set, the second best two-frequency solution is the combination of $f_{\rm H1}$=0.03704 c/d (P$_{\rm H1}$=27.00d) and $f_{\rm H2}$=0.09042c/d (P$_{\rm
H2}$=11.06d). These frequencies account for 95.8% of the total variance. The best combination in the least-squares sense, $f_{\rm H1'}$=0.07738 c/d (P$_{\rm H1'}$=12.92d) and $f_{\rm H 2'}$=0.09044c/d (P$_{\rm
H2'}$=11.06d) does not give a significantly different result (96.4% variance reduction). The frequencies $f_{\rm H1}$ and $f_{\rm H1'}$ could be related aliases since the second one is not recovered after prewhitening with the first frequency and vice-versa, and there is no doubt that the star has a main period of around 28 days.
CORALIE radial velocities {#subsect:snel}
-------------------------
The CORALIE fiber-fed echelle spectrograph at the Swiss 1.2m telescope at La Silla has a resolution of $\sim$50000 at 5000Å and a wavelength coverage from 3880 to 6810Å. Due to the stability of the whole configuration and the possibility of making simultaneously Thorium-Argon calibration spectra in the second fibre, the spectrograph is specifically designed for high-precision radial-velocity measurements.
Since November 1999, 103 radial velocity measurements of have been collected (Fig. \[fig:smcorwim\]). Velocities are deduced by cross-correlating the spectra with an appropriate mask. In the case of , an F0 mask is used. Sample correlation profiles for one observational run are given at the bottom of Fig. \[fig:smcorwim\]. The accuracy of these velocities is $\sim$1.0–1.5kms$^{-1}$. From Fig. \[fig:smcorwim\] it is clear that the period in the radial velocity data must be close to the main period $f_1$ found in the photometric data. The frequency found in the frequency analysis (Table \[tab:sumsnel1stefr\] and Fig. \[fig:snelspectra1stefr\]) is indeed equal to the main frequency found in the Geneva photometry.
----------------------- ------------- ------------- ------------------------------
algorithm $f_{\rm v}$ P$_{\rm v}$ note
(c/d) (days)
[pdm]{} 0.01745 57.31 $\Theta$=0.23
[clean]{} 0.03502 28.56 Amplitude 0.93
[least squares fit]{} 0.01741 57.44 [*Variance reduction*]{} 76%
----------------------- ------------- ------------- ------------------------------
: Frequency analysis of the CORALIE radial velocity data. Note that both [pdm]{} and the [least squares]{} method find $\nicefrac{1}{2}f_1$ instead of $f_1$.[]{data-label="tab:sumsnel1stefr"}
Variation of the H$\alpha$ profile during a pulsational cycle is often used as a powerful diagnostic tool in the determination of the pulsation character. The CORALIE spectra, however, are of too low signal-to-noise (s/n) for a detailed profile variation study, because the first intention of these spectra was only to obtain a radial velocity by cross-correlation. For this purpose, a s/n of around 30 is sufficient, and the mean s/n of all spectra is indeed 40. Nontheless, we [*can*]{} study line profile variations over a cycle by combining spectra in the same phase bin. In such a procedure, a limited number of higher s/n spectra is compiled out of spectra situated in the same phase bin. In the construction of these combined spectra, we chose to use 6 phase bins designated A to F, illustrated in Fig. \[fig:phbns\]. The number of spectra added, the total integration time and the achieved s/n ratios are summarized in Table \[tab:phsbns\]. Spectra were first transformed to rest wavelength, normalised and then added, s/n weighted.
From Fig. \[fig:phbns\] it is clear that H$\alpha$ in is variable during a pulsational cycle: relatively strong emission in the blue wing is seen in bins B and C, while in bin F for example, there is only some emission present in the red wing, and none in the blue. H$\alpha$ emission in the blue wing was already noted by @sasselov85, and later confirmed by @luck90. @sasselov85 gave a $\sim$60kms$^{-1}$ separation between the emission and the photospheric rest velocity of the H$\alpha$ absorption. We find similar values for this separation, but the exact value for each phase binned spectrum depends on the method adopted to determine the central wavelength of both the emission and the absorption peak.
--- ------------------ ---------------- --------------- ------------
\# spectra total integra- achieved s/n
tion time (s) (at 6550Å)
A $-$0.083 - 0.083 14 2968 160
B 0.083 - 0.250 10 1713 100
C 0.250 - 0.417 6 1200 90
D 0.417 - 0.583 22 5632 150
E 0.583 - 0.750 25 7343 180
F 0.750 - 0.917 25 7172 150
--- ------------------ ---------------- --------------- ------------
: Details of the phase binning of the 102 CORALIE spectra into 6 bins. The older measurement of 1999 was not used.[]{data-label="tab:phsbns"}
Geneva photometry with Mercator {#subsect:merc}
-------------------------------
The Mercator telescope is a 1.2 meter telescope located on the Roque de los Muchachos observatory on La Palma, Spain. The Mercator telescope, operated by our Institute, is a twin of the Swiss telescope in La Silla. Two instruments are available for the moment: the refurbished photometer P7 which was in use at the “old” Swiss telescope at La Silla, and a CCD camera.
is in the standard program catalogue for P7, but since the declination of this object is rather low (-11$^{\circ}$36$^m$), the minimum air mass for La Palma is quite high ($F_z$=1.30). Therefore, is always measured in concatenation with the standard star (F4III, V=6.72) and the comparison star (G5, V=8.7). 72 measurements of have been collected with the Mercator telescope. During the reduction, a quality flag is assigned to each observation, ranging from 0 to 4. Observations with the lowest quality (flag 0) were discarded, and 66 measurements remained, covering 787 days. This data set is still quite limited to perform a reliable frequency analysis. However, a preliminary analysis again suggests a frequency close to the main frequency found in the “old” Geneva data: $f_{\rm M}$=0.03524c/d($\pm$0.00005)c/d (P$_{\rm M}$=28.38days). This frequency accounts for 86% of the total variance.
There is a secondary peak in the periodograms near the main frequency and at 0.0897c/d an unrelated frequency is found. The best 3-frequency solution (94% reduction of the initial variance) is with 0.03521, 0.04072 and 0.08970c/d.
Variability {#subsect:puttogether}
-----------
### Photometry and velocity
In Fig. \[fig:merc\] we compare the light curve of with its radial velocity curve. The comparison is made for a time interval for which we have both photometric and velocity data, so no extrapolation is needed. Both curves are fitted with a simple harmonic with the main frequency (0.03501c/d). It is clear that both curves are mirror images from each other.
### Beating in the Mercator data
-.5 cm
To determine whether the beating is seen in the Mercator data, we extended the light curve constructed in Fig. \[fig:f1h3f2f3\] to the time range covered by the Mercator data (upper panel of Fig. \[fig:beatmerc\]). We remind that the amplitude of the light curve seems to be underestimated. Thus, the points of the second observational run on Mercator being quite off the light curve (by $\sim$0.12mag) is not an argument to reject a stable beating. Caution is needed because a small difference in the frequencies will cause a quite large difference in the [*global*]{} behaviour of the extrapolation.
Another and more robust method is depicted in the lower panel of the same figure. There, a simple harmonic fit was constructed through the extreme values (either minimum or maximum) of each year (1982–1988) and with a fixed frequency of $f_b$=$f'_1$$-$$f_1$=0.00034c/d. This fit is then mirrored with respect to the mean m$_{\rm V}$ value, and an enveloping curve is obtained. The data points of the second Mercator run again fall out of the permitted region, so they do not seem to be compatible with the beating. However, with a small change in the beating frequency ($f_b'$=0.00042c/d), we were able to make these points “permitted”.
Thus, a beating phenomenon is not excluded although the values of the frequencies around the main frequency seem not well defined. If a very close duplet or triplet of frequencies is present, the total time span of the observations could still be insufficient to resolve the exact frequency values. Alternatively, it is not excluded that the main frequency $f_1$ changes its value [*and*]{} its amplitude in time in a more or less random way. In this case the nearby frequencies are only “remnants” of the broadened peak in the periodogram not dealt with by the prewhitening.
For the secondary frequency, near 0.09c/d, we found indications of its presence in independent data sets, and it might correspond to the frequency $f$=0.0847c/d that was already found by @fernie86. Here also its value is either variable in time or some beating of closely spaced, so-far unresolved frequencies is present.
Discussion {#sect:dscssn}
==========
Luminosity {#sect:lmnst}
----------
In this first section of the discussion, the luminosity of is estimated. As a first step, the reddening of the object is quantified by calculating synthetic colour indices. Then, two different methods are used to obtain the luminosity of our object.
[**Synthetic colour indices and reddening**]{} The reddening is estimated by comparing the observed Geneva colours of our object with calculated synthetic colours. Such synthetic colours were calculated by combining the digitized passband of the Geneva filters [@rufener88] with the Kurucz model atmospheres [see @reyniers02a for more details]. The synthetic B-V of with model (T$_{\rm eff}$, $\log g$, \[M/H\]) = (6250K, 1.0, $-$1.5) is B-V=$-$0.574. The mean B-V in our Geneva data is B-V=$-$0.318, hence E(B-V)=0.256$\pm$0.05mag. This value is somewhat higher than the value derived by @fernie86: E(B-V)=0.10mag.\
[**Luminosity with trigonometric parallax**]{} The parallax found in the HIPPARCOS catalogue (retrieved via SIMBAD) is $\pi$=2.56($\pm$0.97)mas, which gives a distance of d$=$391pc, with an upper limit of 629pc and a lower limit of 283pc. Taking the reddening into account, we get an absolute visual magnitude of M$_{\rm V}$$=$$-$2.29. However, the uncertainty induced by the parallax is much larger than the uncertainty of the reddening:\
d$=$629pc $\rightarrow$ M$_{\rm V}$$=$$-$3.33\
d$=$283pc $\rightarrow$ M$_{\rm V}$$=$$-$1.60\
Assuming the bolometric correction to be negligible [see e.g. @flower96], we derive a luminosity of L$=$656L$_{\odot}$. An upper limit is obtained for this method by taking d$=$629pc, which yields L$=$1701L$_{\odot}$ or log(L/L$_{\odot}$)$=$3.2.\
[**Luminosity with P-L-\[Fe/H\] relation**]{} @nemec94 derived P-L-\[Fe/H\] relations and distances for several classes of Pop. II variables. Given the pulsational properties of together with its metal deficiency, there is strong evidence that this object belongs to the class of Pop. II Cepheids (see also Sect. \[subsect:wvrclassification\]). The P-L-\[Fe/H\] relation for this class depends on the pulsation mode (either fundamental or first-overtone). With the P-L-\[Fe/H\] relations given in @nemec94, and P$=$28.56d, we obtain\
M$_{\rm V}$(F)$=$$-$2.47($\pm$0.07)\
M$_{\rm V}$(H)$=$$-$2.92($\pm$0.07)\
where the error of 0.07 only accounts for the error on the P-L-\[Fe/H\] relation. This error is, again, probably underestimated, not only because the variables in the formula are uncertain (especially \[Fe/H\]), but also because the relation was derived with data from cluster variables. It is therefore not clear if this relation also holds for field stars like . Note also that with a maximum magnitude of M$_{\rm V}$$=$$-$3, we derive a distance of d$=$673pc (log(L/L$_{\odot}$)$=$3.1 and distance from the galactic plane $|Z|$=247pc).
UU Her classification {#subsect:uhrclassification}
---------------------
UU Her type stars are defined by @sasselov84 as variable F-type supergiants at high galactic latitudes having long periods and low amplitudes. Further characteristics of this group are [@bartkevicius92]:\
- metal deficiency
- high velocities
- very specific variability with relatively small amplitudes and two or three alternating periods form 20 to 120 days
- large infrared (IR) excesses due to circumstellar dust
-.2 cm However, an object does not have to possess all properties mentioned here to be a member of this heterogeneous class. The evolutionary status of these stars is not yet clear, but most characteristics clearly point to a final stage of high luminosity or a post-AGB stage.
Given its low metal content already mentioned by @mcdonald76 and its high galactic latitude, was mentioned as a possible UU Her candidate [@waelkens85; @sasselov85; @fernie86]. However, due to the apparently stable and short period (relative to other UU Her candidates) found by @waelkens85, the genuine character of as UU Her star was doubted. Moreover, has a “normal” radial velocity and no significant infrared excess has been observed by IRAS.
With the observational material presented here, we are able to further contest this UU Her classification. We confirm the stable @waelkens85 period of $\sim$28.6d. Our radial velocity data further confirm the “normal” velocity of . Also the fact that there is no significant reddening is opposite to what is seen in most UU Her type stars. Therefore, a [*post-AGB stage*]{} of our object is less likely. Also,\
- the chemical signature does [*not*]{} point to a 3rd dredge-up scenario (no carbon or s-process enhancement)
- the (however very uncertain) luminosity (not higher than log$L/L_{\odot}$$\simeq$3.2) is too low to support a post-AGB stage
-.2 cm The absence of a clear 3rd dredge-up signature is, however, not enough to exclude a post-AGB scenario [e.g. @vanwinckel97].
W Vir classification {#subsect:wvrclassification}
--------------------
W Virginis variables are pulsating stars of population II located in the Cepheid instability strip. In the HR diagram, W Vir stars occupy a region slightly lower and slightly to the right compared to the classical (Pop. I) Cepheids. Periods are between 12 and about 20 days and amplitudes between 0.3 and 1.2 mag. W Vir stars are in fact part of the larger class of Pop. II Cepheids, containing also the BL Her stars at the shorter period end (P=1$-$5d), and the RV Tauri stars at the longer period end (P=20$-$100d); see @wallerstein02 for a review of Pop. II Cepheids. Spectra of W Vir stars sometimes show hydrogen and helium emission near maximum light, and double metallic absorption lines, usually explained by the propagation of a shock wave through the photosphere.
The metal deficiency, the luminosity (too low for post-AGB) and most importantly the length of the period itself suggest a W Vir classification for . Furthermore, the CNO chemical pattern points to a 1st (post-RGB) rather than a 3rd (post-AGB) dredge up. The (anti-)correlations seen in the colours are reminiscent of variables in the instability strip. Moreover, the luminosity inherent to this classification is compatible (within the error) with the luminosity derived by the trigonometric parallax. On the other hand, the spectroscopic features seen in (strong hydrogen and helium emission and double metallic absorption lines) are not seen in the spectrum (except the weak emission in the wing of H$\alpha$). The photosphere of is seemingly much more stable than that of , and no strong shock wave is developed during the pulsation.
Lithium in
-----------
In Sect. \[sect:abndc\] we derived a Li abundance of log$\epsilon$(Li)=1.9. For an extensive discussion of Li in evolved stars, we refer to Sect. 5.2 in @reyniers01. The Li abundance of is [*not*]{} higher than the interstellar Li abundance of log$\epsilon$(Li)=3.1. However, since the initial Li abundance of (thought to be the Spite plateau) is approximately the present one, and since at least one mixing event has occurred in (as seen from the CNO pattern), the Li content is expected to be much lower than the initial one. Note that a Li dilution of a factor of 20 during the 1st dredge-up is expected for stars in this metallicity range [e.g. @gratton00]. Therefore, Li [*production*]{} is required in the case of .
An obvious Li producing scenario for , which was also invoked by @luck90, is a “hot bottom burning” (HBB) scenario in which the base of the convective envelope of a AGB star is hot enough for the hydrogen burning to ignite and in which fresh $^7$Li is quickly transported to cooler layers before it is destroyed by proton capture. However, only AGB stars of intermediate mass (4-5M$_{\odot}$) or equivalently high luminosity M$_{\rm bol}$$\la$$-$6 are thought to exhibit this HBB, and therefore such a scenario is difficult to reconcile with a W Vir or post-AGB classification. There exists also a class of Li enriched galactic low mass field Carbon stars [@abia93; @abia97; @abia00], but these stars are chemically very different from .
Turning to less evolved stars, Li enrichment is also seen in about one percent of “normal” G-K giants. Some of these giants show Li abundances even larger than that of the interstellar medium. While the exact mechanism responsible for this enrichment is still not known, internal production of $^7$Li is also for this group of stars most favourable [e.g. @delareza97]. Non-standard mixing models are developed to explain these high Li abundances and commonly dubbed “cool bottom processing” [CBP, @wasserburg95; @sackmann99]. An attractive but highly speculative scenario for the Li enhancement in could be a scenario in which turned into a Li-rich K giant during its ascent on the red giant branch and in which it was able to preserve this Li in its outer layers.
The 25.7day Population II Cepheid V42 in the globular cluster M5 [@carney98] is a metal poor (\[Fe/H\]=$-$1.22) star with a surprisingly high Li abundance of log$\epsilon$(Li)$\simeq$1.8. No intrinsic neutron-capture element enhancement was observed either. Using the distance to M5, the authors give an estimate for the mean absolute visual magnitude of M$_V$=$-$3.15. Apart from these characteristics very similar to , some differences are also seen. Oxygen for example is slightly deficient in this star, whereas sodium is enhanced. This could be important evidence that deep mixing has taken place, in which products of the ON part of the CNO cycle are brought from the hydrogen-burning shell to the surface.
Conclusion {#sect:cnclsns}
==========
In this paper an attempt was made to clarify the evolutionary status of the luminous F-type supergiant . The star was previously classified as a post-AGB star, based on its high galactic latitude and metal deficiency. However, no unambiguous [*chemical*]{} indications were found (carbon and s-process enrichment) to support this view.
Here, we confirm the main pulsational period of $\sim$28.6d. The high resolution, high signal-to-noise spectra confirmed the metal deficiency, the C deficiency, the N and O enhancement and the unexpectedly high Li abundance. A detailed abundance study of the s-process elements indicated a slight s-process enhancement which is in our view probably not intrinsic, but a consequence of the (small) scatter in the galactic chemical evolution. We were not able to give a conclusive evolutionary status for this star, but contrary to the post-AGB classification found in the literature, we suggest a W Vir classification instead.
Analysis of the different independent data sets showed that in this star a second periodicity is present near 11.1d. We suggest that the pulsation has a beating with frequency $f_b$=0.00034c/d. This could be a working hypothesis guiding future observations of .
MR acknowledge financial support from the Fund for Scientific Research - Flanders (Belgium). This research has made use of the Vienna Atomic Line Database (VALD), operated at Vienna, Austria, and the SIMBAD database, operated at CDS, Strasbourg, France. Hans Van Winckel, Peter De Cat and Katrien Kolenberg are warmly thanked for many improvements on early versions of the manuscript.
[53]{} natexlab\#1[\#1]{}
, C., [Boffin]{}, H. M. J., [Isern]{}, J., & [Rebolo]{}, R. 1993, , 272, 455
, C. & [Isern]{}, J. 1997, , 289, L11
, C. & [Isern]{}, J. 2000, , 536, 438
, A. 1992, Baltic Astronomy, 1, 194
, E., [Hibbert]{}, A., [Godefroid]{}, M., & [Vaeck]{}, N. 1993, , 412, 431
, E., [Hibbert]{}, A., [Godefroid]{}, M., [Vaeck]{}, N., & [Fawcett]{}, B. C. 1991, , 375, 818
, W. P. 1951, , 113, 304
, D. E., [Shallis]{}, M. J., & [Simmons]{}, G. J. 1980, , 81, 340
, B. W., [Fry]{}, A. M., & [Gonzalez]{}, G. 1998, , 116, 2984
, E., [Gratton]{}, R. G., & [Sneden]{}, C. 2000, , 356, 238
, R., [Drake]{}, N. A., [da Silva]{}, L., [Torres]{}, C. A. O., & [Martin]{}, E. L. 1997, , 482, L77
, P., [Reyniers]{}, M., [Van Winckel]{}, H., [Goriely]{}, S., & [Siess]{}, L. 2005, , submitted
ESA. 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200
, J. D. 1986, , 301, 302
, P. J. 1996, , 469, 355
, S., [Ferro]{}, A., & [Parrao]{}, L. 1997, , 109, 1077
, R. G., [Sneden]{}, C., [Carretta]{}, E., & [Bragaglia]{}, A. 2000, , 354, 169
, N. & [Sauval]{}, A. J. 1998, Space Science Reviews, 85, 161
, A., [Bi[' e]{}mont]{}, E., [Godefroid]{}, M., & [Vaeck]{}, N. 1991, , 88, 505
, H. 2001, in AIP Conf. Proc. 598: Joint SOHO/ACE workshop “Solar and Galactic Composition”, 23
, F., [Piskunov]{}, N., [Ryabchikova]{}, T. A., [Stempels]{}, H. C., & [Weiss]{}, W. W. 1999, , 138, 119
, R. 1993, CD-ROM set, Cambridge, MA : Smithsonian Astrophysical Observatory
, D. L., [Heath]{}, J. E., [Lemke]{}, M., & [Drake]{}, J. 1996, , 103, 183
, J. E., [Bonvallet]{}, G., & [Sneden]{}, C. 2001, , 556, 452
, J. E., [Wickliffe]{}, M. E., [Den Hartog]{}, E. A., & [Sneden]{}, C. 2001, , 563, 1075
, R. E., [Bond]{}, H. E., & [Lambert]{}, D. L. 1990, , 357, 188
, L. 1976, Bull. AAS, 8, 49
, M. H. & [O’Donoghue]{}, D. 1999, Delta Scuti Newsletter, 13, 28
, J. M., [Nemec]{}, A. F. L., & [Lutz]{}, T. E. 1994, , 108, 222
, P. E., [Chen]{}, Y. Q., [Schuster]{}, W. J., & [Zhao]{}, G. 2000, , 353, 722
, J. X. & [McWilliam]{}, A. 2000, , 537, L57
, M. 2002, Ph.D. Thesis, K.U.Leuven
, M. & [Van Winckel]{}, H. 2001, , 365, 465
, M., [Van Winckel]{}, H., [Bi[' e]{}mont]{}, E., & [Quinet]{}, P. 2002, , 395, L35
, M., [Van Winckel]{}, H., [Gallino]{}, R., & [Straniero]{}, O. 2004, , 417, 269
, D. H., [Lehar]{}, J., & [Dreher]{}, J. W. 1987, , 93, 968
, F. 1981, , 45, 207
, F. & [Nicolet]{}, B. 1988, , 206, 357
, I.-J. & [Boothroyd]{}, A. I. 1999, , 510, 217
, D. D. 1984, , 102, 161
, D. D. 1985, Informational Bulletin on Variable Stars, 2776, 1
, C. & [Cuypers]{}, J. 2004, in ASP Conf. Proc. 310 “Variable Stars in the Local Group”, eds. D.W. Kurtz and K.R. Pollard, 283
, A. 1991, , 253, 198
, R. F. 1978, , 224, 953
, W., [Knigge]{}, R., & [Ott]{}, H. 1965, Informational Bulletin on Variable Stars, 81, 1
, C., [Galli]{}, D., [Gallino]{}, R., [et al.]{} 1999, , 521, 691
, C., [Gallino]{}, R., [Arnone]{}, E., [et al.]{} 2004, , 601, 864
, H. 1997, , 319, 561
, H. 2003, , 41, 391
, H. & [Reyniers]{}, M. 2000, , 354, 135
, C. & [Burnet]{}, M. 1985, Informational Bulletin on Variable Stars, 2808, 1
, G. 2002, , 114, 689
, G. J., [Boothroyd]{}, A. I., & [Sackmann]{}, I.-J. 1995, , 447, L37
[^1]: Postdoctoral fellow of the Fund for Scientific Research, Flanders
[^2]: based on observations collected at the European Southern Observatory, La Silla, Chile (programme 61.E-0426), and at the Observatorio del Roque de los Muchachos, La Palma, Spain
|
---
abstract: |
This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere.
We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered.
The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability $p$ of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to *some* Bernoulli measure $B_p$. A notion of “uniform test” for Bernoulli sequences is introduced which allows a quantitative strengthening of this result. Uniform tests are then generalized to arbitrary measures.
Bernoulli measures $B_{p}$ have the important property that $p$ can be recovered from each random sequence of $B_{p}$. The paper studies some important consequences of this orthogonality property (as well as most other questions mentioned above) also in the more general setting of constructive metric spaces.
author:
- |
Laurent Bienvenu[^1], Peter Gács[^2], Mathieu Hoyrup[^3],\
Cristobal Rojas[^4], Alexander Shen[^5]
bibliography:
- 'ait.bib'
- 'publ.bib'
title: Algorithmic tests and randomness with respect to a class of measures
---
Introduction
============
This paper, though intended to be rather self-contained, can be seen as a continuation of [@GacsUnif05] (which itself built on earlier work of Levin) and [@HoyrupRojasRandomness09].
Our enterprise is to develop the theory of randomness beyond the framework where the underlying probability distribution is the uniform distribution or a computable distribution. A randomness test ${\mathbf{t}}(\omega,{P})$ of object $\omega$ with respect to measure ${P}$ is defined to be a function of both the measure ${P}$ and the point $\omega$.
In some later parts of the paper, we will also go beyond the case where the underlying space is the set of finite or infinite sequences: rather, we take a constructive metric space with its algebra of Borel sets.
We will apply the above notion of test to define, following ideas of [@LevinRand73], for a class ${\mathcal{C}}$ of measures having some compactness property, a “class test” ${\mathbf{t}}_{{\mathcal{C}}}(\omega)$. This is a test to decide whether object $\omega$ is random with respect to any one measure ${P}$ in the class ${\mathcal{C}}$. We will show that in case of the class of Bernoulli measures over binary sequences, this notion is equivalent to the class tests introduced by Martin-Löf in [@MLof66art].
In case there is an effective sense in which the elements of the class are mutually orthogonal, we obtain an especially simple separation of the randomness test ${\mathbf{t}}(\omega,{P})$ into two parts: the class test and an arbitrarily simple test for “typicality” with respect to the measure ${P}$. In some natural special cases, the typicality test corresponds to a convergence property of relative frequencies, allowing to apply the theory to any general effectively compact class of ergodic stationary processes.
There are some properties of randomness tests ${\mathbf{t}}(\omega,{P})$ that depend on the measure ${P}$, which our tests do not necessarily possess, for example a kind of monotonicity in ${P}$. It is therefore notable that in case of the orthogonal classes, randomness is equivalent to an “blind” notion of randomness, that only considers randomness tests that do not depend on the measure ${P}$.
Here is an outline of the paper. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of tests. A randomness test provides a quantitative measure of non-randomness, called “randomness deficiency”; it is finite for random sequences and infinite for non-random ones. There are two versions of these tests (“average-bounded” and “probability-bounded” ones); a relation between them is established.
A formula that expresses the (average-bounded) randomness deficiency in terms of prefix complexity is given (in two forms). It implies the Levin-Schnorr criterion of randomness (with prefix complexity, as in the special case first announced in Chaitin’s paper [@Chaitin75]). Some approaches that go in another direction (from deficiency to complexity) are considered.
The notion of Bernoulli sequence (looking like the outcome of independent coin tosses for an asymmetric coin) is defined. It is shown that the set of Bernoulli sequences is the union (over all $p\in{[0,1]}$) of the sets of sequences that are random with respect to $B_p$, the Bernoulli measure with probability $p$; here we assume that $p$ is given as an oracle). A notion of “uniform test” for Bernoulli sequences is introduced. Then the statement above is proved in the following quantitative form: the Bernoulli deficiency is the infimum of $B_{p}$ deficiencies over all $p\in{[0,1]}$.
The notion of general uniform test (not restricted to the class of Bernoulli measures) is introduced. It is shown that it generalizes Martin-Löfs earlier definition of randomness (which was given only for computable measures).
Bernoulli measures $B_{p}$ have the important property that $p$ can be recovered from each random sequence of $B_{p}$. The paper studies some important consequences of this orthogonality property (as well as most other questions mentioned above) also in the more general setting of constructive metric spaces.
The following notation is useful, since inequalities hold frequently only within an additive or multiplicative constant.
We will write $f(x){\stackrel{{}_*}{<}}g(x)$ for inequality between positive functions within a multiplicative constant, that is for the relation $f(x)=O(g(x))$: precisely, if there is a constant $c$ with $f(x){\leqslant}c g(x)$ for all $x$. The relation $f{\stackrel{{}_*}{=}}g$ means $f{\stackrel{{}_*}{<}}g$ and $f{\stackrel{{}_*}{>}}g$. Similarly, $f{\stackrel{{}_+}{<}}g$ and $f{\stackrel{{}_+}{=}}g$ means inequality within an additive constant.
Let $\Lambda$ denote the empty string.
Logarithms are taken, as a default, to base 2. We use ${|x|}$ to denote the length of a string $x$. For finite string, $x$ and finite or infinite string $y$ let $x{\sqsubseteq}y$ denote that $x$ is a prefix of $y$. If $x$ is a finite or infinite sequence then its elements are written as $x(1), x(2),\dots$, and its prefix of size $n$ will be denoted by $x(1:n)$.
Let $\overline{\mathbb{R}}_{+}={[0,\infty]}$ be the set of nonnegative reals, with the special value $\infty$ added. The binary alphabet $\{0,1\}$ will also be denoted by ${\mathbb{B}}$.
Randomness on sequences, for computable measures
================================================
Lower semicomputable functions on sequences
-------------------------------------------
In the first sections, we will study randomness over infinite binary sequences.
\[def:binary-Cantor\] We will denote by $\Omega$ the set of infinite binary sequences, and call it also the [*binary Cantor space*]{}. For a finite string $x$ let $x\Omega$ be the set of all infinite sequences that have finite prefix $x$. These sets will be called [*basic open sets*]{}, the set of all basic open set is called the [*basis*]{} of $\Omega$ (as a topological space). A subset of $\Omega$ is [*open*]{} if it is the union of a set of basis elements.
The set of infinite sequences of natural numbers will be called the [*Baire space*]{}. Basic open sets and open sets can be defined for it analogously.
A notion somewhat weaker than computability will play crucial role.
\[def:lower-semicomp.seqs\] An open set $G\subseteq\Omega$ is called [*effectively open*]{}, or [*lower semicomputable open*]{}, or [*c.e. open*]{}, or [*r.e. open*]{} if it is the union of a computable sequence $x_{i}\Omega$ of basic elements. A set is [*upper semicomputable closed*]{}, or [*effectively closed*]{} if its complement is effectively open.
A set $\Gamma$ is called [*effectively $G_{\delta}$*]{} if there is a sequence of sets $U_{k}$, $k=1,2,\dots$ effectively open uniformly in $k$ such that $\Gamma=\bigcap_{k}U_{k}$.
A function $t:\Omega\to{[0,\infty]}$ is [*lower semicomputable*]{} if
\[i:lower-semicomp.seqs.lower-semicont\] For any rational $r$ the set $
\{\omega : r < t(\omega)\}
$ is open in $\Omega$, that is is a union of intervals $x\Omega$.
Moreover, this set is effectively open uniformly in $r$, that is there exists an algorithm that gets $r$ as input and generates strings $x_0,x_1,\ldots$ such that the union of interval $x_{i}\Omega$ is equal to $\{\omega: r<t(\omega)\}$.
This definition is a constructive version of the classical notion of lower semicontinuous function as in requirement (\[i:lower-semicomp.seqs.lower-semicont\]). The same class of lower semicomputable functions has other (equivalent) definitions; here is one of them.
\[def:basic-func.seqs\] A function $u$ defined on $\Omega$ and having rational values is called [*basic*]{} if the value $u(\omega)$ is determined by some finite prefix of $\omega$.
If this prefix has length $N$, the function can be presented as a table with $2^N$ rows; each row contains $N$ bits (the values of the first $N$ bits of $\omega$) and a rational number (the value of the function). Such a function is a finite object.
The proof of the following proposition is a simple exercise:
\[propo:lower-semi-limit.seqs\] The (pointwise) limits of monotonic sequences of basic functions are exactly the lower semicomputable functions on $\Omega$.
Since the difference of two basic functions is a basic function, we can reformulate this criterion as follows: lower semicomputable functions are (pointwise) sums of computable series made of non-negative basic functions.
One more way to define a lower semicomputable function goes as follows.
\[def:generate-lower\] Let $T$ be a lower semicomputable function on the set $\{0,1\}^{*}$ of finite sequences of zeros and ones with non-negative (finite or infinite) values. This means that the set of pairs ${(x,r)}$ such that $r<T(x)$ is enumerable. Then function $t$ defined as $$t(\omega)=\sup_{x{\sqsubseteq}\omega}T(x)$$ is a lower semicomputable function on $\Omega$: we will say that function $T(\cdot)$ [*generates*]{} function $t(\cdot)$ if it is also monotone: $T(x){\leqslant}T(y)$ if $x{\sqsubseteq}y$.
The monotonicity requirement can always be satisfied by taking $T'(x)=\max_{z{\sqsubseteq}x}T(z)$.
\[propo:generate-lower\] Any lower semicomputable function $t$ on $\Omega$ is generated by an appropriate function $T$ on $\{0,1\}^{*}$ this way.
We may also assume that $T$ is a computable function with rational values. Indeed, since only the supremum of $T$ on all the prefixes is important, instead of increasing $T(x)$ for some $x$ we may increase $T(y)$ for all $y{\sqsupseteq}x$ of large length; this delay allows $T$ to be computable.
For a given lower semicomputable function $t$ on $\Omega$ there exists a maximal monotonic function $T$ on finite strings that generates $t$ (in the sense just described). This maximal $T$ can be defined as follows: $$\label{eq:T-as-inf}
T(x)=\inf_{\omega{\sqsupseteq}x} t(\omega).$$ Let us now exploit the finiteness of the binary alphabet $\{0,1\}$, which implies that the space $\Omega$ is a compact topological space.
\[propo:inf-lower-semicomp.seqs\] The function $T$ defined by is lower semicomputable. In the definition, we can replace $\inf$ by $\min$.
Indeed, $r<\inf_{\omega{\sqsupseteq}x} t(\omega)$ if and only if there exists some rational $r'>r$ with $r'<t(\omega)$ for all $\omega{\sqsupseteq}x$. The last condition can be reformulated: the open set of all sequences $\omega$ such that $t(\omega)>r'$ is a superset of $x\Omega$. This open set is a union of an enumerable family of intervals; if these intervals cover $x\Omega$, compactness implies that this is revealed at some finite stage, so the condition is enumerable (and the existential quantifier over $r'$ keeps it enumerable).
Since the function $t(\omega)$ is lower semicontinuous, it actually reaches its infimum on the compact set $x\Omega$, so $\inf$ can be replaced with $\min$.
Randomness tests
----------------
We assume that the reader is familiar with the basic concepts of measure theory and integration, at least in the space $\Omega$ of infinite binary sequences. A measure ${P}$ on $\Omega$ is determined by the values $$\begin{aligned}
{P}(x)={P}(x\Omega)
\end{aligned}$$ which we will denote by the same letter $P$, without danger of confusion. Moreover, any function ${P}:\{0,1\}^{*}\to{[0,1]}$ with the properties $$\begin{aligned}
\label{eq:measure.Omega}
{P}(\Lambda)=1,\quad{P}(x)={P}(x0)+{P}(x1)
\end{aligned}$$ uniquely defines a measure (this is a particular case of Caratheodory’s theorem).
\[def:computable-measure-Omega\] A real number is called [*computable*]{} if there is an algorithm that for all rational $\varepsilon$ returns a rational approximation of $x$ with error not greater than $\varepsilon$. Computable numbers can also be determined as limits of sequences $x_{1},x_{2},\dots$ for which $|x_{n}-x_{n+k}|{\leqslant}2^{-n}$. An infinite sequence $s_{1},s_{2},\dots$ of real numbers is a [*strong Cauchy*]{} sequence if for all $m<n$ we have $|s_{m}-s_{n}|{\leqslant}2^{-m}$.
A function determined on words (or other constructive objects) is [*computable*]{} if its values are computable uniformly from the input, that is there is an algorithm that from each input and $\varepsilon>0$ returns an $\varepsilon$-approximation of the function value on this input.
Measure ${P}$ over $\Omega$ is said to be [*computable*]{} if the function ${P}:\{0,1\}^{*}\to{[0,1]}$ is computable.
\[def:test-computable-measure\] Let ${P}$ be a computable probability distribution (measure) on $\Omega$. A lower semicomputable function $t$ on $\Omega$ with non-negative (possible infinite) values is an ([*average-bounded*]{}) [*randomness test*]{} with respect to ${P}$ (or [*${P}$-test*]{}) if the expected value of $t$ with respect to ${P}$ is at most $1$, that is $$\int t(\omega)\,d{P}{\leqslant}1.$$ A sequence $\omega$ [*passes*]{} a test $t$ if $t(\omega)<\infty$. A sequence is called [*random*]{} with respect to ${P}$ it is passes all ${P}$-randomness tests (as defined above).
The intuition: when $t(\omega)$ is large, this means that test $t$ finds a lot of “regularities” in $\omega$. Constructing a test, we are allowed to declare whatever we want as a “regularity”; however, we should not find too many of them on average: if we declare too many sequences to be “regular”, the average becomes too big.
This definition turns out to be equivalent to randomness as defined by Martin-Löf (see below). But let us start with the universality theorem:
\[thm:universality.cptable\] For any computable measure ${P}$ there exists a [*universal*]{} (maximal) ${P}$-test $u$: this means that for any other ${P}$-test $t$ there exists a constant $c$ such that $$t(\omega){\leqslant}c\cdot u(\omega)$$ for every $\omega\in\Omega$.
In particular, $u(\omega)$ is finite if and only if $t(\omega)$ is finite for every ${P}$-test $t$, so the sequences that pass test $u$ are exactly the random sequences.
Let us enumerate the algorithms that generate all lower semicomputable functions. Such an algorithm produces a monotone sequence of basic functions. Before letting through the next basic function of this sequence, let us check that its ${P}$-expectation is less than $2$. If the algorithm considered indeed defines a ${P}$-test, this expectation does not exceed $1$, so by computing the values of ${P}$ with sufficient precision we are able to guarantee that the expectation is less than $2$. If this checking procedure does not terminate (or gives a negative result), we just do not let the basic function through.
In this way we enumerate all tests as well as some lower semicomputable functions that are not exactly tests but are at most twice bigger than tests. It remains to sum up all these functions with positive coefficients whose sum does not exceed $1/2$ (say, $1/2^{i+2}$).
Recall the definition of randomness according to Martin-Löf.
Let ${P}$ be a computable distribution over $\Omega$. A sequence of open sets $U_{1},U_{2},\dots$ is called a [*Martin-Löf test*]{} for ${P}$ if the sets $U_{i}$ are effectively open in a uniform way (that is $U_{i}=\bigcup_{j}x_{ij}\Omega$ where the double sequence $x_{ij}$ of strings is computable), moreover ${P}(U_{k}){\leqslant}2^{-k}$ for all $k$.
A set $N$ is called a [*constructive (effective) null set*]{} for the measure ${P}$ if there is a Martin-Löf test $U_{1},U_{2},\dots$ with the property $N=\bigcap_{k}U_{k}$. Note that effective null sets are constructive $G_{\delta}$ sets.
A sequence $\omega\in\Omega$ is said to [*pass*]{} the test $U_{1},U_{2},\dots$ if it is not in $N$. It is [*Martin-Löf-random*]{} with respect to measure ${P}$ if it is not contained in any constructive null set for ${P}$.
The following theorem is not new, see for example [@LiViBook97].
A sequence $\omega$ passes all average-bounded ${P}$-tests (=passes the universal ${P}$-test) if and only if it is Martin-Löf random with respect to ${P}$.
If $t$ is a test, then the set of all $\omega$ such that $t(\omega)>N$ is an effectively open set that can be found effectively given $N$. This set has ${P}$-measure at most $1/N$ (by Chebyshev’s inequality), so the sets of sequences $\omega$ that do not pass $t$ (that is $t(\omega)$ is infinite) is an effectively ${P}$-null set.
On the other hand, let us show that for every effectively null set $Z$ there exists an average-bounded test that is infinite at all its elements. Indeed, for every effectively open set $U$ the function $1_{U}$ that is equal to $1$ inside $U$ and to $0$ outside $U$ is lower semicomputable. Then we can get a test $\sum_{i} 1_{U_i}$. The average of this test does not exceed $\sum_{i} 2^{-i}=1$, while the sum is infinite for all elements of $\bigcap_{i} U_i$.
When talking about randomness for a computable measure, we will write [*randomness*]{} from now on, understanding Martin-Löf randomness, since no other kind will be considered.
Sometimes it is useful to switch to the logarithmic scale.
\[def:deficiency\] For every computable measure ${P}$, we will fix a universal ${P}$-test and denote it by ${\mathbf{t}}_{{P}}(\omega)$. Let ${\mathbf{d}}_{{P}}(\omega)$ be the logarithm of the universal test ${\mathbf{t}}_{{P}}(\omega)$: $${\mathbf{t}}_{{P}}(\omega)=2^{{\mathbf{d}}_{{P}}(\omega)}.$$ With other kinds of test also, it will be our convention to use ${\mathbf{t}}$ (boldface) for the universal test, and ${\mathbf{d}}$ (boldface) for its logarithm.
In a sense, the function ${\mathbf{d}}_{{P}}$ measures the randomness deficiency in bits.
The logarithm, along with the requirement $\int {\mathbf{t}}_{{P}}(\omega)\,d{P}{\leqslant}1$, implies that ${\mathbf{d}}_{{P}}(\omega)$ may have some negative values, and even values $-\infty$. By just choosing a different universal test we can always make ${\mathbf{d}}_{{P}}(\omega)$ bounded below by, say, $-1$, and also integer-valued. On the other hand, if we want to make it nonnegative, we will have to lose the property $\int 2^{{\mathbf{d}}_{{P}}(\omega)}d{P}{\leqslant}1$, though we may still have $\int 2^{{\mathbf{d}}_{P}(\omega)}d{P}{\leqslant}2$. It will still have the following property:
The function ${\mathbf{d}}_{P}(\cdot)$ is lower semicomputable and is the largest (up to an additive constant) among all lower semicomputable functions such that the ${P}$-expectation of $2^{{\mathbf{d}}_{{P}}(\cdot)}$ is finite.
As we have shown, for any fixed computable measure ${P}$ the value ${\mathbf{d}}_{{P}}(\omega)$ (and ${\mathbf{t}}_{{P}}(\omega)$) is finite if and only if the sequence $\omega$ is Martin-Löf random with respect to ${P}$.
\[rem:ML-tests\]
1. \[i:prob-bounded\] Each Martin-Löf’s test ($U_{1},U_{2},\dots$) is more directly related to a lower semicomputable function $F(\omega)=\sup_{\omega\in U_{i}} i$. This function has the property $P{\mathopen[\,F(\omega){\geqslant}k\,\mathclose]}{\leqslant}2^{-k}$. Such functions will be called [*probability-bounded*]{} tests, and were used in [@ZvLe70]. We will return to such functions later (subsection \[subsec:average-probability\]).
2. We have defined ${\mathbf{d}}_{P}(\omega)$ separately for each computable measure ${P}$ (up to a constant). We will later give a more general definition of randomness deficiency ${\mathbf{d}}(\omega,{P})$ as a function of two variables ${P}$ and $\omega$ that coincides with ${\mathbf{d}}_{P}(\omega)$ for every computable ${P}$ up to a constant depending on ${P}$.
Average-bounded and probability-bounded deficiencies {#subsec:average-probability}
----------------------------------------------------
Let us refer for example to [@LiViBook97; @ShenUppsala00] for the definition of and basic properties of plain and prefix (Kolmogorov) complexity. We will define prefix complexity in Definition \[def:prefix\] below, though. We will not use complexities explicitly in the present section, just refer to some of their properties by analogy.
The definition of a test given above resembles the definition of prefix complexity; we can give another one which is closer to plain complexity. For that we use a weaker requirement: we require that the ${P}$-measure of the set of all sequences $\omega$ such that $t(\omega)>N$ does not exceed $1/N$. (This property is true if the integral does not exceed $1$, due to Chebyshev’s inequality.)
In logarithmic scale this requirement can be restated as follows: the ${P}$-measure of the set of all sequences whose deficiency is greater than $n$ does not exceed $2^{-n}$. If we restrict tests to integer values, we arrive at the classical Martin-Löf tests: see also Remark \[rem:ML-tests\], part \[i:prob-bounded\].
While constructing an universal test in this sense, it is convenient to use the logarithmic scale and consider only integer values of $n$. As before, we enumerate all tests and “almost-tests” $d_i$ (where the measure is bounded by twice bigger bound) and then take the weighted maximum in the following way: $${\mathbf{d}}(\omega)=\max_i [d_i(\omega)-i] - c.$$ Then ${\mathbf{d}}$ is less than $d_i$ only by $i+c$, and the set of all $\omega$ such that ${\mathbf{d}}(\omega)>k$ is the union of sets where $d_i(\omega)>k+i+c$. Their measures are bounded by $O(2^{-k-i-c})$ and for a suitable $c$ the sum of measures is at most $2^{-k}$, as required.
In this way we get two measures of non-randomness that can be called “average-bounded deficiency” ${\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}}$ (the first one, related to the tests called “integral tests” in [@LiViBook97]) and “probability bound deficiency” ${\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{prob}}}}}$ (the second one). It is easy to see that they define the same set of nonrandom sequences (=sequences that have infinite deficiency). Moreover, the finite values of these two functions are also rather close to each other:
$${\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}}(\omega){\stackrel{{}_+}{<}}{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{prob}}}}}(\omega)
{\stackrel{{}_+}{<}}{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}} (\omega)+ 2 \log {\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}}(\omega).$$
Any average-bounded test is also a probability-bounded test, therefore ${\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}}(\omega) {\stackrel{{}_+}{<}}{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{prob}}}}}(\omega)$.
For the other direction, let $d$ be a probability-bounded test (in the logarithmic scale). Let us show that $d-2\log d$ is an average-bounded test. Indeed, the probability of the event “$d(\omega)$ is between $i-1$ and $i$” does not exceed $1/2^{i-1}$, the integral of $2^{d-2\log d}$ over this set is bounded by $2^{-i+1}2^{i-2\log i}=2/i^2$ and therefore the integral over the entire space converges.
It remains to note that the inequality $a{\stackrel{{}_+}{<}}b+2\log b$ follows from $b{\stackrel{{}_+}{>}}a-2\log a$. Indeed, we have $b{\geqslant}a/2$ (for large enough $a$), hence $\log a{\leqslant}\log b+1$, and then $a{\stackrel{{}_+}{<}}b+2\log a{\stackrel{{}_+}{<}}b+2\log b$.
In the general case the question of the connection between boundedness in average and boundedness in probability is addressed in the paper [@ShaferShenVereshVovk09]. It is shown there (and this is not difficult) that if $u:{[1,\infty]}\to{[0,\infty]}$ is a monotonic continuous function with $\int_{1}^{\infty}u(t)/t^{2}\,d t{\leqslant}1$, then $u(t(\omega))$ is an average-bounded test for every probability-bounded test $t$, and that this condition cannot be improved. (Our estimate is obtained by choosing $u(t)=t/\log^{2} t$.)
This statement resembles the relation between prefix and plain description complexity. However, now the difference is bounded by the logarithm of the *deficiency* (that is bounded independently of length for the sequences that are close to random), not of the *complexity* (as usual), which would be normally growing with the length.
It would be interesting to understand whether the two tests differ only by a shift of scale or in some more substantial way. For the confirmation of such a more substantial difference could serve two families of sequences $\omega_{i}$ and $\omega'_{i}$ for which $$\begin{aligned}
{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}}(\omega'_{i})-{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{aver}}}}}(\omega_{i}) \to \infty
\end{aligned}$$ for $i\to\infty$, while $$\begin{aligned}
{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{prob}}}}}(\omega'_{i})-{\mathbf{d}}^{{{\text{\rmfamily\mdseries\upshape{prob}}}}}(\omega_{i}) \to -\infty.
\end{aligned}$$ The authors do not know whether such a family exists.
A formula for average-bounded deficiency
----------------------------------------
Let us recall some concepts connected with the prefix description complexity. For reference, consult for example [@LiViBook97; @ShenUppsala00].
\[def:prefix\] A set of strings is called [*prefix-free*]{} if no element of it is a prefix of another element. A computable partial function $T:\{0,1\}^{*}\to\{0,1\}^{*}$ is called a [*self-delimiting interpreter*]{} if its domain of definition is a prefix-free set. We define the complexity ${\mathit{Kp}}_{T}(x)$ of a string $x$ with respect to $T$ as the length of a shortest string $p$ with $T(p)=x$. It is known that there is an [*optimal*]{} (self-deliminiting) interpreter: that is a (self-delimiting) interpreter $U$ with the property that for every self-delimiting interpreter $T$ there is a constant $c$ such that for every string $x$ we have ${\mathit{Kp}}_{U}(x){\leqslant}{\mathit{Kp}}_{T}(x)+c$. We fix an optimal self-delimiting interpreter $U$ and denote ${\mathit{Kp}}(x)={\mathit{Kp}}_{U}(x)$.
We also denote ${\mathbf{m}}(x)=2^{-{\mathit{Kp}}(x)}$, and call it sometimes [*discrete a priori probability*]{}.
The “a priori” name comes from some interpretations of a property that distinguishes the function ${\mathbf{m}}(x)$ among certain “weight distributions” called semimeasures.
\[def:semimeasure\] A function $f:\{0,1\}^{*}\to{[0,\infty)}$ is called a [*discrete semimeasure*]{} if $\sum_{x}f(x){\leqslant}1$.
Lower semicomputable semimeasures arise as the output distribution of a randomized algorithm using a source of random numbers, and outputting some word (provided the algorithm halts; with some probability, it may not halt). It is easy to check that ${\mathbf{m}}(x)$ is a lower semicomputable discrete semimeasure.
Recall the following fact.
\[propo:coding\] Among lower semicomputable discrete semimeasures, the function ${\mathbf{m}}(x)$ is maximal within a multiplicative constant: that is for every lower semicomputable discrete semimeasure $f(x)$ there is a constant $c$ with $c\cdot{\mathbf{m}}(x){\geqslant}f(x)$ for all $x$.
The universal average-bounded randomness test ${\mathbf{t}}_{{P}}1$ (the largest lower semicomputable function with bounded expectation) can be expressed in terms of a priori probability (and therefore prefix complexity):
\[propo:sum-characteriz\] Let ${P}$ be a computable measure and let ${\mathbf{t}}_{{P}}$ be the universal average-bounded randomness test with respect to ${P}$. Then $${\mathbf{t}}_{{P}}(\omega){\stackrel{{}_*}{=}}\sum_{x{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x)}{{P}(x)}.$$ (If ${P}(x)=0$, then the ratio ${\mathbf{m}}(x)/{P}(x)$ is considered to be infinite.)
A lower semicomputable function on sequences is a limit of an increasing sequence of basic functions.
Withouth loss of generality we may assume that each increase is made on some cylinder $x\Omega$. In other terms, we increase the “weight” $w(x)$ of $x$ and let our basic function on $\omega$ be the sum of the weights of all prefixes of $\omega$. The weights increase gradually: at any moment, only finitely many weights differ from zero. In terms of weights, the average-boundedness condition translates into $$\begin{aligned}
\sum_{x}{P}(x)w(x){\leqslant}1,
\end{aligned}$$ so after multiplying the weights by ${P}(x)$, this condition corresponds exactly to the semimeasure requirement. Let us note that due to the computability of ${P}$, the lower semicomputability is conserved in both directions (multiplying or dividing by ${P}(x)$). More formally, the function $$\begin{aligned}
\sum_{x{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x)}{{P}(x)}
\end{aligned}$$ is a lower semicomputable average-bounded test: its integral is exactly $\sum_{x}{\mathbf{m}}(x)$. On the other hand, every lower semicomputable test can be presented in terms of an increase of weights, and the limits of these weights, multiplied by ${P}(x)$, form a lower semicomputable semimeasure. (Note that the latter transformation is not unique: we can redistribute the weights among a string and its continuations without altering the sum over the infinite sequences.)
Note that we used that both ${P}$ (in the second part of the proof) and $1/{P}$ (in the first part) are lower semicomputable.
In Proposition \[propo:sum-characteriz\], we can replace the sum with a least upper bound. This way, the following theorem connects three quantities, ${\mathbf{t}}_{{P}}$, the supremum and the sum, all of which are equal within a multiplicative constant.
\[thm:randomness-complexity\] We have $
{\mathbf{t}}_{{P}}(\omega) {\stackrel{{}_*}{=}}\sup_{x{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x)}{{P}(x)}
{\stackrel{{}_*}{=}}\sum_{x{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x)}{{P}(x)}
$, or in logarithmic notation $$\begin{aligned}
\label{eq:randomness-complexity}
{\mathbf{d}}_{{P}}(\omega) &{\stackrel{{}_+}{=}}\sup_{x{\sqsubseteq}\omega}{{\left( -\log{P}(x)-{\mathit{Kp}}(x)\right)}}.
\end{aligned}$$
The supremum is now smaller, so only the second part of the proof of Proposition \[propo:sum-characteriz\] should be reconsidered.
The lower semicomputable function ${{\lceil {\mathbf{d}}_{{P}}(\omega)\rceil}}$ can be obtained as the supremum of a sequence of integer-valued basic functions of the form $k_{i}g_{x_{i}}(\omega)$, where $g_{x}(\omega)=1_{x\Omega}(\omega)=1$ if $x{\sqsubseteq}\omega$ and 0 otherwise. We can also require that if $i\ne j$, $x_{i}{\sqsubseteq}x_{j}$ then $k_{i}\ne k_{j}$: indeed, suppose $k_{i}=k_{j}$. If $i<j$ then we can delete the $j$th element, and if $i>j$, then we can replace $2^{k_{i}}g_{x_{i}}$ with the sequence of all functions $2^{k_{i}}g_{z}$ where $z$ has the same length as $x_{j}$ but differs from it. We have $$\begin{aligned}
2{\mathbf{t}}_{{P}}(\omega)&{\geqslant}2^{{{\lceil {\mathbf{d}}_{{P}}(\omega)\rceil}}}
=\sup_{i}2^{k_{i}}g_{x_{i}}(\omega)=\sup_{x_{i}{\sqsubseteq}\omega}2^{k_{i}}
{\geqslant}2^{-1}\sum_{i:x_{i}{\sqsubseteq}\omega}2^{k_{i}}=2^{-1}\sum_{i}2^{k_{i}}g_{x_{i}}(\omega).
\end{aligned}$$ The last inequality holds since according to our assumption, all the values $k_{i}$ belonging to prefixes $x_{i}$ of the same sequence $\omega$ are different, and the sum of different powers of 2 is at most twice larger than its largest element. Integrating by ${P}$, we obtain $4{\geqslant}\sum_{i}2^{k_{i}}{P}(x_{i})$, hence $2^{k_{i}}{P}(x_{i}){\stackrel{{}_*}{<}}{\mathbf{m}}(x_{i})$ by the maximality of ${\mathbf{m}}(x)$, so $2^{k_{i}}{\stackrel{{}_*}{<}}\frac{{\mathbf{m}}(x_{i})}{{P}(x_{i})}$. We found $$\begin{aligned}
{\mathbf{t}}_{{P}}(\omega){\stackrel{{}_*}{<}}\sup_{i:x_{i}{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x_{i})}{{P}(x_{i})}
{\leqslant}\sup_{x{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x)}{{P}(x)}.
\end{aligned}$$
Here is a reformulation: $${\mathbf{d}}_{{P}}(\omega) {\stackrel{{}_+}{=}}\sup_{n}{{\left( -\log{P}(\omega(1:n))-{\mathit{Kp}}(\omega(1:n))\right)}}.$$ This reformulation can be generalized:
\[thm:seq-of-prefixes\] Let $n_{1}<n_{2}<\dotsm$ be an arbitrary computable sequence of natural numbers. Then $${\mathbf{d}}_{{P}}(\omega) {\stackrel{{}_+}{=}}\sup_{k}{{\left( -\log{P}(\omega(1:n_{k}))-{\mathit{Kp}}(\omega(1:n_{k}))\right)}}.$$ The constant in the ${\stackrel{{}_+}{=}}$ depends on the sequence $n_{k}$.
Every step of the proof of Theorem \[thm:randomness-complexity\] generalizes to this case straightforwardly.
This theorem has interesting implications of the case when instead of a sequence $\omega$ we consider an infinite two-dimensional array of bits. Then for the randomness deficiency, it is sufficient to compare complexity and probability of squares starting at the origin.
### Historical digression {#historical-digression .unnumbered}
The above formula for randomness deficiency is a quantitative refinement of the following criterion.
\[thm:rand-prefix\] A sequence $\omega$ is random with respect to a computable measure ${P}$ if and only if the difference $-\log{P}(x) - {\mathit{Kp}}(x)$ is bounded above for its prefixes.
(Indeed, the last theorem says that the maximum value of this difference over all prefixes is exactly the average-bounded randomness deficiency.) This characterization of randomness was announced first, without proof, in [@Chaitin75], with the proof attributed to Schnorr. The first proof, for the case of a computable measure, appeared in [@GacsExact80].
The historically first clean characterizations of randomness in terms of complexity, by Levin and Schnorr independently in [@LevinRand73] and [@Schnorr73] have a similar form, but use complexity and a priori probability coming from a different kind of interpreter called “monotonic”. (In the cited work, Schnorr uses a slightly different form of complexity, but later, he also adopted the version introduced by Levin.)
\[def:monotonic\] Let us call to strings [*compatible*]{} if one is the prefix of the other. An enumerable subset $A\subseteq \{0,1\}^{*}\times\{0,1\}^{*}$ is called a [*monotonic interpreter*]{} if for every $p,p',q,q'$, if ${(p,q)}\in A$ and ${(p',q')}\in A$ and $p$ is compatible with $p'$ then $q$ is compatible with $q'$. For an arbitrary finite or infinite $p\in\{0,1\}^{*}\cup\Omega$, we define $$\begin{aligned}
A(p)=\sup{\mathopen\{\,x : \exists p'{\sqsubseteq}p\;{(p',x)}\in A\,\mathclose\}}.
\end{aligned}$$ The monotonicity property implies that this limit, also in $\{0,1\}^{*}\cup\Omega$, is well defined.
We define the (monotonic) complexity ${\mathit{Km}}_{A}(x)$ of a string $x$ with respect to $A$ as the length of a shortest string $p$ with $A(p){\sqsupseteq}x$. It is known that there is an [*optimal*]{} monotonic interpreter, where optimality has the same sense as above, for prefix complexity. We fix an optimal monotonic interpreter $V$ and denote ${\mathit{Km}}(x)={\mathit{Km}}_{V}(x)$.
\[rem:oracles\] A monotonic interpreter is a slightly generalized version of what can be accomplished by a Turing machine with a one-way read-only input tape containing the finite or infinite string $p$. The machine also has a working tape and a one-way output tape. In the process of work, on this tape appears a finite or infinite sequence $T(p)$. The work may stop, if the machine halts or passes beyond the limit of the input word; it may continue forever otherwise. It is easy to check that the map $p\mapsto T(p)$ is a monotonic interpreter (though not all monotonic interpreters correspond to such machines, resulting in a somewhat narrower class of mappings).
These machines can be viewed as the definition of what we will later call [*oracle computation*]{}: namely, a computation that uses $p$ as an oracle.
In our applications, such a machine would have the form $T(p,\omega)$ where the machine works on both infinite strings $p$ and $\omega$ as input, but considers $p$ the oracle and $\omega$ the string it is testing for randomness.
The class of mappings is narrower indeed. Let $S$ be an undecidable recursively enumerable set of integers. Set $T(0^{n}1)=0$ for all $n\in S$, and $T(0^{n}10)=0$ for all $n$. Now after reading $0^{n}1$, the machine $T$ has to decide whether to output a 0 before reading the next bit, which is deciding the undecidable set $S$. It is unknown to us whether this class of mappings yields also a different monotonic complexity.
A monotonic interpreter will also give rise to something like a distribution over the set of finite and infinite strings.
\[def:contin-semim\] Let us feed a monotonic interpreter $A$ a sequence of independent random bits and consider the output distribution on the finite and infinite sequences. Denote ${M}_{A}(x)$ the probability that the output sequence begins with $x$. Denote ${\mathit{KM}}_{V}(x)=-\log {M}_{V}(x)$.
Recall that $\Lambda$ denotes the empty string. A function $\mu:\{0,1\}^{*}\to{[0,1]}$ is called a [*continous semimeasure*]{} over the Cantor space $\Omega$ if $\mu(\Lambda){\leqslant}1$ and $\mu(x){\geqslant}\mu(x0)+\mu(x1)$ for all $x\in\{0,1\}^{*}$.
It is easy to check that ${M}_{V}(x)$ is a lower semicomputable continuous semimeasure. The proposition below is similar in form to the Coding Theorem (Proposition \[propo:coding\]) above, only weaker, since it does not connect to the complexity ${\mathit{Km}}(x)$ defined in terms of shortest programs. (It cannot, as shown in [@GacsRel83].)
(see [@ZvLe70])
Every lower semicomputable continuous semimeasure is the output distribution of some monotonic interpreter.
Among lower semicomputable continuous semimeasures, there is one that is maximal within a multiplicative constant.
\[def:continuous-apriori\] Let us fix a maximal lower semicomputable continuous semimeasure and denote it ${M}(x)$. We call ${M}(x)$ sometimes the [*continuous a priori probability*]{}, or [*apriori probability on a tree*]{}.
Now, the characterization by Levin (and a similar one by Schnorr) is the following. Its proof, technically not difficult, can be found in [@GacsLnotesAIT88; @LiViBook97; @ShenUppsala00].
\[propo:rand-monot\] Let ${P}$ be a computable measure over $\Omega$. Then the following properties of an infinite sequence $\omega$ are equivalent.
1. $\omega$ is random with respect to ${P}$.
2. $\lim\sup_{x{\sqsubseteq}\omega}-\log{P}(x)-{\mathit{Km}}(x)<\infty$.
3. $\lim\inf_{x{\sqsubseteq}\omega}-\log{P}(x)-{\mathit{Km}}(x)<\infty$.
4. $\lim\sup_{x{\sqsubseteq}\omega}-\log{P}(x)-{\mathit{KM}}(x)<\infty$.
5. $\lim\inf_{x{\sqsubseteq}\omega}-\log{P}(x)-{\mathit{KM}}(x)<\infty$.
Theorem \[thm:rand-prefix\] proved above adds to this a next equivalent characterization, namely that $-\log{P}(x)-{\mathit{Kp}}(x)$ is bounded above. It is different in nature from the one in Proposition \[propo:rand-monot\]: indeed, the expressions $-\log{P}(x)-{\mathit{Km}}(x)$ and $-\log{P}(x)-{\mathit{KM}}(x)$ are *always bounded from below* by a constant depending only on the measure ${P}$ (and not on $x$ or $\omega$), while $-\log{P}(x)-{\mathit{Kp}}(x)$ is not.
Moreover, in the latter we cannot replace $\lim\sup$ with $\lim\inf$, as the following example shows. Note that we can add to every word $x$ some bits to achieve ${\mathit{Kp}}(y){\geqslant}{|y|}$ (where ${|y|}$ is the length of word $y$). Indeed, if this was not so, then for the continutations of the word we would have ${\mathbf{m}}(y){\geqslant}2^{-{|y|}}$, and the sum $\sum_{y}{\mathbf{m}}(y)$ would be infinite. Let us build a sequence, adding alternatingly long stretches of zeros to make the complexity substantially less than the length, then bits that again bring the complexity up to the length (as shown, this is always possible). Such a sequence will not be random with respect to the uniform measure (since the $\lim\sup$ of the difference is infinite), but has infinitely many prefixes for which the complexity is not less than the length, making the $\lim\inf$ finite.
The following statement is interesting since no direct proof of it is known: the proof goes through Theorem \[thm:seq-of-prefixes\], and noting that since the permutation of terms of the sequence does not change the coin-tossing distribution, it does not change the notion of randomness. More general theorems of this type, under the name of *randomness conservation*, can be found in [@LevinUnif76; @LevinRandCons84; @GacsUnif05].
Consider the uniform distribution (coin-tossing) ${P}$ over binary sequences. The maximal difference between ${|x|}$ and ${\mathit{Kp}}(x)$ for prefixes $x$ of a random sequence is invariant (up to a constant) under any computable permutation of the sequence terms. (The constant depends on the permutation, but not on the sequence.)
Here is another corollary, a reformulation of Proposition \[propo:sum-characteriz\]:
\[coroll:ample-excess\] A sequence $\omega$ is random with respect to a computable measure ${P}$ if and only if $$\sum_{x{\sqsubseteq}\omega} 2^{-\log{P}(x) - {\mathit{Kp}}(x)}<\infty.$$
This corollary also implies the fact mentioned above already:
\[coroll:extension\] Every finite sequence $x$ has an extension $y$ with ${\mathit{Kp}}(y)>{|y|}$.
Take $\omega$ random, then $x\omega$ is random, and therefore by the Miller-Yu lemma $x\omega$ has arbitrarily long prefixes whose complexity is larger than the length.
Game interpretation
-------------------
The formula for the average-bounded deficiency can be interpreted in terms of the following game. Alice and Bob make their moves having no information about the opponent’s move. Alice chooses an infinite binary sequence $\omega$, Bob chooses a finite string $x$. If $x$ turns out to be a prefix of $\omega$, then Alice pays Bob $2^n$ where $n$ is the length of $x$. (This version of the game corresponds to the uniform Bernoulli measure, in the general case Alice pays $1/{P}(x)$.) Recall the game-theoretic notions of [*pure*]{} strategy, as a deterministic choice by a player, and [*mixed*]{} strategy, as a probability distribution over deterministic choices.
Bob has a trivial strategy (choosing the empty string) that guarantees him $1$ whatever Alice does. Also Alice has a mixed strategy (the uniform distribution, or, in general case, ${P}$) that guarantees her the average loss $1$ whatever Bob does. Bob can devise a strategy that will benefit him in case (for whatever reason) Alice brings a nonrandom sequence.
A randomized algorithm that has no input and produces a string (or nothing) can be considered a mixed strategy for Bob (if the algorithm does not produce anything, Bob gets no money). For any such algorithm $D$ the expected payment (if Alice produces $\omega$ according to distribution ${P}$) does not exceed $1$. Therefore, the set of sequences $\omega$ where the expected payment (averaged over Bob’s random bits) is infinite, is a null set. Observe the following:
1. For every probabilistic strategy of Bob, his expected gain (as a function of Alice’s sequence) is an average-bounded test. (From here already follows that this expected value will be finite, if Alice’s sequence is random in the sense of Martin-Löf.)
2. If $m(x)$ is the probability of $x$ as Bob’s move with algorithm $D$, his expected gain against $\omega$ is equal to $$\begin{aligned}
\sum_{x{\sqsubseteq}\omega} m(x)/{P}(x).
\end{aligned}$$
3. Therefore if we take the algorithm outputting the discrete apriori probability ${\mathbf{m}}(x)$, then Bob’s expected gain will be a universal test (by the proved formula for the universal test).
Using the apriori probability as a mixed strategy enables Bob to punish Alice with an infinite penalty for any non-randomness in her sequence.
One can consider more general strategies for Bob: he can give for a pure strategy, not only a string $x$, but some basic function $f$ on $\Omega$ with non-negative values. Then his gain for the sequence $\omega$ brought by Alice is set to $f(\omega)/\int f(\omega)\,d{P}$. (The denominator makes the expected return equal to $1$.) To the move $x$ corresponds the basic function that assigns $2^{{|x|}}$ to extensions of $x$ and zero elsewhere. This extension does not change anything, since this move is a mixed strategy and we allow Bob to mix his strategies anyway. (After producing $f$, Bob can make one more randomized step and choose some of the intervals on which $f$ is constant, with an appropriate probability.) In this way we get another formula for the universal test: $${\mathbf{t}}_{{P}}(\omega){\stackrel{{}_*}{=}}\sum_{f} \frac{{\mathbf{m}}(f) f(\omega)}{\int f(\omega)\, d{P}},$$ where the sum is taken over all basic functions $f$. This formula might be useful in more general situations (not Cantor space) where we do not work with intervals and consider some class of basic functions instead.
On concluding this part let us point to a similar game-theoretical interpretation of probability theory developed in the book [@ShaferVovkGame01] of Shafer and Vovk. There, the randomness of an object is not its property but, roughly speaking, a kind of guarantee with which it is being sold.
From tests to complexities
==========================
Formula expresses the randomness deficiency (the logarithm of the universal test) of an infinite sequence in terms of complexities of its finite prefixes. A natural question arises: can we go in the other direction? Is it possible to express the complexity of a finite string $x$, or some kind of “randomness deficiency” of $x$, in terms of the deficiencies of $x$’s infinite extensions? Proposition \[propo:generate-lower\] and the discussion following it already brought us from infinite sequences to finite ones. This can also be done for the universal test:
\[def:bar-u\] Fix some computable measure ${P}$, and let $t$ be any (average-bounded) test for ${P}$. For any finite string $x$ let $\bar t(x)$ be the minimal deficiency of all infinite extensions of $x$: $$\bar t(x) = \inf_{\omega{\sqsupseteq}x} t(\omega).$$
By Proposition \[propo:inf-lower-semicomp.seqs\], $\bar t$ is a lower semicomputable function defined on finite strings, and the function $t$ can be reconstructed back from $\bar t$; so if ${\mathbf{t}}_{{P}}$ is our fixed universal test then $\bar{\mathbf{t}}_{{P}}$ can be considered as a version of randomness deficiency for finite strings.
The intuitive meaning is clear: a finite sequence $z$ looks non-random if *all* infinite sequences that have prefix $z$ look non-random.
Kolmogorov [@KolmPPI69] had a somewhat similar suggestion: for a given sequence $z$ we may consider the minimal deficiency (with respect to the uniform distribution, defined as a difference between length and complexity) of all its *finite* extensions. Are there any formal connections?
Let us spell out what we found, in more general terms.
\[def:extended-test.computable\] A lower semicomputable, monotonic (with respect to the prefix relation) function $T:\{0,1\}^{*}\to{[0,\infty]}$ is called an [*extended test*]{} for computable measure $P$ if for all $N$ the average over words of length $N$ is bounded by 1: $$\begin{aligned}
\sum_{x:|x|=N}{P}(x)T(x) {\leqslant}1.\end{aligned}$$
Monotonicity guarantees that the sum over words of a given length can be replaced by the sum over an arbitrary finite (or even infinite) prefix-free set $S$: $$\begin{aligned}
\label{eq:extended-test.computable}
\sum_{x\in S}{P}(x)T(x) {\leqslant}1.\end{aligned}$$ (Indeed, extend the words of $S$ to some common greater length.)
\[propo:extended-test.computable\] Every extended test generates (in the sense of Definition \[def:generate-lower\]) some averge-bounded test on the infinite strings. Conversely, every average-bounded test on the infinite sequences is generated by some extended test.
The first part follows immediately from the definition (and the theorem of monotone convergence under the integral sign). In the opposite direction, we can set for example $T(x)=\bar t(x)$, or refer to Proposition \[propo:lower-semi-limit.seqs\] if we do not want to rely on compactness.
The existence of a universal extended test is proved by the usual methods:
\[propo:extended-univ.computable\] Among the extended tests $T(x)$ for a computable measure ${P}(x)$ there is a maximal one, up to a multiplicative constant.
\[def:extended-test\] Let us fix some dominating extended test and call it the [*universal*]{} extended test.
\[propo:extended-to-bar\] The universal extended test coincides with $\bar{\mathbf{t}}_{{P}}(x)$ to within a bounded factor.
Since $\bar{\mathbf{t}}_{{P}}$ is an extended test, it is not greater than the universal test (to within a bounded factor). On the other hand, the universal extended test generates a test on the infinite sequences, it just remains to compare it with the maximal one.
If our space is not compact (say, it is the set of infinite sequences of integers), then $\bar{\mathbf{t}}_{{P}}(x)$ is not defined, but there is still a universal extended test, which we will denote by ${\mathbf{t}}_{{P}}(x)$.
Warning: not all extended tests generating ${\mathbf{t}}_{{P}}(\omega)$ are maximal. (For example, one can make the test equal to zero on all short words, transferring its values to its extensions.)
The advantage of the function ${\mathbf{t}}_{{P}}(x)$ is that it is defined on finite strings, the condition (for finite sets $S$) imposed on it is also more elementary than the integral condition, but clearly implies that it generates a test.
The method just shown is not the only way to move to tests on prefixes from tests on infinite sequences:
\[def:hat-u\] Assume that the computable measure ${P}$ is positive on all intervals: ${P}(x)>0$ for all $x$. Let $\hat{\mathbf{t}}_{{P}}(x)$ be the conditional expected value of ${\mathbf{t}}_{{P}}(\omega)$ if a random variable $\omega\in\Omega$ has distribution ${P}$ and the condition is $\omega{\sqsupseteq}x$. In other terms: let $\hat{\mathbf{t}}_{{P}}(x)$ be the average of ${\mathbf{t}}_{{P}}$ on the interval $x\Omega$, that is let $\hat {\mathbf{t}}_{{P}}(x)=U(x)/{P}(x)$ where $$\begin{aligned}
U(x)=\int_{x\Omega}{\mathbf{t}}_{{P}}(\omega)\,dP(\omega).
\end{aligned}$$
The function $U$ is a lower semicomputable semimeasure. (It is even a measure, but the measure is not guaranteed to be computable and the measure of the entire space $\Omega$ is not necessarily $1$. In other words, we get a measure on $\Omega$ that has density ${\mathbf{t}}_{{P}}$ with respect to ${P}$.) This implies that the function $\hat{\mathbf{t}}_{{P}}(x)$ is a martingale, according to the following definition.
\[def:martingale\] A function $g:\{0,1\}^{*}\to{\mathbb{R}}$ is called a [*martingale*]{} with respect to the probability measure ${P}$ if $$\begin{aligned}
{P}(x)g(x) = {P}(x0)g(x0)+{P}(x1)g(x1).
\end{aligned}$$ It is a [*supermartingale*]{} if at least the inequality ${\geqslant}$ holds here.
Note that, as a martingale, the function $\hat{\mathbf{t}}_{{P}}(x)$ is *not* monotonically increasing with respect to the prefix relation.
\[thm:long-chain\] $$\label{eq:long-chain}
\frac{{\mathbf{m}}(x)}{{P}(x)}{\stackrel{{}_*}{<}}{\mathbf{t}}_{{P}}(x) {\stackrel{{}_*}{<}}\hat{\mathbf{t}}_{{P}}(x) {\stackrel{{}_*}{<}}\frac{{M}(x)}{{P}(x)},$$ where ${\mathbf{m}}$ is the a priori probability on strings as isolated objects (whose logarithm is minus prefix complexity) and ${M}$ is the a continuous priori probability as introduced in Definition \[def:continuous-apriori\].
In fact, the first inequality can be made stronger: we can replace ${\mathbf{m}}(x)/{P}(x)$ by $\sum_{t{\sqsubseteq}x}{\mathbf{m}}(t)/{P}(t)$. Indeed, this sum is a part of the expression for ${\mathbf{t}}_{{P}}(\omega)$ for every $\omega$ that starts with $x$.
The second inequality uses Proposition \[propo:extended-to-bar\] and relates the minimal and average values of a random variable. The third inequality just compares the lower semicomputable semimeasure $U(x)$ and the maximal semimeasure ${M}(x)$.
Note that while $\hat{\mathbf{t}}_{{P}}(x)$ is a martingale, $\frac{{M}(x)}{{P}(x)}$ is a supermartingale: it is actually maximal within multiplicative constant, among the lower semicomputable supermartingales for ${P}$.
1. We may insert $$\label{eq:inserted}
{\stackrel{{}_*}{<}}\max_{t{\sqsubseteq}x} \frac{{\mathbf{m}}(t)}{{P}(t)}
{\stackrel{{}_*}{<}}\sum_{t{\sqsubseteq}x} \frac{{\mathbf{m}}(t)}{{P}(t)}{\stackrel{{}_*}{<}}$$ between the first and the second terms of .
2. Using the logarithmic scale, we get $$-\log{P}(x)-{\mathit{Kp}}(x) {\stackrel{{}_+}{<}}\log{\mathbf{t}}_{{P}}(x) {\stackrel{{}_+}{<}}\log \hat{\mathbf{t}}_{{P}}(x)
{\stackrel{{}_+}{<}}-\log{P}(x)- {\mathit{KM}}(x).$$
3. The Measure $U$ depends on ${P}$ (recall that $U$ is a maximal measure that has density with respect to ${P}$), so for different ${P}$’s, for example with different supports, like the Bernoulli measures with different parameters, we get different measures. But this dependence is bounded by the inequality above: it shows that the possible variations do not exceed the difference between ${\mathit{Kp}}(x)$ and ${\mathit{KM}}(x)$.
4. The rightmost inequality cannot be replaced by an equality. For example, let ${P}$ be the uniform (coin-tossing) measure. Then the value of $U(x)$ tends to $0$ when $x$ is an increasing prefix of a computable sequence (we integrate over decreasing intervals whose intersection is a singleton that has zero uniform measure). On the other hand, the value ${M}(x)$ is bounded by a positive constant for all these $x$.
5. We used compactness (the finiteness of the alphabet $\{0,1\}$) in proving Proposition \[propo:inf-lower-semicomp.seqs\]. But we could have used Proposition \[propo:generate-lower\] and the discussion following it for a starting point, obtaining analogous results for the Baire space of infinite sequences of natural numbers.
All quantities listed in Theorem \[thm:long-chain\] can be used to characterize randomness: a sequence $\omega$ is random if the values of the quantity in question are bounded for its prefixes. Indeed, the Levin-Schnorr theorem guarantees that for a random sequence the right-hand side is bounded, and for a non-random one the left-hand side is unbounded. The monotonicity of the second term guarantees that all expressions except the first one tend to infinity. As we already mentioned above, one cannot say this about the first quantity.
Some quantities used in the theorem (${\mathbf{t}}_{{P}}(x)$ and two added ones in ) are monotonic (with respect to the prefix partial order of $x$) by definition. We have seen that $\hat{\mathbf{t}}_{{P}}(x)$, as a martingale, is not monotonic. What can be said about $\frac{{M}(x)}{{P}(x)}$?
All these quantities are “almost monotonic” since they do not differ much from the monotonic ones.
Bernoulli sequences
===================
One can try to define randomness not only with respect to some fixed measure but also with respect to some family of measures. Intuitively a sequence is random if we can believe that it is obtained by a random process that respects *one of* these measures. As we show later, this definition can be given for any *effectively compact* class of measures. But to make it more intuitive, we start with a specific example: *Bernoulli measures*.
Tests for Bernoulli sequences
-----------------------------
The Bernoulli measure $B_p$ arises from independent tossing of a non-symmetric coin, where the probability of success $p$ is some real number in ${[0,1]}$ (the same for all trials). Note that we do not require $p$ to be computable.
\[def:Bernoulli-test\] A lower semicomputable function $t$ on infinite binary sequences is a [*Bernoulli test*]{} if its integral with respect to any $B_p$ does not exceed $1$.
There exists a universal (maximal up to a constant factor) Bernoulli test.
A lower semicomputable function is the monotonic limit of basic functions. If the integral of a given basic function with respect to every $B_{p}$ is less or equal than $1$ for all $p$, this fact can be established effectively (indeed, the integral is a polynomial in $p$ with rational coefficients). This allows us to eliminate all functions unfit to be tests, and to list all Bernoulli tests. Adding these up with appropriate coefficients, we obtain a universal one.
\[def:univ.Bernoulli\] We fix a universal Bernoulli test and denote it ${\mathbf{t}}_{{\mathcal{B}}}(\omega)$. Its logarithm will be called [*Bernoulli deficiency*]{} ${\mathbf{d}}_{{\mathcal{B}}}(\omega)$. A sequence is called a [*Bernoulli sequence*]{} if its Bernoulli deficiency is finite.
Again, we may modify the definition to within an additive constant, to make it nonnegative and integer.
The informal motivation is the following: $\omega$ is a Bernoulli sequence if the claim that it is obtained by independent coin tossing (coin symmetry is not required) looks plausible. And this statement is not plausible if one can formulate some property that is true for $\omega$ but defines an “effectively Bernoulli null set” (we did not formally introduce this notion, but could, analogously to effective null sets).
Analogously to the case of computable measures, we can extend the class test to finite sequences:
\[def:extended-test.Bernoulli\] A lower semicomputable monotonic function $T:\{0,1\}^{*}\to{[0,\infty]}$ is called an [*extended Bernoulli tests*]{} if for all natural numbers $N$ and for all $p\in{[0,1]}$ the inequality $\sum_{x: |x|=N}B_{p}(x)T(x) {\leqslant}1$ holds.
As for computable measures, there is a connection between tests for finite and tests for infinite sequences:
\[propo:extended-univ.Bernoulli\] Every extended Bernoulli test generates a Bernoulli test over $\Omega$. On the other hand, every Bernoulli test over $\Omega$ is generated by some extended Bernoulli test.
There is a dominating universal extended Bernoulli test: it generats a universal Bernoulli test on $\Omega$. As earlier, we wil use the same notation ${\mathbf{t}}_{{\mathcal{B}}}$ for the maximal tests on the finite and on the infinite sequences. Of course, it generates a universal Bernoulli test.
Other characterizations of the Bernoulli property
-------------------------------------------------
Just as for the randomness with respect to computable measures, several equivalent definitions exist. One may consider probability-bounded tests (the probability of the event $t(\omega)>N$ on any of the measures $B_{p}$ must be not greater than $1/N$). One may call a test, following Martin-Löf’s definition for the computable measures, any computable sequence of effectively open sets $U_{i}$ with $B_{p}(U_{i}){\leqslant}2^{-i}$ for all $i$ and all $p\in{[0,1]}$. All these variant definitions are equivalent (and this is proved just as for randomness with respect to a computable measures).
Let ${\mathbb{B}}(n,k)$ denote the set of binary strings of length $n$ with $k$ ones (and $n-k$ zeroes).
Martin-Löf defined a Bernoulli test as a family of sets of words $U_{1}\supseteq U_{2}\supseteq U_{3}\supseteq\dotsm$; each of these sets is hereditary upward, that is for every word contains all of its extensions. The following restriction is made on these sets: consider arbitrary integer $n{\geqslant}0$ and $k$ from $0$ to $n$; it is required that for all $i$ the share of words in ${\mathbb{B}}(n,k)$ belonging to $U_{i}$ is not greater than $2^{-i}$.
For convenience of comparison let us replace the sets $U_{i}$ with an integer-valued lower semicomputable function $d$ for which $U_{i}={\mathopen\{\,x : d(i){\geqslant}i\,\mathclose\}}$. The hereditary property of the sets $U_{i}$ implies the monotonicity of this function $d$ with respect to the prefix relation. Besides this, it is required that the event $d{\geqslant}i$ within each set ${\mathbb{B}}(n,k)$ is not greater than $2^{-i}$. Clearly, these requirements correspond to probability-bounded extended tests (in the logarithmic scale), only in place of the class $B_{p}$ on words of length $n$ another set of measures is considered, those concentrated on words of a given length with a given number of ones. The measures in the class $B_{p}$ take equal values on words of equal lengths with equal number of ones, and are therefore representable by a mixture of uniform measures on ${\mathbb{B}}(n,k)$ with some coefficients. Replacing $B_{p}$ with these measures, the condition becomes stronger.
Let us show that nonetheless, the set of Bernoulli sequences does not change from such a replacement; moreover, the universal test (as a function on infinite sequences) does not change (as usual, to within a bounded factor). We will show this for the average-bounded variant of tests (changing Martin-Löf’s definition accordingly); this does not change the class of Bernoulli sequences. The reasoning is analogous for the probability-bounded tests.
\[def:combinat-Bernoulli\] A [*combinatorial Bernoulli test*]{} is a function $f:\{0,1\}^{*}\to{[0,\infty]}$ with the following constraints:
It is lower semicomputable.
It is monotonic with respect to the prefix relation.
For all integer $n,k$ with $0{\leqslant}k{\leqslant}n$ the average of the function $f$ on the set ${\mathbb{B}}(n,k)$ remains below 1: $$\begin{aligned}
\label{eq:combinat-Bernoulli-sum}
|{\mathbb{B}}(n,k)|^{-1}\sum_{x\in{\mathbb{B}}(n,k)} f(x) &{\leqslant}1.
\end{aligned}$$
The last condition says that not only is the average of $f(x)$ bounded by 1 over the set $\{0,1\}^{n}$, as in extended tests for the unbiased coin-tossing measure, but its average is bounded by 1 separately in each set ${\mathbb{B}}(n,k)$ whose union is $\{0,1\}^{n}$.
Having such a test for words of bounded length, it can be continued by monotonicity:
\[propo:Bernoulli-extend\] If a combinatorial Bernoulli test $f(x)$ is given on strings $x$ of length less than $n$, then extending it to longer strings using monotonicity we get a function that is still a combinatorial Bernoulli test.
We extend $f$ to words of length $n$, setting $f(x0)=f(x1)=f(x)$ for words $x$ of length $n-1$. The set ${\mathbb{B}}(n,k)$ consists of two parts: words ending on zero and words ending on one. The first ones are in a one-to-one correspondence with ${\mathbb{B}}(n-1,k)$, the second ones with ${\mathbb{B}}(n-1,k-1)$. The function conserves the values in this correspondence, therefore the average in both parts is not greater than 1. Hence, the average over the whole ${\mathbb{B}}(n,k)$ is not greater than 1.
The following is obtained by standard methods:
Among combinatorial Bernoulli tests, there is one that is maximal to within a bounded factor.
Let us fix a universal combinatorial Bernoulli test ${\mathbf{b}}(x)$ and extend it to infinite sequences $\omega$ by $$\begin{aligned}
{\mathbf{b}}(\omega) = \sup_{x{\sqsubseteq}\omega} {\mathbf{b}}(x).
\end{aligned}$$ We will call the function obtained this way a universal combinatorial test on $\Omega$ and will denote it also by ${\mathbf{b}}$.
(By monotonicity, the least upper bound in this definition can be replaced with a limit.) Let us show that the this test coincides (to within a bounded factor) with the Bernoulli tests introduced earlier in Definition \[def:univ.Bernoulli\].
\[thm:combinat-Bernoulli\] ${\mathbf{b}}(\omega) {\stackrel{{}_*}{=}}{\mathbf{t}}_{{\mathcal{B}}}(\omega)$.
We have already seen that ${\mathbf{b}}(x)$ is an extended Bernoulli test (from the bounds on the average on each part ${\mathbb{B}}(n,k)$ follows the bound on the expected value by the measure $B_{p}$, since this measure is constant on each part). Consequently ${\mathbf{b}}(\omega){\stackrel{{}_*}{<}}{\mathbf{t}}_{{\mathcal{B}}}(\omega)$.
The converse is not true: an extended Bernoulli test may not be a combinatorial test. But it is possible to construct a combinatorial test that takes the same values (to within a bounded factor) on the infinite sequences, and only this is asserted in the theorem.
Here is the idea. Consider an extended Bernoulli test $t$ on words of length $n$ and transfer it to words of much greater length $N$ (applying the old test to its beginnings of length $n$). We obtain a certain function $t'$. We have to show that $t'$ is close to some combinatorial test (that is only exceeds it by a constant factor). For this, $t'$ must be averaged over the set ${\mathbb{B}}(N,K)$ for an arbitrary $K$ between $0$ and $N$. In other words, we must average $t$ by the probability distribution on the $n$-bit prefixes of sequences of length $N$ containing $K$ ones. With $N\gg n$ this distribution will be close to the Bernoulli one with distribution $p=K/N$.
In terms of elementary probability theory, we have an urn with $N$ balls, $K$ of which is black, and take out from it $n$ balls. We must compare the probability distribution with the Bernoulli one that would have been obtained at sampling with replacement. Let us show that
> *for $N=n^{2}$ the distribution without replacement does not exceed the one with replacement more than $O(1)$ times*.
(The inequality does not hold in the other direction: for $K=1$ without replacement we cannot obtain a word with two ones, and with replacement we can. But we only need the inequality in the given direction.)
Indeed, in sampling without replacement the probability that a ball of a given color will be drawn is equal to the quotient $$\begin{aligned}
\frac{\text{the number of remaining balls of this color}}{\text{the number of
all remaining balls}}.
\end{aligned}$$ The number of balls of this color is not more than in the case with replacement, on the other hand the denominator is at least $N-n$. Therefore the probability of any combination during sampling with replacement is at most the probability of the same combination with replacement, multiplied by $N/(N-n)$ to the power $n$. For $N=n^{2}$ the multiplier $(1+O(1/n))^{n}=O(1)$ is obtained.
This way, taking the extended Bernoulli test $t$ and then defining $t'(x)$ on a word $x$ of length $N$ as $t$ on the prefix of $x$ of length ${{\lfloor \sqrt{N}\rfloor}}$, the obtained function $t'$ will be a combinatorial test to within a bounded factor. (Note that its monotonicity follows from that of $t$.)
Criterion for Bernoulli sequences
---------------------------------
It is natural to compare the notion of Bernoulli sequence (those sequences for which the Bernoulli test is finite) with the notion of a sequence random with respect to the measure $B_{p}$. But Martin-Löf definition of randomness assumes that the measure is computable. Therefore it cannot be applied directly to $B_p$ if $p$ is non-computable. But this definition can be relativized, and if (the binary expansion of) $p$ is given as an oracle (see Remark \[rem:oracles\]), then the measure $B_p$ becomes computable and randomness is well defined. The following theorem supports an intuitive idea of Bernoulli sequence as a sequence that is random with respect to some Bernoulli measure:
\[thm:Bernoulli-oracle\] A sequence $\omega$ is a Bernoulli sequence if and only if it is random with respect to some measure $B_{p}$, with oracle $p\in{[0,1]}$.
By “with oracle $p$”, we understand the possiblity to obtain from each $i$ the $i$th bit in the binary expansion of the real number $p$ (which is essentially unique, except in those cases when $p$ is binary-rational, and in these cases both expansions are computable, and the oracle is trivial).
Before proving the theorem (even in a stronger quantitative form), we introduce a new notion, of a test depending explicitly on the parameter $p$ of the Bernoulli measure $B_{p}$, which later will be extended to arbitrary (not just Bernoulli) measures. The required result will be obtained as the combination of the following claims:
Among the “uniform” randomness tests, there exists a maximal test ${\mathbf{t}}(\omega,p)$.
The function $\omega\mapsto\inf_{p}{\mathbf{t}}(\omega,p)$ coincides (as usual, to within a bounded factor) with the universal Bernoulli test.
For a fixed $p$, the function $\omega\mapsto{\mathbf{t}}(\omega,p)$ coincides (to the same precision) with the maximal randomness test for the ($p$-computable) measure $B_{p}$, relativized to $p$.
These three assertions imply Theorem \[thm:Bernoulli-oracle\] easily: sequence $\omega$ is Bernoulli, if the Bernoulli test is finite; the latter is equal to the greatest lower bound of ${\mathbf{t}}(\omega,p)$, hence its finiteness means ${\mathbf{t}}(\omega,p)<\infty$ for some $p$, which is equivalent to the relativized randomness with respect to the measure $B_{p}$.
We need some technical preparation. The randomness tests (as functions of two variables) will also be lower semicomputable, but the definition of this concept needs to be extended, since an additional real parameter is involved. (In what follows we will also consider a more general situation, in which the second argument is a measure.)
\[def:lower-semicomp.product\] In the space $\Omega\times{[0,1]}$, let us call [*basic rectangles*]{} all sets of the form $x\Omega\times{(u,v)}$, where $u<v$ are rational numbers. (A technical point: we allow $u,v$ to be outside ${[0,1]}$, but in this case the rectangle we mean is $x\Omega\times({[0,1]}\cap{(u,v)})$.)
A function $f:\Omega\times{[0,1]}\to{[-\infty,\infty]}$ is called [*lower semicomputable*]{} if there is an algorithm that, given a rational $r$ on its input, enumerates a sequence of basic rectangles whose union is the set of all pairs ${(\omega,p)}$ with $f(\omega,p)>r$.
The notion of [*upper semicomputability*]{} is defined analogously, and is equivalent to the lower semicomputablity of $(-f)$.
A function with finite real values is called [*computable*]{} if it is both upper and lower semicomputable.
This definition, as earlier, requires that the preimage of ${(-\infty,r)}$ be an effective open set uniformly in $r$, only now we consider effectively open sets in $\Omega\times{[0,1]}$, defined in a natural way.
Since the intersection of effective open sets is effective open, the following—more intuitive—formulation is obtained for computability:
A real function $f:\Omega\times{[0,1]}\to{\mathbb{R}}$ is computable if and only if for every rational interval ${(u,v)}$ its preimage is the union of a sequence of basic rectangles that are effectively enumerated, uniformly in $u$ and $v$.
The intuitive meaning of this characterization will become clearer after observing that to “give approximations to $\alpha$ with any given precision” is equivalent to “enumerate all intervals containing $\alpha$”. Therefore for a computable function $f$ we can find approximations to $f(\omega,p)$, if we are given appropriate approximations to $\omega$ and $p$.
We can reformulate the definition of (non-negative) lower semicomputable function, introducing the notion of basic functions. It is important for us that the basic functions are continuous, therefore the dependence on the real argument will be piecewise linear, without jumps.
\[def:basic-functions.Bernoulli\] We define an enumerated list of [*basic*]{} functions ${\mathcal{E}}=\{e_{1},e_{2},\dots\}$ over the set $\Omega\times{[0,1]}$ as follows. For $x\in\{0,1\}^{*}$, positive integer $k$ and rational numbers $u,v$ with $u+2^{-k}<v-2^{-k}$ define the function $g_{x,u,v,k}(\omega,p)$ as follows. If $x\not{\sqsubseteq}\omega$, then it is 0. Otherwise, its value does not depend on $\omega$ and depends piecewise linearly on $p$: it is 0 if $p\not\in{(u,v)}$ and 1 if $u+2^{-k}{\leqslant}p{\leqslant}v-2^{-k}$, and varies linearly in between. Now ${\mathcal{E}}$ is the smallest set of functions containing all $g_{x,u,v,k}$, and closed under maxima, minima and rational linear combination.
Now lower semicomputable functions admit the following equivalent characterization:
\[propo:lower-semicomp-as-limit\] A function $f:\Omega\times{[0,1]}\to{[0,\infty]}$ is lower semicomputable if and only if it is the pointwise limit of an increasing computable sequence of basic functions. (It follows that basic functions are computable.)
This would be completely clear if for basic functions we also allowed the indicator functions of basic rectangles and the maxima of such functions. But we want the basic functions to be continuous (this will be important in what follows). One must note therefore that for $k\to\infty$ the function $g_{x,u,v,k}$ converges to the indicator function of a rectangle.
The continuity of the basic functions guarantees the following important property:
\[propo:integral-computable.Bernoulli\] Let $f:\Omega\times {[0,1]}\to{\mathbb{R}}$ be a basic function. The integral $\int f(\omega,p)\,B_{p}(d\omega)$ is a computable function of the parameter $p$, uniformly in the code of the basic function $f$.
(Computability is understood in the above described sense; we remark that every computable function is continuous. An analogous statement holds for an arbitrary computable function $f$, not only for basic functions, but we do not need this.)
The following fact, proved in [@HoyrupRojasRandomness09], will be used in the present paper a number of times, also in generalizations, but with essentially the same proof.
\[propo:trim\] Let $\varphi:\Omega\times{[0,1]}\to{[0,\infty]}$ be a lower semicomputable function. There is a lower semicomputable function $\varphi'(\omega,p)$ not exceeding $\varphi(\omega,p)$ with the property that for all $p$:
$\int \varphi'(\omega,p)B_{p}(d\omega){\leqslant}2$;
If $\int\varphi(\omega,p)B_{p}(d\omega){\leqslant}1$ then $\varphi'(\omega,p)=\varphi(\omega,p)$ for all $\omega$.
By Proposition \[propo:lower-semicomp-as-limit\], we can represent $\varphi(\omega,p)$ as a sum of a series of basic functions $\varphi(\omega,p)=\sum_{n}h_{n}(\omega,p)$. The integral $\int\sum_{i{\leqslant}n} h_{i}(\omega,p)B_{p}(d\omega)$ is computable by Proposition \[propo:integral-computable.Bernoulli\], as a function of $p$ (uniformly in $n$), therefore the set $S_{n}$ of all $p$ where this integral is less than $2$ is effectively open, uniformly in $n$.
Define now $h'_{n}(\omega,p)$ as $h_{n}(\omega,p)$ for all $p\in S_{n}$, and 0 otherwise. The function $h'_{n}(\omega,p)$ is lower semicomputable, and the integral $\int\sum_{i{\leqslant}n} h'_{i}(\omega,p)B_{p}(d\omega)$ will be less than $2$ for all $p$. Defining $\varphi'=\sum_{n}h'_{n}$ we obtain a lower semicomputable function, and the theorem on the integral of monotonic limits gives that $\int \varphi'(\omega,p)B_{p}(d\omega)$ is less than $2$ for all $p$.
It remains to note that if for some $p$ the integral $\int\varphi(\omega,p)B_{p}(d\omega)$ does not exceed 1, then this $p$ enters all sets $S_{n}$, and the change from $h_{n}$ to $h'_{n}$ as well as the change from $\varphi$ to $\varphi'$ does not change it.
Now we are ready to introduce tests depending explicitly on $p$:
\[def:uniform.Bernoulli\] A [*uniform test for Bernoulli measures*]{} is a function $t$ of two arguments $\omega\in\Omega$ and $p\in{[0,1]}$; informally, $t(\omega,p)$ measures the amount of nonrandomness (“regularity”) in the sequence $\omega$ with respect to distribution $B_p$. We require the following:
\[i:uniform.Bernoulli.joint-lower\] $t(\omega,p)$ is lower semicomputable jointly as a function of the pair ${(\omega,p)}$.
For every $p\in{[0,1]}$ the expected value of $t(\omega,p)$ (that is $\int t(\omega,p)B_p(d\omega)$) does not exceed $1$.
It remains to prove the three assertions promised earlier:
\[lem:univ-unif.Bernoulli\] There exists a universal uniform test ${\mathbf{t}}(\omega,p)$, that is a test that multiplicatively dominates all uniform tests for Bernoulli measures.
\[lem:Bernoulli-from-unif\] For the universal uniform test ${\mathbf{t}}$ of lemma \[lem:univ-unif.Bernoulli\], the function ${\mathbf{t}}'(\omega)=\inf_p {\mathbf{t}}(\omega,p)$ coincides (to within a bounded factor in both directions) with the universal Bernoulli test of Definition \[def:univ.Bernoulli\].
This lemma implies that $\omega$ is a Bernoulli sequence iff ${\mathbf{t}}'(\omega)$ is finite, that is ${\mathbf{t}}(\omega,p)$ is finite for some $p\in{[0,1]}$.
\[lem:Bernoulli-oracle\] For a fixed $p$ the function ${\mathbf{t}}_p(\omega)={\mathbf{t}}(\omega,p)$ coincides (to within a bounded factor) with the universal randomness test with respect to $B_p$ relativized with oracle $p$.
Generate all lower semicomputable functions; using Proposition \[propo:trim\], they can be then trimmed to guarantee that all expectations do not exceed, say, $2$, and all uniform tests should get through unchanged. Sum up all the trimmed functions with coefficients whose sum is less than $1/2$.
Let us show that ${\mathbf{t}}'(\omega)$ is a universal Bernoulli test. The integral of this function with respect to $B_p$ does not exceed $1$ since this function does not exceed ${\mathbf{t}}(\omega,p)$ for that $p$. The statement that this function is lower semicomputable (as a function of $\omega$) is analogous to Proposition \[propo:inf-lower-semicomp.seqs\], and the proof is also analogous, relying on compactness. Both are special cases of the general theorem given in Proposition \[propo:lsc-min\].
Therefore the function $\inf_p {\mathbf{t}}(\omega,p)$ is a Bernoulli test. The universality (maximality) follows obviously, since any Bernoulli test can be considered a uniform Bernoulli test of two variables that does not depend on variable $p$.
Consider first the case when $p$ is a computable real number. Then the function ${\mathbf{t}}_p\colon \omega\mapsto {\mathbf{t}}(\omega,p)$ (where ${\mathbf{t}}$ is a uniform randomness test for Bernoulli measures) is lower semicomputable (we can enumerate all intervals that contain $p$ and combine then with an algorithm for ${\mathbf{t}}$; in this way we represent ${\mathbf{t}}_p$ as the least upper bound of the computable sequence of basic functions).
A similar argument works for an arbitary $p$ and shows that ${\mathbf{t}}_{p}$ is lower semicomputable with a $p$-oracle. Thus, ${\mathbf{t}}_{p}$ does not exceed the universal relativized test with respect to $B_p$.
The reverse implication is a bit more difficult. Assume that $t$ is a lower semicomputable (with oracle $p$) randomness test with respect to $B_p$. We need to find a uniform Bernoulli test $t'$ that majorizes it (for a given $p$). This $t'$ must be lower semicomputable, now (a subtle but important point) using $p$ as an argument of the function $t'$, not as an oracle. In other words, one has to extend a function defined initially only for a single $p$, to all values of $p$, while also guaranteeing the bound on the integral.
As a warmup consider the case of computable $p$. Then no oracle is needed, and $t$ is lower semicomputable. Adding dummy variable $p$ we get a lower semicomputable function of two arguments. But this function may not be a uniform test since its expectation with respect to $B_q$ may be arbitrary if $q\ne p$. However, Proposition \[propo:trim\] helps transform it into a $t'$ (which will now really depend on $q$) with $\int t'(\omega,q)B_{q}(d\omega) {\leqslant}2$ for all $q$ and $t'(\cdot,p)=t(\cdot,p)$. Dividing $t'$ into half provides a uniform test.
Now consider the case of noncomputable $p$. In this case $p$ is irrational, so the bits of its binary expansion can be obtained from any sequence of decreasing rational intervals that converge to $p$. Therefore an oracle machine that enumerates approximations for $t$ from below (having $p$ as an oracle) can be transformed into a machine that enumerates from below some function $\tilde t(\omega,q)$, that coincides with $t(\omega)$ if $q=p$. The function $\tilde t$ may not be a uniform Bernoulli test (its expectations for $q\ne p$ can be arbitrary); but it again can be trimmed with the help of Proposition \[propo:trim\].
Arbitrary measures over binary sequences
========================================
In this section, we generalize the theory to arbitrary measures, not only Bernoulli ones, but still stay in the space $\Omega$ of binary sequences.
The set of all probability measures over the space $\Omega$ is denoted by ${\mathcal{M}}(\Omega)$. (Recall that the measure of the whole space $\Omega$ is equal to 1.)
Uniform randomness tests {#subsec:uniform-tests}
------------------------
\[def:uniform-test.bin-Cantor\] A [*uniform*]{} test is a lower semicomputable function $t(\omega,{P})$ of two arguments ($\omega$ is a sequence, ${P}$ is a measure on $\Omega$) with $$\int t(\omega,{P})\, {P}(d\omega) {\leqslant}1$$ for every measure ${P}$.
However, we have to define carefully the notion of a lower semicomputability in this case. The set ${\mathcal{M}}(\Omega)$ of all measures is a closed subset of the infinite (countable) product $$\label{eq:inf-product}
\Xi={[0,1]}\times {[0,1]}\times {[0,1]}\times\dotsm$$ (the measure is defined by the values ${P}(x)$ for all strings $x$; these values should satisfy the equations , so we get a closed subset).
Let us introduce basic open sets and computability notions for the set $\Omega\times{\mathcal{M}}(\Omega)$.
An (open) [*interval*]{} (basic open set) in the space of measures is given by a finite set of conditions of type $u<{P}(y)<v$ where $y$ is some binary string and $u,v$ are some rational numbers; the basic open set consists of the measures ${P}$ that satisfy these conditions. A [*basic open set*]{} in $\Omega\times{\mathcal{M}}(\Omega)$ has the form $x\Omega\times
\beta$, (product of intervals in $\Omega$ and ${\mathcal{M}}(\Omega)$) where $\beta$ is a basic open set of measures. Now lower and upper semicomputability and computability are defined in terms of these basic open sets just as they were defined for $\Omega\times{[0,1]}$ in Definition \[def:lower-semicomp.product\].
In much of what follows, we will exploit the fact that, due to the finiteness of the alphabet $\{0,1\}$, the space $\Omega$ of infinite binary sequences is compact, and also the set of measures ${\mathcal{M}}(\Omega)$ is compact. Recall that a set $C$ is compact if every cover of $C$ by open sets contains a finite subcover. We need, however, an effective version of compactness:
\[def:effectively-compact\] A compact subset $C$ of ${\mathcal{M}}(\Omega)$ is called [*effectively compact*]{} if the set $$\begin{aligned}
{\mathopen\{\,S : S \text{ is a finite set of basic open sets and } \bigcup_{E\in S}E\supseteq C\,\mathclose\}}
\end{aligned}$$ is enumerable.
The set ${\mathcal{M}}(\Omega)$ itself is, as it is easy to see, compact and effectively compact. It is compact, as said above, as a closed set in the product of compact spaces, and the effectivity follows from the fact that we can check whether some given basic sets cover the whole space (we are dealing with linear equations and inequalities in a finite number of variables, where everything is algorithmically decidable). From here, it also follows:
\[propo:closed-to-compact\] Every effectively closed subset of ${\mathcal{M}}(\Omega)$ is effectively compact.
Let an effectively closed subset $C$ of ${\mathcal{M}}(\Omega)$. be the complement of the union of a list $B_{1},B_{2},\dots$ of basic open sets. Then a finite set $S$ of basic open sets covers $C$ if and only if together with a finite set of the $B_{i}$, it covers the whole space. And this property is decidable.
Effective compactness implies effective closedness. This follows from the following two properties of our space and our basic open sets:
\[i:separation\] For every closed set $F$ and every point $x$ outside $F$ there are two disjoint open sets containing $F$ and $x$.
\[i:disjointness\] For every pair of basic open sets, it is uniformly decidable whether they are disjoint.
Let $F$ be an effectively compact set. We call a basic open set $B$ [*manifestly disjoint*]{} of $F$, if there is a finite set of basic open sets $S$ disjoint of $B$ covering $F$. Due to the effective compactness of $F$ and property , the set of all basic open sets manifestly disjoint of $F$ is enumerable. Property implies that it covers the complement of $F$.
In view of later generalization to cases where the space itself may not be compact, we will refer to some effectively closed sets of ${\mathcal{M}}(\Omega)$ as effectively compact.
Now we introduce a dense set of computable functions called basic functions on the set $\Omega\times{\mathcal{M}}(\Omega)$, similarly to Definition \[def:basic-functions.Bernoulli\]. Their specific form is not too important.
The set of [*basic*]{} functions over the set $\Omega\times{\mathcal{M}}(\Omega)$ is defined analogously to Definition \[def:basic-functions.Bernoulli\], starting from the functions $$\begin{aligned}
g_{x,y,u,v,k}:\Omega\times{\mathcal{M}}(\Omega)\to{[0,1]}
\end{aligned}$$ with $x,y\in\{0,1\}^{*}$ defined as follows. If $x\not{\sqsubseteq}\omega$, then $g_{x,y,u,v,k}(\omega,{P})=0$. Otherwise, its value does not depend on $\omega$ and depends piecewise linearly on ${P}(y)$ in a way that it is 0 if ${P}(y)\not\in{(u,v)}$ and 1 if $u+2^{-k}{\leqslant}{P}(y){\leqslant}v-2^{-k}$.
The analogue of Proposition \[propo:lower-semicomp-as-limit\] holds again: a lower semicomputable function is the monotonic limit of a computable sequence of basic functions (which themselves are computable).
The analogue of Proposition \[propo:integral-computable.Bernoulli\] holds also: the integral $\int f(\omega,{P}){P}(d\omega)$ of a basic function is computable as a function of the measure ${P}$, uniformly in the number of the basic function.
Finally, the analogue of Proposition \[propo:trim\] holds again:
\[thm:trim.Cantor\] Let $\varphi(\omega,{P})$ be a lower semicomputable function. Then there exists a lower semicomputable function $\varphi'(\omega,{P})$ such that for all ${P}$:
$\int \varphi'(\omega,{P}){P}(d\omega){\leqslant}2$,
if $\int \varphi(\omega,{P}){P}(d\omega){\leqslant}1$ then $\varphi'(\omega,{P})=t(\omega,{P})$ for all $\omega$.
The proof is completely analogous to the proof we gave for Proposition \[propo:trim\]. This allows the construction of a universal test as a function of a sequence and an arbitrary measure over $\Omega$:
There exists a maximal (maximal to within a bounded factor) uniform randomness test.
We use the same approach as before: we trim a lower semicomputable function in such a way that it becomes a test (or almost a test) and remains untouched if it were a test in the first place.
Let us fix a universal uniform randomness test ${\mathbf{t}}(\omega,{P})$.
We call a sequence $\omega$ [*uniformly random*]{} with respect to a (not necessarily computable) measure ${P}$ if ${\mathbf{t}}(\omega,{P})<\infty$.
Let us show that for computable measures, the new definition coincides with the old one.
\[propo:unif-comput.Cantor\] Let ${P}$ be a computable measure, let ${\mathbf{t}}_{{P}}(\omega)$ be a universal (average-bounded) randomness test for ${P}$ as, and ${\mathbf{t}}(\omega,{P})$ the universal uniform test defined above. Then there are constants $c_{1},c_{2}>0$ such that $c_{1}{\mathbf{t}}_{{P}}(\omega){\leqslant}{\mathbf{t}}(\omega,{P}){\leqslant}c_{2}{\mathbf{t}}_{{P}}(\omega)$.
The constants $c_{1},c_{2}$ here depend on the choice of measure ${P}$ and of the choice of the test ${\mathbf{t}}_{{P}}$ for this measure (this choice was done in an arbitrary way for each computable measure).
This proposition shows, that in the case of the computable measures, uniform randomness coincides with randomness in the sense of Martin-Löf.
Let us show ${\mathbf{t}}(\omega,{P}){\leqslant}c_{2}{\mathbf{t}}_{{P}}(\omega)$ first. The function $\omega\mapsto {\mathbf{t}}(\omega,{P})$ is lower semicomputable since we can effectively enumerate all intervals in the space of measures that contain ${P}$; therefore it is dominated by ${\mathbf{t}}_{{P}}(\omega)$.
To prove ${\mathbf{t}}(\omega,{P}){\geqslant}c_{1}{\mathbf{t}}_{{P}}(\omega)$, consider the lower semicomputable function $$\begin{aligned}
t(\omega,{Q})={\mathbf{t}}_{{P}}(\omega).
\end{aligned}$$ The function ${(\omega,{Q})}\mapsto t(\omega)$ is not guaranteed to be a uniform randomness test, since its integral can be greater than $1$ if ${Q}\ne{P}$. However, it can be trimmed without changing it at ${P}$, and then it still remains (almost) a test.
We are also interested in tests defined just for one, not necessarily computable, measure ${P}$:
\[def:P-test.Cantor\] We will call a function $f:\Omega\to{[0,\infty]}$ [*lower semicomputable*]{} relatively to measure ${P}$ if it is obtained from a lower semicomputable function on the set $\Omega\times{\mathcal{M}}(\Omega)$ after fixing the second argument at ${P}$.
For a measure ${P}\in{\mathcal{M}}(\Omega)$, a ${P}$-[*test of randomness*]{} is a function $f:\Omega\to{[0,\infty]}$ lower semicomputable from ${P}$ with the property $\int f(\omega)\,d P {\leqslant}1$.
It seems as if a ${P}$-test may capture some nonrandomnesses that uniform tests cannot—however, this is not so, since trimming (see Theorem \[thm:trim.Cantor\]) generalizes:
\[propo:uniformize.Cantor\] Let ${P}_{0}$ be some measure along with some ${P}_{0}$-test $t_{{P}_{0}}(\omega)$. There is a uniform test $t'(\cdot,\cdot)$ with $t_{{P}_{0}}(\omega){\leqslant}2
t'(\omega,{P}_{0})$. On the other hand, the restriction of any uniform test to the measure ${P}$ is a ${P}$-test.
The notion of extended text can be generalized to uniform tests:
\[def:extended-test.uniform\] A lower semicomputable function $T:\{0,1\}^{*}\times{\mathcal{M}}(\Omega)\to{[0,1]}$ monotonic with respect to the prefix relation is called an [*extended uniform test*]{} if for all $n$ and all distributions $P$ we have $\sum_{x: |x|=n}T(x,{P}){P}(x){\leqslant}1$.
As earlier, due to monotonicity, we could sum not only over words of a given length, but over an arbitrary prefix-free set.
The following follows from the analogue of Proposition \[propo:lower-semicomp-as-limit\] (representing a nonnegative lower semicomputable function as a sum of nonnegative basic functions):
\[propo:extended-test.uniform\] Every uniform test $t(\omega,{P})$ can be generated by an extended uniform test in the sense of $t(\omega,{P})=\sup_{x{\sqsubseteq}\omega}T(x,{P})$. Conversely, every extended uniform test $T$ generates a uniform test $t$.
Among the uniform extended tests, it is also possible to select a maximal one (using an analogous trimming method and summing the results). We fix an extended uniform test and denote it ${\mathbf{t}}(x,{P})$ (where $x\in\{0,1\}^{*}$, and ${P}$ is a measure over $\Omega$). It generates a maximal uniform test ${\mathbf{t}}(\omega,{P})$ (to within a bounded factor).
\[rem:seq-of-natural\] Much of the theory worked out at the beginning of this paper for 0-1 sequences holds also for sequences whose elements are arbitrary natural numbers. The extended tests of Definition \[def:extended-test.uniform\] generalize, and the existence of a uniform universal extended test is proven in the same way. But it becomes important to define extended tests directly, and not via tests for infinite sequences, since compactness may not hold.
Proposition \[propo:unif-comput.Cantor\] allows us to generalize a result about Bernoulli measures:
\[thm:oracle2uniform\] Let ${P}$ be a measure computable with some oracle $A$. Assume also that $A$ can be effectively reconstructed as the values of the measure are provided with more and more precision. Then a sequence $\omega$ is uniformly random with respect to ${P}$ if and only if it is random with respect to ${P}$ with oracle $A$.
(Since the oracle $A$ makes ${P}$ computable, the notion of Martin-Löf randomness is well defined.)
Assume that ${\mathbf{t}}(\omega,{P})=\infty$ for the universal uniform test ${\mathbf{t}}$. Note that ${\mathbf{t}}(\cdot,{P})$ is an $A$-lower semicomputable function and is a ${P}$-test, so $\omega$ is nonrandom with respect to ${P}$ with oracle $A$.
On the other hand, let $t(\omega,A)$ be some $A$-lower semicomputable ${P}$-test with $t(\omega,A)=\infty$. That $A$ can be reconstructed from ${P}$ means that there is a computable mapping $f$ from measures to binary sequences (oracles) defined at least over ${P}$, with $A=f({P})$. But then ${(\omega,{P})}\mapsto t(\omega,f({P}))$ is a ${P}$-test. The uniformization theorem \[propo:uniformize.Cantor\] converts it into a uniform test that is infinite on ${(\omega,{P})}$.
Let us note that not all measures ${P}$ satisfy the condition of the theorem (it means that the mass problem of “show approximations to the values of ${P}$” is equivalent to the decision problem of some set; on the degrees of such mass problems, see [@MillerDegreesCont04]). Later, in Theorem \[thm:some-oracle\], we show a characterization of uniform randomness for arbitrary measures (in terms of Martin-Löf randomness with oracle).
Another application of the trimming technique: let us show that the notion of uniform randomness test is indeed a generalization of the notion of an uniform Bernoulli test we introduced earlier in Definition \[def:Bernoulli-test\].
Let ${\mathbf{t}}(\omega,{P})$ be the universal uniform test and let ${\mathbf{t}}(\omega,p)$ be the universal uniform Bernouli test defined in Lemma \[lem:univ-unif.Bernoulli\]. Then ${\mathbf{t}}(\omega,B_{p}){\stackrel{{}_*}{=}}{\mathbf{t}}(\omega,p)$.
(Here $B_p$ is the Bernoulli measure with parameter $p$.)
For the inequality ${\stackrel{{}_*}{<}}$ note that the function ${(\omega,p)}\mapsto {\mathbf{t}}(\omega,B_{p})$ is an uniform Bernoulli test, since the mapping $p\mapsto B_p$ is computable mapping (in a natural sense).
For the other direction, there exists a computable function on measures that maps $B_p$ to $p$ (just take the probability of the one-bit string). Combining this function with ${\mathbf{t}}(\omega,p)$, we get a lower semicomputable function $f(\omega,{P})$ on general measures ${P}$ with $f(\omega,B_p)={\mathbf{t}}(\omega,p)$. The function $f(\omega,p)$ is not a uniform test yet, but again the trimming technique given by Theorem \[propo:trim\] yields the desired result.
Apriori probability with an oracle, and uniform tests {#subsec:uniform-exact}
-----------------------------------------------------
For a computable measure, we had an expression for the universal test via apriori probability in Proposition \[propo:sum-characteriz\]. An analogous expression exists also for the universal uniform test:
\[thm:uniform-test-sum\] $$\begin{aligned}
{\mathbf{t}}(\omega,{P}) {\stackrel{{}_*}{=}}\sum_{x{\sqsubseteq}\omega}\frac{{\mathbf{m}}(x\mid{P})}{{P}(x)}.
\end{aligned}$$
To be honest, we still owe the reader the definition of the concept of apriori probability with respect to a measure, that is the quantity ${\mathbf{m}}(x\mid{P})$. We do this right away, before returning to the proof.
\[def:uniform-semimeasure\] A nonnegative function $t(x,{P})$ whose arguments are the binary word $x$ and the measure ${P}$ will be called a [*uniform lower semicomputable semimeasure*]{}, if it is lower semicomputable and $\sum_{x} t(x,{P}){\leqslant}1$ for all measures ${P}$ over $\Omega$.
\[propo:apriori-universal-uniform\] Among the uniform lower semicomputable semimeasures, there is a largest one to within a multiplicative constant.
This is proved by the same method as the existence of a universal test (and even simpler, since the constraints on the values of the test do not depend on the measure).
We will fix one such largest semimeasure, and call it the [*apriori probability with respect to*]{} ${P}$. We will denote it by ${\mathbf{m}}(x\mid{P})$.
(The vertical bar in place of a comma emphasizes the similarity to the conditional apriori probability normally considered.)
We need to check two things. First we need to convince ourselves that the right-hand side of the formula defines a uniform test. Every member of the sum can be considered to be a function of two arguments, equal to 0 outside the cone of extensions of $x$, and equal to ${\mathbf{m}}(x\mid{P})/{P}(x)$ inside the cone. For every $x$, the functions ${\mathbf{m}}(x\mid{P})$ and $1/{P}(x)$ are lower semicomputable (uniformly in $x$), and the sum gives a lower semicomputable function. The integral of this function by any measure ${P}$ is equal to the sum of the integrals of the members, that is $\sum_{x}{\mathbf{m}}(x\mid{P})$, and therefore does not exceed 1.
There is a special case, when ${P}(x)=0$ for some $x$. In this case the corresponding member of the sum becomes infinite for any $\omega$ extending $x$. But since the measure of this cone is zero, the integral by this measure is by definition zero, and therefore the additive term, if is not equal to ${\mathbf{m}}(x\mid{P})$, is simply smaller. This way, the right-hand side of the formula is a uniform test, and therefore does not exceed the universal uniform test: we proved the inequality ${\stackrel{{}_*}{>}}$.
The second part of the proof is not so simple: observing the increase of the values of the uniform test, we must distribute this increase among the different members of the sum of the right-hand side, while preserving lower semicomputability. The difficulty is that if, say, the lower semicomputable function was 1 on some effectively open set $A$, and outside it was zero, and then this set was changed to a larger set $B$, then the difference (the characteristic function of $B\setminus A$) will not in general be lower semicomputable since in the set of measures (as also on a segment) the difference of two intervals will not be an open The.
This problem is solved by moving to continuous functions. Let us be given an arbitrary uniform test $t(\omega,{P})$. Since it is lower semicomputable, it can be represented as the limit of a nondecreasing sequence of basic functions, or—passing to differences—in the form of a sum of a series of nonnegative basic functions: $t(\omega,{P})=\sum_{i}t_{i}(\omega,{P})$.
Being basic, the function $t_{i}$ of $\omega$ depends only on a finite prefix of the sequence $\omega$; denote the length of this prefix by $n_{i}$. For every word $x$ of length $n_{i}$ we get some lower semicomputable function $t_{i,x}({P})$, where $t_{i}(\omega,{P})=t_{i,x}({P})$ if $\omega$ begins by $x$. Now define $m_{i}(x,{P})=t_{i,x}({P})\cdot{P}(x)$, if $x$ has length $n_{i}$ (for the other lengths, zero). The function $m_{i}$ is lower semicomputable (as the product of two lower semicomputable functions) uniformly in $i$, and therefore the sum $m(x,{P})=\sum_{i}m_{i}(x,{P})$ will be lower semicomputable.
Let us show that $m$ is a semimeasure, that is $\sum_{x}m(x,{P}){\leqslant}1$ for all ${P}$. Indeed, in $\sum_{i}m_{i}(x,{P})$ the nonzero terms correspond to words of length $n_{i}$, and this sum is equal to $\sum_{x}t_{i,x}({P})\cdot{P}(x)$, that is exactly the integral $\int t_{i}(\omega,{P})\,{P}(d\omega)$, and the sum of these integrals does not exceed 1 by our condition.
Moreover, if for all prefixes $x$ of the sequence $\omega$ the measure ${P}(x)$ is not equal to zero, then $$\begin{aligned}
\sum_{x{\sqsubseteq}\omega}\frac{m_{i}(x,{P})}{{P}(x)}=\frac{t_{i,x_{i}}({P})\cdot{P}(x_{i})}{{P}(x_{i})}
= t_{i}(\omega,{P})
\end{aligned}$$ (here $x_{i}$ is the prefix of length $n_{i}$ of $\omega$), hence after summing over $i$ $$\begin{aligned}
\sum_{x{\sqsubseteq}\omega}\frac{m(x,{P})}{{P}(x)}=t(\omega,{P}),
\end{aligned}$$ and it just remains to apply the maximality of the apriori probability to obtain the ${\stackrel{{}_*}{<}}$-inequality for the case that all prefixes of $\omega$ have nonzero ${P}$-measure. On the other hand, if one of these has zero ${P}$-measure, then the right-hand side is infinite, and so here the inequality is also satisfied.
For the universal randomness test with respect to a computable measure, in this formula one could replace the sum with a maximum. Is this possible for uniform tests? (The reasoning applied there encounters difficulties in the uniform case.) Can one define apriori probabilithy on the tree in a reasonable way, and prove a uniform variant of the Levin-Schnorr theorem?
Effectively compact classes of measures
---------------------------------------
We have considered Bernoulli tests, that is lower semicomputable functions $t(\omega)$ that are tests with respect to all Bernoulli measures. In this definition, in place of Bernoulli measures, an arbitrary effectively compact class can be taken:
\[def:test-effectively-compact.Cantor\] Let ${\mathcal{C}}$ be an effectively compact class of measures over $\Omega$. We say that lower semicomputable function $t$ on $\Omega$ is a [*${\mathcal{C}}$-test*]{} if $\int t(\omega) \,d{P}{\leqslant}1$ for every ${P}\in{\mathcal{C}}$.
\[thm:class-test\] Let ${\mathcal{C}}$ be an effectively compact class of measures.
There exists a universal ${\mathcal{C}}$-test ${\mathbf{t}}_{{\mathcal{C}}}(\cdot)$.
${\mathbf{t}}_{{\mathcal{C}}}(\omega)=\inf_{{P}\in{\mathcal{C}}}{\mathbf{t}}(\omega,{P})$.
Both of these statements are proved analogously to Lemmas \[lem:univ-unif.Bernoulli\] and \[lem:Bernoulli-from-unif\].
Since ${\mathcal{C}}$ is compact and the function ${\mathbf{t}}(\omega,{P})$ is lower semicomputable, the $\inf$-operation can be replaced by the $\min$-operation.
Can we give criteria for randomness with respect to natural closed classes of measures (for example in terms of complexity)? How can we describe Bernoulli sequences in terms of complexities of their initial segments? It is known that the main term of the randomness deficiency is $$\begin{aligned}
\log\binom{n}{k}-{\mathit{Kp}}(x\mid n,k).
\end{aligned}$$ The lecture notes [@GacsLnotesAIT88] contains a characterization Bernoulli sequences, but it is rather messy.
What about Markov measures? Shift-invariant measures?
Sparse sequences
----------------
There are several situations closely related to some intuitive understanding of randomness, but not fitting directly into the framework of the question of a randomness of a given outcome $\omega$ to a given model (measure ${P}$). Our example is here a natural notion of sparsity, introduced in [@BienvenuRomashShenSparse08], but another example, online tests, will be considered in Section \[sec:too-strong\].
It is natural to call “$p$-sparse” a sequence $\omega$, when its 1’s come from some $p$-random sequence $\omega'$, but we allow some of its 0’s to also come from the 1’s of $\omega'$. For example, the 1’s of $\omega'$ may be a sequence of miracles, and $\omega$ is the sequence of those miracles that have been reported. The tacit hypothesis is, of course, that all reported miracles actually happened.
Let us introduce a coordinate-wise order between infinite binary sequences (or binary sequences of the same length): we say $\omega{\leqslant}\omega'$ if this is true coordinate-wise, that is $\omega(i){\leqslant}\omega'(i)$ for all $i$: in other words, $\omega'$ is obtained from $\omega$ replacing some $0$’s with $1$s.
Let $B_p$ be a Bernoulli measure with some computable $p$. We say that a binary sequence $\omega$ is $p$-[*sparse*]{} if $\omega{\leqslant}\omega'$ for some $B_p$-random sequence $\omega'$. (In terms of sets, $p$-sparse sets are subsets of $p$-random sets).
We will show that in the definition of sparsity, the existential quantifier can be eliminated, giving a criterion in terms of monotonic tests.
A real function $f$ on $\Omega$ will be called [*monotonic*]{} if $\omega'{\geqslant}\omega$ implies $f(\omega'){\geqslant}f(\omega)$.
A monotonic lower semicomputable function $t:\Omega\to{[0,\infty]}$ is a $p$-[*sparsity test*]{} if $\int t(\omega)\, d B_{p}{\leqslant}1$. A $p$-sparsity test is [*universal*]{} if it multiplicatively dominates all other sparsity tests for $p$.
The monotonicity of tests guarantees, informally speaking, that only the presence of some $1$s is counted as regularity, not their absence. (Note that earlier we spoke of an entirely different kind of monotonicity, while defining extended tests: there we compared the values of a function on a finite word and its extension.)
Consider the universal test ${\mathbf{t}}(\omega,{P})$. The expression $$\begin{aligned}
{r}_{p}(\omega) =\min_{\omega'{\geqslant}\omega}{\mathbf{t}}(\omega',B_{p})
\end{aligned}$$ defines a universal $p$-sparsity test.
Each $p$-sparsity test is by definition a test with respect to the measure $B_{p}$. Using its monotonicity and comparing it with the universal test we obtain that no sparsity test exceeds $r_{p}$ (to within a bounded factor).
In the other direction it must be shown that the minimum in the expression for $r_{p}$ is achieved, and that this function is a $p$-sparsity test. The lower semicomputability is proved usin that the property $\omega{\leqslant}\omega'$ gives an effectively closed set of the effectively compact space $\Omega\times\Omega$. The monotonicity and the integral inequality follow immediately from the definition.
From this follows the following characterization in terms of tests:
A sequence $\omega$ is $p$-sparse (is obtained from a $p$-random by replacing some $1$s with zeros) if and only if the universal sparsity test ${r}_{p}(\omega)$ is finite.
Sparsity is equivalent to randomness with respect to a certain class of measures. To define this class, we introduce the notion of coupling of measures.
For measures ${P},{Q}$ we say ${P}\preceq {Q}$, or that ${P}$ can be [*coupled below*]{} ${Q}$ if there exists a probability distribution $R$ on pairs of sequences ${(\omega,\omega')}$ such that
The first projection (marginal distribution) is ${P}$ and the second one is ${Q}$.
Measure $R$ is entirely concentrated on pairs ${(\omega,\omega')}$ with $\omega{\leqslant}\omega'$ (the probability of this event by the measure $R$ is $1$.
The following characterization of coupling is well known: it has many proofs, but all seem to go back to [@Strassen65] (Theorem 11, p. 436). A proof can be found in [@BienvenuRomashShenSparse08].
\[propo:coupling-characteriz\] The property ${P}\preceq {Q}$ is equivalent to the following: for all monotonic basic functions $f$ the following inequality holds: $$\begin{aligned}
\int f(\omega)\,d{P}{\leqslant}\int f(\omega)\,d {Q}.
\end{aligned}$$
In this characterization, we could have said “all monotonic integrable functions” as well.
Let ${\mathcal{S}}_{p}$ be the set of measures that can be coupled below $B_{p}$.
The set ${\mathcal{S}}_{p}$ of measures is effectively closed (and thus effectively compact).
For each function $f$ in Proposition \[propo:coupling-characteriz\], the condition defines an effectively closed set, and their intersection will also be effectively closed.
\[thm:mon-class-test\] The universal $p$-test ${r}_{p}(\omega)$ is a universal class test for class ${\mathcal{S}}_{p}$.
Thus, a sequence is $p$-sparse if and only if it is random with respect to some measure that can be coupled below $B_{p}$.
The following lemma will be key to the proof.
\[lem:monotonize\] Let $t:\Omega\to{\mathbb{R}}$ be a basic function with $\int t(\omega)\,d {Q}{\leqslant}1$ for all ${Q}\in{\mathcal{S}}_{p}$. Define the monotonic function $\hat t(\omega)=\max_{\omega'{\leqslant}\omega}t(\omega')$ (the maximum is achieved since $t(\omega)$ depends only on finitely many positions of $\omega$). Then $\int\hat t(\omega)\,d B_{p}{\leqslant}1$.
Let function $t$ depend only on the first $n$ coordinates. For each $x\in\{0,1\}^{n}$ fix $x'{\leqslant}x$ for which $t(x')$ reaches the maximum (among all such $x'$). Besides the distribution $B_{p}$ consider a distribution ${Q}$ in which the Bernoulli measure of $x$ is tranferred to $x'$ (the measures of several $x$ may be transferred to the same $x'$ and then be added). We described the behavior of ${Q}$ on the first $n$ bits; the following bits are chosen independently, and the probability of 1 in each position is equal to $p$. Note also that for the expected values of the functions $t$ and $\hat t$ only the first $n$ bits count.
By the construction, ${Q}\preceq B_{p}$ (essentially, we described a measure on pairs), therefore $\int t(\omega)\,d{Q}{\leqslant}1$. But this integral is equal to $\int\hat t(\omega)\, d B_{p}$.
Let us return to the theorem.
Every $p$-sparsity test $t$ is a class test for ${\mathcal{S}}_{p}$. Indeed, its integral by a measure in the class ${\mathcal{S}}_{p}$ does not exceed its integral by the measure $B_{p}$, by the monotonicity of the test and the possibility of coupling.
On the other hand, let us show that for every test $t$ for the class ${\mathcal{S}}_{p}$, there is a a $p$-sparsity test that is not smaller. Indeed, the test $t$ is the limit of an increasing sequence $t_{n}$ of basic functions. Applying to them the monotonization lemma \[lem:monotonize\], we obtain a sequence of basic functions $\hat t_{n}$ that are everywhere greater or equal to $t_{n}$ and have integrals bounded by 1 with respect to the measure $B_{p}$. Their limit is the needed $p$-sparsity test.
Different kinds of randomness
-----------------------------
There are several ways to define randomness with respect to an arbitrary (not necessarily computable) measure. We have already defined uniform randomness. Here are some other ways.
#### Oracles
We can use the Martin-Löf definition (or its average-bounded version) with oracles. We would call a sequence $\omega$ random with respect to ${P}$, if there exists an oracle $A$ that makes ${P}$ computable such that $\omega$ is ML-random with respect to ${P}$ with oracle $A$. (We say “there exists an oracle that makes ${P}$ computable” but not “for all oracles that make ${P}$ computable”: indeed, some powerful oracle can always make $\omega$ computable and therefore non-random, unless $\omega$ is an atom of ${P}$.) As Adam Day and Joseph Miller have shown [@DayMiller10], this definition turns out to be equivalent to uniform randomness. The proof of this equivalence needs some preparation.
First let us look into why is it not possible to take for oracle the measure itself (as was done for the Bernoulli measures, where for oracle we chose a binary expansion of the number $p$). Well, the choice of such a representation is not unique ($0.01111\dots=0.10000\dots$). When all we have is a single number $p$ then this is not important, as the non-uniqueness arises only for rational $p$, and in this case both representations are computable. But for measures this is not so: a measure is represented by a countable number of reals (say, the probabilities of individual words, or the conditional probabilities), and the arbitrariness in the choice of representation is not reduced to a finite number of variants.
\[def:r-test\] Fix some representation of measures by infinite binary sequences, that is a computable (and therefore continuous) mapping $\pi\mapsto{R}_{\pi}$ of $\Omega$ onto the space of measures. For example, we may split the binary sequence $\pi$ into countably many parts and use these parts as binary representations of the probability that the sequence continues with $1$ after a certain prefix.
Define the notion of an [*r-test*]{} (representation-test, test of randomness relative to a given representation of the measure) as a lower semicomputable function $t(\omega,\pi)$ with $\int t(\omega,\pi){R}_{\pi}(d\omega){\leqslant}1$ for all $\pi$.
This notion of r-test depends on the representation method chosen; there are no intuitive reasons to choose one specific representation and declare it to be “natural”, but any representation is good for the argument below and we assume some representation fixed. The following statements can be proven just as similar statements before:
\[i:some-oracle.trim\] Every lower semicomputable function $t(\omega,\pi)$ can be trimmed to make it not greater than twice an r-test (not changing it for those $\pi$ where it already was a r-test).
\[i:some-oracle.univ\] There exists an universal (maximal to within a bounded factor) r-test ${\mathbf{t}}(\omega,\pi)$.
For a fixed $\pi$, the function ${\mathbf{t}}(\cdot,\pi)$ is universal among the $\pi$-computable average-bounded tests with respect to the measure ${R}_{\pi}$. Indeed, it is such a test; on the other hand, any such test can be lower semicomputed by the oracle machine. This machine is applicable to any oracle (though may not give a test), giving a lower semicomputable function $t'(\omega,\pi)$ that is equal to the starting test for the given $\pi$. It remains to apply property .
As a consequence of this simple reasoning we obtain that the quantity ${\mathbf{t}}(\omega,\pi)$ is finite if and only if the sequence $\omega$ is random relative to the oracle $\pi$, with respect to measure ${R}_{\pi}$.
\[thm:some-oracle\] A sequence $\omega$ is uniformly random with respect to measure ${P}$ if and only if there is an oracle computing ${P}$ that makes $\omega$ random (in the original Martin-Löf sense). More precisely, $$\label{eq:some-oracle.inf}
{\mathbf{t}}(\omega,{P}){\stackrel{{}_*}{=}}\inf_{{R}_{\pi}={P}}{\mathbf{t}}(\omega,\pi).$$
Let us prove the equality shown in the theorem. Note that if $t$ is a uniform test, then $t(\omega,{R}_{\pi})$ as a function of $\omega$ and $\pi$ is an r-test, and is therefore dominated by the universal r-test.
The other direction is somewhat more difficult. We have to show that the function on the right-hand side is lower semicomputable as a function of the sequence $\omega$ and the measure ${P}$. (The integral condition is obtained easily afterwards, as the measure ${P}$ has at least one representation $\pi$.) This can be proved using the effective compactness of the set of those pairs ${({P},\pi)}$ with ${P}={R}_{\pi}$. In the general form (for constructive metric spaces) this statement forms the content of Lemma \[lem:push-forward\].
It remains to explain the connection between the given equality and randomness relative to an oracle. If ${\mathbf{t}}(\omega,{P})$ is finite, then by the proved equality a $\pi$ exists with ${R}_{\pi}={P}$ and finite ${\mathbf{t}}(\omega,\pi)$. As we have seen, this in turn means that $\omega$ is random with respect to the measure ${P}$, with an oracle $\pi$ that makes ${P}$ computable. Conversely, if ${\mathbf{t}}(\omega,{P})$ is infinite, and some oracle $A$ makes ${P}$ computable, then the function ${\mathbf{t}}(\cdot,{P})$ becomes $A$-lower semicomputable, and its integral by measure ${P}$ does not exceed 1, hence the sequence $\omega$ will not be random relative to oracle $A$ and with respect to measure ${P}$.
#### Blind (oracle-free) tests
We can define the notion of an effectively null set as before, even if the measure is not computable. The maximal effectively null set may not exist. For example, if measure ${P}$ may be concentrated on some non-computable sequence $\pi$, then all intervals not containing $\pi$ will be effective null sets, and their union (the complement of the singleton $\{\pi\}$) will not be, otherwise $\pi$ would be computable.
However, we still can define random sequence as a sequence that does not belong to *any* effectively null set. Kjos-Hanssen suggested the name “Hippocratic randomness” for this definition (referring to a certain legend about the doctor Hippocrates), but we prefer the more neutral name “blind randomness”.
\[def:blind-test.Cantor\] A lower semicomputable function $t(\omega)$ with integral bounded by $1$ will be called a [*blind, or oracle-free, test*]{} for measure ${P}$. A sequence $\omega$ is [*blindly random*]{} iff $t(\omega)<\infty$ for all blind tests.
As seen, there may not exist a maximal blind test.
This oracle-free notion of randomness can be characterized in the terms introduced earlier:
\[thm:characterize-obliv\] Sequence $\omega$ is blindly random with respect to measure ${P}$ if and only if $\omega$ is random with respect to any effectively compact class of measures that contains ${P}$.
Assume first that $\omega$ is not random with respect to some effectively compact class of measures that contains ${P}$. Then the universal test with respect to this class is a blind test that shows that $\omega$ is not blindly random with respect to ${P}$.
Now assume that there exists some blind test $t$ for measure ${P}$ with $t(\omega)=\infty$. Then just consider the class ${\mathcal{C}}$ of measures ${Q}$ with $\int t(\omega)\, d{Q}{\leqslant}1$. This class is effectively closed, (and thus effectively compact). Indeed, $t$ be the supremum of the computable sequence of basic functions $t_{n}$. The class of measures ${Q}$ with $\int t_{n}(\omega)\,d Q>1$ is effectively open, uniformly in $n$, and ${\mathcal{C}}$ is the complement of the union of these sets.
It is easy to see from the definition (or from the last theorem) that uniform randomness implies blind randomness (either directly or using the last theorem). The reverse statement is not true:
\[thm:nonrandom-blind\] There exists a sequence $\omega$ and measure ${P}$ such that $\omega$ is blindly random with respect to ${P}$ but not uniformly random.
Indeed, oblivous randomness does not change if we change the measure slightly (up to $O(1)$-factor). On the other hand, the changed measure may have much more oracle power that makes a sequence non-random. For example, we may start with uniform Bernoulli random measure $B_{1/2}$ (coin tosses with probabilities $1/2, 1/2, 1/2,\ldots$ and fix some random sequence $\omega=\omega(1)\omega(2)\dots$. Then consider a (slightly) different measure $B'$ with probabilities $1/2+\omega(1)\varepsilon_{1},
1/2+\omega(2)\varepsilon_2,\dots$ where $\varepsilon_1,\varepsilon_2,\dots$ are so small and converge to zero so fast that they do not change the measure more than by $O(1)$-factor while being all positive. Then $B'$ encodes $\omega$, which makes it easy to construct a uniform test $t$ with $t(\omega, B')=\infty$.
However, there are some special cases (including Bernoulli measures) where uniform and blind randomness are equivalent. In order to formulate the sufficient conditions for such a coincidence, let us start with some definitions.
\[def:effectively-orth\] For a probability measure ${P}$, let ${\mathrm{Randoms}}({P})$ denote the set of sequences uniformly random with respect to ${P}$. A class of measures is called [*effectively orthogonal*]{} if ${\mathrm{Randoms}}({P})\cap{\mathrm{Randoms}}({Q})=\emptyset$ for any two different measures in it.
\[thm:orthogonal-blind.bin-Cantor\] Let ${\mathcal{C}}$ be an effectively compact, effectively orthogonal class of measures. Then for every measure ${P}$ in ${\mathcal{C}}$ the uniform randomness with respect to ${P}$ is equivalent to blind randomness with respect to ${P}$.
The statement looks strange: we claim something about randomness with respect to measure ${P}$, but the condition of the claim is that ${P}$ can be included into a class of measures with some properties. (It would be natural to have a more direct requirement for ${P}$ instead.) The theorem implies that the measures of Theorem \[thm:nonrandom-blind\] do not belong to any such class.
We have noted already that in one direction the statement is obviously true. Let us prove the converse. Assume that sequence $\omega$ is blindly random with respect to measure ${P}$. By Theorem \[thm:characterize-obliv\], it is random with respect to the class ${\mathcal{C}}$. So, $\omega$ is uniformly random with respect to some measure ${P}'$ from the class ${\mathcal{C}}$. It remains to show ${P}={P}'$.
Imagine that this is not the case. Then we can construct an effectively compact class of measures ${\mathcal{C}}'$ that contains ${P}$ but not ${P}'$. Indeed, since ${P}$ and ${P}'$ are different, they assign different measures to some finite string, and this fact can be used, in form of a closed condition separating ${P}$ from ${P}'$, to construct ${\mathcal{C}}'$. Consider now the effectively compact class ${\mathcal{C}}\cap{\mathcal{C}}'$. It contains ${P}$, and therefore $\omega$ will be random with respect to this class. Hence the class contains some measure ${P}''$ with respect to which $\omega$ is uniformly random. But ${P}'\ne{P}''$ (one measure is in ${\mathcal{C}}'$, the other one is not), so we get a contradiction with the assumption with the effective orthogonality of the class ${\mathcal{C}}$.
The proved theorem is applicable in particular to the class of Bernoulli measures. It is tempting to think that there is a simpler proof, at least for this case: if $\omega$ is random with respect to $p$ we can compute $p$ from $\omega$ as the limit of relative frequency, and no additional oracle is needed. This is not so: though $p$ is *determined* by $\omega$, it does not even depend continuously on $\omega$. Indeed, no initial segment of the sequence guarantees that its limiting frequency is in some given interval. However, we can apply an analogous reasoning to those sequences $\omega$ with the randomness deficiency bounded by some constant. (See [@HoyrupRojasCiE2009] which introduces the notion of [*layerwise computability*]{}.) In particular, it can be shown that if $\omega$ is random with respect to the measure $B_{p}$ then $p$ is computable with oracle $\omega$.
Neutral measure
===============
The following theorem, first published in [@LevinUnif76] and then again in [@GacsUnif05], points to a curious property of uniform randomness which distinguishes it from randomness using an oracle.
A measure is called [*neutral*]{} if every sequence is uniformly random with respect to it.
There exists a neutral measure; moreover, there is a measure $N$ with ${\mathbf{t}}(\omega,N){\leqslant}1$ for all sequences $\omega$.
Note that a neutral measure cannot be computable. Indeed, for a computable measure there exists a computable sequence that is not an atom (adding bits sequentially, we choose the next bit in such a way that its conditional probability is at most $2/3$). Such a sequence cannot be random with respect to $N$. For the same reason a neutral measure cannot be equivalent to an oracle (for a neutral measure $N$ one cannot find an oracle $A$ that make it computable and at the same time can be uniformly reconstructed from every approximation of $N$). Indeed, in this case uniform randomness (as we have shown) is equivalent to randomness with respect to $N$ with oracle $A$, and the same argument works.
A neutral measure cannot be lower or upper semicomputable either, but this statement does not seem interesting, since here a semicomputable measure over $\Omega$ is also computable. Some more meaningful (and less trivial) versions of this fact are proved in [@GacsUnif05].
Consider the universal test ${\mathbf{t}}(\omega,{P})$. We claim that there exists a measure $N$ with ${\mathbf{t}}(\omega,N){\leqslant}1$ for every $\omega$. In other terms, for every $\omega$ we have a condition on $N$ saying that ${\mathbf{t}}(\omega,N){\leqslant}1$ and we need to prove that these conditions (there is continuum of them) have non-empty intersection. Each of these condition is a closed set in a compact space (recall that ${\mathbf{t}}$ is lower semicontinuous), so it is enough to show that finite intersections are non-empty.
So let $\omega_1,\ldots,\omega_k$ be $k$ sequences. We want to prove that there exists a measure $N$ such that ${\mathbf{t}}(\omega_i,N){\leqslant}1$ for every $i$. This measure will be a convex combination of measures concentrated on $\omega_1,\ldots,\omega_k$. So we need to prove that $k$ closed subsets of a $k$-vertex simplex (corresponding to $k$ inequalities) have a common point. It is a direct consequence of the following classical topology result formulated in Lemma \[lem:Sperner\] below (which is used in the standard proof of Brouwer’s fixpoint theorem).
To show that the lemma gives us what we want, consider any point of some face. For example, let $X$ be a measure that is a mixture of, say, $\omega_1$, $\omega_5$ and $\omega_7$. We need to show that $X$ belongs to $A_1\cup A_5\cup A_7$: in our terms, that one of the numbers ${\mathbf{t}}(\omega_1,X)$, ${\mathbf{t}}(\omega_5,X)$ and ${\mathbf{t}}(\omega_7,X)$ does not exceed $1$. It is easy since we know $\int t(\omega,X)\,d X(\omega){\leqslant}1$ (by the definition of the test), and this integral is a convex combination of the above three numbers with some coefficients (the weights of $\omega_1$, $\omega_5$ and $\omega_7$ in $X$.
\[lem:Sperner\] Let a simplex with vertices $1,\dots,n$ be covered by closed sets $A_1,\ldots,A_{k}$ in such a way that vertex $i$ belongs to $A_i$ (for every $i$), edge $i$-$j$ is covered by $A_i\cup A_j$, and so on (formally, face $(i_1,\dots,i_s)$ of the simplex is a subset of $A_{i_1}\cup\ldots\cup A_{i_s}$; in particular, the union $A_1\cup\dots\cup A_{k}$ is the entire simplex). Then the intersection $A_1\cap\dots\cap A_{k}$ is not empty.
For completeness, let us reproduce the standard proof of this lemma.
Consider a disjoint division $T$ of the simplex into smaller $n$-dimensional simplices (in such a way that every vertex in the division is a vertex of every simplex containing it). Let $S$ be the set of vertices of $T$. A [*Sperner-labeling*]{} is a covering of $S$ by sets $A_{1},\dots,A_{k}$ such that the points of $S$ belonging to each lower-dimensional simplex formed by some vertices $i_1<i_2<\dots <i_r{\leqslant}k$ are covered by $A_{i_1}\cup\dots\cup A_{i_{r}}$. (A point gets label $i$ if it belongs to $A_{i}$.) Sperner’s famous combinatorial lemma (see for example the Wikipedia) implies that in any Sperner labeling, there is a simplex whose vertices are labeled with all $k$ colors.
To apply the Sperner’s lemma, note that our closed sets $A_{i}$ satisfy the rules of Sperner coloring. Sperner’s lemma guarantees the existence of a simplex that has all possible labels on its vertices. In this way we can get arbitrarily small simplices with this property; compactness then shows that all $A_i$ have a common point.
Randomness in a metric space
============================
Most of the theory presented above for infinite binary sequences generalizes to infinite sequences of natural numbers. Much of it generalizes even further, to an arbitrary metric space. In what follows below we not only generalize; some of the results are new also for the binary sequence case.
Constructive metric spaces
--------------------------
We rely on the definition of a constructive metric space, and the space of measures on it, as defined in [@GacsUnif05] and [@HoyrupRojasRandomness09] (the lecture notes [@GacsLnotesAIT88] are also recommended).
A [*constructive metric space*]{} is a tuple ${\mathbf{X}}= (X, d, D, \alpha)$ where $(X,d)$ is a complete separable metric space, with a countable dense subset $D$ and an enumeration $\alpha$ of $D$. It is assumed that the real function $d(\alpha(v),\alpha(w))$ is computable. Open balls with center in $D$ and rational radius are called [*ideal balls*]{}, or [*basic open sets*]{}, or [*basic balls*]{}. The (countable) set of basic balls will also be called the [*canonical basis*]{} in the topology of the metric space.
An infinite sequence $s_{1},s_{2},\dots$ with $s_{i}\in D$ is called a [*strong Cauchy*]{} sequence if for all $m<n$ we have $d(s_{m},s_{n}){\leqslant}2^{-m}$. Since the space is complete, such a sequence always has a (unique) limit, which we will say is [*represented*]{} by the sequence.
We will generally use the notational convention of this definition: if there is a constructive metric space with an underlying set $X$ then the we will use ${\mathbf{X}}$ (boldface) to denote the whole structure $(X,d,D,\alpha)$. But frequently, we just use $X$ when the structure is automatically understood.
\[example:metric\]
1. \[i:example.metric.discrete\] A set $X=\{s_{1},s_{2},\dots\}$ can be turned into a constructive [*discrete*]{} metric space by making the distance between any two different points equal to 1. The set $D$ consists of all points $\alpha(i)=s_{i}$.
2. \[i:example.metric.one-point-compactif\] The set $\overline{\mathbb{N}}={\mathbb{N}}\cup\{\infty\}$ can be turned into a constructive metric space by making the distance between any two different points with the distance function $d(x,y)=|\frac{1}{x}-\frac{1}{y}|$, where of course, $\frac{1}{\infty}=0$. The set $D$ consists of all points of ${\mathbb{N}}$. This metric space is called the [*one-point compactification*]{}, in a topological sense, of the [*discrete metric space*]{} ${\mathbb{N}}$ of Example \[i:example.metric.discrete\].
3. The real line ${\mathbb{R}}$ with the distance $d(x,y) = |x - y|$ is a constructive metric space, and so is ${\mathbb{R}}_{+}={[0,\infty)}$. We can add the element $\infty$ to get $\overline{\mathbb{R}}_{+}={[0,\infty]}$. This is not a metric space now, but is still equipped with a natural constructive topology (see Remark \[rem:constr-top\] below). It could be equipped with a new metric in a way that would not change this constructive topology.
4. \[i:example.metric.L1\] If ${\mathbf{X}},{\mathbf{Y}}$ are constructive metric spaces, then we can define a constructive metric space ${\mathbf{Z}}={\mathbf{X}}\times{\mathbf{Y}}$ with one of its natural metrics, for example the sum of distances in both coordinates. In case when ${\mathbf{X}}={\mathbf{Y}}={\mathbb{R}}$, this is called the $L_{1}$ metric. Let $D_{{\mathbf{Z}}}$ be the product $=D_{{\mathbf{X}}}\times D_{{\mathbf{Y}}}$.
5. \[i:example.metric.Cantor\] Let $X$ be a finite or countable (enumerated) alphabet, with a fixed numbering, and let $X^{{\mathbb{N}}}$ be the set of infinite sequences $x = {( x(1), x(2), \dots)}$ with distance function $d^{{\mathbb{N}}}(x,y) = 2^{-n}$ where $n$ is the first $i$ with $x(i)\ne y(i)$. This space generalizes the binary Cantor space of Definition \[def:binary-Cantor\], to the case mentioned in Remark \[rem:seq-of-natural\]. The balls in it are cylinder sets: for a given finite sequence $z$, we take all continuations of $z$.
Each point $x$ of a constructive metric space ${\mathbf{X}}$ can be viewed as an “approximation mass problem”: the set of total functions that for any given rational $\varepsilon>0$ produce a $\varepsilon$-approximation to $x$ by a point of the canonical dense set $D$. This is a mass problem in the sense of [@MedvedevMass55]. One can also note that this mass problem is Medvedev equivalent to the enumeration problem: enumerate all basic balls that contain $x$.
\[rem:constr-top\] A constructive metric space is special case of a more general concept, which is often useful: a constructive topological space.
A [*constructive topological space*]{} ${\mathbf{X}}= (X, \tau, \nu)$ is a topological space over a set $X$ with a basis $\tau$ effectively enumerated (not necessarily without repetitions) as a list $\tau ={\mathopen\{\nu(1),\nu(2),\dots\mathclose\}}$.
For every nonempty subset $Z$ of the space $X$, we can equip $Z$ with a constructive topology: we intersect all basic sets with $Z$, without changing their numbering. On the other hand, not every subset of a constructive metric space naturally has the structure of a constructive metric space (the everywhere dense set $D$ is not inherited).
But instead of introducing constructive topological spaces formally, we prefer not to burden the present paper with more abstractions, and will speak about some concepts like effective open sets and continuous functions, as defined on an arbitrary subset $Z$ of the constructive metric space $X$.
An open subset of a constructive metric space is [*lower semicomputable open*]{} (or r.e. open, or c.e. open), or [*effectively open*]{} if it is the union of an enumerable set of elements of the canonical basis. It is [*upper semicomputable closed*]{}, or [*effectively closed*]{} if its complement is effectively open. Given any set $A\subseteq X$, a set $U$ is [*effectively open on $A$*]{} if there is an effective open set $V$ such that $U\cap A=V\cap A$.
Note that in the last definition, $U$ is not necessarily part of $A$, but only its intersection with $A$ matters.
Computable functions can be defined in terms of effectively open sets.
Let $X,{Y}$ be constructive metric spaces and $f:X\to {Y}$ a function. Then $f$ is [*continuous*]{} if for each element $U$ of the canonical basis of ${Y}$ the set $f^{-1}(U)$ is an open set. It is [*computable*]{} if $f^{-1}(U)$ is also an effectively open set, uniformly in $U$. A partial function $f:X\to {Y}$ defined at least on a set $A$ is [*computable*]{} if for each element $U$ of the canonical basis of ${Y}$ the set $f^{-1}(U)$ is effectively open on $A$, uniformly in $U$.
An element $x\in X$ is called [*computable*]{} if the function $f:\{0\}\to X$ with $f(0)=x$ is computable.
When $f(x)$ is defined only in a single point $x_{0}$ then we say that the element $y_{0}=f(x_{0})$ is $x_{0}$-[*computable*]{}. When $f:X\times Z\to {Y}$, defined on $X\times{\mathopen\{z_{0}\mathclose\}}$, is computable, then we say that the function $g:X\to {Y}$ defined by $g(x)=f(y, z_{0})$ is $z_{0}$-[*computable*]{}, or computable [*from*]{} $z_{0}$.
There are several alternative characterizations of a computable element.
The following statements are equivalent for an element $x$ of a constructive metric space ${\mathbf{X}}=(X,d,D,\alpha)$.
1. $x$ is computable.
2. the set of basic balls containing $x$ is enumerable.
3. There is a computable sequence $z_{1},z_{2},\dots$ of elements of $D$ with $d(x,z_{n}){\leqslant}2^{-n}$.
The following proposition connects computability with a more intuitive concept based on representation by strong Cauchy sequences.
Let ${\mathbf{X}},{\mathbf{Y}}$ be constructive metric spaces and $f:X\to {Y}$ a function. Then $f$ is computable if and only if there is a computable transformation that turns each strong Cauchy sequence $s_{1},s_{2},\dots$ with $s_{i}\in D_{{\mathbf{X}}}$ converging to a point $x\in X$ into a strong Cauchy sequence $t_{1},t_{2},\dots$ with $t_{i}\in D_{{\mathbf{Y}}}$ converging to $f(x)$.
If $f$ is a partial function with domain $Z$ then $f$ is computable if and only if there is a computable transformation that turns each strong Cauchy sequence $s_{1},s_{2},\dots$ with $s_{i}\in D_{{\mathbf{X}}}$ converging to some point $x\in Z$ into a strong Cauchy sequence $t_{1},t_{2},\dots$ with $t_{i}\in D_{{\mathbf{Y}}}$ converging to $f(x)$.
We omit the—not difficult—proof of this statement.
Though $x_{0}$-computability means computability from a strong Cauchy sequence $s_{1},s_{2},\dots$ converging to $x_{0}$, it should not be considered the same as computability using a machine that treats this sequence as an “oracle”. In case of $x_{0}$-computability, the resulting output must be independent of the strong Cauchy sequence $s_{1},s_{2},\dots$ representing $x_{0}$.
The following definition of lower semicomputability is also a straightforward generalization of the special case in Definition \[def:lower-semicomp.seqs\].
Let ${\mathbf{X}}=(X,d,D,\alpha)$ be a constructive metric space. A function $f:X\to{[-\infty,\infty]}$ is [*lower semicontinuous*]{} if the sets ${\mathopen\{\,x : f(x)>r\,\mathclose\}}$ are open, for every rational number $r$ (from here it follows that they are open for all $r$, not only rational).
It is [*lower semicomputable*]{} if these sets are effectively open, uniformly in the rational number $r$. It is [*upper semicomputable*]{} if $-f$ is lower semicomputable.
A partial function $f:X\to {Y}$ defined at least on a set $A$ is [*lower semicomputable*]{} on $A$ if the sets ${\mathopen\{\,x : f(x)>r\,\mathclose\}}$ are effectively open in $A$, uniformly for every rational number $r$.
It is easy to check that a real function over a constructive metric space is computable if and only if it is lower and upper semicomputable. As before, one can define semicomputability equivalently with the help of basic functions.
Let us introduce an everywhere dense set of simple functions.
\[example:hat-funcs\] We define an enumerated list of [*basic*]{} functions ${\mathcal{E}}=\{e_{1},e_{2},\dots\}$ in the constructive metric space ${\mathbf{X}}=(X,d,D,\alpha)$ as follows. For each point $u\in D$ and positive rational numbers $r,\varepsilon$ let us define the [*hat function*]{} $g_{u,r,\varepsilon}$: its value in point $x$ is determined by the distance of $x$ to $u$ and is equal to $1$, if this distance is at most $r$, equal to zero, if the distance is not less than $r+\varepsilon$, and varies linearly as the distance runs through the segment ${[r,r+\varepsilon]}$: see Figure \[fig:hat-function\].
![A hat function[]{data-label="fig:hat-function"}](hat-function)
Let ${\mathcal{E}}$ be the smallest set of functions containing all hat functions that is closed under $\min,\max$ and rational linear combinations.
\[propo:lower-semi\] A function $f:X\to{[0,\infty]}$ defined on a constructive metric space is lower semicomputable if and only if it is the limit of a computable increasing sequence of basic functions.
Note that the above characterization holds also for lower semicontinous functions, if we just omit the requirement that the sequence $g_{n}$ be computable.
We can introduce the notion of lower semicomputability [*from*]{} $z_{0}$, or [*$z_0$-lower semicomputability*]{}, similarly to the $z_{0}$-computability of Definition \[def:comp-func\], as lower semicomputability of a function defined on the set $X\times \{z_{0}\}$.
Sometimes two metrics on a space are equivalent from the point of view of computability questions. Let us formalize this notion.
\[def:uniform-contin\] Let $X,Y$ be two metric spaces, and $f:X\to Y$ a function. We say that $f$ is [*uniformly continuous*]{} if for each $\varepsilon>0$ there is a $\delta>0$ such that $d_{X}(x,y){\leqslant}\delta$ implies $d_{Y}(f(x),f(y)){\leqslant}\varepsilon$.
If ${\mathbf{X}},{\mathbf{Y}}$ are constructive metric spaces and function $f$ is computable, we will call it [*effectively uniformly continuous*]{} if $\delta$ can be computed from $\epsilon$ effectively.
Two metrics $d_{1},d_{2}$ over the same space are [*(effectively) equivalent*]{} if the identity map is (effectively) continuous in both directions.
For example, the Euclidean metric and the $L_{1}$ metric introduced in Example \[example:metric\].\[i:example.metric.L1\] are equivalent in the space ${\mathbb{R}}^{2}$.
Effective compactness was introduced in Definition \[def:effectively-compact\]: this generalizes immediately to arbitrary metric spaces. A weaker notion, local compactness, also has an effective version.
\[def:effectively-compact-metric\] A compact subset $C$ of a constructive metric space ${\mathbf{X}}=(X,d,D,\alpha)$ is called [*effectively compact*]{} if the set $$\begin{aligned}
{\mathopen\{\,S : S \text{ is a finite set of basic open sets and } \bigcup_{E\in S}E\supseteq C\,\mathclose\}}
\end{aligned}$$ is enumerable.
A subset $C$ of a metric space is called [*locally compact*]{} if it is covered by the union of a set of balls $B$ such that $\overline B\cap C$ is compact. Here $\overline B$ is the closure of $B$. It is [*effectively locally compact*]{} if it is covered by the union of an enumerated sequence of basic balls $B_{k}$ such that $\overline B_{k}\cap C$ is effectively compact, uniformly in $k$.
1. The countable discrete space of Example \[example:metric\].\[i:example.metric.discrete\] is effectively compact if it is finite, and effectively locally compact otherwise.
2. The segment ${[0,1]}$ is effectively compact. The line ${\mathbb{R}}$ is effectively locally compact.
3. If the alphabet $X$ is finite then the space $X^{{\mathbb{N}}}$ of infinite sequences is effectively compact. Otherwise it is not even locally compact.
4. Let $\alpha\in{[0,1]}$ be a lower semicomputable real number that is not computable. (It is known that there are such numbers, for example $\sum_{x\in {\mathbb{N}}} 2^{-{\mathit{Kp}}(x)}$.) The lower semicomputability of $\alpha$ allows to enumerate the rationals less than $\alpha$ and allows for the segment ${[0,\alpha]}$ to inherit the constructive metric (and topology) from the real line. This space is compact, but not effectively so.
The following is a useful characerization of effective compactness.
\[propo:epsilon-net\]
A compact subset $C$ of a constructive metric space ${\mathbf{X}}=(X,d,D,\alpha)$ is effectively compact if and only if from each (rational) $\varepsilon$ one can compute a finite set of $\varepsilon$-balls covering $C$.
For an effectively compact subset $C$ of a constructive metric space, in every enumerable set of basic open sets covering $C$ one can effectively find a finite covering.
Assume that for all $\varepsilon$ we can show a finite covering $S_{\varepsilon}$ of the set $C$ by balls of radius $\varepsilon$. Along with such a covering, we can enumerate all coverings with *guaranteedly* large balls (this means that for all balls $B(x,\varepsilon)$ from the covering $S_{\varepsilon}$ there is a ball $B(y,\sigma)$ from the new covering with $\sigma>\varepsilon+d(x,y)$). The compactness of $C$ guarantees that while $S_{\varepsilon}$ runs through all $\varepsilon$-coverings of $C$, this way all coverings of $C$ will be enumerated. (Indeed, if there is some covering $S'$ not falling into the enumeration, then for all $\varepsilon$ there is a ball of the covering $S_{\varepsilon}$ not guaranteedly contained in any ball of $S'$. Applying compactness and taking a limit point of the centers of these non-contained balls, we obtain contradiction.)
The remaining statements are proved quite easily.
The following statement generalizes Proposition \[propo:closed-to-compact\], with the same proof.
Every effectively closed subset $E$ of an effectively compact set $C$ is also effectively compact.
As earlier, the converse also holds: every effectively compact subset of a constructive metric space is effectively closed. Indeed, we can consider all possible coverings of this set by basic balls, and also outside balls that manifestly (by the relation of the distances of their centerse and their radiuses) are disjoint from the balls of the covering. The union of all these outside balls provide the complement of our effectively compact set.
It is known that a continuous function maps compact sets into compact ones. This statement also has a constructive counterpart, also provable by a standard argument:
\[propo:image-of-eff-compact\] Let $C$ be an effectively compact subset of a constructive metric space $X$, and $f$ a computable function from $X$ into another constructive metric space $Y$. Then $f(C)$ is effectively compact.
The statement that a lower semicontinuous function on a compact set reaches its minimum has also a computable analog (we provide a parametrized variant):
\[propo:lsc-min\] Let ${\mathbf{Y}},{\mathbf{Z}}$ be constructive metric spaces, let $f:Y\times Z\to{[0,\infty]}$ be a lower semicomputable function, and $C$ an effectively closed subset of $Y\times Z$. If it is also effectively compact, then the function $$\begin{aligned}
g(y)=\inf_{z:(y,z)\in C} f(y,z)
\end{aligned}$$ is lower semicomputable from below (and the $\inf$ can be replaced with $\min$ due to compactness).
Instead of effective compactness of $C$, it is sufficient to require that its projection $C_{Y}={\mathopen\{\,y : \exists z\,(y,z)\in C\,\mathclose\}}$ is effectively closed and covered by an enumerated sequence of basic balls $B_{k}$ such that $\overline B_{k}\times Z\cap C$ is effectively compact, uniformly in $k$.
The weaker condition formulated at the end holds for example if $Y$ is effectively locally compact and $Z$ is effectively compact.
For start, we reproduce the classical proof of lower semicontinuity. One needs to check that the set ${\mathopen\{\,y : r<g(y)\,\mathclose\}}$ is open for all $r$. This set can be represented in the form of a union, noting that the condition $r<g(y)$ is equivalent to the condition $$\begin{aligned}
(\exists r'>r)\forall z\,[(y,z)\in C\Rightarrow f(y,z)>r'],
\end{aligned}$$ and it is sufficient to check the openness of the set $$\begin{aligned}
U={\mathopen\{\,y : \forall z\,[(y,z)\in C\Rightarrow f(y,z)>r']\,\mathclose\}}.
\end{aligned}$$ Now, $U=(Y\setminus C_{Y})\cup\bigcup_{k}(B_{k}\cap U)$. Since $Y\setminus C_{Y}$ is assumed to be open, it is sufficient to show that each $B_{k}\cap U$ is open. Let $F_{k}=\overline B_{k}\times Z$, then by the assumptions, $F_{k}\cap C$ is compact. The condition $f(y,z)>r'$ by the assumption defines a certain open set $V$ of pairs, hence $F_{k}\cap C\setminus V$ is closed, and as a subset of a compact set, compact. It follows that its projection ${\mathopen\{\,y\in \overline B_{k} : \exists z\,(y,z)\in F_{k}\cap C\setminus V\,\mathclose\}}$, as a continuous image of a compact set, is also compact, and so closed. Its complement in $B_{k}$, which is $B_{k}\cap U$, is then open.
Now this argument must be translated to an effective language. First of all note that it is sufficient to consider rational $r$ and $r'$. Then the set $V$ is effectively open, the set $F_{k}\cap C\setminus V$ is effectively closed, and as a subset of an effectively compact set, also effectively compact. Its projection, as a computable image of an effectively compact set, is also effectively compact, and as such, effectively closed. The complement of the projection is then effectively open.
The following lemma is an application:
\[lem:push-forward\] Let $X,Z,Z'$ be metric spaces, where $X$ is locally compact and $Z$ is compact. Let $f\colon Z\to Z'$ be continuous and surjective, and $t\colon X\times Z\to {[0,\infty]}$ a lower semicontinuous function. Then the function $t_{f}\colon X\times Z'\to{[0,\infty]}$ defined by the formula $$t_{f}(x,z')=\inf_{z:f(z)=z'} t(x,z)$$ is lower semicontinuous.
If $X,Z,Z'$ are constructive metric spaces, $X$ is effectively locally compact, $Z$ is effectively compact and $f$ is computable, further $t$ is lower semicomputable, then $t_{f}$ is lower semicomputable.
We will prove just the effective version. We will apply Proposition \[propo:lsc-min\] with $Y=X\times Z'$, and $C=X\times{\mathopen\{\,(f(z),z) : z\in Z\,\mathclose\}}$. Then $t_{f}(x,z')=\inf_{(x,z',z)\in C} t(x,z)$. The set $Y$ is effectively locally compact, as the product of an effectively locally compact set and an effectively compact set. The projection of the set $C$ onto $Y$ is the whole set $Y$, and hence it is closed. Hence the proposition is applicable, according to the remark following it.
Measures over a constructive metric space
-----------------------------------------
On a metric space, the [*Borel sets*]{} are the smallest $\sigma$-algebra containing the open sets. We can define measures on Borel sets. These measures have the following [*regularity*]{} property:
\[propo:regular\] Let ${P}$ be a measure over a complete separable metric space. Then every measureable set $A$ can be approximated by large open sets: ${P}(A)=\inf_{G\supseteq A}{P}(G)$, where $G$ is open.
It is possible to introduce a metric over measures:
\[def:prokh\] For a set $A$ and point $x$ let us define the distance of $x$ from $A$ as $d(x, A) = \inf_{y \in A} d(x,y)$. The $\varepsilon$-neighborhood of a set $A$ is defined as $A^{\varepsilon} = {\mathopen\{\,x : d(x, A) < \varepsilon\,\mathclose\}}$.
The [*Prokhorov distance*]{} $\rho({P}, {Q})$ of two measures is the greatest lower bound of all those $\varepsilon$ for which, for all Borel sets $A$ we have ${P}(A) {\leqslant}{Q}(A^{\varepsilon}) + \varepsilon$ and ${Q}(A) {\leqslant}{P}(A^{\varepsilon}) + \varepsilon$.
It is known that $\rho({P},{Q})$ is indeed a metric, and it turns the set of probability measures over metric space $X$ into a metric space. There is a number of other metrics for measures that are equivalent, in the sense of Definition \[def:uniform-contin\].
For a constructive metric space, ${\mathbf{X}}$, let ${\mathbf{M}}={\mathcal{M}}({\mathbf{X}})$ define the metric space of the set of probability measures over ${\mathbf{X}}$, with the metric $\rho({P}, {Q})$. The dense set $D_{{\mathbf{M}}}$ is the set of those probability measures that are concentrated on finitely many points of $D_{{\mathbf{X}}}$ and assign rational values to them. Let $\alpha_{{\mathbf{M}}}$ be a natural enumeration of $D_{{\mathbf{M}}}$, this turns ${\mathbf{M}}$ into a constructive metric space, too.
A probability measure is called [*computable*]{} when it is a computable element of the space ${\mathbf{M}}$.
Computability of measures is a particularly simple property for the Cantor space of binary sequences in Definition \[def:computable-measure-Omega\] (which is easily shown to be equivalent to the definition given here); it is just as simple for the Baire space of sequences over a countable alphabet.
The analogue of Proposition \[propo:integral-computable.Bernoulli\] holds again: the integral $\int f(\omega,{P}){P}(d\omega)$ of a basic function is computable as a function of the measure ${P}$, uniformly in the code of the basic function. Here is a closely related result:
\[propo:integral-computable\] If $f$ is a bounded, effectively uniformly continuous function then its integral by the measure ${P}$ is an effectively uniformly continuous function of ${P}$.
It can be assumed without loss of generality that $f$ is nonnegative (add a constant). Let measures ${P}$ and ${P}'$ be close. Then ${P}'(A){\leqslant}{P}(A_{\varepsilon})+\varepsilon$, where $A_{\varepsilon}$ denotes the $\varepsilon$-neighborhood of $A$. Then $$\begin{aligned}
\int f\, d{P}'{\leqslant}\int f_{\varepsilon}\, d{P}+\varepsilon,
\end{aligned}$$ where $f_{\varepsilon}(x)$ is the least upper bound of $f$ on the $\varepsilon$-neighborhood of $x$. (The integral of a nonnegative function $g$ is defined by the measures of the sets $G_{t}={\mathopen\{\,x : g(x){\geqslant}t\,\mathclose\}}$; by Fubini’s theorem on the change of the order of integration, this measure must be integrated by $t$ as a function of $t$. Now, if $f(x){\geqslant}t$ then $f_{\varepsilon}(x){\geqslant}t$ in the $\varepsilon$-neighborhood of point $x$.) It remains to apply the effective uniform continuity of $f$ to find out the precision by which the measure must be given in order to obtain a given precision in the integral.
On the other hand, the measure of ${P}(B)$ of a basic ball $B$ is not necessarily computable, only lower semicomputable. It is shown in [@HoyrupRojasRandomness09] that this property also characterizes the computability of measures: ${P}$ is computable if and only if ${P}(B)$ is lower semicomputable, uniformly in the basic ball $B$.
It is known that if a complete separable metric space is compact then so is the set of measures with the described metric. The following constructive version is proved by standard means:
If a constructive metric space ${\mathbf{X}}$ is effectively compact then its space of probability measures ${\mathcal{M}}({\mathbf{X}})$ is also effectively compact.
For the binary Cantor space, this was proved in Proposition \[propo:closed-to-compact\]. There, the topology of the space of measures was simply derived from the topology of the space ${[0,1]}\times{[0,1]}\times\dotsm$. It can be seen that the Prokhorov metric leads to the same topology.
\[example:discrete-measures\] Another interesting simple metric space is the infinite discrete space, say on the set of natural numbers ${\mathbb{N}}$. This is not a compact space, and the set of measures, namely the set of all functions ${P}(x){\geqslant}0$ with $\sum_{x\in{\mathbb{N}}}{P}(x)=1$, is not compact either.
On the other hand, the set of semimeasures (see Definition \[def:semimeasure\]) is compact. Indeed, recall that the space ${[0,1]}\times{[0,1]}\times\dotsm$, of functions ${P}:{\mathbb{N}}\to{[0,1]}$ is compact. Hence also for each $n$ the subset $F_{n}$ of this set of consisting of functions ${P}$ obeying the restriction ${P}(0)+{P}(1)+\dots+{P}(n){\leqslant}1$ is compact, as the product of a compact finite-dimensional set ${\mathopen\{\,({P}(0),{P}(1),\dots,{P}(n))\in{[0,1]}^{n} : {P}(0)+\dots+{P}(n){\leqslant}1\,\mathclose\}}$ and the compact infinite product set ${\mathopen\{\,({P}(n+1),{P}(n+1),\dots) :
0{\leqslant}{P}(x){\leqslant}1 \text{ for } x>n\,\mathclose\}}$. The intersection of all sets $F_{n}$ is then also compact, and is equal to the set of semimeasures.
Equivalently, we can consider the one-point compactification $\overline{\mathbb{N}}$ of ${\mathbb{N}}$ given in Example \[example:metric\].\[i:example.metric.one-point-compactif\]. Measures ${P}$ on this space can be identified with semimeasures over ${\mathbb{N}}$: we simply set ${P}(\infty)=1-\sum_{n<\infty}P(n)$.
Randomness in a metric space {#subsec:randomness-metric}
----------------------------
In the Cantor space $\Omega$ of infinite binary sequences we defined
randomness with respect to computable measures (in the sense of Martin-Löf); see Definition \[def:test-computable-measure\];
uniform randomness with respect to arbitrary measures (when the test is a function of the sequence and the measure), Definition \[def:uniform-test.bin-Cantor\];
Randomness with respect to an effectively compact class of measures, Definition \[def:test-effectively-compact.Cantor\];
Blind (oracle-free) randomness in Definition \[def:blind-test.Cantor\];
All these notions carry over with minor changes to an arbitrary constructive metric space. In the present section we discuss these generalizations and their properties, and then consider in more detail randomness with respect to an orthogonal class of measures.
For computable measures, a test is defined as a lower semicomputable function on a constructive metric space, whose integral is bounded by $1$. Among such tests, there is a maximal one to within a multiplicative constant. As earlier, this is proved with the help of trimming: we list all lower semicomputable functions, forcing them into tests or almost tests, and then add them up with coefficients from a converging series.
This is done as before, by considering lower semicomputable functions as monotonic limits of basic ones. It is used that the integral of a basic function by a measure is computable as a function of the measure: see Propositions \[propo:integral-computable\] and the discussion preceding it.
The uniform tests introduced in Definition \[def:uniform-test.bin-Cantor\] generalize immediately to the case of constructive metric spaces. Such a test is a lower semicomputable function of two arguments $t(x,{P})$, where $x$ is a point of our metric space, and ${P}$ is a measure over this space. The integral condition has the same form as earlier: $\int t(x,{P})\,{P}(d x){\leqslant}1$.
As earlier, there exists a universal test, and this can be proved by the technique of trimming:
\[thm:trim\] Let $u(x,{P})$ be a lower semicomputable function whose first argument is a point of a constructive metric space, and the second one is measure over this space. Then there exists a uniform tests $t(x,{P})$ satisfying $u(x,{Q}){\leqslant}2 t(x,{Q})$ for all ${Q}$ such that the function $u_{{Q}}:x\mapsto u(x,{Q})$ is a test by the measure ${Q}$, that is $\int u(x,{Q})\,{Q}(d x){\leqslant}1$.
The proof repeats the reasoning of the proof of Theorem \[thm:trim.Cantor\], while using the fact that for a basic function $b(x,{P})$ on the product space the integral $\int b(x,{P})\,{P}(d x)$ is a computable (continuous) function of ${P}$ (which is proved analogously to our above argument on the computability of the integral).
We will denote the universal uniform test again by ${\mathbf{t}}(x,{P})$. Strictly speaking, it depends also on the constructive metric space on which it is defined, but in general it is evident, which space is being considered, therefore it is not shown in the notation.
Definition \[def:P-test.Cantor\] and Proposition \[propo:uniformize.Cantor\] extend without difficulty.
\[def:P-test\] Let ${\mathbf{X}}= (X, d, D, \alpha)$ be a constructive metric space. For a measure ${P}\in{\mathcal{M}}({\mathbf{X}})$, a ${P}$-[*test of randomness*]{} is a function $f:X\to{[0,\infty]}$ lower semicomputable from ${P}$ with the property $\int f(x)\,d P {\leqslant}1$.
It seems as if a ${P}$-test may capture some nonrandomnesses that uniform tests cannot—however, this is not so, since trimming (see Theorem \[thm:trim.Cantor\]) generalizes:
\[propo:uniformize\] Let ${P}$ be some measure over a constructive metric space $X$, along with some ${P}$-test $t_{{P}}(x)$. There is a uniform test $t'(\cdot,\cdot)$ with $t_{{P}}(x){\leqslant}2 t'(x,{P})$.
Theorem \[thm:some-oracle\] generalizes to the case of constructive metric spaces. Let us mention one of the facts that generalize to uniform tests.
\[propo:Kurtz-uniform\] Let $S$ be an effectively open subset of the space $X\times{\mathcal{M}}(X)$. If the set $S_{{P}}={\mathopen\{\,x : {(x,{P})}\in S\,\mathclose\}}$, has ${P}$-measure $1$ for some measure ${P}$, then the set $S({P})$ contains all uniformly ${P}$-random points.
The indicator function $1_{S}(x,{P})$ of the set $S$, that is equal to unity on $S$ and to zero outside, is lower semicomputable. According to Proposition \[propo:lower-semi\], it can be written as the limit of a computable increasing sequence of basic functions $0{\leqslant}g_{n}(x,{P}){\leqslant}1$. The sequence $G_{n}:{P}\mapsto\int g_{n}(x,{P})\,d{P}$ is an increasing sequence of functions computable uniformly in $n$. The motonone convergence theorem implies $G_{n}({P})\to 1$ for all ${P}\in{\mathcal{C}}$. Let us define for each measure ${P}$ the numbers $n_{K}({P})$ as the minimal values of $n$ for which $G_{n}({P})>1-2^{k}$. These numbers are upper semicomputable as functions of ${P}$ (in a natural sense; for measures ${P}$ with ${P}(S_{{P}})<1$, some of these $n_{k}({P})$ are infinite). Correspondingly, the functions $1-g_{n_{k}({P})}(x,{P})$, as functions of $x$ and ${P}$ (define such a function to be zero for infinite $n_{k}({P})$, independently of $x$) are lower semicomputable, uniformly in $k$. Then $t(x,{P}) = \sum_{k>0}(1-g_{n_{k}({P})}(x,{P}))$ is a uniform test, since at a given ${P}$, if its $k$th addend is zero if $n_{k}({P})$ is infinite, and is not greater than $2^{-k}$ for finite $n_{k}({P})$.
The conditions of the theorem talk about a measures ${P}$ with ${P}(S_{{P}})=1$. Then all numbers $n_{k}({P})$ are finite. Consider an $x$ outside $S_{{P}}$: then $g_{n_{k}({P})}(x,{P})=0$ by definition. Therefore all addends of the test sum are equal to unity, thus $x$ is is not ${P}$-random point. Consequently, $S_{{P}}$ includes all uniformly ${P}$-random points.
Apriori probability, with an oracle
-----------------------------------
In Section \[subsec:uniform-exact\] we defined apriori probability with a condition whose role was played by a measure over the Cantor space $\Omega$. Now, having introduced the notion of a constructive metric space, we can note that this definition extends naturally to an arbitrary such space ${\mathbf{X}}$: we consider nonnegative lower semicomputable functions $m:{\mathbb{N}}\times X\to{[0,\infty]}$ for which $\sum_{i}m(i,x){\leqslant}1$, for all $x\in X$.
Among such functions, there is a maximal one to within a multiplicative constant. This is proved by the method of trimming: the lower semicomputable function $m(i,x)$ can be obtained as a sum of a series of basic functions each of which differs form zero only for one $i$; these basic functions must be multiplied by correcting coefficients that depend on the sum over all $i$. (In each stage, this sum has only finitely many members.)
We will call the maximal function of this kind [*apriori probability with condition $x$*]{}, and denote it ${\mathbf{m}}(i\mid x)$. We consider the first argument a natural number, but this is not essential: it is possible to consider words (or any other discrete constructive objects). As a special case we obtain the definition of apriori probability conditioned on a measure (Section \[subsec:uniform-exact\]), and also the standard notions of apriori probability with an oracle (which corresponds to ${\mathbf{X}}=\Omega$, the Cantor space of infinite sequences), and the conditional apriori probability (corresponding to ${\mathbf{X}}={\mathbb{N}}$).
In analogy with Martin-Löf’s theorem, the apriori probability with a condition is expressible, in an arbitrary *effectively compact* constructive metric space ${\mathbf{X}}$ by apriori probability with an oracle.
\[propo:day-miller-apriori\] Let $F:\Omega\to X$ be a computable map whose image is the whole space $X$. Then $$\begin{aligned}
{\mathbf{m}}(i\mid x) {\stackrel{{}_*}{=}}\min_{\pi: F(\pi)=x}{\mathbf{m}}(i\mid\pi).
\end{aligned}$$
We reason as in the proof of Theorem \[thm:some-oracle\]. The function ${(i,\pi)}\mapsto {\mathbf{m}}(i\mid F(\pi))$ is lower semicomputable on ${\mathbb{N}}\times\Omega$, hence the ${\stackrel{{}_*}{<}}$-inequality.
In order to obtain the reverse inequality, we use Lemma \[lem:push-forward\] and note that the function on the right-hand side is correctly defined (the minimum is achieved) and is lower semicomputable.
Note that ${\mathbf{m}}(i\mid\pi)$, the apriori probability with an oracle on the right-hand side of Proposition \[propo:day-miller-apriori\], is expressible by prefix complexity with an oracle. For the case of prefix complexity with condition in metric spaces it is not clear, how to define prefix complexity with such a condition (one can speak of functions whose graph is enumerable with respect to $x$, but it is not clear how to build a universal one). But one can define formally ${\mathit{Kp}}(i\mid x)$ as $\max_{\pi:F(\pi)=x}{\mathit{Kp}}(i\mid
\pi)$, and then ${\mathit{Kp}}(i\mid x){\stackrel{{}_+}{=}}-\log{\mathbf{m}}(i\mid x)$, but it is questionable whether this can be considered a satisfactory definition of prefix complexity (say, the usual arguments using the self-delimiting property of programs are not applicable at such a definition). It is more honest to simply speak of the logarithm of apriori probability. Many results still stay true: for example the formula ${\mathit{Kp}}(i,j\mid x){\stackrel{{}_+}{<}}{\mathit{Kp}}(i\mid x) + {\mathit{Kp}}(j\mid x)$ can be proved, without introducing self-delimiting programs, just reasoning about probabilities.
Analogously, it is possible to supply points in constructive metric spaces as conditions in some of our other definitions. For example, we can consider uniform tests over the Cantor space $\Omega$ of infinite binary sequences, with condition in an arbitrary constructive metric space $X$: these will be lower semicomputable functions $t(\omega,{P},x)$ with $\int t(\omega,{P},x)\,{P}(d\omega){\leqslant}1$ for all ${P},x$. It is also possible to fix a computable measure ${P}$, say the uniform one, and define tests with respect to this measure with conditions in $X$.
Classes of orthogonal measures
==============================
The definition of a class test for an effectively compact class of measures, as well as Theorem \[thm:class-test\] about the expression of a class test, generalizes, with the same proof.
The set of Bernoulli measures has an important property shared by many classes considered in practice: namely that a random sequence determines the measure to which it belongs. A consequence of this was spelled out in Theorem \[thm:orthogonal-blind.bin-Cantor\]. This section explores the topic in a more general setting.
There are some examples naturally generalizing the Bernoulli case: finite or infinite ergodic Markov chains, and ergodic stationary processes. Below, we will dwell a little more on the latter, since it brings up a rich complex of new questions.
We will consider orthogonal classes in the general setting of metric spaces: from now on, our measureable space is the one obtained from a constructive metric space ${\mathbf{X}}= (X,d,D,\alpha)$. The following classical concept is analogous to effective orthogonality, introduced in Definition \[def:effectively-orth\].
\[def:orthogonal\] Let ${P},{Q}$ be two measures over a measureable space $(X,{\mathcal{A}})$, that is a space $X$ with a $\sigma$-algebra ${\mathcal{A}}$ of measureable sets on it. We say that they are [*orthogonal*]{} if the space can be partitioned into measureable sets $U,V$ with the property ${P}(V)={Q}(U)=0$.
Let ${\mathcal{C}}$ be a class of measures. We say that ${\mathcal{C}}$ is [*orthogonal*]{} if there is a measureable function $\varphi:X\to{\mathcal{C}}$ with the property ${P}(\varphi^{-1}({P}))=1$.
Note that the space ${\mathcal{M}}(X)$, as a metric space, also allows the definition of Borel sets, and it is in this sense that we can talk about $f$ being measureable.
\[example:orthogonal\]
1. In an orthogonal class, any two (different) measures ${P}$ and ${Q}$ are orthogonal. Indeed, the sets $\{{P}\}$ and $\{{Q}\}$ are Borel (since closed), hence their preimages are measureable (and obviously disjoint). The converse statement is false: A class ${\mathcal{C}}$ of mutually orthogonal probability measures is not necessarily orthogonal, even if the class is effectively compact. For example, let $\lambda$ be the uniform distribution over the interval ${[0,1]}$, and let for each $x\in{[0,1]}$ the probability measure $\delta_{x}$ be concentrated on $x$. Then the class $\{\lambda\}\cup{\mathopen\{\,\delta_{x} : x\in{[0,1]}\,\mathclose\}}$ is effectively compact, and its elements are mutually orthogonal. But the whole class is not orthogonal: the orthogonality condition requires $\phi(x)=\delta_{x}$, but then $\phi^{-1}(\lambda)$ will be empty.
2. \[i:example.orthogonal.rand\] Let ${P},{Q}$ be two probability measures. Of course, if ${\mathrm{Randoms}}({P})$ and ${\mathrm{Randoms}}({Q})$ are disjoint, then ${P}$ and ${Q}$ are orthogonal. The converse is not always true: for example it fails if $\lambda,\delta_{x}$ are as above, where $x$ is random with respect to $\lambda$.
The following definition introduces the important example of stationary ergodic processes.
The Cantor space $\Omega$ of infinite binary sequences is equipped with an operation $T: \omega(1)\omega(2)\omega(3)\dots\mapsto
\omega(2)\omega(3)\omega(4)\dots$ called the [*shift*]{}. A probability distribution ${P}$ over $\Omega$ is [*stationary*]{} if for every Borel subset $A$ of $\Omega$ we have ${P}(A)={P}(T^{-1}(A))$. It is easy to see that this property is equivalent to requiring $$\begin{aligned}
{P}(x)={P}(0x)+{P}(1x)
\end{aligned}$$ for every binary string $x$.
A Borel set $A\subseteq\Omega$ is called [*invariant*]{} with respect to the shift operation if $T(A)\subseteq A$. For example the set of all sequences in which the relative frequency converges to $1/2$ is an invariant set. A stationary distribution is called [*ergodic*]{} if every invariant Borel set has measure 0 or 1.
Here is a new example of a stationary process (all Bernoulli measures and stationary Markov chains are also examples).
Let $Z_{1},Z_{2},\dots$ be a sequence of independent, identically distributed random variables taking values $0,1$ with probabilities $0.9$ an $0.1$ respectively. Let $X_{0},X_{1},X_{2},\dots$ be defined as follows: $X_{0}$ takes values $0,1,2$ with equal probabilities, and independently of all $Z_{i}$, further $X_{n}=X_{0}+\sum_{i=1}^{n}Z_{i}\bmod 3$. Finally, let $Y_{n}=0$ if $X_{n}=0$ and $1$ otherwise. The process $Y_{0},Y_{1},\dots$ is clearly stationary, and can also be proved to be ergodic. As a function of the Markov chain $X_{0},X_{1},\dots$, it is also called a [*hidden Markov chain*]{}.
The following theorem is a consequence of Birkhoff’s pointwise ergodic theorem. For each binary string $x$ let $$\begin{aligned}
g_{x}(\omega)=1_{x\Omega}(\omega)
\end{aligned}$$ be the indicator function of the set $x\Omega$: it is 1 if and only if $x$ is a prefix of $\omega$.
\[propo:Birkhoff\] Let ${P}$ be a stationary process over the Cantor space $\Omega$.
\[i:Birkoff.converge\] With probability $1$, the average $$\begin{aligned}
\label{eq:Axn}
A_{x,n}(\omega)=\frac{1}{n}(g_{x}(\omega)+g_{x}(T\omega)+\dots+g_{x}(T^{n-1}\omega))
\end{aligned}$$ converges.
\[i:Birkhoff.ergodic\] If the process is ergodic then the sequence converges to ${P}(x)$.
(For non-ergodic processes, the limit may depend on $\omega$.) Birkhoff’s theorem is more general, talking about more general spaces and measure-preserving transformations $T$, arbitrary integrable functions in place of $g_{x}$, and convergence to the expected value in the ergodic case. But the proposition captures its essence (and can also be used in the derivation of the more general versions).
Part of Proposition \[propo:Birkhoff\] implies that the class ${\mathcal{C}}$ of ergodic measures is an orthogonal class. Indeed, let us call a sequence $\omega$ “stable” if for all strings $x$, the averages $A_{x,n}(\omega)$ of converge. It is easy to see that in this case, the numbers ${P}(x)$ determine some probability measure ${Q}_{\omega}$. Now, let $\varphi:\Omega\to{\mathcal{C}}$ be a function that assigns to each stable sequence $\omega$ the measure ${Q}_{\omega}$ provided ${Q}_{\omega}$ is ergodic. If the sequence is not stable or ${Q}_{\omega}$ is not ergodic, then let $\varphi(\omega)$ be some arbitrary fixed ergodic measure. It can be shown that $\varphi$ is a measureable function: here, we use the fact that the set of stable sequences is a Borel set. By part of Proposition \[propo:Birkhoff\], the relation ${P}(\varphi^{-1}({P}))=1$ holds for all ergodic measures.
Note that the class of all ergodic measures is not closed, but we did not rely on the closedness of this class in the definition.
Example \[example:orthogonal\].\[i:example.orthogonal.rand\] shows that two measures can be orthogonal and still have common random sequences. But, for computable measures, as we will show right away, this is not possible.
We called a class of measures ${P}$ effectively orthogonal in Definition \[def:effectively-orth\], if all sets of random sequences ${\mathrm{Randoms}}({P})$ for measures ${P}$ in the class are disjoint from each other.
Two computable probability measures on a constructive metric space are orthogonal if and only they are effectively orthogonal.
Speaking of the effective orthogonality of two measures, we mean that they have no common (uniform) random sequences. In the effective case, pairwise orthogonality within the class and the orthogonality of the whole class are equivalent by definition.
We only need to prove one direction. Assume that ${P},{Q}$ are orthogonal, that is there is a measureable set $A$ with ${P}(A)=1$, ${Q}(A)=0$. By Proposition \[propo:regular\], these measures are regular, so there is a sequence $G_{n}\supseteq A$ of open sets with ${Q}(G_{n})< 2^{-n}$. Then for every $n$ there is also a finite union $H_{n}$ of basic balls with ${P}(H_{n})>1-2^{-n}$ and ${Q}(H_{n})<2^{-n}$; moreover, there is a computable sequence $H_{n}$ with this property. Let $U_{m}=\bigcup_{n>m}H_{n}$. By Proposition \[propo:Kurtz-uniform\], $\bigcap_{m} U_{m}$ contains all random points of ${P}$. On this other hand, the sets $U_{m}$ form a Martin-Löf test for measure ${Q}$, so the intersection contains no random points of ${Q}$.
We have shown above that ergodic measures form an orthogonal class. Careful analysis shows that this is also true effectively.
\[thm:effective-Birkhoff\] The set of ergodic measures over the Cantor set $\Omega$ forms an effectively orthogonal class.
The paper [@VyuginErgodic98] (more precisely, an analysis of it that will create uniform tests) shows that
\[i:effective-Birkhoff.stable\] Sequences uniformly random with respect to some stationary measure are stable (in the sense that the above indicated limit of averages exists for them).
\[i:effective-Birkhoff.ergodic\] Uniformly random sequences with respect to an ergodic measure are “typical” in the sense that these averages converge to ${P}(x)$.
To show , the paper introduces the function $$\begin{aligned}
\sigma(\omega,\alpha,\beta)
\end{aligned}$$ for rationals $0<\alpha<\beta$, which is the maximum number of times that $A_{x,n}(\omega)$ crosses from below $\alpha$ to above $\beta$. This function is lower semicomputable, uniformly in the rationals $\alpha,\beta$. Then it shows $$\begin{aligned}
(1+\alpha^{-1})(\beta-\alpha)\int\sigma(\omega,\alpha,\beta)\,d{P}{\leqslant}1,
\end{aligned}$$ that is that $(1+\alpha^{-1})(\beta-\alpha)\sigma(\omega,\alpha,\beta)$ is an average-bounded test, implying that for Martin-Löf-random sequences, the average $A_{x,n}(\omega)$ crosses from below $\alpha$ to above $\beta$ only a finite number of times. Now one can combine all these tests, for all strings $x$ and all rational $0<\alpha<\beta$, into a single test. This test is uniform in ${P}$: we did not rely on the computability of ${P}$.
To express , in view of part , it is sufficient, for each $x$, to prove $$\begin{aligned}
\label{eq:liminf-limsup}
\lim\inf_{n} A_{x,n}(\omega){\leqslant}{P}(x){\leqslant}\lim\sup_{n} A_{x,n}(\omega)
\end{aligned}$$ for random $\omega$. Take for example the statement for the lim inf. It is sufficient to show for each $k,m$ that $\inf_{n{\geqslant}m} A_{x,n}(\omega){\leqslant}{P}(x)+2^{-k}$ for a random $\omega$. The set $$\begin{aligned}
S_{x,k,m} &={\mathopen\{\,{(\omega,{P})} : \exists{n{\geqslant}m}\;A_{x,i}(\omega)<{P}(x)+2^{-k}\,\mathclose\}}
\end{aligned}$$ is effectively open, and the Birkhoff theorem implies ${P}(S_{x,k,m}({P}))=1$ for for all ergodic measures ${P}$, for the set $S_{x,k,m}({P})={\mathopen\{\,x : {(x,{P})}\in S_{x,k,m}\,\mathclose\}}$. Proposition \[propo:Kurtz-uniform\] implies that then for each ${P}$, the set $S_{x,k,m}({P})$ contains all ${P}$-random points.
Another approach is a proof that just shows for computable ergodic measures (in a relativizable way), without an explicit test, as done in [@BienvDayHoyrMezhShen10]. Then a reference to Theorem \[thm:oracle2uniform\] allows us to conclude the same about uniformly random sequences.
It is convenient to treat orthogonality of a class in terms of separator functions. For this, note that by a measureable real function we mean a Borel-measureable real function, that is a function with the property that the inverse images of Borel sets are Borel sets.
Let ${\mathcal{C}}$ be a class of measures over the metric space $X$. A measureable function ${s}:X\times{\mathcal{M}}(X)\to{[0,\infty]}$, is called a [*separator function*]{} for the class ${\mathcal{C}}$ if for all measures ${P}$ we have $\int {s}(x,{P})\,d{P}{\leqslant}1$, further for ${P},{Q}\in{\mathcal{C}}$, ${P}\ne{Q}$ implies that only one of the values ${s}(x,{P})$, ${s}(x,{Q})$ is finite.
In case we have a constructive metric space ${\mathbf{X}}$, a separator function ${s}(x,{P})$ is called a [*separator test*]{} if it is lower semicomputable in ${(x,{P})}$.
We could have required the integral to be bounded only for measures on the class, since trimming allows the extension of the boundedness property to all measures, just as in the remark after Definition \[def:P-test\].
The following observation connects orthogonality with separator functions and also shows that in case of effective orthogonality, each measure can be effectively reconstructed from any of its random elements.
\[thm:eff-orthog\] Let ${\mathcal{C}}$ be a class of measures.
\[i:thm.orthog.sep-fun\] If class ${\mathcal{C}}$ is Borel and orthogonal then there is a separator function for it.
\[i:thm.orthog.sep-test\] Class ${\mathcal{C}}$ is effectively orthogonal if and only if there is a separator test for it.
The converse of part might not hold: this needs further investigation.
Let us prove . If $\varphi(x)$ is a measureable function assigning measure ${P}\in{\mathcal{C}}$ to each element $x\in X$ as required in the definition of orthogonality, then by a general theorem of topological measure theory (see [@KuratowskiTop]), its graph is measureable. This allows the following definition: for ${P}\not\in{\mathcal{C}}$ set ${s}(x,{P})=1$, further for ${P}\in{\mathcal{C}}$, set ${s}(x,{P})=1$ if $\varphi(x)={P}$, and ${s}(x,{P})=\infty$ otherwise.
Let us prove now . If ${\mathcal{C}}$ is effectively orthogonal then the uniform test ${\mathbf{t}}(x,{P})$ is a separator test for the class ${\mathcal{C}}$. Suppose now that there is a separator test $s$ for the class ${\mathcal{C}}$, and let ${P},{Q}\in{\mathcal{C}}$, ${P}\neq{Q}$, $x\in{\mathrm{Randoms}}({P})$. Since ${s}$ is a randomness test, ${s}(x,{P})<\infty$, which implies ${s}(x,{Q})=\infty$, hence $x\not\in{\mathrm{Randoms}}({Q})$.
The following result is less expected: it shows that if the class of measures is effectively compact then the existence of a lower semicontinuous separator function implies the existence of a lower semicomputable one (that is a separator test).
If for an effectively compact class of measures there is a lower semicontinuous separator function ${s}(x,{P})$, then this class is effectively orthogonal.
Let ${\mathcal{C}}$ be an effectively compact class of measures on a constructive metric space. We need to show that under the conditions of the theorem, for any two distinct measures ${P}_{1},{P}_{2}$ in ${\mathcal{C}}$, the sets of random sequences are disjoint: $${\mathrm{Randoms}}({P}_{1})\cap{\mathrm{Randoms}}({P}_{2})=\emptyset.$$ Take two disjoint closed basic balls $B_{1}$ and $B_{2}$ in the constructive metric space ${\mathbf{M}}$ of measures, containing the measures ${P}_{1},{P}_{2}$. The classes ${\mathcal{C}}_{i}={\mathcal{C}}\cap B_{i}$, $i=1,2$ of measures are disjoint effectively compact classes of measures, containing ${P}_{1}$ and ${P}_{2}$. Consider the functions $$\begin{aligned}
t_{i}(x)=\inf_{{P}\in{\mathcal{C}}_{i}}{s}(x,{P}).
\end{aligned}$$ For all $x$ at least one of the values $t_{1}(x)$, $t_{2}(x)$ is infinite. By (a version of) Proposition \[propo:lsc-min\], the functions $t_{i}(x)$ are lower semicontinuous, and hence ${\mathcal{C}}_{1}$- and ${\mathcal{C}}_{2}$-tests respectively.
Now we follow some of the reasoning of the proof of Proposition \[propo:Kurtz-uniform\]. For integer $k>1$, consider the open set $S_{k}={\mathopen\{\,x : t_{1}(x)>2^{k}\,\mathclose\}}$. Since $t_{1}$ is a ${\mathcal{C}}_{1}$-test, then ${P}(S_{k})<2^{-k}$ for all ${P}\in{\mathcal{C}}_{1}$. On the other hand, since for all $x$ one of the two values $t_{1}(x)$, $t_{2}(x)$ is infinite, ${P}(S_{k})=1$ for all ${P}\in{\mathcal{C}}_{2}$. The indicator function $1_{S_{k}}(x)$ of the set $S_{k}$ is lower semicontinuous, therefore it can be written as the limit of an increasing sequence (now not necessarily computable!) of basic functions $g_{k,n}(x)$. We conclude as in the proof of Proposition \[propo:Kurtz-uniform\], that for each ${P}$ there is an $n=n_{k}({P})$ with $\int g_{k,n}(x)\,d{P}>1-2^{-k}$ for all $P\in{\mathcal{C}}_{2}$. The effective compactness of ${\mathcal{C}}$ implies then that there is an $n$ independent of ${P}$ with the same property. In summary, for each $k>0$ a basic function $h_{k}$ is found with $$\begin{aligned}
\int h_{k}\,d{P}&< 2^{-k} \text{ for all }{P}\in{\mathcal{C}}_{1},
\\ \int h_{k}\,d{P}&> 1-2^{-k} \text{ for all }{P}\in{\mathcal{C}}_{2}.
\end{aligned}$$ Such a basic function $h_{k}$ can be found effectively from $k$, by complete enumeration. Now we can construct a lower semicomputble function $$\begin{aligned}
t'_{1}(x)=\sum_{k} h_{k}(x).
\end{aligned}$$ It is a test for the class ${\mathcal{C}}_{1}$, while $t'_{2}(x)=\sum_{k}(1-h_{k}(x))$ is a test for all ${P}\in{\mathcal{C}}_{2}$ for the same reasons. These tests must be finite for elements random for ${P}_{1}$ and ${P}_{2}$, and this cannot happen simultaneously for both tests.
The meaning of separator tests introduced above introduced notion of can be clarified as follows. Due to effective orthogonality of ${\mathcal{C}}$, the universal uniform test ${\mathbf{t}}(\omega,{P})$ allows to separate the sequences into random ones according to different measures of the class ${\mathcal{C}}$: looking at a sequence $\omega$, random with respect to some measure of this class (=random with respect to the class), we are looking for a ${P}\in{\mathcal{C}}$ for which ${\mathbf{t}}(\omega,{P})$ is finite. This measure is unique in the class ${\mathcal{C}}$ (by the definition of effective orthogonality).
This separation property, however, can be satisfied also by a non-universal test, and we called such tests separator tests. The non-universal test is less demanding about the idea of randomness, giving it, so to say, a “first approximation”: it might accept a sequence as random that will be rejected by a more serious test. (The converse is impossible, since the universal test is maximal.) What matters is only that this preliminary crude triage separates the measures of the class ${\mathcal{C}}$, that is that no sequence should appear “random” even “in first approximation”, with respect to two measures at the same time.
For brevity, just for the purposes of the present paper, we will call “typicality” this “randomness in first approximation”:
Given a separator test ${s}(x,{P})$ we call an element $x$ [*typical*]{} for ${P}\in{\mathcal{C}}$ (with respect to the test $s$) if ${s}(x,{P})<\infty$.
A typical element determines uniquely the measure ${P}$ for which it is typical.
For an example, consider the class of ${\mathcal{B}}$ of Bernoulli measures. For a test in “first approximation”, we may recall von Mises, who called the first property of a random sequence (“Kollektiv” in his words) the stability of its relative frequencies. The stability of relative frequencies (strong law of large numbers in today’s terminology) means $S_{n}(\omega)/n\to p$. Here $S_{n}(\omega)$ is the number of ones in the initial segment of length $n$ of the sequence $\omega$, and $p$ is the parameter of the Bernoulli measure $B_{p}$.
There are several requirements close to this in this spirit:
(1) \[i:typical.converge-fast\] $S_{n}(\omega)/n\to p$ with a certain convergence speed.
(2) \[i:typical.converge\] $S_{n}(\omega)/n\to p$.
(3) \[i:typical.Vyugin\] For the case when ${\mathcal{C}}$ is the class of all ergodic stationary measures over the Cantor space $\Omega$, convert the proof of Theorem \[thm:effective-Birkhoff\] into a test, implying $A_{x,n}(\omega)\to {P}(x)$ for all $x$.
Among these requirements, the one that seems most natural to a mathematician, namely , is not expressible in a semicomputable way. Requirement has many possible formulations, depending on the convergence speed: we will show an example below.
Requirement is significantly more complicated to understand, but is still much simpler than a universal test. It *does not* imply a computable convergence speed directly; indeed, as Vyugin showed in [@VyuginErgodic98], a computable convergence speed does not exist for the case of computable non-ergodic measures. But later works, starting with [@AvigadGerhardyTowsner2010], have shown that the the convergence for ergodic measures has a speed computable from ${P}$.
Here is an example of a test expressing requirement . (For simplicity, we obtain the convergence of relative frequencies not on all segments, only on lengths that are powers of two. With more care, one could obtain similar bounds on all initial segments.) By Chebyshev’s inequality $$\begin{aligned}
B_{p}({\mathopen\{\,x\in\{0,1\}^{n} : |\sum_{i}x(i)-n p|{\geqslant}\lambda n^{1/2}(p(1-p))^{1/2}\,\mathclose\}}) {\leqslant}\lambda^{-2}.
\end{aligned}$$ Since $p(1-p){\leqslant}1/4$, this implies $$\begin{aligned}
B_{p}({\mathopen\{\,x\in\{0,1\}^{n} : |\sum_{i}x(i)-n p>\lambda n^{1/2}/2\,\mathclose\}}) < \lambda^{-2}.
\end{aligned}$$ Setting $\lambda=n^{0.1}$ and ignoring the factor $1/2$ gives $$\begin{aligned}
B_{p}({\mathopen\{\,x\in\{0,1\}^{n} : |\sum_{i}x(i)-n p|> n^{0.6}\,\mathclose\}})< n^{-0.2}.
\end{aligned}$$ Setting $n=2^{k}$: $$\begin{aligned}
\label{eq:2^k-Cheb}
B_{p}({\mathopen\{\,x\in\{0,1\}^{2^{k}} : |\sum_{i}x(i)-2^{k}p|> 2^{0.6 k}\,\mathclose\}})< 2^{-0.2k}.
\end{aligned}$$ Now, for a sequence $\omega$ in ${\mathbf{B}}^{{\mathbb{N}}}$, and for $p\in{[0,1]}$ let $$\begin{aligned}
g(\omega, B_{p}) = \sup{\mathopen\{\,k : |\sum_{i=1}^{2^{k}}\omega(k)-2^{k}p|> 2^{0.6k}\,\mathclose\}}.
\end{aligned}$$ Then $$\begin{aligned}
\int g(\omega,B_{p})\,B_{p}(d\omega) {\leqslant}\sum_{k} k\cdot 2^{-0.2 k} = c<\infty.
\end{aligned}$$ Dividing by $c$, we obtain a test. This is a separator test, since $g(\omega,B_{p})<\infty$ implies that $2^{-k}S_{2^{k}}(\omega)$ converges to $p$, and this cannot happen for two different $p$.
Theorem \[thm:orthogonal-blind.bin-Cantor\] generalizes, with essentially the same proof (using basic balls instead of initial sequences): it says that in an effectively compact, effectively orthogonal class of measures, blind randomness is the same as uniform Martin-Löf randomness. This raises the question whether every ergodic measure belongs to some effectively compact class. The answer is negative:
\[thm:ergodic-non-eff\] Consider stationary measures over $\Omega$ (with the shift transformation). Among these, there are some ergodic measures that do not belong to any effectively compact class of ergodic measures.
Before proving the theorem, let us prove some preparatory statements.
\[propo:ergodic-nonergodic-dense\] Both the ergodic measures and the nonergodic measures are dense in the set of stationary measures ${\mathcal{M}}(\Omega)$ over $\Omega$.
First we will show how to approximate an arbitrary stationary measure ${P}$ by ergodic measures. Without loss of generality assume that all probabilities ${P}(x)$ for finite strings $x$ are positive. (If not, then we can mix in a little of the uniform measure.) For a fixed $n$, consider the values ${P}(x)$ on strings $x$ of length at most $n$. There is a process that reproduces these probabilities and that is isomorphic to an ergodic Markov process on $\{0,1\}^{n-1}$. In this process, for an arbitrary $x\in\{0,1\}^{n-2}$, $b,b'\in\{0,1\}$ the transition probability from $bx$ to $xb'$ is ${P}(bxb')/{P}(bx)$. Since both transition probabilities are positive, this Markov process is ergodic.
Now we show how to approximate an arbitrary ergodic measure by nonergodic measures. Let ${P}$ be ergodic. Let us fix some $n>0$ and $\varepsilon>0$. By the pointwise ergodic theorem, there is a sequence in which the limiting frequencies of all words converge to the measure (almost all sequences—with respect to this measure—are such). Taking a long piece of this sequence and repeating it leads to a periodic sequence in which the frequencies of words of length not exceeding $n$ differ from the measure ${P}$ by at most $\varepsilon$ (for any given $n$ and $\varepsilon>0$). (The repetition forms new words on the boundaries, but at a large length, this effect is negligible.) Consider now the measure concentrated on the shifts of this sequence, assigning the same weight to each of them (their number is equal to the minimum period). This measure is not ergodic, but is close to ${P}$.
\[propo:ergodic-G-delta\] The set of ergodic measures is a $G_{\delta}$ set in the metric space of all stationary measures over $\Omega$.
We can restrict attention to the (closed) set of stationary measures. Let ${P}$ be a stationary probability measure over $\Omega$. Consider the function $A_{x,n}$ over $\Omega$, defining $A_{x,n}(\omega)$ to be equal to the average number of occurrences of the word $x$ in the $n$ first possible positions of $\omega$. By the ergodic theorem, the sequence of functions $A_{x,1},A_{x,2},\dots$ converges in the $L_{1}$ sense. Moreover, the stationary measure ${P}$ is ergodic if and only if the limit of this convergence is the constant function with value ${P}(x)$.
Since the limit exists for all stationary measures, it is sufficient to check that the constant ${P}(x)$ is a limit point. For each $x,N$ and each rational $\varepsilon$ the set $S_{x,N,\varepsilon}$ of those ${P}$ for which there is an $n{\geqslant}N$ with $$\begin{aligned}
\int|A_{x,n}(\omega)-{P}(x)|{P}(d\omega)<\varepsilon
\end{aligned}$$ is open, and set of ergodic stationary measures is the intersection of these sets for all $x,N,\varepsilon$.
The union of all effectively compact classes of ergodic measures is $F_{\sigma}$. Suppose that it is equal to the set of all ergodic measures. Then the set of nonergodic measures is a $G_{\delta}$ set which is also dense by Proposition \[propo:ergodic-nonergodic-dense\].
As shown in Propositions \[propo:ergodic-nonergodic-dense\] and \[propo:ergodic-G-delta\], the set of ergodic measures is a dense $G_{\delta}$ set. But by the Baire category theorem, two dense $G_{\delta}$ sets cannot have an empty intersection. This contradiction proves the theorem.
The following question still remains open:
Is there an ergodic measure over $\Omega$ for which uniform and blind randomness are different?
Returning to arbitrary effectively compact, effectively orthogonal classes, we can connect the universal tests with class tests of Theorem \[thm:class-test\] and separator tests.
\[thm:test-sep\] Let ${\mathcal{C}}$ be an effectively compact, effectively orthogonal class of measures, let ${\mathbf{t}}(x,{P})$ be the universal uniform test and let ${\mathbf{t}}_{{\mathcal{C}}}(x)$ be a universal class test for ${\mathcal{C}}$. Assume that ${s}(x,{P})$ is a separator test for ${\mathcal{C}}$. Then we have the representation $$\begin{aligned}
{\mathbf{t}}(x,{P}) {\stackrel{{}_*}{=}}\max({\mathbf{t}}_{{\mathcal{C}}}(x), {s}(x,{P}))
\end{aligned}$$ for all ${P}\in{\mathcal{C}}$, $x\in X$.
Let is note first that ${\mathbf{t}}_{{\mathcal{C}}}(x)$ and ${s}(x,{P})$ do not exceed the universal uniform test ${\mathbf{t}}(x,{P})$. Indeed ${s}(x,{P})$ is a uniform test by definition. Also by definition, the universal class test ${\mathbf{t}}_{{\mathcal{C}}}(x)$ is a uniform test.
On the other hand, let us show that if ${\mathbf{t}}_{{\mathcal{C}}}(x)$ and ${s}(x,{P})$ are finite, then ${\mathbf{t}}(x,{P})$ does not exceed the greater one of them (to within a multiplicative constant). The finiteness of the first test guarantees that $\min_{{Q}\in{\mathcal{C}}}{\mathbf{t}}(x,{Q})$ is finite: this minimum is equal to ${\mathbf{t}}_{{\mathcal{C}}}(x)$ to within a constant factor. If this minimum was achieved on some measure ${Q}\not={P}$, then both values ${s}(x,{Q})$ and ${s}(x,{P})$ would be finite, contradicting to the definition of a separator. (Note that we proved a statement slightly stronger than promised: in place of “greater of the two”, one can write “the first of the two, if the second one is finite”.)
The above theorem separates the randomness test into two parts (points at two possible causes of non-randomness). First, we must convince ourselves that $x$ is random with respect to the class ${\mathcal{C}}$. For example in the case of a measure $B_{p}$, in the class ${\mathcal{B}}$ of Bernoulli measures, we must first be convinced that ${\mathbf{t}}_{{\mathcal{B}}}(\omega)$ is finite. This encompasses all the irregularity criteria. If the independence of the sequence is taken for granted, we may assume that the class test is satisfied.
After this, we know that our sequence is Bernoulli, and some kind of simple test of the type of the law of large numbers is sufficient to find out, by just which Bernoulli measure is it random: $B_{p}$ or some other one. This second part, typicality testing, is analogous to parameter testing in statistics.
Separation is the only requirement of the separator test: its numerical value is irrelevant. For example in the Bernoulli test case, no matter how crude the convergence criterion expressed by the separator test ${s}(x,{P})$, the maximum is always (essentially) the same universal test.
Are uniform tests too strong? {#sec:too-strong}
=============================
Monotonicity and/or quasi-convexity {#subsec:monotonicity}
-----------------------------------
Uniform tests may seem too strong, in case ${P}$ is a non-computable measure. In particular, randomness with respect to computable measures (in the sense of Martin-Löf or in the uniform sense, they are the same for computable measures) has certain intuitively desireable properties that uniform randomness lacks. One of these is monotonicity: roughly, if ${Q}$ is greater than ${P}$ then if $x$ is random with respect to ${P}$, it should also be random with respect to ${Q}$.
\[propo:test-computable-mon\] For computable measures ${P},{Q}$, for all rational $\lambda>0$, if $\lambda{P}(A){\leqslant}{Q}(A)$ for all $A$, then $$\begin{aligned}
\label{eq:test-computable-mon}
{\mathbf{m}}(\lambda)\cdot\lambda{\mathbf{t}}(x,{Q}){\stackrel{{}_*}{<}}{\mathbf{t}}(x,{P}).
\end{aligned}$$ Here ${\mathbf{m}}(\lambda)$ is the discrete apriori probability of the rational $\lambda$. To make the constant in ${\stackrel{{}_*}{<}}$ independent of ${P},{Q}$, one needs also to multiply the left-hand side by ${\stackrel{{}_*}{=}}{\mathbf{m}}({P},{Q})$.
We have $1{\geqslant}\int{\mathbf{t}}(x,{Q})\,d{Q}{\geqslant}\int\lambda{\mathbf{t}}(x, {Q})\,d{P}$, hence $\lambda{\mathbf{t}}(x,{Q})$ is a ${P}$-test. Using the trimming method of Theorem \[thm:trim\] in finding universal tests, one can show that the sum $$\begin{aligned}
\sum_{\lambda: \lambda\int{\mathbf{t}}(x,{Q})\,d{P}<2} {\mathbf{m}}(\lambda)\cdot\lambda{\mathbf{t}}(x,{Q})
\end{aligned}$$ is a ${P}$-test, and hence ${\stackrel{{}_*}{<}}{\mathbf{t}}(x,{P})$. Therefore this is true of each member of the sum, which is just what the theorem claims. It is easy to see that the multiplicative constants depend here on ${P},Q$ only via inserting a factor ${\mathbf{m}}({P},{Q})$.
The intuitive motivation for monotonicity is this: if there are two devices with internal randomness generators, outputting numbers with distributions ${P}$ and ${Q}$, and if $\lambda{P}{\leqslant}{Q}$, then it can be imagined that the second device simulates the first one with probability $\lambda$, and does its own thing otherwise. Then every outcome intuitively plausible as the outcome of the first device, must also be deemed a plausible outcome of the second one, since this could have simulated the first one by chance. (The numerical value of the randomness deficiency may be, of course, somewhat larger, since we must believe in addition that the $\lambda$-probability event occurred.)
Uniform randomness violates, alas, this property: if measure ${Q}$ is larger, but computationally more complex, then the randomness tests with respect to ${Q}$ can exploit this additional information, to make nonrandom some outcomes that were random with respect to ${P}$ (see the proof of Theorem \[thm:nonrandom-blind\]). This is just the reason of the difference between uniform and blind (oracle-free) randomness, for which the analogous monotonicity property is obviously satisfied.
Another situation for which we have some intuition on randomness is the mixture (convex combination) of measures. Imagine two devices with output measures ${P}$ and ${Q}$, and an outer box which triggers one of them with some probabilities $\lambda$, $1-\lambda$. As a whole, we obtain a system whose outcome is distributed by the measure $\lambda{P}+ (1-\lambda){Q}$. About which outcomes can we assert that are obtained randomly as a result of this experiment? Clearly both the outcomes random with respect to ${P}$ and those random with respect to ${Q}$ must be accepted (with the understanding that if the coefficient is small then some additional, but finite suspicion is added). And there should not be any other outcomes. A quantitative elaboration of this result (which in one direction follows from monotonicity) is given below.
\[propo:convexity\] Let ${P}$ and ${Q}$ be two computable measures.
\[i:convexity.qconvex\] For a rational $0<\lambda<1$, $$\begin{aligned}
{\mathbf{m}}(\lambda)\cdot{\mathbf{t}}(x,\lambda{P}+(1-\lambda){Q}){\stackrel{{}_*}{<}}\max({\mathbf{t}}(x,{P}),{\mathbf{t}}(x,{Q})).
\end{aligned}$$
\[i:convexity.qconcave\] For arbitrary $0<\lambda<1$, $$\begin{aligned}
{\mathbf{t}}(x,\lambda{P}+(1-\lambda){Q}){\stackrel{{}_*}{>}}\min({\mathbf{t}}(x,{P}),{\mathbf{t}}(x,{Q})).
\end{aligned}$$
The constants in ${\stackrel{{}_*}{<}}$ depend on the length of the shortest programs defining ${P}$ and ${Q}$ (their complexities), but not on $\lambda$ (or other aspects of ${P},{Q}$).
Statement (\[i:convexity.qconvex\]) could be called the [*quasi-convexity*]{} of randomness tests (to within a multiplicative constant). For a test with an exact quasi-convexity property (without any multiplicative constants) there is a lower semicomputable semimeasure that is neutral (after extending tests to semimeasures see [@LevinUnif76; @GacsExact80]).
Statement (\[i:convexity.qconcave\]) implies that no other random outcomes exist for the mixture of ${P}$ and ${Q}$. This could be called the [*quasi-concavity*]{} of randomness tests (to within a multiplicative constant).
Part (\[i:convexity.qconvex\]) follows from Proposition \[propo:test-computable-mon\]. Indeed, if $\lambda{\geqslant}1/2$ then Proposition \[propo:test-computable-mon\] implies $ {\mathbf{m}}(\lambda)\cdot{\mathbf{t}}(x,\lambda{P}+(1-\lambda){Q}){\stackrel{{}_*}{<}}{\mathbf{t}}(x,{P})$ (absorbing $1/2$ into the ${\stackrel{{}_*}{<}}$). If $\lambda<1/2$ then it implies $ {\mathbf{m}}(1-\lambda)\cdot{\mathbf{t}}(x,\lambda{P}+(1-\lambda){Q}){\stackrel{{}_*}{<}}{\mathbf{t}}(x,{Q})$ similarly, and we just recall ${\mathbf{m}}(\lambda){\stackrel{{}_*}{=}}{\mathbf{m}}(1-\lambda)$.
Part (\[i:convexity.qconcave\]) follows from the fact that the right-hand side is a test with respect to an arbitrary mixture of the measures ${P}$ and ${Q}$, and trimming can convert it into uniform test.
It is easy to see that all these statements exploit the computability of the measures and the mixing coefficients in an essential way. The corresponding counterexamples are easy to build once it is recognized that the mixture of measures can be stronger from an oracle-computational point of view than any of them, as well as in the other way. For example, let us divide the segment ${[0,1]}$ into two halves and consider the measures ${P}$ and ${Q}$ that are uniformly distributed over these halves. Their mixture with coefficients $\lambda$ and $1-\lambda$ will make the number $\lambda$ obviously non-random (since it can be computed from this measure), though with respect to one of the measures in can very well be random. Taking instead of ${P}$ and ${Q}$ their mixtures, say, with coefficients $1/3$ and $2/3$ and then reversed, one can make $\lambda$ random with respect to both measures.
In this example the mixture contains more information than each of the original measures. It can also be the other way: bend the interval ${[0,1]}$ with the uniform measure into a circle, and cut it into two half-circles by the points $p$ and $p+1/2$. Then the uniform measures on these half-circles make $p$ computable with respect to them and thus non-random, while the average of these measures is the uniform measure on the circle, with respect to which $p$ can very well be random.
Let us note that for blind (oracle-free) randomness, we can guarantee without any restrictions that the set of points random with respect to the mixture of ${P}$ and ${Q}$ is the union of points random with respect to ${P}$ and ${Q}$. (In one direction this follows from monotonicity, which we already mentioned. In the other one: if an outcome is not random with respect to ${P}$ and not random with respect to ${Q}$, then there are two tests proving this, and their minimum will a lower semicomputable test proving its non-randomness with respect to the mixture.)
These are strong motives to modify the concept of randomness test in order to reproduce these properties, while conserving other desireable properties (say the existence of a universal test and with it the notion of a deficiency of randomness). Some such modifications can be seen in [@LevinUnif76; @GacsExact80; @LevinRandCons84].
Locality
--------
Imagine that some sequence $\omega$ is uniformly random with respect to measure ${P}$ and starts with $0$. Change the values of the measure on sequences that start with $1$. It is not guaranteed that $\omega$ remains uniformly random since now the measure may become stronger as an oracle (allowing to compute more). But this looks strange since the changes in measure are in the part of the universe that does not touch $\omega$.
For blind (oracle-free) randomness, specifically this example is impossible (one can force the test to zero on sequences beginning with unity), but in principle the concept of test depends not only on the measure along the sequence (not only on the probabilities of occurrences of nulls and ones after its start).
For randomness with respect to computable measures, the situation is again better.
Let ${P},{Q}$ be two computable measures on the space $\Omega$ of binary sequences, coinciding on all initial segments of some sequence $\omega$. Then this sequence is simultaneously random or non-random with respect to ${P}$ and ${Q}$.
This follows immediately from the randomness criterion in terms of the complexity of the inital segments (Levin-Schnorr Theorem) in any of its variants (Theorem \[thm:rand-prefix\], Proposition \[propo:rand-monot\], Corollary \[coroll:ample-excess\]).
On the other hand, it is easy to modify one of the counterexamples in \[subsec:monotonicity\] to violate prequentiality as well.
In case of an arbitrary constructive metric space an analogous statement holds, though with a stronger requirement: we assume that two computable measures are equal on all sets contained in some neighborhood of the outcome $\omega$. (In this case it is possible to multiply the test by a basic function without changing it in $\omega$, and making it zero outside the neighborhood of coincidence).
Here is yet another way to obtain a clearly prequential definition of randomness, in which the randomness deficiency is a function of the sequence itself and the measures of its initial segments. For a given sequence $\omega$ and a given sequence $\{q(i)\}$ of real numbers with $1=q(0){\geqslant}q(1){\geqslant}q(2){\geqslant}\dotsm{\geqslant}0$, let $$\begin{aligned}
{\mathbf{t}}'(\omega,q)=\inf{\mathbf{t}}(\omega,{P}),
\end{aligned}$$ where the minimum is taken over all measures ${P}$ with ${P}(\omega(1:n))=q(n)$. The corresponding sets are effectively compact, so that this minimum will be a lower semicomputable function of $\omega$ and the sequence $q$. If for the sequence $\omega$ and the measures $q(i)$ of its initial segments, the value ${\mathbf{t}}'(\omega,q)$ is finite, then the sequence $\omega$ can be called [*prequentially random*]{}.
In other words, sequence $\omega$ is prequentially random with respect to measure ${P}$ if there is a (in general different) measure ${Q}$ with respect to which $\omega$ is random and which coincides with ${P}$ on all initial segments of $\omega$.
The requirement of prequentiality has been invoked in connection with a theory that extends probability theory and statistics to models of forecasting: see for example [@ChernovShenVereshVovk08] and [@ShenVovkPrequential10]. An example situation is the following. Let $\omega(n)=1$ mean that there is rain on day $n$ and $0$ otherwise. Suppose that a forecasting office makes daily forecasts $p(1),p(2),\dots$ of the probability of rain. It is not necessarily proposing a coherent probability model of global weather (a global probability distribution). It just provides forecasts for the conditional probabilities along the path corresponding to the weather that actually takes place.
Is it possible to estimate the quality of the forecast? It seems that in some situations, yes: if say, all forecasts are close to zero (say, less than $10\%$), and the majority of days (say more than $90\%$) is rainy. (It is said that the forecast is poorly [*calibrated*]{}.) Naturally, there are other possible inconsistencies, not related to the frequencies: the general question is whether the given sequence can be accepted as randomly obtained with the predicted probabilities. (Such a question arises also in the situation of estimating the quality of a random number generator each of whose output values is claimed to occur with whatever distribution the customer requires at that time of the process, for that particular bit.)
An additional circumstance to consider at the estimation of the quality of forecasts is that the forecaster can use a variety of information accessible to her at the moment of prediction (say, the evening of the preceding day), and not only the members of the sequence $\omega$. The presence of such information must also be taken into account at the estimation of the quality of the forecast.
The paper [@ShenVovkPrequential10] proposes several different approaches to this question, which turn out to be equivalent. One involves a generalization of the notion of martingale (see Definition \[def:martingale\]). It would be interesting to establish a connection with uniform randomness tests in the spirit of the above defined prequential deficiency. (Admittedly, in place of probabilities of initial segments, one must deal here with conditional probabilities, which is not quite the same, if these are not separated from zero.)
Questions for future discussion
===============================
We have already noted some questions that (in our view) would be interesting to study. In this section we collected a few more such questions.
1. Consider the following method for generating a sequence $\omega\in\Omega$ using an arbitrary distribution ${P}$ on $\Omega$ in which the probabilities of all words are positive. Take a random sequence $\rho$ of independent reals $\rho(1),\rho(2),\dots$ uniformly distributed over ${[0,1]}$. At stage $n$, after outputting $\xi(1:n-1)$, set $\xi(n)=1$ if $$\begin{aligned}
\rho(n)<{P}(\xi(1:n-1)1)/{P}(\xi(1:n-1)).
\end{aligned}$$ Considering this as a random process, the output distribution will be exactly ${P}$. What sequences can be obtained on the ouput, from a Martin-Löf-random sequence of real numbers on the input? (It can be verified that for computable measures ${P}$ one gets exactly the sequences that are Martin-Löf-random with respect to ${P}$.)
2. Recall the formula for the deficiency for computable measures: $$\label{eq:sum-quotient-def}
{\mathbf{t}}(\omega,{P}) {\stackrel{{}_*}{=}}\sum_{x{\sqsubseteq}\omega} \frac{{\mathbf{m}}(x)}{{P}(x)}.$$ Both sides make sense for non-computable ${P}$, but this formula is no more true. Indeed, the right-hand side does not change significantly if a measure ${P}$ is replaced by some other one that is close to ${P}$ but is much more powerful as an oracle; and the left-hand side can become infinite while it was finite for ${P}$.
Denote the righ-hand side by $t'(\omega,{P})$. Does it make sense to take the finiteness of $t'(\omega,{P})$ as a definition of randomness by a non-computable measure? It will be at least monotonic (an increase of the measure will only increase randomness). With respect to mixtures of measures, we can say that it is quasi-convex; moreover, it is proved in [@GacsDiss78] that $1/t'(\omega,{P})$ is a concave function of ${P}$.
Another possibility is to define the randomness deficiency for an infinite sequence $\omega$ as $\log\sup_{x{\sqsubseteq}\omega}{M}(x)/{P}(x)$ (and consider the corresponding definition of randomness). For computable measures we obtain a definition equivalent to Martin-Löf’s standard one. Paper [@GacsUnif05] shows that the uniform tests defined by this expression (whether to use ${\mathbf{m}}(x)$ or ${M}(x)$) do not obey randomness conservation, while the universal uniform test does. The work [@GacsDiss78] shows that, on the other hand, an expression related to the right-hand side of , whith the summation running over all positive basic functions instead of only the functions $1_{x\Omega}(\omega)$, obeys randomness conservation.
3. Can we define a reasonable class of tests with the property in Proposition \[propo:test-computable-mon\] holding for all measures ${P}$ (or some stronger version of it) so that there exists an universal class? For example, one may require $${P}{\leqslant}c\cdot {Q}\Rightarrow t(\omega,{P}){\geqslant}t(\omega,{Q})/c$$ (motivation: this is true for the right-hand side of formula . Could one also require the quasi-convexity, as in Proposition \[propo:convexity\]? Papers [@LevinUnif76] and [@GacsExact80] provide some such examples, as well as [@LevinRandCons84].
How about the quasi-concavity of Proposition \[propo:convexity\]? A uniform test with this property seems less likely, since our counterexample seems more robust.
4. Relativization in recursion theory means that we take some set $A$ and artificially declare it “decidable” by adding some oracle that tells us whether $x\in A$ for any given $x$. Almost all the theorems of classical recursion theory can be relativized. It is more delicate to declare some set $E$ “enumerable”. This means that we have some enumeration-oracle that enumerates the set $E$. The problem is, of course, that there are many enumerations. Still we can give the definition of an $E$-enumerable set. Let $W$ be a set of pairs of the form ${(x,S)}$ where $x$ is an integer and $S$ is a finite set of integers; assume $W$ to be enumerable in the classical sense. Then consider the set $S(E,W)$ of all $x$ such that ${(x,S)}\in W$ for some $S\subset E$. The sets $S(E,W)$ (for fixed $E$ and all enumerable $W$) are called [*enumerable with respect to the enumeration-oracle $E$*]{}. (The relation ${(x,S)}\in E$ means that we add $x$ to the $E$-enumeration as soon as we see all elements of $S$ in $E$.) A standard (decision) oracle for a set $A$ can be considered a special case of an enumeration oracle (say, for the set $\{2n: n\in A\}\cup \{2n+1: n\notin A\}$.
For some purposes, an enumeration oracle is as meaningful as a decision oracle: for example, we can speak about a lower semicomputable function with respect to enumeration oracle $E$, since it can be defined in terms of enumerable sets. But what can be proved for this kind of relativized notions?
For example, is there (for an arbitrary $E$) a maximal lower $E$-semicomputable semimeasure? Can one define prefix complexity with oracle $E$, and will it coincide with the logarithm of the maximal semimeasure lower semicomputable relative to $E$ (if the latter exists)? What if we assume, in addition, that $E$ is the set of all basic balls in a constructive metric space, containing a given point?
(For comparison: we could define an $E$-computable function as a function whose graph is $E$-enumerable. Then some familiar properties will hold; say, the composition of $E$-computable functions is again $E$-computable. On the other hand, we cannot guarantee that every non-empty $E$-enumerable set is the range of a total $E$-computable function: for some $E$ this is not so.)
5. We may try do extend the definition of randomness in a different direction: to lower semicomputable semimeasures (that is output distributions of probabilistic machines that generate output sequence bit by bit). Levin’s motivation for his definition was his goal to define the independence of the pair ${(x,\eta)}$ of infinite sequences as randomness with respect to the semimeasure ${M}\times {M}$. Correspondingly, the the deficiency of randomness of the pair ${(\xi,\eta)}$ with respect to ${M}\times {M}$ could be called the quantity of mutual information between the sequences $\xi$ and $\eta$. This is motivated by the fact that the algorithmic mutual information $$\begin{aligned}
{\mathit{Kp}}(x)+{\mathit{Kp}}(y)-{\mathit{Kp}}(x,y)=-\log({\mathbf{m}}(x)\times{\mathbf{m}}(y))-{\mathit{Kp}}(x,y)
\end{aligned}$$ between finite objects $x,y$ indeed looks like deficiency of randomness with respect to ${\mathbf{m}}\times{\mathbf{m}}$.
One possibility is to require that ${M}(z)/{Q}(z)$ is bounded, where ${M}$ is a priori probability on the tree and ${Q}$ is the semimeasure in question. The other possibility is to use random sequences for unbiased coin tossing and consider the output sequences in all these cases. It is not clear whether these two definitions coincide or if the second notion is well-defined (that is for two different machines with the same output distribution the image of the set of random sequences is the same). For *computable measures* it is indeed the case.
6. (Steven Simpson) Can we use uniform tests (modified in a proper way) for defining, say, $2$-randomness? (The standard definition uses non-semicontinuous tests, but maybe it can be reformulated.)
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are thankful to their colleagues with them they have discussed the questions considered in the paper: first to L. Levin with whom many of the concepts of the paper originate, to A. Bufetov and A. Klimenko, and also to V. V’yugin and other participants of the Kolmogorov seminar (at the Mechanico-Mathematical school of Moscow State University). The paper was written with the financial support of the grants NAFIT ANR-08- EMER-008-01, RFBR 0901-00709-a.
[^1]: LIAFA, CNRS & Université Paris Diderot, Paris 7, Case 7014, 75205 Paris Cedex 13, France, e-mail: `Laurent dot Bienvenu at liafa dot jussieu dot fr`
[^2]: Department of Computer Science, Boston University, 111 Cummington st., Room 138, Boston, MA 02215, e-mail: `gacs at bu dot edu`
[^3]: LORIA – B248, 615, rue du Jardin Botanique, BP 239, 54506 Vandœuvre-lès-Nancy, France, e-mail: `Mathieu dot Hoyrup at loria dot fr`
[^4]: Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Toronto, Ontario, Canada, M5S 2E4, e-mail: `crojas at math dot utoronto dot ca`
[^5]: LIF, Universitè Aix – Marseille, CNRS, 39, rue Joliot-Curie, 13453 Marseille cedex 13, France, on leave from IITP RAS, Bolshoy Karetny, 19, Moscow. Supported by NAFIT ANR-08-EMER-008-01, RFBR 0901-00709-a grants. e-mail: `sasha dot shen at gmail dot com`.
|
---
abstract: |
We present a search for the standard model Higgs boson using events with two oppositely charged leptons and large missing transverse energy as expected in $H\rightarrow WW$ decays. The events are selected from data corresponding to 8.6 of integrated luminosity in $p
\overline{p}$ collisions at $\sqrt{s}=1.96$ TeV collected with the D0 detector at the Fermilab Tevatron Collider. No significant excess above the standard model background expectation in the Higgs boson mass range this search is sensitive to is observed, and upper limits on the Higgs boson production cross section are derived.
date: 'July 04, 2012'
title: 'Search for Higgs boson production in oppositely charged dilepton and missing energy events in $\boldsymbol{p\bar{p}}$ collisions at $\boldsymbol{\sqrt{s} =}$ 1.96 TeV'
---
author\_list.tex
\[sec:conclusion\]CONCLUSIONS
=============================
We have performed a search for SM Higgs boson production using final states with two oppositely charged leptons and large missing transverse energy in the *, , and channels. After imposing all selection criteria, no significant excess in data over expected SM backgrounds is observed. We set upper limits on Higgs boson production at the 95% C.L. The sensitivity of the search reaches an expected exclusion of 159 $< M_{H} <$ 169 GeV. The best observed limit is obtained at 160 GeV, where it reaches 1.1 times the SM expectation. This channel is the single most sensitive channel when the $H
\rightarrow WW$ branching ratio is dominant $(M_H > 135$ GeV$)$, and for lower masses at $M_H=125$ GeV, this search still has a similar sensitivity as a single major low mass channel ($WH$ or $ZH$) with an expected limit of 3.8 times the SM expectation [@bib:low_mass]. The results and the analysis techniques are validated through an independent measurement of the $WW$ production cross section, which agrees with the NNLO calculation.*
\[sec:ackn\]ACKNOWLEDGMENTS
===========================
acknowledgement.tex
[99]{}
R. Barate [*et al.*]{}, Phys. Lett. B [**565**]{}, 61 (2003).
T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**108**]{}, 151803 (2012).
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett. [**108**]{}, 151804 (2012).
The LEP Electroweak Working group, “Status of March 2012”, http://lepewwg.web.cern.ch/LEPEWWG.
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Nucl. Instrum. Meth. in Phys. Res. A [**565**]{}, 463 (2006).
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett. [**104**]{}, 061804 (2010).
T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**104**]{}, 061803 (2010).
G. Aad [*et al.*]{} \[ATLAS Collaboration\], submitted to Phys. Lett. B, arXiv:1206.0756 \[hep-ex\] (2012).
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**710**]{}, 91 (2012).
G. Aad [*et al.*]{} \[ATLAS Collaboration\], submitted to Phys. Rev. D, arXiv:1207.0319 \[hep-ex\] (2012)
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**710**]{}, 26 (2012).
T. Aaltonen [*et al.*]{} \[CDF and D0 Collaborations\], Phys. Rev. Lett. [**104**]{}, 061802 (2010).
T. Sjöstrand, S. Mrenna, and P. Skands, J. High Energy Phys. [**05**]{}, 026 (2006)
J. Pumplin [*et al.*]{}, J. High Energy Phys. [**07**]{}, 12 (2002).
D. de Florian and M. Grazzini, Phys. Lett. B [**674**]{}, 291 (2009).
J. Baglio and A. Djouadi, J. High Energy Phys. [**10**]{}, 064 (2010).
P. Bolzoni [*et al.*]{}, Phys. Rev. Lett. [**105**]{}, 011801 (2011).
A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C [**63**]{}, 189 (2009).
S. Alekhin [*et al.*]{} \[PDF4LHC Working Group\], arXiv:1101.0536 \[hep-ph\]; M. Botje [*et al.*]{} \[PDF4LHC Working Group\], arXiv:1101.0538 \[hep-ph\].
C. Anastasiou, G. Dissertori, M. Grazzini, F. Stöckli, and B. R. Webber, J. High Energy Phys. **08**, 099 (2009).
A. Djouadi, J. Kalinowski, and M. Spira, Comput. Phys. Commun. [**108**]{}, 56 (1998).
G. Bozzi, S. Catani, D. de Florian, and M. Grazzini, Phys. Lett. B [**564**]{}, 65 (2003); Nucl. Phys. [**B737**]{}, 73 (2006).
M. L. Mangano [*et al.*]{}, J. High Energy Phys. [**07**]{}, 001 (2003); we use version 2.11.
R. Hamberg, W. L. van Neerven, and T. Matsuura, Nucl. Phys. [**B359**]{}, 343 (1991) \[Erratum-ibid. [**B644**]{}, 403 (2002)\].
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett. [**100**]{}, 102002 (2008).
K. Melnikov and F. Petriello, Phys. Rev. D [**74**]{}, 114017 (2006).
S. Moch and P. Uwer, Phys. Rev. D [**78**]{}, 034003 (2008); we use $\sigma(t\bar{t})$ = 7.88 pb.
J.M. Campbell and R.K. Ellis, Phys. Rev. D [**60**]{}, 113006 (1999); we use $\sigma(WW)$ = 11.66 pb, $\sigma(WZ)$ = 3.45 pb, and $\sigma(ZZ)$ = 1.37 pb.
S. Frixione and B.R. Webber, J. High Energy Phys. [**06**]{}, 029 (2002).
R. Brun and F. Carminati, CERN Program Library Long Writeup W5013, 1993 (unpublished).
M. Abolins [*et al.*]{}, Nucl. Instrum. Meth. in Phys. Res. A [**584**]{}, 75 (2008)
R. Angstadt [*et al.*]{}, Nucl. Instrum. Meth. in Phys. Res. A [**622**]{}, 298 (2010).
The pseudorapidity is defined as $\eta = - \ln[\tan(\theta/2)]$, where $\theta$ is the polar angle relative to the proton beam direction.
G. C. Blazey [*et al.*]{}, arXiv:hep-ex/0005012v2.
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Nucl. Instrum. Meth. in Phys. Res. A [**620**]{}, 490 (2010).
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. D [**85**]{}, 052006 (2012).
J. Alwall [*et al.*]{}, Eur. Phys. J. C [**53**]{}, 473 (2008).
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Lett. B [**669**]{} (2008), 278.
C. G. Lester and D. J. Summers, Phys. Lett. B [**463**]{}, 99 (1999); H. Cheng and Z. Han, J. High Energy Phys. [**12**]{}, 063 (2008).
L. Breiman [*et al.*]{}, [*Classification and Regression Trees*]{} (Wadsworth, Belmont, CA, 1984); Proceedings of the Thirteenth International Conference, Bari, Italy, 1996, edited by L. Saitta (Morgan Kaufmann, San Francisco, 1996), p. 148.
I. W. Stewart and F. J. Tackmann, Phys. Rev. D [**85**]{}, 034011 (2012).
C. Anastasiou, R. Boughezal and F. Petriello, J. High Energy Phys. [**04**]{}, 003 (2009)
J. M. Campbell, R. K. Ellis, and C. Williams, Phys. Rev. D **81**, 074023 (2010).
T. Binoth, M. Ciccolini, N. Kauer, and M. Krämer, J. High Energy Phys. [**03**]{}, 065 (2005); J. High Energy Phys. [**12**]{}, 046 (2006).
T. Junk, Nucl. Instrum. Meth. in Phys. Res. A [**434**]{}, 435 (1999); A. Read, J. Phys. G [**28**]{}, 2693 (2002). W. Fisher, FERMILAB-TM-2386-E (2006).
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], arXiv:1207.0422 \[hep-ex\] (2012).
|
---
abstract: 'In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green’s tensors. It is well known in the literature that standard $\mathcal{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. $\mathcal{H}^2$-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard $\mathcal{H}$-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the $\mathcal{H}$-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.'
author:
- 'Stéphanie Chaillat[^1]'
- 'Luca Desiderio[^2]'
- 'Patrick Ciarlet[^3]'
bibliography:
- 'Hmat.bib'
title: 'Theory and implementation of $\mathcal{H}$-matrix based iterative and direct solvers for [Helmholtz and elastodynamic]{} oscillatory kernels'
---
Time-harmonic elastic waves; $\mathcal{H}$-matrices, Low-rank approximations; Estimators; Algorithmic complexity; fast BEMs.
Introduction
============
The 3D linear isotropic elastodynamic equation for the displacement field ${\boldsymbol{ u}}$ (also called Navier equation) is given by $$\label{NE}
{\operatorname{\mathrm{div}}}\upsigma({\boldsymbol{ u}})+ \rho\omega^2{\boldsymbol{ u}}=0$$ where $\omega>0$ is the circular frequency. It is supplemented with appropriate boundary conditions which contain the data. The stress and strain tensors are respectively given by $\upvarsigma({\boldsymbol{ u}})=\lambda({\operatorname{\mathrm{div}}}{\boldsymbol{ u}}){\mathbf{I}}_{3}+2\mu\upvarepsilon({\boldsymbol{ u}})$ and $\upvarepsilon({\boldsymbol{ u}})=\dfrac{1}{2}\big([{\boldsymbol{\nabla }}{\boldsymbol{ u}}]+{{\vphantom{[{\boldsymbol{\nabla }}{\boldsymbol{ u}}]}}{[{\boldsymbol{\nabla }}{\boldsymbol{ u}}]}^{\sf T}}\big)$, where ${\mathbf{I}}_{3}$ is the 3-by-3 identity matrix and $[{\boldsymbol{\nabla }}{\boldsymbol{ u}}]$ is the 3-by-3 matrix whose $\beta$-th column is the gradient of the $\beta$-th component of ${\boldsymbol{ u}}$ for $1\le \beta \le 3$, $\mu$ and $\lambda$ are the Lamé parameters and $\rho$ the density. Denoting $\kappa_{p}^2=\rho\omega^2(\lambda+2\mu)^{-1}$ and $\kappa_{s}^2=\rho\omega^2\mu^{-1}$ the so-called P and S wavenumbers, the Green’s tensor of the Navier equation is a 3-by-3 matrix-valued function expressed by $${\bf U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})=\dfrac{1}{\rho\omega^2}\left({\operatorname{\boldsymbol{curl}}}{\operatorname{\boldsymbol{curl}}}_{{\boldsymbol{x}}} \left[\dfrac{e^{i\kappa_{s}|{\boldsymbol{x}}-{\boldsymbol{y}}|}}{4\pi|{\boldsymbol{x}}-{\boldsymbol{y}}|}\,{ {\mathbf{I}}}_{3}\right]-{\boldsymbol{\nabla }}_{{\boldsymbol{x}}}\boldsymbol{ {\operatorname{\mathrm{div}}}}_{{\boldsymbol{x}}}\,\left[\dfrac{e^{i\kappa_{p}|{\boldsymbol{x}}-{\boldsymbol{y}}|}}{4\pi|{\boldsymbol{x}}-{\boldsymbol{y}}|}{\mathbf{I}}_{3}\right]\right)
\label{elasto_U}$$ where the index ${\boldsymbol{x}}$ means that differentiation is carried out with respect to ${\boldsymbol{x}}$ and $\boldsymbol{ {\operatorname{\mathrm{div}}}_x} \mathbb{A}$ corresponds to the application of the divergence along each row of $\mathbb{A}$. One may use this tensor to represent the solution of (\[NE\]). Alternately, one may use the tensor ${\bf{T}}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})$, which is obtained by applying the traction operator $${\boldsymbol T}=2\mu\dfrac{\partial}{\partial{\boldsymbol{n}}}+\lambda{\boldsymbol{n}}{\operatorname{\mathrm{div}}}+\mu\,{\boldsymbol{n}}\times{\operatorname{\boldsymbol{curl}}}\label{elasto_T}$$ to each column of $\bf{U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})$: ${\bf{T}}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})=[ {\boldsymbol T}_{{\boldsymbol{y}}} {\bf U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})]$ where the index ${\boldsymbol{y}}$ means that differentiation is carried out with respect to ${\boldsymbol{y}}$.
We consider the fast solution of dense linear systems of the form $$\mathbb{ A} {\bf p}={\bf b}, \quad {\mathbb A}\in \mathbb{C}^{3N_c \times 3N_c}
\label{syst_a}$$ where ${\mathbb A}$ is the matrix corresponding to the discretization of the 3-by-3 Green’s tensors ${\bf U}_{\omega}({\boldsymbol{x}}_i,{\boldsymbol{y}}_j)$ or ${\bf T}_{\omega}({\boldsymbol{x}}_i,{\boldsymbol{y}}_j)$ for two clouds of $N_c$ points $ {({\boldsymbol{x}}_i)_{ 1 \le i \le N_c}}$ and $ {({\boldsymbol{y}}_j)_{1 \le j \le N_c}}$. Here ${\bf p}$ is the unknown vector approximating the solution at $ {({\boldsymbol{x}}_i)_{ 1 \le i \le N_c}}$ and ${\bf b}$ is a given right hand side that depends on the data. Such dense systems are encountered for example in the context of the Boundary Element Method [@bonnet1999boundary; @sauter2010boundary].
If no compression or acceleration technique is used, the storage of such a system is of the order $O(N_c^2)$, the iterative solution (e.g. with GMRES) is $O(N_{\operatorname{iter}}N_c^2)$ where $N_{\operatorname{iter}}$ is the number of iterations, while the direct solution (e.g. via LU factorizations) is $O(N_c^3)$. In the last decades, different approaches have been proposed to speed up the solution of dense systems. The most known method is probably the fast multipole method (FMM) proposed by Greengard and Rokhlin [@greengard1987fast] which enables a fast evaluation of the matrix-vector products. We recall that the matrix-vector product is the crucial tool in the context of an iterative solution. Initially developed for N-body simulations, the FMM has then been extended to oscillatory kernels [@greengard1998accelerating; @darve2000fast]. The method is now widely used in many application fields and has shown its capabilities in the context of mechanical engineering problems solved with the BEM [@chaillat2013recent; @takahashi2012wideband].
An alternative approach designed for dense systems is based on the concept of hierarchical matrices ($\mathcal{H}$-matrices) [@bebendorf2008hierarchical]. The principle of $\mathcal{H}$-matrices is to partition the initial dense linear system, and then approximate it into a data-sparse one, by finding sub-blocks in the matrix that can be accurately estimated by low-rank matrices. In other terms, one further approximates the matrix $\mathbb{A}$ from (\[syst\_a\]). The efficiency of hierarchical matrices relies on the possibility to approximate, under certain conditions, the underlying kernel function by low-rank matrices. The approach has been shown to be very efficient for asymptotically smooth kernels (e.g. Laplace kernel). On the other hand, oscillatory kernels such as the Helmholtz or elastodynamic kernels, are not asymptotically smooth. In these cases, the method is not optimal [@banjai2008hierarchical]. To avoid the increase of the rank for high-frequency problems, directional $\mathcal{H}^2$-methods have been proposed [@borm2015approximation; @borm2015directional]. $\mathcal{H}^2$-matrices are a specialization of hierarchical matrices. It is a multigrid-like version of $\mathcal{H}$-matrices that enables more compression, by factorizing some basis functions of the approximate separable expansion [@borm2006matrix].
Since the implementation of $\mathcal{H}^2$ or directional methods is much more involved than the one of the standard $\mathcal{H}$-matrix, it is important to determine the frequency-range within which the $\mathcal{H}$-matrices are efficient for elastodynamic problems and what can be expected of such an approach to solve problems encountered in mechanical engineering. Previous works on $\mathcal{H}$-matrices for oscillatory kernels have mainly be devoted to the direct application to derive fast iterative solvers for 3D acoustics [@brunner2010comparison; @stolper2004computing], a direct solver for 3D electromagnetism [@lize] or iterative solvers for 3D elastodynamics [@milazzo2012hierarchical; @MESSNER2010]. There is no discussion in these references on the capabilities and limits of the method for oscillatory kernels. We show in this work that even though the method is not optimal (in the sense that more efficient approaches can be proposed at the cost of a much more complex implementation effort), an efficient solver is easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.
The article is organized as follows. After reviewing general facts about $\mathcal{H}$-matrices in Section 2, $\mathcal{H}$-matrix based solvers and an estimator to certify the results of the direct solver are given in Section 3. Section 4 gives an overview of the Boundary Element Method for 3D elastodynamics. In Section 5, theoretical estimates for the application of $\mathcal{H}$-matrices to elastodynamics are derived. Section 6 presents the implementation issues for elastodynamics. Finally in Sections 7 and 8, we present numerical tests with varying number of points $N_c$ for both representations of the solution ($\bf{U}_{\omega}$ or $\bf{T}_{\omega}$). In Section 7, numerical tests for the low frequency regime i.e. for a fixed frequency are reported and discussed. In Section 8, similar results are given for the high frequency regime i.e. for a fixed density of points per S-wavelength.
General description of the $\mathcal{H}$-LU factorization
=========================================================
$\mathcal{H}$- matrix representation \[section\_tree\]
------------------------------------------------------
Hierarchical matrices or $\mathcal{H}$-matrices have been introduced by Hackbusch [@hackbusch1999sparse] to compute a data-sparse representation of some special dense matrices (e.g. matrices resulting from the discretization of non-local operators). The principle of $\mathcal{H}$-matrices is (i) to partition the matrix into blocks and (ii) to perform low-rank approximations of the blocks of the matrix which are known *a priori* (by using an admissibility condition) to be accurately approximated by low-rank decompositions. With these two ingredients it is possible to define fast iterative and direct solvers for matrices having a hierarchical representation. Using low-rank representations, the memory requirements and costs of a matrix-vector product are reduced. In addition, using $\mathcal{H}$-matrix arithmetic it is possible to derive fast direct solvers.
#### Illustrative example
To illustrate the construction of an $\mathcal{H}$-matrix, we consider the matrix ${\mathbb G}_e$ resulting from the discretization of the 3D elastic Green’s tensor ${\bf U}_{\omega}$ for a cloud of points located on the plate $(x_1,x_2,x_3)\in [-a,a]\times[-a,a]\times \{0\}$. The plate is discretized uniformly with a fixed density of $10$ points per S-wavelength $\lambda_s=2\pi/\kappa_s$. We fix the Poisson’s ratio $\nu$ to $\nu=\frac{\lambda}{2(\lambda+ \mu)}=1/3$ and a non-dimensional frequency $\eta_S=\kappa_s a=5 \pi$ (e.g. $\rho\mu^{-1}=1$, $\omega=5\pi$, $\kappa_s=5 \pi$, $\kappa_p=\kappa_s/2$ and $a=1$). As a result, the discretization consists of $N_d=50$ points in each direction leading to $N_c=2500$ points and a matrix of size $7500 \times 7500$. We recall that the numerical rank of a matrix ${\mathbb A}$ is $$r(\varepsilon):=\mbox{min} \{ r \quad | \quad ||{\mathbb A} -{\mathbb A}_r|| \le \varepsilon ||{\mathbb A}|| \}
\label{num_rank}$$ where ${\mathbb A}_r$ defines the singular value decomposition (SVD) of ${\mathbb A}$ keeping only the $r$ largest singular values and $\varepsilon>0$ is a given parameter. In Fig. \[svd\_all\]a, we report the decrease of the singular values of the matrix ${\mathbb G}_e$. As expected, the decay of the singular values is very slow such that the matrix cannot be approximated by a low-rank decomposition. Now, we partition the plate into two parts of equal area (see Fig. \[geo\_illu\]) and subdivide the matrix accordingly into 4 subblocks. Figure \[svd\_all\]b gives the decrease of the singular values both for diagonal and off-diagonal blocks. It illustrates the possibility to accurately represent off-diagonal blocks by low-rank matrices while diagonal blocks cannot have low-rank representations.
----- -----
(a) (b)
----- -----
![Illustrative example: partition of the degrees of freedom. []{data-label="geo_illu"}](./binary_tree_geo1)
If we keep subdividing the full rank blocks in a similar manner, we observe in Fig. \[ranks\] that diagonal blocks are always full rank numerically, i.e. with respect to (\[num\_rank\]) where we have chosen $\varepsilon=10^{-4}$, while off-diagonal blocks become accurately approximated by a low-rank decomposition after some iterations of the subdivision process. This academic example illustrates the concept of data-sparse matrices used to derive fast algorithms. Using the hierarchical structure in addition to low-rank approximations, significant savings are obtained both in terms of computational times and memory requirements.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Computed numerical ranks of each block of the matrix ${\mathbb G}_e$ to achieve an accuracy of $10^{-4}$, i.e. if singular values smaller than $10^{-4}$ are neglected in the singular value decomposition.[]{data-label="ranks"}](./kernel_lev1 "fig:") ![ Computed numerical ranks of each block of the matrix ${\mathbb G}_e$ to achieve an accuracy of $10^{-4}$, i.e. if singular values smaller than $10^{-4}$ are neglected in the singular value decomposition.[]{data-label="ranks"}](./kernel_lev2 "fig:") ![ Computed numerical ranks of each block of the matrix ${\mathbb G}_e$ to achieve an accuracy of $10^{-4}$, i.e. if singular values smaller than $10^{-4}$ are neglected in the singular value decomposition.[]{data-label="ranks"}](./kernel_lev3 "fig:")
[Entire matrix]{} [2 levels]{} [3 levels]{}
Size block=$7500$ Size block=$3750$ Size block=$1875$
\[-2ex\]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Clustering of the unknowns
The key ingredient of hierarchical matrices is the recursive block subdivision. On the illustrative example, the subdivision of the matrix is conducted by a recursive subdivision of the geometry, i.e. a recursive subdivision of the plan into partitions of equal areas. The first step prior to the partition of the matrix is thus a partitioning based on the geometry of the set of row and column indices of the matrix ${\mathbb A}$. The purpose is to permute the indices in the matrix to reflect the physical distance and thus interaction between degrees of freedom. Consecutive indices should correspond to DOFs that interact at close range. For the sake of clarity, in this work ${\mathbb A}$ is defined by the same set of indices $I=\{1, \ldots, n\}$ for rows and columns. A binary tree $\mathcal{T}_I$ is used to drive the clustering. Each node of the tree defines a subset of indices $\sigma \subset I$ and each subset corresponds to a part in the partition of the domain, see Figure \[binary\_tree\] for the case of the illustrative example. There exist different approaches to perform the subdivision [@hackbusch2015hierarchical]. We consider the simplest possible one : based on a geometric argument. For each node in the tree, we determine the box enclosing all the points in the cloud and subdivide it into 2 boxes, along the largest dimension. The subdivision is stopped when a minimum number of DOFs per box is reached ($N_{\operatorname{leaf}}=100$ in the following). For uniform meshes, this strategy defines a balanced binary tree [@bebendorf2008hierarchical] such that the number of levels in the tree $\mathcal{T}_I$ is given by $L(I)= \lceil \log_2(\frac{|I|}{N_{\operatorname{leaf}}}) \rceil \le \log_2 |I| -\log_2 N_{\operatorname{leaf}} +1$. Note that for this step only geometrical information is needed.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illustration of the clustering of the degrees of freedom: (a) partition of the degrees of freedom in the domain and (b) corresponding binary tree. []{data-label="binary_tree"}](./binary_tree_geo2 "fig:") ![Illustration of the clustering of the degrees of freedom: (a) partition of the degrees of freedom in the domain and (b) corresponding binary tree. []{data-label="binary_tree"}](./binary_tree2 "fig:")
\(a) Partition of the physical domain \(b) Binary cluster tree $\mathcal{T}_I$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Subdivision of the matrix
After the clustering of the unknowns is performed, a block cluster representation $\mathcal{T}_{I\times I}$ of the matrix ${\mathbb A}$ is defined by going through the cluster tree $\mathcal{T}_{I}$. Each node of $\mathcal{T}_{I\times I}$ contains a pair $(\sigma,\tau)$ of indices of $\mathcal{T}_{I}$ and defines a block of ${\mathbb A}$ (see Figure \[def\_tree\]). This uniform partition defines a block structure of the matrix with a full pattern of $4^{L(I)-1}$ blocks, in particular every node of the tree at the leaf level is connected with all the other nodes at the leaf level (Figure \[def\_tree\_uni\]a). Figure \[svd\_all\]b shows that this partition is not optimal. As a matter of fact, some parts of the matrix ${\mathbb A}$ can accurately be approximated by a low-rank matrix at a higher level (i.e. for larger clusters). Such blocks are said to be *admissible*. A hierarchical representation $\mathcal{P}\subset \mathcal{T}_{I \times I}$ that uses the cluster tree $\mathcal{T}_{I}$ and the existence of *admissible* blocks is more appropriate (Figure \[def\_tree\_uni\]b). Starting from the initial matrix, each block is recursively subdivided until it is either *admissible* or the leaf level is achieved. In the illustrative example, only diagonal blocks are subdivided due to the very simple 2D geometry (see again Fig. \[ranks\], red-colored blocks). For more complex 3D geometries, an admissibility condition based on the geometry and the interaction distance between points is used to determine *a priori* the *admissible* blocks. For more details on the construction of the block cluster tree, we refer the interested reader to [@borm2003introduction]. The partition $\mathcal{P}$ is subdivided into two subsets $\mathcal{P}^{\operatorname{ad}}$ and $\mathcal{P}^{\operatorname{non-ad}}$ reflecting the possibility for a block $\tau \times \sigma$ to be either *admissible*, i.e. $\tau \times \sigma \in \mathcal{P}^{\operatorname{ad}}$; or *non-admissible*, i.e. $\tau \times \sigma \in \mathcal{P}^{\operatorname{non-ad}}$. It is clear that $\mathcal{P}=\mathcal{P}^{\operatorname{ad}}\cup \mathcal{P}^{\operatorname{non-ad}}$. To sum up, the blocks of the partition can be of 3 types: at the leaf level a block can be either an *admissible* block or a *non-admissible* block, at a non-leaf level a block can be either an *admissible* block or an $\mathcal{H}$-matrix (i.e a block that will be subsequently hierarchically subdivided).
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illustration of the construction of the block cluster tree: (a) Clustering of the unknowns on the geometry and (b) corresponding block clustering in the matrix. []{data-label="def_tree"}](./cluster_In "fig:"){width="7cm"} ![Illustration of the construction of the block cluster tree: (a) Clustering of the unknowns on the geometry and (b) corresponding block clustering in the matrix. []{data-label="def_tree"}](./mat_clus2 "fig:"){width="4cm"}
(a) (b)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ (a) Block cluster representation $\mathcal{T}_{I \times I}$ for the illustrative example (full structure); (b) Hierarchical partition $\mathcal{P} \subset \mathcal{T}_{I \times I}$ of the same matrix based on the admissibility condition (sparse structure). []{data-label="def_tree_uni"}](./mat_uni "fig:"){width="4cm"} ![ (a) Block cluster representation $\mathcal{T}_{I \times I}$ for the illustrative example (full structure); (b) Hierarchical partition $\mathcal{P} \subset \mathcal{T}_{I \times I}$ of the same matrix based on the admissibility condition (sparse structure). []{data-label="def_tree_uni"}](./mat_hiera "fig:"){width="4cm"}
(a) (b)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
At this point, we need to define the sparsity pattern introduced by L. Grasedyck [@grasedyck2001theorie]. For sparse matrices, the sparsity pattern gives the maximum number of non-zero entries per row. Similarly for a hierarchical matrix defined by the partition $\mathcal{P} \subset \mathcal{T}_{I\times I}$, the sparsity pattern of a row cluster $\tau \in \mathcal{T}_I$, resp. a column cluster $\sigma \in \mathcal{T}_I$, is $$C_{sp}(\tau)=|\{ \sigma \in \mathcal{T}_I: \tau \times \sigma \in \mathcal{P}\}|, \mbox{ resp. } C_{sp}(\sigma)=|\{ \tau \in \mathcal{T}_I: \tau \times \sigma \in \mathcal{P}\}|.$$ It is convenient to define the overall sparsity pattern (for row and column clusters): $$\displaystyle{ C_{sp}=\mbox{max} \{ \mbox{max}_{\tau \in \mathcal{T}_I} C_{sp}(\tau), \ \mbox{max}_{\sigma \in \mathcal{T}_I} C_{sp}(\sigma) \}.}$$
#### Special case of asymptotically smooth kernels
$\mathcal{H}$-matrix representations have been derived for some specific problems and will not result in efficient algorithms for all equations or matrices. The crucial point is to know *a priori* (i) if savings will be obtained when trying to approximate *admissible* blocks with a sum of separated variable functions and (ii) which blocks are *admissible* since the explicit computation of the rank of all the blocks would be too expensive. In the case of asymptotically smooth kernels $G({\boldsymbol{x}},{\boldsymbol{y}})$, it is proved that under some *a priori* condition on the distance between ${\boldsymbol{x}}$ and ${\boldsymbol{y}}$, the kernel is a degenerate function (see Section \[theory\]). After discretization, this property is reformulated as the efficient approximation of blocks of the matrix by low-rank matrices. The Laplace Green’s function is an example of asymptotically smooth kernel for which $\mathcal{H}$-matrix representations have been shown to be very efficient. Since the 3D elastodynamics Green’s tensor, similarly to the Helmholtz Green’s function, is a linear combination of derivatives of the 3D Laplace Green’s function with coefficients depending on the circular frequency $\omega$, this work is concerned with the determination of the frequency range for which hierarchical representations can be successful for 3D elastodynamics.
Algorithms to perform low-rank approximations
---------------------------------------------
Once the *admissible* blocks are determined, an accurate rank-revealing algorithm is applied to determine low-rank approximations. Such an algorithm must be accurate (i.e. its result, the computed numerical rank, must be as small as possible) to avoid unnecessary computational costs. The truncated Singular Value Decomposition (SVD) [@golub2012matrix] gives the best low-rank approximation (Eckart-Young theorem) for unitary invariant norms (e.g. Frobenius or spectral norm). Thus it produces an approximation with the smallest possible numerical rank for a given prescribed accuracy. But the computation of the SVD is expensive, i.e. in the order of $O(\max(m,n) \times \min(m,n)^2)$ for an $m \times n$ matrix, and in addition it requires the computation of all the entries of ${\mathbb A}$. In the context of the $\mathcal{H}$-matrices, the use of the SVD would induce the undesired need to assemble the complete matrix.
The adaptive cross approximation (ACA) [@bebendorf2015wideband; @bebendorf2003adaptive] offers an interesting alternative to the SVD since it produces a quasi-optimal low-rank approximation without requiring the assembly of the complete matrix. The starting point of the ACA is that every matrix of rank $r$ is the sum of $r$ matrices of rank $1$. The ACA is thus a greedy algorithm that improves the accuracy of the approximation by adding iteratively rank-1 matrices. At iteration $k$, the matrix is split into the rank $k$ approximation ${\mathbb B}_k= \sum_{\ell=1}^k {\bf u}_{\ell} {\bf v}_{\ell}^{*}=\mathbb{U}_k \mathbb{V}_k^*$ with $\mathbb{U}_k \in \mathbb{C}^{m \times k}$, $ \mathbb{V}_k \in \mathbb{C}^{n \times k}$, and the residual ${\mathbb R}_k= {\mathbb A} - \sum_{\ell=1}^k {\bf u}_{\ell} {\bf v}_{\ell}^*$; ${\mathbb A}={\mathbb B}_k+{\mathbb R}_k$. The information is shifted iteratively from the residual to the approximant. A stopping criterion is used to determine the appropriate rank to achieve the required accuracy. A straightforward choice of stopping criterion is $$\mbox{until} \quad || {\mathbb A} - {\mathbb B}_k||_F \le \varepsilon_{\footnotesize\operatorname{ACA}} || {\mathbb A}||_F
\label{stop_full}$$ where $\varepsilon_{\footnotesize\operatorname{ACA}}>0$ is a given parameter, and $||.||_F$ denotes the Frobenius norm. In the following, we denote $r_{\operatorname{ACA}}$ the numerical rank obtained by the ACA for a required accuracy $\varepsilon_{\footnotesize\operatorname{ACA}}$. The complexity of this algorithm to generate an approximation of rank $r_{\operatorname{ACA}}$ is $O(r_{\operatorname{ACA}}mn)$.
There are various ACAs that differ by the choice of the best pivot at each iteration. The simplest approach is the so-called fully-pivoted ACA and it consists in choosing the pivot as the largest entry in the residual. But similarly to the SVD, it requires the computation of all the entries of ${\mathbb A}$ to compute the pivot indices. It is not an interesting option for the construction of $\mathcal{H}$-matrices.
The partially-pivoted ACA proposes an alternative approach to choose the pivot avoiding the assembly of the complete matrix. The idea is to maximize alternately the residual for only one of the two indices and to keep the other one fixed. With this strategy, only one row and one column is assembled at each iteration. More precisely, at iteration $k$, given ${\mathbb B}_k$ and assuming the row index $\hat{i}$ is known the algorithm is given by the following six steps:
1. Generation of the rows ${\bf a}:= {\mathbb A}^* {\bf e}_{\hat{i}}$ and $\begin{small}{\mathbb R}_k^* {\bf e}_{\hat{i}}={\bf a}- \sum_{\ell=1}^k ({ u}_{\ell})_{\hat{i}}{\bf v}_{\ell}\end{small}$
2. Find the column index $\hat{j}:=\mbox{argmax}_{j}|({\mathbb R}_k)_{\hat{i}j}|$ and compute $\gamma_{k+1}= ({\mathbb R}_k)_{\hat{i},\hat{j}}^{-1}$
3. Generation of the columns: ${\bf a}:= {\mathbb A} {\bf e}_{\hat{j}}$ and ${\mathbb R}_k {\bf e}_{\hat{j}}={\bf a}- \sum_{\ell=1}^k {\bf u}_{\ell} ({ v}_{\ell})_{\hat{j}}$
4. Find the next row index $\hat{i}:=\mbox{argmax}_{i}|({\mathbb R}_k)_{i\hat{j}}|$
5. Compute vectors ${\bf u}_{k+1}:=\gamma_{k+1} {\mathbb R}_k {\bf e}_{\hat{j}}$, ${\bf v}_{k+1}:= {\mathbb R}_k^* {\bf e}_{\hat{i}}$
6. Update the approximation ${\mathbb B}_{k+1}={\mathbb B}_{k} + {\bf u}_{k+1} {\bf v}_{k+1}^*$
With this approach however, approximates and residuals are not computed explicitly nor stored, so the stopping criteria (\[stop\_full\]) needs to be adapted. The common choice is a stagnation-based error estimate which is computationally inexpensive. The algorithm stops when the new rank-1 approximation does not improve the accuracy of the approximation. Since at each iteration $k$, ${\mathbb B}_k-{\mathbb B}_{k-1}= {\bf u}_{k}{\bf v}_{k}^*$, the stopping criteria now reads: $$\mbox{until } ||{\bf u}_{{k}}||_2 ||{\bf v}_{{k}}||_2 \le\varepsilon_{\operatorname{ACA}} ||{\mathbb B}_{{k}} ||_F.
\label{stop_part}$$ The complexity of the partially-pivoted ACA is reduced to $O(r_{\operatorname{ACA}}^2 (m+n))$. Since the partially-pivoted ACA is a heuristic method, there exist counter-examples where ACA fails [@Borm2003] and variants have been designed to improve the robustness of the method. [Nevertheless, in all the numerical examples we have performed, we do not need them]{}. It is worth mentioning that other approaches exist such as fast multipole expansions [@rokhlin1985rapid; @greengard1987fast], panel clustering [@sauter2000variable; @hackbusch1989fast], quadrature formulas [@borm2016approximation] or interpolations [@messner2012fast]. These approaches combine approximation techniques and linear algebra. The advantages of the ACA are to be purely algebraic and easy to implement.
From now on, we shall write the result $\mathbb{B}_{r_{\operatorname{ACA}}}$ as $\mathbb{B}$ for short, and $\mathbb{B}\approx \mathbb{A}$ or $\mathbb{A}\approx \mathbb{B}$.
Extension to problems with vector unknowns \[ACA\_vect\]
--------------------------------------------------------
One specificity of this work is to consider $\mathcal{H}$-matrices and ACA in the context of systems of partial differential equations with vector unknowns. There exist a lot of works both theoretical and numerical on the ACA for scalar problems, in particular on the selection of non-zero pivots (since they are used to normalize the new rank-1 approximation). Indeed for scalar problems, it is straightforward to find the largest non-zero entry in a given column. For problems with vector unknowns in $\mathbb{R}^d$, the system has a block structure, i.e. each pair of nodes on the mesh does not define a single entry but rather a $d \times d$ subblock in the complete matrix. This happens for example for 3D elastodynamics where the Green’s tensor is a $3 \times 3$ subblock.
Different strategies can be applied to perform the ACA on matrices with a block structure. The first strategy consists in ordering the system matrix such that it is composed of $9=3^2 $ subblocks of size $N_c \times N_c$ where $N_c$ is the number of points in the cloud. In 3D elastodynamics, this corresponds to the following partitioning of the matrix (below the solution is represented with ${\bf U}_{\omega}$): $$\left[
\begin{array}{ccc}
\mathbb{A}_{11} & \mathbb{A}_{12} & \mathbb{A}_{13}\\
\mathbb{A}_{21} & \mathbb{A}_{22} & \mathbb{A}_{23}\\
\mathbb{A}_{31} & \mathbb{A}_{32} & \mathbb{A}_{33}\\
\end{array}
\right], \quad (\mathbb{A}_{\alpha \beta})_{ij} = ({\bf U}_{\omega})_{\alpha \beta}({\boldsymbol{x}}_i, {\boldsymbol{y}}_j) \ \ 1 \le \alpha,\beta \le 3, \ 1 \le i,j \le N_c.$$ Then each submatrix is approximated independently with the conventional scalar ACA. This strategy is used in [@MESSNER2010]. It is well suited for iterative solvers. But it cannot be adapted straightforwardly in the context of direct solvers since the recursive $2 \times 2$ block structure inherited from the binary tree is used. Indeed even though it is possible to determine the LU decomposition of each block of $\mathbb{A}$ (see Section \[lu\_solve\]), such that $$\left[
\begin{array}{ccc}
\mathbb{A}_{11} & \mathbb{A}_{12} & \mathbb{A}_{13}\\
\mathbb{A}_{21} & \mathbb{A}_{22} & \mathbb{A}_{23}\\
\mathbb{A}_{31} & \mathbb{A}_{32} & \mathbb{A}_{33}\\
\end{array}
\right] \approx
\left[
\begin{array}{ccc}
\mathbb{L}_{11}\mathbb{U}_{11} & \mathbb{L}_{12} \mathbb{U}_{12}& \mathbb{L}_{13} \mathbb{U}_{13}\\
\mathbb{L}_{21}\mathbb{U}_{21} & \mathbb{L}_{22} \mathbb{U}_{22}& \mathbb{L}_{23} \mathbb{U}_{23}\\
\mathbb{L}_{31} \mathbb{U}_{31}& \mathbb{L}_{32} \mathbb{U}_{32} & \mathbb{L}_{33} \mathbb{U}_{33}\\
\end{array}
\right],$$ on the other hand it would be very expensive to deduce the LU decomposition of $\mathbb{A}$ from these LU decompositions since what we are looking for is a decomposition of the kind $$\left[
\begin{array}{ccc}
\mathbb{L}_{1} & \mathbb{O} & \mathbb{O}\\
\mathbb{L}_{2} & \mathbb{L}_{3} & \mathbb{O}\\
\mathbb{L}_{4} & \mathbb{L}_{5} & \mathbb{L}_{6} \\
\end{array}
\right]
\left[
\begin{array}{ccc}
\mathbb{U}_{1} & \mathbb{U}_{2}& \mathbb{U}_{3}\\
\mathbb{O} & \mathbb{U}_{4}& \mathbb{U}_{5}\\
\mathbb{O} & \mathbb{O} & \mathbb{U}_{6}\\
\end{array}
\right],$$ i.e. 12 factors instead of 18. A solution would be to replace the binary by a ternary tree but in that case the clustering process would be more complex.
The second naive approach consists in considering the complete matrix as a scalar matrix, i.e. to forget about the block structure. While appealing this approach fails in practice for 3D elastodynamics, whatever ordering is used, due to the particular structure of the matrix. Rewriting the Green’s tensors (\[elasto\_U\])-(\[elasto\_T\]) component-wise [@bonnet1999boundary] and using the Einstein summation convention, it reads $$\begin{gathered}
\hspace*{-0.5cm} {({\bf U}_{\omega})_{\alpha \beta}({{\boldsymbol{x}}},{{\boldsymbol{y}}})=\frac{1}{4 \pi \varrho} (a_1 \delta_{\alpha \beta}+a_2 \varrho_{,\alpha} {\varrho}_{,\beta}),} \\
{ ({\bf T}_{\omega})_{\alpha \beta}({{\boldsymbol{x}}},{{\boldsymbol{y}}})=\frac{1}{4 \pi \varrho} \Big[2a_3 {\varrho}_{,\alpha} {\varrho}_{,\gamma} {\varrho}_{,\beta} +a_4(\delta_{\alpha \beta} \varrho_{,\gamma}
+ \delta_{\gamma \beta} \varrho_{,\alpha}) +a_5 \delta_{\alpha \gamma}\varrho_{,\beta} \Big] { n}_{\gamma}({\boldsymbol{y}})}\end{gathered}$$ where the constants $a_1, ..., \ a_5$ depend only on the mechanical properties, $ {\varrho}=||{{\boldsymbol{x}}}-{{\boldsymbol{y}}}||$ and $ {\varrho}_{,\alpha}=\frac{\partial }{\partial y_\alpha}\varrho({{\boldsymbol{x}}},{{\boldsymbol{y}}})$. As a result, as soon as ${{\boldsymbol{x}}}$ and ${{\boldsymbol{y}}}$ belong to the same plane (let’s say $x_3=y_3$ for simplicity) the Green’s tensors simplify to $$\hspace*{-0.4cm}{\bf U}_{\omega}({{\boldsymbol{x}}},{{\boldsymbol{y}}}) = \frac{1}{4 \pi {\varrho}}\left[
\begin{array}{ccc}
a_{1}+a_2 {\varrho}^2_{,1} & a_{2} {\varrho}_{,1} {\varrho}_{,2} & 0\\
a_{2} {\varrho}_{,1} {\varrho}_{,2} & a_{1}+a_2 {\varrho}^2_{,1} & 0\\
0 & 0 & a_{1}\\
\end{array}
\right], \ {\bf T}_{\omega}({{\boldsymbol{x}}},{{\boldsymbol{y}}}) = \frac{1}{4 \pi {\varrho}}\left[
\begin{array}{ccc}
0 & 0& a_4 {\varrho}_{,1} \\
0 & 0& a_4 {\varrho}_{,2}\\
a_4 {\varrho}_{,1} & a_4 {\varrho}_{,2}&0\\
\end{array}
\right].
\label{FSplane}$$ It is then clear that the complete matrix ${\mathbb A}$ composed of such subblocks is a reducible matrix (there are decoupled groups of unknowns after discretization). While the fully-pivoted ACA will succeed in finding the best pivot to perform low-rank approximations, the partially pivoted ACA will only cover parts of the matrix resulting in a non accurate approximation of the initial matrix. As a result, this approach cannot be applied for 3D elastodynamics. Another strategy would be to adapt the choice of the pivot to reducible matrices, i.e. to cover all the decoupled groups of unknowns. This is possible with the use of the ACA+ algorithm that defines a reference column and a reference row along whose the pivots are looked for [@borm2003introduction]. Even though this approach could fix the problem it does not seem to be the best suited one since it does not use the vector structure of 3D elastodynamic problems.
To sum up, it is now important to use an algorithm that takes into account the particular structure of our matrix such that the vector ACA does not rely anymore on a rank-1 update but instead on a rank-3 update. The central question is then how to find the pivot used for this rank-3 update instead. There are 3 possible strategies:
1. To look for the largest scalar pivot, determine the corresponding point in the cloud and update all 3 DOFs linked to this point simultaneously. This approach is not stable since the $3 \times 3$ subblock pivot may not be invertible (cf. a discretized version of (\[FSplane\])).
2. The second strategy is to look for the $3\times 3$ subblock with the largest norm. Again this approach fails in practice since the $3 \times 3$ subblock pivot may not be invertible.
3. The third strategy used in this work is to compute the singular values $\sigma_1\ge \sigma_2 \ge \sigma_3$ of every candidate subblock. In order to achieve convergence and to avoid singular pivots, the safest approach consists in choosing the pivot corresponding to the subblock with the largest $\sigma_3$. A similar approach is used for electromagnetism in [@rjasanow2016matrix].
It is worth noting that for some specific configurations the 3D elastodynamic double layer potential ${\bf T}_{\omega}$ may lead to a matrix with only singular subblocks. In such cases, the randomized SVD [@liberty2007randomized] is preferred to the vector ACA.
$\mathcal{H}$-matrix based iterative and direct solvers
=======================================================
$\mathcal{H}$-matrix based iterative solver \[iter\_solver\]
------------------------------------------------------------
Once the $\mathcal{H}$-matrix representation of a matrix is computed, it is easy to derive an $\mathcal{H}$-matrix based iterative solver. The only operation required is an efficient matrix-vector product. It is performed hierarchically by going through the block cluster tree $\mathcal{P}$. At the leaf level, there are two possibilities. If the block of size $m \times n$ does not admit a low-rank approximation (*non-admissible* block), then the standard matrix-vector product is used with cost $O(mn)$. Otherwise, the block is marked as *admissible* such that a low-rank approximation has been computed : ${\mathbb A}_{\tau \times \sigma} \approx \mathbb{B}_{\tau \times \sigma}$. The cost of this part of the matrix-vector product is then reduced from $O(mn)$ to $O(r_{\operatorname{ACA}}(m+n))$ where $r_{\operatorname{ACA}}$ is the numerical rank of the block $\mathbb{B}_{\tau \times \sigma}$ computed with the ACA.
$\mathcal{H}$-LU factorization and direct solver \[lu\_solve\]
--------------------------------------------------------------
One of the advantages of the $\mathcal{H}$-matrix representation is the possibility to derive a fast direct solver. Due to the hierarchical block structure of a $\mathcal{H}$-matrix, the LU factorization is performed recursively on $2 \hspace*{-0.1cm} \times \hspace*{-0.1cm}2$ block matrices of the form $$\left(
\begin{array}{ c c }
{\mathbb A}_{\tau_1 \times \sigma_1} & {\mathbb A}_{\tau_1 \times \sigma_2}\\
{\mathbb A}_{\tau_2 \times \sigma_1} & {\mathbb A}_{\tau_2 \times \sigma_2}
\end{array}
\right)=\left(
\begin{array}{ c c }
{\mathbb L}_{\tau_1 \times \sigma_1} & \mathbb{O}\\
{\mathbb L}_{\tau_2 \times \sigma_1} & {\mathbb L}_{\tau_2 \times \sigma_2}
\end{array}
\right) \left(
\begin{array}{ c c }
{\mathbb U}_{\tau_1 \times \sigma_1} & {\mathbb U}_{\tau_1 \times \sigma_2}\\
\mathbb{O} & {\mathbb U}_{\tau_2 \times \sigma_2}
\end{array}
\right)$$ For such block matrices, we recall that the LU factorization is classically decomposed into 4 steps:
1. LU decomposition to compute ${\mathbb L}_{\tau_1 \times \sigma_1}$ and ${\mathbb U}_{\tau_1 \times \sigma_1}$: ${\mathbb A}_{\tau_1 \times \sigma_1} = {\mathbb L}_{\tau_1 \times \sigma_1} {\mathbb U}_{\tau_1 \times \sigma_1}$;
2. Compute $ {\mathbb U}_{\tau_1 \times \sigma_2}$ from $ {\mathbb A}_{\tau_1 \times \sigma_2}={\mathbb L}_{\tau_1 \times \sigma_1} {\mathbb U}_{\tau_1 \times \sigma_2}$;
3. Compute $ {\mathbb L}_{\tau_2 \times \sigma_1} $ from $ {\mathbb A}_{\tau_2 \times \sigma_1}= {\mathbb L}_{\tau_2 \times \sigma_1} {\mathbb U}_{\tau_1 \times \sigma_1}$;
4. LU decomposition to compute ${\mathbb L}_{\tau_2 \times \sigma_2}$ and ${\mathbb U}_{\tau_2 \times \sigma_2}$: $ {\mathbb A}_{\tau_2 \times \sigma_2}- {\mathbb L}_{\tau_2 \times \sigma_1} {\mathbb U}_{\tau_1 \times \sigma_2}= {\mathbb L}_{\tau_2 \times \sigma_2} {\mathbb U}_{\tau_2 \times \sigma_2}$.
The obvious difficulty is the fact that the process is recursive (cf steps 1 and 4). In addition, Step 4 requires the addition and multiplication of blocks which may come in different formats. In more general terms, the difficulty is to evaluate $${\mathbb A}^{'''}_{\tau \times \sigma} \leftarrow {\mathbb A}_{\tau \times \sigma} + {\mathbb A}^{'}_{\tau \times \sigma'} {\mathbb A}^{''}_{\tau' \times \sigma}.$$ Since each block may belong to three different classes this results into $27$ cases to take into account. For example, if ${\mathbb A}_{\tau \times \sigma} $ is a full matrix, $ {\mathbb A}^{'}_{\tau \times \sigma'} $ an $\mathcal{H}$-matrix, i.e. a *non-admissible* block at a non-leaf level and ${\mathbb A}^{''}_{\tau' \times \sigma}$ a matrix having a low-rank representation, then the computation is efficiently decomposed into the following steps. First, ${\mathbb A}^{''}_{\tau' \times \sigma} $ being a low-rank matrix it is decomposed into ${\mathbb A}^{''}_{\tau' \times \sigma} \approx \mathbb{B}^{''}_{\tau' \times \sigma}= \mathbb{U} \mathbb{V}^*$ and using the fast $\mathcal{H}$-matrix / vector product the product between ${\mathbb A}^{'}_{\tau \times \sigma'}$ and each column of ${\mathbb U}$ is evaluated. In a second step, the result is multiplied to ${\mathbb V}^{*}$ and the final matrix is directly a full matrix. A more detailed review of the 27 cases is presented in [@desiderio].
It is known that the numerical rank obtained by the ACA is not optimal. Similarly the numerical rank for blocks obtained after an addition and a multiplication and stored in a low-rank or $\mathcal{H}$-matrix format is not optimal. All these blocks are further recompressed to optimize the numerical ranks. Starting from the low-rank approximation $\mathbb{B}_{\tau \times \sigma}={\mathbb U} {\mathbb V}^{*} $, the reduced QR-decompositions of ${\mathbb U}$ and $ {\mathbb V}$ are computed, i.e. $$\mathbb{B}_{\tau \times \sigma} = {\mathbb U} {\mathbb V}^{*} = {\mathbb Q}_U {\mathbb R}_U {\mathbb R}_V^* {\mathbb Q}_V^*.$$ Recall that the only decomposition that yields the optimal numerical rank is the truncated singular value decomposition. It is thus applied to the reduced matrix ${\mathbb R}_U {\mathbb R}_V^* \approx \mathbb{PSL}^*$ such that the cost of this step is reasonable. $$\mathbb{B}_{\tau \times \sigma} = {\mathbb Q}_U {\mathbb R}_U {\mathbb R}_V^* {\mathbb Q}_V^*= ({\mathbb Q}_U {\mathbb P} {\mathbb S}^{1/2}) ({\mathbb S}^{1/2} {\mathbb L}^* {\mathbb Q}_V^*)= {\mathbb U'} {\mathbb V'}^{*}.$$ During the construction of the $\mathcal{H}$-matrix representation, the parameter $\varepsilon_{\operatorname{ACA}}$ is also used to determine the number of singular values selected during the recompression process. Moreover, in the case of a recompression after an addition and a multiplication, i.e. when the format of a block is modified, we introduce the parameter $\varepsilon_{\operatorname{LU}}$ to determine the level of accuracy required.
Proposition of an estimator to certify the results of the $\mathcal{H}$-LU direct solver \[esti\_direct\]
---------------------------------------------------------------------------------------------------------
Since the $\mathcal{H}$-LU direct solver is based on a heuristic method (the ACA) to perform the low-rank approximations and due to the various modifications on the $\mathcal{H}$-matrix performed during the LU factorization, it is relevant to propose a simple and efficient way to certify the obtained results. To this aim, we propose an estimator. We consider the initial system $$\mathbb{A}{\bf x}={\bf b}.$$ We denote ${\mathbb A}_{\mathcal{H}}$ the $\mathcal{H}$-matrix representation of ${\mathbb A}$ (in which low-rank approximations have been performed) and ${\mathbb L}_{\mathcal{H}}{\mathbb U}_{\mathcal{H}}\approx {\mathbb A}_{\mathcal{H}}$ its $\mathcal{H}$-LU factorization. The solution of the approximated system is ${\bf x}_0$ : ${\mathbb L}_{\mathcal{H}}{\mathbb U}_{\mathcal{H}} {\bf x}_0={\bf b}$. Our aim is to give an upper bound of $||{\bf b}- {\mathbb A}{\bf x}_0||_2$. Since $${\bf b}-{\mathbb A}{\bf x}_0={\bf b}-{\mathbb A}_{\mathcal{H}}{\bf x}_0+{\mathbb A}_{\mathcal{H}}{\bf x}_0-{\mathbb A}{\bf x}_0$$ this yields for $\alpha\in \{2,F\}$[^4] $$\begin{array}{rl}
\displaystyle \frac{||{\bf b}- {\mathbb A}{\bf x}_0||_2}{||{\bf b}||_2}& \displaystyle\le \frac{1}{||{\bf b}||_2} (||{\bf b}- {\mathbb A}_{\mathcal{H}} {{\bf x}_0}||_2+|| {\mathbb A}_{\mathcal{H}}-{\mathbb A}||_{\alpha} ||{{\bf x}_{0}}||_2) { = \frac{1}{||{\bf b}||_2} (\delta+\delta_{\mathcal{H},\alpha} ||{\bf x}_{0}||_2)}\\[2ex]
& \mbox{where } \delta=||{\bf b}- {\mathbb A}_{\mathcal{H}}{{\bf x}_0}||_2 \mbox{ and } \delta_{\mathcal{H},\alpha}=||{\mathbb A}_{\mathcal{H}}-{\mathbb A}||_{\alpha}.
\end{array}
\label{est_direct}$$ It is worth noting that $\delta_{\mathcal{H},\alpha}=||{\mathbb A}_{\mathcal{H}}-{\mathbb A}||_{\alpha} $ estimates the accuracy of the $\mathcal{H}$-matrix representation (i.e. the influence of the parameter $\varepsilon_{\operatorname{ACA}}$ in the computation of the low-rank approximations) while $\delta=||{\bf b}- {\mathbb A}_{\mathcal{H}}{{\bf x}_0}||_2$ accounts for the stability of the $\mathcal{H}$-LU factorization (i.e. the influence of the parameter $\varepsilon_{\operatorname{LU}}$ in the LU factorization). The evaluations of $\delta$, $||{\bf b}||_2$ and $||{\bf x}_0||_2$ reduce to the computation of the norm of a vector. [The most expensive part is $\delta_{\mathcal{H},\alpha}$ but its evaluation is performed with the Frobenius norm to reduce the cost.]{} Also this term does not depend on the right hand side meaning that for multiple right hand sides this error estimate is not expensive.
3D Elastodynamics Boundary Element Method
=========================================
Let’s consider a bounded domain $\Omega^-$ in ${{\mathbb R}}^3$ representing an obstacle with a closed Lipschitz boundary $\Gamma:=\partial \Omega^-$. Let $\Omega^+$ denote the associated exterior domain ${{\mathbb R}}^3\backslash\overline{\Omega^-}$ and ${\boldsymbol{n}}$ its outward unit normal vector field on its boundary $\Gamma$. We consider the propagation of time-harmonic waves in a three-dimensional isotropic and homogeneous elastic medium modeled by the Navier equation (\[NE\]). The following results about traces of vector fields and integral representations of time-harmonic elastic fields can be found in [@darbas2015well].
Traces, integral representation formula and integral equations
--------------------------------------------------------------
We denote by $H^s_{loc}({\Omega^\pm})$ and $H^s(\Gamma)$ the standard (local in the case of the exterior domain) complex valued Hilbertian Sobolev spaces of order $s\in{{\mathbb R}}$ ($|s|\le 1$ for $H^s(\Gamma)$) defined on ${\Omega^\pm}$ and $\Gamma$ respectively (with the convention $H^0=L^2$). Spaces of vector functions will be denoted by boldface letters, thus ${\boldsymbol{ H}}^s=(H^s)^3$. We set $\mathbf{\mathbf{\Updelta}}^*{\boldsymbol{ u}}:={\operatorname{\mathrm{div}}}\upsigma({\boldsymbol{ u}})=(\lambda+2\mu){\boldsymbol{\nabla }}{\operatorname{\mathrm{div}}}{\boldsymbol{ u}}-\mu{\operatorname{\boldsymbol{curl}}}{\operatorname{\boldsymbol{curl}}}{\boldsymbol{ u}}$ and introduce the energy spaces ${\boldsymbol{ H}}^1_{+}(\mathbf{\Updelta}^*) := \big\{{\boldsymbol{ u}}\in {\boldsymbol{ H}}^1_{loc}({\Omega^+}):\;\mathbf{\Updelta}^*{\boldsymbol{ u}}\in {\boldsymbol{L}}_{loc}^2({\Omega^+})\big\}$ and ${\boldsymbol{ H}}^1_{-}(\mathbf{\Updelta}^*) := \big\{{\boldsymbol{ u}}\in {\boldsymbol{ H}}^1({\Omega^-}):\;\mathbf{\Updelta}^*{\boldsymbol{ u}}\in {\boldsymbol{L}}^2({\Omega^-})\big\}$. The traction trace for elastodynamic problems is defined by ${\boldsymbol{ t}}_{|\Gamma}:={\bf{T}}{\boldsymbol{ u}}$ where ${\bf{T}}$ is the traction operator defined in (\[elasto\_T\]). We recall that we have ${\boldsymbol{ u}}_{|\Gamma}\in{\boldsymbol{ H}}^{\frac{1}{2}}(\Gamma)$ and ${\boldsymbol{ t}}_{|\Gamma}\in{\boldsymbol{ H}}^{-\frac{1}{2}}(\Gamma)$ for all ${\boldsymbol{ u}}\in{\boldsymbol{ H}}^1_{\pm}(\mathbf{\Updelta}^*)$.
For a solution ${\boldsymbol{ u}}\in{\boldsymbol{ H}}^1_{+}(\mathbf{\Updelta}^*)$ to the Navier equation in $\Omega^+$, that satisfies the Kupradze radiation conditions, the Somigliana integral representation of the field is given by $$\label{somigliana}
{\boldsymbol{ u}}({\boldsymbol{x}})=\,\mathcal{D} {\boldsymbol{ u}}_{|\Gamma}({\boldsymbol{x}})\,-\,\mathcal{S}{\boldsymbol{ t}}_{|\Gamma}({\boldsymbol{x}}), \quad {\boldsymbol{x}}\in \Omega^+,$$ where the single- and double-layer potential operators are respectively defined by $$\label{potential}
\mathcal{S} {\boldsymbol{\varphi}}({\boldsymbol{x}})=\int_{\Gamma}{\bf U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}}) {\boldsymbol{\varphi}}({\boldsymbol{y}})ds({\boldsymbol{y}})\;\;\text{ and }\;\; \mathcal{D} {{\boldsymbol{\psi}}}({\boldsymbol{x}})=\int_{\Gamma}{{\vphantom{\left[{\bf{T}}_{{\boldsymbol{y}}}{\bf U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})\right]}}{\left[{\bf{T}}_{{\boldsymbol{y}}}{\bf U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})\right]}^{\sf T}}{{\boldsymbol{\psi}}}({\boldsymbol{y}})ds({\boldsymbol{y}}) \quad {\boldsymbol{x}}\in \mathbb{R}^3 \backslash \Gamma.$$
The potentials $\mathcal{S}$ (resp. $\mathcal{D}$) are continuous from ${\boldsymbol{ H}}^{-1/2}(\Gamma)$ to ${\boldsymbol{ H}}^1_{-}(\mathbf{\Updelta}^*) \cup {\boldsymbol{ H}}^1_{+}(\mathbf{\Updelta}^*)$ (resp. from ${\boldsymbol{ H}}^{1/2}(\Gamma)$ to ${\boldsymbol{ H}}^1_{-}(\mathbf{\Updelta}^*) \cup {\boldsymbol{ H}}^1_{+}(\mathbf{\Updelta}^*)$). For any ${\boldsymbol{\varphi}}\in {\boldsymbol{ H}}^{-1/2}(\Gamma)$ and ${{\boldsymbol{\psi}}}\in {\boldsymbol{ H}}^{1/2}(\Gamma)$, the potentials $\mathcal{S}{\boldsymbol{\varphi}}$ and $\mathcal{D}{{\boldsymbol{\psi}}}$ solve the Navier equation in $\Omega^+$ and $\Omega^-$, and satisfy the Kupradze radiation condition. The exterior and interior Dirichlet $(\gamma^{\pm}_0)$ and traction $(\gamma^{\pm}_1)$ traces of $\mathcal{S}$ and $\mathcal{D}$ are given by $$\gamma^{\pm}_0 \mathcal{S} =S, \quad \gamma^{\pm}_1 \mathcal{S} =\mp \frac{1}{2}I + D', \quad \gamma^{\pm}_0 \mathcal{D} =\pm \frac{1}{2}I + D$$ where the operators $S$ (resp. $D$) are continuous from ${\boldsymbol{ H}}^{-1/2}(\Gamma)$ to ${\boldsymbol{ H}}^{1/2}(\Gamma)$ (resp. continuous from ${\boldsymbol{ H}}^{1/2}(\Gamma)$ to ${\boldsymbol{ H}}^{-1/2}(\Gamma)$) and are given by $${S} {\boldsymbol{\varphi}}({\boldsymbol{x}})=\int_{\Gamma}{\bf U}_{\omega}({\boldsymbol{x}}\,,{\boldsymbol{y}}) {\boldsymbol{\varphi}}({\boldsymbol{y}})ds({\boldsymbol{y}})\;\;\text{ and }\;\; {D} {{\boldsymbol{\psi}}}({\boldsymbol{x}})=\int_{\Gamma}{{\vphantom{\left[{\bf{T}}_{{\boldsymbol{y}}}{\bf U}_{\omega}({\boldsymbol{x}}\,,{\boldsymbol{y}})\right]}}{\left[{\bf{T}}_{{\boldsymbol{y}}}{\bf U}_{\omega}({\boldsymbol{x}}\,,{\boldsymbol{y}})\right]}^{\sf T}}{{\boldsymbol{\psi}}}({\boldsymbol{y}})ds({\boldsymbol{y}}), \ {\boldsymbol{x}}\in \Gamma.$$ The scattering problem is formulated as follows : Given an incident wave field ${{\boldsymbol{ u}}^{\mathrm{inc}}}$ which is assumed to solve the Navier equation in the absence of any scatterer, find the displacement ${\boldsymbol{ u}}$ solution to the Navier equation in $\Omega^+$ which satisfies the Dirichlet boundary condition on $\Gamma$ $$\label{Dirichlet}{\boldsymbol{ u}}_{\vert\Gamma}+ {\boldsymbol{ u}}^{\mathrm{inc}}_{\vert\Gamma} = 0.$$ Applying the potential theory, the elastic scattering problem reduces to a boundary integral equation $$\label{EFIE}
S ({\boldsymbol{ t}}_{| \Gamma}+{\boldsymbol{ t}}^{inc}_{| \Gamma}) ({\boldsymbol{x}})=
{\boldsymbol{ u}}^{inc}_{| \Gamma}({{\boldsymbol{x}}}) , \quad {{\boldsymbol{x}}} \in \Gamma$$ $$\label{MFIE}
\mbox{or} \quad (\frac{I}{2} + D') ({\boldsymbol{ t}}_{| \Gamma}+{\boldsymbol{ t}}^{inc}_{| \Gamma}) ({\boldsymbol{x}})=
{\boldsymbol{ t}}^{inc}_{| \Gamma}({{\boldsymbol{x}}}) , \quad {{\boldsymbol{x}}} \in \Gamma.$$ In the following, $\mathcal{H}$-matrix based solvers are applied and studied in the special case of elastodynamic scattering problems but the method can be applied to the solution of any boundary integral equation defined in terms of the single and double layer potential operators.
Classical concepts of the Boundary Element Method
-------------------------------------------------
The main ingredients of the Boundary Element Method are a transposition of the concepts developed for the Finite Element Method [@bonnet1999boundary]. First, the numerical solution of the boundary integral equations (\[EFIE\]) or (\[MFIE\]) is based on a discretization of the surface $\Gamma$ into $N_{E}$ isoparametric boundary elements of order one, i.e. [three-node triangular elements]{}. Each physical element $E_e$ on the approximate boundary is mapped onto a reference element $\Delta_e$ via an affine mapping $${\boldsymbol \xi} \in \Delta_e \rightarrow {\boldsymbol{y}}({\boldsymbol \xi}) \in E_e, \quad 1 \le e \le N_e.$$ $ \Delta_e$ is the reference triangle in the $(\xi_1,\xi_2)$-plane. The $N_c$ interpolation points ${\boldsymbol{y}}_1, \ldots, {\boldsymbol{y}}_{N_c}$ are chosen as the vertices of the mesh. Each component of the total traction field is approximated with globally continuous, piecewise-linear shape functions $(v_i({\boldsymbol{y}}))_{1\le i\le N_c}$: $v_i({\boldsymbol{y}}_j)=\delta_{ij}$ for $1 \le i,j \le N_c$. A boundary element $E_e$ contains exactly $3$ interpolation nodes $({\boldsymbol{y}}_k^{e})_{1 \le k \le 3}$ associated with $3$ basis functions $(v^e_k)_{1 \le k \le 3}$. These basis functions are related to the canonical basis $(\hat{v}_k)_{1 \le k \le 3}$ defined on the reference element $\Delta_e$ by $v_k^e({\boldsymbol{y}}({\boldsymbol \xi}))=\hat{v}_k({\boldsymbol \xi})$. Each component of the total traction field ${\boldsymbol p}({\boldsymbol{y}}) = ({\boldsymbol{ t}}_{| \Gamma}+{\boldsymbol{ t}}^{inc}_{| \Gamma}) ({\boldsymbol{y}})$ is approximated on the element $E_e$ by $${\boldsymbol p}_{\alpha}({\boldsymbol{y}}) \approx \sum_{k=1}^{3} p^{k}_{\alpha} v_k^e({{\boldsymbol{y}}}) \quad (1 \le \alpha \le 3),$$ where $p^k_{\alpha}$ denotes the approximation of the nodal value of the component $\alpha$ of the vector ${\boldsymbol p}({\boldsymbol{y}}_k)$. To discretize the boundary integral equations (\[EFIE\]) or (\[MFIE\]) we consider the collocation approach. It consists in enforcing the equation at a finite number of collocation points ${\boldsymbol{x}}$. To have a solvable discrete problem, one has to choose $N_c$ collocation points. The $N_{c}$ traction approximation nodes thus defined also serve as collocation points, i.e. $({\boldsymbol{x}}_i)_i=({\boldsymbol{y}}_j)_j$. This discretization process transforms (\[EFIE\]) or (\[MFIE\]) into a square complex-valued linear system of size $3N_{c}$ of the form $${\mathbb A} {\bf p} = {\bf b}, \label{discre_syst}$$ where the $(3N_c)$-vector ${\bf p}$ collects the degrees of freedom (DOFs), namely the nodal traction components, while the $(3N_c)$-vector ${\bf b}$ arises from the imposed incident wave field. Assembling the matrix ${\mathbb A}$ classically [@bonnet1999boundary] requires the computation of all element integrals for each collocation point, thus requiring a computational time of order $O(N_c^{2})$.
It is interesting to note that a data-sparse representation of the matrix ${\mathbb A}$ is a direct consequence of the discretization of the Green’s tensor. For example with the defined process, the discretization of equation (\[EFIE\]) leads to the system $$\hspace*{-1cm}
{\mathbb U}_{\omega}
{\mathbb W}_y
{\mathbb V}_y {\bf p}={\bf b}$$ where the *diagonal* matrix ${\mathbb W}_y$ corresponds to the weights used to evaluate numerically the integrals, the matrix ${\mathbb V}_y$ corresponds to shape functions evaluated at the quadrature points of the reference element and the matrix ${\mathbb U}_{\omega}$ corresponds to the evaluation of the Green’s tensor at the collocation points $({\boldsymbol{x}}_i)_{i=1,\ldots,N_c}$ and interpolation points $({\boldsymbol{y}}_j)_{j=1,\ldots,N_c}$. The relevant question is then how the $\mathcal{H}$-matrix representation of ${\mathbb U}_{\omega}$ can be transmitted to $\mathbb{A}$. The key point is to remark that the matrix ${\mathbb V}_y$ is a sparse matrix whose connectivity can be described as follows: it has a non-zero entry if there exists one triangle that has the two corresponding nodes as vertices. From that argument, it appears that if no efficient $\mathcal{H}$-matrix representation of $\mathbb{U}_{\omega}$ is available, we cannot expect to find an efficient $\mathcal{H}$-matrix representation of $\mathbb{A}$. On the other hand, since the structure of the matrix ${\mathbb V}_y$ is based on a notion of distance similarly to the construction of the binary tree $\mathcal{T}_I$, we can expect that if an efficient $\mathcal{H}$-matrix representation of $\mathbb{U}_{\omega}$ is available, an efficient $\mathcal{H}$-matrix representation of $\mathbb{A}$ will be found. However, the interface separating two subdomains used to compute the binary tree can also separate nodes belonging to the same elements. It is thus very likely that the ranks of the subblocks of $\mathbb{A}$ will be larger than the ranks of the same blocks in $\mathbb{U}_{\omega}$. Importantly, this property is independent of the order of approximation used in the BEM.
As a consequence, although the behavior of the $\mathcal{H}$-matrix based solvers for 3D elastodynamics are presented in the context of the BEM, the observations can nevertheless be extended to other configurations where a Green’s tensor is discretized over a cloud of points.
Application of $\mathcal{H}$-matrices to oscillatory kernels: theoretical estimates \[theory\]
==============================================================================================
The efficiency of $\mathcal{H}$-matrix based solvers depends on the possible storage reduction obtained by low-rank approximations. It is thus important to estimate the memory requirements of the method in the context of 3D elastodynamics. We follow the proof of [@hackbusch2015hierarchical] proposed in the context of asymptotically smooth kernels.
Storage estimate
----------------
We use the notations introduced in Section \[section\_tree\]. Since we are using a balanced binary tree $\mathcal{T}_I$, the number of levels is $L(I)\le \log_2|I| - \log_2 N_{\operatorname{leaf}}+1$. We denote by $N_c=|I|$ the number of points in the cloud. For a *non-admissible* block $\tau \times \sigma \in \mathcal{P}^{\operatorname{non-ad}}$, the memory requirements are $$N_{st}= |\tau| \ |\sigma| = \mbox{min} \{|\tau|, |\sigma| \} \ \mbox{max} \{|\tau|, |\sigma| \} \mbox{ for } \tau \times \sigma \in \mathcal{P}^{\operatorname{non-ad}}.$$ Since *non-admissible* blocks can only appear at the leaf level, the memory requirements are bounded by $$N_{st} \le N_{\operatorname{leaf}} \ \max \{|\tau|, |\sigma| \} \le N_{\operatorname{leaf}} (|\tau|+ |\sigma|) \mbox{ for } \tau \times \sigma \in \mathcal{P}^{\operatorname{non-ad}}.$$ For *admissible* blocks, the memory requirements are $$N_{st}= r_{\tau \times \sigma} (|\tau| + |\sigma|) \le r_{\operatorname{ACA}}^{\max} (|\tau|+ |\sigma|) \mbox{ for } \tau \times \sigma\in \mathcal{P}^{\operatorname{ad}},$$ where $r_{\tau \times \sigma}$ is the rank obtained by the ACA, resp. $r_{\operatorname{ACA}}^{\max}$ denotes the maximum rank obtained by the ACA among all the *admissible* blocks. The total memory requirement is given by $$N_{st}({\mathbb A})=\sum_{\tau \times \sigma \in \mathcal{P}^{\operatorname{ad}} } r_{\tau \times \sigma} (|\tau| + |\sigma|) + \sum_{\tau \times \sigma \in \mathcal{P}^{\operatorname{non-ad}} } |\tau| \ |\sigma| \le \mbox{max} \{N_{leaf}, r_{ACA}^{\max}\}\sum_{\tau \times \sigma\in \mathcal{P} } (|\tau|+ |\sigma|).$$ It is clear that for a binary tree $$\sum_{\tau \in \mathcal{T}_i} |\tau| = \sum_{\ell=0}^{L(I)-1} |I^{(\ell)}|=L(I) |I|.$$ Then, using the definition of the sparsity pattern we obtain $$\sum_{\tau \times \sigma \in \mathcal{P}}|\tau| = \sum_{\tau \in \mathcal{T}_I} \Big[ |\tau| \sum_{\sigma | \tau \times \sigma \in \mathcal{P} } 1 \Big] \le C_{sp} \sum_{\tau \in \mathcal{T}_I} |\tau| \le C_{sp} \ L(I) \ |I|.$$ A similar bound is found for $\sum_{\tau \times \sigma \in \mathcal{P}}|\sigma| $ such that the storage requirement is bounded by $$N_{st}({\mathbb A}) \le 2 C_{sp} \max \{r_{\operatorname{ACA}}^{\max},N_{\operatorname{leaf}}\} N_c (\log_2 N_c - \log_2 N_{\operatorname{leaf}} +1).$$ In this storage estimate, the parameters $N_c$ and $ N_{\operatorname{leaf}}$ are known. The sparsity pattern $C_{sp}$ depends on the partition $\mathcal{P}$ and will be discussed later (Section \[impl\_issues\]). In the context of oscillatory kernels, $r_{\operatorname{ACA}}^{\max}$ is known to depend on the frequency. The aim of the rest of this section is to characterize this dependence. We consider first the 3D Helmholtz Green’s function.
Convergence of the Taylor expansion of the 3D Helmholtz Green’s function
------------------------------------------------------------------------
It is well-known that $\mathcal{H}$-matrices are efficient representations for matrices coming from the discretization of asymptotically smooth kernels, i.e. kernels satisfying Definition \[asym\_kern\] below. For such matrices the method is well-documented and estimates are provided for the Taylor expansion and interpolation errors.
A kernel $s(.,.)~: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R} $ is asymptotically smooth if there exist two constants $c_1,c_2$ and a singularity degree $\sigma \in \mathbb{N}_0$ such that $\forall z \in \{x_{\alpha},y_{\alpha}\}$, $ \forall n \in \mathbb{N}_0$, $\forall {\boldsymbol{x}}\neq {\boldsymbol{y}}$ $$|\partial_z^n s({\boldsymbol{x}},{\boldsymbol{y}})| \le n! c_1 (c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-n -\sigma}.$$ \[asym\_kern\]
The capability to produce low rank approximations is closely related to the concept of degenerate functions [@bebendorf2008hierarchical]. A kernel function is said to be degenerate if it can be well approximated (under some assumptions) by a sum of functions with separated variables. In other words, noting $X$ and $Y$ two domains of $\mathbb{R}^3$, we are looking for an approximation $s^r $ of $s$ on $X \times Y$ with $r$ terms such that it writes $$s^r({\boldsymbol{x}},{\boldsymbol{y}})=\sum_{\ell=1}^{r} u^{(r)}_{\ell}({\boldsymbol{x}}) v^{(r)}_{\ell}({\boldsymbol{y}}) \quad {\boldsymbol{x}}\in X, \ \ {\boldsymbol{y}}\in Y.$$ $r$ is called the *separation rank* of $s^r $ and $s({\boldsymbol{x}},{\boldsymbol{y}})=s^r({\boldsymbol{x}},{\boldsymbol{y}})+R_s^r({\boldsymbol{x}},{\boldsymbol{y}})$ with $R^r_s$ the remainder. Such an approximation is obtained for instance with a Taylor expansion with $r:=| \{{\boldsymbol \alpha}\in \mathbb{N}^3_0: |{\boldsymbol \alpha}| \le m\}|$ terms, i.e. $$s^r({\boldsymbol{x}},{\boldsymbol{y}})=\sum_{{\boldsymbol \alpha}\in \mathbb{N}^3_0, |{\boldsymbol \alpha}| \le m } ({\boldsymbol{x}}-{\boldsymbol{x}}_0)^{\boldsymbol \alpha} \frac{1}{{\boldsymbol \alpha}!} \partial_x^{\boldsymbol \alpha}s({\boldsymbol{x}}_0,{\boldsymbol{y}}) + R^r_s \quad {\boldsymbol{x}}\in X, \ \ {\boldsymbol{y}}\in Y$$ where ${\boldsymbol{x}}_0\in X$ is the centre of the expansion. A kernel function is said to be a separable expression in $X \times Y$ if the remainder converges quickly to zero. The main result for asymptotically smooth kernels is presented in [@bebendorf2008hierarchical Lemma 3.15] and [@hackbusch2015hierarchical Theorem 4.17]. For such kernels, it can be easily shown that $$|R^r_s ({\boldsymbol{x}},{\boldsymbol{y}})| \le C' \sum_{\ell=m}^{\infty} \Big( \frac{ ||{\boldsymbol{x}}-{\boldsymbol{x}}_0||}{c_2 ||{\boldsymbol{x}}_0-{\boldsymbol{y}}||}\Big)^{\ell}$$ where $c_2$ is the constant appearing in Definition \[asym\_kern\]. The convergence of $R^r_s$ is thus ensured for all ${\boldsymbol{y}}\in Y$ such that $$\gamma_x:= \frac{\max_{{\boldsymbol{x}}\in X} ||{\boldsymbol{x}}-{\boldsymbol{x}}_0||}{c_2 ||{\boldsymbol{x}}_0-{\boldsymbol{y}}||}< 1.
\label{admi_cond_taylor_h}$$ Provided that the condition (\[admi\_cond\_taylor\_h\]) holds, the remainder is bounded by $$\label{error_est_remain}
\begin{array}{rl}
|R^r_s({\boldsymbol{x}},{\boldsymbol{y}})|\le &\displaystyle{C'
\frac{\gamma_x^m}{1-\gamma_x} \xrightarrow[m \to \infty]{} 0.}
\end{array}$$ In this configuration, the Taylor expansion of the asymptotically smooth kernel converges exponentially with convergence rate $\gamma_x$. Since [$r$ is the cardinal of the set $\{{\boldsymbol \alpha}\in \mathbb{N}^3_0: |{\boldsymbol \alpha}| \le m\}$, it holds ]{} $m \sim r^{1/3}$, and then $r\approx |\log \varepsilon|^3$ is expected to achieve an approximation with accuracy $\varepsilon>0$.
In other words, the exponential convergence of the Taylor expansion $s^r$ over $X$ and $Y$, with center ${\boldsymbol{x}}_0 \in X$ is constrained by a condition on ${\boldsymbol{x}}_0$, $X$ and $Y$. One can derive a sufficient condition, independent of ${\boldsymbol{x}}_0$, be observing that $$\max_{{\boldsymbol{x}}\in X} ||{\boldsymbol{x}}- {\boldsymbol{x}}_0|| \le \operatorname{diam}X \quad \mbox{and} \quad \operatorname{dist}({\boldsymbol{x}}_0,Y) \ge \operatorname{dist}(X,Y)$$ where $\operatorname{dist}$ denotes the euclidian distance, i.e. $$\mbox{dist}(X,Y)=\inf \{ ||{\boldsymbol{x}}-{\boldsymbol{y}}||, \ {\boldsymbol{x}}\in X, \ {\boldsymbol{y}}\in Y\}
\label{dist}$$ and $\mbox{diam}$ denotes the diameter of a domain (Fig. \[distances2\]), i.e. $$\mbox{diam}(X)=\sup \{ ||{\boldsymbol{x}}_1-{\boldsymbol{x}}_2||, \ {\boldsymbol{x}}_1, \ {\boldsymbol{x}}_2 \in X\}.
\label{diam}$$
---------------------------------------------------------------------------------------------
![Definition of the distance and diameter. []{data-label="distances2"}](./distances "fig:")
---------------------------------------------------------------------------------------------
Let us introduce the notion of $\eta$-admissibility. $X$ and $Y$ are said to be $\eta$-*admissible* if $\operatorname{diam}X \le \eta \operatorname{dist}(X,Y)$ where $\eta>0$ is a parameter of the method. In this configuration, it follows that $$\label{admi_cond_taylor}
\max_{{\boldsymbol{x}}\in X} ||{\boldsymbol{x}}- {\boldsymbol{x}}_0|| \le \operatorname{diam}X \le \eta \operatorname{dist}(X,Y) \le \eta \operatorname{dist}({\boldsymbol{x}}_0,Y) \quad \mbox{and so} \quad \gamma_x \le \frac{\eta}{c_2}.$$ For asymptotically smooth kernels, we thus conclude that the exponential convergence is ensured if $X$ and $Y$ are $\eta$-*admissible* and if $\eta$ is chosen such that $\eta < c_2$. We observe that in [@bebendorf2008hierarchical], the condition is more restrictive on $\eta$ since it reads $\eta \sqrt{3} < c_2$.
The Helmholtz Green’s function $G_{\kappa}({\boldsymbol{x}},{\boldsymbol{y}}) =\frac{\exp (i \kappa ||{\boldsymbol{x}}-{\boldsymbol{y}}||)}{4 \pi ||{\boldsymbol{x}}-{\boldsymbol{y}}||}$ is not asymptotically smooth [@banjai2008hierarchical]. On the other hand, noting that it holds $
G_{\kappa}({\boldsymbol{x}},{\boldsymbol{y}}) =\exp (i \kappa ||{\boldsymbol{x}}-{\boldsymbol{y}}||)G_0({\boldsymbol{x}},{\boldsymbol{y}}) $ and since $G_0({\boldsymbol{x}},{\boldsymbol{y}}) $ is asymptotically smooth [@bebendorf2008hierarchical; @hackbusch2015hierarchical], the Helmholtz Green’s function satisfies the following estimate [@banjai2008hierarchical]: $\exists c_1,c_2$ and $\sigma \in \mathbb{N}_0$ (singularity degree of the kernel $G_0(.,.)$) such that $\forall z \in \{x_{\alpha},y_{\alpha}\}$, $\forall n \in \mathbb{N}_0$, $\forall {\boldsymbol{x}}\neq {\boldsymbol{y}}$ $$\label{est_Helmholtz}
|\partial_z^n G_{\kappa}({\boldsymbol{x}},{\boldsymbol{y}})| \le n! c_1 (1+\kappa ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^n (c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-n -\sigma}.$$ From inequality (\[est\_Helmholtz\]), it is clear that in the so-called *low-frequency regime*, i.e. when $\kappa \ \max_{{\boldsymbol{x}},{\boldsymbol{y}}\in X \times Y} (\kappa ||{\boldsymbol{x}}-{\boldsymbol{y}}||)$ is small, the Helmholtz Green’s function behaves similarly to an asymptotically smooth kernel because $\kappa ||{\boldsymbol{x}}-{\boldsymbol{y}}||$ is uniformly small for all ${\boldsymbol{x}},{\boldsymbol{y}}\in X \times Y$. There exists a lot of works on the $\mathcal{H}$-matrix representation [@bebendorf2000approximation; @borm2005hybrid] of asymptotically smooth kernels such as $G_0$ but much less attention has been devoted to the case of the 3D Helmholtz equation. We can mention the work [@stolper2004computing] where a collocation approach is considered for problems up to $20 \ 480$ DOFs and where encouraging numerical results are presented.
We denote $G^r_{\kappa}$ the Taylor expansion with $r:=| \{{\boldsymbol \alpha}\in \mathbb{N}^3_0: |{\boldsymbol \alpha}| \le m\}|$ terms of the Helmholtz Green’s function $G_{\kappa}$, i.e. $$G^r_{\kappa}({\boldsymbol{x}},{\boldsymbol{y}})=\sum_{{\boldsymbol \alpha}\in \mathbb{N}^3_0, |{\boldsymbol \alpha}| \le m } ({\boldsymbol{x}}-{\boldsymbol{x}}_0)^{\boldsymbol \alpha} \frac{1}{{\boldsymbol \alpha}!} \partial_x^{\boldsymbol \alpha}G_{\kappa}({\boldsymbol{x}}_0,{\boldsymbol{y}}) + R^r_{\kappa} \quad {\boldsymbol{x}}\in X, \ \ {\boldsymbol{y}}\in Y
\label{taylor_exp}$$ where ${\boldsymbol{x}}_0\in X$ is the centre of the expansion. According to the previous discussion, we know that, for the asymptotically smooth kernel $G_0$, the asymptotic smoothness allows to prove the exponential convergence of the Taylor series if $X$ and $Y$ are $\eta$-*admissible*. On the other hand for the Helmholtz kernel $G_{\kappa}$, the estimate (\[est\_Helmholtz\]) now leads to $$|R^r_{\kappa} ({\boldsymbol{x}},{\boldsymbol{y}})|\le C' \sum_{\ell=m}^{\infty} \Big((1 + \kappa ||{\boldsymbol{x}}_0 -{\boldsymbol{y}}||) \frac{ ||{\boldsymbol{x}}-{\boldsymbol{x}}_0||}{c_2 ||{\boldsymbol{x}}_0-{\boldsymbol{y}}||}\Big)^{\ell}.$$ Thus to assure convergence of the series, $X$ and $Y$ must now be chosen in such a way that $$\gamma_{\kappa,x}:= \displaystyle \ (1+\kappa ||{\boldsymbol{x}}_0- {\boldsymbol{y}}||)
\frac{\max_{{\boldsymbol{x}}\in X}||{\boldsymbol{x}}-{\boldsymbol{x}}_0||}{c_2 ||{\boldsymbol{x}}_0-{\boldsymbol{y}}||} <1.
\label{admi_cond_taylor2}$$ From now on, we say that $X$ and $Y$ are $\eta_{\kappa}$-*admissible* if (\[admi\_cond\_taylor2\]) is satisfied. Provided that the condition (\[admi\_cond\_taylor2\]) holds, the remainder is bounded $$\label{error_est_remain}
\begin{array}{rl}
|R^r_{\kappa}({\boldsymbol{x}},{\boldsymbol{y}})|\le &\displaystyle{C'
\frac{\gamma_{\kappa,x}^m}{1-\gamma_{\kappa,x}}}.
\end{array}$$ Remark that in the low-frequency regime, one has the estimate $\gamma_{\kappa,x} \sim
\frac{\max_{{\boldsymbol{x}}\in X}||{\boldsymbol{x}}-{\boldsymbol{x}}_0||}{c_2 ||{\boldsymbol{x}}_0-{\boldsymbol{y}}||}
$ such that the $\eta_{\kappa}$-admissibility condition is similar to the case of the asymptotically smooth kernel $s=G_0$. Within this framework, for higher frequencies, $\gamma_{\kappa,x}$ depends *linearly* on the wavenumber $\kappa$ and thus on the circular frequency $\omega$.
We do not discuss in this work on the choice of the optimal ${\boldsymbol{x}}_0$. It is very likely that the notions of $\eta$- and $\eta_{\kappa}$-admissibility can be improved by determining the optimal ${\boldsymbol{x}}_0$.
It is worth noting that a Taylor expansion with centre ${\boldsymbol{y}}_0\in Y$ instead of ${\boldsymbol{x}}_0\in X$ can be performed. In that case, a similar error estimate is obtained: $$\label{error_est_remain2}
\begin{array}{rl}
|R^r_{\kappa}({\boldsymbol{x}},{\boldsymbol{y}})|\le &\displaystyle{C'
\frac{\gamma_{\kappa,y}^m}{1-\gamma_{\kappa,y}}}, \quad \mbox{ with } \quad \gamma_{\kappa,y}:= \displaystyle \ (1+\kappa ||{\boldsymbol{y}}_0- {\boldsymbol{x}}||)
\frac{\max_{{\boldsymbol{y}}\in Y}||{\boldsymbol{y}}-{\boldsymbol{y}}_0||}{c_2 ||{\boldsymbol{y}}_0-{\boldsymbol{x}}||}.
\end{array}$$
Theoretical estimate of the rank of 3D elastodynamics single-layer and double-layer potentials \[taylor\_elasto\]
-----------------------------------------------------------------------------------------------------------------
The results on the convergence of the Taylor expansion of the 3D Helmholtz Green’s function are easily extended to 3D elastodynamics. From (\[elasto\_U\]) it is clear that the 3D elastodynamic Green’s tensor is a combination of derivatives of the 3D Helmholtz Green’s tensor. In particular, rewriting (\[elasto\_U\]) component by component, we have $$({\bf U}_{\omega})_{\alpha \beta}({\boldsymbol{x}},{\boldsymbol{y}})=\frac{1}{\kappa_s^2 \mu} \Big((\delta_{\tau \sigma} \delta_{ \alpha \beta}-\delta_{ \beta \tau}\delta_{\alpha \sigma}) \frac{\partial}{\partial x_{\tau}} \frac{\partial}{\partial y_\sigma}G_{\kappa_s}({\boldsymbol{x}},{\boldsymbol{y}})+ \frac{\partial}{\partial x_{\alpha}} \frac{\partial}{\partial y_{\beta}}G_{\kappa_p}({\boldsymbol{x}},{\boldsymbol{y}}) \Big)$$ $$({\bf T}_{\omega})_{\alpha \beta}({\boldsymbol{x}},{\boldsymbol{y}})= C_{\alpha \gamma \tau \sigma} \frac{\partial}{\partial y_{\sigma}} \Big({\bf U}_{\omega}({\boldsymbol{x}},{\boldsymbol{y}})\Big)_{\beta \tau} n_{\gamma}({\boldsymbol{y}})$$ where the Einstein summation convention is used. Since $\kappa_p \le \kappa_s$, it follows that each component of the 3D elastodynamics Green’s tensor satisfies an inequality similar to (\[est\_Helmholtz\]) with $\kappa=\kappa_s$, i.e. $$\label{est_elasto}
|\partial_z^n ({\bf P})_{\alpha \beta}({\boldsymbol{x}},{\boldsymbol{y}})| \le n! c_1 (1+\kappa_s ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^n (c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-n -\sigma}$$ where ${\bf P}={\bf U}_{\omega}$ or ${\bf P}={\bf T}_{\omega}$. As a result, provided that (\[error\_est\_remain\]) or (\[error\_est\_remain2\]) holds with $\kappa=\kappa_s$ the Taylor expansion of the 3D elastodynamic Green’s tensor converges exponentially with convergence rate $\gamma_{\kappa_s,x}$ or $\gamma_{\kappa_s,y}$. However again two regimes can be distinguished. In the *low-frequency* regime, the Taylor expansion behaves similarly to the case of the asymptotically smooth kernel $G_0$. The exponential convergence is ensured if $X$ and $Y$ are $\eta$-*admissible* and if $\eta$ is chosen such that $\eta < c_2$. For higher frequencies, $\gamma_{\kappa_s,x}$ and $\gamma_{\kappa_s,y}$ defined in (\[error\_est\_remain\]) and (\[error\_est\_remain2\]) depend linearly on the circular frequency $\omega$.
The definition of the [$\eta_{\kappa}$-admissibility]{} for oscillatory kernels could be modified in order to keep the separation rank constant while the frequency increases. It is the option followed for example in directional approaches [@messner2012fast; @delamotte2016etude]. Another approach to avoid the rank increase is the concept of $\mathcal{H}^2$-matrices [@banjai2008hierarchical]. The option we consider is to work with the $\eta$-admissibility condition defined for the asymptotically smooth kernels $G_0$. In other words, we choose to keep $\eta$ constant for all frequencies and use the admissibility condition defined by: $$\operatorname{diam}X \le \eta \operatorname{dist}(X,Y) \quad \mbox{ with } \eta<c_2.
\label{admi_G}$$ Our goal is to determine what is the actual behavior of the algorithm in that configuration. Following the study of the remainder of the Taylor expansion, it is clear that the method will not be optimal since the separation rank to achieve a given accuracy will increase with the frequency. Nevertheless, we are interested in determining the limit until which the low-frequency approximation is viable for oscillatory kernels such as the 3D Helmholtz Green’s function or the 3D elastodynamic Green’s tensors.
The relevant question to address is then to determine the law followed by the growth of the numerical rank if the $\eta$-admissibility condition (\[admi\_G\]) is used in the context of oscillatory kernels. Importantly, [ the framework of ]{} Taylor expansions is only used to illustrate the degenerate kernel in an abstract way [and to determine the optimal admissibility conditions in the low and high-frequency regimes within this framework]{}. In practice the ACA [which is a purely algebraic method]{} is used after discretization. [As a result, we will use linear algebra arguments to study the growth of the numerical rank.]{} As a matter of fact, one expects a small numerical rank provided that the entries of the matrix are approximated by a short sum of products. From the Taylor expansion (\[taylor\_exp\]) this can be achieved if the discretization is accurate enough. Consider two clusters of points $X=({\boldsymbol{x}}_i)_{i \in \tau}$ and $Y=({\boldsymbol{y}}_j)_{j \in \sigma}$ and the corresponding subblock $\mathbb{A}_{\tau \times \sigma}$. Then using (\[taylor\_exp\]) we can build an approximation of $\mathbb{A}_{\tau \times \sigma}$ of the form $\mathbb{A}_{\tau \times \sigma} \approx \mathbb{B}_{\tau \times \sigma} =\mathbb{U}_{\tau } \mathbb{V}^*_{ \sigma}$ where $\mathbb{U}_{\tau }$ corresponds to the part that depends on ${\boldsymbol{x}}$ and $\mathbb{V}_{ \sigma}$ to the part that depends on ${\boldsymbol{y}}$.
In the following, $\kappa$ denotes either the wavenumber in the context of the 3D Helmholtz equation, or also the S-wavenumber $\kappa_s$ in the context of 3D elastodynamics.
To start, let us have a look at two clusters of points $X$ and $Y$ which are $\eta$-*admissible* under (\[admi\_G\]). At the wavenumber $\kappa$ two cases may follow, whether condition (\[admi\_cond\_taylor2\]) is also satisfied or not.
Let’s consider first that the condition (\[admi\_cond\_taylor2\]) is not satisfied at the wavenumber $\kappa$, i.e. if the frequency is too large, with respect to the diameters of the clusters, to consider that this block is $\eta_{\kappa}$-*admissible*. From the study of the Taylor expansion, it follows that the numerical rank $r(\kappa)$ of the corresponding block in the matrix at wavenumber $\kappa$ will be large and the algorithm will not be efficient.
Let’s consider now the case where $X$ and $Y$ are such that the condition (\[admi\_cond\_taylor2\]) is also satisfied at the wavenumber $\kappa$. In that case, the numerical rank $r(\kappa)$ will be small.
These two cases illustrate the so-called *low-frequency* regime (latter case), i.e. when $(1+\kappa ||{\boldsymbol{x}}_0- {\boldsymbol{y}}||) \sim 1$ and *high-frequency* regime (former case), i.e. when $(1+\kappa ||{\boldsymbol{x}}_0- {\boldsymbol{y}}||) \sim \kappa ||{\boldsymbol{x}}_0- {\boldsymbol{y}}||$.
[In this work however, we are not interested by only one frequency but rather by a frequency range. The key point is to remark that in the $\eta_{\kappa}$-admissibility condition, the important quantity is $\kappa ||{\boldsymbol{x}}_0- {\boldsymbol{y}}||$, i.e. the product between the wavenumber and the diameter of the blocks (we recall that the $\eta$-admissibility (\[admi\_G\]) relates the distances between blocks and the diameters of the blocks). Let’s now consider the following configuration: at the wavenumber $\kappa$ all the blocks of a matrix are such that the $\eta$-admissibility includes the $\eta_{\kappa}$-admissibility while at the wavenumber $2\kappa$, it is not the case. This configuration represents the transition regime after the *low-frequency* regime. From (\[error\_est\_remain\]) and (\[error\_est\_remain2\]) it follows that if the frequency doubles, and as a result the wavenumber doubles from $\kappa$ to $\kappa'=2\kappa$, an $\eta_{\kappa}$-*admissible* block ${\mathbb A}_{\tau \times \sigma}$ should be decomposed into a $2 \times 2$ block matrix at wavenumber $\kappa'$ to reflect the reduction by a factor $2$ of the leading diameter. Thus the subblocks ${\mathbb A}_{\tau_i \times \sigma_j}$ are shown by the Taylor expansion to be of the order of the small rank $r(\kappa)$ at the wavenumber $\kappa'$, i.e. $${\mathbb A}_{\tau \times \sigma}=\left[ \begin{array}{cc}
{\mathbb A}_{\tau_1 \times \sigma_1} &{\mathbb A}_{\tau_1 \times \sigma_2}\\
{\mathbb A}_{\tau_2 \times \sigma_1} & {\mathbb A}_{\tau_2 \times \sigma_2}\\
\end{array}
\right] \approx
\left[ \begin{array}{cc}
{\mathbb B}_{\tau_1 \times \sigma_1} &{\mathbb B}_{\tau_1 \times \sigma_2}\\
{\mathbb B}_{\tau_2 \times \sigma_1} & {\mathbb B}_{\tau_2 \times \sigma_2}\\
\end{array}
\right]$$ where each ${\mathbb B}_{\tau_i \times \sigma_j}$ ($1 \le i,j \le 2$) is a low-rank approximation of rank $r(\kappa)$ that can be written as ${\mathbb B}_{\tau_i \times \sigma_j}={\mathbb U}_{\tau_i \times \sigma_j} {\mathbb S}_{\tau_i \times \sigma_j} {\mathbb V}_{\tau_i \times \sigma_j}^*$ and each $ {\mathbb S}_{\tau_i \times \sigma_j}$ is a $r(\kappa) \times r(\kappa)$ matrix. What is more, the Taylor expansion (\[taylor\_exp\]) says that ${\mathbb U}_{\tau_i \times \sigma_j} $ is independent of $ \sigma_j$ such that ${\mathbb U}_{\tau_i \times \sigma_j} ={\mathbb U}_{\tau_i } $: $${\mathbb A}_{\tau \times \sigma}\approx \left[ \begin{array}{cc}
{\mathbb U}_{\tau_1} & {\mathbb O} \\
{\mathbb O} & {\mathbb U}_{\tau_2}\\
\end{array}
\right]
\left[ \begin{array}{cc}
{\mathbb S}_{\tau_1 \times \sigma_1} {\mathbb V}^*_{\tau_1 \times \sigma_1} & {\mathbb S}_{\tau_1 \times \sigma_2} {\mathbb V}^*_{\tau_1 \times \sigma_2}\\
{\mathbb S}_{\tau_2 \times \sigma_1} {\mathbb V}^*_{\tau_2 \times \sigma_1} & {\mathbb S}_{\tau_2 \times \sigma_2} {\mathbb V}^*_{\tau_2 \times \sigma_2}\\
\end{array}
\right].$$ Since one can perform the Taylor expansion also with respect to ${\boldsymbol{y}}$, ${\mathbb A}_{\tau \times \sigma}$ is in fact of the form $${\mathbb A}_{\tau \times \sigma}\approx \left[ \begin{array}{cc}
{\mathbb U}_{\tau_1} & {\mathbb O} \\
{\mathbb O} & {\mathbb U}_{\tau_2}\\
\end{array}
\right]
\left[ \begin{array}{cc}
{\mathbb S}_{\tau_1 \times \sigma_1} & {\mathbb S}_{\tau_1 \times \sigma_2}\\
{\mathbb S}_{\tau_2 \times \sigma_1} & {\mathbb S}_{\tau_2 \times \sigma_2} \\
\end{array}
\right]
\left[ \begin{array}{cc}
{\mathbb V}^*_{\sigma_1} & {\mathbb O} \\
{\mathbb O} & {\mathbb V}^*_{\sigma_2}\\
\end{array}
\right]$$ As a result, ${\mathbb A}_{\tau \times \sigma}$ is numerically at most of rank $2r(\kappa)$ at the wavenumber $\kappa'$. ]{} The same reasoning can be extended to higher frequencies such that the maximum rank of all the blocks of a given matrix is at most increasing linearly with the frequency.
So far, we have assumed that all the blocks of a matrix satisfy the same conditions. However due to the hierarchical structure of $\mathcal{H}$-matrix representations (i.e. subblocks in the matrix at different levels in the partition $\mathcal{P}$ corresponding to clusters with different diameters), it is very likely that the blocks of a matrix at a given wavenumber will not be all in the *low-frequency* regime nor all in the *high-frequency* regime. In particular assuming the frequency is not *too high* and provided the hierarchical subdivision is deep enough, there will always be some blocks in the *low-frequency* regime, i.e. blocks for which the rank is low numerically since the $\eta_{\kappa}$-*admissibility* condition is satisfied. Then for $\eta$-*admissible* blocks at higher levels, the previous discussion on the transition regime can be applied [by induction]{} such that the numerical rank of these blocks is expected to grow linearly through the various levels. [Note that this linear increase depends on the choice of the number of points at the leaf level. The crux is to initialize the induction process: for a given choice, one has a certain limit for the transition regime. If one changes the number of points at the leaf level, the limit changes too. As a result, one cannot continue the process to infinity. To a given choice of number of points per leaf, there is a limit frequency.]{}
From all these arguments, we conclude that if we consider a fixed geometry at a fixed frequency and we increase the density of discretization points per wavelength, and as a consequence the total number of discretization points, the maximum rank of the blocks can be either small or large depending on the value of the product between the frequency and the diameter of the blocks (fixed by the geometry). However we expect that this rank will remain fixed if the density of discretization points per wavelength increases: as a matter of fact, the Taylor expansion with $r$ given, is better and better approximated, so the rank should converge to $r$. This conclusion will be confirmed by numerical evidences in Section 7. On the other hand, if we consider a fixed geometry with a fixed density of points per S-wavelength, the maximum rank of the blocks is expected to grow linearly with the frequency until the *high-frequency regime* is achieved. This transition regime is in some sense a *pre-asymptotic regime* and we demonstrate numerically in Section 8 its existence. In the *high-frequency* regime, the $\mathcal{H}$-matrix representation is expected to [be suboptimal with a rank rapidly increasing.]{}
[This paper presents an original algebraic argument to intuit the existence of this pre-asymptotic regime. It would be interesting in the future to compare this pre-asymptotic regime with other more involved hierarchical matrix based approaches proposed for oscillatory kernels, e.g. directional $\mathcal{H}^2$-matrices and with the multipole expansions used in the Fast Multipole Method. Another interesting but difficult question is to determine theoretically the limits of the pre-asymptotic regime.]{}
Implementation issues \[impl\_issues\]
======================================
Efficient implementation of the $\eta$-admissibility condition\[impl\_issues1\]
-------------------------------------------------------------------------------
A key tool in hierarchical matrix based methods is the efficient determination of *admissible* blocks in the matrix. The first remark from an implementation point of view comes from the possibility to perform the Taylor expansion of 3D elastodynamic Green’s tensors either with respect to ${\boldsymbol{x}}\in X=({\boldsymbol{x}}_i)_{i \in \tau}$ or with respect to ${\boldsymbol{y}}\in Y=({\boldsymbol{y}}_j)_{j \in \sigma}$. When a Taylor expansion with respect to ${\boldsymbol{x}}$ is performed, the clusters $X$ and $Y$ are said to be $\eta$-*admissible* if $\operatorname{diam}X \le \eta \operatorname{dist}(X,Y)$. On the other hand, when a Taylor expansion with respect to ${\boldsymbol{y}}$ is performed, the clusters $X$ and $Y$ are said to be $\eta$-*admissible* if $\operatorname{diam}Y \le \eta \operatorname{dist}(X,Y)$. Thus the admissibility condition at the discrete level becomes $$\mbox{The block } \tau \times \sigma \mbox{ is \emph{admissible}} \quad \mbox{ if } \mbox{min}\Big(\mbox{diam}(X), \mbox{diam}(Y) \Big) < \hat{\eta} \times \mbox{dist}(X,Y)
\label{admissibility_cond}$$ where $\hat{\eta}$ is a numerical parameter of the method discussed in the next section but subjected to some constraints highlighted in Section \[taylor\_elasto\].
Since the computation of the diameter of a set $X$ with (\[diam\]) is an expensive operation [@hackbusch2015hierarchical], it is replaced by computing the diameter of the bounding box with respect to the cartesian coordinates (Fig. \[distances\]b) $$\mbox{diam}(X) \le \Big( \sum_{\alpha=1}^3 (\max_{{\boldsymbol{x}}\in X} {\boldsymbol{x}}_{\alpha}- \min_{{\boldsymbol{x}}\in X} {\boldsymbol{x}}_{\alpha})^2 \Big)^{1/2}=: \mbox{diam}_{\operatorname{box}}(X)$$ which is an upper bound. Similarly the distance between two sets is replaced by the distance between the closest faces of the bounding boxes $$\mbox{dist}(X,Y) \ge \Big( \sum_{\alpha=1}^3 (\min_{{\boldsymbol{x}}\in X}{\boldsymbol{x}}_{\alpha} -\max_{{\boldsymbol{y}}\in Y}{\boldsymbol{y}}_{\alpha} )^2+ (\min_{{\boldsymbol{y}}\in Y}{\boldsymbol{y}}_{\alpha} -\max_{{\boldsymbol{x}}\in X}{\boldsymbol{x}}_{\alpha} )^2 \Big)^{1/2}=: \mbox{dist}_{\operatorname{box}}(X,Y)$$ which is a lower bound. The practical admissibility condition writes $$\mbox{The block } \tau \times \sigma \mbox{ is \emph{admissible}} \quad \mbox{ if } \mbox{min}\Big(\mbox{diam}_{\operatorname{box}}(X), \mbox{diam}_{\operatorname{box}}(Y) \Big) < \hat{\eta} \times \mbox{dist}_{\operatorname{box}}(X,Y).
\label{admissibility_cond_bis}$$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Efficient implementation of the admissibility condition: (a) true condition and (b) more efficient approach using upper and lower bounds. []{data-label="distances"}](./distances "fig:") ![Efficient implementation of the admissibility condition: (a) true condition and (b) more efficient approach using upper and lower bounds. []{data-label="distances"}](./distances2 "fig:")
(a) (b)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The admissibility conditions (\[admissibility\_cond\]) or (\[admissibility\_cond\_bis\]) imply that $X$ and $Y$ must be disjoint and that their distance is related to their diameters. Also, one can check [@hackbusch2015hierarchical Th. 6.16] that the sparsity pattern $C_{sp}$ is in fact bounded by a constant number that depends only on the geometry.
In addition, the choice of the parameter $\hat{\eta}$ influences the shape of the *admissible* blocks and the number of *admissible* blocks. On the one hand, from the study of the Taylor expansion of the Green’s function, it is clear that the smaller the parameter $\hat{\eta}$, the faster the remainder tends to zero. On the other hand, a parameter $\hat{\eta}$ that tends to zero means that only blocks of very small diameter (with respect to the distance) are *admissible*. The choice of a large $\hat{\eta}$ means in practice that very elongated blocks are allowed. From an algorithmic point of view, our goal is to find a good compromise: i.e. not only to have a large number of *admissible* blocks but also to ensure the convergence of the Taylor expansion. As a result, we want to choose $\hat{\eta}$ as large as possible.
Since an upper bound of the diameter and a lower bound of the distance are used, it is clear that even though a theoretical bound on $\eta$ is found from the study of (\[admi\_G\]), larger values of $\hat{\eta}$ can be used and still lead to accurate low rank block approximations in practice. Then, the distances and diameters are defined according to the cartesian coordinates cf. (\[admissibility\_cond\_bis\]). But depending on the choice of clustering method adopted, it is possible that the distance between some blocks will be very small even though the blocks are disjoints. This remark weighs in favor of the use of a numerical parameter $\hat{\eta}$ larger than the one predicted by the theory.
Choice of an acceptable parameter $\hat{\eta}$
----------------------------------------------
To determine the adequate parameter $\hat{\eta}$, it is necessary to determine the constant $c_2$ in (\[admi\_cond\_taylor2\]) that gives the upper bound on $\eta$. This constant comes exclusively from the asymptotically smooth kernel $G_0$. It is easy to determine that $$\forall z \in \{x_{\alpha},y_{\alpha}\} \quad \partial_z^1 G_0({\boldsymbol{x}},{\boldsymbol{y}}) \le \frac{1}{||{\boldsymbol{x}}-{\boldsymbol{y}}||^2} \quad \mbox{ such that } \quad 1\le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-2}.$$ Using Maple, it follows similarly that $$2\le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-3}, \ 4\le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-4},\ \frac{17}{2} \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-5} ,\ \frac{37}{2} \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-6},$$ $$\begin{aligned}
41 \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-7}, \ 92 \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-8}, \ \frac{5001}{24} \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-9},\\
\ \frac{3803}{8} \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-10}, \ \frac{4363}{4} \le c_1(c_2 ||{\boldsymbol{x}}-{\boldsymbol{y}}||)^{-11}.
\end{aligned}$$ In Table \[reg\_const\], we report the values of $c_1$ and $c_2$ obtained if a power regression up to the order $m$ is performed with $2 \le m \le 10$. We observe that $c_2$ is smaller than $0.5$ and decreases if $m$ is increased. [As a result, in theory we should use $\eta<c_2<0.5$.]{} Note that we have limited our study up to $m=10$ because $m\sim r^{1/3}$, where we recall that $r$ is the number of terms in the Taylor expansion (\[taylor\_exp\]). We expect in practice to consider cases with a maximum numerical rank lower than $1000$ such that the current study is representative of the encountered configurations.
m 2 3 4 5 6 7 8 9 10
--------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
$ c_1$ $0.250$ $0.250$ $0.238$ $0.226$ $0.216$ $0.206$ $0.198$ $0.191$ $0.184$
$ c_2 $ $0.500$ $0.500$ $0.490$ $0.483$ $0.476$ $0.470$ $0.465$ $0.461$ $0.458$
: Values of $c_1$ and $c_2$ obtained after a power regression for values of $m$ between $2$ and $10$.[]{data-label="reg_const"}
[In Section \[impl\_issues1\], we have explained why the use of an upper bound of the diameter and a lower bound of the distance in the admissibility condition permits to expect in practice good compression rates even though a larger value $\hat{\eta} >\eta$ is used.]{} Indeed in [@lize], good numerical results for various geometries are obtained with $\hat{\eta}=3$. This is the parameter we use in all our numerical examples for 3D elastodynamics.
Behavior for low frequency elastodynamics \[fixed\_freq\]
=========================================================
Definition of the test problem
-------------------------------
In Sections \[fixed\_freq\] and \[fixed\_dens\], we consider the diffraction of vertical incident plane P waves by a spherical cavity. The material properties are fixed to $\mu=\rho=1$ and $\nu=1/3$. We recall that in all the numerical examples, the binary tree $\mathcal{T}_I$ is built with a stopping criteria $N_{\mbox{leaf}}=100$ and the constant in the admissibility condition (\[admissibility\_cond\]) is set to $\hat{\eta}=3$. Unless otherwise indicated, the required accuracy is fixed to $\varepsilon_{\operatorname{ACA}}=10^{-4}$. In the following, we denote by $N$ the number of degrees of freedom such that $N=3N_c$.
The admissibility condition (\[admissibility\_cond\]) depends only on the geometry of the domain. In Figure \[rep\_mat\], we illustrate the repartition of the expected low-rank (green blocks) and full blocks (red blocks) for the sphere geometry with $N= 30 \ 726$.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illustration of the expected repartition of the full and low-rank blocks for the sphere with $N= 30 \ 726$.[]{data-label="rep_mat"}](./mesh_sphere "fig:"){width="5cm"} ![Illustration of the expected repartition of the full and low-rank blocks for the sphere with $N= 30 \ 726$.[]{data-label="rep_mat"}](./matrix_sphere "fig:")
$N = 30 \ 726$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------
The goal of Section 7 is to determine the behavior of the algorithm for a fixed frequency while the number of discretization points is increased (i.e. in the *low-frequency* regime). In the remainder of this section, the circular frequency $\omega$ is thus fixed. As a result, the density of points per S-wavelength increases as the number of discretization points increases.
Storage requirements for the single and double-layer potentials for a fixed frequency
-------------------------------------------------------------------------------------
In Section \[theory\], we have seen that the memory requirements are expected to be of the order of $O(\max(r^{\max}_{\operatorname{ACA}}, N_{\operatorname{leaf}} )N \log_2 N)$. In addition, for a fixed frequency, $r^{\max}_{\operatorname{ACA}}$ is expected to be constant. In Tables \[max\_rank\_frequency\] and \[max\_rank\_frequency2\], we report the maximum numerical rank observed among all the *admissible* blocks for the case of a sphere for a fixed circular frequency ($\omega=3$ or $\omega=14$) and for various mesh sizes, for the single-layer and double-layer operators. The first rank corresponds to the numerical rank obtained by the partially pivoted ACA. The second rank corresponds to the numerical rank obtained after the recompression step (presented in Section \[iter\_solver\]). The two ranks are seen to be almost constant while the density of points is increasing, as intuited by the study of the Taylor expansion. The numerical rank without the recompression step is much larger than after the optimisation step. The explanation is that the partially pivoted ACA is a heuristic method, the rank is thus not the optimal one. The recompression step permits to recover the intrinsic numerical rank of the blocks. In addition, as expected the maximum numerical rank increases with the frequency. We will study in more details this dependence on the frequency in Section 8. Finally also as expected, since the two Green’s tensors satisfy the inequality (\[est\_elasto\]), similar numerical ranks are observed for the single and double layer potentials.
$N$ $7 \ 686$ $30 \ 726$ $122 \ 886$ $183 \ 099$ $ 490 \ 629$ $763 \ 638$ $ 985 \ 818$
------------- ----------- ------------ ------------- ------------- --------------- ------------- --------------
$\omega=3$ 63/39 72/39 75/39 69/37 75/39 69/40 78/39
$\omega=14$ 99/73 105/75 108/76 99/67 114/76 111/76 111/76
: Maximum numerical rank observed (before/after the recompression step) for a fixed frequency, i.e. while increasing the density of points per S-wavelength, for $\varepsilon_{\operatorname{ACA}}=10^{-4}$ and the single-layer potential (\[EFIE\]).[]{data-label="max_rank_frequency"}
$N$ $7 \ 686$ $30 \ 726$ $122 \ 886$ $183 \ 099$ $ 490 \ 629$ $763 \ 638$ $ 985 \ 818$
------------- ----------- ------------ ------------- ------------- --------------- ------------- --------------
$\omega=3$ 66/37 66/37 72/38 66/36 69/38 72/39 75/39
$\omega=14$ 102/70 117/74 117/74 102/66 114/75 114/75 114/76
: Maximum numerical rank observed (before/after the recompression step) for a fixed frequency, i.e. while increasing the density of points per S-wavelength, for $\varepsilon_{\operatorname{ACA}}=10^{-4}$ and the double-layer potential (\[MFIE\]).[]{data-label="max_rank_frequency2"}
Ultimately we are not interested in the maximum numerical rank but rather on the memory requirements of the algorithm. In Figures \[graph\_mem\_frequency\][**a-b**]{}, we report the memory requirements $N_s$ and the compression rate $\tau(\mathcal{H}):= N_s / N^{2}$ with respect to the number of degrees of freedom $N$ for $\varepsilon_{\operatorname{ACA}}=10^{-4}$ and the single-layer operator. Since the rank is constant, $N_s$ is expected to be of the order of $N \log_2 N $ and $\tau(\mathcal{H})$ of the order of $\log_2 N / N$. We observe for the two frequencies a very good agreement between the expected and observed complexities of the compression rate and memory requirements. Note that the storage is reduced by more than $95 \%$ as soon as $N \ge 10^{5}$.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Observed and estimated ([**a**]{}) memory requirements $N_s$ and ([**b**]{}) compression rate $\tau(\mathcal{H})$ with respect to the number of degrees of freedom $N$ for a fixed frequency ($\omega=3$ or $\omega=14$) for the single layer-operator and $\varepsilon_{\operatorname{ACA}}=10^{-4}$.[]{data-label="graph_mem_frequency"}](./res_freq3b "fig:") ![ Observed and estimated ([**a**]{}) memory requirements $N_s$ and ([**b**]{}) compression rate $\tau(\mathcal{H})$ with respect to the number of degrees of freedom $N$ for a fixed frequency ($\omega=3$ or $\omega=14$) for the single layer-operator and $\varepsilon_{\operatorname{ACA}}=10^{-4}$.[]{data-label="graph_mem_frequency"}](./res_freq3 "fig:")
([**a**]{}) ([**b**]{})
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Certification of the results of the direct solver
-------------------------------------------------
While the previous subsection was dedicated to the study of the numerical efficiency of the method with respect to memory requirements, this subsection is more focused on the accuracy of the direct solver presented in Section \[lu\_solve\]. Since the $\mathcal{H}$-LU factorization is based on various levels of approximations, it is important to check the accuracy of the final solution. In Section \[esti\_direct\], we have derived a simple estimator to certify the results of the $\mathcal{H}$-LU direct solver.
Before we study the accuracy, we recall that there are two important parameters in the method that correspond to two sources of error: $\varepsilon_{\operatorname{ACA}}$ and $\varepsilon_{\operatorname{LU}}$. $\varepsilon_{\operatorname{ACA}}$ is tuned to the desired accuracy of the low-rank approximations of *admissible* blocks, performed with the ACA described in Section \[ACA\_vect\]. $\varepsilon_{\operatorname{LU}}$ is the parameter used to adapt the accuracy when recompressions are performed to modify the format of a block (Section \[lu\_solve\]). It is clear that the best accuracy that can be achieved by the direct solver is driven by $\varepsilon_{\operatorname{ACA}}$. For this reason, the two parameters are set to the same value in the following numerical results.
In Table \[error\_freq\_single\], we report the value of the estimator $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ and the observed normalized residual $\frac{||{\bf b}-\mathbb{A}{\bf x}_0||_2}{ ||{\bf b} ||_2}$ if (\[EFIE\]) is solved with the $\mathcal{H}$-LU based direct solver. Two frequencies ($\omega=3$ and $\omega=14$) and four meshes are considered consistently with the previous subsection. When no compression is possible ($\tau(\mathcal{H})=1$ for $N=1 \ 926$) the estimated and observed residuals are, as expected, of the order of the machine accuracy. For larger problems, the difference between the two residuals is not negligible. But importantly, when the required accuracy is reduced by two orders of magnitude, the two residuals follow the same decrease. In addition, for a fixed value of $\varepsilon_{\operatorname{ACA}}$, the accuracy of the solver and of the estimator does not deteriorate much while the number of degrees of freedom increases.
To understand the lack of accuracy of our estimator, we add in Table \[error\_freq\_single\] the three components of the estimator: $\delta_{\mathcal{H},F}$ which corresponds to the level of error introduced by the low-rank approximation in the hierarchical representation of the system matrix (computed with the Frobenius norm), the ratio $\frac{||{\bf x}_0||_2}{||{\bf b}||_2}$ and the normalized ratio $\frac{\delta}{||{\bf b}||_2}$ that accounts for the stability of the $\mathcal{H}$-LU factorization. On the one hand the contribution from $\frac{\delta}{||{\bf b}||_2}$ is lower that the normalized residual that we try to estimate. On the other hand, $ \delta_{\mathcal{H},F}$ is of the order of the required accuracy $\varepsilon_{\operatorname{ACA}}$. But since this term is multiplied by the ratio $\frac{||{\bf x}_0||_2}{||{\bf b}||_2}$ the part in the estimator accounting for the error introduced by the low-rank approximations is overestimated.
$\#$ DOFs $\omega$ $\varepsilon_{\operatorname{ACA}}=\varepsilon_{\operatorname{LU}}$ $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ $\frac{||{\bf b}-\mathbb{A}{\bf x}_0||_2}{ ||{\bf b} ||_2}$ $\delta_{\mathcal{H},F}$ $\frac{||{\bf x}_0||_2}{||{\bf b}||_2}$ $\frac{\delta}{||{\bf b}||_2}$
------------ ---------- -------------------------------------------------------------------- ---------------------------------------------- ------------------------------------------------------------- -------------------------- ----------------------------------------- --------------------------------
$1 \ 926$ $3$ $10^{-4}$ $3.25 \ 10^{-15}$ $3.64 \ 10^{-15}$ $9.16 \ 10^{-17}$ $6.40$ $2.66 \ 10^{-15}$
$7 \ 686$ $3$ $10^{-4}$ $4.64 \ 10^{-4}$ $9.94 \ 10^{-6}$ $7.21 \ 10^{-5}$ $6.37$ $4.62 \ 10^{-6}$
$30 \ 726$ $3$ $10^{-4}$ $6.37 \ 10^{-4}$ $1.18 \ 10^{-5}$ $9.93 \ 10^{-5}$ $6.36$ $5.53 \ 10^{-6}$
$122\ 886$ $3$ $10^{-4}$ $8.14 \ 10^{-4}$ $1.47 \ 10^{-5}$ $1.27 \ 10^{-4}$ $6.36$ $5.21 \ 10^{-6}$
$1 \ 926$ $3$ $10^{-6}$ $3.25 \ 10^{-15}$ $3.64 \ 10^{-15}$ $9.16 \ 10^{-17}$ $6.40$ $2.66 \ 10^{-15}$
$7 \ 686$ $3$ $10^{-6}$ $5.07 \ 10^{-6}$ $ 7.35 \ 10^{-8}$ $ 7.92 \ 10^{-7}$ $6.37$ $ 2.48 \ 10^{-8}$
$30 \ 726$ $3$ $10^{-6}$ $ 7.44 \ 10^{-6}$ $ 9.38 \ 10^{-8}$ $ 1.16\ 10^{-6}$ $6.36 $ $ 2.93 \ 10^{-8}$
$122\ 886$ $3$ $10^{-6}$ $9.25 \ 10^{-6}$ $9.82 \ 10^{-8}$ $1.45 \ 10^{-6}$ $6.36 $ $2.76 \ 10^{-8}$
$1 \ 926$ $14$ $10^{-4}$ $5.69 \ 10^{-15}$ $4.69 \ 10^{-15}$ $7.16 \ 10^{-17}$ $29.29$ $3.59 \ 10^{-15}$
$7 \ 686$ $14$ $10^{-4}$ $3.89 \ 10^{-3}$ $1.04 \ 10^{-4}$ $1.36 \ 10^{-4}$ $28.15$ $4.73 \ 10^{-5}$
$30 \ 726$ $14$ $10^{-4}$ $5.49 \ 10^{-3}$ $1.29 \ 10^{-4}$ $1.95 \ 10^{-4}$ $27.88$ $5.48 \ 10^{-5}$
$122\ 886$ $14$ $10^{-4}$ $6.24 \ 10^{-3}$ $1.49 \ 10^{-4}$ $2.22 \ 10^{-4}$ $27.82$ $5.86 \ 10^{-5}$
$1 \ 926$ $14$ $10^{-6}$ $5.69 \ 10^{-15}$ $4.69 \ 10^{-15}$ $7.16 \ 10^{-17}$ $29.29$ $3.59 \ 10^{-15}$
$7 \ 686$ $14$ $10^{-6}$ $4.59 \ 10^{-5}$ $9.15 \ 10^{-7}$ $1.62 \ 10^{-6}$ $28.15$ $2.03 \ 10^{-7}$
$30 \ 726$ $14$ $10^{-6}$ $5.95 \ 10^{-5}$ $1.15 \ 10^{-6}$ $2.12 \ 10^{-6}$ $27.88$ $2.44 \ 10^{-7}$
$122\ 886$ $14$ $10^{-6}$ $6.42 \ 10^{-5}$ $1.26 \ 10^{-6}$ $2.30 \ 10^{-6}$ $27.82$ $2.52 \ 10^{-7}$
: Accuracy of the direct solver for a fixed frequency ($\omega=3$ or $\omega=14$) for the single-layer potential.[]{data-label="error_freq_single"}
To accelerate the computation of the estimator, we have used the Frobenius norm. To check the effect of the use of this norm, we report in Table \[error\_freq\_single2\] the values of the estimator $\mathcal{I}(\delta_{\mathcal{H},2},\delta)$ i.e. when the 2-norm is used. We consider only moderate size problems due to the need of the computation of the singular value decomposition of the complete matrix. As expected, the estimator $\mathcal{I}(\delta_{\mathcal{H},2},\delta)$ is much more accurate than $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ but its cost is prohibitive. A good compromise consists in checking only the two main components of the estimator: $ \delta_{\mathcal{H},F}$ and $\frac{\delta}{||{\bf b}||_2}$ to check that the level of error introduced in the different parts of the solver is consistent with the user parameter $\varepsilon_{\operatorname{ACA}}=\varepsilon_{\operatorname{LU}}$.
$\#$ DOFs $\omega$ $\varepsilon_{\operatorname{ACA}}=\varepsilon_{\operatorname{LU}}$ $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ $\mathcal{I}(\delta_{\mathcal{H},2},\delta)$ $\frac{||{\bf b}-\mathbb{A}{\bf x}_0||_2}{ ||{\bf b} ||_2}$ $\delta_{\mathcal{H},F}$ $\delta_{\mathcal{H},2}$
------------ ---------- -------------------------------------------------------------------- ---------------------------------------------- ---------------------------------------------- ------------------------------------------------------------- -------------------------- --------------------------
$7 \ 686$ $3$ $10^{-4}$ $4.64 \ 10^{-4}$ $7.91 \ 10^{-5}$ $9.94 \ 10^{-6}$ $7.21 \ 10^{-5}$ $1.17 \ 10^{-5}$
$30 \ 726$ $3$ $10^{-4}$ $6.37 \ 10^{-4}$ $5.46 \ 10^{-5}$ $1.18 \ 10^{-5}$ $9.93 \ 10^{-5}$ $7.72 \ 10^{-6}$
$7 \ 686$ $3$ $10^{-6}$ $5.07 \ 10^{-6}$ $7.32 \ 10^{-7}$ $ 7.35 \ 10^{-8}$ $ 7.92 \ 10^{-7}$ $1.11 \ 10^{-7}$
$30 \ 726$ $3$ $10^{-6}$ $ 7.44 \ 10^{-6}$ $8.50 \ 10^{-7}$ $ 9.38 \ 10^{-8}$ $ 1.16\ 10^{-6}$ $1.29 \ 10^{-7}$
$7 \ 686$ $14$ $10^{-4}$ $3.89 \ 10^{-3}$ $6.32 \ 10^{-4}$ $1.04 \ 10^{-4}$ $1.36 \ 10^{-4}$ $2.08 \ 10^{-5}$
$30 \ 726$ $14$ $10^{-4}$ $5.49 \ 10^{-3}$ $9.50 \ 10^{-4}$ $1.29 \ 10^{-4}$ $1.95 \ 10^{-4}$ $3.21 \ 10^{-5}$
$7 \ 686$ $14$ $10^{-6}$ $4.59 \ 10^{-5}$ $1.07 \ 10^{-5}$ $9.15 \ 10^{-7}$ $1.62 \ 10^{-6}$ $3.72 \ 10^{-7}$
$30 \ 726$ $14$ $10^{-6}$ $5.95 \ 10^{-5}$ $1.20 \ 10^{-5}$ $1.15 \ 10^{-6}$ $2.12 \ 10^{-6}$ $4.20 \ 10^{-7}$
: Accuracy of the direct solver for a fixed frequency ($\omega=3$ or $\omega=14$) for the single-layer potential when the 2-norm is used instead of the Frobenius norm.[]{data-label="error_freq_single2"}
In Table \[error\_freq\_double\], we consider now the case where the double-layer potential based on equation (\[MFIE\]) is used and solved with the $\mathcal{H}$-LU direct solver. It is clear that the behaviors of the single and double layer potentials are similar also in terms of numerical accuracy.
$\#$ DOFs $\omega$ $\varepsilon_{\operatorname{ACA}}=\varepsilon_{\operatorname{LU}}$ $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ $\frac{||{\bf b}-\mathbb{A}{\bf x}_0||_2}{ ||{\bf b} ||_2}$ $\delta_{\mathcal{H},F}$ $\frac{||{\bf x}_0||_2}{||{\bf b}||_2}$ $\frac{\delta}{||{\bf b}||_2}$
------------ ---------- -------------------------------------------------------------------- ---------------------------------------------- ------------------------------------------------------------- -------------------------- ----------------------------------------- --------------------------------
$1 \ 926$ $14$ $10^{-4}$ $3.35 \ 10^{-15}$ $3.28 \ 10^{-15}$ $5.22 \ 10^{-16}$ $1.26$ $2.69 \ 10^{-15}$
$7 \ 686$ $14$ $10^{-4}$ $2.49 \ 10^{-3}$ $7.16 \ 10^{-5}$ $1.94 \ 10^{-3}$ $1.27$ $2.58 \ 10^{-5}$
$30 \ 726$ $14$ $10^{-4}$ $3.18 \ 10^{-3}$ $6.92 \ 10^{-5}$ $2.43 \ 10^{-3}$ $1.30$ $3.05 \ 10^{-5}$
$122\ 886$ $14$ $10^{-4}$ $9.23 \ 10^{-3}$ $5.70 \ 10^{-3}$ $2.66 \ 10^{-3}$ $1.33$ $5.70 \ 10^{-3}$
$1 \ 926$ $14$ $10^{-6}$ $3.35 \ 10^{-15}$ $3.28 \ 10^{-15}$ $5.22 \ 10^{-16}$ $1.26$ $2.69 \ 10^{-15}$
$7 \ 686$ $14$ $10^{-6}$ $2.43 \ 10^{-5}$ $4.77 \ 10^{-7}$ $1.90 \ 10^{-5}$ $1.27$ $9.85 \ 10^{-8}$
$30 \ 726$ $14$ $10^{-6}$ $3.31 \ 10^{-5}$ $5.25 \ 10^{-7}$ $2.54 \ 10^{-5}$ $1.30$ $1.15 \ 10^{-7}$
$122\ 886$ $14$ $10^{-6}$ $5.74 \ 10^{-3}$ $5.70 \ 10^{-3}$ $3.17 \ 10^{-5}$ $1.33$ $5.70 \ 10^{-3}$
: Accuracy of the direct solver for a fixed frequency ($\omega=14$) for the double-layer potential.[]{data-label="error_freq_double"}
All these numerical experiments confirm the theoretical study of the Taylor expansion, which yields that $\mathcal{H}$-matrix based solvers are very efficient tools for the *low-frequency* elastodynamics.
Towards highly oscillatory elastodynamics \[fixed\_dens\]
=========================================================
The goal of Section 8 is to determine the behavior of the algorithm for a given fixed density of points per S-wavelength while the number of discretization points is increased (i.e. in the *higher frequency* regime). The circular frequency $\omega$ is thus fitted to the mesh length to keep the density of points per S-wavelength fixed for the different meshes. In this case, an increase of the number of degrees of freedom $N$ leads to an increase of the size of the body to be approximated. In Table \[table\_corr\] we report the number of degrees of freedom, corresponding circular frequencies and number of S-wavelengths spanned in the sphere diameter used in the examples.
$\omega$ 8.25 16.5 33 40 66 84 92
------------------ ----------- ------------ ------------- ------------- --------------- ------------- --------------
$N$ $7 \ 686$ $30 \ 726$ $122 \ 886$ $183 \ 099$ $ 490 \ 629$ $763 \ 638$ $ 985 \ 818$
$n_{ \lambda_s}$ 2.6 5.2 10.5 12.6 21 26.6 29.1
: Number of degrees of freedom, corresponding number of wavenumbers spanned in the sphere diameter $D$ ($D=n_{ \lambda_s} \lambda_s$) on the different meshes of spheres and corresponding circular frequencies used in the examples of Section 8 (i.e for a fixed density of points per S-wavelength).[]{data-label="table_corr"}
Storage requirements for the single and double-layer potentials for a fixed density of points per S-wavelength
--------------------------------------------------------------------------------------------------------------
In Section \[taylor\_elasto\], it has been shown that for a fixed geometry with a fixed density of points per S-wavelength, the maximum rank of the $\eta$-*admissible* blocks is expected to grow linearly with the frequency until the *high-frequency regime* is achieved. The goal of this section is to validate the existence of this pre-asymptotic regime. In Figure \[rank\_fixed\_freq\][**a**]{}, we report the maximum numerical rank observed among all the $\hat{\eta}$-*admissible* blocks before and after the recompression step with respect to the circular frequency $\omega$, for both the single and double layer potentials. In accordance to the study of the Taylor expansion, the growth of the numerical rank after the recompression step (i.e. the intrinsic rank of the blocks) is seen to be linear with respect to the circular frequency. The behavior of the single layer and double layer operators are again very similar.
In practice however, we are interested in the complexity with respect to the number of degrees of freedom. In this section, the number of discretization points is fitted to the circular frequency in order to keep 10 discretization points per S-wavelength. Since the density of points is fixed, the mesh size is $h=O(\lambda_S)=O(\omega^{-1})$. In addition since the mesh for the BEM is a surface mesh, the mesh size is $h=O(N^{-2})$. Hence, the maximum numerical rank is expected to be of the order of $O(N^{1/2})$. In Figure \[rank\_fixed\_freq\][**b**]{}, we report the maximum numerical rank observed among all the $\hat{\eta}$-*admissible* blocks before and after the recompression step with respect to the number of DOFs $N$, for both the single and double layer potentials. The growth of the numerical rank after the recompression step (i.e. the intrinsic rank of the blocks) is indeed seen to be of the order of $O(N^{1/2})$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Maximum numerical rank observed (before/after the recompression step) for a fixed density of points per wavelength, i.e. while increasing the circular frequency with the number of discretization points, for $\varepsilon_{\operatorname{ACA}}=10^{-4}$: [**(a)**]{} with respect to the circular frequency and [**(b)**]{} with respect to the number of degrees of freedom.[]{data-label="rank_fixed_freq"}](./max_rank_fixed_dens_freq "fig:") ![Maximum numerical rank observed (before/after the recompression step) for a fixed density of points per wavelength, i.e. while increasing the circular frequency with the number of discretization points, for $\varepsilon_{\operatorname{ACA}}=10^{-4}$: [**(a)**]{} with respect to the circular frequency and [**(b)**]{} with respect to the number of degrees of freedom.[]{data-label="rank_fixed_freq"}](./max_rank_fixed_dens "fig:")
[**(a)**]{} [**(b)**]{}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Ultimately we are not interested in the maximum numerical rank but rather on the memory requirements of the algorithm. In Figures \[mem\_fixed\_freq\][**a-b**]{}, we report the memory requirements $N_s$ and compression rate $\tau(\mathcal{H})$, with respect to the number of degrees of freedom $N$ for $\varepsilon_{\operatorname{ACA}}=10^{-4}$. Since the rank is of the order of $O(N^{1/2})$, $N_s$ is expected to be of the order of $N^{3/2} \log_2 N $ and $\tau(\mathcal{H})$ of the order of $\log_2 N / N^{1/2}$. In practice, observed complexities are lower than the expected one. The reason is that the estimation of the memory requirements gives only an upper bound based on the estimation of the maximum numerical rank. But, all the *admissible* blocks do not have the same numerical rank such that the complexity is lower than the estimated one.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Observed and estimated ([**a**]{}) memory requirements $N_s$ and ([**b**]{}) compression rate $\tau(\mathcal{H})$ with respect to the number of degrees of freedom $N$ for a fixed density of points per S-wavelength for the single and double layer-operators and $\varepsilon_{\operatorname{ACA}}=10^{-4}$.[]{data-label="mem_fixed_freq"}](./mem_req_fixed_dens "fig:") ![Observed and estimated ([**a**]{}) memory requirements $N_s$ and ([**b**]{}) compression rate $\tau(\mathcal{H})$ with respect to the number of degrees of freedom $N$ for a fixed density of points per S-wavelength for the single and double layer-operators and $\varepsilon_{\operatorname{ACA}}=10^{-4}$.[]{data-label="mem_fixed_freq"}](./mem_req_fixed_dens_tau "fig:")
[**(a)**]{} [**(b)**]{}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Certification of the results of the direct solver for higher frequencies
------------------------------------------------------------------------
Once the existence of the *pre-asymptotic regime* is established, and thus the efficiency of the hierarchical representation of the matrix to reduce computational times and memory requirements, is demonstrated, this subsection is devoted to the study of the accuracy of the method in this regime.
In Table \[error\_dens\_single\], we report the values of the estimator $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ and the observed normalized residual $\frac{||{\bf b}-\mathbb{A}{\bf x}_0||_2}{ ||{\bf b} ||_2}$ if equation (\[EFIE\]) is solved with the $\mathcal{H}$-LU based direct solver. Four meshes are considered and the density of points per S-wavelength is fixed consistently with the previous subsection. When no compression is possible ($\tau(\mathcal{H})=1$ for $N=1 \ 926$) the estimated and observed residuals are, as expected, of the order of the machine accuracy. When the required accuracy is reduced by two orders of magnitude, the two residuals follow the same decrease. The estimator computed in Frobenius norm is again not very sharp and the solver could appear to be less accurate when the frequency increases. But the discrepancy between the true system matrix and its $\mathcal{H}$-matrix representation is seen to be of the order of $\varepsilon_{\operatorname{ACA}}$ even though computed in Frobenius norm. The increase of the estimator is due to the increase of the ratio $\frac{||{\bf x}_0||_2}{||{\bf b}||_2}$. In addition, the ratio $\frac{\delta}{||{\bf b}||_2}$ indicates the very good stability of the $\mathcal{H}$-LU factorization.
$\#$ DOFs $\omega$ $\varepsilon_{\operatorname{ACA}}=\varepsilon_{\operatorname{LU}}$ $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ $\frac{||{\bf b}-\mathbb{A}{\bf x}_0||_2}{ ||{\bf b} ||_2}$ $\delta_{\mathcal{H},F}$ $\frac{||{\bf x}_0||_2}{||{\bf b}||_2}$ $\frac{\delta}{||{\bf b}||_2}$
------------ ---------- -------------------------------------------------------------------- ---------------------------------------------- ------------------------------------------------------------- -------------------------- ----------------------------------------- --------------------------------
$1 \ 926$ $4$ $10^{-4}$ $3.12 \ 10^{-15}$ $3.42 \ 10^{-15}$ $9.07 \ 10^{-17}$ $8.37$ $2.36 \ 10^{-15}$
$7 \ 686$ $8.25$ $10^{-4}$ $1.73 \ 10^{-3}$ $4.28 \ 10^{-5}$ $1.02 \ 10^{-4}$ $16.83$ $2.23 \ 10^{-5}$
$30 \ 726$ $16.5$ $10^{-4}$ $6.42 \ 10^{-3}$ $1.68 \ 10^{-4}$ $1.94 \ 10^{-4}$ $32.66$ $7.04 \ 10^{-5}$
$1 \ 926$ $4$ $10^{-6}$ $3.12 \ 10^{-15}$ $3.42 \ 10^{-15}$ $9.07 \ 10^{-17}$ $8.37$ $2.36 \ 10^{-15}$
$7 \ 686$ $8.25$ $10^{-6}$ $1.79 \ 10^{-5}$ $2.61 \ 10^{-7}$ $1.06 \ 10^{-6}$ $16.83$ $9.63 \ 10^{-8}$
$30 \ 726$ $16.5$ $10^{-6}$ $7.14 \ 10^{-5}$ $1.36 \ 10^{-6} $ $2.18 \ 10^{-6}$ $32.66$ $2.93 \ 10^{-7}$
: Comparison of the estimation of the error and the actual error of the direct solver for a fixed density of ten points per S-wavelength for the single-layer potential.[]{data-label="error_dens_single"}
These results confirm the numerical efficiency and accuracy of $\mathcal{H}$-matrix based solvers even for higher frequencies. The estimator $\mathcal{I}(\delta_{\mathcal{H},F},\delta)$ gives a rough idea of the accuracy of the solver. But it is better to check separately the two components of the error.
The computational times of the iterative and direct solvers have not been compared due to the difficulty to perform a fair and exhaustive comparison. On the one hand, the iterative solver can easily be optimized and parallelized but its performances depend largely on the quality of the preconditioner used to reduce the number of iterations. On the other hand, the optimization of the direct solver is a more involved operation that requires a task-based parallelization. Our experience with the two solvers is that the computational time of the direct solver is increasing very rapidly as a function of the number of DOFs if no parallelization is performed. However, the preconditioning of iterative BEM solvers for oscillatory problems is still an open question.
Conclusion and future work
==========================
In this work, $\mathcal{H}$-matrix based iterative and direct solvers have been successfully extended to 3D elastodynamics in the frequency domain. To perform the low-rank approximations in an efficient way, the Adaptive Cross Approximation has been modified to consider problems with vector unknowns. In addition, the behavior of the hierarchical algorithm in the context of oscillatory kernels has been carefully studied. It has been shown theoretically and then numerically confirmed that the maximum numerical rank among all the $\eta$-*admissible* blocks is constant if the frequency is fixed and the density of discretization points per wavelength is increased (i.e. for the low-frequency regime). On the other hand, if the density of points per S-wavelength is fixed, as it is usually the case for higher frequency problems, the growth of the maximum numerical rank among all the $\eta$-*admissible* blocks is limited to $O(N^{1/2})$. Then, since the $\mathcal{H}$-LU factorization is based on various levels of approximations, a simple estimator to certify the results of the $\mathcal{H}$-LU direct solver has been derived. It has been shown that not only the accuracy of the $\mathcal{H}$-matrix representation can be monitored by a simple parameter but also the additional error introduced by the $\mathcal{H}$-LU direct solver. Globally, when combined with the Boundary Element Method formulation, the $\mathcal{H}$-matrix representation of the system matrix permits to drastically reduce computational costs in terms of memory requirements and computational times for the frequency range that can be considered with the current available computational resources.
Applications of the present $\mathcal{H}$-matrix accelerated BEM to 3D realistic cases in seismology are underway. Moreover, a natural extension of this work will be to determine the efficiency of this approach for isotropic and anisotropic visco-elastic problems.
Acknowledgments {#acknowledgments .unnumbered}
================
The authors would like to thank Shell for funding the PhD of Luca Desiderio.
References {#references .unnumbered}
==========
[^1]: stephanie.chaillat@ensta-paristech.fr
[^2]: luca.desiderio@yahoo.it
[^3]: patrick.ciarlet@ensta-paristech.fr
[^4]: Recall that $||{\mathbb B}||_2 \le ||{\mathbb B}||_F$ for all square matrices.
|
---
author:
- Pierre Suret
- Rebecca El Koussaifi
- Alexey Tikan
- Clément Evain
- Stéphane Randoux
- Christophe Szwaj
- Serge Bielawski
title: Direct observation of Rogue Waves in optical turbulence using Time Microscopy
---
[**The formation of coherent structures in noise driven phenomena and in Turbulence is a complex and fundamental question [@Sirovich:87]. A particulary important structure is the so-called Rogue Wave (RW) that arises as the sudden appearance of a localized and giant peak [@Onorato:01; @Pelinovskybook2; @Onorato:13]. First studied in Oceanography, RWs have been extensively investigated in Optics since 2007 [@Solli:07], in particular in optical fibers experiments on supercontinua [@Solli:07; @Erkintalo:09; @Mussot:09] and optical turbulence [@Walczak:15; @Onorato:13]. However the typical timescales underlying the random dynamics in those experiments prevented –up to now– the direct observation of isolated RWs. Here we report on the direct observation of RWs, using an ultrafast acquisition system equivalent to microscope in the time domain [@Kolner:89; @Bennett:99; @Foster:08; @Okawachi:12]. The RWs are generated by nonlinear propagation of random waves inside an optical fiber, and recorded with $\sim
250$ fs resolution. Our experiments demonstrate the central role played by “breathers-like” solutions of the one-dimensional nonlinear Schrödinger equation (1D-NLSE) in the formation of RWs [@Akhmediev:09].**]{}
Common oceans waves are weakly nonlinear random objects having nearly Gaussian statistics, while at the same time, RWs are waves of extremely large amplitude that occur more frequently than expected from the normal law [@Onorato:01; @Pelinovskybook2; @Onorato:13]. The mechanisms underlying the generation of coherent structures such as RWs from the nonlinear propagation of random waves i.e., in turbulent flows, is a subject of very active debates and still represents an open question [@Hammani:10; @Walczak:15; @Agafontsev:15; @Toenger:15].
From the theoretical point of view, the so-called [*focusing*]{} 1D-NLSE (see Eq. \[eq:NLS1D\]), which is a generic equation having an ubiquitous importance in Physics, plays a central role in this debate [@Onorato:01; @Onorato:13; @Akhmediev:13; @Dudley:14]. In particular, the 1D-NLSE describes at leading order the physics of deep-water wave trains and nonlinear propagation in optical fibers [@Chabchoub:15]. The breather-like solutions of the 1D-NLSE, also called solitons on finite background, are now considered as being prototypes of RWs [@Akhmediev:09; @Akhmediev:13; @Kibler:10; @Chabchoub:11; @Kibler:12].
Despite the numerous experimental works devoted to optical RWs, [@Solli:07; @Erkintalo:09; @Mussot:09; @Hammani:08; @Walczak:15; @Onorato:13; @Dudley:14] the direct observation of these coherent structures in the time domain has never been reported in the context of the nonlinear propagation of [*random waves*]{}. Randomness of the initial condition is known to play a crucial role in the generation of RWs as it has been pointed out in the supercontinuum driven by noise [@Solli:07; @Mussot:09; @Erkintalo:09; @Dudley:14] or in optical turbulence [@Walczak:15; @Onorato:13; @Dudley:14; @Picozzi:14]
Contrary to the experiments performed in the spatial domain [@Bromberg:10; @Pierangeli:15], the fast time scales of fluctuations (picoseconds or less) involved in single-mode fiber experiments makes single-shot recording of RWs a particularly challenging task. Pioneer works hence naturally provided [*indirect evidences*]{} of RWs, using e.g. spectral filtering [@Solli:07; @Erkintalo:09; @Mussot:09] or statistical measurement from optical sampling techniques [@Walczak:15].\
In this Letter, we present direct single-shot recordings of optical RWs by using a specially-designed [*Time Microscope*]{} (TM) ultrafast acquisition system [@Kolner:89; @Bennett:99; @Okawachi:12]. The temporal resolution of $\sim$250 fs of our TM (see Methods and supplementary material) allows us to investigate the fast dynamics arising from the nonlinear propagation of random waves in an optical fiber (upper part of Fig. \[fig:1\]). Observations performed with the TM at the output of the fiber immediately reveals the emergence of intense peaks, with powers frequently exceeding the average power $\langle P \rangle$ by factors of 10-50 \[see Fig \[fig:1\](b-d) and supplementary movie 1\]. Starting from random fluctuations having typical time scale around $5-10$ps \[Fig \[fig:1\](a)\], those extreme peaks are also found to be extremely narrow, with time scales of the order of several hundreds of femtoseconds \[Fig \[fig:1\](b-d) and Fig \[fig:dynamics\](b-d)\].\
{width="17cm"}
More precisely, the random waves used as initial conditions in our experiments are partially coherent light waves emitted by a high power Amplified Spontaneous Emission (ASE) light source at a wavelength $\lambda\sim1560$ nm (see Fig. \[fig:1\]). Using a programmable optical filter, the optical spectrum of the partially coherent light is precisely designed to assume a Gaussian shape having a full width at half maximum that is adjusted either to $\Delta \nu =$[0.1]{}THz or $\Delta \nu =$[0.05]{}THz. The partially coherent waves are launched into a 500 m-long single mode polarization maintaining fiber at a wavelength falling into the anomalous (focusing) regime of dispersion. The light at the output of the nonlinear fiber is then directed to the TM (detailed in Figure \[fig:setup\]), which acquires traces (optical power versus time over a $\approx {20}$ ps-long window) at a rate of 500 per second, and displays the signals in real time.
As in a standard spatial imaging microscope, the TM is composed of an objective and a tube lens \[see Fig. \[fig:setup\].(b)\]. The objective is a time lens [@Kolner:89; @Bennett:99; @Okawachi:12] operating from sum-frequency generation (SFG) between the 1560 nm signal and a chirped pump pulse (at 800 nm) (see Sec. Methods). The observation in the focal plane of the tube lens is achieved by a spectrum analyser (composed of a diffraction grating, a lens and a camera). [The use of this TM strategy enables to easily reach extremely high dynamical ranges (up to $40$ dB, see Methods), which is a crucial point for analyzing extreme events embedded in moderate power fluctuations.]{}
{width="17cm"}
In order to quantify the emergence of RWs, we compute statistical distributions from a large amount of data recorded with the TM at the input and at the output ends of the fiber (see Sec. Methods). As expected, the probability density function (PDF) of the optical power emitted by the ASE source is systematically very close to the exponential distribution that correspond to a Gaussian statistics for the field \[see Fig. \[fig:dynamics\].(a)\] [@Goodman:85; @Walczak:15; @Agafontsev:15]. On the contrary, the PDF of light power at the output of the nonlinear fiber is found to exhibit heavy-tailed deviations from the exponential distribution, thus confirming the generation of RWs \[see Fig. \[fig:dynamics\].(a)\]. Moreover the PDFs demonstrate that the number of RWs having high peak power increases while the mean power $\langle P \rangle$ of random optical waves ([*i.e.*]{} the strength of nonlinearity) increases \[see Fig. \[fig:dynamics\].(a)\].
{width="17cm"}
To the best of our knowledge, experimental signals plotted in Figs. \[fig:1\].(a-d) and \[fig:dynamics\].(b-h) represent the first direct and accurate observation of the RWs underlying these heavy tailed statistics. Starting from random light propagating with a mean power of 4 W in the fiber, huge RWs having peak power that exceeds 300 W can be observed at the output of the fiber \[Fig.\[fig:dynamics\].(d) \]. From a careful analysis of the data, two typical shapes can be distinguished : isolated breather-like RWs \[see Fig. \[fig:dynamics\].(b)\] and more complicated structures composed of several peaks \[see Fig. \[fig:dynamics\].(c)\]. In order to illustrate the process of emergence of breather like RWs, we plot in Fig. \[fig:dynamics\].(e-h) different structures observed after the propagation of partially coherent waves having an initial spectral width $\Delta \nu = {0.05}$THz and an average power $\langle P \rangle= 0.3$W. We have selected these structures because their shapes are strikingly similar to those found in the scenario leading to the formation of the Peregrine soliton (PS) while starting from a single hump at initial stage [@Bertola:13]. Remarkably, the [*power*]{} profile of the exact analytical PS coincide very well with experimental structures \[see green dashed line in \[fig:dynamics\].(h)\]. However note that the precise identification of the breathers-like structures (as PS or Akhmediev breather or other more complex solutions of 1D-NLSE) would require a simultaneous measurement of the phase dynamics [@Randoux:15].\
Behaviors observed in experiments can be well reproduced from numerical simulations of the 1D-NLSE (see Sec. Methods). First of all, the PDFs of optical power \[see Fig. \[fig:num\].(a)\] and the optical spectra (see Supplementary Material) are well reproduced by numerical simulations. Fig. \[fig:num\] shows a picture of typical random fluctuations of the optical power that are found at the input and output ends of the optical fiber. Taking a partially-coherent light field having a bandwidth of ${0.1}$ THz at initial stage, the typical time scale for power fluctuations is around a few picoseconds \[Fig. \[fig:num\](b)\]. The scenari observed in the experiments are also found in numerical simulations. In particular, either breather-like structures appear and disappear along the propagation \[see Fig.\[fig:num\](g)\], either several pulses simultaneously emerge together from the random background \[see Fig. \[fig:num\](c)\]. Our experiments provide snapshots randomly recorded while the numerical simulations allow to follow the dynamics of nonlinear random waves along the propagation. In this respect, the numerical simulations reveal that the breather-like structures often emerge on the top of the initial power fluctuations (see Fig. \[fig:num\](d-g) and video in Supplementary Material).\
{width="17cm"}
In the last years, the common and shared conjecture is that breather-like solutions of 1D-NLSE such as PS or Akhmediev breathers represent prototypes of RWs [@Akhmediev:09; @Akhmediev:13; @Kibler:10; @Chabchoub:11; @Kibler:12; @Dudley:14; @Toenger:15]. This has motivated very nice experiments in which these solitons on finite background have been generated in a deterministic way in optical fibers [@Kibler:10; @Kibler:12; @Frisquet:13] and in a one-dimensional water tank [@Chabchoub:11]. These experiments make use of carefully-designed [*coherent initial conditions*]{}.
On the contrary, the initial conditions in our experiments are designed to be “ocean-like” [*random waves*]{} [@Onorato:13; @Janssen:03]. In this context of nonlinear propagation of random waves, previous experimental works performed in a 1D water tank [@Onorato:04] and in optical fibers [@Walczak:15] have revealed heavy-tailed deviations from Gaussian statistics. For the first time, our time-resolved observations correlate in an unambiguous way the occurrence of this heavy-tailed statistics with the frequent occurence of breather-like coherent structures. Our experimental observations favor a well-known scenario in which a PS emerges from a single real hump [@Bertola:13]. However it must be emphasized that the precise identification of coherent structures requires the precise knowledge of the phase evolution. The simultaneaous fast measurement of phase and amplitude fluctuations therefore represents the next experimental bottleneck for the careful identification of optical RWs.\
The emergence of coherent structures is a general and mysterious feature of stochastically driven processes such as turbulence, supercontinuum generation or pattern formation [@Solli:12; @Dudley:14; @Sirovich:87]. The time-resolved direct observation of RWs presented in the letter opens the way to numerous studies on the relationship betwen coherent structures and noise driven phenomena. In particular, shot by shot [*spectral*]{} measurements in pulsed experiments recently revealed the fascinating complexity of the statistical features associated to the so called modulation instability [@Solli:12]. By using our TM, the underlying dynamics of the so-called process of noise-driven modulational instability and its nonlinear stage is an open fundamental question that can now be studied [@Solli:12; @Agafontsev:15].
Methods {#methods .unnumbered}
=======
The partially coherent light (i.e., the initial condition) is generated by an Erbium fiber broadband Amplified Spontaneous Emission (ASE) source (Highwave), which is spectrally filtered (with programmable shape and linewidth) using a programmable optical filter ( Waveshaper 1000S, Finisar). The output is then amplified by an Erbium-doped fiber amplifier (Keopsys). This random light is launched into a single-mode polarization maintaining fiber (Fibercore HB-1550T), with 500 m length and a dispersion $\beta_2=-20$ ps$^2$km$^{-1}$ (measured). For a given spectral width, the power of the light launched inside the fiber is controlled using a half wavelength plate and a polarizing cube.
For the single-shot acquisition of the sub-picosecond optical signals, we realized an [*upconversion time-microscope*]{}, largely based on the work of Ref. [@Bennett:99]. From the input-output point of view, the time microscope encodes the temporal shape of the optical signal onto the spectrum of a chirped pulse (i.e., spectral encoding). Then the spectrum is recorded using a simple spectrometer composed of a 1800 l/mm grating and a sCMOS camera. A region of interest (of typically 2048x8 pixels) is selected for recording the image (a raw image is presented in Fig. 2 of the supplementary material).
For reaching high temporal resolution, a key element is the time-lens [@Kolner:89], which is composed by a BBO crystal, pumped by a chiped 800 nm pulse. Before entering the time lens, the 1560 nm signal experiences anomalous dispersion in a classic Treacy grating compressor (see Fig. \[fig:setup\]).
As in other time-lens systems [@Kolner:89; @Bennett:99; @Okawachi:12; @Foster:08] high resolution requires proper adjustement of the 1560 nm compressor (see supplemental material for adjustement detail, and performances of the setup). Conceptually, this is exactly analog to the tuning of the object-microscope objective distance in classical microscopes. For all results presented in this paper, the temporal resolution is 250 fs FWHM, and the field of view is of the order of 20 ps.
As another crucial point, the time-microscope strategy leads to an extremely high dynamical range (i.e., the ratio between maximal recordable signal and dark noise). This directly stems from the choice of employing a camera for the recording. More precisely, our 16 bit sCCD camera has an RMS dark noise of $\approx 2$ electrons and a saturation value of 30000 electrons, leading to a $\approx
40$ dB dynamical range.
The 800 nm pump is provided by an amplified Titanium-Sapphire laser (Spectra Physics Spitfire, 2 mJ, 40 fs, a spectral bandwidth of about 25 nm), operated at 500 Hz, and only 20 nJ are typically used here. For inducing (normal) dispersion on the 800 nm pulses we simply adjusted the amplifier’s output compressor. The dispersion was fixed to $0.23~$ps$^2$, leading to chirped pulses of duration of about $20~$ps. The 1560 nm grating compressor uses two 600 l/mm gratings, operated at an angle of incidence of 40 degrees, and whose planes are separated by 42 mm. The BBO crystal has 8 mm length and is cut for noncollinear type-I SFG. Focusing of the 800 nm and 1560 nm signals on the BBO crystal are performed by two lenses with 20 cm focal lengths. In order to improve the rejection of the 800 nm and the 1560 nm and to keep only the SFG at $528$ nm, a 40-nm bandpass filter around 531 nm (FF01 531/40-25 Semrock) is added after the crystal. The camera is a sCMOS Hamamatsu Orca flash 4.0 V2 (C11440-22U), equiped with a 80 mm lens (Nikkor Micro 60 mm f/2.8 AF-D). The objective is focused at infinity and the waist of the SFG in the BBO crystal is imaged on the camera sensor. The camera is synchronized on the 800 nm laser pulses, and the integration time is adjusted to 1 ms, thus enabling single-shot operation of the time-microscope. PDFs of optical power are computed with $75.10^6$ samples ($10^2$ points taken in the center of the temporal field of the TM from $75.10^4$ frames) for a given set of parameters.
Numerical simulations are performed by integrating the 1D-NLSE : $$\label{eq:NLS1D}
i\frac{\partial \psi}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 \psi}{\partial t^2}-\gamma|\psi|^2\psi,$$ where $\psi$ is complex envelope of the electric field, normalized so that $|\psi|^2$ is the optical power, $z$ is the longitudinal coordinate in the fiber, and $t$ is the retarded time. $\beta_2=-20$ ps$^2$km$^{-1}$ is the second-order dispersion coefficient of the fiber and $\gamma=2$ W$^{-1}$km$^{-1}$ is the Kerr coupling coefficient. All numerical integrations are performed using an adaptive stepsize pseudospectral method, using a mesh of 2048 points, over a temporal window of $\Delta T= 250$ps.
In numerical simulations presented in this letter, we neglect linear losses ($\simeq 0.5$dB) and stimulated Raman scattering. These approximations provide quantitative agreement between experiments and numerical simulations at moderate powers ($<2$W). Additionnal numerical simulations show that stimulated Raman scattering has to be taken into account in order to reproduce very precisely the experimental PDFs and optical spectra at high values of the mean power ([*i.e.*]{} $\langle P \rangle=4$W). However the main physical results (formation of RWs, emergence of breather-like structures and heavy-tailed PDFs) are not affected by stimulated Raman scattering.
The random complex field $\psi(t,z=0)$ used as initial condition in numerical simulations is made from a discrete sum of Fourier components : $$\label{ini_field}
\psi(z=0,t)=\sum\limits_{m} \widehat{X_{m}} e^{i m \Delta \omega t}.$$ with $\widehat{X_{m}}=\frac{1}{\Delta T} \int_0^{\Delta T} \psi(z=0,t) e^{-i m \Delta \omega t} dt$ and $\Delta \omega=2 \pi/\Delta T$. The Fourier modes $\widehat{X_{m}}=|\widehat{X_{m}}|e^{i \phi_{m}}$ are complex variables. We have used the so-called random phase (RP) model in which only the phases $\phi_{m}$ of the Fourier modes are considered as being random [@Nazarenko]. In this model, the phase of each Fourier mode is randomly and uniformly distributed between $-\pi$ and $\pi$. Moreover, the phases of separate Fourier modes are not correlated so that $<e^{i\phi_{n}}e^{i\phi_{m}}>= \delta_{nm}$ where $\delta_{nm}$ is the Kronecker symbol ($\delta_{nm}=0$ if $n\ne m$ and $\delta_{nm}=1$ if $n=m$). With the assumptions of the RP model above described, the statistics of the initial field is stationary, which means that all statistical moments of the complex field $\psi(z=0,t)$ do not depend on $x$ [@Picozzi:14]. In the RP model, the power spectrum $n_0(\omega)$ of the random field $\psi(z=0,t)$ reads as : $$\label{power_spectrum}
<\widehat{X_{n}}\widehat{X_{m}}>=n_{0n} \, \delta_{nm}=n_0(\omega_n).$$ with $\omega_n=n\, \Delta \omega$. In our simulations, we have taken a random complex field $\psi(z=0,t)$ having a Gaussian optical power spectrum that reads $$\label{gaussian_ci}
n_0(\omega)=n_0 \, \exp \left[- \left( \frac{\omega^2}{\Delta \omega^2} \right)
\right]$$ where $\Delta \omega=2 \pi \Delta \nu$ is the half width at $1/e$ of the power spectrum. Statistical properties of the random wave have been computed from Monte Carlo simulation made with an ensemble of $10^5$ realizations of the random initial condition.
[10]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
. ** ****, ().
, , & . ** ****, ().
, & ** (, ).
, , , & . ** ****, ().
, , & . ** ****, ().
, & . ** ****, ().
*et al.* . ** ****, ().
, & . ** ****, (). <http://link.aps.org/doi/10.1103/PhysRevLett.114.143903>.
& . ** ****, ().
& . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, (). <http://ol.osa.org/abstract.cfm?URI=ol-37-23-4892>.
, & . ** ****, ().
, , & . ** ****, ().
& . ** ****, (). <http://stacks.iop.org/0951-7715/28/i=8/a=2791>.
*et al.* . ** **** ().
, , & . ** ****, ().
, , & . ** ****, ().
*et al.* . ** ****, (). [http://www.sciencedirect.com/science/article/pii/S00034916150%
02687](http://www.sciencedirect.com/science/article/pii/S00034916150%
02687).
*et al.* . ** ****, ().
, & . ** ****, ().
*et al.* . ** **** ().
, , & . ** ****, ().
*et al.* . ** ****, ().
, , & . ** ****, ().
, , , & . ** ****, (). <http://link.aps.org/doi/10.1103/PhysRevLett.115.093901>.
** ().
& . ** ****, (). <http://dx.doi.org/10.1002/cpa.21445>.
, & . ** ().
, & . ** ****, ().
. ** ****, ().
*et al.* . ** ****, ().
, , & . ** ****, ().
**. Lecture Notes in Physics (, ).
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the French National Research Agency (ANR-12-BS04-0011 OPTIROC) and the BQR Émergence-Innovation of Lille 1 University. The authors are grateful to Francois Anquez and the Biophysics of Cellular Stress Response group of the PhLAM for the fruitful discussions, their crucial help, and for providing the sCMOS Camera. The authors are also grateful to Arnaud Mussot, Rémi Habert and the photonics group of the PhLAM for fruitful discussions, for the equipments (the ps laser), and for the measurement of the GVD of the fiber. The authors thank Nunzia Savoia for the everyday work on the femto laser and Marc Le Parquier for his crucial contribution in the development of the time-lens.
|
---
abstract: 'We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants $|R|^2$ and $|{{\operatorname{{Ric}}}}|^2$ of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó.'
address:
- 'Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain'
- 'Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany'
author:
- 'Teresa Arias-Marco'
- Dorothee Schueth
title: Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres
---
[^1]
Introduction
============
\[sec:intro\]
For a compact closed Riemannian manifold $S$ the [*spectrum*]{} of $S$ is the eigenvalue spectrum, including multiplicities, of the associated (positive semi-definite) Laplace operator $\Delta$ acting on smooth functions. A central question of inverse spectral geometry asks to which extent the geometry of $S$ is determined by its spectrum. The so-called [*heat invariants*]{} $a_k(S)$ of $S$ are examples of geometric invariants which are determined by the spectrum of $S$; indeed, they are the coefficients in the famous asymptotic expansion by Minakshisundaram-Pleijel, $${{\operatorname{Tr}}}(\exp(-t\Delta))\sim(4\pi t)^{-\dim(S)/2}\sum_{k=0}^\infty a_k(S) t^k$$ for $t\downarrow0$. The first few of these coefficients are given by $$a_0(S)={{\operatorname{vol}}}(S),\mbox{\ \ }
a_1(S)=\frac16\int_S{{\operatorname{{scal}}}}\,{{\operatorname{\textit{dvol}}}}_S\,,\mbox{\ \ }
a_2(S)=\frac1{360}\int_S(5{{\operatorname{{scal}}}}^2-2|{{\operatorname{{Ric}}}}|^2+2|R|^2){{\operatorname{\textit{dvol}}}}_S\,,$$ where ${{\operatorname{{scal}}}}$, ${{\operatorname{{Ric}}}}$, and $R$ denote the scalar curvature, the Ricci operator, and the Riemannian curvature operator of $S$, respectively. In general, each $a_k(S)$ is the integral over $S$ of certain curvature invariants; see [@Gi] for more information.
Nevertheless, there exist many examples of pairs or families of [*isospectral*]{} Riemannian manifolds (i.e., sharing the same spectrum) which are not isometric, sometimes not even locally isometric; see, for example, the survey article [@Go00]. Still, many questions remain open; for example, it is not known whether a locally symmetric compact closed Riemannian manifold can be isospectral to a locally nonsymmetric Riemannian manifold.
On the other hand, the geometry of [*geodesic spheres*]{} plays an interesting role in Riemannian geometry. Chen and Vanhecke [@CV] formulated the following general question: To what extent do the properties of small geodesic spheres determine the Riemannian geometry of the ambient space? For example, Gray and Vanhecke [@GV] studied the information contained in the volume function of small geodesic spheres and investigated the question whether a Riemannian manifold whose geodesic spheres have the same volumes as spheres in euclidean space must necessarily be flat (answering this question in the positive under various choices of additional assumptions).
In the context of inverse spectral geometry, an interesting special version of the above question is: To what extent do the spectra of small geodesic spheres in a (possibly noncompact) Riemannian manifold $M$ determine the geometry of $M$? For example, Theorem 6.18 in [@CV] uses the information contained in the heat invariants $a_0$ and $a_1$ of small geodesic spheres (viewed as functions of the radius) and concludes local isometry of manifolds with adapted holonomy to certain model spaces under the assumption that all small geodesic spheres around each point are isospectral to the corresponding geodesic spheres in those model spaces.
In order to arrive at such and similar results, one uses radial power series expansions of curvature invariants, both of the ambient space and of the geodesic spheres. In general, even the first few coefficients of such expansions become very complicated; see, for example, the various formulas in [@GV] or [@CV]. One setting in which a quite restrictive geometric assumption on the ambient space makes the calculations considerably easier is the setting of [*harmonic*]{} ambient spaces.
A manifold is called harmonic if the volume density function of the geodesic exponential map is radial around each point. The notion of harmonicity was first introduced by Copson and Ruse [@CR] and intensively studied by Lichnerowicz [@Li44]; see also [@RWW]. Chapter 6 of the book by Besse [@Be] gives a useful survey of properties of harmonic spaces. One of the important facts about harmonic spaces is that they are Einstein [@Be] and hence analytic [@DK] (the latter result was not yet known when [@Be] was written). A locally symmetric manifold is harmonic if and only if it is flat or of rank one; the famous Lichnerowicz conjecture postulated that, conversely, each harmonic space is locally symmetric; i.e., satisfies $\nabla R=0$ (this condition is classically known to be equivalent to the condition that the local geodesic symmetries around each point be isometries). For the case of compact manifolds with finite fundamental group the Lichnerowicz conjecture was proved by Szabó [@Sz90]; however, Damek and Ricci gave examples of noncompact homogeneous harmonic manifolds which are not locally symmetric in infinitely many dimensions greater or equal to seven [@DR]. These spaces are usually referred to as [*Damek-Ricci spaces*]{}; see [@BTV] for more information.
Specializing the above question about the information contained in the spectra of small geodesic spheres to the setting of harmonic spaces, we are able to prove in the present paper that the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric (see Corollary \[cor:main\] below). More precisely, we obtain:
\[thm:main\] Let $M_1$ and $M_2$ be harmonic spaces, and let $p_1\in M_1$, $p_2\in M_2$. If there exists ${{\varepsilon}}>0$ such that for each $r\in(0,{{\varepsilon}})$ the geodesic spheres $S_r(p_1)$ and $S_r(p_2)$ are isospectral, then $|\nabla R|_{p_1}^2=|\nabla R|_{p_2}^2$.
\[cor:main\] Let $M_1$ and $M_2$ be harmonic spaces. Assume that the hypothesis of Theorem \[thm:main\] is satisfied for *each* pair of points $p_1\in M_1$, $p_2\in M_2$. Then $M_1$ is locally symmetric if and only if $M_2$ is locally symmetric.
In particular, note that in the case of [*locally homogeneous*]{} harmonic spaces $M_1$ and $M_2$, the hypothesis of Theorem \[thm:main\] implies that $M_1$ is locally symmetric if and only if $M_2$ is locally symmetric. Actually, all known examples of harmonic spaces are locally homogeneous; it is an open question whether there exist harmonic spaces which are not locally homogeneous.
Interestingly, our result implies that certain pairs of geodesic spheres which were claimed to be isospectral by Szabó in [@Sz01], [@Sz05] are actually [*not*]{} isospectral. In fact, Szabó considered (as the featured examples in a more general construction) geodesic spheres in certain symmetric spaces $M_1$ (namely, quaternionic hyperbolic space of real dimension $4m\ge12$) and in certain associated locally nonsymmetric Damek-Ricci spaces $M_2$ of the same dimension (see also Remark \[rem:damekricci\]). He stated that every pair of geodesic spheres $S_r(p_1)\subset M_1$ and $S_r(p_2)\subset M_2$ of the same radius was isospectral. Since these ambient manifolds $M_1$ and $M_2$ are harmonic and homogeneous, and $M_1$ is locally symmetric while $M_2$ is not, Corollary \[cor:main\] immediately implies that Szabó’s result cannot be correct. Note that it was Fürstenau [@Fu] who first discovered that actually there was a gap in Szabó’s isospectrality argument. The question of whether that proof could be repaired or not had since remained open; our result settles this question in the negative.
The incorrect examples of geodesic spheres mentioned above had the notable property that one is homogeneous and the other not. While it remains unknown whether a homogeneous metric on a sphere can be isospectral to a non-homogeneous one, Szabó in an earlier article [@Sz99] did construct a pair of isospectral metrics, only one of which is homogeneous, on the product of a sphere and a torus (those results are not affected by the error in the later papers).
We obtain analogs of our above results for geodesic balls endowed with either Dirichlet or Neumann boundary conditions; see Theorem \[thm:balls\] and Corollary \[cor:balls\]. Similarly as above, this implies that Szabó’s examples in [@Sz01], [@Sz05] of isospectral geodesic balls (of any given radius) in quaternionic hyperbolic space of real dimension at least $12$ and in certain associated locally nonsymmetric Damek-Ricci spaces were erroneous.
Note that nevertheless there do exist isospectral pairs and even continuous families of isospectral metrics on spheres and balls; the first such examples were due to Gordon [@Go01].
In order to prove Theorem \[thm:main\] we use the heat invariants $a_0(S_r(p))$ and $a_2(S_r(p))$ of geodesic spheres in harmonic spaces. In particular, we study the coefficients of $r^2$ in the radial power series expansions of $\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)}|{{\operatorname{{Ric}}}}^S|^2{{\operatorname{\textit{dvol}}}}_{S_r(p)}$ and $\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)}|R^S|^2{{\operatorname{\textit{dvol}}}}_{S_r(p)}$, where ${{\operatorname{{Ric}}}}^S$ and $R^S$ denote the Ricci operator and the Riemannian curvature operator of $S_r(p)$. From the form of these coefficients (see Proposition \[prop:coeffs\] and its mean value version Proposition \[prop:intcoeffs\]), we are able to conlude that the heat invariants $a_0$ and $a_2$ of $S_r(p)$, viewed as functions of $r$, together determine the value of $|\nabla R|^2$ at the midpoint $p$. Note that the same is not true for $a_0(S_r(p))$ alone; see Remark \[rem:damekricci\]. Moreover, in the harmonic setting, the function $r\mapsto a_1(S_r(p))=\frac16\int_{S_r(p)}{{\operatorname{{scal}}}}^S\,{{\operatorname{\textit{dvol}}}}_{S_r(p)}$ does actually not contain more information than $r\mapsto a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))$; see Remark \[rem:volscal\]. So it is indeed necessary for our purpose to consider $a_2(S_r(p))$.
Our computations rely heavily on the harmonicity of the ambient space. Note that they are related to certain more general computations in [@GV] and [@CV]; for example, Theorem 8.1 of [@CV] actually includes a kind of analog to our Proposition \[prop:coeffs\], and this even for general, not only for harmonic manifolds; however, that theorem contains information only on the coefficients of $r^j$ with $j\le0$, while we need the coefficients of $r^2$. In fact, in the harmonic case, the lower order coefficients turn out to be determined already by the function $r\mapsto a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))$; see Proposition \[prop:coeffs\] and Remark \[rem:volscal\](i).
This paper is organized as follows:
In Section \[sec:prelim\] we gather the necessary background on harmonic spaces, mostly following [@Be]; in particular, we recall Ledger’s recursion formula for the power series expansion of the second fundamental form of geodesic spheres, and the resulting curvature identities in harmonic spaces.
In Section \[sec:curv\], we study the coefficients of $r^j$ for $j\le2$ in $|{{\operatorname{{Ric}}}}^{S_r(p)}|^2_{\exp(ru)}$ and $|R^{S_r(p)}|^2_{\exp(ru)}$ for unit tangent vectors $u$ of harmonic spaces, using the power series expansion of the second fundamental form and its radial covariant derivative, as well as the Taylor series expansion of the Riemannian curvature tensor. Proposition \[prop:coeffs\] is the main result of this section.
Section \[sec:proof\] is devoted to the proof of the Main Theorem \[thm:main\]. In preparation for this, we first derive a mean value version of Proposition \[prop:coeffs\]; see Proposition \[prop:intcoeffs\].
Finally, in Section \[sec:balls\], we prove the analog of Theorem \[thm:main\] for geodesic balls. We consider the heat coefficients of geodesic balls in harmonic spaces and show that the functions $r\mapsto a_0(B_r(p))$ and $r\mapsto a_2(B_r(p))$ (either for Dirichlet or for Neumann boundary conditions) together determine the value of $|\nabla R|^2$ at the midpoint $p$ of the balls. More precisely, we show that the coefficient of $r^3$ in the radial power series expansion of the quotient $a_2(B_r(p))/a_0(B_r(p))$ is a sum of a nonzero multiple of $|\nabla R|^2_p$ and of terms determined by the function $r\mapsto a_0(B_r(p))$.
Preliminaries
=============
\[sec:prelim\]
Volume density and the shape operator of geodesic spheres
---------------------------------------------------------
In the following, let $M$ be a complete, connected, $n$-dimensional Riemannian manifold. For $p\in M$, let $\exp_p=\exp{\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{T_pM}$}}:T_pM\to M$ denote the associated geodesic exponential map. For a vector $v\in T_pM$ we denote by $\gamma_v$ the geodesic with initial velocity $v$. Identifying $T_v(T_pM)$ with $T_pM$, we regard the differential $d(\exp_p)_v$ as a linear map from $T_pM$ to $T_{\exp v}M$. We denote parallel translation along $\gamma_v$ by $P_{\gamma_v}^{s,t}:T_{\gamma_v(s)}M\to T_{\gamma_v(t)}M$. Given any unit vector $u\in S_1(0_p):=\{u\in T_pM\mid |u|=1\}$ and $r\in{{\mathbb R}}$, we consider the volume density $$\theta_u(r):=
\det\bigl(P_{\gamma_u}^{r,0}\circ d(\exp_p)_{ru}\bigr).$$ Note that $\theta_u(r)$ is the infinitesimal volume distortion of the map $\exp_p$ at the point $ru\in T_pM$. Recall the Gauss lemma: The vector $d(\exp_p)_{ru} u$ is a unit vector perpendicular to each $d(\exp_p)_{ru} w$ with $w\perp u$. Thus, for each $r\in(0,i(p))$, where $i(p)$ denotes the injectivity radius of $M$ at $p$, $$\label{eq:vtheta}
v_u(r):=r^{n-1}\theta_u(r)$$ is the infinitesimal volume distortion at $u$ of the map $$S_1(0_p)\ni u\mapsto\gamma_u(r)=\exp(ru)\in S_r(p),$$ where $S_r(p)\subset M$ denotes the geodesic sphere of radius $r$ around $p$. Let $\sigma_u(r)$ denote the shape operator of $S_r(p)$ at $\exp(ru)$; that is, $$\sigma_u(r):=(\nabla\nu){\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{T_{\exp(ru)}M}$}}\,,$$ where $\nabla$ is the Levi-Civita connection of $M$ and $\nu$ denotes the outward pointing unit normal vector field on the geodesic ball $B_{i(p)}(p)\setminus\{p\}$. In particular, $\nu\circ\gamma_u=\dot\gamma_u$, $\sigma_u\nu=0$, and the image of $\sigma_u(r)$ is contained in $T_{\gamma_u(r)}S_r(p)$. It is well-known that for all $r\in(0,i(p))$, $$\label{eq:volgrowth}
v'_u(r)/v_u(r)={{\operatorname{Tr}}}(\sigma_u(r)),$$ and that the covariant derivative $\sigma'_u$ of the endomorphism field $\sigma_u$ along $\gamma_u{\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{(0,i(p))}$}}$ satisfies the so-called Riccati equation $$\label{eq:riccati}
\sigma'_u=-\sigma_u^2-R_{\dot\gamma_u},$$ where $R$ is the Riemannian curvature tensor of $M$, given by $R(x,y)z=-\nabla_x\nabla_yz+\nabla_y\nabla_xz+\nabla_{[x,y]}z$, and where $R_\nu:=R(\nu,\,.\,)\nu$. (Note that here we use the same sign for $R$ as Besse [@Be].) Let $C_u(r):=r\sigma_u(r)$. This endomorphism field along $\gamma_u{\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{(0,i(p))}$}}$ is smoothly extendable to $r=0$ by $C_u(0):=I_u$, where $I_u$ is defined by $I_u(u)=0$ and $I_u{\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{\{u\}^\perp}$}}={{\operatorname{Id}}}_{\{u\}^\perp}$. Moreover, from (\[eq:riccati\]) one can derive Ledger’s recursion formula for the covariant derivatives of $C_u$ at $r=0$ (see, e.g., [@CV]): $$(k-1)C_u^{(k)}(0)=-k(k-1)R_u^{(k-2)}-\sum_{\ell=0}^k\binom k\ell C_u^{(\ell)}(0)C_u^{(k-\ell)}(0)$$ for all $k\in{{\mathbb N}}$, where $R_u^{(k)}$ is the $k$-th covariant derivative of the endomorphism field $R_{\dot\gamma_u}$ along $\gamma_u$ at $r=0$. This formula allows one to successively compute the $C_u^{(k)}(0)$ in terms of the endomorphisms $R_u^{(k)}$ of $T_pM$. Forming the Taylor series of $C_u$ and dividing by $r$, one obtains (see, e.g., [@Be], [@CV]): $$\label{eq:pow}
\begin{split}
P^{r,0}_{\gamma_u}\circ\sigma_u(r)\circ P^{0,r}_{\gamma_u}={}&\frac1r I_u-\frac r3 R_u
-\frac{r^2}4 R'_u -\bigl(\frac1{10}R''_u+\frac1{45}R_uR_u\bigr)r^3\\
&-\bigl(\frac1{36}R'''_u+\frac1{72}R_uR'_u
+\frac1{72}R'_uR_u\bigr)r^4\\
&-\bigl(\frac1{168}R^{(4)}_u+\frac1{210}R_uR''_u+\frac1{210}R''_uR_u+\frac1{112}R'_uR'_u
+\frac2{945}R_uR_uR_u\bigr)r^5\\
&+O(r^6).
\end{split}$$
Curvature identities in harmonic spaces
---------------------------------------
The manifold $M$ is called a *harmonic space* if for every $p\in M$ the above function $\theta_u$ does not depend on $u\in S_1(0_p)$. An equivalent condition is that for all $r\in(0,i(p))$, the geodesic spheres $S_r(p)$ have constant mean curvature (recall equations (\[eq:vtheta\]), (\[eq:volgrowth\])). For more information on harmonic spaces see [@RWW] or [@Be]. If $M$ is harmonic then the function $\theta_u$ does in fact not even depend on $p$; that is, there exists $\theta:[0,\infty)\to{{\mathbb R}}$ such that $$\theta_u(r)=\theta(r)$$ for all $u\in TM$ with $|u|=1$. Moreover, even the local or infinitesimal versions of the above condition imply that the manifold is Einstein [@Be] and therefore analytic [@DK]. Hence, the local or infinitesimal versions of the above conditions are equivalent to the global versions. Since $\theta_u(r)$ depends only on $r$, so does $v_u(r)$ and hence ${{\operatorname{Tr}}}(\sigma_u(r))$. From this one can successively derive, using the expansion (\[eq:pow\]):
\[see [@Be], Chapter 6\] \[prop:constants\] If $M$ is harmonic then there exist constants $C,H,L\in{{\mathbb R}}$ such that for all $p\in M$ and all $u\in T_pM$ with $|u|=1$[:]{}
- ${{\operatorname{Tr}}}(R_u)=C$; in particular[:]{}
- ${{\operatorname{Tr}}}(R^{(k)}_u)=0$ for all $k\in{{\mathbb N}}$.
- ${{\operatorname{Tr}}}(R_uR_u)=H$; in particular[:]{}
- ${{\operatorname{Tr}}}(R_uR'_u)=0$ and
- ${{\operatorname{Tr}}}(R_uR''_u)=-{{\operatorname{Tr}}}(R'_uR'_u)$.
- ${{\operatorname{Tr}}}(32R_uR_uR_u-9R'_uR'_u)=L$.
In fact, taking traces in (\[eq:pow\]), one has in the harmonic case: $$\label{eq:powtrharm}
{{\operatorname{Tr}}}(\sigma_u(r))=(n-1)\frac1r-\frac 13 Cr
-\frac1{45}H r^3-\frac1{15120}Lr^5+O(r^7)$$ for $r\downarrow0$ and all $u\in TM$ with $|u|=1$. Note that Proposition \[prop:constants\](i) just says that the Einstein constant of $M$ is $C$; that is, ${{\operatorname{{Ric}}}}=C{{\operatorname{Id}}}$ on each $T_pM$. Recall that the Ricci operator is defined by ${\langle}{{\operatorname{{Ric}}}}(x),y{\rangle}={{\operatorname{Tr}}}(R(x,\,.\,)y)$ for all $x,y\in T_pM$ and all $p\in M$. From Proposition \[prop:constants\] one can further derive:
\[see [@Be], Chapter 6\] \[prop:reqs\] If $M$ is harmonic, then for the above constants $C,H,L$ and each $p\in M$[:]{}
- ${\langle}R(x,\,.\,)\,.\,,R(y,\,.\,)\,.\,{\rangle}=\frac23((n+2)H-C^2){\langle}x,y{\rangle}$ for all $x,y\in T_pM$; in particular[:]{}
- $|R|_p^2=\frac23n((n+2)H-C^2)$.
- $32\bigl(nC^3+\frac92C|R|_p^2+\frac72\hat R(p)-{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p)\bigr)-27|\nabla R|_p^2=n(n+2)(n+4)L$.
Here, the functions $\hat R, {{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}\in C^\infty(M)$ are certain curvature invariants of order six which are defined as follows: If $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $T_pM$ and $R_{ijk\ell}:={\langle}R(e_i,e_j)e_k,e_\ell{\rangle}$, then $${{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p):=\sum_{i,j,k,\ell,a,b} R_{ijk\ell}R_{ja\ell b}R_{aibk},\;\;\;\;
\hat R(p):=\sum_{i,j,k,\ell,a,b} R_{ijk\ell}R_{k\ell ab}R_{abij}.$$ Note that the term $nC^3$ in Proposition \[prop:reqs\](iii) reads $nC^2$ in the corresponding equation 6.67 in [@Be], but this was obviously a misprint (note that curvature terms of different order cannot occur here); see also formula (3.1) in [@Wa].
Proposition \[prop:reqs\](iii) will be used in Section \[sec:proof\], together with the following formula which actually holds in any Einstein manifold; see formula (6-7) in [@Li58] or formula (11.3) in [@GV]: $$\label{eq:lichn}
-\frac12\Delta(|R|^2)=2C|R|^2-\hat R-4{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}+|\nabla R|^2,$$ where $\Delta$ denotes the Laplace operator on functions, that is, $\Delta f=-\sum_i \bigl(e_i(e_if)-(\nabla_{e_i}e_i)f\bigr)$ for local orthonormal frames $\{e_1,\ldots,e_n\}$. (Again, there is a misprint in two of the coefficients in the corresponding formula 6.65 in [@Be].) If $M$ is harmonic, then the left hand side of (\[eq:lichn\]) is zero by Proposition \[prop:reqs\](ii). Finally, we recall the following well-known observations which will be used in Section \[sec:proof\]:
\[rem:volscal\] Let $M$ be an $n$-dimensional harmonic space with volume density function $\theta$ as above.
\(i) For any $p\in M$, the volume of the geodesic sphere $S_r(p)$ with $0<r<i(p)$ equals the volume $\omega_{n-1}$ of the standard unit sphere $S^{n-1}$ in ${{\mathbb R}}^n$ multiplied by the factor $$v(r):=r^{n-1}\theta(r)$$ Note that $v(r)=v_u(r)$ for each unit vector $u\in TM$, where $v_u$ is the function defined in (\[eq:vtheta\]). The function $v$ determines the volume growth function $v'/v$ of the geodesic spheres, and thus it determines, by (\[eq:volgrowth\]), the function ${{\operatorname{Tr}}}(\sigma_u(r))$ (which is independent of $u$). By (\[eq:powtrharm\]), the function which associates to small values of $r$ the volume of geodesic spheres of radius $r$ in a given harmonic space $M$ determines the constants $C,H,L$ (and of course $n$) associated with $M$.
\(ii) Let ${{\operatorname{{scal}}}}=nC$ denote the scalar curvature of $M$. Let $p\in M$, fix some $r\in(0,i(p))$, and let ${{\operatorname{{scal}}}}^S$ denote the scalar curvature function of $S_r(p)$. A routine calculation using the Gauss equation shows that for each unit vector $u\in T_pM$ we have $${{\operatorname{{scal}}}}^S(\exp(ru))= {{\operatorname{{scal}}}}-2{\langle}{{\operatorname{{Ric}}}}(\dot\gamma_u(r)),\dot\gamma_u(r){\rangle}+({{\operatorname{Tr}}}(\sigma_u(r)))^2
-{{\operatorname{Tr}}}(\sigma_u(r)^2)$$ which by the Einstein condition and equations (\[eq:volgrowth\]) and (\[eq:riccati\]) implies $$\begin{aligned}
{{\operatorname{{scal}}}}^S(\exp(ru)) &= (n-2)C+(v'(r)/v(r))^2+{{\operatorname{Tr}}}(\sigma'_u(r))+{{\operatorname{Tr}}}(R_{\dot\gamma_u(r)})\\
&= (n-2)C+(v'(r)/v(r))^2+(v'/v)'(r)+C = (n-1)C+v''(r)/v(r).\end{aligned}$$ Therefore, geodesic spheres in $M$ have constant scalar curvature, and the respective constant depends only on the radius, not on the midpoint. Finally, using (i) one concludes that the function which associates to small values of $r$ the scalar curvature of geodesic spheres of radius $r$ is determined already by the function which associates to small values of $r$ the volume of geodesic spheres of radius $r$.
\[rem:damekricci\] As mentioned in the Introduction, the aim of this paper is to show that in harmonic spaces, the heat invariants $a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))$ and $a_2(S_r(p))$, viewed as functions of $r$, together determine $|\nabla R|^2_p$. This is not the case for $a_0$ alone, as manifested by certain pairs of Damek-Ricci spaces. A Damek-Ricci space $AN$ is a certain type of solvable Lie groups with left invariant metric, namely, the standard $1$-dimensional solvable extension of a simply connected Riemannian nilmanifold $N$ of Heisenberg type. The volume density function of $AN$ is radial and depends only on the dimensions of $N$ and its center [@DR]; see also the book [@BTV]. Within the class of Damek-Ricci spaces, there exist pairs of symmetric spaces $AN$ and locally nonsymmetric spaces $AN'$ where $N$ and $N'$ have the same dimension and so do their centers. (In fact, certain such pairs $AN$ and $AN'$ were the ambient manifolds used by Szabó in [@Sz01], [@Sz05]; recall the Introduction.) In particular, geodesic spheres of the same radius in $AN$ and $AN'$ have the same volume. This shows that in harmonic spaces, the function $r\mapsto{{\operatorname{vol}}}(S_r(p))$ alone does not determine $|\nabla R|^2_p$. In turn, Remark \[rem:volscal\] shows that in harmonic spaces, the function $r\mapsto a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))$ already determines the function $r\mapsto a_1(S_r(p))=\frac16\int_{S_r(p)}{{\operatorname{{scal}}}}^S\,{{\operatorname{\textit{dvol}}}}_{S_r(p)}$. Therefore, we need to consider $a_2(S_r(p))$. The next section gives some necessary preparations for this.
Radial expansions of $|{{\operatorname{{Ric}}}}|^2$ and $|R|^2$ for geodesic spheres in harmonic spaces
=======================================================================================================
\[sec:curv\]
In this section we will describe a certain coefficient in the radial power series expansions of the curvature invariants $|{{\operatorname{{Ric}}}}|^2$ and $|R|^2$ of geodesic spheres in harmonic spaces. First we need the following lemma.
\[lem:norms\] Let $M$ be an $n$-dimensional harmonic space, and let $C$ and $H$ be the constants from Proposition \[prop:constants\]. Let $p\in M$, and let $S:=S_r(p)$ be a geodesic sphere around $p$ with radius $r\in(0,i(p))$, endowed with the induced Riemannian metric. Let $u$ be a unit vector in $T_pM$, let $\sigma:=\sigma_u(r)$ be as in Section \[sec:prelim\], and write $\sigma':=\sigma'_u(r)$. Let $R^S$ and ${{\operatorname{{Ric}}}}^S$ denote the curvature tensor, resp. the Ricci operator, of $S$. Then in the point $q:=\exp(ru)\in S$ we have[:]{} $$\begin{aligned}
{\rm(i)}&\
|{{\operatorname{{Ric}}}}^S|_q^2=(n-1)C^2+2C({{\operatorname{Tr}}}(\sigma))^2+({{\operatorname{Tr}}}(\sigma))^2{{\operatorname{Tr}}}(\sigma^2)
+2C{{\operatorname{Tr}}}(\sigma')+2{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma\sigma')+{{\operatorname{Tr}}}(\sigma'\sigma'),\\
{\rm(ii)}&\
|R^S|_q^2=\frac23(n-4)\bigl((n+2)H-C^2\bigr)+4H+2({{\operatorname{Tr}}}(\sigma^2))^2-2{{\operatorname{Tr}}}(\sigma^4)
+4\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(\sigma\circ R(e_i,\,.\,) \sigma e_i\bigr),\\
&\ \mbox{where $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $T_q M$}.\end{aligned}$$
\(i) Let $\nu$ be the outward pointing radial unit vector field as in Section \[sec:prelim\]. From the Gauss equation one easily derives the following formula whose analog is valid for submanifolds of codimension one in arbitrary Riemannian manifolds: $${{\operatorname{{Ric}}}}^S_q=({{\operatorname{{Ric}}}}-R_{\nu_q}+{{\operatorname{Tr}}}(\sigma)\sigma-\sigma^2){\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{T_qS}$}}$$ Using the Einstein condition and the Riccati equation (\[eq:riccati\]), this formula becomes in our situation: $${{\operatorname{{Ric}}}}^S_q=(C{{\operatorname{Id}}}+{{\operatorname{Tr}}}(\sigma)\sigma+\sigma'){\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{T_qS}$}}$$ (see also [@NV], p. 67). Now one obtains the desired formula immediately, keeping in mind that both $\sigma$ and $\sigma'$ are symmetric and annihilate $\nu_q$.
\(ii) Choose an orthonormal basis $\{e_1,\ldots,e_n\}$ of $T_q M$ such that $e_1=\nu_q$. For all $i,j,k,\ell\in\{2,\ldots,n\}$ we have by the Gauss equation (recall our sign convention for $R$): $${\langle}R^S(e_i,e_j)e_k,e_\ell{\rangle}={\langle}R(e_i,e_j)e_k,e_\ell{\rangle}+{\langle}\sigma e_i,e_k{\rangle}{\langle}\sigma e_j,e_\ell{\rangle}-{\langle}\sigma e_j, e_k{\rangle}{\langle}\sigma e_i,e_\ell{\rangle}.$$ Squaring both sides and forming the sum over $i,j,k,\ell$, while recalling that $\sigma$ is symmetric and annihilates $e_1$, we get $$\begin{aligned}
|R^S|_q^2={}&\sum_{i,j,k,\ell=2}^n{\langle}R(e_i,e_j)e_k,e_\ell{\rangle}^2
+|\sigma|^2|\sigma|^2+|\sigma|^2|\sigma|^2\\
&-2|\sigma^2|^2
+2\sum_{i,j=1}^n {\langle}R(e_i,e_j)\sigma e_i,\sigma e_j{\rangle}-2\sum_{i,j=1}^n {\langle}R(e_i,e_j)\sigma e_j,\sigma e_i{\rangle}\\
={}&\sum_{i,j,k,\ell=2}^n {\langle}R(e_i,e_j)e_k,e_\ell{\rangle}^2+2({{\operatorname{Tr}}}(\sigma^2))^2-2{{\operatorname{Tr}}}(\sigma^4)
+4\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(\sigma\circ R(e_i,\,.\,)\sigma e_i\bigr).\end{aligned}$$ The desired formula now follows from the fact that the first sum on the right hand side is equal to $|R|_q^2-4|R(e_1,\,.\,)\,.\,|^2+4|R_{e_1}|^2$ which by Proposition \[prop:reqs\](i), (ii) and Proposition \[prop:constants\](iii) becomes $\frac23(n-4)\bigl((n+2)H-C^2\bigr)+4H$.
Using the radial power series expansion of $\sigma$ together with the previous lemma, we will make conclusions concerning the first few coefficients of the radial expansions of $|{{\operatorname{{Ric}}}}^S|^2$ and $|R^S|^2$. The following proposition will be the key of the proof of the Main Theorem \[thm:main\]. Actually we will use only the statements about $\alpha_2$ and $\beta_2$ in this proposition.
\[prop:coeffs\] Let $M$ be an $n$-dimensional harmonic space, and let $C$, $H$, and $L$ be the constants from Proposition \[prop:constants\]. Let $p\in M$, and let $u$ be a unit vector in $T_p M$. Then $$\begin{aligned}
|{{\operatorname{{Ric}}}}^{S_r(p)}|_{\exp(ru)}^2&=\alpha_{-4}r^{-4}+\alpha_{-2} r^{-2}+\alpha_0
+\alpha_2(u) r^2+O(r^3)\mbox{ and}\\
|R^{S_r(p)}|_{\exp(ru)}^2&=\beta_{-4}r^{-4}+\beta_{-2} r^{-2}+\beta_0+\beta_2(u) r^2+O(r^3)\end{aligned}$$ for $r\downarrow 0$, where the coefficients $\alpha_i$ and $\beta_i$ for $i\in \{-4,-2,0\}$ are constants depending only on $n$, $C$, and $H$. Moreover, $$\begin{aligned}
\alpha_2(u)&=\hat\alpha_2+\frac1{16}{{\operatorname{Tr}}}(R'_uR'_u)\mbox{\ \ and}\\
\beta_2(u)&=\hat\beta_2
+\frac 49\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_u e_i\bigr),\end{aligned}$$ where $\hat\alpha_2$ and $\hat\beta_2$ are constants depending only on $n$, $C$, $H$, and $L$, and where $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $T_p M$.
We use Lemma \[lem:norms\] together with the power series expansions (\[eq:pow\]), (\[eq:powtrharm\]) of $\sigma:=\sigma_u(r)$ and ${{\operatorname{Tr}}}(\sigma)$. Let us first consider $|{{\operatorname{{Ric}}}}^{S_r(p)}|_{\exp(ru)}^2$ and the individual contributions of the nonconstant terms in Lemma \[lem:norms\](i) to its expansion. By (\[eq:powtrharm\]) we have $$({{\operatorname{Tr}}}(\sigma))^2=\bigl((n-1)\frac1r-\frac13Cr-\frac1{45}Hr^3-\frac1{15120}Lr^5\bigr)^2+O(r^6)$$ for $r\downarrow0$. Moreover, from the expansion (\[eq:pow\]) and Proposition \[prop:constants\] one gets $$\label{eq:trsigmasq}
{{\operatorname{Tr}}}(\sigma^2)=(n-1)\frac1{r^2}-\frac23 C+\frac1{15}Hr^2+\frac1{3024}Lr^4+O(r^5).$$ Further, $${{\operatorname{Tr}}}(\sigma')=\frac d{dr}{{\operatorname{Tr}}}(\sigma)=-(n-1)\frac1{r^2}-\frac13C-\frac1{15}Hr^2+O(r^4)$$ by (\[eq:powtrharm\]), and $$2{{\operatorname{Tr}}}(\sigma\sigma')=\frac d{dr}{{\operatorname{Tr}}}(\sigma^2)=-2(n-1)\frac1{r^3}+\frac2{15}Hr+\frac1{756}Lr^3+O(r^4)$$ by (\[eq:trsigmasq\]). Using these expansions and (\[eq:powtrharm\]), one easily checks that each of the the four individual terms $2C({{\operatorname{Tr}}}(\sigma))^2$, $({{\operatorname{Tr}}}(\sigma))^2{{\operatorname{Tr}}}(\sigma^2)$, $2C{{\operatorname{Tr}}}(\sigma')$, and $2{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma\sigma')$ appearing on the right hand side of Lemma \[lem:norms\](i) has the property that the corresponding coefficients of $r^{-4}, r^{-2},r^0$ depend only on $n,C,H$, the coefficient of $r^2$ depends only on $n,C,H,L$, and the coefficients of $r^{-3},r^{-1},r$ vanish.
It remains to consider the term ${{\operatorname{Tr}}}(\sigma'\sigma')$ in Lemma \[lem:norms\](i). From (\[eq:pow\]) we get $$\begin{split}
P^{r,0}_{\gamma_u}\circ\sigma'\circ P^{0,r}_{\gamma_u}={}&-\frac1{r^2} I_u-\frac 13 R_u
-\frac r2 R'_u -\bigl(\frac3{10}R''_u+\frac1{15}R_uR_u\bigr)r^2\\
&-\bigl(\frac19R'''_u+\frac1{18}R_uR'_u
+\frac1{18}R'_uR_u\bigr)r^3\\
&-\bigl(\frac5{168}R^{(4)}_u+\frac1{42}R_uR''_u+\frac1{42}R''_uR_u+\frac5{112}R'_uR'_u
+\frac2{189}R_uR_uR_u\bigr)r^4\\
&+O(r^5)
\end{split}$$ and thereby, using Proposition \[prop:constants\]: $$\begin{split}
{{\operatorname{Tr}}}(\sigma'\sigma')={}&(n-1)\frac1{r^4}+\frac23\frac C{r^2}+\frac{11}{45}H\\
&+\Bigl(\bigl(-\frac2{21}+\frac5{56}-\frac15+\frac14\bigr){{\operatorname{Tr}}}(R'_uR'_u)
+\bigl(\frac4{189}+\frac2{45}\bigr){{\operatorname{Tr}}}(R_uR_uR_u)\Bigr)r^2+O(r^3).
\end{split}$$ The coefficient of $r^2$ in the latter expansion is $$\frac{37}{840}{{\operatorname{Tr}}}(R'_uR'_u)+\frac{62}{945}{{\operatorname{Tr}}}(R_uR_uR_u)$$ which by Proposition \[prop:constants\](vi) turns out to be $$\frac{62}{32\cdot945}L+\bigl(\frac{37}{840}+\frac{9\cdot 62}{32\cdot945}\bigr){{\operatorname{Tr}}}(R'_uR'_u)
=\frac{31}{15120}L+\frac1{16}{{\operatorname{Tr}}}(R'_uR'_u).$$ This concludes the proof of the statements concerning the expansion of $|{{\operatorname{{Ric}}}}^{S_r(p)}|^2_{\exp(ru)}$.
We now turn to $|R^{S_r(p)}|^2_{\exp(ru)}$ and study the individual contributions of the nonconstant terms in Lemma \[lem:norms\](ii) to its expansion. Squaring (\[eq:trsigmasq\]), we see that in the expansion of the term $2({{\operatorname{Tr}}}(\sigma^2))^2$ the coefficients of $r^{-4},r^{-2},r^0$ depend only on $n,C,H$, the coefficient of $r^2$ depends only on $n,C,H,L$, and the coefficients of $r^{-3},r^{-1},r$ vanish.
Regarding the term $-2{{\operatorname{Tr}}}(\sigma^4)$ we obtain from (\[eq:pow\]): $$\begin{gathered}
P^{r,0}_{\gamma_u}\circ\sigma^4\circ P^{0,r}_{\gamma_u}=
\frac1{r^4}I_u-\frac 4{3r^2} R_u-\frac 1r R'_u +\bigl(-\frac25R''_u+\frac{26}{45}R_u^2\bigr)
+\bigl(-\frac 19 R'''_u+\frac 49 (R_uR'_u+R'_uR_u)\bigr)r\\
+\Bigl(-\frac1{42}R^{(4)}_u +\bigl(-\frac 2{105}+\frac 15\bigr)(R''_uR_u+R_uR''_u)
+\bigl(-\frac 1{28}+\frac 38\bigr)R'_uR'_u
+\bigl(-\frac 8{945}+\frac4{45}-\frac 4{27}\bigr)R_u^3\Bigr)r^2\\
+O(r^3)\end{gathered}$$ for $r\downarrow0$. Using Proposition \[prop:constants\] we get $$\begin{aligned}
-2{{\operatorname{Tr}}}(\sigma^4)={}&-2(n-1)\frac1{r^4}+\frac 83 \frac C{r^2}-\frac{52}{45}H\\
&+\Bigl(\bigl(-\frac8{105}+\frac 45+\frac1{14}-\frac 34\bigr){{\operatorname{Tr}}}(R'_uR'_u)
+\bigl(\frac{16}{945}-\frac8{45}+\frac8{27}\bigr){{\operatorname{Tr}}}(R_uR_uR_u)\Bigr)r^2+O(r^3).\end{aligned}$$ The coefficient of $r^2$ in the latter expansion is $$\frac{19}{420}{{\operatorname{Tr}}}(R'_uR'_u)+\frac{128}{945}{{\operatorname{Tr}}}(R_uR_uR_u)$$ which by Proposition \[prop:constants\](vi) equals $$\label{eq:1over12}
\frac{128}{32\cdot 945}L+\bigl(\frac{19}{420}+\frac{9\cdot128}{32\cdot945}\bigr)
{{\operatorname{Tr}}}(R'_uR'_u)=\frac4{945}L+\frac1{12}{{\operatorname{Tr}}}(R'_uR'_u).$$
It remains to consider the term $4\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(\sigma\circ R(e_i,\,.\,)\sigma e_i\bigr)$ in Lemma \[lem:norms\](ii). We make some preliminary observations. For $k\in{{\mathbb N}}_0$, let $R^{(k)}$, resp. ${{\operatorname{{Ric}}}}^{(k)}$ denote the $k$-th covariant derivative of the curvature tensor, resp. the Ricci operator, along $\gamma_u$ at $r=0$. We will use the the Taylor series expansion of the Riemannian curvature tensor along $\gamma_u$ (recall that $M$ is analytic): $$\label{eq:taylorR}
P_{\gamma_u}^{r,0}\circ R_{\gamma_u(r)}\circ P_{\gamma_u}^{0,r}=
\sum_{k=0}^\infty\frac{r^k}{k!}R^{(k)}$$ Moreover, ${{\operatorname{{Ric}}}}^{(k)}=0$ for $k\ge1$ since $M$ is Einstein. Note that ${{\operatorname{{Ric}}}}{\lower0.4ex\hbox{$|$}\lower0.7ex
\hbox{$\scriptstyle{T_pM}$}}=\sum_{i=1}^n R_{e_i}$ and similarly on each $T_{\gamma_u(r)}M$ if we extend $\{e_1,\ldots,e_n\}$ parallelly along $\gamma_u$. For any $k\in{{\mathbb N}}_0$ we have, using Proposition \[prop:constants\]: $$\label{eq:IuIu}
\begin{split}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(I_u\circ R^{(k)}(e_i,\,.\,)I_ue_i\bigr)
&={{\operatorname{Tr}}}(I_u\circ{{\operatorname{{Ric}}}}^{(k)})-{{\operatorname{Tr}}}(I_u\circ R^{(k)}_u)\\
&={{\operatorname{Tr}}}({{\operatorname{{Ric}}}}^{(k)})-{\langle}{{\operatorname{{Ric}}}}^{(k)}u,u{\rangle}-{{\operatorname{Tr}}}(R^{(k)}_u)
=\begin{cases} (n-2)C,& k=0,\\0,& k\ge1.
\end{cases}
\end{split}$$ Moreover, $$\begin{split}
\label{eq:RuIu}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_u\circ R^{(k)}(e_i,\,.\,)I_ue_i\bigr)
&={{\operatorname{Tr}}}(R_u\circ{{\operatorname{{Ric}}}}^{(k)})-{{\operatorname{Tr}}}(R_uR^{(k)}_u)\\
&=\begin{cases} C^2-H,&k=0,\\-{{\operatorname{Tr}}}(R_uR^{(k)}_u),&k\ge1,
\end{cases}
\end{split}$$
$$\begin{split}
\label{eq:RpuIu}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R'_u\circ R^{(k)}(e_i,\,.\,)I_ue_i\bigr)
&={{\operatorname{Tr}}}(R'_u\circ{{\operatorname{{Ric}}}}^{(k)})-{{\operatorname{Tr}}}(R'_uR^{(k)}_u)\\
&=\begin{cases}0,&k=0,\\-{{\operatorname{Tr}}}(R'_uR^{(k)}_u),&k\ge1,
\end{cases}
\end{split}$$
$$\begin{aligned}
\label{eq:RppuIu}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R''_u\circ R(e_i,\,.\,)I_ue_i\bigr)
&={{\operatorname{Tr}}}(R''_u\circ{{\operatorname{{Ric}}}})-{{\operatorname{Tr}}}(R''_uR_u)=0+{{\operatorname{Tr}}}(R'_uR'_u),
\\
\label{eq:RuRu_Iu}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_uR_u\circ R(e_i,\,.\,)I_ue_i\bigr)
&={{\operatorname{Tr}}}(R_uR_u\circ{{\operatorname{{Ric}}}})-{{\operatorname{Tr}}}(R_uR_uR_u)=CH-{{\operatorname{Tr}}}(R_uR_uR_u).\end{aligned}$$
Note that for any pair of symmetric endomorphisms $F,G$ of $T_pM$ we have $$\label{eq:FG}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(F\circ R^{(k)}(e_i,\,.\,)Ge_i\bigr)
=\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(G\circ R^{(k)}(e_i,\,.\,)Fe_i\bigr)$$ by the symmetries of the curvature operator. Keeping the expansions (\[eq:pow\]) and (\[eq:taylorR\]) in mind, we see that the expression in (\[eq:IuIu\]) contributes only to the coefficient of $r^{-2}$ in the expansion of $\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(\sigma\circ R(e_i,\,.\,)\sigma e_i\bigr)$, the expression in (\[eq:RuIu\]) contributes to the coefficients of $r^0$ and $r^2$ (and higher order), the expressions in (\[eq:RpuIu\]), (\[eq:RppuIu\]), (\[eq:RuRu\_Iu\]) contribute to the coefficient of $r^2$ (and higher order). The only additional contribution to the coefficient of $r^2$ is given by the sum of ${{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_ue_i\bigr)$. Recalling (\[eq:FG\]) (and multiplying $R^{(k)}$ by $1/k!$), we obtain from (\[eq:pow\]), \[prop:constants\](iv), (\[eq:taylorR\]), and the above observations: $$\begin{split}
4\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(\sigma\circ R(e_i,\,.\,)\sigma e_i\bigr)
={}&4\Bigl((n-2)\frac C{r^2} -\frac 23 (C^2-H)\\
&+\Bigl[\frac 2{3\cdot2!}{{\operatorname{Tr}}}(R_uR''_u)+\bigl(\frac 24-\frac 2{10}\bigr){{\operatorname{Tr}}}(R'_uR'_u)
-\frac2{45}CH+\frac2{45}{{\operatorname{Tr}}}(R_uR_uR_u)\\
&\hphantom{\Big[\frac13{{\operatorname{Tr}}}}+\frac19\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_ue_i\bigr)\Bigr]r^2\Bigr)+O(r^3).
\end{split}$$ By Proposition \[prop:constants\](v), the coefficient of $r^2$ in the latter expansion is $$-\frac8{45}CH-\frac2{15}{{\operatorname{Tr}}}(R'_uR'_u)+\frac8{45}{{\operatorname{Tr}}}(R_uR_uR_u)+\frac49
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_ue_i\bigr)$$ By Proposition \[prop:constants\](vi), the two terms involving ${{\operatorname{Tr}}}(R'_uR'_u)$ and ${{\operatorname{Tr}}}(R_uR_uR_u)$ become $$\frac8{32\cdot 45}L+\bigl(-\frac2{15}+\frac{9\cdot 8}{32\cdot 45}\bigr){{\operatorname{Tr}}}(R'_uR'_u)
=\frac1{180}L-\frac1{12}{{\operatorname{Tr}}}(R'_uR'_u).$$ Combining this with the result for the $r^2$-coefficient of $-2{{\operatorname{Tr}}}(\sigma^4)$ from (\[eq:1over12\]), we conclude that the terms involving ${{\operatorname{Tr}}}(R'_uR'_u)$ in the coefficient of $r^2$ in the power series expansion of $|R^{S_r(p)}|_{\exp(ru)}^2$ cancel each other, and the only remaining term apart from those which depend solely on $n,C,H,L$ is $\frac49\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_ue_i\bigr)$, as claimed.
\[rem:tracesv\] For the purpose of the proof of the Main Theorem \[thm:main\] in Section \[sec:proof\], which we will perform using the heat invariants $a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))$ and $a_2(S_r(p))
=\frac1{360}\int_{S_r(p)}(5({{\operatorname{{scal}}}}^S)^2-2|{{\operatorname{{Ric}}}}^S|^2+2|R^S|^2){{\operatorname{\textit{dvol}}}}_{S_r(p)}$, we would actually not have needed the exact statement of the previous proposition – which might, however, be interesting in its own right. Rather, we could have restricted our attention to the term ${{\operatorname{Tr}}}(\sigma'\sigma')$ in the expression of $|{{\operatorname{{Ric}}}}^S|_{\gamma_u(r)}^2$ in Lemma \[lem:norms\](i), and to the last two terms in the expression of $|R^S|_{\gamma_u(r)}^2$ in Lemma \[lem:norms\](ii). In fact, even without the explicit calculation of the expansion of the other terms, one easily sees that those are determined by the volume function $r\mapsto a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))$ of the geodesic spheres (which is just the function $v$ multiplied by the volume of the standard unit sphere, see Remark \[rem:ave\] below). More precisely, in the spirit of Remark \[rem:volscal\] we obtain $$\begin{aligned}
2C({{\operatorname{Tr}}}(\sigma))^2&=2C(v'/v)^2,\\
({{\operatorname{Tr}}}(\sigma))^2{{\operatorname{Tr}}}(\sigma^2)&=(v'/v)^2(-(v'/v)'-C),\\
2C{{\operatorname{Tr}}}(\sigma')&=2C{{\operatorname{Tr}}}(\sigma)'=2C(v'/v)',\\
2{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma\sigma')&=2v'/v\cdot\tfrac12({{\operatorname{Tr}}}(\sigma^2))'
=v'/v\cdot(-(v'/v)''),\\
2({{\operatorname{Tr}}}(\sigma^2))^2&=2(-(v'/v)'-C)^2.\end{aligned}$$
Proof of the Main Theorem
=========================
\[sec:proof\]
In this section we will first derive an integrated version of Proposition \[prop:coeffs\]. Using this and the heat invariants $a_0,a_1,a_2$ of geodesic spheres in harmonic spaces we will then prove our Main Theorem \[thm:main\]. We need the following general remark on mean values.
\[rem:ave\] In any harmonic space $M$, the average (or mean value) of a smooth function $f$ on a geodesic sphere $S_r(p)$ (with $0<r<i(p)$) is the same as the average of $f(\exp(r\,.\,))$ over the unit sphere $S_1(0_p)$ in $T_pM$. More explicitly: Let $\omega_{n-1}$ denote the volume of the $(n-1)$-dimensional standard sphere. In particular, $\omega_{n-1}$ is the volume of $S_1(0_p)$. Recall from Section \[sec:prelim\] that $\theta(r)=\theta_u(r)$ is independent of $u$ (and even of $p$) by harmonicity. We have $${{\operatorname{vol}}}(S_r(p))=r^{n-1}\theta(r)\omega_{n-1}=v(r)\omega_{n-1}\,,$$ and for any smooth function $f$ on $S_r(p)$, $$\begin{split}
\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)}f\,{{\operatorname{\textit{dvol}}}}_{S_r(p)}
&=\frac1{v(r)\omega_{n-1}}
\int_{S_1(0_p)}f(\exp(ru))v(r)\,du\\
&=\frac1{\omega_{n-1}}\int_{S_1(0_p)}f(\exp(ru))\,du.
\end{split}$$
Now we can give an “integrated” version of Proposition \[prop:coeffs\].
\[prop:intcoeffs\] Let $M$ be an $n$-dimensional harmonic space, and let $C$, $H$, and $L$ be the constants from Proposition \[prop:constants\]. Let $p\in M$. Then $$\begin{aligned}
\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)} |{{\operatorname{{Ric}}}}^{S_r(p)}|^2 {{\operatorname{\textit{dvol}}}}_{S_r(p)}&=
\alpha_{-4}r^{-4}+\alpha_{-2} r^{-2}+\alpha_0
+\overline\alpha_2 r^2+O(r^3)\mbox{ and}\\
\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)} |R^{S_r(p)}|^2 {{\operatorname{\textit{dvol}}}}_{S_r(p)}&=
\beta_{-4}r^{-4}+\beta_{-2} r^{-2}+\beta_0+\overline\beta_2 r^2+O(r^3)\end{aligned}$$ for $r\downarrow 0$, where the coefficients $\alpha_i$ and $\beta_i$ for $i\in \{-4,-2,0\}$ are the constants from Proposition \[prop:coeffs\] depending only on $n$, $C$, and $H$. Moreover, $$\begin{aligned}
\overline\alpha_2&=\tilde\alpha_2+\frac3{16n(n+2)(n+4)}|\nabla R|_p^2\mbox{\ \ and}\\
\overline\beta_2&=\tilde\beta_2
+\frac1{8n(n+2)}|\nabla R|_p^2,\end{aligned}$$ where $\tilde\alpha_2$ and $\tilde\beta_2$ are constants depending only on $n$, $C$, $H$, and $L$.
For any unit vector $u$ in $T_pM$, let $\alpha_2(u)$ and $\beta_2(u)$ be the coefficients from Proposition \[prop:coeffs\]. Using that proposition and Remark \[rem:ave\], we only need to show that $$\begin{aligned}
\overline\alpha_2&:=\frac1{\omega_{n-1}}\int_{S_1(0_p)}\alpha_2(u)\,du
=\hat\alpha_2+\frac1{\omega_{n-1}}\int_{S_1(0_p)}\frac1{16}{{\operatorname{Tr}}}(R'_uR'_u)\,du\mbox{\ \ and}
\\
\overline\beta_2&:=\frac1{\omega_{n-1}}\int_{S_1(0_p)}\beta_2(u)\,du
=\hat\beta_2+\frac1{\omega_{n-1}}\int_{S_1(0_p)}\frac 49\sum_{i=1}^n
{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_ue_i\bigr)\,du\end{aligned}$$ are of the claimed form, where $\hat\alpha_2,\hat\beta_2$ are as in Proposition \[prop:coeffs\]. For $\overline\alpha_2$ this follows immediately (with $\tilde\alpha_2:=\hat\alpha_2$) from the following formula (see the proof of Theorem 5.7 of [@NV]; details of the computation can be found on p. 170 of [@GV]): $$\label{eq:int16}
\int_{S_1(0_p)}{{\operatorname{Tr}}}(R'_uR'_u)\,du=\frac{3\omega_{n-1}}{n(n+2)(n+4)}|\nabla R|_p^2$$ This confirms the statement concerning $\overline\alpha_2$.
We now consider $\overline\beta_2$. Writing $u=\sum_{i=1}^n u_ie_i$ and $R_{ijk\ell}={\langle}R(e_i,e_j)e_k,e_\ell{\rangle}$ we have $$\begin{split}
\label{eq:int49}
\sum_{i=1}^n{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_u e_i\bigr)
&=\sum_{i,j,k,\ell=1}^n {\langle}R(e_i,e_j)e_k,e_\ell{\rangle}{\langle}R_ue_i,e_k{\rangle}{\langle}R_ue_j,e_\ell{\rangle}\\
&=\sum_{a,b,c,d=1}^n\Bigl[\sum_{i,j,k,\ell=1}^n R_{ijk\ell} R_{aibk} R_{cjd\ell}\Bigr]
u_au_bu_cu_d.
\end{split}$$ Note that the integral of $u_au_bu_cu_d$ over $S_1(0_p)$ is zero whenever $\{a,b,c,d\}$ contains at least three different elements. Abbreviating $A_{abcd}:=\sum_{i,j,k,\ell=1}^n R_{ijk\ell} R_{aibk} R_{cjd\ell}$ we have, using the Einstein condition and recalling the definition of $\hat R$ and ${{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}$ from Section \[sec:prelim\]: $$\begin{aligned}
\sum_{a,b=1}^n A_{aabb}&=\sum_{a,b,i,j,k,\ell=1}^n R_{ijk\ell} R_{aiak} R_{bjb\ell}
= C^2\sum_{i,j,k,\ell=1}^n R_{ijk\ell} \delta_{ik} \delta_{j\ell}
= C^2\sum_{i,j=1}^n R_{ijij} = nC^3,\\
\sum_{a,b=1}^n A_{abab}&=\sum_{a,b,i,j,k,\ell=1}^n R_{ijk\ell} R_{aibk} R_{ajb\ell}
= \sum_{a,b,i,j,k,\ell=1}^n R_{ijk\ell} R_{aibk} R_{ja\ell b}
= {{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p),\\
\sum_{a,b=1}^n A_{abba}&=\sum_{a,b,i,j,k,\ell=1}^n R_{ijk\ell} R_{aibk} R_{bja\ell}
=\sum_{a,b,i,j,k,\ell=1}^n R_{ijk\ell} R_{aibk} R_{jb\ell a} = {{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p)-\frac14\hat R(p),\end{aligned}$$ where for the last equality we have used formula (2.7)(vi) of [@Sa]; see also formula (2.15) of [@GV]. Let $S^{n-1}\subset R^n$ denote the $(n-1)$-dimensional standard sphere. Note that $\int_{S^{n-1}}u_1^2u_2^2\,du=\frac{\omega_{n-1}}{n(n+2)}$ and $\int_{S^{n-1}}u_1^4\,du=\frac{3\omega_{n-1}}{n(n+2)}$. From the above equations and (\[eq:int49\]) we thus obtain $$\begin{aligned}
\int_{S_1(0_p)}\sum_{i=1}^n&{{\operatorname{Tr}}}\bigl(R_u\circ R(e_i,\,.\,)R_u e_i\bigr)\,du\\
&=\sum_{ \genfrac{}{}{0pt}{} {a,b\in\{1,\ldots,n\}} {a\ne b} }
[A_{aabb}+A_{abab}+A_{abba}]\int_{S^{n-1}}
u_1^2u_2^2\,du+
\sum_{a=1}^n A_{aaaa}\int_{S^{n-1}}u_1^4\,du\\
&=\sum_{a,b=1}^n [A_{aabb}+A_{abab}+A_{abba}]\frac{\omega_{n-1}}{n(n+2)}
=\bigl(nC^3+2{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p)-\frac14\hat R(p)\bigr)\frac{\omega_{n-1}}{n(n+2)}\end{aligned}$$ Hence, $$\label{eq:b2}
\overline\beta_2=\hat\beta_2+\frac49\bigl(nC^3+2{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p)-\frac14\hat R(p)\bigr)\frac1{n(n+2)}
=\hat\beta_2+\frac{4C^3}{9(n+2)}+\bigl(\frac89{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p)-\frac19\hat R(p)\bigr)\frac1{n(n+2)}$$ Recall from Proposition \[prop:reqs\](ii), (iii) and equation (\[eq:lichn\]) that $$\begin{aligned}
112\hat R(p)-32{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p) &= 27|\nabla R|_p^2\,+\,\mbox{some constant depending only on }n,C,H,L\mbox{ and }\\
\hphantom{112}\hat R(p)+\hphantom{32}\llap{4}{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p)
&=\hphantom{27} |\nabla R|_p^2\,+\,\mbox{some constant depending only on }n,C,H,\end{aligned}$$ using which one easily computes that $$-\frac19\hat R(p)+\frac89{{\operatorname{\overset{\hphantom{i}_\circ}{\textit R}}}}(p) =
\frac18 |\nabla R|_p^2\,+\,\mbox{some constant depending only on }n,C,H,L.$$ Thus we conclude from (\[eq:b2\]): $$\overline\beta_2=\tilde\beta_2+\frac1{8n(n+2)}|\nabla R|_p^2\,,$$ where $\tilde\beta_2$ is a constant depending only on $n,C,H,L$.
[**Proof of the Main Theorem \[thm:main\]:**]{}
Let $M_1$, $M_2$ be harmonic spaces, $p_1\in M_1$, $p_2\in M_2$, and assume there exists ${{\varepsilon}}$ in the interval $(0,\min\{i(p_1),i(p_2)\})$ such that for each $0<r<{{\varepsilon}}$ the geodesic spheres $S_r(p_1)$ and $S_r(p_2)$ are isospectral. Then $\dim M_1=:n=\dim M_2$, and the heat invariants of the geodesic spheres coincide: $$a_k(S_r(p_1))=a_k(S_r(p_2))$$ for each $r\in(0,{{\varepsilon}})$ and all $k\in{{\mathbb N}}_0$. We want to deduce that $|\nabla R|_{p_1}^2=|\nabla R|_{p_2}^2$. Actually this will follow using just $a_0$ and $a_2$.
Reformulating the problem, let $M$ be an $n$-dimensional harmonic space and $p\in M$. We want to show that for any ${{\varepsilon}}\in(0,i(p))$, the two functions $${{\varphi}}_k:(0,{{\varepsilon}})\ni r\mapsto a_k(S_r(p))\in{{\mathbb R}}$$ with $k\in\{0,2\}$ together determine the value of $|\nabla R|_p^2$. By Remark \[rem:volscal\](i), the function $${{\varphi}}_0:r\mapsto a_0(S_r(p))={{\operatorname{vol}}}(S_r(p))=v(r)\omega_{n-1}$$ determines the constants $C,H,L$ associated with $M$ (see Section \[sec:prelim\]). Recall that the scalar curvature ${{\operatorname{{scal}}}}^S=:{{\operatorname{{scal}}}}^{S_r}$ of $S_r(p)$ is constant on the manifold $S_r(p)$, and that the function $v:(0,{{\varepsilon}})\to{{\mathbb R}}$ determines, by Remark \[rem:volscal\](ii), the function $(0,{{\varepsilon}})\ni r\mapsto{{\operatorname{{scal}}}}^{S_r}\in{{\mathbb R}}$. In particular, the function ${{\varphi}}_0=v\omega_{n-1}$ also determines the function $(0,{{\varepsilon}})\ni r\mapsto\int_{S_r(p)}({{\operatorname{{scal}}}}^S)^2{{\operatorname{\textit{dvol}}}}_{S_r(p)}={{\varphi}}_0(r)\cdot({{\operatorname{{scal}}}}^{S_r})^2\in{{\mathbb R}}$. By $$\begin{aligned}
{{\varphi}}_2(r)=a_2(S_r(p))&=\frac1{360}\int_{S_r(p)}\bigl
(5({{\operatorname{{scal}}}}^S)^2-2|{{\operatorname{{Ric}}}}^S|^2+2|R^S|^2\bigr){{\operatorname{\textit{dvol}}}}_{S_r(p)}\end{aligned}$$ it follows that the functions ${{\varphi}}_0$ and ${{\varphi}}_2$ together determine the function $$(0,{{\varepsilon}})\ni r\mapsto\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)}\bigl(|R^S|^2-|{{\operatorname{{Ric}}}}^S|^2\bigr)
{{\operatorname{\textit{dvol}}}}_{S_r(p)}\in{{\mathbb R}}.$$ By Proposition \[prop:intcoeffs\], the $r^2$-coefficient in the power series expansion of this function is the sum of the term $$\Bigl(\frac1{8n(n+2)}-\frac3{16n(n+2)(n+4)}\Bigr)|\nabla R|_p^2
=\frac{2n+5}{16n(n+2)(n+4)}|\nabla R|_p^2$$ and $\tilde\beta_2-\tilde\alpha_2$. Recall that the latter is a constant depending only on $n,C,H,L$, and is thus determined by ${{\varphi}}_0$. We conclude that the functions ${{\varphi}}_0$ and ${{\varphi}}_2$ together determine $|\nabla R|_p^2$, as claimed.
Geodesic balls
==============
\[sec:balls\]
In this section we will prove the following version of the Main Theorem \[thm:main\] for geodesic balls:
\[thm:balls\] Let $M_1$ and $M_2$ be harmonic spaces, and let $p_1\in M_1$, $p_2\in M_2$. If there exists ${{\varepsilon}}>0$ such that for each $r\in(0,{{\varepsilon}})$ the geodesic balls $B_r(p_1)$ and $B_r(p_2)$ are Dirichlet isospectral, then $|\nabla R|_{p_1}^2=|\nabla R|_{p_2}^2$. The same holds if the assumption of Dirichlet isospectrality is replaced by the assumption of Neumann isospectrality.
This theorem implies the corresponding analog of our Main Corollary \[cor:main\]:
\[cor:balls\] Let $M_1$ and $M_2$ be harmonic spaces. Assume that the Dirichlet isospectrality hypothesis of Theorem \[thm:balls\] is satisfied for *each* pair of points $p_1\in M_1$, $p_2\in M_2$. Then $M_1$ is locally symmetric if and only if $M_2$ is locally symmetric. The same holds if the assumption of Dirichlet isospectrality is replaced by the assumption of Neumann isospectrality.
For the proof of Theorem \[thm:balls\] we will use the heat invariants for manifolds with boundary. Let $M$ be an $n$-dimensional Riemannian manifold, and let $B\subset M$ be a compact domain with smooth boundary. If $\Delta$ denotes the Laplace operator on $B$ with Dirichlet boundary conditions then there is an asymptotic expansion $${{\operatorname{Tr}}}(\exp(-t\Delta))\sim(4\pi t)^{-n/2}\sum_k a_k^D(B) t^k$$ for $t\downarrow0$, where $k=0, 0.5, 1, 1.5, \ldots$ ranges over the nonnegative half integers (see [@BG]). For the Laplace operator on $B$ with Neumann boundary conditions the analog of this formula holds with certain coefficients $a_k^N(B)$. The coefficients $a_k^D(B)$ (resp. $a_k^N(B)$) are given by certain curvature integrals over $B$ and $\partial B$. One has $a_0^D(B)=a_0^N(B)={{\operatorname{vol}}}(B)$ and $a_{0.5}^D(B)=-a_{0.5}^N(B)=-\frac{\sqrt{\pi}}2
{{\operatorname{vol}}}(\partial B)$ (see [@BG]). In the proof of Theorem \[thm:balls\] we will use the explicit formulas for $a_2^D(B)$ and $a_2^N(B)$ from [@BG]. Let $\nu$ denote the outward pointing unit vector field on the boundary $\partial B$ of $B$, and let $\sigma=\nabla\nu$ be the associated shape operator. Let ${{\operatorname{{scal}}}}$, ${{\operatorname{{Ric}}}}$, $R$ always refer to the usual objects on $M$ (not to the ones associated with the induced metric on $\partial B$). Then $$\begin{aligned}
a_2^D(B)&=\frac1{360}\biggl[\int_B \bigl(-12\Delta({{\operatorname{{scal}}}})+5{{\operatorname{{scal}}}}^2-2|{{\operatorname{{Ric}}}}|^2+2|R|^2\bigr){{\operatorname{\textit{dvol}}}}_B\\
&+\int_{\partial B}\Bigl(18\nu({{\operatorname{{scal}}}})+20{{\operatorname{{scal}}}}\cdot{{\operatorname{Tr}}}(\sigma)
-4{{\operatorname{Tr}}}(R_\nu){{\operatorname{Tr}}}(\sigma)+12{{\operatorname{Tr}}}(R_\nu\circ\sigma)\\
&\hphantom{{}+\int_{\partial B}\Bigl(}-4{{\operatorname{Tr}}}(({{\operatorname{{Ric}}}}-R_\nu)\circ\sigma)
+\frac{40}{21}({{\operatorname{Tr}}}(\sigma))^3-\frac{88}7{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma^2)
+\frac{320}{21}{{\operatorname{Tr}}}(\sigma^3)\Bigr){{\operatorname{\textit{dvol}}}}_{\partial B}\biggr],\end{aligned}$$ $$\begin{aligned}
a_2^N(B)&=\frac1{360}
\biggl[\int_B \bigl(-12\Delta({{\operatorname{{scal}}}})+5{{\operatorname{{scal}}}}^2-2|{{\operatorname{{Ric}}}}|^2+2|R|^2\bigr){{\operatorname{\textit{dvol}}}}_B\\
&+\int_{\partial B}\Bigl(-42\nu({{\operatorname{{scal}}}})+20{{\operatorname{{scal}}}}\cdot{{\operatorname{Tr}}}(\sigma)
-4{{\operatorname{Tr}}}(R_\nu){{\operatorname{Tr}}}(\sigma)+12{{\operatorname{Tr}}}(R_\nu\circ\sigma)\\
&\hphantom{{}+\int_{\partial B}\Bigl(}-4{{\operatorname{Tr}}}(({{\operatorname{{Ric}}}}-R_\nu)\circ\sigma)
+\frac{40}{3}({{\operatorname{Tr}}}(\sigma))^3+8{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma^2)
+\frac{32}3{{\operatorname{Tr}}}(\sigma^3)\Bigr){{\operatorname{\textit{dvol}}}}_{\partial B}\biggr].\end{aligned}$$
If $M$ is harmonic then, by the results of Section \[sec:prelim\], the previous formulas simplify to $$\label{eq:a2d}
\begin{split}
a_2^D(B)={}&\frac1{360}\Bigl[{{\operatorname{vol}}}(B)\cdot\bigl(5(nC)^2-2nC^2+\frac43n((n+2)H-C^2)\bigr)\\
&+\int_{\partial B}\Bigl(20nC{{\operatorname{Tr}}}(\sigma)
-8C{{\operatorname{Tr}}}(\sigma)+16{{\operatorname{Tr}}}(R_\nu\circ\sigma)\\
&\hphantom{{}+\int_{\partial B}\bigl(}
+\frac{40}{21}({{\operatorname{Tr}}}(\sigma))^3-\frac{88}7{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma^2)
+\frac{320}{21}{{\operatorname{Tr}}}(\sigma^3)\Bigr){{\operatorname{\textit{dvol}}}}_{\partial B}\Bigr],
\end{split}$$ $$\begin{split}
\label{eq:a2n}
a_2^N(B)&=\frac1{360}
\Bigl[{{\operatorname{vol}}}(B)\cdot\bigl(5(nC)^2-2nC^2+\frac43n((n+2)H-C^2)\bigr)\\
&+\int_{\partial B}\Bigl(20nC{{\operatorname{Tr}}}(\sigma)
-8C{{\operatorname{Tr}}}(\sigma)+16{{\operatorname{Tr}}}(R_\nu\circ\sigma)\\
&\hphantom{{}+\int_{\partial B}\bigl(}
+\frac{40}3({{\operatorname{Tr}}}(\sigma))^3+8{{\operatorname{Tr}}}(\sigma){{\operatorname{Tr}}}(\sigma^2)
+\frac{32}3{{\operatorname{Tr}}}(\sigma^3)\Bigr){{\operatorname{\textit{dvol}}}}_{\partial B}\Bigr].
\end{split}$$
In the proof of Theorem \[thm:balls\] we will follow a similar strategy as in the proof of Theorem \[thm:main\]. To this end, we need some preliminary results in the special case that $B=B_r(p)$ with $r\in(0,i(p))$ and $M$ is harmonic. We remark – without going into details this time – that one can compute in this case, using equation (\[eq:pow\]), Proposition \[prop:constants\], and the Taylor series expansion $$P_{\gamma_u}^{r,0}\circ(R_\nu)_{\gamma_u(r)}\circ P_{\gamma_u}^{0,r}=\sum_{k=0}^\infty
\frac{r^k}{k!}R^{(k)}_u,$$ that the $r^3$-coefficient in the power series expansion of ${{\operatorname{Tr}}}(R_\nu\circ\sigma)$ equals $-\frac1{1440}L+\frac1{96}{{\operatorname{Tr}}}(R'_uR'_u)$, and that the $r^3$-coefficient in the power series expansion of ${{\operatorname{Tr}}}(\sigma^3)$ equals $\frac1{30240}L-\frac1{96}{{\operatorname{Tr}}}(R'_uR'_u)$. (That the contributions of ${{\operatorname{Tr}}}(R'_uR'_u)$ in these terms are negatives of each other can also be checked as follows: Using (\[eq:riccati\]) twice, we have ${{\operatorname{Tr}}}(R_\nu\circ\sigma)+{{\operatorname{Tr}}}(\sigma^3)=
-{{\operatorname{Tr}}}(\sigma'\sigma)=-\frac12{{\operatorname{Tr}}}(\sigma^2)'=\frac12{{\operatorname{Tr}}}(R_\nu+\sigma')'
=\frac12{{\operatorname{Tr}}}(\sigma)''$ whose $r^3$-coefficient indeed depends only on $L$ by (\[eq:powtrharm\]).) Using (\[eq:int16\]), we conclude that the $r^3$-coefficient in the power series expansion of $r\mapsto\frac1{vol(S_r(p))}\int_{S_r(p)}{{\operatorname{Tr}}}(R_\nu\circ\sigma){{\operatorname{\textit{dvol}}}}_{S_r(p)}$ is $$\label{rsigcoeff}
-\frac1{1440}L+\frac1{32n(n+2)(n+4)}|\nabla R|_p^2\,,$$ Similarly, the $r^3$-coefficient in the power series expansion of $r\mapsto\frac1{vol(S_r(p))}\int_{S_r(p)}{{\operatorname{Tr}}}(\sigma^3){{\operatorname{\textit{dvol}}}}_{S_r(p)}$ is $$\label{sig3coeff}
\frac1{30240}L-\frac1{32n(n+2)(n+4)}|\nabla R|_p^2\,.$$
[**Proof of Theorem \[thm:balls\]:**]{}
Let $M_1$, $M_2$ be harmonic spaces, $p_1\in M_1$, $p_2\in M_2$, and assume there exists ${{\varepsilon}}$ in the interval $(0,\min\{i(p_1),i(p_2)\})$ such that for each $0<r<{{\varepsilon}}$ the geodesic spheres $B_r(p_1)$ and $B_r(p_2)$ are Dirichlet isospectral (resp. Neumann isospectral). Then $\dim M_1=:n=\dim M_2$, and the heat invariants of the geodesic spheres coincide: $$a^D_k(B_r(p_1))=a^D_k(B_r(p_2)),\mbox{\ \ resp.\ \ }a^N_k(B_r(p_1))=a^N_k(B_r(p_2))$$ for each $r\in(0,{{\varepsilon}})$ and all $k\in{{\mathbb N}}_0$. We want to deduce that $|\nabla R|_{p_1}^2=|\nabla R|_{p_2}^2$. Actually this will follow using just $a_0$ and $a_2$. (We remark without proof here that, viewed as functions of $r$, the heat coefficients $a_{0.5}$, $a_1$, and $a_{1.5}$ do actually not contain more information than $a_0$ in our situation.)
We first consider the case of Dirichlet conditions. Similarly as in the proof of Theorem \[thm:main\], we reformulate the problem as follows: Let $M$ be an $n$-dimensional harmonic space and $p\in M$. We want to show that for any ${{\varepsilon}}\in(0,i(p))$, the two functions $$\psi^D_k:(0,{{\varepsilon}})\ni r\mapsto a^D_k(B_r(p))\in{{\mathbb R}}$$ with $k\in\{0,2\}$ together determine the value of $|\nabla R|_p^2$. Note that the function $$\psi^D_0:r\mapsto a^D_0(B_r(p))={{\operatorname{vol}}}(B_r(p))$$ determines its own derivative which is just $$r\mapsto{{\operatorname{vol}}}(S_r(p))=v(r)\omega_{n-1}$$ (see the previous section). By Remark \[rem:volscal\](i), we conclude that $\psi^D_0$ again determines the constants $C,H,L$ associated with $M$. Moreover, the function $v:(0,{{\varepsilon}})\to{{\mathbb R}}$ determines the radial functions ${{\operatorname{Tr}}}(\sigma)=v'/v$ and ${{\operatorname{Tr}}}(\sigma^2)=-(v'/v)'-C$ (compare Remark \[rem:tracesv\]). By (\[eq:a2d\]) it now follows that $\psi^D_0$ and $\psi^D_2$ together determine the function $$(0,{{\varepsilon}})\ni r\mapsto\frac1{{{\operatorname{vol}}}(S_r(p))}\int_{S_r(p)}\bigl(
16{{\operatorname{Tr}}}(R_\nu\circ\sigma)+\frac{320}{21}{{\operatorname{Tr}}}(\sigma^3)\bigr)
{{\operatorname{\textit{dvol}}}}_{S_r(p)}\in{{\mathbb R}}.$$ Recalling (\[rsigcoeff\]) and (\[sig3coeff\]), we see that the $r^3$-coefficient in the power series expansion of the latter function is the sum of $$\frac1{32n(n+2)(n+4)}\bigl(16-\frac{320}{21}\bigr)|\nabla R|_p^2
=\frac1{42n(n+2)(n+4)}|\nabla R|_p^2$$ and a term depending only on $L$. Since $L$ is determined by $\psi^D_0$, we conclude that $\psi^D_0$ and $\psi^D_2$ together determine $|\nabla R|_p^2$, as claimed.
In the Neumann case, letting $\psi^N_k(r):=a^N_k(B_r(p))$, we again have $\psi^N_0(r)={{\operatorname{vol}}}(B_r(p))=\psi^D_0(r)$. Proceeding exactly as in the Dirichlet case, using (\[eq:a2n\]) this time, we see that $\psi^N_0$ and $\psi^N_2$ together determine the sum of $$\frac1{32n(n+2)(n+4)}\bigl(16-\frac{32}3\bigr)|\nabla R|^2_p
=\frac1{6n(n+2)(n+4)}|\nabla R|_p^2$$ and a term depending only on $L$. Hence they determine $|\nabla R|_p^2$, as claimed.
[99]{}
J. Berndt, F. Tricerri, L. Vanhecke, *Generalized Heisenberg groups and Damek-Ricci harmonic spaces*, , Springer-Verlag, Berlin/Heidelberg/New York, 1995.
A.L. Besse, *Manifolds all of whose geodesics are closed*, , Springer-Verlag, Berlin/New York, 1978.
T. Branson, P.B. Gilkey, *The asymptotics of the Laplacian on a manifold with boundary*, **15** (1990), no. 2, 245–272.
B.-Y. Chen, L. Vanhecke, *Differential geometry of geodesic spheres*, **325** (1981), 28–67.
E.T. Copson, H.S. Ruse, *Harmonic Riemannian spaces*, **60** (1940), 117–133.
E. Damek, F. Ricci, *A class of nonsymmetric harmonic Riemannian spaces*, **27** (1992), no. 1, 139–142.
D. DeTurck, J. Kazdan, *Some regularity theorems in Riemannian geometry*, **14** (1981), 249–260.
H. Fürstenau, *Über Isospektralität von topologischen Bällen*, Diploma thesis, Universität Bonn, 2006.
P. Gilkey, *Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem*, , Publish or Perish, Wilmington, Del., 1984.
C. Gordon, *Survey of isospectral manifolds*, , Vol. I, 747–778, North-Holland, Amsterdam, 2000.
C. Gordon, *Isospectral deformations of metrics on spheres*, **145** (2001), 317–331.
A. Gray, L. Vanhecke, *Riemannian geometry as determined by the volumes of small geodesic balls*, **142** (1979), no. 3-4, 157–198.
A. Lichnerowicz, *Sur les espaces Riemanniens complètement harmoniques*, **72** (1944), 146–168.
A. Lichnerowicz, *Géométrie des groupes de transformations*, Dunod, Paris, 1958.
L. Nicolodi, L. Vanhecke, *The geometry of $k$-harmonic manifolds*, **6** (2006), no. 1, 53–70.
H.S. Ruse, A.G. Walker, T.J. Willmore, *Harmonic Spaces*, , Rome, 1961.
T. Sakai, *On eigen-values of Laplacian and curvature of Riemannian manifolds*, **23** (1971), 589–603.
Z.I. Szabó, *The Lichnerowicz conjecture on harmonic manifolds*, **31** (1990), no. 1, 1–28.
Z.I. Szabó, *Locally non-isometric yet super isospectral spaces*, **9** (1999), no. 1, 185–214.
Z.I. Szabó, *Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries*, **154** (2001), no. 2, 437–475.
Z.I. Szabó, *A cornucopia of isospectral pairs of metrics on spheres with different local geometries*, **161** (2005), no. 1, 343–395.
Y. Watanabe, *On the characteristic function of harmonic Kählerian spaces*, **27** (1975), 13–24.
[^1]: The authors were partially supported by DFG Sonderforschungsbereich 647. The first author’s work has also been supported by D.G.I. (Spain) and FEDER Project MTM2010-15444, by Junta de Extremadura and FEDER funds, and the program “Estancias de movilidad en el extranjero ‘José Castillejo’ para jóvenes doctores” of the Ministry of Education (Spain).\
to*Dedication.* Dorothee Schueth would like to dedicate this article to her former high school teacher, Martin Berg. It was he who first made her see the beauty of Mathematics.
|
---
abstract: 'We use the very high resolution, fully cosmological simulations from the [*Aquarius*]{} project, coupled to a semi-analytical model of galaxy formation, to study the phase-space distribution of halo stars in “solar neighbourhood”-like volumes. We find that this distribution is very rich in substructure in the form of stellar streams for all five stellar haloes we have analysed. These streams can be easily identified in velocity space, as well as in spaces of pseudo-conserved quantities such as $E$ vs. $L_{z}$. In our best-resolved local volumes, the number of identified streams ranges from $\approx 300$ to 600, in very good agreement with previous analytical predictions, even in the presence of chaotic mixing. The fraction of particles linked to (massive) stellar streams in these volumes can be as large as $84\%$. The number of identified streams is found to decrease as a power-law with galactocentric radius. We show that the strongest limitation to the quantification of substructure in our poorest-resolved local volumes is particle resolution rather than strong diffusion due to chaotic mixing.'
author:
- |
Facundo A. Gómez$^{1,2}$[^1], Amina Helmi$^{3}$, Andrew P. Cooper$^{4}$, Carlos S. Frenk$^{5}$, Julio F. Navarro$^{6}$, Simon D. M. White$^{7}$\
$^{1}$ Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA\
$^{2}$ Institute for Cyber-Enabled Research, Michigan State University, East Lansing, MI 48824, USA\
$^{3}$ Kapteyn Astronomical Institute, University of Groningen,P.O. Box 800, 9700 AV Groningen, The Netherlands\
${4}$ National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang, Beijing 100012, China\
${5}$ Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK\
${6}$ CIfAR Senior Fellow, Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada V8P 5C2\
${7}$ Max-Planck-Institut f[ü]{}r Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany
bibliography:
- 'aquarius.bib'
title: Streams in the Aquarius stellar haloes
---
\[firstpage\]
galaxies: formation – galaxies: kinematics and dynamics – methods: analytical – methods: N-body simulations
Introduction
============
In the last decade the characterization of the phase-space distribution of stars in the vicinity of the Sun has become a subject of great interest. It is now well-established that by studying the phase-space distribution function of the Milky Way we may be able to unveil its formation history [@FBH; @hreview].
According to the current paradigm of galaxy formation, the stellar haloes of large galaxies such as our own are mostly formed through the accretion and mergers of smaller objects [@sz78], rather than by a rapid collapse of a large and pristine gas cloud [@ELS]. During the process of accretion, these systems are tidally disrupted and contribute gas and stars to the final object. Whereas the gas component rapidly forgets its dynamical origin due to its dissipative nature, stars should retain this information for much longer time scales because their density in phase-space is conserved. This memory is especially easily accessible in the outer regions of galaxies, or outer stellar haloes, where the dynamical mixing time scales are very long. A large amount of observed stellar streams in the outer regions of the Milky Way supports this picture for the formation of its stellar halo. The Sagittarius stream [@ibata94; @ibata01b] and the Orphan stream [@belu07] are just two examples among many other overdensities recently discovered [see also @new02; @ibata03; @yanny03; @belu06; @grill; @else09].
This formation model also predicts that for Milky Way-like galaxies, the portion of the accreted stellar halo that dominates in the solar neighbourhood is dynamically old. [e.g. @cooper; @tiss12]. Therefore, fossil signatures of the most ancient accretion events that a galaxy has experienced should be buried in their inner regions or, in the case of the Milky Way, close to or in the Solar Neighbourhood. However, due to the very short dynamical time scales characteristic of these regions, debris from accretion events is expected to be spatially well-mixed and therefore rather difficult to detect. Furthermore, in cold dark matter models, dark matter haloes are expected to be strongly triaxial [see e.g. @Allgood06; @vera10]. As a consequence, chaotic orbits may be quite important and result in much shorter mixing time-scales [see, e.g., @vol08], which leads to a phase-space distribution that is effectively smooth. On the other hand, the dissipative condensation of baryons can induce a transformation of the structure of dark haloes to a more oblate and axisymmetric shape in the inner regions while preserving their triaxial shape in the outer parts [@deba; @abadi; @val10; @bryan].
The first attempt to quantify the amount of substructure in the form of stellar streams expected in the Solar Neighbourhood was carried out by @hw. They suggested that $\sim 300$ - $500$ kinematically-coherent substructures should be present in a local volume around the Sun. Although they considered a fixed Galactic potential, their results were later confirmed by @hws03, who analysed a full cosmological simulation of the formation of a cluster dark matter halo scaled down to galaxy size. They also showed that debris from accretion events appeared to mix on time-scales comparable to those expected for integrable potentials. However, their results were based on a single N-body simulation and were limited by its numerical resolution (despite the 66 million particles used).
Stellar streams in the inner regions of the Galactic halo cannot be detected simply by looking for overdensities in the distribution of stars on the sky. Thus, several different spaces have been proposed and explored to identify substructure. Examples are the velocity space [@hw; @hwzz99; @W11], metallicity, colour and chemical abundance spaces [e.g. @font; @schlau; @anto; @val13] or spaces defined by integrals of motion and their associated orbital frequencies [e.g. @hz00; @bj05; @knebe05; @mcmillan; @klement09; @morri09; @gh10a; @gh10b]. At first sight, these studies as well as that by @hws03 might be in tension with the results reported by @val13. These authors found no substructure in the integral of motion space of halo stars formed in a fully cosmological hydrodynamical simulation. They attributed the lack of substructure to the strong chaotic mixing present in this simulation. Their results were based on samples of $\approx 10^{4}$ stellar particles distributed across the entire halo. This likely imposes a strong limitation on the ability to identify and quantify substructure. If, as discussed by @hws03, there are 300-500 streams crossing the Solar neighbourhood a resolution of at least a few thousand stars in a local small volume (of a few kpc across) is required. Hence a much larger number across the whole halo. Therefore samples of a total of 10,000 halo stars would really be too small to find or characterize substructure. We confirm this intuition below, where we show that substructure is clearly visible when the numerical resolution (particle number) is large enough.
Thanks to current surveys such as RAVE [@rave] and SEGUE [@segue], as well as the forthcoming astrometric satellite [ *Gaia*]{} [@perry], model predictions are becoming testable. Although to date only a handful of stellar streams have been detected in the Solar Neighbourhood [e.g. @hwzz99; @klement09; @smith09], with the full 6-D phase-space catalogues that are already becoming available [@maarten; @bond10; @carollo10; @zwitter; @burnett; @beers12] it may soon be possible to start deciphering the formation history of the Milky Way.
In this work we revise the predictions made by @hws03. Our goal is to characterize and quantify the amount of substructure inside solar neighbourhood like volumes obtained from the fully cosmological very high resolution simulation of galactic stellar haloes modelled by @cooper. In Section \[sec:methods\] we briefly describe these models. We characterize the phase-space distribution of stellar particles in “solar neighbourhood” spheres in Section \[sec:charac\]. In Section \[sec:quanti\] we measure the number of stellar streams, and explore the relevance of orbital chaos for halo stars near the Sun, as well as the impact of particle resolution on the quantification of substructure. Finally, in Section \[sec:conclu\] we summarise our results.
Throughout this work we use the term substructure *only* to refer to substructure in the form of stellar streams.
------ ----------- -------------- --------------- -------------- ----------------- ---------------- ---------------- -------------- ----------- --------------- ---------------------
Name $r_{200}$ $N_{200}$ $V_{\rm max}$ $m_{\rm p}$ $c_{{\rm NFW}}$ $b/a^{\it sn}$ $c/a^{\it sn}$ $M_{*}$ $r_{1/2}$ $M_{\rm gal}$ $\log_{10}~(m_{*})$
\[$10^{6}$\] \[$10^{3}$\] \[$10^{8}$\] \[$10^{10}$\] RMS
A-2 $246$ $135$ $209$ $13.7$ $16.2$ $0.65$ $0.53$ $3.8$ $20$ $1.88$ $1.06$
B-2 $188$ $127$ $158$ $6.4$ $9.7$ $0.46$ $0.39$ $5.6$ $2.3$ $1.49$ $1.30$
C-2 $243$ $127$ $222$ $14.0$ $15.2$ $0.55$ $0.46$ $3.9$ $53$ $7.84$ $1.22$
D-2 $243$ $127$ $203$ $14.0$ $9.4$ $0.67$ $0.58$ $11.1$ $26$ $0.72$ $2.18$
E-2 $212$ $124$ $179$ $9.6$ $8.3$ $0.67$ $0.46$ $18.5$ $1.0$ $0.45$ $1.90$
------ ----------- -------------- --------------- -------------- ----------------- ---------------- ---------------- -------------- ----------- --------------- ---------------------
Masses are in M$_{\odot}$, distances in kpc and velocities in km s$^{-1}$.
Note that our stellar halo masses ($M_{*}$) also includes the mass assigned to the bulge component in @cooper.
The Simulations {#sec:methods}
===============
In this work we analyse the suite of high-resolution $N$-body simulations from the [*Aquarius Project*]{} [@springel2008a; @springel2008b], coupled with the [GALFORM]{} semi-analytic model [@cole94; @cole00; @bower06] as described by @cooper.
N-body simulations
------------------
![Cumulative stellar mass fractions obtained from our solar neighbourhood-like spheres, centred at 8 kpc from the galactic centre. The step-like cumulative fractions show the contribution from the five most massive contributors to each sphere in rank order. The most massive contributor is ranked as number 1. The red dashed line shows the $90\%$ level.[]{data-label="fig:mass_halos"}](fig1.eps){width="84mm"}
The [*Aquarius Project*]{} has simulated the formation of six Milky Way-like dark matter haloes in a $\Lambda$CDM cosmology. The simulations were carried out using the parallel Tree-PM code GADGET-3 [an upgraded version of GADGET-2, @springel2005a]. Each halo was first identified in a lower resolution version of the Millennium-II Simulation [@bk09] which was carried out within a periodic box of side 125 h$^{-1}$ Mpc in a cosmology with parameters $\Omega_{m} = 0.25$, $\Omega_{\Lambda}= 0.75$, $\sigma_{8} = 0.9$, $n_{s} = 1$, and Hubble constant $H_{0} = 100~h$ km s$^{-1}$ Mpc$^{-1}$ = 73 km s$^{-1}$ Mpc$^{-1}$. The haloes were selected to have masses comparable to that of the Milky Way and to be relatively isolated at $z=0$. By applying a multi-mass particle ‘zoom-in’ technique, each halo was re-simulated at a series of progressively higher resolutions. The results presented in this work are based on the simulations with the second highest resolution (Aq-2) available, with a Plummer equivalent softening length of 65.8 pc. In all cases 128 outputs, starting from redshift $z \approx 45$[^2], were stored. In each snapshot, dark matter haloes were identified using a Friends-of-Friends [@fof] algorithm while subhaloes were identified with SUBFIND [@SUBFIND]. For more details about the simulations, we refer the reader to @springel2008a [@springel2008b]. The main properties of the resulting haloes are summarised in Table \[table:aquarius\]. Note that we exclude from our analysis the halo Aq-F. This is due to its assembly history; more than $95\%$ of its accreted stellar halo mass comes from a single galaxy that was accreted very late, at $z \approx
0.7$.
Semi-analytic model
-------------------
------ ----------------- ----------------------- -------------- ----------------- --------------
Name $\rho_{0}/10^4$ $N_{\rm ms}^{\it sn}$ $\sigma_{R}$ $\sigma_{\phi}$ $\sigma_{Z}$
A-2 1.54 5 149.4 130.9 90.5
B-2 15.4 3 85.3 51.1 45.5
C-2 2.55 3 148.2 98.1 80.7
D-2 8.24 5 175.1 88.5 75.7
E-2 3.44 3 93.8 62.3 55.0
------ ----------------- ----------------------- -------------- ----------------- --------------
: Properties of the distribution of stellar particles inside our “solar neighbourhood” ([*sn*]{}) spheres of 2.5 kpc radius. The first column labels the simulation. From left to right, the columns give the local stellar density, $\rho_{0}$, the number of most significant stellar mass contributors, $N^{\it sn}_{\rm ms}$ (see text) and the three components of the velocity ellipsoid.[]{data-label="table:solar_volume"}
To study the growth and properties of stellar haloes, @cooper coupled the semi-analytic model [GALFORM]{} to the [*Aquarius*]{} dark matter $N$-body simulations. The basic idea behind semi-analytic techniques is to model the evolution of the baryonic components of galaxies through a set of observationally and/or theoretically motivated analytic prescriptions. Critical physical processes that govern galaxy formation, such as gas cooling, star formation feedback, supernovae, winds of massive stars and AGN, etc. are taken into account. The parameters that control these processes were fixed on the basis of an implementation of [GALFORM]{} on the Millennium Simulation [@springel2005b], as this set of choices allows one to successfully reproduce the properties of galaxies on large scales as well as those of the Milky Way [@bower06; @cooper].
{width="175mm"}\
{width="175mm"}
{width="176mm"}
{width="175mm"}\
{width="175mm"}
Building up stellar haloes {#sec:build}
--------------------------
Within the context of CDM, stellar haloes may be formed via hierarchical aggregation of smaller objects, each one of them imprinting their own chemical and dynamical signatures on the final halo’s properties [@FBH]. As previously explained, the [ GALFORM]{} model provides a detailed description of the composite stellar population of every galaxy in the simulation at any given time. However it does not follow the dynamics of these systems fully. To study the dynamical properties of the resulting galactic stellar haloes @cooper assumed that the most strongly bound dark matter particles in progenitor satellites could be used to trace the phase-space evolution of their stars. In every snapshot of the simulation a fixed fraction of the most-bound dark matter particles were selected to trace any newly formed stellar population in each galaxy in the simulation. This implies that each tagged particle has a different final stellar mass associated to it. The fraction of selected bound particles is set such that properties of the satellite population at $z=0$, like their luminosities, surface brightness, half-light radii and velocity dispersions, are consistent with those observed for the Milky Way and M31 satellites. For a detailed description of this procedure we refer the reader to Section 3 of @cooper. The properties of the resulting stellar haloes obtained in each simulation, such as total mass, half-light radius and the root-mean-square (RMS) scatter in the logarithm of the stellar mass assigned to the individual particles are listed in Table \[table:aquarius\]. Note that, although the method introduced in @cooper leads to a successful match of various observables regarding the structure and characteristics of the Galactic stellar halo and its satellite population, the dynamical evolution of the baryonic components of galaxies is much simplified. This likely has an effect on e.g. the efficiency of satellite mass loss due to tidal stripping, or the satellite’s internal structural changes due to adiabatic contraction, and possibly even on its final radial distribution. [@LY10; @RS10; @SM11; @G13].
Characterization of Substructure in solar volumes {#sec:charac}
=================================================
In this Section we characterize the phase-space distribution of stellar particles inside a “solar neighbourhood” sphere located at 8 kpc from the galactic centre of each [*Aquarius*]{} stellar halo. Following @gh10b, we chose for the spheres a radius of 2.5 kpc as this is approximately the distance within which the astrometric satellite [*Gaia*]{} [@perry] will be able to provide extremely accurate 6D phase-space measurements for an unprecedentedly large number of stars. The final configuration of the host dark matter haloes is, in all cases, strongly triaxial. Therefore, to allow a direct comparison between the “solar neighbourhood” spheres from different haloes, we have rotated each halo to its set of principal axis and placed the sphere along the direction of the major axis. In all cases, the ratios and directions of the principal axis were computed using dark matter particles located within 6 to 12 kpc.
Table \[table:solar\_volume\] shows that $90\%$ of the total stellar mass enclosed in these spheres comes, in all cases, from 3-5 significant contributors, $N_{\rm ms}^{\it sn}$. This is in agreement with @lh [@cooper], who find that stellar haloes are predominantly built from fewer than 5 satellites with masses comparable to the brightest classical dwarf spheroidals of the Milky Way. Figure \[fig:mass\_halos\] shows the stellar mass fractions contributed by the five most significant contributors to the total stellar mass enclosed in each sphere.
In Figure \[fig:vela\] we present two different projections of velocity space. The different colours indicate particles coming from different satellites that have contributed with at least five particles (while those from satellites contributing fewer particles are shown in black). It is interesting to observe how these distributions in velocity space vary from halo to halo. Essentially, less massive (dark) haloes have smaller velocity dispersions and thus the distribution of particles in velocity space is more compact. The values of the velocity dispersions of these distributions are listed in Table \[table:solar\_volume\]. A comparison with the estimated values of the velocity ellipsoid of the local stellar halo [see @chibabeers] $(\sigma_{R},\sigma_{\phi},\sigma_{Z}) =
(141 \pm 11, 106 \pm 9, 94 \pm 8)$ km s$^{-1}$ shows that the ellipsoids of haloes Aq-A-2, -C-2 and -D-2 have amplitudes comparable to those observed for the Milky Way. Note, however, that the dynamics of the stellar particles in these simulations are only determined by the underlying dark matter halo potential. If the mass associated to the disc and the bulge were to be included, the velocity dispersions would be significantly increased, since we may relate the velocity dispersion $\sigma^{2}$ to the circular velocity $V_c^2$ and $V_c^2
\propto M(<r) = M_{\rm disc} + M_{\rm halo}$, and $M_{\rm disc} \sim
M_{\rm halo}$ near the Sun [@bt]. In Table \[table:solar\_volume\] we also show the local stellar halo average densities, $\rho_{0}$. We find that, except for halo Aq-B-2, the values obtained are in reasonable agreement with the estimates for the Solar Neighbourhood, $\rho_{0} = 1.5 \times
10^{4}$ M$_{\odot}$ kpc$^{-3}$ [see, e.g., @fuchs]. However, one should bear in mind that, if a disc component were to be included, this should lead to a contraction of the halo, and hence to a larger density, especially on the galactic disc plane [e.g., @abadi; @deba]. The highest local density is found for halo Aq-B-2. This halo has a low stellar mass compared to haloes Aq-D-2 and E-2, but is much more centrally concentrated than Aq-D-2. In comparison to Aq-E-2, which is also very centrally concentrated, Aq-B-2 has a strongly prolate shape, which explains its much higher value of $\rho_{0}$ on the major axis, where our “solar neighbourhood” sphere is placed.
From Figure \[fig:vela\] we can also appreciate the very large amount of substructure, coming from many different objects, that is present in these volumes. This substructure, in the form of stellar streams, can be seen as groups of unicoloured star particles. Recall that these stellar haloes are built up in a fully cosmological scenario. Therefore, effects such as violent variation of the host potential due to merger events, and chaotic orbital behaviour induced by the strongly triaxial dark matter haloes [@vera10] are naturally accounted for in these simulations. Nonetheless, these physical processes have not been efficient enough to completely erase the memory of the origin of the stellar halo particles. Note that, for this analysis, we have treated all tagged particles equally. As described in Section \[sec:build\], the stellar-to-total mass ratio varies from particle to particle. Thus, the relative masses of the identified streams as judged by their particle number could be quite different from their relative stellar mass. We will explore this further in the following Section.
Evidence for this can also be seen in Figure \[fig:ener\], which shows the distribution of stellar particles in the space of the pseudo-conserved quantities $E_{\rm norm}$ and $L_{z}$, where $$E_{\rm norm}= \dfrac{E - E_{\min}}{E_{\rm max} - E_{\rm min}},$$ with $E_{\rm max}$ and $E_{\rm min}$ are the energies of the most and the least bound stellar particles inside the volume under analysis, respectively. To compute the energy we assume a smooth and spherical representation of the underlying gravitational potential, as given by @nfw $$\label{NFW_prf}
\Phi(r)=-\frac{GM_{\rm 200}}{r\left[\ln(1+c_{\rm NFW})-c_{\rm
NFW}/(1+c_{\rm NFW})\right]} \\ \ln\left(1+\frac{r}{r_{s}}\right),$$ with values for the parameters at redshift $z=0$ as listed in Table \[table:aquarius\]. As expected, substructure in this space is much better defined than in velocity space [see also e.g. @hz00; @knebe05; @font; @gh10a]. In this projection of phase-space streams from the same satellite tend to cluster together and can be observed as well defined clumps. Note however that within a given clump substructure associated with the various streams crossing the “solar neighbourhood” may be apparent [see, e.g., @gh10a].
To search for stellar streams in the Solar Neighbourhood without assuming an underlying Galactic potential, @klement09 introduced the space of $\nu$, $V_{\Delta E}$ and $V_{\rm az}$, where $$\begin{array}{lll}
\displaystyle
\label{Klements}
\nu = \arctan\left(\dfrac{V + V_{\rm LSR}}{W}\right), \\
\\
\displaystyle
V_{\Delta{\rm E}} = \sqrt{U^2 + 2(V_{\rm LSR} - V_{\rm az})^2}, \\
\\
\displaystyle
V_{\rm az} = \sqrt{(V + V_{\rm LSR})^2 + W^2}.
\end{array}$$ These are based only on kinematical measurements. For a star in the Galactic plane, $\nu$ would be the angle between the orbital plane and the direction towards the North Galactic Pole, $V_{\rm az}$ is related to the angular momentum and $V_{\Delta {\rm E}}$ is a measure of a star’s eccentricity. Under the assumption of a spherical potential and a flat rotation curve, stars in a given stellar stream should be distributed in a clump when projected onto this space. The distribution of stellar particles located inside our “solar neighbourhood” spheres projected onto the spaces of $V_{\rm az}$ vs. $V_{\Delta {\rm E}}$ and $\nu$ vs. $V_{\Delta {\rm E}}$ are shown in Figure \[fig:klm\]. Although perhaps less sharply defined than in $E$ vs. $L_{z}$ space, substructure stands out in these two projections of phase-space.
Quantification of Substructure in solar volumes {#sec:quanti}
===============================================
In the previous Section we found that the phase-space distribution of stellar particles inside a “solar neighbourhood” sphere is very rich in substructure in all of our haloes. We will now quantify the number of stellar streams crossing the “solar neighbourhoods” of our [*Aquarius*]{} haloes and characterize their evolution in time. Our goal is to compare the results of this analysis with previous studies [@hw; @hws03] that have analytically estimated this number and concluded that the merger history of the Milky Way could be recovered from the phase-space distribution of Solar Neighbourhood stars. Furthermore, we will address what fraction of the stellar particles that appear to be smoothly distributed in phase-space are actually in streams that could not be resolved due to the high but limited particle resolution of our $N$-body simulations. We will also establish the importance of chaotic mixing for the quantification of substructure.
Resolved substructure {#sec:resolved_subs}
---------------------
To quantify the number of streams inside our spheres we first need to specify how to identify them. Our definition of a stream must ensure that particles in the same stream share the same orbital phase and progenitor. Following @hws03, a stream is identified when [ *i)*]{} two or more particles from the same parent satellite are found within one of our 2.5 kpc solar neighbourhood spheres at redshift $z=0$ and [*ii)*]{} they share at least one particle from the same parent satellite that has never been separated by more than a distance $r_{\rm stream}$ from either of them[^3]. The value of $r_{\rm stream}$ is set to account for the stellar streams that, due to numerical resolution, are poorly resolved in the “solar neighbourhood”. In this work we adopt $r_{\rm stream} = 0.8 \times
r_{\rm apo}$ for each particle, where $r_{\rm apo}$ is its apocenter at $z=0$, estimated from the outputs of the $N$-body simulations. Varying the value of $r_{\rm stream}$ between $0.6$ - $0.9 \times
r_{\rm apo}$ did not significantly affect our results. This is required because a parent satellite can contribute with multiple stellar streams that may cross each other in configuration space [see, e.g., @hw]. Note that using $r_{\rm apo}$ as a scale allows us to naturally adapt the algorithm to different orbital configurations.
{width="170mm"}\
{width="170mm"}\
{width="170mm"}
In practice, we proceed as follows:
- For each particle inside our solar neighbourhood sphere of 2.5 kpc radius at redshift $z=0$ we identify its parent satellite.
- We measure the time $t_{\rm form}$ (prior the time of accretion) when the spatial extent of all particles associated with this parent takes its smallest value (as measured by the root-mean-square dispersion of distance from the centre of mass of the satellite).
- At $t=t_{\rm form}$ we identify all particles located in a sphere of 4 kpc radius, centred on the selected particle. This radius has to be large enough to initially contain all potential neighbouring members of this given particle. We have performed various tests and found the results to be robust to the sphere’s extent.
- We follow these particles forward in time until $z=0$, and discard those that, at any time, are more distant than $r_{\rm
stream}$ from the selected particle.
- The selected particle is considered to be in a stream if, at final time, another particle from the same parent satellite can be found within our solar neighbourhood sphere and, moreover, they have in common at least one of the original neighbouring particles lying within their respective $r_{\rm stream}$.
Name $N_{\rm sat}^{\it sn}$ $n_{*}$ $f_{\rm stream}^{n_{*}}$ $f_{\rm stream}^{m_{*}}$ $N_{\rm stream}$ $f_{\rm stream}^{5\rm{mm}}$
------ ------------------------ --------- --------------------------- -------------------------- ------------------ -----------------------------
A-2 $85$ $1400$ $20.2\%$ $31.1\%$ $83$ $63\%$
B-2 $38$ $10740$ $82.0\%$ $92.4\%$ $582$ $74\%$
C-2 $105$ $2334$ $46.9\%$ $62.7\%$ $223$ $68\%$
D-2 $63$ $3853$ $68.2\%$ $85.5\%$ $301$ $65\%$
E-2 $53$ $1957$ $36.7\%$ $62.7\%$ $125$ $84\%$
: Properties of the distribution of particles tagged with stars inside “solar neighbourhood” spheres of 2.5 kpc radius located at 8 kpc from the galactic centre. The first column labels the simulation. From left to right, the remaining columns give the total number of contributing satellites, $N_{\rm sat}^{\it sn}$; the total number of star particles, $n_{*}$; the fraction of star particles in resolved streams; $f_{\rm stream}^{n_{*}}$; the fraction of stellar mass in resolved streams, $f_{\rm stream}^{m_{*}}$; the total number of streams, $N_{\rm stream}$; and the fraction of streams coming from the five most significant stellar mass contributors, $f_{\rm stream}^{5\rm{mm}}$.[]{data-label="table:streams"}
{width="175.5mm"}\
{width="175mm"}
{width="175.1mm"}
Figure \[fig:streams\_pos\_1\] shows, as an example, the time evolution of one of the streams crossing the “solar neighbourhood” of the halo Aq-A-2, at $z=0$. The corresponding progenitor is the second most massive contributor to the stellar halo and, prior accretion, it had a stellar mass of $\approx 1.25 \times
10^{8}$ M$_{\odot}$. Its redshift of infall is $z \approx 2.3$ and by $z = 0$ it has been fully disrupted. The stream is indicated with an arrow in Figure \[fig:streams\_vel\]. The black dots correspond to the particles that have always been neighbours (i.e. within $r_{\rm
stream}$) of a reference particle in this stream, which is indicated with a red dot. This figure clearly shows the full spatial extent of the stream, which probes regions far beyond the “solar neighbourhood” volume.
The numbers of streams with at least two particles found inside the “solar neighbourhood” spheres located at 8 kpc from the galactic centre are listed in Table \[table:streams\]. These numbers are well in the range of the $\sim$ 300 - 500 stellar streams around the Sun predicted by the models of @hw [see also @gh10b]. Halo Aq-A-2 has the smallest number of resolved streams, as well as the lowest fraction of particles associated with them (see Table \[table:streams\]), and the largest associated particle fraction are found in halo Aq-B-2. Table \[table:streams\] also shows the total number of accreted particles found within each sphere as well as the total number of progenitor satellites that have contributed them. Note that the fraction of stellar mass in resolved streams is always larger than the fraction of star particles in resolved streams. Thus, the star particles with high stellar-to-total mass ratio, which come from the denser and inner parts of the original satellites, are more likely to be found in resolved streams.
From this table we see that disrupted satellites contribute many more star particles at 8 kpc in Aq-B-2 than in the other haloes, and as a result the streams are much better resolved. The next best resolved “solar neighbourhood” is found in Aq-D-2. For the other haloes, a significant fraction of the particles have not been associated with any substructure and, therefore the total number of streams (which would include those unresolved), could be significantly larger. We will explore this further in Section \[sec:unresolved\].
In Figure \[fig:streams\_vel\] we compare the velocity distributions of stellar particles in streams (top panels) to those not associated with any structure (bottom panels) according to our algorithm. Different columns correspond to distributions of stellar particles inside “solar neighbourhoods” of different haloes. As in Figure \[fig:vela\], the different colours represent different satellites. In addition to a very large number of inconspicuous substructures, we can see that the most prominent streams have been recovered in all cases. Note that the particles in streams tend to populate the wings of the distribution. In general, particles in the core of the velocity distribution will have been accreted earlier and tend to have shorter periods (and hence to mix faster). Therefore, streams will be more difficult to identify because of the limited resolution. In addition, and as explained below, it is possible that due to the triaxiality of the dark matter potential some of these orbits exhibit chaotic mixing. The associated streams would then be too convoluted to be observable. Note, however, that thanks to the much larger number of star particles and the smaller velocity dispersion (see Section \[sec:charac\]), a significant fraction of the particles in the core of the distribution for Aq-B-2 are in resolved streams. Figure \[fig:streams\_ener\] shows the distribution of stellar particles in streams in integrals of motion space. Substructure in this space is clearly visible. We emphasise that this is the case even in the presence of violent variation of the host potential due to merger events and chaotic orbital behaviour induced by the strongly triaxial dark matter haloes.
In the top panel of Figure \[fig:tot\_str\] we explore how the number of streams found inside spheres of 2.5 kpc radius changes as a function of galactocentric distance along the direction of the major axis. It is interesting to see that in all cases the number of streams decreases as a power-law with radius. Note, however, that haloes Aq-B-2 and E-2 have steeper profiles than the other haloes. As described by @cooper, the majority of the stellar mass in these two cases comes from one or two objects that deposit most of their mass in the inner regions of the haloes. Interestingly, these two haloes have the largest fraction of streams coming from the five most significant stellar mass contributor (see Table \[table:streams\]). As shown in the bottom panel of Figure \[fig:tot\_str\], the same behaviours is observed in the local density profiles of these haloes, $\rho_{0}(r)$, measured along the major axis. Note that, due to the triaxial nature of the resulting haloes, varying azimuthally the location of our spheres results in local stellar densities that are, in general, an order of magnitude smaller than the observed value in the Solar Neighbourhood. For example, at a distance of 8 kpc along the intermediate axis, halo Aq-A-2 has $\rho_{0} = 4.9 \times 10^3$ M$_{\odot}$ kpc$^{3}$ and a total number of streams $N_{\rm streams} =
37$. An isodensity contour defined by the value of $\rho_{0}$ at 8 kpc along the major axis places the solar neighbourhood-like sphere at distance of $\approx 4$ kpc on the intermediate axis. At this location $N_{\rm streams}$ raises to 68, indicating that azimuthal variations in the number of streams mainly reflect changes in local stellar density.
![Top panel: Number of streams as a function of galactocentric radius found within spheres of 2.5 kpc radii. Bottom panel: Spherically averaged stellar density profiles. The different colours indicate different haloes.[]{data-label="fig:tot_str"}](fig8.eps){width="82mm"}
Smooth/Unresolved component {#sec:unresolved}
---------------------------
In addition to the stellar particles associated with streams with at least two particles, our “solar neighbourhood” spheres contain an important number of stellar particles that are not linked to any substructure, i.e., they appear to be smoothly distributed in phase-space. The reason for this could be either of dynamical origin or, as posited before, simply due to the numerical resolution of the simulation. It is well known that phase-space regions of triaxial dark matter haloes may exist that are occupied by chaotic orbits [e.g., @vol08]. On such orbits, stellar particles that are initially nearby diverge in space exponentially with time. As a result, substructure that is present in localised volumes of phase-space left by accretion events is rapidly erased. This process is known as chaotic mixing. On the other hand, initially nearby particles in phase-space on regular orbits diverge in space as a power-law in time. Therefore, the time-scale in which these particles fully phase-mix may be much larger.
Based on the previous discussion, we will now analyse the evolution in time of the local (stream) density around the particles found at the present day in each “solar neighbourhood” sphere. Our goal is to assess the influence of chaotic mixing on the underlying phase-space distributions and to estimate the fraction of particles that, due to the high but nonetheless limited numerical resolution of our simulations, were not associated with any stream.
As in Section \[sec:resolved\_subs\], we place a sphere of 4 kpc radius around each particle at $t = t_{\rm form}$ and tag all the surrounding neighbours. We track these particles forward in time until redshift $z=0$ and those neighbours that depart from our particles by more than $r_{\rm stream}$ at any time are discarded. We compute the local spatial density of the selected particle, $\rho(t)$, by counting the number of neighbours within a distance $r_{\rm dens} = 0.5 \times
r_{\rm apo}$. In Figure \[fig:dens\_evol\] we show the time evolution of the spatial density of particles from two different satellites that contribute to the “solar neighbourhood” of the halo Aq-A-2. From this figure we can appreciate that the rate at which the spatial density of a stream decreases with time varies strongly (likely as a consequence of the different orbits). We can also observe that in many cases, when a particle becomes unbound from its parent satellite, the density of the newly formed stream exhibits initially a very rapid and quasi-exponential decrease. As shown by @g10 [see also @hg07], this initial transient is also present in regular orbits and does not necessarily imply a manifestation of chaotic behaviour. It is therefore necessary to follow the evolution of the stream’s density for long time scales (much longer than a crossing time) to disentangle regular from chaotic behaviour.
Our approach to characterizing the type of mixing consists in fitting a power-law function to the time evolution of the local density of a stream, $$\label{eq:rho_t}
\hat{\rho}(t) = \alpha t^{-n},$$ and determining the value of $n$ for each stream. As shown by @vol08 [see also @hw] for a stream on a regular orbit, $n=1$, 2 or 3 depending on the number of fundamental orbital frequencies. Although we know that the density of a stream on a chaotic orbit does not evolve as a power law in time, we nonetheless fit the functional form given by Eq. (\[eq:rho\_t\]), and expect larger values of $n$ than for the regular case. We warn the reader that, given our simple approach, the resulting distributions of $n$-values should be considered as an estimate of the rate at which diffusion is acting in a given volume, rather than a precise characterization of the underlying phase-space structure.
Figure \[fig:dens\_evol\] shows the results of applying such a fit to the stream densities as a function of time. Since we are interested in the behaviour of the density on long time scales, we do not include the initial quasi-exponential transient in the fits. We proceed as follows:
- We define the formation time of a stream, $t_{\rm stream}$, as the time when its density decreases from one snapshot to the next one by $50\%$.
- Starting from the snapshot associated with $t_{\rm stream}$, we iteratively fit ten times a power-law function to the stream’s density by increasing in each iteration the starting time snapshot by snapshot. The fits are always normalized to the value of the density at the starting point. Furthermore, each data point is weighted according to the number of particles used to measure the corresponding density.
- We estimate the goodness of each fit by computing the root mean squared error, $RMSE$, defined as: $$\displaystyle
RMSE = \sqrt{\dfrac{\sum_{i=1}^{m}\left(\hat{\rho}(t_{i}) - \rho(t_{i})\right)^{2}}{m-2}}$$ where $\rho(t_{i})$ is the density estimated from the particle count and $m$ is the number of data points used in the fit.
- Of all these experiments, we only keep the value of $n$ obtained from the fit where the $RMSE$ is smallest.
The results of applying this procedure to all the particles inside 2.5 kpc radius spheres of different haloes are shown in Figure \[fig:histo\]. We explore spheres located at three different galactocentric radii, namely 8 , 15 and 20 kpc. The black solid histograms show the distribution of $n$-values for all the particles in each “solar neighbourhood” sphere. At ${\rm R} = 8$ kpc (top panels), with the exception of haloes Aq-B-2 and E-2, we find narrow distributions peaking at values slightly larger than $n=3$. Note, however, that as we move outward from the galactic centre, the peak of all distributions tend to shift towards values much closer to $n=3$. A clear example is halo Aq-B-2, where the peak of the distribution shifts from $n \approx 7$ at ${\rm R} = 8$ kpc to $n
\lessapprox 3$ at ${\rm R} = 20$ kpc (bottom panel).
{width="85mm"} {width="85mm"}
Figure \[fig:histo\] also shows in black dashed histograms the distribution $n$-values for those particles associated with “resolved” stellar streams, while the grey-dashed histograms correspond to the remaining particles. From this figure we can observe that although the two distributions overlap significantly, there is a clear offset between them. In each panel we show the median values of $n$, $\tilde{n}$. Stellar particles in streams (which are more likely to be on regular orbits) tend to have the smallest values of $n$ and, in general, their distribution peaks at a value much closer to $n = 3$ than their unresolved counterparts. Let us recall that this is the value of $n$ expected for the dependence of density on time for regular orbits in a three dimensional time independent gravitational potential [see, e.g. @hw; @vol08].
It is interesting to compare the $n$-distributions obtained from haloes Aq-A-2 and B-2 at ${\rm R}=8$ kpc. As previously discussed, while halo B-2 has the largest number of streams and the greatest fraction of accreted particles associated with them, the opposite is found for halo Aq-A-2. The value of $\tilde{n} \approx 7$ obtained from the resolved component in halo B-2 suggests that these particles are undergoing chaotic mixing. Nonetheless, thanks to the high particle resolution, substructure can be efficiently identified. Note the large difference between the medians obtained from the resolved and unresolved components. For halo Aq-A-2 , the distribution of $n$-values associated with the two components show a very similar value of $\tilde{n} \approx 4.3$, considerably closer to what is expected for regular orbits. This suggests that a significant fraction of the particles that have not been associated with streams are on (nearly) regular orbits, and that because of the limited numerical resolution, they are not found in (massive) streams.
Interestingly, we find that, on average, particles in resolved streams in Aq-A-2 were released from their parent satellite[^4] 2.5 Gyr later and have more extended orbits than the particles in the unresolved component. The average apocentric distances for these two subsets are 32 and 12 kpc, respectively. Thus, the shorter orbital periods and longer times since release partially explain why the latter are not found to be part of (massive) streams. A lower limit to the fraction of accreted particles in resolved stellar streams, corrected by resolution effects, can be obtained by quantifying the fraction of particles in the unresolved component with $n$-values smaller than the median of the resolved component, $\tilde{n}_{\rm
res}$. We find in halo Aq-A-2 a total of 411 particles that satisfy this condition. In addition to the 283 particles associated with the resolved component, we estimate that at least $50\%$ of all accreted particles in this halo should be part of a resolved stellar stream. Furthermore, the corresponding stellar mass fraction should at least be as large as $65.7\%$.
In general, particles in resolved streams were released later (than those in the unresolved component) for all haloes. We find for haloes Aq-B-2, C-2, D-2 and E-2 a time difference of 1.9, 1, 2.2, and 2 Gyr, respectively. However, differences in the mean apocentric distances are found only in halo Aq-A-2. As for halo Aq-A-2, we can obtain for the remaining haloes an estimate for the lower limit of particles in resolved stellar streams corrected by resolution effects. We find a total of $(380,348,268,350)$ particles with $n$-values smaller than $\tilde{n}_{\rm res}$ in haloes (B-2, C-2, D-2, E-2), respectively. Thus, we estimate that the corresponding fractions of particles and stellar mass in resolved streams for these haloes should be, at least, as large as $(55\%, 75\%, 62\%,84\%)$ and $(96\%, 80\%,
91\%, 84\%)$, respectively.
{width="180mm"}\
{width="180mm"}\
{width="180mm"}
Summary {#sec:conclu}
=======
In this work we have characterized the phase-space distribution as well as the time evolution of debris for stars found inside “solar neighbourhood” volumes. Our analysis is based on simulations of the formation of galactic stellar haloes, as described by @cooper. The haloes were obtained by combining the very high resolution fully cosmological $\Lambda$CDM simulations of the [*Aquarius*]{} project with a semi-analytic model of galaxy formation.
We find that, for haloes Aq-A-2, -C-2 and -D-2, the measured velocity ellipsoid at 8 kpc on the major axis of the halo is in good agreement with the estimate for the local Galactic stellar halo [@chibabeers]. Similarly, for all haloes but Aq-B-2, the local stellar halo average density on the major axis is in reasonable agreement with the observed value [@fuchs]. However, we note that if the contribution of the disc and bulge were to be included, the resulting velocity dispersions as well as the local stellar halo average density would be significantly increased. This could suggest that these dark haloes are too massive to host the Milky Way [in agreement with @giusy; @s07; @mcs; @wang; @vera13]. On the other hand, if located in a prolate low mass halo such as Aq-B-2, the measured local stellar density would imply that the Sun should be on the intermediate or minor axis, i.e. the Galactic disc’s angular momentum would be aligned with the major axis of the dark halo [@ah04]. On the minor axis, haloes B and E have velocity ellipsoids of smaller amplitude and hence could perhaps be made consistent with those observed near the Sun, if the effects of the contraction induced by the disc were taken into account. However, for haloes A, C, and D, the dispersions on the minor axis are larger than those on the major axis at the equivalent of the solar circle.
In agreement with previous work, we find that $90\%$ of the stellar mass enclosed in our “solar neighbourhoods” comes, in all cases, from 3 to 5 significant contributors. These “solar neighbourhood” volumes contain a large amount of substructure in the form of stellar streams. In the five analysed stellar haloes, substructure can be easily identified as kinematically cold structures, as well as in well defined lumps of stellar particles in spaces of pseudo-conserved quantities.
By applying a simple algorithm to follow the time evolution of local densities in the neighbourhood of reference particles, we have quantified the number of (massive) streams that can be resolved in solar neighborhood-like volumes. In haloes where solar neighbourhood-like volumes are best resolved we find that the number of resolved streams is in very good agreement with the predictions given by @hw. As we move outward from the galactic centre the number of streams decreases as a power-law, in a similar fashion to the density profiles.
To explore whether the halo-to-halo scatter in the total number of resolved streams is due to poor particle resolution or to chaotic mixing, we have estimated the rate at which local (stream) densities decrease with time. Interestingly, we find that in our best-resolved solar neighbourhood-like volume (Aq-B-2) a large amount of substructure can be identified, even in the presence of chaotic mixing. In this halo, up to $82\%$ of the star particles (and $92\%$ of the stellar mass) can be associated with a resolved stream. On the other hand, our poorest-resolved solar neighbourhood-like volume (Aq-A-2) shows a smaller amount of substructure but significantly longer mixing time-scales (a rate that is close to what is expected from regular orbits). In this case, only $20\%$ of the accreted star particles (and $31\%$ of the accreted stellar mass) can be linked to streams. A comparison of the orbital properties of resolved and unresolved stream-populations shows that, in general, particles in resolved streams are released later and have longer orbital periods. Moreover, diffusion in both populations occurs at very similar rates. Thus, this analysis suggests that our strongest limitation to quantifying substructure is mass resolution rather than diffusion due to chaotic mixing. We have estimated a lower limit to the fraction of accreted particles in resolved stellar streams by correcting for resolution effects. We find that this fraction should be, at least, as large as $50\%$ by number and $65\%$ by stellar mass in all haloes.
The results presented in this work suggest that the phase-space structure in the vicinity of the Sun should be quite rich in substructure. Recall that our analysis is based on fully cosmological simulations and therefore chaos and violent changes in the gravitational potential that are so characteristic of cold dark matter model, are naturally included.
With the imminent launch of the astrometric satellite [*Gaia*]{} [@perry], we will be able for the first time to robustly quantify the amount of substructure in the form of stellar streams present within $\approx 2$ kpc from the Sun. This mission from the European Space Agency will provide accurate measurements of positions, proper motions, parallaxes and radial velocities of an unprecedentedly large number of stars[^5]. In addition, recent and ongoing surveys such as SEGUE [@segue], RAVE [@rave], LAMOST [@lamost], HERMES [@hermes] and Gaia-ESO [@2012Msngr.147...25G] will allow us to complement these [ *Gaia*]{} observations. A direct comparison of the degree of substructure found in the Solar Neighbourhood with that found in numerical models such as those presented here will allow us to establish directly, for example, how much of the stellar halo has been accreted and how much has been formed in-situ.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank the anonymous referee for his/her very useful comments and suggestions that helped improve the clarity of this paper. FAG would like to thank Brian W. O’Shea for providing useful comments and suggestions and Monica Valluri for valuable discussions. FAG was supported through the NSF Office of Cyberinfrastructure by grant PHY-0941373, and by the Michigan State University Institute for Cyber-Enabled Research (iCER). This work was initiated thanks to financial support from the Netherlands Organisation for Scientific Research (NWO) through a VIDI grant. AH acknowledges the European Research Council for ERC-StG grant GALACTICA-240271. APC acknowledges support from the Natural Science Foundation of China, grant No. 11250110509.
\[lastpage\]
[^1]: Email:fgomez@msu.edu
[^2]: Note however that the simulations were started at redshift $z=127$
[^3]: Note that this particle does not need to be located within the solar neighbourhood-like spheres at $z=0$, as $r_{\rm stream}$ can be larger than 2.5 kpc. Note further that throughout this section when we refer to “particles” we mean "particles tagged with stars”
[^4]: As measured by the stream’s formation time, $t_{\rm stream}$
[^5]: for details see: http://www.rssd.esa.int/gaia
|
---
abstract: 'Matrix elements of a two-body interaction between states of the $j^n$ configuration (*n* fermions in the *j*-orbit) are functions of two-body energies. In many cases, diagonal matrix elements are linear combinations of two-body energies, whose coefficients are rational and positive numbers. It is shown that if in the $j^n$ configuration there is only one state with a given spin *J*, its eigenvalue (the diagonal matrix element) has rational and positive coefficients of two-body energies. The situation in cases with several *J*-states in the $j^n$ configuration is more complicated and is discussed in detail. States with this property are identified and constructed. In particular, the relevance to states in the seniority scheme is shown.'
author:
- |
Igal Talmi\
Weizmann Institute of Science,\
Rehovot, Israel
title: 'Rational coefficients of two-body energies in diagonal matrix elements of $j^n$ configurations'
---
Introduction
============
A system of *n* identical fermions in a given *j*-orbit is considered. Such $j^n$ configurations of protons or neutrons occur, for instance, in the shell model. Any configuration contains a certain number of fully antisymmetric independent states allowed by the Pauli principle. We consider shell model Hamiltonians which contain mutual interactions between the particles. The interactions considered are rotationally invariant and hence, eigenstates of the Hamiltonian have definite values of angular momentum *J*. The mutual interaction is considered to be a perturbation. Hence, only the sub-matrix of the Hamiltonian defined by the states of the $j^n$ configuration is taken into account. The states considered here are the states which define this sub-matrix.
If there is only one state with given *J* in the $j^n$ configuration, it is an eigenstate of the sub-matrix of the Hamiltonian. Its eigenvalue is given by the corresponding diagonal element of the sub-matrix. If there are several independent states with the same value of *J*, they define a smaller sub-matrix whose diagonalization yields the eigenstates for a given mutual interaction. If two-body mutual interactions are considered, elements of the Hamiltonian sub-matrix of the $j^n$ configuration are functions of two-body matrix elements. Due to the rotational invariance of the interaction, the only non-vanishing two-body matrix elements are between two-body states with the same value of *J*. Thus, these two-body matrix elements are the energies of the two-body configuration.
In a recent paper \[1\], Zamick and Van Isacker consider partial dynamical symmetries of certain states of the (9/2)$^4$ configuration. They used their results to calculate also the energies of the $J=2$ and $J=4$ states of the (7/2)$^4$ configuration. They express these energies as linear combinations of two-body energies and in their arXiv: 0803.1569v1 (nucl-th) version they “note that this derivation constitutes a proof that the coefficients...in the energy expression...must be rational numbers”.
This feature is quite general. There are many cases where diagonal matrix element in $j^n$ configurations are linear combinations of two-body energies whose coefficients are rational and positive numbers. This feature has little relevance to numerical calculations but it still has theoretical interest. The aim of this paper is to study this property of states, to find the cases in which diagonal elements are linear combinations of two-body energies whose coefficients are positive rational numbers.
There is a simple way to calculate the diagonal matrix elements of a two-body interaction in a system of *n* fermions in a *j*-orbit. It is based on the use of *m*-scheme wave functions. These are determinantal wave functions characterized by a set of quantum numbers $m_1,m_2,\ldots,m_n$, which are all different. These wave functions are discussed in great detail in ref.\[2\] where matrix elements of a two body interaction between such states are derived. It was shown there that diagonal matrix elements are equal to the $\sum<m_im_k|V|m_im_k>$ where the summation is over all pairs of *m*-values in the state considered.
Matrix elements of a rotationally invariant two-body interaction between *m*-states with given total $M=\sum m_i$ form a sub-matrix. Its trace is equal to the sum of diagonal elements taken between all states with given *M*. States of the $j^n$ configuration with given values of *J* and *M*, $J\geq
M$, are linear combinations of these *m*- states. The linear transformation, from *m*-states to states with definite *J* and *M*, is unitary or rather orthogonal. Hence, the trace of the sub-matrix is invariant and may be used to calculate diagonal matrix elements of a rotationally invariant, *M*-independent interaction in the *J*-scheme.
Subtracting the trace of the $M=J-1$ sub-matrix from the one of $J=M$, yields the sum $\sum <j^n a J,M|V|j^n a J,M>$ where *a* characterizes the state if there are several states with given *J* in the $j^n$ configuration. If there is only one such state, this procedure yields its interaction energy. Otherwise, only the sum of energies of all states with given *J* is obtained. This case will be discussed in detail in the following. The diagonal matrix elements in the *m*- scheme may be expressed in terms of matrix elements of two particles coupled to a given *J* value. The two-particle matrix elements in the *m*-scheme may be transformed by using Clebsch-Gordan coefficients as follows. $$\begin{aligned}
&<m_im_k|V|m_im_k>=\cr
&\sum <j,m_i,j,m_k|j,j,J,M=m_i+m_k>^2<j^2JM|V|j^2JM>\end{aligned}$$ Since the *m*-states in (1) are antisymmetric, the sum on the r.h.s. of (1) is only over *even* values of *J* (states of two fermions with *odd J* values are symmetric). Thus, the coefficients of the $V(J)= <j^2JM|V|j^2JM>$ in the diagonal elements in the *m*-scheme are equal to sums of squares of Clebsch-Gordan coefficients. From formulae for these coefficients (e.g. (15.36), ref.\[2\]) it is seen that their squares are rational functions of their arguments. These arguments are the 3 *J*-values and 3 *m*-values.
Thus, the sum $\sum_{\substack{J}}<j^n a J,M|V|j^n a J,M>$ is a linear combination of two-body matrix elements $V(J)=<j^2JM|V|j^2JM>$ which are independent of *M*, whose coefficients are rational and positive numbers. If there is only one state with given *J* in the $j^n$ configuration, it has the “rational property” considered here. Its eigenvalue of a two-body interaction is a linear combination of the *V(J)* with positive rational coefficients. If there are several such states, it was shown above that this property holds only for the sum of their diagonal elements. It will be shown in this paper which of these states has this property. It will also be shown below, that in such cases, not any arbitrary state may have this property.
States in the $j^n$ configuration with given *J* must be considered individually. This may be achieved by using coefficients of fractional parentage (c.f.p.), introduced by Bacher and Goudsmit \[3\] and intensively applied to atomic spectra by Racah\[4\]. An antisymmetric wave function with *n* particles and given *J* may be expanded as $$\begin{aligned}
\psi(j^n a J)=\Sigma [j^{n-1}(bJ_1)j J|\}j^n a J] \psi (j^{n-1}(b J_1)j_n a J)\end{aligned}$$ In (2), *a* and *b* are used to specify uniquely *J* and $J_1$ states if there are several of them. The wave functions on the r.h.s. of (2) are fully antisymmetric functions of particles 1 to *n*-1. They are usually not antisymmetric in all *n* particles but their linear combination in (2) is.
Using the expansion (2), matrix elements in states of the $j^n$ configuration may be expressed in terms of those in the $j^{n-1}$ configuration. Due to the full antisymmetry of (2), it is possible to calculate matrix elements of $\Sigma V_{ik}$ for $i,k<n$ whose number is (*n*-1)(*n*-2)/2 and multiply the result by *n*(*n*-1)/(*n*-1)(*n*-2)=*n*/(*n*-2) to obtain the full value of the matrix element. This leads to $$\begin{aligned}
&<j^n a_n J| \Sigma V_{ik}|j^n b_n J>= \cr
&n/(n-2)\sum_{a_{n-1}J_1} [j^{n-1}(a_{n-1}J_1)jJ|\}j^n a_nJ][j^{n-1}(b_{n-1}J_1)jJ|\}j^n
b_nJ]
\times\cr &<j^{n-1}a_{n-1}J_1|\Sigma V_{ik}|j^{n-1} b_{n-1}J_1>\end{aligned}$$ Using c.f.p. the matrix elements in (3) may be further reduced to those in the $j^{n-2}$ configuration and so on until the $j^2$ configuration is reached. A direct result of (3) follows if there is only one state with given *J* in the configurations involved. Then, diagonal matrix elements have the property considered here if the squares of the c.f.p. are rational numbers. Since this was proved above, the case with several states with the same value of *J* should be considered. This will be carried out in the following, starting with the simple case of *n=3* and proceeding to higher *n* values.
A simple case: the $j^3$ configuration
======================================
A state with given *J* may be obtained by starting from a *principal parent* with given $J_0$ in the $j^2$ configuration, coupling to it $j_3$ and antisymmetrizing the function $\psi(j^2(J_0)j_3J)$. If the resulting state is allowed, the wave function does not vanish and its c.f.p. are given by (e.g. (26.11), ref.\[2\] or (15.10), ref.\[5\]) $$\begin{aligned}
&[j^2(J_1)j J|\}j^3[J_0]J]=\cr &N(J_0,J)[\delta (J_1,J_0)+
2[(2J_0+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0\}]\end{aligned}$$ where the normalization factor is given by $$\begin{aligned}
N(J_0,J)=[3+6(2J_0+1)\{jjJ_0/JjJ_0\}]^{-1/2}\end{aligned}$$ and *{abc/def*} are 6*j*-symbols (Racah coefficients). The c.f.p. of the principal parent is obtained by putting $J_1=J_0$ in (4). Then from (5) follows the special $n=3$ case of the identity proved in ref.\[2\], $$\begin{aligned}
n N(J_0,J)[j^2(J_0)j J|\}j^3[J_0]J]=1\end{aligned}$$
To find the properties of the c.f.p., it is worth while to look at the form of the 6*j*-symbols. When Racah introduced these symbols \[6\] (in a slightly different form and symmetry properties), he also derived a closed formula of them in terms of their 6 arguments. In (15.38),ref.\[2\], it is expressed as $$\begin{aligned}
\{{abc/def}\}=\Delta (abc) \Delta (aef) \Delta (dbf) \Delta
(dec)R(abcdef)\end{aligned}$$ where $$\begin{aligned}
\Delta(abc)=[(a+b-c)!(b+c-a)!(c+a-b)!/(a+b+c+1)!]^{1/2}\end{aligned}$$ and *R* is a rational function of the 6 arguments. Hence, the square of a 6*j*-symbol is a rational function of its arguments. If a state obtained by antisymmetrization is not allowed by the Pauli principle, the wave function vanishes. This always happens for states with *J*=3*j*-4 or *J*=${1\over2}$. The unnormalized c.f.p. all vanish due to the properties of the rational function in(7). From (7) and (8) follow some simple properties of 6*j*-symbols, like $$\begin{aligned}
\{jjJ_0/JjJ_0\}=\Delta(jjJ_0)^2\Delta (JjJ_0)^2 R(jjJ_0JjJ_0)\end{aligned}$$ From (9) follows that the c.f.p. of the principal parent, with $J_1=J_0$, is a square root of a rational function of *j*, $J_0$ and *J*. This is true also for the normalization coefficient (5). Therefore, also the other c.f.p., with $J_1\neq
J_0$, share this feature since their squares are equal to $$\begin{aligned}
4N(J_0,J)^2(2J_0+1)(2J_1+1)\{j,j,J_1/J,j,J_0 \}^2\end{aligned}$$ Thus, the squares of the c.f.p. of a state obtained by antisymmetrization of a principal parent are rational functions of the angular momenta. Hence, due to (3), such states have the property considered here. Their eigenvalues of a two-body interaction are linear combinations of the $V(J)= <j^2J|V|j^2J>$ with coefficients which are positive rational numbers. For *n*=3, the relation (3) yields directly these linear combinations as follows $$\begin{aligned}
&<j^3 a J|\Sigma V_{ik}|j^3 a J>=\cr
&3\Sigma [j^2(J_1)jJ|\}j^3aJ][j^2(J_1)jJ|\}j^3aJ] <j^2J_1|V|j^2J_1>\end{aligned}$$
If there is only one state with given *J* in the $j^3$ configuration, the proof that it has the property considered, is complete as already shown in the preceding section. In that case, starting with all possible principal parents, those with $J_0$ which can yield *J* when coupled with *j*, the same state is obtained after antisymmetrization (with a possible change of overall sign). If there are several states with given *J* in the $j^3$ configuration, they may be constructed by antisymmetrizing *several* principal parents. The diagonal matrix element of each of them is given by the relation (11). Still, the states obtained in this way need not be orthogonal. It will be shown now that states obtained in the orthogonalization procedure have also the property considered.
The orthogonalization of states with the same value of *J* constructed from principal parents is simplified by the following property of such c.f.p. Due to (4), they may be expressed, for $J_1\neq J_0$, as $$\begin{aligned}
&[j^2(J_1)j J|\}j^3[J_0]J]=\cr
&N(J_0,J)2[(2J_0+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0\}= \cr
&N(J_0)(2J_0+1)^{1/2}\Delta (jjJ_0)\Delta (JjJ_0)(2J_1+1)^{1/2} \Delta(jjJ_1)\Delta
(JjJ_1)R\end{aligned}$$ where $N(J_0)= N(J_0,J)$ (also in the following) and $R=R(jjJ_1JjJ_0)$. Thus, these c.f.p. are products of a rational function which multiplies a term which depends only on $J_0$ and another one which depends only on $J_1$. The latter term is the same for all states with various values of $J_0$.
To make a state $|b>$ orthogonal to a given state $|a>$, the latter multiplied by $<a|b>$ should be subtracted from the former. The orthogonalized state $$\begin{aligned}
|b^\prime >=|b>-<a|b>|a>\end{aligned}$$ should then be normalized. Using c.f.p., the scalar product of the two states is given by $$\begin{aligned}
<j^n a J|j^n b
J>=\Sigma_{a_1J_1}[j^{n-1}(a_1J_1)jJ|\}j^naJ][j^{n-1}(a_1J_1)jJ|\}j^nbJ]\end{aligned}$$ The state $|a\!>$ is taken to be constructed from the principal parent $\psi(j^2(J_0)j_3J)$ and the $|b>$ state from $\psi(j^2(J_0^\prime)j_3J)$. Making use of the c.f.p. (4) we find that every term in the summation in (14), for $J_1\neq J_0, J_1\neq
J_0^\prime$ is given by $$\begin{aligned}
&N(J_0)N(J_0^\prime)2[(2J_0+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0\}\times\cr
&2[(2J_0^\prime+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0^\prime\}\end{aligned}$$
From the properties (7) of 6*j*-symbols follows that the product of the two of them in (15) can be expressed as $$\begin{aligned}
&\{jjJ_1/JjJ_0\}\{jjJ_1/JjJ_0^\prime \}= \Delta (jjJ_1)\Delta (jjJ_0)\Delta (JjJ_1)\Delta
(JjJ_0)\times\cr
&\Delta(jjJ_1)\Delta (jjJ^\prime_0) \Delta (JjJ_1)
\Delta(JjJ_0^\prime )R(jjJ_1JjJ_0)R(jjJ_1JjJ_0^\prime)=\cr
&\Delta(jjJ_0)\Delta (JjJ_0)\Delta (jjJ_0^\prime)\Delta
(JjJ_0^\prime)\Delta (jjJ_1)^2\Delta (JjJ_1)^2 \times\cr
&R(jjJ_1JjJ_0)R(jjJ_1JjJ_0^\prime)\end{aligned}$$ Due to the form (7) of 6j-symbols, the r.h.s. of (16) may be expressed as $$\begin{aligned}
&({jjJ_0/JjJ_0^\prime}/R(jjJ_0JjJ_0^\prime))\Delta(jjJ_1)^2
\Delta(JjJ_1)^2\times\cr
&R(jjJ_1JjJ_0)R(jjJ_1JjJ^\prime_0)\end{aligned}$$ This expression is a 6*j*-symbol multiplied by a rational function, R($J_1$). Thus, every term in (14) becomes equal to $$\begin{aligned}
4N(J_0)N(J_0^\prime
)[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime\}\times \cr&(2J_1+1)R(J_1)\end{aligned}$$ The summation of terms with values of $J_1$ different from $J_0^\prime$ and from $J_0$ may now be carried out and yields a rational function multiplying $$\begin{aligned}
N(J_0)N(J_0^\prime )[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime \}\end{aligned}$$
The term in (14) with $J_1=J_0$ is equal to $$\begin{aligned}
&N(J_0)N(J_0^\prime
)[1+2[(2J_0+1)(2J_0+1)]^{1/2}\{jjJ_0/JjJ_0\}]\times\cr
&2[(2J_0^\prime+1)(2J_0+1)]^{1/2}\{jjJ_0/JjJ_0^\prime \}\end{aligned}$$ Due to the properties of 6*j*-symbols (7), this expression is equal to a rational function multiplying the expression (19). Also the term with $J_1=J_0^\prime$ is equal to (19) multiplied by a (different) rational function. Thus, the summation in (14) yields a rational function multiplied by the factor (19).
To obtain the state which is orthogonal to the one characterized by $J_0^\prime$ , eq.(13) is used. The unnormalized c.f.p. of that state are defined by $$\begin{aligned}
& N(J_0^\prime)[\delta (J_1,J_0^\prime )+
2[(2J_0^\prime+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0^\prime
\}]-\cr
&N(J_0)N(J_0^\prime
)[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime \}\times\cr
&N(J_0)[\delta (J_1,J_0)+ 2[(2J_0+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0\}]R\end{aligned}$$ where *R* is a rational function. In the case that $J_1$ is different from $J_0$ and $J_0^\prime$, (21) assumes the form $$\begin{aligned}
N(J_0^\prime)2[(2J_0^\prime+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0^\prime\}
-\cr
N(J_0)N(J_0^\prime
)[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime \} \times \cr
N(J_0)2[(2J_0+1)(2J_1+1)]^{1/2}\{jjJ_1/JjJ_0\}R\end{aligned}$$ In (22) there are several square roots which appear twice. The product of two 6*j*-symbols is equal, according to (16) and (17) to $\{jjJ_1/jjJ_0^\prime \}$ multiplied by a rational function. Hence, the unnormalized c.f.p. of the orthogonalized state, given by (22) are equal to the original c.f.p. multiplied by rational functions.
In the case $J_1=J_0$, that c.f.p. is equal to $$\begin{aligned}
N(J_0^\prime)2[(2J_0^\prime+1)(2J_0+1)]^{1/2}\{jjJ_0/JjJ_0^\prime \}
-\cr
N(J_0)N(J_0^\prime
)[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime \} \times\cr
N(J_0)[1+2[(2J_0+1)(2J_0+1)]^{1/2}\{jjJ_0/JjJ_0\}R\end{aligned}$$ In (23), $N(J_0)^2[1+2[(2J_0+1)(2J_0+1)]^{1/2}\{jjJ_0/JjJ_0\}]$ is a rational function and hence, (23) is equal to the original c.f.p. multiplied by a rational function, as are the other c.f.p. Also in the case $J_1=J_0^\prime$ , the situation is similar. In that case, (21) is equal to $$\begin{aligned}
N(J_0^\prime)[1+ 2[(2J_0^\prime+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0^\prime/JjJ_0^\prime \}]
-\cr
N(J_0)N(J_0^\prime
)[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime\} \times\cr
N(J_0)2[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0^\prime/JjJ_0\}R\end{aligned}$$ The c.f.p. of the principal parent of the original state is equal to $N(J_0^\prime)$ multiplied by a rational function. The term substracted from it is equal to $N(J_0^\prime)$ multiplied by the product of the rational function *R* and the square of a product of square roots. Hence, all c.f.p. of the unnormalized orthogonalized state are equal to the original ones multiplied by various rational functions. The resulting state should be normalized. It should be divided by the square root of the sum of squares of these c.f.p. Since the squares of the latter are rational, the normalization coefficient is the square root of a rational function. Thus, the squares of the c.f.p. of the normalized and orthogonalized state are rational and positive.
The c.f.p. of the orthogonalized and normalized states are equal, apart from a common normalization factor, to the original c.f.p. multiplied by rational functions. It may happen that certain c.f.p. vanish after this procedure. For instance, starting from a certain $J_0$, in the state with $J_0^\prime$ which was made orthogonal to it, the c.f.p. for $J_1=J_0$ must vanish. If further orthogonalizaions are necessary, due to existence of more independent states with the same total *J*, It is possible to repeat the steps taken above.
Looking at the derivations above, it is clear that not all linear combinations of states obtained from principal parents have the property considered here. The c.f.p. may become sums of square roots of relatively prime numbers and their squares will then not be rational numbers. In fact, there are states whose diagonal matrix elements are not linear combinations of two-body energies. This may happen to eigenstates obtained by diagonalization of the submatrix, with given *J*, of a two-body interaction. An example is shown at the end of this section. Naturally, it is possible to construct linear combinations, other than those discussed above, which have the “rational property”.
Non-diagonal matrix elements have also a simple form when taken between states sharing the property discussed here. We consider the matrix element (3) for *n=3* taken between different states. We may take them to be the state obtained by antisymmetrizing the principal parent with $J_0$ and the one obtained from the $J_0^\prime$ principal parent, orthogonalized as above. To calculate the non-diagonal matrix element it is possible to make use of the calculation of the overlap of the two states above. We start from the formula (3) for the special case of *n=3* $$\begin{aligned}
&<j^3[J_0]J|\Sigma V_{ik}|j^3[J_0^\prime]J>=\cr
&3\Sigma[j^2(J_1)j
J|\}j^3[J_0]J][j^2(J_1)jJ|\}j^3[J_0^\prime]J]<j^2J_1|V|j^2J_1>\end{aligned}$$ Every term in the summation in (25) is equal to a corresponding terms in(14) multiplied by $<j^2J_1|V|j^2J_1>$. Every term (14) is transformed into (19), which is $$\begin{aligned}
N(J_0)N(J_0^\prime )[(2J_0+1)(2J_0^\prime+1)]^{1/2}\{jjJ_0/jjJ_0^\prime \}\end{aligned}$$ multiplied by a rational function. Hence, the coefficient of $<j^2J_1|V|j^2J_1>$ in (25) is (26) multiplied by a rational function. Thus, the non-diagonal matrix elements in the scheme of states described above, are linear combinations of two-body energies with rational functions multiplied by a common square root of a rational function.
Before proceeding to higher values of *n*, it it worth-while to look at some simple examples in $j^3$ configurations. States with *J=j* (for *n* odd) and seniorities *v=1* of the $j^n$ configuration are obtained from principal parents with $J_0=0$ and $v_0=0$ in the $j^{n-1}$ configuration. The seniority scheme is explained in detail in refs.\[2\] and \[5\]. It is also reviewed in the last section of this paper. In the case of *n=3*, putting $J_0=0$ in (4) the following results were obtained $$\begin{aligned}
&[j^2(0)j J=j|\}j^3 v=1J=j]=\{1+2\{jj0/jj0\}N(0)=\cr
&[(2j-1)/3(2j+1)]^{1/2}\cr
&[j^2(J_1)j J=j|\}j^3 v=1 J=j]=\cr
&2(2J_1+1)^{1/2}\{jjJ_1/jj0\}N(0)=-2[(2J_1+1)/3(2j-1)(2j+1)]^{1/2}\end{aligned}$$ As long as $j<9/2$, there is only one state with *J=j* in the $j^3$ configuration and that state has c.f.p. given above. For higher values of *j* there are other states with *J=j* and they all have seniorities *v=3*. Their number is $[(2j-3)/6]$ where $[x]$ is the largest integer not exceeding $x$. They may be constructed from principal parents with even values of $J_0>0$ and then orthogonalized to the *v=1* state and to each other. The first part of this procedure, for any *j*, is carried out in ref.\[5\], Section 20, pages 396 - 398.
The two states with *J*=9/2 in the $j^3$ configuration offer an opportunity to demonstrate a statement made above. Not all states have the property considered in this paper. There are states whose diagonal matrix elements are not equal to linear combinations of two-body energies. It was mentioned that such cases occur in diagonalization of the Hamiltonian sub-matrix defined in the $j^n$ configuration by the states with a given value of *J*.
Consider the 2x2 sub-matrix defined by the *v*=1 and *v*=3 states with *J*=9/2. Both diagonal matrix elements $V_1$ and $V_3$ are linear combinations of the *V(J)* with positive rational coefficients. The non-diagonal matrix element *V* is also a linear combination of the *V(J)* with rational coefficients multiplied by the square root of a rational function. The two eigenvalues with *J*=9/2 are the roots of the quadratic secular equation $$\begin{aligned}
x^2-(V_1+V_3)x+V_1V_3-V^2=0\end{aligned}$$ They are equal to $$\begin{aligned}
& x={1\over 2}\{V_1+V_{3}\pm [(V_1+V_3)^2-4V_1V_3+4V^2]^{1/2}\}=\cr
&{1\over 2}\{V_1+V_{3}\pm [(V_1-V_3)^2+4V^2]^{1/2}\}\end{aligned}$$ Both eigenvalues are sums of linear combinations with rational coefficients and square roots of quadratic functions of the *V(J)*.
In some special cases, the square root may turn out to be a linear function of the *V(J)*. In the case of the $J=9/2$ described above, this cannot happen. The $v=3$ state is orthogonal to the $v=1$ state whose principal parent is the $j^2(J_1=0)$ state. Hence, the c.f.p. with $J_1$=0 of the *v*=3 state vanishes. Thus, in the linear combination $V_3$ the coefficient of *V*(0) vanishes. This coefficient vanishes also in the non-diagonal matrix element *V*. The term $V_1-V_3$ may be expressed as a linear combination of *V*(0) and terms with $V(J), J>0$ even, $c_0V(0)+\Sigma c_JV(J)$. Its square is equal to $c_0^2V(0)^2+2c_0V(0)\Sigma c_JV(J)+(\Sigma c _JV(J))^2$. The term with $V^2$ has the form $(\Sigma d_JV(J))^2$ with no *V*(0) terms, and adding it cannot change the quadratic and linear terms in *V*(0). If *V* does not vanish, adding 4$V^2$ to $(V_1-V_3)^2$ cannot yield a square of a rational function of the *V*(*J*). Hence, in the example considered here, the eigenvalues of a two-body interaction are definitely not linear combinations of the two-body energies.
There is no contradiction between this result and (11). There, the diagonal matrix element of a state is a linear combination of the *V*(*J*) with coefficients which are squares of the c.f.p.(multiplied by 3). The states constructed in (4), as well as all states constructed in this paper, are states which form a basis of all states with given *J*. Any antisymmetric state may be expanded in terms of c.f.p. as in (2). The states constructed in (4), as well as others, are based on the use of principal parents. Hence, the c.f.p. are functions of the angular momenta and are independent of the two-body interaction. If there is only one state with given *J*, it is an eigenstate and its eigenvalue is a linear combination of the *V*(*J*). If there are several such states and the Hamiltonian sub-matix has non-vanishing non-diagonal elements, this is no longer the case. Eigenstates may be expanded in terms of c.f.p. but the latter may well be (irrational) functions of the two-body matrix elements. This is indeed the case in the example shown above.
General $j^n$ configurations
============================
In the preceding section it was shown that for the *n=3* case, states constructed by antisymmetrizing principal parents have the property considered in this paper. Diagonal matrix elements of a two-body interaction are linear combinations of two-body matrix elements (energies) *V(J)* whose coefficients are rational numbers. This property is still shared by such states which were orthogonalized to each other. These features follow from demonstrating that for *n=3*, the c.f.p. of these states are square roots of rational functions of the angular momenta involved. It was shown that the normalization factor is a square root of a rational function. That the c.f.p. of the principal parent is equal to the normalization factor multiplied by a rational function. There is a general relation between this c.f.p. and the normalization factor. It was proved in ref.\[2\], (26.23), that $$\begin{aligned}
n[j^{n-1}(a,J_0)jJ|\}j^n[a,J_0]J] N(J_0,J)=1\end{aligned}$$ In the following it will be shown by induction that these features are shared also by c.f.p. of states in $j^n$ configurations for $n>3$. Starting from *n=3* it will be shown that if these features hold for c.f.p. in the $j^{n-1}$ configuration, they hold also for c.f.p. in the $j^n$ configuration. The property considered in this paper, that the expectation value of a two-body interaction is then a linear combination of two-body energies whose coefficients are rational functions then follows as will be shown below.
Consider a state of the $j^n$ configuration obtained by antisymmetrizing a principal parent $\psi (j^{n-1}(a J_0)j_n J)$. If the result does not vanish, its c.f.p. can be expressed by c.f.p. of states in the $j^{n-1}$ configuration. All states considered in the following are taken to be allowed states whose wave functions do not vanish. This relation (ref.\[2\] eq.(26.30) or (15.29) of ref.\[5\]) is $$\begin{aligned}
&n[
j^{n-1}(a_{n-1},_0J_{n-1},_0)jJ_n|\}j^n[a_{n-1},_0J_{n-1},_0]J_n]\times\cr
&[j^{n-1}(a_{n-1},_1J_{n-1},_1)jJ_n|\}j^n[a_{n-1},_0J_{n-1},_0]J_n]=\cr
&\delta( a_{n-1},_1,a_{n-1},_0) \delta(J_{n-1},_1,J_{n-1},_0)+\cr
&(n-1)
\Sigma(-1)^{J_{n-1,0}}+J_{n-1,1}[(2J_{n-1,0}+1)(2J_{n-1,1}+1)]^{1/2}\cr
&\{J_{n-2}jJ_{n-1,1}/J_njJ_{n-1,0}\}[j^{n-2}(a_{n-2}J_{n-2})jJ_{n-1,0}|\}j^{n-1}a_{n-1,0}J_{n-1,0}]\times\cr
&[j^{n-2}(a_{n-2}J_{n-2})jJ_{n-1,1}|\}j^{n-1}a_{n-1,1}J_{n-1,1}]\end{aligned}$$ The first step in calculating the c.f.p. is to put in (31), $a_{n-1,1}=a_{n-1,0}$ and $J_{n-1,1}=J_{n-1,0}$. This leads to the result $$\begin{aligned}
&n[J^{n-1}(a_{n-1,0}J_{n-1,0})jJ_n| \}j^n[a_{n-1,0}J_{n-1,0}]J_n]^2
=\cr
&1+(n-1)(-1)^{2J_0}\Sigma (2J_{0+1})\{J_2jJ_0/JjJ_0\}\times\cr
&[j^{n-2}(a_{n-2}J_{n-2})jJ_{n-1,0}|\}j^{n-1}a_{n-1,0}J_{n-1,0}]^2\end{aligned}$$ As in (9), the 6 *j*-symbol in (32) is a rational function of its arguments. If the squares of the n-2 $\rightarrow $ n-1 c.f.p. are rational, as assumed for the induction, then the c.f.p. on the l.h.s. of (32) is indeed the square root of a rational function for any *n*. According to (30), the normalization coefficient is also a square root of a rational function.
The c.f.p. of all states are given by the r.h.s. of (31) divided by *n* multiplying the c.f.p. of the principal parent. Due to (30), this amounts to multiplying the r.h.s. of (31) by the normalization coefficient $N(J_{n-1,0}J_n)$. Starting from the known c.f.p. for the *n*=3 case, it seems that in general the c.f.p. $[j^{n-1}(a_{n-1,1}J_{n-1,1})jJ_n|\}j^n[a_{n-1,0}J_{n-1,0}]J_n]$ is equal to a rational function of the various angular momenta multiplied by $$\begin{aligned}
N(J_{n-1,0}J_n)\Pi N(J_{r0}J_{r+1,0})\Pi
N(J_{r1}J_{r+1,1})\Pi [(2J_{r0}+1)(2J_{r1}+1)]^{1/2}\times\cr
\Pi \Delta (J_{r0}jJ_{r+1,0}) \Pi\Delta (J_{r1}jJ_{r+1,1}) \Delta
(J_{n-1,0} j J_n)\Delta (J_{n-1,1} j J_n)\end{aligned}$$ In the products in (33), $J_{1,0}=J_{1,1}=j$ and *r* goes up to *n*-2. In the following, this form of c.f.p. will be proved by induction. Apart from the first factor, (33) is a product of two expressions which are the same functions of the various $J_{r0}$ and the various $J_{r1}$. The former define the state considered and the latter, the component whose c.f.p. is selected. This feature was noticed in the *n*=3 case and is in agreement with the property of c.f.p. given by (26.21) of ref\[2\]. The various angular momenta belong to wave functions which were obtained by antisymmetrization of principal parents and by orthogonalization if necessary. The intermediate angular momenta supply the additional quantum numbers, denoted above by $a_{r0}$ and $a_{r1}$, needed to specify the states considered. This will be the case for all states considered below.
It is worth-while to point out that the form (33) is the correct one also for the c.f.p. of the principal parent. Putting $J_{r1}=J_{r0}$ for all *r*, the result is a rational function multiplying the normalization factor $$\begin{aligned}
N(J_{n-1,0}J_n)=n N(J_{n-1,0}J_n)^2/n
N(J_{n-1,0}J_n)\cr\end{aligned}$$ According to (30) this is equal to a rational function multiplying the c.f.p. of the principal parent. Hence, no special care should be paid to that c.f.p. The fact that (33) is the correct expression of all c.f.p. will be proved by induction with respect to *n*. Before doing it, let us draw some conclusions from the structure (33) of c.f.p.
The first conclusion which follows from (33) is that the squares of these c.f.p. are positive rational functions of the various angular momenta. This was shown above for the c.f.p. of the principal parent and this feature is shared by all of them. The aim of this paper is to show for which states the expectation values of a two-body interaction are linear combinations of two-body energies with rational coefficients. When (33) will be shown to be correct, the answer to this question will be given. States with this property are those obtained by consecutive constructions from principal parents and antisymmetrizations.
Another property of c.f.p. which follows from (33) is important for the construction of orthogonal sets of states with the same total spin *J*. As in the case of *n*=3, we construct such a set by choosing one state due to a principal parent. If there are other such independent states, obtained from other principal parents, they are made orthogonal to it. The orthogonalization is carried out by using (13). First, the scalar product (14) is calculated, for the state (33) defined by $J^\prime_{r0}$ and the one defined by $J_{r0}$. It is a linear combination and each of its terms is a rational function multiplied by $$\begin{aligned}
&N(J_{n-1},_0J_n)\Pi N(J_{r0}J_{r+1,0})\Pi
N(J_{r1}J_{r+1,1})\Pi[(2J_{r0}+1)(2J_{r1}+1)]^{1/2}\times\cr
&\Pi \Delta(J_{r0}jJ_{r+1,0}) \Pi \Delta (J_{r1}jJ_{r+1,1})\Delta
(J_{n-1,0} j J_n)\Delta (J_{n-1,1} j J_n)\times \cr
&N(J_{n-1,0}^\prime J_n)\Pi N(J_{r0}^\prime J_{r+1,0}^\prime)\Pi
N(J_{r1}J_{r+1,1})\Pi[(2J_{r0}^\prime+1)(2J_{r1}+1)]^{1/2}\times \cr
&\Pi\Delta (J_{r0}^\prime jJ_{r+1,0}^\prime )\Pi\Delta (J_{r1}j
J_{r+1,1}) \Delta(J_{n-1,0}^\prime jJ_n) \Delta(J_{n-1,1} j J_n)\end{aligned}$$ In (34) all terms with $J_{r1}$ appear twice and thus contribute only to the rational function multiplying it. Hence, the summation over $J_{n-1,1}$ in (14) may be carried out yielding for the scalar product a rational function multiplying $$\begin{aligned}
&N(J_{n-1,0}J_n) N(J_{n-1,0}^\prime J_n)\Pi N(J_{r0}J_{r+1,0}) \Pi
N(J_{r0}^\prime J_{r+1,0}^\prime)\times \cr
&\Pi[(2J_{r0}^\prime+1)(2J_{r0}+1)]^{1/2}\Pi\Delta
(J_{r0}jJ_{r+1,0}) \Pi\Delta (J_{r0}^\prime jJ_{r+1,0}^\prime)\times
\cr &\Delta(J_{n-1,0} j J_n) (J_{n-1,0}^\prime jJ_n)\end{aligned}$$
According to (13), the state with $J_{r0}$ should be multiplied by the scalar product (35) and subtracted from the state with $J_{r0}^\prime$ which is to be orthogonalized to the other state. The factor (35) multiplied by the rational function is independent of $J_{n-1,1}$. Multiplying it by a c.f.p. of the state with $J_{r0}$, we obtain according to (35) and (33), a rational function multiplying $$\begin{aligned}
&N(J_{n-1,0}J_n) N(J_{n-1,0}^\prime J_n)\Pi N(J_{r0}J_{r+1,0}) \Pi N(J_{r0}^\prime
J_{r+1,0}^\prime)\times \cr
&\Pi[(2J_{r0}^\prime+1)(2J_{r0}+1)]^{1/2}\Pi\Delta
(J_{r0}jJ_{r+1,0}) \Pi\Delta (J_{r0}^\prime jJ_{r+1,0}^\prime
)\times \cr &\Delta(J_{n-1,0} j J_n)\Delta (J_{n-1,0}^\prime
jJ_n)\times \cr &N(J_{n-1,0}J_n)\Pi N(J_{r0}J_{r+1,0})\Pi
N(J_{r1}J_{r+1,1})\Pi[(2J_{r0}+1)(2J_{r1}+1)]^{1/2}\times \cr
&\Pi\Delta (J_{r0}jJ_{r+1,0}) \Pi \Delta (J_{r1}jJ_{r+1,1})\Delta
(J_{n-1,0} j J_n)\Delta (J_{n-1,1} j J_n)\end{aligned}$$ In (36), all terms with the various $J_{r0}$ are squared and may be absorbed in the rational function which multiplies the following expression $$\begin{aligned}
&N(J_{n-1,0}^\prime J_n)\Pi N(J_{r0}^\prime J_{r+1,0}^\prime )\Pi
N(J_{r1}J_{r+1,1})\times \cr
&\Pi[(2J_{r0}^\prime+1)(2J_{r1}+1)]^{1/2} \Pi\Delta (J_{r0}^\prime
jJ_{r+1,0}^\prime )\Pi\Delta (J_{r1}jJ_{r+1,1})\times \cr
&\Delta(J_{n-1,0}^\prime jJ_n) \Delta(J_{n-1,1} j J_n)\end{aligned}$$
The expression (37) is identical with the corresponding c.f.p. of the state with $J_{r0}^\prime$ multiplied by a rational function. Hence, also the c.f.p. of the orthogonalized state are square roots of rational functions and have the form given by (33). All states with the same value of $J_n$ have c.f.p. given by (33). They are different by a common factor to all c.f.p. of each state which is a square root of a rational function. They are orthogonal to each other due to the different rational functions multiplying the expression (33) to obtain the full c.f.p. Some of these rational functions may vanish. The states orthogonalized in this way are not normalized and must be explicitely normalized.
We turn now to the form (33) of c.f.p. of states constructed consecutively from principal parents and antisymmetrized. Obviously, it holds for the case *n=3*. To demonstrate that it is correct also for c.f.p. with $n>3$ we use the recursion relation (31) for induction by *n*. We show that if (31) is the correct form for a given *n-1*, it holds also for *n*. According to the assumption, every term in the summation in (31) is given by a rational function multiplying the expression $$\begin{aligned}
&N(J_{n-2,0}J_{n-1,0})\Pi N(J_{r0}J_{r+1,0})\Pi
N(J_{r1}J_{r+1,1})\Pi[(2J_{r0}+1)(2J_{r1}+1)]^{1/2}\times \cr
&N(J^\prime _{n-2,0} J^\prime _{n-1,0})\Pi N(J_{r0}^\prime
J_{r+1,0}^\prime)\Pi
N(J_{r1}J_{r+1,1})\Pi[(2J_{r0}^\prime+1)(2J_{r1}+1)]^{1/2} \times
\cr &\Pi\Delta (J_{r0}jJ_{r+1,0}) \Pi \Delta(J_{r1}jJ_{r+1,1})\Delta
(J_{n-2,0} j J_{n-1,0}) \Delta(J_{n-2,1} j J_{n-1,0})\times \cr
&\Pi\Delta (J_{r0}^\prime jJ_{r+1,0}^\prime)\Pi\Delta (J_{r1}jJ
_{r+1,1}) \Delta(J_{n-2,0}^\prime j J_{n-1,0}^\prime)
\Delta(J_{n-2,1} j J_{n-1,0}^\prime)\times \cr
&\{J_{n-2,1}jJ_{n-1,0}^\prime/J_njJ_{n-1,0}\}\end{aligned}$$
This expression may be simplified by noting that there are several identical factors which may be absorbed in the rational function. Thus, we obtain instead of (38) the form $$\begin{aligned}
&\Pi N(J_{r1}J_{r+1,1})^2\Pi(2J_{r1}+1) \Pi
\Delta(J_{r1}jJ_{r+1,1})^2 \times \cr
&\Pi N(J_{r0}J_{r+1,0})\Pi N(J_{r0}^\prime J^\prime_{r+1,0})
\Pi[(2J_{r0}+1)(2J_{r0}^\prime+1)]^{1/2}\times \cr &\Pi\Delta
(J_{r0}jJ_{r+1,0}) \Pi \Delta(J_{r0}^\prime
jJ_{r+1,0}^\prime)\{J_{n-2,1}jJ_{n-1,0}^\prime/J_njJ_{n-1,0}\}\times
\cr &\Delta(J_{n-2,1} j J_{n-1,0})\Delta (J_{n-2,1} j
J_{n-1,0}^\prime)\end{aligned}$$
Whereas in the products in (38) *r* goes up to *n*-3, in (39) it goes up to *n*-2. Using the formula (7) for the 6*j*-symbol, the product $$\begin{aligned}
\Delta(J_{n-2,1}j J_{n-1,0}^\prime)\Delta (J_{n-2,1} j J_{n-1,0})\{J_{n-2,1}jJ_{n-1,0}^\prime/J_njJ_{n-1,0}\}\end{aligned}$$ in (39) becomes equal to a rational function multiplied by $$\begin{aligned}
\Delta(J_{n-2,1}j J_{n-1,0})^2\Delta (J_{n-2,1} j J_{n-1,0}^\prime)^2\Delta (J_{n-1,0}j J_n) \Delta (J_{n-1,0}^\prime j J_n)\end{aligned}$$ Thus the $J_{n-2,1}$ angular momentum appears as an argument only of rational functions and the summation over it can be directly carried out for all terms. The result of this summation is a rational function multiplied by $$\begin{aligned}
\Pi N(J_{r0}J_{r+1,0})\Pi N(J_{r0}^\prime J_{r+1,0}^\prime
)\Pi[(2J_{r0}+1)(2J_{r0}^\prime +1)]^{1/2}\times \cr
\Pi\Delta (J_{r0}jJ_{r+1,0}) \Pi\Delta (J_{r0}^\prime
jJ_{r+1,0}^\prime)\Delta (J_{n-1,0}j J_n)\Delta (J_{n-1,0}^\prime j
J_n)\end{aligned}$$ According to (31), this expression should be multiplied by $N(J_{n-1,0}J_n)$ and is then identical with (33).
Thus, if the form (33) is correct for *n*-1 it is correct also for *n*. Since it is correct for *n=3*, it holds for any value of *n*. From (33) follows that the squares of c.f.p. obtained from principal parents and antisymmetrization are rational functions. According to (3), this is sufficient to guarantee that if in every $j^r$ configuration, $r\leq n$, there is only one state with a given value of $J_r$, the diagonal matrix elements have the property considered in this paper. This special case was discussed in the Introduction in a simpler way. Relation (3) becomes more complicated if there are several states with the same value of *J*. Unlike the situation in the simple *n=3* case, using relation (3) for diagonal matrix elements in the $j^n$ configuration may well lead to non-diagonal matrix elements in $j^r$ configurations with $r\leq n$. Non-diagonal matrix elements are not given by squares of $n-1\rightarrow n$ c.f.p. There is, however, a way to make use of the structure (33) of these c.f.p. to show that the states constructed in this way have the property considered in this paper.
Any antisymmetric wave function in the $j^n$ configuration may be expanded in $n-2\rightarrow n$ c.f.p. which are discussed in detail in ref.\[2\]. They are defined by $$\begin{aligned}
\psi(j^naJ)= \Sigma[j^{n-2}(bJ_1)j^2(J^\prime )J|\}j^naJ]\psi (j^{n-2}(bJ_1)j_{n,n-1}^2 (J^\prime )aJ)
\end{aligned}$$ Such an expansion is always possible since the wave functions on the r.h.s. of (43) form a complete basis for the space of $j^n$ states with given *J*, which are antisymmetric in particles 1 to *n-2* and in particles *n,n-1*. Fully antisymmetric $j^n$ wave functions with given *J* are included in this space and hence, may be expanded as in (43). The calculation of two-body matrix elements of a scalar operator, like the mutual interaction between particles, is very simple if such c.f.p. are used. Since these states are fully antisymmetric, it is possible to calculate matrix elements of one term in the interaction, $V_{n,n-1}$ say, and multiply the result by the number of such terms, $n(n$-1)/2. Thus, using the expansion (43), we obtain the result $$\begin{aligned}
&<j^naJ|\Sigma V_{ik}|j^naJ>=n(n-1)<j^naJ|V_{n,n-1}|j^naJ>/2=\cr
&[n(n-1)/2]\Sigma [j^{n-2}(bJ_1)j^2(J^\prime )J|\}j^naJ]
[j^{n-2}(b"J_1")j^2(J ")J|\}j^naJ]\times \cr
&<j^{n-2}(bJ_1)j^2(J^\prime )J|V_{n,n-1}|j^{n-2}(b"J_1" )j^2(J")J>
\end{aligned}$$
Integration of (44) over coordinates 1 to *n*-2 yields zero unless $J_1=J_1"$ and $b"\equiv b$. Since $V_{n,n-1}$ is a scalar operator, its matrix element on the r.h.s. of (44) vanishes if $J"$ is not equal to $J^\prime$. If they are equal, the two-body matrix element in (44) is equal to $<j^2J^\prime|V |j^2J^\prime
>$. Hence, the matrix element is given by $$\begin{aligned}
&<j^n a J|\Sigma V_{ik}|j^n a J >=\cr
&[n(n-1)/2]\Sigma_{b J_1 J^1} [j^{n-2}(bJ_1)j^2(J^\prime )J|\}j^naJ]^2
<j^2J^\prime|Vj^2J^\prime >
\end{aligned}$$ Hence, using the $n-2\rightarrow n$ c.f.p. in(43), diagonal matrix elements of a two-body interaction are given by the linear combination (45). In it, the coefficients of the two-body energies are squares of the c.f.p. multiplied by $n(n$-1)/2. The expectation value in the state $|j^naJ>$ of a two-body interaction is a linear combination of two-body energies with coefficients that are positive rational functions if the squares of these c.f.p. are rational.
To find out which states have $n-2\rightarrow n$ c.f.p. whose squares are rational functions, it is convenient to express them in terms of $n-1\rightarrow n$ c.f.p. This expression is given by (26.44) in ref.\[2\] as $$\begin{aligned}
&[j^{n-2}(a_{n-2,0}J_{n-2,0})j^2(J^\prime )J_n|\}j^n a J_n]=\cr
&\Sigma[j^{n-2}(a_{n-2,0}J_{n-2,0})jJ_{n-1,1}|\}j^{n-1}a_{n-1,1}J_{n-1,1}]\times
\cr &[J^{n-1}(a_{n-1,1}J_{n-1,1})jJ_n|\}j^{n}aJ_n]\times \cr
&[(2J_{n-1,1}+1)(2J ^\prime +1)]^{1/2}
\{J_{n-2,0}jJ_{n-1,1}/jJ_nJ^\prime \}\end{aligned}$$ The form (33) was proved above to be valid for c.f.p. of certain states constructed from principal parents. Using such states in (46), the form of their $n-1\rightarrow n$ c.f.p. may be taken from (33). Thus, the $n-2\rightarrow n-1$ c.f.p. $[j^{n-2}(a_{n-2,0}J_{n-2,0})jJ_{n-1,1}|\}j^{n-1}[a_{n-2,1},J_{n-2,1}]
J_{n-1,1}]$ is expressed as a rational function multiplying $$\begin{aligned}
&N(J_{n-2,1}J_{n-1,1})\Pi N(J_{r1}J_{r+1,1})\Pi
N(J_{r0}J_{r+1,0})\Pi[(2J_{r1}+1)(2J_{r0}+1)]^{1/2}\times \cr
&\Pi\Delta (J_{r1}jJ_{r+1,1})\Pi\Delta (J_{r0}jJ_{r+1,0})\Delta
(J_{n-2,1} j J_{n-1,1})\Delta (J_{n-2,0} j J_{n-1,1})\end{aligned}$$ The $n-1\rightarrow n$ c.f.p. ,$[j^{n-1}([a_{n-2,1},J_{n-2,1}]
J_{n-1,1})jJ_n|\}j^{n}[J_{n-1,0}^\prime]J_n]$ in (46) can be similarly expressed, as a rational function multiplying the following expression which is written in more detail than in (33) $$\begin{aligned}
&N(J_{n-1,0}^\prime J_n)\Pi N(J_{r0}^\prime J_{r+1,0}^\prime)\Pi
N(J_{r1}J_{r+1,1})N(J_{n-2,1}J_{n-1,1})\times \cr
& \Pi[(2J_{r0}^\prime+1)(2J_{r1}+1)]^{1/2}
[(2J_{n-1,0}^\prime+1)(2J_{n-1,1}+1)]^{1/2}\times \cr &\Pi\Delta
(J_{r0}^\prime jJ_{r+1,0}^\prime ) \Pi\Delta
(J_{r1}jJ_{r+1,1})\Delta (J_{n-2,1}jJ_{n-1,1})\times \cr
&\Delta(J_{n-1,0}^\prime j J_n)\Delta (J_{n-1,1} j J_n)\end{aligned}$$
Substituting these expressions into (46) yields for each term in the summation a rational function multiplied by $$\begin{aligned}
&N(J_{n-1,0}^\prime J_n) N(J_{n-2,1}J_{n-1,1})^2\times
\Pi N(J_{r1}J_{r+1,1})^2\Pi N(J_{r0}J_{r+1,0})\cr &\Pi
N(J_{r0}^\prime J_{r+1,0}^\prime) \Pi[(2J_{r0}+1)(2J_{r0}^\prime
+1)]^{1/2} \Pi(2J_{r1}+1)\times\cr &[(2J_{n-1,0}^\prime +1)(2J
^\prime +1)]^{1/2} \Pi \Delta(J_{r0}jJ_{r+1,0}\Pi
\Delta(J_{r0}^\prime jJ_{r+1,0}^\prime )\times\cr\ &\Pi\Delta
(J_{r1}jJ_{r+1,1})^2 \Delta(J_{n-2,0} j J_{n-1,1})
\Delta(J_{n-2,1}jJ_{n-1,1})^2 \Pi \Delta(J_{r0}jJ_{r+1,0})\times\cr
&\Delta(J_{n-1,0}^\prime j J_n) \Delta(J_{n-1,1} j
J_n)(2J_{n-1,1}+1)\{J_{n-2,0}jJ_{n-1,1}/jJ_nJ ^\prime \}\end{aligned}$$ Several terms in (49) are rational functions. Using the form (7) of 6*j*-symbols, the following product $$\begin{aligned}
\Delta(J_{n-2,0}jJ_{n-1,1})\Delta(J_{n-1,0}^\prime j J_n) \Delta(J_{n-1,1} j J_n)\{J_{n-2,0}jJ_{n-1,1}/jJ_nJ ^\prime \}\end{aligned}$$ becomes equal to a rational function multiplying $$\begin{aligned}
&\Delta(J_{n-1,0}^\prime j J_n) \Delta(J_{n-1,1} j J_n)^2
\Delta(J_{n-2,0} j J_{n-1,1})^2\times \cr
&\Delta(j j J ^\prime )
\Delta(J_{n-2,0} J^\prime J_n)\end{aligned}$$ Thus, (49) may be expressed as a rational function multiplied by $$\begin{aligned}
&N(J_{n-1,0}^\prime J_n)\Pi N(J_{r0}J_{r+1,0})\Pi
N(J_{r0}^\prime J_{r+1,0}^\prime)\Pi[(2J_{r0}+1)(2J_{r0}^\prime+1)]^{1/2}\cr
&[(2J_{n-1,0}^\prime+1)(2J ^\prime+1)]^{1/2} \Pi \Delta
(J_{r0}^\prime jJ_{r+1,0}^\prime ) \Pi \Delta (J_{r1}jJ_{r+1,1})\cr
&\Delta(J_{n-1,0}^\prime j J_n) \Delta(j j J
^\prime)\Delta(J_{n-2,0} J^\prime J_n)\end{aligned}$$ The summation in (46) is over $J_{n-1,1}$ which does not appear in (52) and hence, it appears only in the rational function multiplying it. Therefore, the summation over it in (46) can be simply carried out yielding another rational function multiplying the expression (52).
Thus, the $n-2\rightarrow n$ c.f.p.$[j^{n-2}(a_{n-2,0}J_{n-2,0})j^2(J ^\prime)J_n|\}j^n a J_n]$ is the square root of a rational function and its square is a rational function. According to (45), the expectation value of a state with these c.f.p. of a two-body interaction is a linear combination of two-body energies with rational coefficients.
The $n-2\rightarrow n$ c.f.p. obtained above, may be used to derive expressions of non-diagonal matrix elements of a two-body interaction. Instead of (45), such non-diagonal elements are defined by $$\begin{aligned}
&<j^n a J|\Delta V_{ik}|j^n a^\prime J >=\cr &[n(n-1)/2]
\Sigma[j^{n-2}(bJ_1)j^2(J^\prime )J|\}j^naJ]\times \cr
&[j^{n-2}(bJ_1)j^2(J^\prime )J|\}j^na^\prime J]<j^2J^\prime |Vj^2J^\prime >\end{aligned}$$ Only non-vanishing terms are written on the r.h.s. of (53). The expression (52) is taken to be the non-rational factor in the c.f.p. of the state *a*. The c.f.p. of the state $a^\prime $ is the same expression in which $J_{r0}^\prime $ is replaced by $J_{r0}"$. All angular momenta $J_{r0}$ which define the$|j^{n-2}bJ_{n-2,0}>$, as well as $J^\prime$, appear in the same expressions in both c.f.p. and hence, contribute in (53) only rational functions. Each term in the summation in (53) is thus equal to a rational function multiplying the expression $$\begin{aligned}
&N(J_{n-1,0}^\prime J_n^\prime)N(J_{n-1,0}"J_n) \Pi N(J_{r0}^\prime J_{r+1,0}^\prime)\Pi
N(J_{r0}"J_{r+1,0}")\times \cr
&\Pi[(2J_{r0}^\prime +1)(2J_{r0}"+1)]^{1/2}\Pi\Delta (J_{r0}^\prime
jJ_{r+1,0}^\prime ) \Pi \Delta (J_{r0}"jJ_{r+1,0}")\times \cr
&\Delta(J_{n-1,0}^\prime j J_n) \Delta (J_{n-1,0}"j J_n)V(J^\prime)\end{aligned}$$ The factor of $V(J^\prime)$in (54) is a common factor of all terms in the summation in (53). Hence, the non-diagonal matrix element (53) is a linear combination of two-body energies whose coefficients are rational functions multiplied by a common, $J^\prime$ independent factor. The square of this factor as seen in (54) is a rational function of the angular momenta involved.
The seniority scheme
====================
The seniority quantum number *v* measures in some sense the amount of pairing of particle pairs in *J*=0 states. In the seniority scheme, states are eigenstates of the pairing interaction defined by $$\begin{aligned}
<j^2JM\mid q_{12}\mid j^2JM>=\delta_{J0}\delta_{M0}(2j+1)\end{aligned}$$ The eigenvalues of this interaction in the $j^n$ configuration are equal to $$\begin{aligned}
(n-v)(2j+3-n-v)/2\end{aligned}$$ Thus, loosely speaking, *v* is the number of unpaired particles. There is a very large class of interactions, not just the pairing interaction, which are diagonal in the seniority scheme. Also for general interactions, the seniority scheme is a convenient one to use. Here, this scheme will be discussed only in relation to the theme of the paper. The aim is to see which of the states of this scheme have diagonal matrix elements which are linear combinations of two-body energies with rational coefficients.
According to (56), states with seniority *v* in the $j^v$ configuration have no *J*=0 pairing. States with the same seniority in higher configurations are obtained by starting from those states, adding $(n-v)$/2 pairs with *J*=0 and antisymmetrizing. Adding one particle to a state with seniority *v* and antisymmetrizing, the state obtained has seniority *v*+1 or *v*-1. It may also be a linear combination of states with both seniorities. The state with no *j*-particles has seniority *v*=0 whereas the one particle state, with *J=j*, has *v*=1. States with two particles have states with *J*=0, *v*=0 or $J>$0, *even* and seniority *v*=2.
Consider a state in the $j^n$ configuration, $n<2j$ *even* with *J*=0 and *v*=0. Adding one particle to this principal parent, a state with *J*=*j* and *v*=1 is obtained. A state, with $n<2j$+1 odd, with $J=j$, *v*=1, is a principal parent of a *J*=0, *v*=0 state. Hence, such states have the property considered in this paper. The expectation values of any two-body interaction of states with lowest seniorities, *v*=0 or *v*=1, are given explicitly by (30.1) ref.\[2\] or (20.13) ref.\[5\] as $$\begin{aligned}
an(n-1)/2+[n/2]b\end{aligned}$$ where \[*n*\] is the largest integer not exceeding *n*. In (57), *a* and *b* are linear combinations of the *V*(*J*) which appear only in two linear combinations
$ V_0=V(J=0)\quad $ and $\quad \bar{V}_2=\Sigma (2J+1)V(J)/ \Sigma(2J+1)\, \qquad J>0,
even$ The coefficients *a* and *b* are given by these combinations as $$\begin{aligned}
a=\{(2j+2)\bar{V}_2-{V}_0 \}/(2j+1)\quad b=(2j+2)(V_0-{\bar{V}}_2)/(2j+1)\end{aligned}$$
In the case of an interaction which is diagonal in the seniority scheme, the expressions (57) are the eigenvalues of these states. Unlike the cases of $j<$7/2, there are no closed formulae for diagonal matrix elements of states with higher values of *v*. Similar expressions exist only for the average values of diagonal matrix elements of all states with given *v* in the $j^n$ configuration $v=n, n-2,\ldots$. It will be shown, however, that states of the seniority scheme may be constructed from principal parents, antisymmetrizations and orthogonalization if necessary. As shown in the preceding section, this guarantees that they have the property considered here. Diagonal matrix elements of a two-body interaction of states in the seniority scheme, constructed as explained above, are linear combinations of two-body energies with rational coefficients.
In the $j^2$ configuration, the *J*=0 state has seniority *v*=0 as mentioned above. The other antisymmetric states have even values of $J>$0 and seniority *v*=2. As mentioned above, adding a *j*-particle to a principal parent with *J*=0, *v*=0, the state $j^n$ $J=j$ with seniority *v*=1 is obtained. Since *n-v* is always an even integer, adding a *j*-particle to a state with seniority *v* in the $j^n$ configuration (and antisymmetrizing) yields a state of the $j^{n+1}$ configuration with seniority *v*+1 or *v*-1 or both. The added particle either completes a *J*=0 pair in which case the seniority is reduced or destroys a *J*=0 pair which raises the seniority.
If a *j*-particle is added to a principal parent with $J_0>0$, *v*=2, the state obtained has *v*=3 if $J\neq
j$. If a state with $J=j$ is constructed, it is a linear combination of states with seniorities *v*=1 and *v*=3, if there is a state with $J=j$, *v*=3 in the $j^3$ configuration. Such a linear combination will have a non-vanishing c.f.p. $[j^2(v=0,J=0)J=j|\}j^3J=j]$. The $J=j$, *v*=1 state obtained earlier, may be projected out by adding it with a coefficient that was defined by (13). The remaining state will be a pure $J=j$, *v*=3 state, orthogonal to the $J=j$, *v*=1 state. This procedure was described in detail in the preceding section. If there are several $J=j$ states in the $j^3$ configuration, they may be constructed from several principal parents. They should then be orthogonalized to the $J=j$, *v*=1 state and among themselves.
This procedure may be carried out in any $j^n$ configuration. If *n* is even, the steps described above may be taken. States with seniorities *v*=4 to which a *j*-particle is added, may be used to construct *v*=5 states or linear combinations of *v*=3 and *v*=5 states. In that case, the components with *v*=3 may be projected out by using the *v*=3 states constructed in the preceding step. At the end, states with seniority $v=n$ may be used to construct states with seniorities $v=n$+1 in the $j^{n+1}$ configuration.
If *n* is odd, the state with $J=j$, *v*=1 may be used to construct a state with *J*=0 which has seniority *v*=0, there are no states with *J*=0 and *v*=2. States with seniority *v*=2 may be constructed from the same $J=j$, *v*=1 state by coupling an additional *j*-particle to obtain a state with $2j
>J
>0$, *even*. From states with *v*=3, states with *v*=4 may be obtained. If a linear combination of *v*=2 and *v*=4 is obtained, the *v*=2 components may be projected out by using *v*=2 states obtained earlier. This procedure may be further carried out until all states in the $j^{n+1}$ configuration are constructed. They all have well defined seniorities and their diagonal matrix elements have the property described in this paper.
An example of the procedure described above is offered in ref.\[1\]. The authors calculated the eigenvalues of *J*=2 and *J*=4 states with seniority *v*=4 in the (7/2)$^4$ configuration. They found that the coefficients of two-body energies in both cases are rational numbers. In that configuration, there are also *J*=2 and *J*=4 states with seniority *v*=2. The latter may be obtained from the state in where a 7/2 particle is coupled to the principal parent with *J*=7/2 in the (7/2)$^3$ configuration. There is only one such state in that configuration and hence, it has seniority *v*=1. The *J*=2 and *J*=4 states obtained from it have seniority *v*=2 since the only state with *v*=0 has *J*=0. The other *J*=2,4 states must be obtained from another principal parent. In that one, a 7/2 particle should be coupled with any *v*=3 state in the (7/2)$^3$ configuration. From the resulting states, the *v*=2 components should be projected out as described above. Hence, the *v*=4 states are examples of states which have the property considered in the present paper.
A more complicated situation occurs in the (9/2)$^4$ configuration considered in ref.\[1\]. In that case there are 3 states with *J*=4 and 3 states with *J*=6. One of the *J*=4 (and *J*=6) states has seniority *v*=2 and may be constructed from a state where a 9/2 particle is coupled to the *J*=9/2, *v*=1 state. One of the remaining *J*=4 states may be constructed from a principal parent with *J*=9/2, *v*=3 coupled to a 9/2 particle. The resulting state,$|a\!>$, is a linear combination of *J*=4, *v*=4 and *J*=4, *v*=2 states. Another independent *J*=4 state,$|b\!>$, may be constructed from another principal parent with *v*=3 and is also a combination of *v*=4 and *v*=2 states. The *J*=4, *v*=4 states which are discussed in ref.\[1\] may be obtained as follows. First, the state $|b>$ is orthogonalized to state $|a>$ as indicated by vanishing of the c.f.p. $[(9/2)^3(J_0=9/2,v=3)(9/2)J=4|\}(9/2)^4bJ=4]$ of the orthogonalized $|b>$ state. Then, from both states the *J*=4, *v*=2 components are projected out leaving them as pure *v*=4 states. In each resulting state the c.f.p. corresponding to that of the principal parent of the *v*=2 state, $[(9/2)3(J_0=9/2,v=1)(9/2)J=4|\}(9/2)^4J=4]$, vanishes. It follows that these *v*=4 states have the property considered in this paper. The same arguments apply also to the *J*=6, *v*=4 states. Their diagonal matrix elements are linear combinations of two-body energies with rational coefficients.
Summary
=======
In the present paper, matrix elements of a two-body interaction between states of $j^n$ configurations were considered. The two-body interaction is taken to be a perturbation and hence, only the sub-matrix of the Hamiltonian, defined by states of the $j^n$ configuration, is considered. The matrix elements are functions of two-body energies $V(J)=<j^2JM|V|j^2JM>$. There are states whose expectation values (diagonal matrix elements) are linear combinations of the $V(J)$ with positive rational coefficients. In this paper, it was shown that this is always the case for states which are the only ones with given *J* in the $j^n$ configuration considered. Such states are eigenstates of the sub-matrix considered.
When there are several states with the same value of *J*, the situation is more complicated. It was shown above that it is possible to construct in such cases, a complete basis of states which have this simple feature. The diagonal matrix elements of these states are linear combinations of the *V(J)* whose coefficients are positive rational numbers. The construction of these states was carried out by using coefficients of fractional parentage (c.f.p.). These are obtained by starting from states of the $j^3$ configuration which serve as principal parents. This procedure is repeated by adding more particles until the $j^n$ configuration is reached. In every step, orthogonalization (and normalization) of the states thus obtained, is performed. It was shown that the squares of the c.f.p. are rational functions of the angular momenta used in the construction and hence, are independent of the particular two-body interaction.
It was shown that states of the seniority scheme form bases of this kind. If the interaction is diagonal in the seniority scheme, non-diagonal matrix elements between states with different seniorities vanish. This, however, is not true for the most general two-body interaction. In any case, there are usually non-vanishing non-diagonal matrix elements between states with the same seniority *v* and the same spin *J*. Irrespective of the nature of the basis states for a given value of *J*, they define a sub-matrix of the Hamiltonian. This sub-matrix should be diagonalized in order to find the eigenstates and corresponding eigenvalues.
The eigenstates are linear combinations of the basis states but the coefficients strongly depend on the eigenvalues. The latter are functions of the *V(J)* but are usually far from linear functions. The eigenvalues are the roots of the secular equation whose degree is limited only be the order of the sub-matrix. The eigenstates may be expanded in terms of c.f.p. but unlike the basis states constructed in this paper, their c.f.p. are functions of the *V(J)*.
It is convenient to use the basis states to set up the sub-matrix to be diagonalized. In addition to the rather simple structure of diagonal elements, also the non-diagonal elements have a simple structure. Each is a linear combination of the $V(J)$ with rational coefficients multiplied by a factor which is the square root of a rational function
From the discussion above, a simple conclusion follows that states with the rational property are the exception rather than the rule. In any given $j^n$ configuration, states which are the only ones with a given value of *J* are rather rare. They may be found among the states with the highest values of *J* (certainly the state $J=nj-n(n-1)/2)$ as well as among those with the lowest *J* values.
Still, there may be special cases like those mentioned in ref.\[1\]. In the case of $j=7/2$, the non-diagonal matrix element of any two body interaction between the *J*=2 states with *v*=2 and *v*=4 vanishes. The same is true also for the *v*=2 and *v*=4 states with *J*=4. This follows from general properties of the seniority scheme. More surprising is the fact that in the $(9/2)^4$ configuration, there is a *v*=4, *J*=4 state whose matrix element of any two-body interaction, vanishes not only with the *v*=2, *J*=4 state but also with the orthogonal *v*=4, *J*=4 state. Its eigenvalue is a linear combination of two-body interactions with positive rational coefficients as given in ref.\[1\]. The same situation occurs also for the corresponding *J*=6 states.
[99]{} L.Zamick and P.Van Isacker, Phys. Rev. **78**, 044327 (2008). A.de Shalit and I.Talmi, Nuclear Shell Theory, Academic Press (1963); Reprinted by Dover Publications (2003). R.F.Bacher and S.Goudsmit, Phys. Rev. **46** (1934) 948. G.Racah, Phys. Rev. **63** (1943) 367. I.Talmi, Simple Models of Complex Nuclei, Harwood Academic Publishers (1993). G.Racah, Phys. Rev. **62** (1942) 438.
|
epsf
The standard cosmology provides a whole description from a few minutes after “big bang” to now [@KoTu]. One of the evidence which supports it is the nucleosynthesis, which successfully predicts cosmological abundances of all light elements. It requires that there is a small asymmetry of the baryons in the universe: $\eta_B=n_B/n_{\gamma} \sim 10^{-10}$, where $n_B$ and $n_{\gamma}$ denote the number density of the baryon and photon, respectively. This and other observations show that our universe is made almost entirely of matters and devoid of antimatters. Such matter-antimatter asymmetry is produced by baryogenesis, which takes place nonthermally through baryon and CP violating interactions in the very early universe.
On the other hand, inflation solves many problems which cannot be explained in the standard cosmology, such as the homogeneity, flatness, and monopole problems. Inflation thus predicts that $\Omega_{{\rm tot}}= 1$, where $\Omega$ is the density parameter, the ratio of the density to the critical density $\rho_c \simeq 1.9 \times 10^{-29} h^2$ g/cm$^3$, and $h$ is the Hubble parameter normalized by 100 km/sec/Mpc. However, $\eta_B \sim 10^{-10}$ corresponds to $\Omega_B h^2 \simeq 0.02$, far less than the prediction from the inflation. Even if one does not consider the inflation, $\Omega_{{\rm matter}} \gtrsim 0.2$ is expected from observations for dynamical properties of galaxies and clusters of galaxies. Therefore, most of the density of the universe has to be in the form of dark matter.
Several mechanisms for creating baryons have been proposed, but none of them explain directly why the universe has almost the same amount of baryons and the dark matter. Their answer is that it is a numerical coincidence. However, the Q ball provides a natural scenario for explaining both of them simultaneously [@KuSh; @EnMc].
In the supersymmetric standard models, Affleck-Dine (AD) mechanism [@AfDi] is the most promising procedure for the baryogenesis. In the minimal supersymmetric standard model (MSSM), there are many flat directions consist of squarks and sleptons [@DiRaTh], which can be identified as the AD field. Its potential is almost flat but slightly lifted up by effects of supersymmetry (SUSY) breaking. For the mechanism of SUSY breaking, there are two famous scenarios: the gravity- and gauge-mediated SUSY breakings. It was believed that the AD field stays at large field value at the inflationary stage, and, when the Hubble parameter becomes as small as the AD scalar mass after inflation, rolls down homogeneously its potential with rotation, making the baryon number of the universe.
Recently, however, it was revealed that the AD field does not evolve homogeneously, but feels spatial instabilities, which grow nonlinear and form into Q balls [@KuSh]. A Q ball is a kind of the nontopological soliton, whose stability is guaranteed by the nonzero charge $Q$ [@Coleman; @Kusenko1]. In the context of the AD baryogenesis, the charge $Q$ is the baryon number $B$. In the gauge-mediated SUSY breaking, a Q ball is stable against the decay into nucleons, provided that its charge is large enough so that its energy per unit charge is less than nucleon mass: $E_Q/Q \simeq m_{\phi}Q^{-1/4} < 1$ GeV [@DvKuSh; @KuSh], where $m_{\phi}\sim 1$ TeV, is the mass of AD field. Therefore, the Q ball itself can be a candidate for the dark matter. On the other hand, in the gravity-mediated SUSY breaking, the energy of a Q ball per unit charge is essentially constant: $E_Q/Q \simeq m_{\phi} >$ 1 GeV [@EnMc]. Thus, it should decay into nucleons, and the dark matter will be lightest supersymmetric particles (neutralinos) produced in Q-ball decays. In either case, the dark matter and the baryon number of the universe can be explained simultaneously by the Q-ball formation through the AD mechanism.
In all the previous studies of Q balls in the context of SUSY breaking, the effects of gauge- and gravity-mediations are considered separately. However, it is natural to have both effects in the gauge-mediated SUSY breaking scenario, since the gravity-mediation effects will dominate over the gauge ones at the large field value. Cosmology including AD baryogenesis in such more realistic SUSY breaking scenario was considered in Ref. [@GoMoMu]. There, AD field is regarded as a homogeneously rotating condensate, but we notice that it will form Q balls due to the instabilities of the field. Particular interest is the smallness of the gravitino mass comparing with that in the gravity-mediation scenario. It usually ranges between 100 keV and 1 GeV. Therefore, we can imagine a new type of a stable Q ball: the profile is the same as that in the gravity-mediation, but its energy per unit charge is less that 1 GeV because of the small gravitino mass. In this Letter, we study the cosmological consequences of Q balls (baryogenesis and the dark matter) in the gauge-mediated SUSY breaking, taking into account the gravity-mediation effects at large field value.
To be concrete, let us assume the following potential for the AD field, $$\begin{aligned}
V(\Phi) & = & m_{\phi}^4
\log\left(1+\frac{|\Phi|^2}{m_{\phi}^2} \right)
\nonumber \\
& & + m_{3/2}^2|\Phi|^2 \left[ 1 + K\log \left(
\frac{|\Phi|^2}{M_*^2}\right)\right],\end{aligned}$$ where $m_{3/2}$ is the gravitino mass, $K(<0)$ term a one-loop correction, [^1] $M_*$ the renormalization scale, and we assume that the second term should be neglected for small field value. This is nothing but the sum of the potentials for the gauge- and gravity-mediation mechanisms studied previously [@KuSh; @EnMc; @KK1; @KK2]. However, as we mentioned earlier, the gravitino mass is considerably smaller. The second term will dominate the potential when $$\label{eq}
\phi \gtrsim \phi_{{\rm eq}}
\equiv \sqrt{2}\frac{m_{\phi}^2}{m_{3/2}},$$ where $\Phi=\phi\exp(i\omega t)/\sqrt{2}$. In this case, a new type of a stable Q ball is produced. Its property is very similar to that in the gravity-mediation, such as [@EnMc; @KK3] $$\begin{aligned}
R_Q \simeq |K|^{-1/2}m_{3/2}^{-1},
& \qquad & \omega \simeq m_{3/2}, \nonumber \\
\phi \simeq |K|^{3/4}m_{3/2}Q^{1/2},
& \qquad & E_Q \simeq m_{3/2} Q,
\label{gravity}\end{aligned}$$ but, as can be seen from the last equation, it is stable against the decay into nucleons. In the opposite case, the Q-ball properties are the same as in the gauge-mediation only [@KuSh]: $$\begin{aligned}
R_Q \simeq m_{\phi}^{-1}Q^{1/4},
& \qquad & \omega \simeq m_{\phi}Q^{-1/4}, \nonumber \\
\phi \simeq m_{\phi}Q^{1/4},
& \qquad & E_Q \simeq m_{\phi} Q^{3/4}.
\label{gauge}\end{aligned}$$ The energy per unit charge can be treated from unified viewpoint. The largest charge of the Q ball formed depends linearly on the initial charge density of the AD field as [@KK1; @KK2] $$Q \simeq \beta \frac{q(0)}{m_{3/2}^3}
\simeq \beta \left(\frac{\phi(0)}{m_{3/2}}\right)^2,$$ where $\beta \lesssim 1$ [@KK3], and we use $q=\omega\phi^2\simeq m_{3/2}\phi^2$. It can be understood by estimating Q-ball charge in the standard way. The charge is given by $$Q = \int d^3 x \omega \phi^2
= \left(\frac{\pi}{2}\right)^{3/2}\omega \phi_0^2 R^3
= \beta' \left( \frac{\phi_0}{m_{3/2}}\right)^2,$$ where we assume the Gaussian profile ansatz [@EnMc], $\phi = \phi_0 \exp(-r^2/R^2)$, which is a very good approximation, and use relations (\[gravity\]), and $\beta' \simeq 2\times 10^3 (|K|/0.01)^{-3/2}$. The discrepancy of coefficients $\beta$ and $\beta'$ comes from the fact that $\phi(0)\neq\phi_0$ and there are more than one Q balls with charges of the same order of magnitude as the largest.
Inserting it into Eq.(\[eq\]), we obtain $$\label{mgravitino}
m_{3/2} \gtrsim (2\beta)^{1/4} m_{\phi}Q^{-1/4}.$$ The right hand side of this equation is identical to the expression for the energy per unit charge of the gauge-mediation besides the factor of order unity. The energy per unit charge of the Q ball is written as $$\frac{E_Q}{Q} =
\left\{
\begin{array}{ll}
m_{\phi}Q^{-1/4} & \qquad \phi \lesssim \phi_{{\rm eq}} \\[2mm]
m_{3/2} & \qquad \phi \gtrsim \phi_{{\rm eq}}
\end{array}
\right. \,,$$ which is shown in Fig. \[e-q\]. The gap on the boundary should disappear and both sides of the curves will be smoothly connected because Q balls formed in this region are not the exact type of either (\[gravity\]) or (\[gauge\]), but will show properties between them.
\
Since Q balls are stable even for $\phi \gtrsim \phi_{{\rm eq}}$, where the gravity-mediation effect is crucial, the baryon number in the universe should be explained by the baryons evaporated from Q balls, as is the same as for the gauge-mediation type [@LaSh]. The evaporation rate of the Q ball is [@LaSh] $$\label{evap}
\Gamma_{{\rm evap}}\equiv\frac{dQ}{dt}
=-\alpha\mu T^2 4\pi R_Q^2,$$ where $\mu$ is a chemical potential of the Q ball, which is estimated as $\mu \simeq \omega$ because $\omega$ is energy of $\phi$-field inside the Q ball. Although the mass of the (free) AD particle $m_{\phi}$ is affected by thermal corrections, which should be changed as $m_{\phi} \rightarrow m_{\phi}(T) \sim T$, at $T\gg m_{\phi}$, the gravitino mass is not affected, since particles coupled to the AD field are decoupled from thermal bath when the AD field has a large vacuum expectation value. At $T\gtrsim m_{\phi}$, large numbers of $\phi$-particles are in thermal bath outside Q balls, so $\alpha \sim
1$. On the other hand, since only light quarks are in thermal bath at $T\lesssim m_{\phi}$, the corresponding cross section is highly suppressed by the heavy gluino exchanges, and $\alpha \simeq (T/m_{\phi})^2$.
However, if the rate of the charge diffusion from the “atmosphere” of the Q ball is smaller than the evaporation rate, chemical equilibrium will established there, which results in the suppression of the evaporation [@BaJe]. The diffusion rate is [@BaJe] $$\label{diff}
\Gamma_{{\rm diff}}\equiv\frac{dQ}{dt}
=-4\pi\zeta R_Q D \mu_Q T^2,$$ where $D=a/T$ with $a\simeq 4-6$, and $\zeta \sim 1$.
The time scale of charge transportation is determined by the diffusion when $\Gamma_{{\rm diff}}\lesssim\Gamma_{{\rm evap}}$. It holds for $T\gtrsim T_* \equiv a^{1/3}|K|^{1/6}(m_{3/2}m_{\phi}^2)^{1/3}$. In this case, using Eqs.(\[gravity\]) and (\[diff\]), and assuming the radiation-dominated universe, $t=AM/T^2$, where $A\approx 0.2$ and $M\simeq 2.4\times 10^{18}$ GeV, we obtain $$\label{high-T}
\frac{dQ}{dT} = \frac{8\pi aAM}{|K|^{1/2}T^2}.$$ On the other hand, when $T\lesssim T_*$, the diffusion effect is negligible, and Eq.(\[high-T\]) should be replaced by $$\frac{dQ}{dT} = \frac{8\pi AMT}{|K|m_{3/2}m_{\phi}^2}.$$ Therefore, total amount of the charge evaporated from the Q ball is $$\Delta Q \simeq 12\pi AM
\left( \frac{a}{|K|} \right)^{2/3}
( m_{3/2}m_{\phi}^2)^{-1/3}.$$ Provided that the initial charge of the Q ball is larger than the evaporated charge, we regard that the Q ball survives from evaporation, and contributes to the dark matter of the universe: $$\label{survive}
Q_{{\rm init}} \gtrsim \Delta Q
\simeq
9.8\times 10^{18}
\left( \frac{m_{3/2}}{{\rm GeV}} \right)^{-1/3}
\left( \frac{m_{\phi}}{{\rm TeV}} \right)^{-2/3},$$ where we set $a=4$ and $|K|=0.01$.
Now we can relate the baryon number and the amount of the dark matter in the universe. As mentioned above, the baryon number of the universe should be explained by the amount of the charge evaporated from Q balls, $\Delta Q$, and the survived Q balls become the dark matter. If we assume that Q balls do not exceed the critical density of the universe, i.e., $\Omega_Q \lesssim 1$, and the baryon-to-photon ratio as $\eta_B \sim 10^{-10}$, $$\label{dm}
Q \lesssim 3.2h^2\times 10^{21}
\left( \frac{m_{3/2}}{{\rm GeV}} \right)^{-4/3}
\left( \frac{m_{\phi}}{{\rm TeV}} \right)^{-2/3}.$$ Rewriting Eq.(\[mgravitino\]), we have $$\label{Q-phi-eq}
Q \gtrsim 2\beta \times 10^{12}
\left( \frac{m_{\phi}}{{\rm TeV}} \right)^4
\left( \frac{m_{3/2}}{{\rm GeV}} \right)^{-4}.$$ Combining this constraint with Eqs.(\[survive\]) and (\[dm\]), together with $m_{3/2} \lesssim 1$ GeV, which implies that the gravity-mediation type of the Q ball is stable against the decay into nucleons, we obtain the allowed region for the new type of the stable Q ball explaining the baryon number of the universe. Figure \[Q-m\] shows the allowed region on $(Q,m_{3/2})$ plane for $m_{\phi}=1$ TeV. The shaded regions represent that the new type of stable Q balls are created, and the baryon number of the universe can be explained by the mechanism mentioned above. Furthermore, the new type of stable Q balls contribute crucially to the dark matter of the universe at present, if the Q balls have the charge given by the thick line in the figure. Notice that Q balls with very large charge are not allowed because they will overclose the density of the universe. [^2] Therefore, the initial conditions for the AD field is restricted severely [@KK3].
=5.5cm\
One may wonder if these new type of stable Q balls can be detected. When a Q ball collides with nucleons, they enter the surface layer of the Q ball, and dissociate into quarks, which are converted into squarks. In this process, Q balls release $\sim 1$ GeV energy per collision by emitting soft pions. This process is the basis for the Q-ball detections [@KuKuShTi; @ArYoNaOg], which is called (Kusenko-Kuzmin-Shaposhnikov-Tinyakov) KKST process in the literature. It occurs for electrically neutral Q balls (ENQB). For electrically positively charged Q balls (EPCQB), the KKST process is strongly suppressed by Coulomb repulsion, and only electromagnetic processes will take place. For electrically negatively charged Q balls (ENCQB), the both KKST and electromagnetic processes occur, but the former is dominant, which is essentially the same as for ENQBs.
In either case, the detection is more difficult than for the gauge-mediation type of Q balls, since the geometrical cross section is smaller for large $Q$, and the Q-ball mass is larger for the same $Q$, which results in small flux. With the discussions similar to Ref. [@ArYoNaOg], we can restrict the parameter space $(Q,m_{3/2})$ by the several experiments. Fig. \[SENS\] shows the results for ENQBs. Lower left regions are excluded by the various experiments. The present available data prohibit the stable gravity-mediation type Q balls with large gravitino mass to be both the dark matter and the source of the baryons, and future experiments such as the Telescope Array Project or the OWL-AIRWATCH detector may detect this type of Q balls with an interesting gravitino mass $\sim 100$ keV.
=7.4cm\
For EPCQBs with $Z=1$, similar constraints as for ENQBs are put by the MACRO experiment [@KK3].
In summary, we have obtained a new type of a stable Q ball in the context of gauge-mediated SUSY breaking in MSSM. Many properties are the same as the gravity-mediation type of Q ball, but it is stable against the decay into nucleons, since the energy per unit charge is equal to the gravitino mass $m_{3/2}$, which can be smaller than nucleon mass of 1 GeV in the gauge-mediation mechanism. We have considered the cosmological consequences in this new Q-ball scenario. Because of its stability, it can be a nice candidate for the dark matter of the universe. In the present case, the baryons are produced only by evaporation from Q balls, since (almost) all the baryons are trapped in Q balls during their formation. We have found that the Q ball with $Q \sim 10^{25}-10^{22}$ can account for both the dark matter and the baryon number of the present universe for $m_{3/2}\simeq 10^{-4}-10^{-1}$ GeV and $m_{\phi}=1$ TeV, and such Q balls may be detected by the future experiments.
The authors are grateful to R. Banerjee for useful comments and T. Yoshida for helpful discussions. M.K. is supported in part by the Grant-in-Aid, Priority Area “Supersymmetry and Unified Theory of Elementary Particles”($\#707$).
For a review, see, e.g., E.W. Kolb and M.S. Turner, [*The Early Universe*]{}, Addison Wesley, (1990). A. Kusenko and M. Shaposhnikov, Phys. Lett. [**B418**]{}, 46 (1998). K. Enqvist and J. McDonald, Phys. Lett. [**B425**]{}, 309 (1998); Nucl. Phys. [**B538**]{}, 321 (1999); Phys. Lett. [**B440**]{}, 59 (1998). I. Affleck and M. Dine, Nucl. Phys. [**B249**]{}, 361 (1985). M. Dine, L. Randall, and S. Thomas, Nucl. Phys. [**B458**]{}, 291 (1996). S. Coleman, Nucl. Phys. [**B262**]{}, 263 (1985). A. Kusenko, Phys. Lett. [**B404**]{}, 285 (1997). G. Dvali, A. Kusenko, and M. Shaposhnikov, Phys. Lett. [**B417**]{}, 99 (1998). A. de Gouvêa, T. Moroi, and H. Murayama, Phys. Rev. [**D56**]{}, 1281 (1997). S. Kasuya and M. Kawasaki, Phys. Rev. [**D61**]{}, 041301 (2000). S. Kasuya and M. Kawasaki, to appear in Phys. Rev. [**D**]{} (2000). S. Kasuya and M. Kawasaki (in preparation). M. Laine and M. Shaposhnikov, Nucl. Phys. [**B532**]{}, 376 (1998). R. Banerjee and K. Jedamzik, Phys. Lett. [**B484**]{}, 278 (2000). A. Kusenko, V. Kuzmin, M. Shaposhnikov, and P.G. Tinyakov, Phys. Rev. Lett. [**80**]{}, 3185 (1998). J. Arafune, T. Yoshida, S. Nakamura, and K. Ogure, hep-ph/0005103, and references therein.
[^1]: Usually, gaugino contributions to $K$ is dominated and $K$ becomes negative. However, if $K>0$ by some large Yukawa couplings, the AD field can be stabilized and Q balls can be created only for $\phi \lesssim \phi_{{\rm eq}}$, where $\phi_{{\rm eq}}$ is defined in Eq.(\[eq\]).
[^2]: If the baryons are produced by other mechanism, larger Q balls can be allowed. In this case, however, a nice relation between the densities of the baryon and the dark matter does not hold.
|
---
abstract: 'Six-dimensional $\mathcal{N}=(1,0)$ superconformal field theories can be engineered geometrically via F-theory on elliptically-fibered Calabi-Yau 3-folds. We include torsional sections in the geometry, which lead to a finite Mordell-Weil group. This allows us to identify the full non-Abelian group structure rather than just the algebra. The presence of torsion also modifies the center of the symmetry groups and the matter representations that can appear. This in turn affects the tensor branch of these theories. We analyze this change for a large class of superconformal theories with torsion and explicitly construct their tensor branches. Finally, we elaborate on the connection to the dual heterotic and M-theory description, in which our configurations are interpreted as generalizations of discrete holonomy instantons.'
bibliography:
- 'biblio.bib'
---
CERN-TH-2020-081\
UUITP-14/20
\
Markus Dierigl$^{\,\text{a}}$, Paul-Konstantin Oehlmann$^{\,\text{b}}$, Fabian Ruehle$^{\,\text{c,d}}$
Introduction
============
Six-dimensional superconformal field theories (SCFTs) have played a prominent role in high energy physics in recent years. Since there are no SCFTs in dimensions beyond six [@Nahm:1977tg], and those in dimension six are necessarily strongly coupled, these theories are highly interesting already in their own rights, see e.g. [@Witten:1995gx; @Witten:1995ex; @Strominger:1995ac; @Seiberg:1996vs; @Ganor:1996mu]. Moreover, 6d SCFTs can be used to derive a vast network of lower-dimensional SCFTs via compactification, see [@Ohmori:2015pua; @Ohmori:2015pia; @DelZotto:2015rca; @Morrison:2016nrt; @Razamat:2016dpl; @Apruzzi:2016nfr; @Bah:2017gph; @DelZotto:2017pti; @Apruzzi:2018oge; @Bhardwaj:2018yhy; @Bhardwaj:2018vuu; @Bhardwaj:2019fzv] for a partial list of compactifications of 6d $\mathcal{N} = (1,0)$ theories. In this sense, the 6d SCFTs can be understood as master theories whose investigation is paramount in order to understand the generation and connection of SCFTs in general.
Of great importance in the investigation of 6d SCFTs is their construction within string theory frameworks and especially F-theory [@Vafa:1996xn; @Morrison:1996na; @Morrison:1996pp]. In fact, F-theory has lead to a vast number of 6d SCFTs using a classification of base geometries describing the tensor branch deformation of the singular theories and taking into account the possibility of singular fibers [@Heckman:2013pva; @Heckman:2015bfa], see also [@Bhardwaj:2015oru; @Bertolini:2015bwa; @Bhardwaj:2015xxa; @Bhardwaj:2018jgp] and the review [@Heckman:2018jxk]. In most of these models, the construction of the 6d SCFTs was achieved by tuning a Weierstrass model: The appearance of non-Higgsable clusters on rational curves with negative self-intersection [@Morrison:2012np], as well as the collisions of singular divisors, lead to superconformal field theories [@DelZotto:2014hpa]. While the Weierstrass model, together with monodromy data [@Grassi:2011hq], is sufficient to extract the non-Abelian part of the gauge and flavor *algebras*,[^1] it is usually not enough to determine the full gauge and flavor *groups* on the tensor branch. For works discussing also Abelian flavor symmetries see [@Lee:2018ihr; @Apruzzi:2020eqi].
In order to determine the full non-Abelian group structure, one needs to access additional information encoded in the presence of torsional sections, see e.g. [@Aspinwall:1998xj; @Mayrhofer:2014opa; @Cvetic:2017epq; @Hajouji:2019vxs] and also [@Monnier:2017oqd]. These impose certain factorization constraints on the parameters in the Weierstrass model. Equivalently, in these models the full $\text{SL}(2,\mathbb{Z})$ monodromy is reduced to an orbit of an appropriate congruence subgroup, see [@Hajouji:2019vxs]. With the torsional sections enforcing a modding out of a finite discrete group embedded into the center of the non-Abelian group factors, this leads to a restriction on the gauge and flavor representations that appear on the tensor branch of the corresponding SCFTs. Moreover, the resulting non-Abelian groups are no longer simply-connected. Therefore, SCFTs respecting the presence of a non-trivial Mordell-Weil torsion will have restrictions on their tensor branch and other deformations, see e.g. [@Heckman:2015ola; @Mekareeya:2016yal; @Heckman:2016ssk; @Heckman:2018pqx].
In this paper, we will extend the classification of 6d SCFTs to models with non-trivial Mordell-Weil torsion. In this way, we can classify the gauge and flavor symmetry groups appearing on the tensor branch of 6d SCFTs rather than their algebras. As in the unrestricted case, the major building blocks are given by non-Higgsable clusters and the collision of two singular divisors, now, in the presence of torsional sections. Scanning different theories and their compatible singularity enhancements, we find a class of allowed deformations which respect the Mordell-Weil torsion. This can be regarded as a starting point for the development of constructive approaches as described for example in [@Heckman:2018pqx] and has the potential for a full classification of 6d SCFTs with respect to their global group structure, harnessing the full power of F-theory.
The global structure also modifies the possibility of compactifications to lower dimensions, since reducing the group structure (see e.g. [@Anderson:2019kmx] for compact examples) naturally leads to a large possibility for background fluxes [@tHooft:1979rtg; @Aharony:2013hda], see also [@Tachikawa:2013hya; @Garcia-Etxebarria:2019cnb] for considerations with more supersymmetry and the connection of the global group in 4d to flux data in the 6d theory. In fact, some of the theories we find here have already been anticipated in [@Ohmori:2018ona] and used to compactify from six to four dimensions with fluxes of non-trivial Stiefel-Whitney class. While in [@Ohmori:2018ona] the authors use a field-theoretic intuition based on the operator spectrum, our construction sheds light on the UV-complete origin of the global data in terms of the string theory construction within F-theory.
The rest of this work is organized as follows. In Section \[sec:rev\] we review the construction of F-theory models with non-trivial and finite Mordell-Weil group related to the presence of torsional sections or the restriction of the monodromy groups. The approach to the construction of general 6d SCFTs in F-theory is briefly described in Section \[sec:SCFT\]. While not essential for our later analysis, we further point out alternative approaches in string constructions to access the global realization of gauge and flavor groups, which should be understood as a staring point for further investigations in the M-theory and heterotic realization of 6d SCFTs. In Section \[sec:NHC\] we begin our systematic approach for the construction of globally constrained 6d SCFTs by investigating non-Higgsable clusters in the presence of torsional sections. These become important building blocks for the analysis of non-simply-connected 6d SCFTs, which is carried out in Section \[sec:NSCFTs\] and \[sec:zoo\]. We conclude in Section \[sec:concl\] and present details of some of the results, as well as exotic models, in the Appendices.
Mordell-Weil Torsion and Restricted Monodromies {#sec:rev}
===============================================
In this section we briefly review F-theory models with non-trivial Mordell-Weil group. In F-theory on Calabi-Yau 3-folds, singularities over non-compact curves in the base lead to flavor symmetries, whereas singularities over compact curves lead to gauge symmetries of the 6d effective theory. In the presence of non-trivial Mordell-Weil torsion, the structure of these groups is modified. More precisely, the presence of torsional sections in the geometry of the elliptically-fibered 3-folds leads to a restriction of the allowed fiber monodromies[^2]. These restrictions impose constraints on the available matter representations and lead in general to a non-simply-connected non-Abelian structure of the symmetries in the model. For a discussion of Mordell-Weil torsion in F-theory see also [@Aspinwall:1998xj; @Mayrhofer:2014opa; @Hajouji:2019vxs] as well as the review [@Weigand:2018rez].
F-Theory and Weierstrass Models
-------------------------------
F-theory compactifications can be thought of as a non-perturbative generalization of type IIB string constructions, where the axio-dilaton is identified with the complex structure parameter $\tau$ of an auxiliary torus $\mathcal{E}$, [@Vafa:1996xn; @Morrison:1996na; @Morrison:1996pp]. The generalization proceeds by allowing the axio-dilaton to vary over the physical compactification space $B$, sourced by the back-reaction of generalized 7-branes, so-called $(p,q)$-branes. The whole geometry can be described as an elliptic fibration $X$ $$\begin{aligned}
\begin{array}{rcl}
\mathcal{E} & \rightarrow & X \\
&&\,\downarrow \pi \\
&& B \, ,
\end{array} \end{aligned}$$ which, in order to preserve the minimal amount of supersymmetry, needs to be Calabi-Yau. In the following, we are interested in the description of 6d $\mathcal{N} = (1,0)$ supersymmetric field theories decoupled from gravity. For this purpose, we choose $B$ to be a non-compact, complex two-dimensional Kähler manifold.
The main tool for the description of the elliptically-fibered Calabi-Yau 3-fold $X$ is going to be the Weierstrass model, which defines $X$ by the hypersurface equation $$\begin{aligned}
y^2 = x^3 + f x z^4+ g z^6 \,.\end{aligned}$$ Here, $x$, $y$, and $z$ are projective coordinates in $\mathbb{P}_{2,3,1}$, and the coefficients $f,g$ are sections of multiples of the anti-canonical class $-K$ of the base, $$\begin{aligned}
f \sim - 4 K \qquad g \sim - 6 K \, .\end{aligned}$$ They parametrize the local complex structure $\tau$ up to SL$(2,\mathbbm{Z})$ transformations. By construction, the Weierstrass model has a smooth section, called the zero-section, given by $s_0:~[1:1:0]$. Along the discriminant locus $\{\Delta=0\}$ with $$\begin{aligned}
\Delta = 4f^3 + 27 g^2 \sim -12 K \,,\end{aligned}$$ the elliptic fiber degenerates. The type of singularity is determined by the vanishing order of $f$, $g$, and $\Delta$, as well as possible monodromy actions [@Grassi:2011hq]. This classification is summarized in Table \[tab:KodTate\]. For generalization see also [@Katz:2011qp].
Fiber Type ord (f) ord(g) $ord(\Delta) $ monodromy cover algebra
--------------------- ------------ ------------ ---------------- ----------------------------------------------------------------- ------------------------------------------------------------------
$I_0$ $\geq 0$ $\geq 0 $ $0$ - -
$I_1$ $ 0$ $ 0 $ $1$ - -
$I_m$ $0$ $0 $ $m$ $\psi^2+(2 g/2f)|_{z=0}$ $ \mathfrak{sp}([m/2])$ or $\mathfrak{su}(m)$
$II$ $\geq 1$ $1 $ $2$ $-$ -
$III$ $ 1$ $\geq 2 $ $3$ $-$ $\mathfrak{su}(2) $
$IV$ $ \geq 2$ $2 $ $4$ $ \psi^2 - (g/z^2)|_{z=0}$ $\mathfrak{sp}(1) $ or $\mathfrak{su}(3) $
$I_0^*$ $ \geq 2$ $ \geq 3 $ $6$ $ \psi^3 + \psi (f/z^2)|_{z=0} \,\psi +(g/z^3)|_{z=0}$ $\mathfrak{g}(2) $ or $\mathfrak{so}(7) $ or $\mathfrak{so}(8) $
$I_{2n-5}^*$, $n>2$ $ 2$ $ 3 $ $2n+1$ $ \psi^2 + \frac{1}{4} (\Delta/z^{2n+1})(2 z f/ 9 g)^3|_{z=0}$ $\mathfrak{so}(4n-3) $ or $\mathfrak{so}(4n-2) $
$I_{2n-4}^*$, $n>2$ $ 2$ $ 3 $ $2n+2$ $ \psi^2 + (\Delta/z^{2n+2})(2 z f/9 g)^2|_{z=0} $ $\mathfrak{so}(4n-1) $ or $\mathfrak{so}(4n ) $
$IV^*$ $ \geq 3$ $4 $ $8$ $ \psi^2 - (g/z^4)|_{z=0}$ $\mathfrak{f}_4 $ or $\mathfrak{e}_6 $
$III^*$ $ 3$ $\geq 5 $ $9$ $-$ $\mathfrak{e}_7 $
$II^*$ $ \geq 4$ $ 5 $ $10$ $-$ $\mathfrak{e}_8 $
non-min $ \geq 4$ $ \geq 6 $ $ \geq 12$ $-$ -
: \[tab:KodTate\] Kodaira-Tate classification of singular fibers and local gauge algebras.
In codimension one, these degenerations of the fiber precisely correspond to the loci of $(p,q)$ 7-brane stacks in $B$, whose world-volume gauge algebras correspond to their Tate fiber type. Since the gauge-kinetic function of a 7-brane is proportional to the volume of the divisor it wraps, we distinguish the cases where they have finite or infinite volume:
- Branes that wrap compact divisors admit dynamical gauge fields and encode gauge symmetries.
- Branes that wrap non-compact divisors encode non-dynamical background gauge connections and host flavor symmetries.
To engineer a consistent 6d supersymmetric field theory with the described ingredients, cancellation of various anomalies must be taken care of by the 6d version of the Green-Schwarz mechanism [@Green:1984bx; @Sagnotti:1992qw]. Since we work in a non-gravitational setup, anomaly cancellation only needs to be imposed for the gauge part of the symmetries, while the flavor sector (or would-be gravitational sector prior to decoupling) can be anomalous. For a general account of anomaly cancellation in F-theory see [@Park:2011ji].
In the type IIB picture, matter multiplets naturally arise from open strings stretching between two stacks of 7-branes. These states become massless where the branes intersect. In the F-theory picture, the 7-branes wrap codimension-one loci in the base, over which the fiber degenerate. Brane intersections hence correspond to codimension-two loci in the base. Since two loci over which the fiber degenerates intersect, the fiber singularity becomes worse at the intersections.
From the type IIB picture, matter multiplets transform in the bi-fundamental representation of the two 7-brane stacks. However, since in F-theory more general $(p,q)$ branes and strings are possible, more general representations can occur. To deduce the precise representations $\mathbf{R}$ of the matter multiplets, one can employ two different techniques: The first way is to fully resolve the geometry and use the duality to M-theory on the Coulomb branch to geometrically extract the weights from M2-branes wrapping holomorphic curves (e.g. see [@Marsano:2011hv]). The second approach is due to Katz and Vafa [@Katz:1996xe] and is closer to the IIB picture by interpreting the two intersecting 7-brane stacks as a deformation of the configuration where they were parallel and on top of each other. Even though Kodaira’s (or Tate’s) classification is strictly speaking only valid in codimension one, one can usually start from the enhanced algebra $\mathfrak{g}_{IJ}$ dictated by the codimension-two vanishing orders, and break it by tilting one part of the brane stack such that two stacks with gauge algebra $\mathfrak{g}_I$ and $\mathfrak{g}_J$ arise. In field theory, such a tilting corresponds to a VEV in the adjoint representation and therefore the representation at codimension two can be read off from the decomposition $$\begin{aligned}
\text{adj}(\mathfrak{g}_{IJ})\to\bigoplus_{I,J} (\mathbf{R}_{I},\mathbf{R}_{J})\oplus (\text{adj}(\mathfrak{g}_{I}),\mathbf{1})\oplus (\mathbf{1},\text{adj}(\mathfrak{g}_{J})) \,.\end{aligned}$$ In most cases, this is enough to deduce matter representations. More care has to be taken when considering non-simply laced groups. Here the additional source of monodromy has to be taken into account when deducing the resulting representations [@Aspinwall:2000kf; @Grassi:2011hq].
Non-Simply-Connected Groups from Mordell-Weil Torsion
-----------------------------------------------------
In this work we focus on the global structure of non-Abelian gauge and flavor symmetries in the six-dimensional F-theory models. In general, these groups will be non-simply-connected which means that $$\begin{aligned}
\pi_1 (\mathcal{G}) = T \,,
\label{eq:globalT}\end{aligned}$$ where $\mathcal{G}$ describes the flavor and gauge symmetries of the model. In the following, we focus on models without Abelian gauge group factors, thus $T$ is a finite group. By starting with the simply-connected group $\mathcal{G}^*$ related to the algebras determined by the codimension-one singularities, can be understood as modding out part of the center, see e.g. [@Aharony:2013hda], of the flavor and gauge symmetries, $$\begin{aligned}
\mathcal{G} = \frac{\mathcal{G}^*}{T}\,.\end{aligned}$$ Note that if $T$ is only a subgroup of the centers of the individual factors in $\mathcal{G}^*$, the group action has to be further specified. One way to deduce it is to study the matter fields of the theory, which have to transform trivially under $T$.
In F-theory, the appearance of a non-simply-connected gauge group is related to the presence of extra sections in the elliptic fibration. In general, additional sections are related to the Mordell-Weil group of the elliptic fiber. Extra rational sections lead to Abelian $\text{U}(1)$ symmetries [@Cvetic:2018bni; @Weigand:2018rez] and would correspond to the free part of the Mordell-Weil group, which we do not discuss here. Extra torsional sections are related to the torsion part of the Mordell-Weil group, which is encoded in the discrete group $T$, see e.g. [@Aspinwall:1998xj; @Mayrhofer:2014opa]. Adding a $k$-torsional section $k$ times to itself (addition is meant as an element of the Mordell-Weil group) will map it back to the zero-section of the elliptic fibration. Therefore, there are no extra degrees of freedom associated to torsional sections.
The presence of these torsional sections imposes restrictions on the coefficients in the Weierstrass model [@Aspinwall:1998xj]. For compact Calabi-Yau 3-folds[^3] with no extra rational sections the possibilities for the Mordell-Weil group are quite constrained. The only possibilities are: $$\begin{aligned}
\begin{split}
T&= \mathbb{Z}_N \, , \qquad\qquad\!\! N \in \{2,3,4,5,6 \}\,, \\
T&= \mathbb{Z}_2 \times \mathbb{Z}_{2 N}\,, \quad N \in \{1,2 \} \, ,\\
T&= \mathbb{Z}_3 \times \mathbb{Z}_{3} \, .
\end{split}\end{aligned}$$ The most general Weierstrass models for these torsions have been constructed in [@Aspinwall:1998xj] and proven to be exhaustive for compact 3-folds in [@Hajouji:2019vxs]. For convenience we reproduce the restricted Weierstrass coefficients for all possibilities in in Appendix \[sec:EnhancedWSFs\]. In local models, also more exotic torsion models can be constructed, as illustrated in an example in Appendix \[sec:ExoticTorsion\].
Mordell-Weil torsion can also be understood in terms of the allowed monodromies of the axio-dilaton field, i.e. the complex structure of the elliptic fiber. While this can be any $\text{SL}(2,\mathbb{Z})$ element in a generic F-theory model, it is restricted to a proper subgroup thereof (or an $\text{SL}(2,\mathbb{Z})$ orbit of the subgroup) in the presence of torsion. Since the monodromies directly correspond to stacks of 7-branes in the base, one can already anticipate that this will restrict the allowed symmetry groups as well as matter representations. Indeed, one finds that the simply-connected group $\mathcal{G}^*$ must have a center which is compatible[^4] with the discrete Mordell-Weil torsion $T$. In addition, the matter representations have to respect the structure as well and thus have to be uncharged under $T$.
As a simple example consider the symmetry algebra $\mathfrak{su}_n$. If $T$ is trivial, all matter representation are allowed and there is no restriction. However, if the group is $\text{SU}(n) / \mathbb{Z}_k$ where $k$ divides $n$, representations that transform under the $\mathbb{Z}_k$ subgroup of the center are forbidden. This is not only relevant for the presence of dynamical charged states, but also on the level of line operators [@Aharony:2013hda; @Gaiotto:2014kfa]. For product groups, the structure can be even more versatile. For example, if the global group structure is given by $[\text{SU}(n) \times \text{SU}(n)] / \mathbb{Z}_k$, fields in the bi-fundamental representation are allowed (among others), while fundamentals of one group which are singlets of the other are forbidden.
Beyond restricting the allowed symmetry groups in the six-dimensional theory, the restricted Weierstrass models in also lead to singularities of the fiber that appear generically. For example in the $\mathbb{Z}_2$ case, $a_4$ appears quadratically in $\Delta$ and the zeros of this section will directly lead to $I_2$ fibers and hence to $\mathfrak{su}_2$ algebras. These generic singularities can also be seen by studying the modular curve of the restricted monodromy groups [@Hajouji:2019vxs].
6d SCFTs {#sec:SCFT}
========
In this section we review different ways of generating 6d SCFTs form string theory. We start with a discussion of the construction via F-theory, which we will then extend to include Mordell-Weil torsion in later sections. We further describe some of the elements in M-theory as well as heterotic string theories. While we do not claim to have a full understanding of the global characterization of the symmetry group in M-theory, we hint at some interesting possibilities for future investigations in that direction.
6d SCFTs and F-Theory
---------------------
6d SCFTs are non-gravitational theories that are strongly coupled and contain tensionless strings. In the F-theory framework, decoupling gravity is associated to decompactifying the base manifold $B$, as already mentioned in Section \[sec:rev\]. The anti-selfdual strings originate from D3 branes that wrap curves in $B$. Since the tension of the 6d strings is associated to the volume of the curves these have to collapse in the SCFT limit. For that to be possible at finite distance in moduli space they have to be contractible, imposing strong constraints on the overall geometry.
Using this reasoning, it is possible to classify all configurations that lead to 6d SCFTs in terms of a collection of (possibly intersecting but still contractible) curves in a complex two-dimensional, non-compact Kähler manifolds, [@Heckman:2013pva]. For all these models the SCFT point can be reached by a continuous deformation of the curve volumes, which are given in terms of the VEVs of scalar fields in tensor multiplets of the 6d theory. This is where the name tensor branch of the theory comes from. On generic points of the tensor branch the base manifold $B$ is smooth.
Relevant information about the base is contained in the intersection matrix of the compact curves $C_a$, given by $$\begin{aligned}
\label{eq:ChargeMatrix}
\Omega_{a b} = C_a \cdot C_b \,,\end{aligned}$$ which has to be negative definite in order for the model to lead to a 6d SCFT, see [@Mumford1961]. Simultaneously, $\Omega$ describes the string charge lattice of anti-selfdual strings in the 6d effective theory, which couple to the 2-form fields in the tensor multiplets. The allowed charge lattices for such configurations is quite restricted and plays a central role in the classification of 6d SCFTs [@Heckman:2015bfa].
The intersection pattern can enforce certain singularities in the elliptic fiber. These singularities can be further enhanced by tuning complex structure parameters, cf. also [@Heckman:2015bfa]. We focus on theories without frozen singularities, see [@Tachikawa:2015wka; @Bhardwaj:2018jgp] for a discussion of the latter. Of course many models of this type are connected by deformations with subsequent RG flow, see e.g. [@Heckman:2015ola; @Heckman:2016ssk; @Mekareeya:2016yal; @Heckman:2018pqx].
The essential building blocks of the F-theory construction are non-Higgsable clusters (NHCs) [@Morrison:2012np] and superconformal matter (SCM) theories [@DelZotto:2014hpa; @Heckman:2014qba]. The former can be separated into single-curve NHCs and multi-curve NHCs. Both involve curves with self-intersection smaller than $(-2)$, as will be discussed in section \[sec:NHC\]. Superconformal matter theories can be understood as the collision of two irreducible components of the discriminant locus $\Delta$. Usually this would lead to hypermultiplets in certain matter representations[^5] [@Katz:1996xe], but in the case of SCM collisions, the vanishing order at the intersection point is $$\begin{aligned}
(4,6,12) \leq \text{ord}(f,g,\Delta) < (8,12,24) \,.
\label{eq:vanord}\end{aligned}$$ Therefore, a crepant resolution of these models require resolving not only in the fiber, but also blowing up the base manifold[^6], which corresponds to a deformation that moves into the tensor branch of the theory as stated above. The upper limit in is imposed by the fact that codimension-two singularities with $\text{ord}(f,g,\Delta) \geq (8,12,24)$ do not allow for a crepant resolution: after blowing up such a codimension-two locus in the base, which reduces the vanishing order in codimension one by $(4,6,12)$, one still finds a codimension-one locus (i.e. the blow-up divisor) of vanishing order $(4,6,12)$ or worse. Often a single blow-up is not enough and one uncovers a full set of new tensor multiplets forming the SCM sector.
For illustrational purposes, we discuss a simple example of an SCM theory given by the collision of a $II^*$ fiber with $\mathfrak{e}_8$ algebra over the non-compact divisor defined by $\{ u = 0 \}$ and an $I_1$ fiber on $\{ v = 0 \}$, which intersect at the origin $$\begin{aligned}
y^2 = x^3 + x u^4(v-3) + u^5 (2 u-v) \, .\end{aligned}$$ The collision leads to a non-minimal $(4,6,12)$ singularity that can be resolved with a single blow-up, which introduces a curve of self-intersection $(-1)$. More explicitly, the blow-up along $\{ e_1 = 0 \}$ reads $$\begin{aligned}
\begin{split}
u& \rightarrow \tilde{u}\, e_1 \, , \quad v\rightarrow \tilde{v}\, e_1\,, \quad y \rightarrow \tilde{y}\, e_1^3 \, , \quad x \rightarrow \tilde{x}\, e_1^2 \,.
\end{split}\end{aligned}$$ After taking the proper transform, the new Weierstrass equation is $$\begin{aligned}
\widetilde{y}^2 = \widetilde{x}^3 + \widetilde{x}\, \widetilde{u}^4( \widetilde{v}\, e_1-3) + \widetilde{u}^5 (2 \widetilde{u}-\widetilde{v}) \,.\end{aligned}$$ The resulting geometry is smooth since now $\tilde{u} \tilde{v}$ is in the Stanley-Reisner ideal, indicating the fact that both coordinates cannot vanish at the same time. The resulting theory is the tensor branch of the SCM with just a single tensor. This is the minimal example of a 6d SCFT and it is usually referred to as the E-string theory due to its flavor group being $\text{E}_8$.
The full set of 6d SCFTs can be composed of several of the above components connected by $(-1)$ curves, leading to a vast set of 6d SCFTs. The goal of this paper is to use the geometric F-theory description of these 6d SCFTs and their fundamental building blocks and include torsional sections, which leads to a global specification of non-Abelian gauge and flavor symmetries on the tensor branch. The global constraints can already be anticipated by studying close relatives of the E-string theory above, namely discrete holonomy instanton theories (which also feature prominently in [@Aspinwall:1998xj]).
The E-string theory admits deformations which are described by finite Abelian subgroups $T$ of $\text{E}_8$. The $\text{E}_8$ flavor symmetry in turn is broken to the centralizer of $T$. The full list of such possibilities is given in Table \[tab:E8Broke\]. The discrete holonomy instanton theories are understood as collisions of two or multiple flavor branes, constituents of the original brane configuration. These lead to flavor groups forming a subgroup of $\text{E}_8$ with a $(4,6,12)$ singularity at the collision point. For example, the $\mathbb{Z}_5$ discrete holonomy instanton theory corresponds to a breaking[^7] of $\text{E}_8$ to the subgroup $[\text{SU}(5) \times \text{SU}(5)]/\mathbb{Z}_5$. This exact setup will appear in the discussion of SCFTs respecting $\mathbb{Z}_5$ Mordell-Weil torsion.
Let us mention that flavor symmetries might get modified at the SCFT point. Moreover, occasionally only part of the full flavor symmetry expected from field theory arguments is realized geometrically, see e.g. [@Bertolini:2015bwa] for the rank one cases. Note that in rare cases the flavor symmetry can also be reduced, as was e.g. discussed in [@Ohmori:2015pia]. The same can happen in the case of 6d SCFTs with non-trivial Mordell-Weil torsion. In cases where the enhancement is realized geometrically, we can track the global group structure throughout the RG flow. Often these situations can be engineered as the collision of two stacks of flavor branes which can explicitly be performed within the Mordell-Weil torsion models we discuss. As a consequence, also the flavor-enhanced models are affected by the action of the torsion $T$. Sometimes, however, these enhancements lead to non-minimal singularities in codimension one that violate the Calabi-Yau condition and the corresponding flavor group enhancement is impossible in the presence of torsional sections.
In cases where the enhancements are not realizable by tuning the geometry, one needs to employ field theoretic arguments. One possibility is to investigate the allowed flavor symmetry fluxes on non-trivial spacetime manifolds, as done e.g. in [@Ohmori:2018ona]. Since a full understanding of the different possible flavor enhancements requires an inclusion of Abelian flavor symmetries, which we do not treat here, we will stick to a discussion of the geometric realization of the group structure and leave a discussion of the field theoretic arguments to future work. Nevertheless, we often find differences of the tensor branches in theories which only differ by their flavor groups (but their algebra), which confirms that global data is preserved under the RG flow.
$T$ Flavor group
------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathbb{Z}_2$ $\text{Spin}(16)/\mathbb{Z}_2$, $\quad[\text{E}_7 \times \text{SU}(2) ]/\mathbb{Z}_2$
$\mathbb{Z}_3$ $\text{SU}(9)/\mathbb{Z}_3$, $\quad[\text{E}_6 \times \text{SU}(3)]/\mathbb{Z}_3$
$\mathbb{Z}_4$ $[\text{SU}(8)\times \text{SU}(2)] /\mathbb{Z}_4$, $\quad[\text{Spin}(10) \times \text{Spin}(6)] /\mathbb{Z}_4$
$\mathbb{Z}_5$ $[\text{SU}(5) \times \text{SU}(5)]/\mathbb{Z}_5$
$\mathbb{Z}_6$ $[\text{SU}(6) \times \text{SU}(3) \times \text{SU}(2)] /\mathbb{Z}_6$
$\mathbb{Z}_2 \times \mathbb{Z}_2 $ $[\text{Spin}(12) \times \text{Spin}(4)]/[\mathbb{Z}_2 \times \mathbb{Z}_2]$, $\quad[\text{Spin}(8) \times \text{Spin}(8)]/[\mathbb{Z}_2 \times \mathbb{Z}_2]$
$\mathbb{Z}_2 \times \mathbb{Z}_4 $ $[\text{SU}(2)^2 \times \text{SU}(4)^2]/[\mathbb{Z}_2 \times \mathbb{Z}_4]$
$\mathbb{Z}_3 \times \mathbb{Z}_3 $ $\text{SU}(3)^4 / [\mathbb{Z}_3 \times \mathbb{Z}_3]$
: \[tab:E8Broke\]Flavor groups of $\text{E}_8$ discrete holonomy instanton. The surviving flavor group is given by the centralizer of the discrete subgroup in $\text{E}_8$ [@Aspinwall:1998xj].
Before we initiate the investigation of 6d SCFTs with Mordell-Weil torsion in Sections \[sec:NHC\], \[sec:NSCFTs\], and \[sec:zoo\], we first want to mention alternative descriptions of 6d SCFTs in heterotic string theory as well as M-theory constructions.
The Heterotic Perspective
-------------------------
The heterotic duals of F-theories with discrete holonomy instantons can be understood in the following way. F-theory on a genus-one fibered 3-fold is dual to a heterotic theory on a genus-one fibered K3 surface. Let us describe the case where the base of $X$ is a $\mathbb{P}_F^1$ fibration over[^8] $\mathbb{P}_H^1$, i.e. a Hirzebruch surface $\mathbb{F}_n$. This structure induces a K3 fibration of $X$ in addition to the genus-one fibration. The heterotic limit is recovered when the volume of $\mathbb{P}_F^1$ becomes small compared to that of $\mathbb{P}_H^1$. This leads to a stable degeneration of the 3-fold $X$, where each generic K3 fiber of $X$ degenerates into a union of two rational elliptic surfaces that intersect along an elliptic curve. The heterotic K3 is then the elliptic fibration over $\mathbb{P}_H^1$ whose fiber is isomorphic to the elliptic fiber at the intersection of the ruled surfaces. Thinking of $\mathbb{P}_F^1$ as a circle fibration over an interval, the $\mathbb{P}_F^1$ turns into the Hořava-Witten interval of the M-theory picture with the two $\text{E}_8$ walls sitting at the zeros of the two homogeneous coordinates of $\mathbb{P}_F^1$.
It is now possible to take an instanton on one of the $\text{E}_8$ walls and pull it off into the bulk as an NS5-brane via a small instanton transition. In terms of the heterotic Bianchi identity for the Kalb-Ramond field $B$ with field strength $H$, $$\begin{aligned}
\text{ch}_2(V) - \text{ch}_2(TX) = n\,,\end{aligned}$$ where $\text{ch}_2(\cdot)$ denotes the second Chern character and $n$ denotes the number of NS5-branes, the process can be thought of as lowering $\text{ch}_2(V)$ by one and balancing it with an NS5-brane on the right hand side. We will restrict to unbroken supersymmetry, which means that we cannot use anti-NS5-branes, i.e. $n\geq0$
If the heterotic K3 is singular, one can also consider discrete holonomy instantons on the $\text{E}_8$ walls with fractional instanton numbers. These cases are discussed for K$3/\mathbbm{Z}_n$ and for $T^4/\mathbbm{Z}_n$ in [@Aspinwall:1998xj] and [@Ludeling:2014oba], respectively. One should note here that the genus-one fiber of the Calabi-Yau 3-fold, together with the $\mathbb{P}_H^1$, correspond to the heterotic K3 compactification geometry.[^9] Hence, the singularities lead to extra components in the discriminant of the the elliptic fibration of the ruled surfaces in the stable degeneration. This means that the discrete holonomy instantons are accompanied by extra (non-perturbative) gauge groups and matter representations. One can also coalesce point-like instantons with these fractional instantons, or coalesce several fractional instantons; if the charges of the coalesced instantons sum up to one, they can become a standard point-like instanton which can be moved away from the orbifold singularity. The occurrence of extra matter or tensor multiplets in such transitions is somewhat subtle, since extremal transitions can be blocked by a $B$ field (which corresponds to RR modes in the dual Type IIA) [@Aspinwall:1998he].
The M-Theory Perspective
------------------------
In the following, we describe the M-theory perspective of 6d SCFTs and their building blocks. The two main ingredients for their discussion are the M-theory realization of (higher-rank) E-string theories as well as the SCM theories.
The higher-rank E-string can be formulated in terms of M-theory on a half-space. The spacetime boundary is associated to an $\text{E}_8$ wall. Instantons in this $\text{E}_8$ background connection can be shrunk to a point, a small instanton, as in the heterotic description above.
![M-theory setup for a higher-rank E-string theory and its F-theory geometry.[]{data-label="fig:highrkE"}](High_rank_E.pdf){width="80.00000%"}
From this singular point in moduli space, which is associated to the origin of the Higgs branch, the small instanton can be dragged into the bulk as an M5-brane. The situation is depicted in Figure \[fig:highrkE\]. The corresponding graph of the dual F-theory geometry[^10] is also given in Figure \[fig:highrkE\]. It has a curve with self-intersection $(-1)$ connected to a flavor algebra given by $\mathfrak{e}_8$ and a chain of curves with self-intersection $(-2)$.
The second configuration we will need in the following is the $\mathfrak{g}$-type superconformal matter theory. In M-theory, it can be engineered by placing an M5-brane on an ADE-singularity of type $\mathfrak{g}$, see Figure \[fig:SCM\].
![M-theory setup for a superconformal matter.[]{data-label="fig:SCM"}](SCM.pdf){width="95.00000%"}
Depending on the specific type of the ADE-singularity, the M5-brane can split up into fractions corresponding to a deformation onto the tensor branch of the theory. In the process, one finds compact curves of negative self-intersection and the corresponding gauge algebras in the F-theory picture. The flavor symmetry is described by a non-dynamical seven-dimensional super-Yang-Mills theory, which appears naturally in M-theory on an ADE-singularity.
One can also combine the two ingredients, which leads to a setup that was called an *orbi-instanton* theory in [@Heckman:2018pqx]. It is formulated as the intersection of an ADE-singularity in the bulk that intersects the $\text{E}_8$-wall bounding spacetime. Pulling M5-branes into the bulk along the singularity, one finds the theory depicted in Figure \[fig:orbiinst\].
![Realization of the orbi-instanton theory in M-theory on the partial tensor branch.[]{data-label="fig:orbiinst"}](Orbi_instanton.pdf){width="80.00000%"}
In this way the gauge algebra $\mathfrak{g}$ is realized over each of the compact curves of negative self-intersection in the F-theory geometry and further appears as a flavor symmetry on the far right. In order to obtain a smooth geometry, one might have to further blow up the base manifold. As above, this corresponds to a fractionalization of the M5-branes on the ADE-singularity.
With these theories as building blocks, one can engineer six-dimensional SCFTs by switching on deformations that break the flavor symmetries to subgroups. These deformations can be associated to homomorphisms $\mathfrak{su}_2 \rightarrow \mathfrak{g}$, as well as group homomorphisms $\Gamma_{\text{ADE}} \rightarrow \text{E}_8$. The former objects can be interpreted as semi-simple and nilpotent deformations, and thus contains T-brane data [@Heckman:2015ola; @Mekareeya:2016yal; @Heckman:2016ssk; @Heckman:2018pqx]. The latter are associated to the possibility of having a non-trivial $\text{E}_8$ background connection on the boundary of spacetime, and are therefore directly related to discrete holonomy instantons. Of course, once a global group structure is imposed, these deformations of the theory are also subject to consistency requirements, which will be investigated elsewhere. Here, we just illustrate these restrictions in a simple example corresponding to M5-branes on an $\text{A}_5$ singularity, as e.g. in Figure \[fig:SCM\].
In the case of $\mathfrak{su}_n$-type flavor symmetries, the nilpotent deformations are classified by a partition of $n$. For the example $n = 6$, there are 11 possible partitions given by $$\begin{aligned}
\mathcal{P}(6) = \{ [6], [5,1], [4,2], [4,1^2], [3^2], [3,2,1], [3,1^3], [2^3], [2^2,1^2], [2,1^5], [1^6] \} \,.\end{aligned}$$ Denoting a partition by $[ \mu_1^{d_1}, \dots, \mu_k^{d_k}]$, the preserved flavor symmetry is given by $$\begin{aligned}
\mathfrak{g} = \mathfrak{s} \big( \bigoplus_j \mathfrak{u}_{d_j} \big) \,.\end{aligned}$$ If we want to mod out a global $\mathbb{Z}_3$ quotient, we see that only those $d_j$ which are a multiple of three are allowed. This reduces the allowed partitions to $$\begin{aligned}
\mathcal{P}_{\mathbb{Z}_3}(6) = \{ [2^3], [1^6] \} \,.\end{aligned}$$ For the case $[ 1^6 ]$, the flavor group is not broken at all. For the other case, we can work out the effects on the gauge algebras close to the deformed flavor group. The corresponding part of the quiver is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{Z3_deform.pdf}\end{array}\end{aligned}$$ We see that this configuration is indeed consistent with a global $\mathbb{Z}_3$ quotient.
Moreover, we have seen above that the discrete holonomy is only possible if the $\text{E}_8$ wall contains an orbifold singularity, which is associated to the singular geometry in the heterotic dual. This singularity naturally extends into the bulk as an ADE-singularity and introduces further gauge degrees of freedom in the M-theory description. It is therefore suggestive to assume that the induced group structure also extends into the bulk and in F-theory corresponds to the existence of a torsional section. While the global realization of the gauge group is not well understood in general, there are some interesting advances in higher supersymmetric setups [@Garcia-Etxebarria:2019cnb]. Before we go to the F-theory analysis, we want to suggest how to access some of the global data in the M-theory formulation. The full analysis of these systems, however, is left for future work.
Consider an $\text{E}_8$ wall with a $\mathbb{Z}_3$ singularity, which corresponds to the collision with an $\text{A}_2$ singularity from the bulk. On this background, one can switch on a discrete holonomy instanton that breaks the $\text{E}_8$ flavor symmetry to $[\text{E}_6 \times \text{SU}(3)] / \mathbb{Z}_3$. The corresponding discrete holonomy instanton carries a charge whose fractional part is $\tfrac{2}{3}$, see [@Aspinwall:1998xj]. Now consider the 4-chain $\Sigma$ depicted in Figure \[fig:4chaininst\], which surrounds the discrete holonomy instanton.
![4-chain on which the M-theory flux $G_4$ evaluates to the fractional instanton number, i.e. the number of M5-branes.[]{data-label="fig:4chaininst"}](4chain_instanton.pdf){width="25.00000%"}
In this way one finds that $$\begin{aligned}
\int_{\Sigma} G_4 = \oint_{S^3/\Gamma_{\text{ADE}}} C_3 \,.\end{aligned}$$ Since we choose $\Sigma$ such that no additional M5-branes in the bulk contribute, one finds fractional values of the integral of the M-theory 3-form $C_3$ over the Lens space at infinity also in the bulk. Usually, one would say that $\text{A}$-type singularities cannot host a fractional holonomy in $C_3$, which is related to the allowed instanton numbers in the 7d super-Yang-Mills theory [@deBoer:2001wca; @Tachikawa:2015wka; @Ohmori:2015pua]. However, if the gauge group on the $\text{A}_2$ singularity is $\text{SU}(3) / \mathbb{Z}_3$ rather than $\text{SU}(3)$, there are fractional instantons of instanton number $\tfrac{k}{3}$. These can be traced back to holonomies which commute in $\text{SU}(3) / \mathbb{Z}_3$ but not in $\text{SU}(3)$ and thus have a direct connection to the construction of triples for the known cases of fractional holonomies in $C_3$. Therefore, it seems that the fractional instantons of non-simply-connected gauge groups have a direct relation to the fractionalization of branes, their induced fluxes, and holonomies. They are also directly related to frozen singularities, cf. [@Tachikawa:2015wka; @Bhardwaj:2018jgp].
Non-Higgsable Clusters {#sec:NHC}
======================
In this section we begin with our investigation of 6d SCFTs with non-trivial Mordell-Weil torsion in the simple setups of non-Higgsable clusters. The properties of the geometry of NHCs enforces the presence of gauge factors on these compact curves with a minimal matter spectrum preventing further Higgs transitions. In the following, we present a brief recap of the derivation of NHCs and then turn to their realization with Mordell-Weil torsion.
Non-Higgsable Clusters Without Torsion {#subsec:NHCnotorsion}
--------------------------------------
Given an irreducible, effective curve $C$ with negative self-intersection $C \cdot C = -m < 0$ in a base $B$ of an elliptically-fibered Calabi-Yau 3-fold, any other effective curve $D$ on $B$ with $D \cdot C < 0$ is necessarily non-reduced and contains $C$ as an irreducible component, $$\begin{aligned}
D = C + D' \,,\end{aligned}$$ with $D'$ effective [@Morrison:2012np]. Knowing that the anti-canonical class of the base, $-K$, has to be effective and that for genus $g = 0$ curves one has $$\begin{aligned}
- K \cdot C = 2 + C \cdot C \,,
\label{eq:genus}\end{aligned}$$ we deduce that for $C \cdot C < -2$ the anti-canonical class needs to contain $C$. Since the discriminant locus of an elliptic fibration is a section of $\Delta = - 12 K$, it also has to contain $C$. This indicates that the fiber degenerates over $C$ and in general there are non-trivial gauge degrees of freedom localized on $C$. The same logic holds for $f = - 4 K$ and $g = - 6 K$ in the Weierstrass form of the elliptic fibration, leading to a minimal degeneration of the fiber over $C$ and an associated gauge algebra. In general, one can make the ansatz $$\begin{aligned}
- n K = k C + D \,,\end{aligned}$$ with $C \cdot D \geq 0$. Plugging this into one has $$\begin{aligned}
- n K \cdot C = 2 n - n m = - k m + D \cdot C \,,\end{aligned}$$ and can solve for the smallest possible $k$, which in turn determines $D \cdot C$. Defining $C$ to be given by $\{ z = 0 \}$ in the base and parametrized by the coordinates $v_1$ and $v_2$ with $$\begin{aligned}
v_1 \sim v_2 \,, \quad z \sim v_1^{-m} \,,\end{aligned}$$ this can be rephrased as $$\begin{aligned}
- n K \sim z^k P_r (v_1, v_2) \,.\end{aligned}$$ Here, $P_r$ is a general degree $r = 2n - nm + k m$ polynomial in the coordinates $v_1$ and $v_2$. For example for $r=2$ this is $P_2 = P_{2,1} v_1^2 + P_{2,2} v_1 v_2 + P_{2,3} v_2^2$ with $P_{2,i} \in \mathbb{C}$. The minimal gauge algebra and matter content derived in this way is summarized in Figure \[fig:NHCs\].
![Gauge algebra and matter content for single-curve NHCs.[]{data-label="fig:NHCs"}](singleNHC.pdf){width="90.00000%"}
![Gauge algebra and matter content for multiple curve NHCs.[]{data-label="fig:NHC_multiple"}](multiNHC.pdf){width="75.00000%"}
Similarly, such non-Higgsable gauge theories can appear on multiple intersecting curves $C_i$ of negative self-intersection where neighboring curves intersect exactly once, as summarized in Figure \[fig:NHC\_multiple\]. This can be deduced in a similar manner as above by finding the minimal number of times the curves with negative self-intersection are contained in the divisors associated to $f$, $g$, and $\Delta$. For the details of the derivation we refer to the original work in [@Morrison:2012np]. Our notation is such that the compact curves are given by $\{ u = 0 \}$, $\{ v = 0 \}$, and $\{ w = 0 \}$ from left to right. We further introduce the non-compact base divisors $\{ x_u = 0 \}$, $\{ x_v = 0 \}$, and $\{ x_w = 0 \}$ that intersect only the indicated compact curve exactly once. Sections of $-nK$ are then to leading order given by $$\begin{aligned}
- n K \sim u^{k_1} v^{k_2} w^{k_3} x_u^{r_1} x_v^{r_2} x_w^{r_3} \,,\end{aligned}$$ with higher order terms containing larger powers in $u$, $v$, and $w$.
We see that some of the NHCs above actually do have matter states, albeit in half-hypermultiplets. Since one cannot give a D-flat vacuum expectation value to a single half-hyper, the gauge theories can nevertheless not be Higgsed (geometrically) while preserving supersymmetry to a subgroup.
Non-Higgsable Clusters With Torsion {#subsec:NHCtorsion}
-----------------------------------
For models with non-trivial Mordell-Weil torsion, the Weierstrass coefficients have certain factorization properties given in . The corresponding coefficients $a_n, b_n, c_n$ transform as section of $- n K$ and we can apply the same logic as above. Indeed one finds that the configurations of curves with non-trivial implications for the fiber degeneration are identical to the discussion above. Note that while the original NHCs do not possess any flavor symmetries, the presence of torsional sections often enforces flavor factors. We highlight them in the following discussions.
Single Curve NHCs With Torsion {#single-curve-nhcs-with-torsion .unnumbered}
------------------------------
In the case of single curve NHCs we compute the leading vanishing order in $z$ for the coefficients in models with non-trivial Mordell-Weil torsion, cf. Table \[tab:torsion\_NHCs\]. This can be used to compute the NHC gauge algebras and matter contents for models with Mordell-Weil-torsion.
$m$ $a_1$ $a_2$ $a_3$ $a_4$ $b_1$ $b_2$ $c_2$
------ --------- ----------- ------------ ------------ ----------------- ------------------- --------------------
$3$ $z P_2$ $z P_1$ $z P_0$ $z^2 P'_2$ $z \tilde{P}_2$ $z \tilde{P}_1$ $z \tilde{P}'_1$
$4$ $z P_2$ $z P_0$ $z^2 P'_2$ $z^2 P'_0$ $z \tilde{P}_2$ $z \tilde{P}_0$ $z \tilde{P}'_0$
$5$ $z P_2$ $z^2 P_4$ $z^2 P_1$ $z^3 P_3$ $z \tilde{P}_2$ $z^2 \tilde{P}_4$ $z^2 \tilde{P}'_4$
$6$ $z P_2$ $z^2 P_4$ $z^2 P_0$ $z^3 P'_2$ $z \tilde{P}_2$ $z^2 \tilde{P}_4$ $z^2 \tilde{P}'_4$
$7$ $z P_2$ $z^2 P_4$ $z^3 P_6$ $z^3 P_1$ $z \tilde{P}_2$ $z^2 \tilde{P}_4$ $z^2 \tilde{P}'_4$
$8$ $z P_2$ $z^2 P_4$ $z^3 P_6$ $z^3 P_0$ $z \tilde{P}_2$ $z^2 \tilde{P}_4$ $z^2 \tilde{P}'_4$
$12$ $z P_2$ $z^2 P_4$ $z^3 P_6$ $z^4 P_8$ $z \tilde{P}_2$ $z^2 \tilde{P}_4$ $z^2 \tilde{P}'_4$
: Leading behavior of the sections $a_i, b_i, c_i$ in $z$ on a curve $C$ with $C\cdot C=-m$.[]{data-label="tab:torsion_NHCs"}
Let us illustrate this procedure for the curve with self-intersection $(-3)$ and $\mathbb{Z}_2$ Mordell-Weil torsion. From Table \[tab:torsion\_NHCs\] we read off the leading order behavior for $a_2$ and $a_4$ $$\begin{aligned}
a_2 = z P_1 \,, \quad a_4 = z^2 P_2 \,,\end{aligned}$$ from which we find, cf. , $$\begin{aligned}
f = z^2 \big( P_2 - \tfrac{1}{3} P_1^2 \big) \,, \quad g = \tfrac{1}{27} z^3 P_1 (2 P_1^2 - 9 P_2) \,, \quad \Delta = z^6 P_2^2 (4 P_2 - P_1^2) \,.\end{aligned}$$ This indicates a fiber degeneration of type $I_0^*$. In order to determine the monodromy type we consider the monodromy cover, [@Grassi:2011hq], $$\begin{aligned}
\begin{split}
\mu (\psi) & = \psi^3 + \Big( \frac{f}{z^2} \Big)\Big|_{z=0} \psi + \Big( \frac{g}{z^3} \Big)\Big|_{z=0} = \psi^3 + \big( P_2 - \tfrac{1}{3} P_1^2 \big) \psi + \tfrac{1}{27} P_1 \big( 2 P_1^2 - 9 P_2 \big) \\
& = \tfrac{1}{27} \big( 3 \psi - P_1 \big) \big( 9 \psi^2 + 3 P_1 \psi + 9 P_2 - 2 P_1^2 \big) \,,
\end{split}\end{aligned}$$ which fixes the fiber to be $I_0^{*, \text{ss}}$ with gauge algebra $\mathfrak{so}_7$. The discriminant of the corresponding monodromy cover is given by $$\begin{aligned}
\Delta_0 = \frac{\Delta}{z^6} \Big|_{z = 0} = P_2^2 (4 P_2 - P_1^2) \,.\end{aligned}$$ There are hypermultiplets in the spinor representation of $\mathfrak{so}_7$ located at the zeros of $P_2$ and one correspondingly finds two hypermultiplets $\mathbf{8}$. The remaining factor $4 P_2 - P_1^2$ defines a curve of the monodromy cover that can give rise to non-localized matter in the vector representation. The number of vector multiplets is given by the genus of the curve. However, since the genus in the present case is zero[^11], there is no non-local matter transforming as $\mathbf{7}$.
The given matter spectrum solves the anomaly condition for a $\mathfrak{so}_7$ theory on a $(-3)$ curve. The anomaly polynomial receives contributions from the Green-Schwarz term as well as from vector- and hypermultiplets. For $n_{\mathbf{8}}$ hypermultiplets in the spinor representation and $n_{\mathbf{7}}$ in the vector representation, one finds the anomaly polynomial $$\begin{aligned}
\mathcal{I}_8 = \mathcal{I}_8^{\text{GS}} + \mathcal{I}_8^{\text{v}} + \mathcal{I}_8^{\text{h}} = \tfrac{1}{32} \big( 1 - \tfrac{1}{2} n_{\mathbf{8}} \big) \big( \text{tr} F^2 \big)^2 - \tfrac{1}{24} \big( 1 + n_{\mathbf{7}} - \tfrac{1}{2} n_{\mathbf{8}} \big) \text{tr}F^4 \,,\end{aligned}$$ which vanishes identically for $n_{\mathbf{8}}=2$ and $n_{\mathbf{7}}=0$.
Taken by itself, this matter content would be incompatible with $\mathbb{Z}_2$ torsion, since the spinor representation is not invariant with respect to the $\mathbb{Z}_2$ center of a $\text{Spin}(7)$ gauge group. However, the geometry with $\mathbb{Z}_2$ Mordell-Weil torsion automatically contains two additional $I_2$ fibers over the two roots of $P_2$, leading to a $\mathfrak{su}_2 \oplus \mathfrak{su}_2$ flavor algebra. Therefore, we find that the $\mathfrak{so}_7$ matter is contained in two half-hypermultiplets transforming in the $(\mathbf{1}, \mathbf{8}, \mathbf{2})\oplus(\mathbf{2}, \mathbf{8}, \mathbf{1})$ representation (note that $\mathbf{2}$ is pseudo-real and $\mathbf{8}$ is real), which makes the matter content consistent with a symmetry group $$\begin{aligned}
\mathcal{G} = \frac{\text{SU}(2) \times \text{Spin}(7) \times \text{SU}(2)}{\mathbb{Z}_2} \,.\end{aligned}$$ Giving a vacuum expectation value to the matter states one breaks the gauge as well as the flavor symmetries. This in turn violates the restriction imposed by the $\mathbb{Z}_2$ Mordell-Weil torsion as well. An explicit geometric realization of this breaking in terms of a deformation is given by $$\begin{aligned}
g \rightarrow g + \epsilon z^2 \,.\end{aligned}$$ However, within the class of models respecting the $\mathbb{Z}_2$ torsion, the theory has to preserve the flavor symmetries, and the model cannot be Higgsed.
Multi-Curve NHCs With Torsion {#multi-curve-nhcs-with-torsion .unnumbered}
-----------------------------
Next, we consider base configurations with several intersecting curves of negative self-intersection. Again, the only relevant configurations are the ones already appearing for the generic NHCs, i.e. one configuration with two curves which intersect once and which have self-intersections $(-3,-2)$, as well as two chains of three curves of self-intersections $(-3, -2,-2)$ and $(-2,-3,-2)$.
In the case of two curves, which we describe by $\{u = 0\}$ and $\{v = 0\}$, one arrives at the form for the anti-canonical class, which necessarily contains the curves with negative self-intersections, given by $$\begin{aligned}
- n K \sim u^{k_1} v^{k_2} x_u^{r_1} x_v^{r_2} \,,\end{aligned}$$ from which we deduce the leading order behavior of the coefficients with Mordell-Weil torsion $$\begin{aligned}
\begin{array}{| c | c | c | c | c | c | c |}
\hline
a_1 & a_2 & a_3 & a_4 & b_1 & b_2 & c_2 \\ \hline \hline
a_{1,0} \, u v x_u x_v & a_{2,0} \, u v x_v & a_{3,0} u^2 v x_u^2 & a_{4,0} u^2 v x_u & b_{1,0} \, u v x_u x_v & b_{2,0} \, u v x_v & a_{2,0} \, u v x_v \\ \hline
\end{array}
\label{tab:twonodeorders}\end{aligned}$$ Again, $a_n,b_n,c_n$ are sections of $-nK$ and $a_{n,0},b_{n,0},c_{n,0}$ are complex constants. The same exercise can be repeated for three intersecting curves, leading to $$\begin{aligned}
\begin{array}{| c | c | c | c | c |}
\hline
\text{configuration} & a_1 & a_2 & a_3 & a_4 \\ \hline \hline
(2,3,2) & a_{1,0} \, u v w x_u x_w & a_{2,0} \, u v^2 w x_v^2 & a_{3,0} \, u v^2 w x_v & a_{4,0} \, u v^2 w \\ \hline
(3,2,2) & a_{1,0} \, u v w x_u x_w & a_{2,0} \, u v w x_u^2 x_w & a_{3,0} \, u^2 v^2 w x_u x_v & a_{4,0} \, u^2 v^2 w x_v \\ \hline
\end{array}
\label{tab:threenodeorders}\end{aligned}$$
In many situations the intersection of the compact curves with each other or with a non-compact divisor leads to non-minimal fiber singularities, i.e. $\text{ord}(f,g,\Delta) \geq (4,6,12)$. In these cases, one has to blow-up the intersection point, thus reducing the self-intersection of the central curve. In the description below we also include the gauge algebras on the blow-up divisors. This is very similar to what happens in the standard non-Higgsable clusters for $m \in \{ 9, 10, 11\}$. It can also happen that the singularity on the compact curve itself has $\text{ord}(f,g,\Delta) \geq (4,6,12)$ in which case there is no resolution and we denote the model as non-minimal.
$\boldsymbol{\mathbb{Z}_2}$ Torsion {#boldsymbolmathbbz_2-torsion .unnumbered}
-----------------------------------
The single curve configurations with $\mathbb{Z}_2$ torsion are given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{singleNHCZ2.pdf}\end{array}
\label{eq:singleNHCZ2}\end{aligned}$$ Curves with self-intersection smaller than $(-8)$ become too singular for a crepant resolution. For configurations with curves of self-intersection $(-5)$, $(-6)$, and $(-7)$ one encounters instead non-minimal singularities in codimension two which need to be blown up. The resulting configurations are given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{singleNHCZ2blow.pdf}\end{array}\end{aligned}$$ In all cases the final geometry contains a $(-8)$ curve with an $\mathfrak{e}_7$ algebra.
The configurations with multiple curves are $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{multiNHCZ2.pdf}\end{array}\end{aligned}$$ The remaining multi-curve NHC requires a blow-up an the resulting theory is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{multiNHCZ2blow.pdf}\end{array}
\label{eq:3nodeZ2}\end{aligned}$$ The matter is identical to the two-curve cluster above and we see that essentially this results in the two-curve cluster connected to the NHC on a $(-4)$ curve by a $(-1)$ curve.
In all the cases above, the group structure is given by the simply-connected group induced from the algebras modded out by the discrete torsion group, i.e. for two-curve cluster it is given by $$\begin{aligned}
\mathcal{G} = \frac{\widehat{\text{SU}}(2) \times \text{Spin}(7) \times \text{SU}(2)}{\mathbb{Z}_2} \,,\end{aligned}$$ where we distinguish the flavor symmetry by a hat.
$\boldsymbol{\mathbb{Z}_3}$ Torsion {#boldsymbolmathbbz_3-torsion .unnumbered}
-----------------------------------
With the orders for $a_1$ and $a_3$ given in Table \[tab:torsion\_NHCs\], we find for the single curve case[^12] $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{singleNHCZ3.pdf}\end{array}\end{aligned}$$ where we already performed the necessary blow-ups for $m = 4,5$. Beyond $m = 6$ one has non-minimal singularities already in codimension one.
All clusters with multiple curves require blow-ups and the final result is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{2nodeNHCZ3.pdf}\end{array}\end{aligned}$$ for two curves and $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{3nodeNHCZ3.pdf}\end{array}\end{aligned}$$ for three. All of these are variations of the $\mathfrak{e}_6$ superconformal matter as we will see later.
In all clusters, the group structure is uniquely defined. It is given by the product of all gauge and flavor group factors modded out by $\mathbb{Z}_3$.
$\boldsymbol{\mathbb{Z}_4}$ Torsion {#boldsymbolmathbbz_4-torsion .unnumbered}
-----------------------------------
For $\mathbb{Z}_4$ torsion the single curve clusters are given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{singleNHCZ4.pdf}\end{array}\end{aligned}$$ originating from $m = 3,4$. There are half-hypers in representation $(\mathbf{2}, \mathbf{10})$ between the $\mathfrak{so}_{10}$ and $\mathfrak{su}_2$’s. The group structure is given by the $\mathbb{Z}_4$ quotient of the simply-connected realization.
As for the example in section \[subsec:NHCtorsion\], these two theories can be Higgsed to the classical NHC on $(-4)$ curve with $\mathfrak{so}_8$ algebra, breaking the $\mathbb{Z}_4$ torsion structure alongside the flavor symmetries. Note that $\mathbb{Z}_4$ torsion does not allow for NHCs on multiple curves or configurations with $m>4$; these turn out to have no crepant resolution due to $(4,6,12)$ singularities in codimension one.
$\boldsymbol{\mathbb{Z}_2 \times \mathbb{Z}_2}$ Torsion {#boldsymbolmathbbz_2-times-mathbbz_2-torsion .unnumbered}
-------------------------------------------------------
As for $\mathbb{Z}_4$ torsion, there are only single curve clusters with $m = 3,4$, given by $$\begin{aligned}
\label{eq:nhc3z2z2}
\begin{array}{c}
\includegraphics[scale=0.6]{singleNHCZ2Z2.pdf}\end{array}\end{aligned}$$ The $(-4)$ curve does not have any matter. The matter for the $(-3)$ curve is given by $$\begin{aligned}
\tfrac{1}{2} (\mathbf{2}, \mathbf{1},\mathbf{1}, \mathbf{8}_{\text{v}}) \oplus \tfrac{1}{2} (\mathbf{1}, \mathbf{2}, \mathbf{1}, \mathbf{8}_{\text{s}}) \oplus \tfrac{1}{2} (\mathbf{1}, \mathbf{1}, \mathbf{2}, \mathbf{8}_{\text{co}}) \,,\end{aligned}$$ where the subscripts (v, s, co) indicate the vector, spinor, and co-spinor representation of $\mathfrak{so}_8$ Interestingly, if one tries to tune two of the intersections with the $\mathfrak{su}_2$ flavor factors on top of each other, the third $\mathfrak{su}_2$ factor automatically intersects at the same point as well. Simultaneously, this intersection point enhances to $(f, g, \Delta) = (4, 6, 12)$ and one has to perform a blow-up, thus reducing the self-intersection of the curve to $(-4)$ and getting rid of all the matter.
In the case with multiple curves only the two curve case leads to a valid configuration $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{multiNHCZ2Z2.pdf}\end{array}\end{aligned}$$ The configurations with three curves do not have a crepant resolution.
Remaining Cases {#remaining-cases .unnumbered}
---------------
The remaining possibilities for the Mordell-Weil torsion, i.e. $\mathbb{Z}_5$, $\mathbb{Z}_6$, $\mathbb{Z}_4 \times \mathbb{Z}_2$ and $\mathbb{Z}_3 \times \mathbb{Z}_3$, immediately lead to $(4,6,12)$ singularities in codimension one.
Non-Simply-Connected SCM {#sec:NSCFTs}
========================
The next main ingredient in our discussion is going to be non-simply connected superconformal matter. On an intuitive level it is clear why superconformal matter must be revisited: Since a non-simply-connected (gauge-)group comes with a modified charge lattice that restricts certain matter representations, one expects that a similar logic should hold for their non-perturbative extension, the superconformal matter. Indeed, in Section \[sec:SCFT\] we have already seen that the presence of torsion can lead to a breaking of the $\text{E}_8$ flavor symmetry of the E-string theory to a subgroup consistent with the global quotient. Before we discuss the various superconformal matter theories for each quotient factor in detail, we want to comment on the following three generic features that appear in these models:
- **Classical SCM**, as discussed e.g. in [@DelZotto:2014hpa], often already comes with compatible centers, and therefore a non-simply-connected flavor group, as anticipated in [@Ohmori:2018ona]. In Section \[ssec:Classic\] we show that these models indeed admit torsional sections that are compatible with the respective centers.
- **Discrete jumps in the ramp of gauge groups** appear on the tensor branch. This feature results from the restricted monodromy, which forbids various gauge group and matter factors and hence heavily modifies the tensor branch, as we show in simple examples in Section \[ssec:Flavor\].
- **Additional singular $\boldsymbol{{I}_1}$ loci** intersect one or multiple flavor branes in a single point. As we discuss in Section \[ssec:DiscComp\], these curves are often singular themselves, which severely affects the tensor branch of these theories.
Torsion and SCM {#ssec:Classic}
---------------
Some of the superconformal matter theories discussed in the literature (see e.g. [@DelZotto:2014hpa]) in fact admit a non-simply-connected flavor group. This has already been anticipated by the authors of [@Ohmori:2018ona], who noted that in fact superconformal matter of type $(G,G)$ is changed to be of type $(G,G)/\text{center}(G)$. In this section, we present the geometric realization of this statement, which is due to the presence of an $n$-torsional section in the respective Weierstrass model, with $\mathbb{Z}_n$ the center of the simply-connected cover $G^*$. We will mainly consider the $\mathfrak{e} \times \mathfrak{e}$ and $\mathfrak{so} \times \mathfrak{so}$ type superconformal matter.
We start with $\mathbb{Z}_2$. The $\mathfrak{e}_7 \times \mathfrak{e}_7$ SCM can be directly engineered in the $\mathbb{Z}_2$ torsion model of by setting $a_2 = u^2 v^2\,,~a_4 = u^3 v^3$. The tensor branch is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E7_SCM.pdf} \end{array}\end{aligned}$$ where each individual gauge factor has a $\mathbb{Z}_2$ center, as enforced geometrically by the Weierstrass model. Note also that the $(2,3,2)$ NHC which appears here is unmodified when $\mathbb{Z}_2$ torsion is present, cf. .
Similarly, the $\mathfrak{e}_7 \times \mathfrak{su}_{2,III}$ collision[^13] is obtained from $a_2 = u^2 v\,,~a_4 = u^3 v$ and does in fact correspond to one of the $\mathbb{Z}_2$ discrete holonomy instanton theories as discussed in Section \[sec:SCFT\]. Moreover, superconformal matter of type $\mathfrak{so}_{8+4n} \times \mathfrak{so}_{8+4m}$ can be engineered by factorizing $a_2 = u v\,,~a_4 = u^{2+n} v^{2+m}$. The tensor branches of those theories are $$\begin{aligned}
\begin{array}{c}
\label{eq:sonsomZ2SCM}
\includegraphics[scale=0.7]{SO_SCM.pdf} \end{array}\end{aligned}$$ with matter transforming in the bi-fundamental representation of two adjacent algebras. In cases where either $n$ or $m$ are zero (but not both), the $\mathfrak{so}_8$ flavor factor is reduced to $\mathfrak{so}_7$.
We note that the torsion never allows collisions of type $\mathfrak{so}_{8+4n}\times \mathfrak{so}_{10+4m}$. This can be understood from the fact that anomaly cancelation on the tensor branch requires the presence of a single fundamental hypermultiplet of the $\mathfrak{sp}$ gauge group. However, this is incompatible with the $\mathbb{Z}_2$ torsion. Geometrically, this is ensured in the $\mathbb{Z}_2$ torsion model by an automatic enhancement of the flavor symmetry. A similar effect appears for $[\text{Spin}(10+4n) \times \text{Spin}(10+4m)]/\mathbb{Z}_4$ models.[^14]
We show in Section \[sec:Z4TorsionSCM\] that, even when two extended flavor factors have compatible centers, it is not guaranteed that a modding by that group exists: The modding can be obstructed by the presence of incompatible gauge group factors or representations on the tensor branch. An example for this is superconformal matter of type $\mathfrak{e}_7 \times \mathfrak{so}$ which includes a single $\mathfrak{sp}$ fundamental, that is not allowed by the $\mathbb{Z}_2$ embedding. In Section \[ssec:Z2SCM\], we will see that $(\text{E}_7 \times \text{SO}(2n))/\mathbb{Z}_2 $ superconformal matter is possible, but it comes with a modified tensor branch that is consistent with the global $\mathbb{Z}_2$.
The $\mathbb{Z}_3$ case can be handled analogously. For example, $\mathfrak{e}_6 \times \mathfrak{e}_6$ type superconformal matter can be engineered from with $a_1=0$ (which sets $f\equiv0$), and $a_3=u^2 v^2$. The tensor branch is given by $$\begin{aligned}
\label{eq:e6e6Z3}
\begin{array}{c}
\includegraphics[scale=0.7]{E6_SCM.pdf} \end{array}
\end{aligned}$$ without any matter, which is trivially consistent with the the global $\mathbb{Z}_3$. Also here we note that the non-Higgsable cluster of a $(-3)$ curve is unmodified by the $\mathbb{Z}_3$ torsion.
To summarize, we find that indeed some superconformal matter theories that have already been constructed in the literature admit a non-simply-connected flavor group, which is geometrically manifest by Mordell-Weil torsion in the Weierstrass model. In general, this is an important lesson that needs to be taken into account, when constructing superconformal matter within a certain torsion model. When starting with a given torsion, specific flavor configurations might further enhance the Mordell-Weil group beyond the starting configuration. Such an enhancement e.g. occurs when one chooses $a_2=u v$ and $a_4=u^2 v^2$ in an $\mathbb{Z}_2$ torsion model, which engineers an $\mathfrak{so}_8 \times \mathfrak{so}_8$ flavor algebra. After a coordinate shift, one easily sees that this model in fact admits a $\mathbb{Z}_2 \times \mathbb{Z}_2$ torsion, which is what one expects from the center of $\mathfrak{so}_8$. In Appendix \[sec:EnhancedWSFs\], we show the conditions under which such a torsion enhancement is possible. Note that this can also be read in reverse, showing which deformations preserve possible torsion subgroups.
Example: Flavor Groups vs Algebras {#ssec:Flavor}
----------------------------------
After having illustrated that some superconformal matter theories in fact already admit a torsion factor, we want to compare two superconformal matter theories that admit the same flavor algebra, but different flavor groups and hence different tensor branches. We pick the $(\text{E}_7 \times \text{SU}(2n))/\mathbb{Z}_2$ theory which we compare with its simply-connected cover. This type of theory is engineered in the $\mathbb{Z}_2$ torsion model of by setting $a_2= u^2$ and $a_4 = u^3 v^n$, resulting in the Weierstrass functions $$\begin{aligned}
f = \tfrac13 (-u + 3 v^n) u^3 \,, \quad g=\tfrac{1}{27} ( 2 u-9 v^n)u^5 \,, \quad
\Delta = u^9 v^{ 2 n } (-u + 4 v^n) \, .
\label{eq:E7dicinst}\end{aligned}$$ For $n>0$, the collision leads to a $(4,6,10+2n)$ non-minimal singularity. Note that there is also an $I_1$ component intersecting the origin as well. Performing the first resolution by blowing up $u=v=0$ with an exceptional divisor $e_1$ and taking the proper transform results in the Weierstrass functions $$\begin{aligned}
f = \tfrac13 (-\tilde{u} + 3 \tilde{v}^n e_1^{(n-1)}) \tilde{u}^3 \,, \quad \tfrac{1}{27} ( 2 \tilde{u}-9 \tilde{v}^n e_1^{(n-1)})\tilde{u}^5 \, , \quad
\Delta = \tilde{u}^9 \tilde{v}^{ 2 n } e_1^{ 2(n-1)} (-\tilde{u} + 4 \tilde{v}^n) \, .\end{aligned}$$ Over $e_1$, we thus find an $\mathfrak{su}_{2n-2}$ gauge algebra factor, as needed from compatibility with the prescribed center. Continuing this process $n-1$ times results in the chain $$\begin{aligned}
\label{eq:e7su2n_Z2}
\begin{array}{c}
\includegraphics[scale=0.7]{E7_Z2.pdf} \end{array}\end{aligned}$$ All gauge algebra and the bi-fundamental $(\boldsymbol{2k},\overline{\boldsymbol{2k+2}})$ matter factors are compatible with the $\mathbb{Z}_2$ center that is modded out in addition to gauge anomaly cancellation. Note that the chain above ends on one of the $\mathbb{Z}_2$ discrete holonomy instanton theory.
Let us compare this to the $\text{E}_7 \times \text{SU}(2n)$ theory constructed in a similar fashion, but in the most general Weierstrass model without any torsion. The resulting tensor branch is given by [@DelZotto:2014hpa] $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E7.pdf} \end{array}\end{aligned}$$ The direct comparison shows how the non-simply-connected version of the $\mathfrak{e}_7 \times \mathfrak{su}_{2n}$ theory differs from its cover. We find that every algebra factor and representation that is incompatible with the $\mathbb{Z}_2$ modding, is indeed missing in . This includes the $\mathfrak{su}_{2k+1}$ factors as well as the $\mathbf{56}$ half-hyper. This results in a ramp of $\mathfrak{su}_{2k}$ gauge factors that runs towards $\mathfrak{e}_7$ twice as fast as without the quotient, which requires less blow-ups and consequently tensor multiplets. We present an overview of the two theories with and without torsion in Table \[tab:e7\_su2n\_comparison\]. As already pointed out in [@Ohmori:2018ona], a characteristic feature of these theories is a modified matter spectrum and a jump in the rank of the gauge groups in the ramp between the flavor factors.
$\text{E}_7 \times \text{SU}(2n)$ $[\text{E}_7 \times \text{SU}(2n)]/\mathbb{Z}_2$ Difference
------------- ------------------------------------ -------------------------------------------------- ---------------------------------
Tensors $2n-1$ $n$ $n-1$
rank($G$) $(n-1)(2n-1)$ $(n-1)^2$ $n(n-1)$
\# Vectors $\frac13 (n-1) ( 2 n-1) (4 n + 3)$ $\frac13 (n-1) ( 4 n^2-2n-3)$ $\frac43 n (n^2-1)$
\# Hypers $\frac23 n ( 4 n^2-1)+ \frac12 56$ $\frac43 n ( n^2-1)$ $\frac23 n(1+2 n^2)+\frac12 56$
5d dim($C$) $n (2 n-1)$ $n+ ( n-1)^2$ $n^2 - 1$
: Comparison of the $\mathfrak{e}_7 \times \mathfrak{su}_{2n}$ SCM theories with and without torsion. $G$ refers to the gauge algebras. In the last column we give the dimension of the Coulomb branch dim$(C)$ of the 5D circle reduced theory.[]{data-label="tab:e7_su2n_comparison"}
In a very similar fashion, theories of type $(\text{E}_6 \times \text{SU}(3n))/\mathbb{Z}_3$ differ from their simply-connected version by omitting all $\mathfrak{su}_{3k \pm 1}$ algebra factors (and the corresponding tensors and hypermultiplets, see Section \[sssec:Z3SCM\]).
Singular Discriminant Components and Resolutions {#ssec:DiscComp}
------------------------------------------------
A distinctive feature of Weierstrass models with torsion points is that the Weierstrass functions $f$ and $g$ are highly tuned as compared to the standard Weierstrass form. Recall for example the $\mathbb{Z}_2$ torsion model and its Weierstrass functions $$\begin{aligned}
f = a_4 - \tfrac{1}{3} a_2^2 \,, \quad g = \tfrac{1}{27} a_2 (2 a_2^2 - 9 a_4) \,, \quad \Delta = a_4^2 (4 a_4 - a_2^2) \,, \end{aligned}$$ Note that in addition to the $\mathfrak{su}_2$ locus, there is an $I_1$ component that includes the $a_4$ polynomial of the respective $\mathfrak{su}_2$. When tuning this model, we therefore often obtain an $I_1$ divisor with a double-, triple- or higher-point singularity. For example for the model above with $a_2 = u^n\,,~a_4 = v^m$, the $I_1$ component becomes singular, too. Moreover, care needs to be taken when $m$ is even, since then the $I_1$ divisor becomes reducible and can be split into two (possibly singular) divisors. Perturbative examples for this phenomenon are given e.g. by choosing $n=1$ and $m=3$, which results in the discriminant $$\begin{aligned}
\Delta = v^6 (-u^2 + 4 v^3) \,.\end{aligned}$$ This corresponds to an $\mathfrak{sp}_3$ gauge algebra over $v=0$ and an $I_1$ locus with a double-point singularity at $u=v=0$. At this locus, the Weierstrass model enhances to an $(2,3,8)$ singularity with a local $\mathfrak{so}_{12}$ algebra contributing a hypermultiplet in the two-fold antisymmetric representation $\mathbf{14}$ of $\mathfrak{sp}_3$, compatible with the center factor.
Similarly, for $n=2\,,~m=3$, the discriminant is given by $$\begin{aligned}
\Delta= v^6 (-u^4 + 4 v^3) \,.\end{aligned}$$ Now, the $\mathfrak{sp}_3$ fiber becomes split and supports an $\mathfrak{su}_6$ algebra, and the $I_1$ locus has a triple-point singularity at the origin. At the intersection with the $\mathfrak{su}_6$, this results in an enhancement to a $(3,5,9)$ singularity, i.e. that of an $\mathfrak{e}_7$ algebra, which gives again rise to the two-fold antisymmetric $\mathbf{15}$ of $\mathfrak{su}_6$.[^15]
In a similar vein, tuning the $I_1$ locus can lead to extra superconformal matter, e.g. when choosing $n=2$ and $m=4$, in which case the discriminant becomes $$\begin{aligned}
\Delta =v^8 (-u^2 + 2 v^2) (u^2 + 2 v^2)=v^8(u - \sqrt{2} v) (u + \sqrt{2} v)(u - i\sqrt{2} v) (u + i\sqrt{2} v) \, .\end{aligned}$$ This leads to an $\mathfrak{su}_8$ over $v=0$ and four $I_1$ loci all intersecting at the origin. In fact, all five curves meet at the origin, producing a $(4,6,12)$ singularity. This configuration is an E-string theory whose flavor group got broken to $\text{SU}(8)/\mathbb{Z}_2$. Upon blowing up, the singularity can be removed leading to a $(-1)$ curve intersected by the four $I_1$ curves at different points. The corresponding diagram is given by $$\begin{aligned}
\label{eq:su8_I14_1}
\begin{array}{c}
\includegraphics[scale=0.5]{Special_I.pdf} \end{array}\end{aligned}$$ Degenerating the $I_1$ locus further by setting $n=4$, the discriminant becomes $$\begin{aligned}
\begin{split}
\Delta &= v^8 (-u^4 + 2 v^2) (u^4 + 2 v^2)\\ &= -v^8 (u^2 - \sqrt{2} v) (u^2 + \sqrt{2} v)(u^2 - i\sqrt{2} v) (u^2 + i\sqrt{2} v)\, .
\end{split}\end{aligned}$$ Now, there is still the $\mathfrak{su}_8$ factor over $v=0$ but the $I_1$ curves have slightly changed. After two blow-ups in the base, this results in the chain $$\begin{aligned}
\label{eq:su8_I14_2}
\begin{array}{c}
\includegraphics[scale=0.5]{Special_II.pdf} \end{array} \end{aligned}$$ which is simply the same $\text{SU}(8)/\mathbb{Z}_2$ theory as before, but with additional unpaired tensors. Note that can be reached from along an RG flow that decompactifies the $(-2)$ curve.
Similarly, when taking the factorization $a_2 = u v^{l+1}\, ,~ a_4 = u ^4 v $, $$\begin{aligned}
\begin{split}
f=&\frac13 u^2 v (3 u^2 - v^{(1 + 2 r)}) \, , \qquad g= \frac{1}{27}u^3 v^{(
2 + r)} (-9 u^2 + 2 v^{(1 + 2 r)}) \\ \Delta=& u^{10} v^3 (4 u^2 - v^{(1 + 2 r)}) \, .
\end{split}\end{aligned}$$ one obtains a $\mathfrak{su}_{2,III} \times \mathfrak{so}_{16}$ flavor group, where $l$ only affects the $I_1$ locus and its intersection at the origin. For $l=0$, a single resolution with a $(-1)$ curve suffices, resulting in $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{SU_SO16.pdf} \end{array} \end{aligned}$$ However, for $l>0$ the $I_1$ locus develops a double-point singularity at the origin and one needs $l$ additional blow-ups that host $\mathfrak{su}_{2,III}$ factors to get a fully non-singular model, $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{SU_SO16_mod.pdf} \end{array} \end{aligned}$$ Hence, we find a chain that repeats $l$ times an $\mathfrak{su}_2$ gauge factor from the right, similar to the orbi-instanton discussed in Section \[sec:SCFT\]. From another perspective, the above model can be interpreted as an $\mathfrak{su}_2$ flavor brane that collides with a discrete holonomy instanton at a $\mathbb{Z}_{l-1}$ orbifold singularity. In the Weierstrass model, this is all encoded in the structure of the $I_1$ locus. Features like these require taking extra care when analyzing the theory, since the $I_1$ loci can influence the structure of the tensor branch substantially.
In the process of studying non-minimal singular Weierstrass models, we use that traverse intersections of two divisors that lead to singularities of vanishing order less than $(8,12,24)$ can be reduced by a sequence of blow-ups in the base. Since some of the the models we analyze can have non-transverse intersections, we need to be careful. In particular, it can happen that a singularity of type $(8,12,24)$ (or worse) only becomes apparent after performing some blow-ups. We demonstrate this with the $\mathbb{Z}_2$ torsion model where we tune the coefficients $a_2 = u^3\,,~a_4 = u^3 v^6$. This leads to the Weierstrass model $$\begin{aligned}
\begin{split}
f=\tfrac13 u^3 (u^3 - 3 v^6)\, , \quad g=\tfrac{1}{27} u^6 (2 u^3 - 9 v^6)\,,\quad \Delta= -u^9 v^{12} (u^3 - 4 v^6)\,.
\end{split}\end{aligned}$$ At first glance this model seems to admit a crepant resolution: It has a simple $\mathfrak{e}_7 \times \mathfrak{sp}_6 $ collision, but also an $I_1$ curve with a triple-point singularity at the origin. The overall vanishing orders at the origin sum up to $(6,9,24)$ which is just below the bound of a non-crepant resolution. Upon performing the first blow-up, one obtains a model with $$\begin{aligned}
\begin{split}
f=\tfrac13 e_1^2 u^3 (-u^3 + 3 e_1^3 v^6) \, , \quad g= \tfrac{1}{27} e_1^3 u^6 (2 u^3 - 9 e_1^3 v^6)\,,\quad \Delta= e_1^{12} u^9 v^{12} (-u^3 + 4 e_1^3 v^6)\,.
\end{split}\end{aligned}$$ This has an $I_{6}^{*,\text{s}}$ fiber along the blowup divisor $\{e_1=0\}$, which corresponds to an $\mathfrak{so}_{20}$ gauge algebra. However, now we see a $(8,12,24)$ singularity at the intersection of the $\mathfrak{so}_{20}$ divisor $\{ e_1 = 0 \}$ and the $\mathfrak{e}_7$ divisor $\{ u = 0 \}$, revealing that no crepant resolution is possible. Cases like these are relatively generic and complicate a systematic study of the non-simply-connected superconformal matter theories.
Non-Simply-Connected Conformal Matter Zoo {#sec:zoo}
=========================================
In this section we discuss superconformal matter theories with various torsion factors, as well as their tensor branches, in more detail. The theories are constructed by systematically engineering singularities in the restricted Weierstrass models compatible with the Mordell-Weil torsion factors. In order to obtain superconformal matter theories, the singularities are engineered such that the vanishing orders are below $(4,6,12)$ in codimension one, and between $(4,6,12)$ and $(8,12,24)$ in codimension two. These conditions mean that a crepant resolution by blowing up the base is possible. This allows us to determine all gauge group factors and matter representations of each theory. In order to present our results in a systematic way, we discuss each torsion theory using the following approach:
1. For each torsion factor, consider all E$_8$ breaking patterns of a discrete holonomy instanton, which can be inferred from Table \[tab:E8Broke\]. Subsequently, consider all further breakings that can be engineered in a torsion-preserving way (see e.g. [@Aspinwall:1998xj]).
2. From this, we obtain non-compact flavor branes at the coordinates of the $\mathbb{C}^2$ base ($u=0$ and $v=0$), which host the E-string theory at the origin.
3. For each flavor group over a codimension one locus, we engineer higher and higher singularities, until we hit the limit of crepantly resolvable theories.
4. We perform all resolutions in the base, which allows us to deduce the 6d tensor branch. We then determine the gauge groups and matter content, check anomaly cancellation, as well as compatibility with torsion.
Note that some theories cannot be reached via this procedure of starting from a discrete holonomy instanton and engineering further singularities. We call these “outlier theories”. We present a few interesting outliers in the following sections, but the bulk of them is discussed in Appendix \[app:outlier\].
Let us briefly recap our notational conventions:
- Square nodes denote flavor algebras and circle nodes denote gauge algebras.
- We specify these algebras inside the squares and above the circles. We follow the convention $\mathfrak{su}_2\simeq\mathfrak{sp}_1$, $\mathfrak{su}_1=\mathfrak{su}_0=\emptyset$. Likewise, nodes without a gauge algebra above them signal trivial algebras.
- The number in the circle denotes the negative self-intersection of the corresponding divisor.
- Unless specified otherwise, all matter transforms in the bi-fundamental representation of two adjacent nodes. For groups other than $\mathfrak{su}_{n}$, the “fundamental” is the lowest-dimensional non-trivial irreducible representation.[^16]
Single Factors Theories: Zn, n=2...6.
-------------------------------------
We start by considering theories with Mordell-Weil torsion given by a single $\mathbb{Z}_n$ factor and employ the strategy outlined above. It turns out that $\mathbb{Z}_2$ torsion is the most versatile in terms of different models, since a wide range of compatible gauge group factors exist. Typically, one obtains fewer possibilities when the order of the torsion is enhanced. However for non-prime torsion, there is the possibility that the gauge and flavor groups are only affected by a subgroup of the full torsion, which in turn increases the number of possibilities again.
### $\boldsymbol{\mathbb{Z}_2}$ Torsion: Conformal Matter {#ssec:Z2SCM}
Superconformal matter with $\mathbb{Z}_2$ Mordell-Weil torsion is described by the Weierstrass model in . Note that the model generically has an $\mathfrak{su}_2$ singularity at the zeros of $a_4$. The possible breakings of E-string theories by a $\mathbb{Z}_2$ discrete holonomy instanton are[^17] $$\begin{aligned}
\label{eq:z2Instantons}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{}&\text{ Flavor Group} & a_2 & a_4 \\ \hline
1& [\text{E}_7 \times \text{SU}(2)] /\mathbb{Z}_2 & u^2 & u^3 v \\[0pt]
2& [\text{Spin}(8) \times \text{Spin}(8)] / [\mathbb{Z}_2 \times \mathbb{Z}_2] & u v & u^2 v^2 \\[0pt]
3& [\text{Spin}(16)] /\mathbb{Z}_2 & u v & u^4 \\ \hline
4& \text{SU}(8)/\mathbb{Z}_2 & u^2 & v^4 \\[0pt]
5& [\text{SU}(6) \times \text{SU}(2)_{III}] / \mathbb{Z}_2 & u^2 & u v^3 \\[0pt]
6& [\text{Spin}(8) \times \text{SU}(4)]/\mathbb{Z}_2 & u^2 & u^2 v^2 \\[0pt]
7& [\text{Spin}(12) \times \text{SU}(2)_{III}] /\mathbb{Z}_2 & u v & u v^3 \\ \hline \hline
8& \text{Sp}(4)/\mathbb{Z}_2 & u^3 & v^4 \\[0pt]
9& [\text{Sp}(3) \times \text{SU}(2)_{III}]/\mathbb{Z}_2 & u^3 & u v^3 \\[0pt]
10& [\text{Spin}(8) \times \text{Sp}(2)]/\mathbb{Z}_2 & u^3 & u^2 v^2 \\ \hline
\end{array}\end{aligned}$$ For the first three theories, the E$_8$ flavor group is broken to one of its maximal subgroups, while theories 4-7 have rank lower than 8. The last three theories have an additional monodromy that folds the flavor group to a non-simply laced group.
In the following, we will discuss the first seven of these theories and defer the last three to the Appendix. Starting with theory 1, we can enhance the $\mathfrak{su}_2$ flavor symmetry over $v=0$ to an $\mathfrak{su}_{2n}$ factor simply by replacing $u^3v\rightarrow u^3v^n$ in $a_4$. This example has already been discussed in Section \[ssec:Flavor\], and its tensor branch is summarized in . The same is true for theory 2, for which we can further enhance the ranks of the $\mathfrak{so}$ algebras by setting $a_4=u^{2+n} v^{2+m}$, leading to well known theories [@DelZotto:2014hpa] whose tensor branches have already been discussed in .
We continue our discussion with theory 3, which is an $\mathfrak{so}_{16}$ theory intersected by an $I_1$ component. Enhancing its flavor algebra leads to theories with tensor branch $$\begin{aligned}
\label{eq:Z2Theory4}
\begin{array}{c}
\includegraphics[scale=0.7]{SO_I0_Z2.pdf} \end{array} \end{aligned}$$ with-half hypermultiplets in the bi-fundamental representation. This cancels the $\mathfrak{sp}_n$ gauge anomalies.
Moving on to theory 4, we have the E-string theory from an $\mathfrak{su}_8 \times I_1^4$ collision. We can enhance the $\mathfrak{su}_8$ factor to $\mathfrak{su}_{8+2n}$ by changing $a_4$ in the Weierstrass form to $a_4 = u^{4+n}$. Note that for these choices the $I_1$ locus is irreducible when $n$ is odd, factors into two components if $n$ is divisible by 2, and into 4 components if $n$ is divisible by 4. The form of tensor branch depends on $2n$ mod $8$, which specifies the rank of the gauge algebra on the final $(-1)$ curve, $$\begin{aligned}
\label{eq:SU8nZ2}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_Z2_mod.pdf} \end{array} \end{aligned}$$ Note that the $(-1)$ curve does not carry any gauge algebra if $n$ is a multiple of 4. All matter multiplets are bi-fundamental. However, in the case that $n=3$ mod $4$, the chain ends on an $\mathfrak{su}_6$ over the $(-1)$ curve and the $I_1$ curves do not split. In such a case, one finds another antisymmetric $\mathbf{15}$-plet representation at the collision with the $I_1$ curve. The $\mathbf{15}$-plet can be seen to arise from an enhancement to an $(3,5,9)$ singularity in codimension two, encoding an $\mathfrak{e}_7$ fiber (note that an $\mathfrak{e}_7$ enhancement is consistent with the restricted $\mathbb{Z}_2$ monodromy). The matter representation can then be inferred from decomposing the adjoint $\mathbf{133}$ [@Katz:1996xe] into irreducible $\mathfrak{su}_6$ representations, which gives rise to a $\mathbf{15}$-plet. In the case $n=2$ mod $4$, on the other hand, the chain ends on an $\mathfrak{su}_4$ over the $(-1)$ curve which intersects two $I_1$ components. The $\mathfrak{su}_4 $ comes with fundamentals, but also with two $\mathbf{6}$-plet half-hypermultiplets (this is the two-fold antisymmetric representation), originating from the intersections with the two $I_1$ components. This is again confirmed by noting that the $(-1)$ curve intersects the $I_1$ curve in an enhanced $(2,3,6)$ singularity, which yields a codimension two enhancement to $\mathfrak{so}_{8}$. This local enhancement is compatible with the desired center in its simply-connected realization as one expects from the $\mathbb{Z}_2$ torsion. By using the Katz-Vafa rule and branching the adjoint $\mathbf{28}$ of $\mathfrak{so}_8$ into $\mathfrak{su}_4$ representations, one finds the required $\mathbf{6}$-plets as necessary for anomaly cancellation.
Next we consider theory 5, which is an $\mathfrak{su}_6 \times \mathfrak{su}_{2,III}$ broken E-string theory, and enhance the $\mathfrak{su}_6$ factor to $\mathfrak{su}_{2n+6}$ by setting $a_4=u v^{3+n}$. This results in the chain $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_Z2.pdf} \end{array} \end{aligned}$$ Note that the rank of the $\mathfrak{su}$ factors above jumps by $6$ in the quiver. Again, if the chain ends on an $\mathfrak{su}_4$ on the last $(-1)$ curve, there are additional states to the usual bi-fundamentals. Indeed, we find a half-hypermultiplet in the $(\mathbf{6,2})$ representation, but also a full hypermultiplet in the $(\mathbf{4,2})$ representation at the intersection with the $\mathfrak{su}_2$ flavor brane. This can be explicitly verified from its intersection with the $\mathfrak{su}_2$, which enhances to an $\mathfrak{e}_7$ singularity in codimension two, from which we get the two-fold antisymmetric representation required for anomaly cancellation.
We can also enhance the $\mathfrak{su}_2$ to an $\mathfrak{so}_8$, and additionally keep enhancing the $\mathfrak{su}_{2n+6}$ factors further, which gives the chain $$\begin{aligned}
\label{eq:susoz2}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SO_Z2_mod.pdf} \end{array}\end{aligned}$$ Depending on $n$, the above theory can either end on the broken E-string theory 6 (for $n$ even), or on a theory with a gauged $\mathfrak{su}_2$ on the $(-1)$ curve (for $n$ odd). In the latter case, there are $(\mathbf{2 ,6})$ bi-fundamentals hypermultiplets and $(\mathbf{8,2})$ half-hypermultiplets under the flavor $\mathfrak{so}_8$, as required by anomalies. As noted, the above chain also includes the enhanced version of the $[\text{SO}(8) \times \text{SU}(4)]/\mathbb{Z}_2$ of theory 6.
We continue with theory 7, where we enhance the $\mathfrak{so}_{12}$ factor by setting $a_4=u v^{3 + n}$. This results in the tensor branch $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU2_SO_Z2.pdf} \end{array} \end{aligned}$$ Enhancing the other flavor group, i.e. the $\mathfrak{su}_2$ (which is actually a type $III$ singularity) to an $\mathfrak{so}_{8+4n}$, basically results in a theory we have already discussed before in . Remember, however, that it is often impossible to enhance multiple flavor factors to arbitrarily high vanishing order, as this typically leads to non-minimal singularities. A couple of theories where multiple flavor factors can be enhanced simultaneously (but not necessarily both indefinitely), are presented in Appendix \[app:outlier\].
Another class of models can be obtained by taking theory 1 and enhancing the $\mathfrak{su}_2$ factor to an $\mathfrak{so}_{8+4n}$ group. Concretely this is achieved by setting $a_2 = u^2 v \, ,~a_4 = u^3 v^{4+n}$. This branch is relatively close to the standard conformal matter, but there is also an $I_1$ component intersecting the origin as well, changing the tensor branch in a subtle way, $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E7_SO_Z2.pdf} \end{array} \end{aligned}$$ Here, the $\mathfrak{su}_{2}$ over the $(-2)$ curve is of type $III$. Note that for $n=0$ the chain terminates at the $\mathfrak{so}_{12}$ factor and for higher $n$ multiple chains of $(-1)\,(-4)$ curves are appended, as shown above. It is important to point out that the above type of theories differ from the standard superconformal matter (see e.g. [@DelZotto:2014hpa]), which has $\mathfrak{so}_{2n+1}$ gauge algebras over the $(-4)$ curves. The $\mathbb{Z}_2$ factor furthermore changes the spectrum with respect to the non-torsion case, such that with torsion only bi-fundamental matter is present. Note that the last “bi-fundamental” between the $\mathfrak{so}_7$ and the $\mathfrak{su}_2$ factor is in the eight-dimensional spinor representation of $\mathfrak{so}_7$ as for the $\mathbb{Z}_2$ NHCs above.
Before we continue our discussion with other torsion groups, we want to emphasize that the tensor branch strongly depends on the form of the additional $I_1$ component. To illustrate this, consider theory 4 and set $a_2=v^4$. This does not change the flavor algebra at $v=0$. Taking $n=3$ or $n=4$, corresponding to $a_4 = u^6$ or $a_4= u^7$. From the perspective of the flavor brane, these simply look like an $\mathfrak{su}_{12}$ or $\mathfrak{su}_{14}$ flavor symmetry intersecting the $I_1$ component and from that perspective one might have expected theories like those presented in . However, with this modification, the $I_1$ locus develops a triple-point singularity for $n=3$ and a septuple-point singularity for $n=4$ at the origin. As a consequence, the tensor branches are actually quite different from , $$\begin{aligned}
&\begin{array}{c}
\includegraphics[scale=0.7]{SU_SO_Z2.pdf} \end{array}\\
&\begin{array}{c}
\includegraphics[scale=0.7]{SU_E7_Z2.pdf} \end{array} \end{aligned}$$ Indeed, the resulting tensor branches are closer to theories like and , where the $\mathfrak{so}_8$ or $\mathfrak{e}_7$ flavor algebra is gauged. More theories of similar type can be found in the Appendix \[app:outlier\].
### $\boldsymbol{\mathbb{Z}_3}$ Torsion: Conformal Matter {#sssec:Z3SCM}
F-theory models with $\mathbb{Z}_3$ torsion are constructed from the Weierstrass model . It is easy to see that this model only allows fibers of type $I_{3n}$, $IV$, and $IV^{*,\text{s}}$, which are consistent with the restricted monodromy.
We start with presenting the $\mathbb{Z}_3$ discrete holonomy instanton theories: $$\begin{aligned}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{}&\text{ Flavor } & a_1 & a_3 \\ \hline
1&[\text{E}_6 \times \text{SU}(3)] /\mathbb{Z}_3 & u & u^2 v \\[0pt]
2&\text{SU}(9)/\mathbb{Z}_3 & u & v^3 \\[0pt]
3&[\text{E}_6 \times \text{SU}(3)_{IV}] /\mathbb{Z}_3 & u v & u^2 v \\ \hline
4& [\text{SU}(6) \times \text{SU}(3)_{IV}]/\mathbb{Z}_3 & u & u v^2 \\ \hline
\end{array}\end{aligned}$$ Note that the last entry above is not a maximal commutant in $\text{E}_8$, but can instead be viewed as originating from a deformed $ \mathfrak{e}_6 \times \mathfrak{su}_{3}$ theory.
Theory 1 is somewhat analogous to theory 1 of the $\mathbb{Z}_2$ case. New types of conformal matter arise from colliding an $\mathfrak{e}_6$ with an $\mathfrak{su}_{3n}$ algebra. Setting $a_1 = u$ and $a_3 = u^2 v^n$, in theory 1 leads to theories with a tensor branch $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E6_SU_Z3.pdf} \end{array}
\label{eq:Z3chain}\end{aligned}$$ This is similar to the classic $\mathfrak{e}_6 \times \mathfrak{su}$ collision, but the ranks of the gauge algebras on the tensor branch increase in steps of three. This necessarily had to happen, since they have to be compatible with the $\mathbb{Z}_3$ torsion factor.
Let us move on to theory 2 in the table above and enhance the $I_9$ fiber to an $I_{3n}$ fiber by setting $a_1 = u\,,~ a_3 = v^n$. From the Weierstrass model $$\begin{aligned}
\begin{split}
f =& \tfrac{1}{48} \kappa_1 u (-\kappa_1^3 u^3 + 24 v^n)\, ,\qquad g= \tfrac{1}{864} (\kappa_1^6 u^6 - 36 \kappa_1^3 u^3 v^n + 216 v^{2 n}) , \\ \Delta=& \tfrac{1}{16} v^{3 n} (-\kappa_1^3 u^3 + 27 v^n)
\end{split}\end{aligned}$$ we find that the $I_1$ locus develops a triple-point singularity at the origin for $n>2$. Each blow-up in the base reduces the rank of the gauge algebra over the blow-up divisor by $9$, which results in the tensor branch $$\begin{aligned}
\label{eq:Su3nZ3}
\begin{array}{c}
\includegraphics[scale=0.7]{I_SU_Z3.pdf} \end{array} \end{aligned}$$ The matter sector consists of bi-fundamentals. If there is an $\mathfrak{su}_6$ factor over the final $(-1)$ curve, the intersection with the $I_1$ component gives an additional half-hyper in the $\mathbf{20}$-dimensional triple-antisymmetric representation. To see this note that the $\mathfrak{su}_6$ divisor intersects the $I_1$ direction in an $\mathfrak{e}_6$ point. Similarly, for $n=0$ mod $3$, the theory has no gauge group over the $(-1)$ curve, which we identify as the E-string theory where the $\mathfrak{su}_9$ flavor factor is enhanced further. The three types of theories defined by $n=0,1,2$ mod $3$ are in precise agreement with the constructions found in [@Ohmori:2018ona].
Let us continue with theory 3. The flavor algebra at the $\mathfrak{su}_3$ end can only be enhanced once by setting $a_3=u^2 v^{n}$ with $n=0,1$. Higher values of $n$ lead to non-minimal singularities. For $n=1$, we obtain the classical $\mathfrak{e}_6 \times \mathfrak{e}_6$ conformal matter theory discussed in . Let us stress at this point that tuning the $I_1$ fiber of the generic model decides whether we have perturbative matter, or whether the collision of the flavor branes leads to superconformal matter theories. For example, for $a_1 =\kappa$ with $\kappa\in\mathbb{C}$ and $a_3 = u^n v^m$, we can engineer theories with $\mathfrak{su}_{3n} \times \mathfrak{su}_{3m}$ groups. The codimension-two vanishing order at their intersection $u=v=0$ is benign (i.e. below $(4,6,12)$) and hence leads to ordinary, perturbative theories rather than superconformal matter theories. However, by choosing $a_1 = \kappa_1 u + \kappa_2 v$ and $a_3 = u^n v^m$ the discriminant becomes $$\begin{aligned}
\label{eq:nonPertenhanceZ3}
\Delta= u^{3 n} v^{3 m} (-\kappa_1^3 u^3 - 3 \kappa_1^2 \kappa_2 u^2 v - 3 \kappa_1 \kappa_2^2 u v^2 - \kappa_2^3 v^3 + 27 u^n v^m)\end{aligned}$$ where the collision of the $\mathfrak{su}_{3n} \times \mathfrak{su}_{3m}$ branes appears at the triple-point singularity of the $I_1$ factor. There, the singularity enhances to $(4,6,3(m+n+1))$ if $m+n > 2$. Blowing up the intersection point introduces a $(-1)$ curve with an $\mathfrak{su}_{3(n+m-1)}$ gauge algebra that is intersected three times by the $I_1$ curve. The tensor branch is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_Z3.pdf} \end{array} \end{aligned}$$ Notably, in the case where the central node has an $\mathfrak{su}_6$ gauge factor (i.e. for $n+m=5$), more matter is required to cancel the local gauge anomaly. In total we require $15$ hypermultiplets in the fundamental $\mathbf{6}$ of $\mathfrak{su}_6$, which arise naturally as bi-fundamentals for the three distinct combinations of $n$ and $m$, which have $\mathfrak{su}_{15} \times I_1$, $\mathfrak{su}_{12} \times \mathfrak{su}_3$ and $\mathfrak{su}_{9} \times \mathfrak{su}_6$ flavor groups. In addition, anomaly cancellation requires a half-hyper in the triple-antisymmetric $\mathbf{20}$ is required. This can indeed be found by analyzing the discriminant carefully and noting that over $\{ \kappa_1 u + \kappa_2 v = 0 \}$ there is a collision with the exceptional divisor, which enhances the $I_6$ to an $\mathfrak{e}_6$ singularity. From the branching of the $\mathfrak{e}_6$ adjoint one finds the triple-antisymmetric representation as expected. Note that this is the only type of enhancement that can produce the required representation while being consistent with the restricted monodromy. For $n+m>5$, further blow-ups are required that lead to the chain $$\begin{aligned}
\label{eq:SU_SU_SU_Z3}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_SU_Z3.pdf}\end{array}\end{aligned}$$ Notably, this theory has a similar tensor branch structure to the one given in . Hence it appears plausible that the theory is a deformation of which splits the $\mathfrak{su}_{3(n+m)}$ flavor algebras into $\mathfrak{su}_{3m}\times\mathfrak{su}_{3n}$.
Let us move on to theory 4. While a collision of an $I_{3m}$ with an $I_{3n}$ fiber leads to a perturbative theory for any $n$ and $m$, we get superconformal theories if we replace one of the $I_{3m}$ by an $\mathfrak{su}_3$ of type $IV$, which takes us to theory 4. Enhancing the other $I_{3m}$ factor by setting $a_3=u v^n$, we obtain the Weierstrass function $$\begin{aligned}
\begin{split}
f=& \tfrac{1}{48} u^2 (-u^2 + 24 v^n) \,,\qquad g= \tfrac{1}{864} u^2 (u^4 - 36 u^2 v^n + 216 v^{2 n}) \\ \Delta=& \tfrac{1}{16} u^4 v^{3 n} (-u^2 + 27 v^n)\,,
\end{split}\end{aligned}$$ where the $I_1$ component develops a double point singularity at the origin. For $n>2$ and even, the tensor branch is given by $$\begin{aligned}
\begin{array}{c}
\label{eq:su3su3n_even}
\includegraphics[scale=0.7]{SU3_SU_Z3.pdf} \end{array} \end{aligned}$$ where the left-most part is again the $\mathbb{Z}_3$ discrete holonomy instanton theory. For $n>2$ and odd, one has $$\begin{aligned}
\begin{array}{c}
\label{eq:su3su3n_odd}
\includegraphics[scale=0.7]{SU3_SU_Z3_mod.pdf} \end{array} \end{aligned}$$ Note that the two theories and could also been obtained by enhancing , setting $n=1$, and deforming to $\kappa_2 =0$. Various other outlier theories can be found in Appendix \[app:outlier\].
### $\boldsymbol{\mathbb{Z}_4}$ Torsion: Conformal Matter {#sec:Z4TorsionSCM}
For $\mathbb{Z}_4$ conformal matter we analyze theories whose Weierstrass model is given in . One could have expected to find only groups with a $\mathbb{Z}_4$ center. However, this is not the case: it is also possible to find groups with a $\mathbb{Z}_2$ subcenter of the full $\mathbb{Z}_4$ as already being evident for the most generic model, which has an $\mathfrak{su}_4 \times \mathfrak{su}_2$ algebra. Group factors with $\mathbb{Z}_4$ and $\mathbb{Z}_2$ center also appear when considering the three different discrete holonomy instanton configurations $$\begin{aligned}
\label{eq:Z4DHInst}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{}&\text{ Flavor } & a_1 & a_2 \\ \hline
1&[\text{SU}(8) \times \text{SU}(2)]/\mathbb{Z}_4 & u & v^2 \\[0pt]
2&[\text{Spin}(10) \times \text {SU}(4)]/\mathbb{Z}_4 & u & u v \\[0pt] \hline
3& [\text{SU}(4) \times \text{SU}(4) \times \text{SU}(2)]/\mathbb{Z}_4 & u+v & u v \\[0pt] \hline
\end{array}\end{aligned}$$
We start our discussion by enhancing the $\mathfrak{su}_2$ flavor algebra in the first discrete holonomy instanton theory by setting $a_1 = u^n$. From this, we obtain the following chain of theories $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU8_SU_Z4.pdf} \end{array} \end{aligned}$$ If we instead further enhance the $\mathfrak{su}_8$ flavor symmetry by setting $a_2 = v^{2n}$, we get $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU2_Z4.pdf} \end{array} \end{aligned}$$ As in the examples above, the jump in the gauge group rank proceeds along the gauge factor coming from the (broken) flavor theory of the E-string theory. Note that a similar theory exists when the power of $v$ in $a_2$ is odd, $a_2=v^{2n+1}$, which leads to $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU2_SU_Z4.pdf} \end{array} \end{aligned}$$ The $\mathfrak{su}_4$ above the $(-1)$ curve next to the $\mathfrak{su}_2$ flavor algebra comes with a half-hypermultiplet in the $(\mathbf{2,6})$ representation, as required by anomaly cancellation and consistent with a $\mathbb{Z}_2$ center charge. Geometrically, the representation arises from a $(2,3,7)$ singularity at the collision point, which corresponds to an $\mathfrak{so}_{10}$ enhancement. This is also consistent with the full $\mathbb{Z}_4$ center. Branching its adjoint into $\mathfrak{su}_2 \times \mathfrak{su}_4$ representations indeed gives rise to the required representation.
For the second theory in , one can enhance the $\mathfrak{su}_4$ factor to $\mathfrak{su}_{4n}$ by setting $a_2 = u v^n$, which results in the chain $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SO10_SU_Z4.pdf} \end{array} \end{aligned}$$ Note the order 4 jump in between the $\mathfrak{su}_{4k}$ factors. Enhancing the $\mathfrak{so}_{10}$ side on the other hand (by setting $a_1=u^n$) leads to $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SO_SU4_Z4.pdf} \end{array} \end{aligned}$$
Finally, we consider the third theory in , which represents the only non-maximal $\mathfrak{e}_8$ subgroup with $\mathfrak{su}_4^2 \times \mathfrak{su}_2$ flavor group. Note that this theory can in fact be seen as a deformation obtained from the $\mathbb{Z}_4 \times \mathbb{Z}_2$ discrete holonomy instanton case that we consider in Section \[sssec:Z2Z4SCM\], which reduces the global flavor group by an $\mathfrak{su}_2$ factor. Starting from that theory, there is the possibility to either enhance the $\mathfrak{su}_4$ factors or the $\mathfrak{su}_2$ factor. The former case is obtained by setting $a_2=u^n v^m$ and results in $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_SU2_Z4.pdf} \end{array} \end{aligned}$$ From a technical point of view, the above chain can be obtained by performing a coordinate shift after the first blow-up. Setting $w = u-v$ puts the $\mathfrak{su}_2$ over the toric locus $w=0$. For $(n+m)=1$ mod $2$, there is again an $\mathfrak{su}_4$ attached to the $\mathfrak{su}_2$ flavor factor, with half-hypers in the $(\mathbf{2,6})$ representation, as required by anomaly cancellation. This is directly seen by noting that the intersection locus is a $(2,3,7)$ singularity and hence yields the desired representations from the decomposition of the adjoint as argued before. For the case $n+m=3$, one obtains a single $\mathfrak{su}_4$ gauge algebra on the $(-1)$ curve $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_SU2_new_Z4.pdf} \end{array} \end{aligned}$$ We can also enhance the non-toric $\mathfrak{su}_2$ flavor group to $\mathfrak{su}_{2k}$ by setting $a_1=(u+v)^k$. Note that the $\mathfrak{su}_2$ flavor is only modded by a $\mathbb{Z}_2$ subgroup of the full $\mathbb{Z}_4$. If we enhance this flavor group further, we obtain the following tensor branch: $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU4_SU4_Z4.pdf} \end{array} \end{aligned}$$ Here, we find collisions of orders $(2,3,3+2n )$ where the $\mathfrak{sp}_{n-1}$ intersects the $\mathfrak{su}_4$ flavor factors. Hence, we expect half-hypermultiplets that transform in the two-fold antisymmetric of $\mathfrak{su}_4$, consistent with anomaly cancellation. In order to keep the appearing singularities crepantly resolvable, one cannot enhance the non-toric $\mathfrak{su}_2$ and the other $\mathfrak{su}_4$ factors arbitrarily. In fact, the only consistent configuration where one enhances multiple sides is by setting $a_1 = (u+v)^2$ and $a_2 = u v^2$ which results in $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU8_SU4_Z4.pdf} \end{array} \end{aligned}$$ Note that one can interpret the above configuration in part as theory 2 glued to another instanton configuration along the $\mathfrak{so}_{10}$ algebra, and further enhancing the $\mathfrak{su}_4$.
Finally we present the possibility to engineer an $\mathfrak{so}_{10+4n} \times \mathfrak{so}_{10+4m}$ superconformal matter collision which appeared already in [@DelZotto:2014hpa]. Our geometric construction shows that the groups will be modded by a $\mathbb{Z}_4$ quotient factor. The theory is obtained by setting $a_1=u^{1+n} v^{1+m}$ and $a_2=u v$ and reads $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SO_SO_Z4.pdf} \end{array} \end{aligned}$$ This concludes the simple theories which can be obtained by enhancing flavor groups of $\mathbb{Z}_4$ discrete holonomy instanton theories. Various other outlier theories can be found in Appendix \[app:outlier\].
### $\boldsymbol{\mathbb{Z}_5}$ Torsion: Conformal Matter
Next, we focus on the $\mathbb{Z}_5$ torsion model. Monodromies of this type only allow for fibers of type $I_{5n}$, i.e. $\mathfrak{su}_{5n}$ algebras, as one expects from the field theory side. The Weierstrass model is given in Eqn. . The only discrete holonomy instanton theory is $$\begin{aligned}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|} \hline
\text{Flavor Group} & a_1 & b_1 \\ \hline
[\text{SU}(5) \times \text{SU}(5)]/\mathbb{Z}_5 & u & v \\ \hline
\end{array}\end{aligned}$$ Enhancing one of the $\mathfrak{su}_5$ flavor factor is the only non-trivial possibility for superconformal matter of this kind (enhancing both factors simultaneously leads to non-minimal models), which is obtained by setting $a_1 = u\, ,~ b_1 = u^n$. This results in a theory with an $[\text{SU}(5) \times \text{SU}(5n)]/\mathbb{Z}_5$ flavor group. This class is the analog of $\mathfrak{e}_8 \times \mathfrak{su}$ superconformal matter restricted to geometries with a 5-torsion section. Its tensor branch reads $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU_SU_Z5.pdf} \end{array} \end{aligned}$$ At the end of the chain, we have the regular $\mathbb{Z}_5$ discrete holonomy instanton theory. This class of theories has also been constructed from a field theory perspective in [@Ohmori:2018ona].
### $\boldsymbol{\mathbb{Z}_6}$ Torsion: Conformal Matter
Finally, the $\mathbb{Z}_6$ restricted model given in demands a similarly strong tuning as the $\mathbb{Z}_5$ case, which leads to a single consistent discrete holonomy instanton theory, $$\begin{aligned}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|} \hline
\text{Flavor Group} & a_1 & b_1 \\ \hline
[\text{SU}(6) \times \text{SU}(3) \times \text{SU}(2)]/\mathbb{Z}_6 & u & v \\ \hline
\end{array}\end{aligned}$$ Note that there are groups with centers $\mathbb{Z}_2$, $\mathbb{Z}_3$, and $\mathbb{Z}_6$. Since $\mathbb{Z}_6\simeq\mathbb{Z}_2\times\mathbb{Z}_3$, this means that also subgroups of $\mathbb{Z}_6$ appear as centers.
We have the choice to extend the flavor groups of the E-string theory from several directions, which leaves a chain with jumps in their respective orders. Technically, this is best achieved by performing a coordinate shift in the above Weierstrass model and then increasing the vanishing order of the shifted coordinate. Remember that $$\begin{aligned}
\Delta = \tfrac{1}{2^{24}} (3a_1 - 5 b_1) (3 a_1 + b_1)^2 (a_1 + b_1)^3 (a_1 - b_1)^6\,.\end{aligned}$$ Setting $w=a_1-b_1=(u-v)$ and increasing the vanishing order to $w^n$ allows us to construct the chain $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU6n_Z6.pdf} \end{array} \end{aligned}$$ as was also deduced in [@Ohmori:2018ona] from field theory arguments. However, we can also construct extended chains along other directions: Setting $w=a_1+b_1=(u+v)$ and then sending $w\to w^n$ leads to $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU3n_Z6.pdf} \end{array} \end{aligned}$$ while setting $w=3a_1+b_1=(3u+v)$ and then increasing the vanishing order to $w^n$ gives $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU2n_Z6.pdf} \end{array} \end{aligned}$$ Simultaneous enhancement of two gauge factors will result in non-minimal singularities.
Double Factors Theories: Z2xZ2, Z2xZ4, Z3xZ3
--------------------------------------------
This section continues the investigation of superconformal matter theories with non-simply-connected groups that admit two quotient factors. These theories have similar features to the ones we have already discussed. However, tuning the torsion points severely restricts the possible breaking patterns of the E-string theory by discrete holonomy instantons. Moreover, as e.g. in the $\mathbb{Z}_4$ case, the center of some of the group factors is only modded out by a subgroup of the full torsion group.
### $\boldsymbol{\mathbb{Z}_2 \times \mathbb{Z}_2}$ Torsion: Conformal Matter
Here we summarize the general models with $\mathbb{Z}_2 \times \mathbb{Z}_2$ torsion given by the Weierstrass models spelled out in Eqn. . These models admit a couple of $\text{E}_8$ breaking patterns: $$\begin{aligned}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{}&\text{ Flavor Group } & b_2 & c_2 \\ \hline
1& [\text{SU}(4)^2 \times \text{SU}(2)^2]/ [\mathbb{Z}_2 \times \mathbb{Z}_2] & u^2 & v^2 \\ \hline
2& [\text{Spin}(12) \times \text{SU}(2)^2] / [\mathbb{Z}_2 \times \mathbb{Z}_2] & u v & v^2 \\ \hline
3& [\text{Spin}(8) \times \text{Spin}(8)] / [\mathbb{Z}_2 \times \mathbb{Z}_2] & u v & \frac12 u v \\ \hline
\end{array}\end{aligned}$$ Again, the rank of the resulting gauge algebra factors can be further increased by tuning higher vanishing orders. Note that the first broken instanton theory carries a $\mathbb{Z}_4 \times \mathbb{Z}_2$ torsion factor which we discuss later.[^18]
Enhancing one of the $\mathfrak{su}_4$ factors to $\mathfrak{su}_{4n}$ by setting $b_2=u^{2n}$ (or $c_2=v^{2m}$), we get the tensor branch $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU4n_Z4Z2.pdf} \end{array} \end{aligned}$$ This enhancement also preserves the $\mathbb{Z}_4$ torsion point, which is why the group symmetry is modded by the full $\mathbb{Z}_4 \times \mathbb{Z}_2$.
This, however, is not the case when enhancing the $\mathfrak{su}_4$ to an $\mathfrak{su}_{4n+2}$ by setting $b_2=u^{2n-1}$. In that case, the outer $\mathfrak{su}_2$ factors do not split anymore but become a single $\mathfrak{su}_2$ flavor factor. Moreover, the $\mathfrak{su}_4$ is folded to an $\mathfrak{sp}_2$. The rank of the gauge groups on the tensor branch is still going to jump by steps of four for the $\mathfrak{su}$ factors, but it ends with an $\mathfrak{su}_2$ on a $(-1)$ curve intersecting the $\mathfrak{su}_2$ and $\mathfrak{sp}_2$ flavor factor at a double-point singularity. The full tensor branch reads $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU4n+2_Z2Z2.pdf} \end{array} \end{aligned}$$ Note that similarly, we can enhance the original $\mathfrak{su}_2$ factors, which yields a similar tensor branch but only jumps by two in the rank of the $\mathfrak{su}$ gauge factors.
Turning to the second discrete holonomy instanton theory above, we can further enhance the $\mathfrak{so}$ factor by setting $c_2=v^{2+n}$, which results in theories of the type $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SO_SU2_SU2_Z2Z2.pdf} \end{array} \end{aligned}$$ as well as $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SP_SP1_Z2Z2.pdf} \end{array} \end{aligned}$$
Alternatively, we can enhance an $\mathfrak{su}_2$ factor on the other end and obtain a chain very similar to that of $\mathbb{Z}_2$ matter with modified end part: $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SO12_SU_Z2Z2.pdf} \end{array} \end{aligned}$$
A tuning that sits in between the second and first E-string theory can be obtained by using $\mathfrak{su}_4\simeq\mathfrak{so}_6$ and enhance it to an $\mathfrak{so}_{12} \times \mathfrak{so}_{8+4n}\times \mathfrak{su}_2$ collision by setting $b_2 = u^2 v^1 \, , \, c_2 = u^k v^n$. This, however, cannot be done beyond $k=1$, since one eventually ends up with $(4,6,12)$ singularities in codimension one (or $(8,12,24)$ singularities in codimension two). However, one is free to send $n$ to any other value which results in the chain $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SO_SO_Z2Z2.pdf} \end{array}\end{aligned}$$ Note that for $n \leq 1$, the additional $\mathfrak{su}_2$ flavor factor decouples. The $\mathfrak{so}_8$ on the $(-3)$ curve is just the $\mathbb{Z}_2 \times \mathbb{Z}_2$ NHC as given in , with three half-hypers in the bi-fundamental representation of vector, spinor and co-spinor representation, as required by anomaly cancellation. The rest of the matter are half-hypers in the bi-fundamental representations.
### $ \boldsymbol{\mathbb{Z}_2 \times \mathbb{Z}_4}$ Torsion: Conformal Matter {#sssec:Z2Z4SCM}
The $\mathbb{Z}_2 \times \mathbb{Z}_4$ model breaks $\text{E}_8$ to $[ \text{SU}(4)^2 \times \text{SU}(2)^2] / [\mathbb{Z}_2 \times \mathbb{Z}_4]$ generically, which can be observed in the Weierstrass model given in Eqn. . This means that the simplest discrete holonomy instanton theory is $$\begin{aligned}
\label{eq:E8z2z4}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|}\hline
$Flavor Group$ & a_1 & b_1 \\ \hline
[\text{SU}(4)^2 \times \text{SU}(2)^2] / [\mathbb{Z}_2 \times \mathbb{Z}_4] & u & v \\ \hline
\end{array} \end{aligned}$$ We can enhance one of the $\mathfrak{su}_2$ flavor factors by e.g. setting $b_1 = v^n$, which results in the tensor branch $$\begin{aligned}
\label{eq:SU2n_Z4Z2}
\begin{array}{c}
\includegraphics[scale=0.7]{SU2n_Z4Z2.pdf} \end{array} \end{aligned}$$ Note that we are not able to simultaneously enhance the second $\mathfrak{su}_2$ factor in the model without introducing a codimension-two $(8,12,24)$ point at the origin. Alternatively we can enhance the $\mathfrak{su}_4$ factor by setting $a_1 = u\,,~ b_1 = \frac14 (v^n+u)$, which gives an analogous chain $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU4n_Z4Z2.pdf} \end{array} \end{aligned}$$
### $\boldsymbol{\mathbb{Z}_3 \times \mathbb{Z}_3}$ Torsion: Conformal Matter
The final model we are discussing preserves two 3-torsion sections and has the restricted Weierstrass model given in . The simplest factorization is also its generic gauge group in a compact setup, which is identical to the associated discrete holonomy instanton theory $$\begin{aligned}
\renewcommand{{1.4}}{1.3}
\begin{array}{|c|c|c|}\hline
$Flavor Group$ & a_1 & b_1 \\ \hline
\text{SU}(3)^4/[\mathbb{Z}_3 \times \mathbb{Z}_3] & u & v \\ \hline
\end{array}\end{aligned}$$ We can now increase the rank of any of the $\mathfrak{su}_3$ algebras. Since the $b_1$ locus is toric, it is easiest to set $b_1 = v^n$. From this, we obtain the analog of the $\text{E}_8 - \text{A}$ superconformal matter theory for this restricted monodromy, with tensor branch $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SU3n_Z3Z3.pdf} \end{array}
\label{eq:Z3Z3model}\end{aligned}$$ Enhancing more than one $\mathfrak{su}_3$ factor leads to non-minimal singularities.
Construction of General SCFTs with Torsion {#subsec:SCFTconstr}
------------------------------------------
With the above building blocks at hand we can outline the construction of general 6d SCFTs with Mordell-Weil torsion along the lines of [@Heckman:2018pqx][^19]. This constructive approach proceeds in two steps.
First, one performs a fission process for which one switches on relevant deformations that trigger an RG flow. These deformations can have two effects:
1. They deform the gauge and flavor algebras on the tensor branch. \[itm:Deformation1\]
2. They can lead to a flow in which one of the curves of negative self-intersection. decompactifies.\[itm:Deformation2\]
Concerning effect \[itm:Deformation1\], we have already seen in Section \[sec:SCFT\] that the deformations which modify the group structure are restricted to a subclass by a non-trivial Mordell-Weil torsion. Effect \[itm:Deformation2\] might split the compact part of the geometry in two pieces and is not problematic from the point of view of the global group structure: the resulting disconnected parts all respect the restricted Weierstrass form and consequently the action of $T$ is preserved.
Second, one performs a fusion operation which connects two previously disconnected parts, either via a $(-1)$ curve or by gauging a common flavor group factor in the UV. This step is more problematic in the presence of Mordell-Weil torsion, since the gauging has to be performed in a way that is compatible with the global group structure. Having multiple parts with respective torsional groups $T_i$, a fusion operation can only preserve the maximal common subgroup. Even if there is a non-trivial common subgroup, the fusion process still has to be performed in a torsion-preserving way, since otherwise the torsional part is lost in the process. This can be easily understood in terms of the restricted monodromies in models with Mordell-Weil torsion. The individual pieces before the fusion process only allow for monodromies in certain subgroups of $\text{SL}(2,\mathbb{Z})$. After the fusion process, all monodromies appear in the same configuration and one might find a different subgroup of $\text{SL}(2,\mathbb{Z})$ or even the full duality group. The torsion of the fusion product accordingly preserves or destroys the torsional sections of the individual pieces.
We want to demonstrate the above considerations in two relatively simple examples. The first possibility for a fusion process is to connect two distinct pieces via a curve with self-intersection $(-1)$. For that consider for example the $\mathbb{Z}_2 \times \mathbb{Z}_2$ NHC on a $(-4)$ curve and the $\mathbb{Z}_2$ two-curve cluster. Connecting the two pieces via the exceptional curve will at least break the torsion group to the maximal common subgroup $\mathbb{Z}_2$. Indeed, we see that the fusion product can actually be identified with the resolved $\mathbb{Z}_2$ three curve cluster in . $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{Fusion.pdf}\end{array}\end{aligned}$$ This shows that the fusion process can in fact preserve the $\mathbb{Z}_2$ torsion.
The second way to connect disconnected theories is by gauging a common flavor symmetry. If the individual parts do contain compatible torsion sections, this gauging of the flavor symmetry can be performed in a way to retain at least part of the torsion group. As an example, consider a fission product of the $\mathbb{Z}_3 \times \mathbb{Z}_3$ theory given in derived from decompactifying the first $(-2)$ curve and the $\mathbb{Z}_3$ discrete holonomy instanton theory. Gauging the two $\mathfrak{su}_3$ flavor algebras one finds $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.6]{Fusionalt.pdf} \end{array}\end{aligned}$$ which is identical to the $\mathbb{Z}_3$ theory and shows that a torsional $\mathbb{Z}_3$ subgroup can be preserved in the fusion process.
Conclusion and Outlook {#sec:concl}
======================
In this article we construct a wide range of 6D SCFTs with non-simply connected non-Abelian flavor groups by tuning the geometry in an F-theory setup. The tensor branch of these theories inherits this non-simply-connected structure.
In more detail, we proceed by analyzing singular non-compact Calabi-Yau 3-folds which are elliptically-fibered with extra torsional sections. These models feature a non-trivial, finite Mordell-Weil group $T$. The construction restricts the full $\text{SL}(2,\mathbb{Z})$ duality of type IIB string theory to a congruence subgroup. As a consequence, not all $(p,q)$ 7-branes are allowed and the flavor and gauge groups of the models are constrained. Moreover, the Mordell-Weil torsion mods out a part of the center of the simply-connected cover $\mathcal{G} = \mathcal{G}^*/T$. This restricts the possible flavor and gauge groups to those Lie groups whose center admit an action of $T$, or a subgroup thereof.
Within this set of restricted Weierstrass models, we construct the essential building blocks for 6d SCFTs and their tensor branches for all possible $T$. We start by analyzing non-Higgsable clusters with Mordell-Weil torsion. Many of the original gauge algebras of [@Morrison:2012np] on curves with negative self-intersection need to be enhanced in order to respect the global group structure. This enhancement is automatic once we restrict the geometry such that the elliptic fibration has Mordell-Weil torsion.
We then turn to the E-string theory in the presence of Mordell-Weil torsion, which can be interpreted as theories with discrete holonomy instanton in the heterotic/M-theory framework. These theories result in flavor groups which are subgroups of $\text{E}_8$ with an associated action of $T$ as described above.
Finally, we investigate the collision of two non-compact components of the discriminant locus which lead to non-minimal singularities in codimension two, also known as superconformal matter theories. The non-trivial group $T$ imposes strong restrictions on the realization of the tensor branch as well as the deformations of this class of theories. This results in important differences for theories with the same flavor algebra but different flavor group. We further confirm some field theoretic constructions in [@Ohmori:2018ona] as a subclass of models derived in this way and complement the explicit geometric construction. We observe that singularities of vanishing order $(8,12,24)$ (or worse) in codimension two often appear when trying to enhance flavor group factors. If blown up, such singularities lead to $(4,6,12)$ singularities in codimension one over the blow-up divisor. Interestingly, in many cases colliding two flavor groups looks benign. However, after blowing up the collision, we find that $(8,12,24)$ singularities appear between the blown up divisor and one of the flavor group factors.
There are several interesting avenues to pursue further. In Section \[sec:SCFT\] we already hinted at the modifications of allowed deformations and RG flows in the presence of Mordell-Weil torsion. It would be interesting to explore these ideas and obtain a complete network of 6d SCFTs with torsion. Beside the techniques employed in [@Heckman:2015ola; @Heckman:2016ssk; @Heckman:2018pqx], an alternative approach to classify the deformations associated to T-brane data is via $(p,q)$ string junctions, see e.g. [@Hassler:2019eso], which can also be utilized in geometries with restricted monodromies, i.e. Mordell-Weil torsion.
We also suggested an M-theory interpretation of models with non-trivial torsion in Section \[sec:SCFT\]. Exploring this further would not only shed some light on the M-theory picture of the group structure in 6d SCFTs, but on M-theory models in general. The appearance of fractional fluxes might further elucidate the interplay between the gauge group of the 7d super-Yang-Mills theory on an ADE-singularity and the worldvolume theory of M5-branes probing it, with possible effects in holographic descriptions. Compactifying the resulting 6d theories leads to additional possibilities in the flavor backgrounds due to the global group structure. An investigation involving these might reveal further structure in the lower-dimensional superconformal field theories, as discussed e.g. in [@Ohmori:2018ona].
Of course another way to investigate the implications of the global group structure in M-theory is to use M-/F-theory duality. For that one compactifies the 6d model F-theory model on a circle (including possible twists), and deforms the 5d theory onto the Coulomb branch [@Bhardwaj:2018yhy; @Bhardwaj:2018vuu; @Bhardwaj:2019fzv]. It is suggestive that the modification of the group structure in the 6d setup also affects the allowed twists and resulting 5d theories.
Beyond this, a pure 5d approach to M-theory on elliptically-fibered Calabi-Yau 3-fold with torsion, similar to [@Apruzzi:2019vpe; @Apruzzi:2019opn; @Apruzzi:2019enx], could be used to study possible effects of restrictions imposed by the presence of torsional sections. In this way, one can fully benefit from the powerful geometrical F-theory construction of 6d SCFTs acting as seeds for lower-dimensional theories via compactification.
Furthermore, the explicit geometries of this work can serve as a starting point for the constructions of various other theories in six and lower dimensions. One promising direction, presented in [@Anderson:2018heq; @Anderson:2019kmx] is to combine the fiber translation encoded in the finite Mordell-Weil group with an automorphism in the base. This allows to combine quotients in the base, such as in [@Apruzzi:2017iqe], with a non-trivial action on the fiber to obtain new smooth local Calabi-Yau quotients, with new F- and M-theory duals.
Finally, a full classification of non-simply-connected flavor groups also includes additional $\text{U}(1)$ factors in the flavor sector. In [@Apruzzi:2020eqi], it was shown that these do not necessarily originate from the free part of the Mordell-Weil group, but can descend from deformed non-Abelian symmetries. In our analysis we often found additional $I_1$ loci whose tuning strongly influenced the overall tensor branch structure. It is tempting to connect these $I_1$ singularities of the fiber, or combinations thereof dictated by their ABJ anomalies, to the additional Abelian flavor factors. Since these also contribute to the group structure, their inclusion might increase the allowed possibilities. Moreover, they impact the embedding of the discrete torsion group $T$ into $\mathcal{G}^*$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Jonathan Heckman, Thorsten Schimannek, Gianluca Zoccarato for valuable discussions. We further thank Jonathan Heckman for his comments on the draft. The work of M.D. is supported by the individual DFG grant DI 2527/1-1. The work of P.K.O. is supported by a Swedish grant of the Carl Trygger Foundation for Scientific Research.
Enhanced Weierstrass Models {#sec:EnhancedWSFs}
===========================
In this appendix we list the specific factorizations of the Weierstrass coefficients that ensure the presence of torsional sections. We then further discuss the possibility to enhance the Mordell-Weil torsion by further tunings.
Weierstrass Models with Mordell-Weil Torsion
--------------------------------------------
The enhanced Weierstrass models for all allowed torsion groups are given by
\[eq:torsionfrak\] $$\begin{aligned}
\renewcommand{{1.4}}{1.5}
\mathbb{Z}_2:
& \quad f = a_4 - \tfrac{1}{3} a_2^2 \,,
\quad g = \tfrac{1}{27} a_2 (2 a_2^2 - 9 a_4) \,,
\quad \Delta = a_4^2 (4 a_4 - a_2^2) \,, \label{eq:WSFTuning_Z2}\\[6pt]
\mathbb{Z}_3: & \quad f = \tfrac{1}{2} a_1 a_3 - \tfrac{1}{48} a_1^4 \,, \quad g = \tfrac{1}{4} a_3^2 + \tfrac{1}{864} a_1^6 - \tfrac{1}{24} a_1^3 a_3 \,,\nonumber\\
&\quad \Delta = \tfrac{1}{16} a_3^3 (27 a_3 - a_1^3) \,, \label{eq:WSFTuning_Z3}\\[6pt]
\mathbb{Z}_4:
& \quad f = - \tfrac{1}{48} a_1^4 + \tfrac{1}{3} a_1^2 a_2 - \tfrac{1}{3} a_2^2 \,,
\quad g = \tfrac{1}{864} \big( a_1^2 - 8 a_2 \big) \big( a_1^4 - 16 a_1^2 a_2 - 8 a_2^2 \big) \,, \nonumber\\
& \quad \Delta = - \tfrac{1}{16} a_1^2 a_2^4 \big( a_1^2 - 16 a_2 \big) \,, \label{eq:WSFTuning_Z4}\\[6pt]
\mathbb{Z}_5:
& \quad f = \tfrac{1}{48} (-a_1^4-8 a_1^3 b_1+16 a_2^2 b_1^2+8 a_1 b_1^3-16 b_1^4)\,,\nonumber\\
&\quad g = \tfrac{1}{864} (a_1^2 - 2 a_1 b_1 + 2 b_1^2) (a_1^4 + 14 a_1^3 b_1 + 26 a_1^2 b_1^2 - 116 a_1 b_1^3 + 76 b_1^4)\,,\nonumber\\
& \quad \Delta = \tfrac{1}{16} (a_1 - b_1)^5 b_1^5 (a_1^2 + 9 a_1 b_1 - 11 b_1^2)\,,\label{eq:WSFTuning_Z5}\\[6pt]
\mathbb{Z}_6:
& \quad f = \tfrac{1}{192} b_1 (3 a_1^3 - 3 a_1^2 b_1 - 3 a_1 b_1^2 - b_1^3)\,,\nonumber\\
&\quad g = \tfrac{1}{2^{12} 3^3} (3 a_1^2 - 6 a_1 b_1 - b_1^2) (9 a_1^4 - 6 a_1^2 b_1^2 - 24 a_1 b_1^3 - 11 b_1^4)\,,\nonumber\\
&\quad \Delta = \tfrac{1}{2^{24}} (3a_1 - 5 b_1) (3 a_1 + b_1)^2 (a_1 + b_1)^3 (a_1 - b_1)^6\,,\label{eq:WSFTuning_Z6}\\[6pt]
\mathbb{Z}_2 \times \mathbb{Z}_2:
& \quad f = \tfrac13 (b_2 c_2 - b_2^2 - c_2^2)\,,
\quad g = -\tfrac{1}{27} (b_2 + c_2) (b_2 - 2 c_2) (2 b_2 - c_2)\,,\nonumber\\
& \quad \Delta = -b_2^2 c_2^2 (b_2 - c_2)^2\,,\label{eq:WSFTuning_Z2xZ2}\\[6pt]
\mathbb{Z}_2 \times \mathbb{Z}_4:
& \quad f = -\tfrac{1}{768} a_1^4 - 7/24 a_1^2 b_1^2 - 1/3 b_1^4\,,\nonumber\\
& \quad g = \tfrac{1}{2^{11}3^3}(a_1^2 + 16 b_1^2) (a_1^2 - 24 a_1 b_1 + 16 b_1^2) (a_1^2 + 24 a_1 b_1 + 16 b_1^2)\,,\nonumber\\
& \quad \Delta = -\tfrac{1}{2^{16}} a_1^2 b_1^2 (a_1 - 4 b_1)^4 (a_1 + 4 b_1)^4\,, \label{eq:WSFTuning_Z2xZ4}\\[6pt]
\mathbb{Z}_3 \times \mathbb{Z}_3:
& \quad f = -\tfrac{1}{48} a_1 (a_1 - 2 b_1) (a_1 - 2 \omega b_1) (a_1 - 2 \omega^2 b_1)\,,\nonumber\\
& \quad g = \tfrac{1}{864} (a_1^2 + 2 a_1 b_1 - 2 b_1^2) (a_1^2 + 2 \omega a_1 b_1 - 2 \omega^2 b_1^2) (a_1^2 + 2 \omega^2 a_1 b_1 - 2 \omega b_1^2)\,,\nonumber\\
& \quad \Delta = \tfrac{1}{432} (a_1 + b_1)^3 (a_1 + \omega b_1)^3 (a_1 + \omega^2 b_1)^3 b_1^3\,,\label{eq:WSFTuning_Z3xZ3}\end{aligned}$$
where $\omega$ corresponds to a non-trivial cube-root of unity and the index of the coefficients $a_i, b_i, c_i$ indicates the degree in terms of the anti-canonical class of the base, i.e. $a_n \sim - n K$.
Torsion Enhancements
--------------------
Next, we discus factorizations that enhance the torsion in the enhanced Weierstrass models. In Table \[fig:torsiontuning\], we have determined all tunings that further enhance the torsion. Read in reverse, this can also be understood as deformations (such as Higgsing) that preserve a certain subgroup of the initial torsion.
\
start torsion tuned torsion tuning
------------------------------------- -------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathbb{Z}_2 $ $ \mathbb{Z}_4$ $\begin{array}{l}a_{2} \rightarrow \frac{a_{1}^2}{4}-2 \tilde{a}_{2}\, , \\ a_4 \rightarrow \tilde{a}_{2}^2 \end{array}$
$\mathbb{Z}_2 $ $ \mathbb{Z}_6$ $\begin{array}{l}a_2 \rightarrow \frac{1}{16} (b_1^2 + 6 a_1 b_1 - 3 a_1^2)\, , \\ a_4 \rightarrow \frac{1}{256} (a_1 - b_1)^3 (3 a_1 + b_1) \end{array}$
$\mathbb{Z}_2 $ $ \mathbb{Z}_2 \times \mathbb{Z}_2$ $\begin{array}{l}a_2 \rightarrow -b_2 + 2c_2\, , \\ a_4 \rightarrow -b_2 c_2+c_2^2 \end{array}$
$\mathbb{Z}_2 $ $ \mathbb{Z}_2 \times \mathbb{Z}_4$ $\begin{array}{l} a_2 \rightarrow \frac{1}{16} (a_1^2-24 a_1 b_1+16 b_1^2)\,, \\ a_4 \rightarrow -\frac{1}{16} a_1 b_1 (a_1-4 b_1)^2 \end{array} $
$\mathbb{Z}_3 $ $ \mathbb{Z}_6 $ $\begin{array}{l} a_1 \rightarrow b_1 \\ a_3 \rightarrow \frac{1}{32} (\tilde{a}_1 - b_1)^2 (\tilde{a}_1 + b_1) \end{array}$
$\mathbb{Z}_3 $ $ \mathbb{Z}_3 \times \mathbb{Z}_3 $ $\begin{array}{l} a_1 \rightarrow i \sqrt{3}\; \tilde{a}_1 \\ a_3 \rightarrow -\frac{i}{3 \sqrt{3}}(\tilde{a}_1^3+b_1^3) \end{array}$
$\mathbb{Z}_4 $ $ \mathbb{Z}_2 \times \mathbb{Z}_4$ $\begin{array}{l} a_1 \rightarrow -4i b_1 \\ a_2 \rightarrow \frac{1}{16} (a_1^2 - 16 b_1^2) \end{array}$
$\mathbb{Z}_2 \times \mathbb{Z}_2 $ $ \mathbb{Z}_2 \times \mathbb{Z}_4$ $\begin{array}{l} b_2 \rightarrow a_1 b_1 \\ c_2 \rightarrow -\frac{1}{16} (a_1-4b_1)^2 \end{array}$
: Chain of torsion enhancements and their explicit globally defined Weierstrass tunings.[]{data-label="fig:torsiontuning"}
Models with $\mathbb{Z}_2$, $\mathbb{Z}_3$, and $\mathbb{Z}_2\times\mathbb{Z}_2$ torsion admit a tuning $f\equiv0$ or $g\equiv0$ while preserving the torsion points. Those configurations have a constant $J$-invariant and are hence strongly coupled. The allowed tunings are $$\begin{aligned}
\renewcommand{{1.3}}{1.3}
\begin{array}{|c|c|c|c|c|}\hline
\text{Factor} & \text{Tuning} &f & g & \Delta \\ \hline
\mathbb{Z}_2 & a_4 = a_2^2/3& 0& -(a_2^3/27)& a_2^6/27 \\ \hline
\mathbb{Z}_2 & a_4 = (2 a_2^2)/9 & -(a_2^2/9) & 0 & -((4 a_2^6)/729) \\ \hline \hline
\mathbb{Z}_2 \times \mathbb{Z}_2 & b_2= -e^{(2\pi i/3)} c_2 & 0& -i/(3\sqrt{3}) c_2^3 & -c_2^6 \\ \hline
\mathbb{Z}_2 \times \mathbb{Z}_2 & b_2 = -c_2 & -c_2^2 & 0 & -4 c_2^6 \\ \hline \hline
\mathbb{Z}_3 & a_1 = 0 & 0 & a_3^2/4 & (27 a_3^4)/16 \\ \hline
\end{array}\end{aligned}$$ Note that the models in line $1$ and $3$ become identical, as do the models in line $2$ and $4$.
The $\mathbb{Z}_3$ model has a type $IV$ fiber with vanishing orders $(\infty,2,4)$, whereas all other models have a type $I_0^{*,s}$ fiber. The latter are very restricted and only allow for very few further specializations. These include $\mathfrak{so}_8 \times \mathfrak{so}_8 $ collisions which can be resolved by a simple $(-1)$ curve, or by having a more general polynomial Ansatz $$\begin{aligned}
a_2=(u-\kappa_1 v)(u-\kappa_2 v)(u-\kappa_3 v)\,, \end{aligned}$$ with $\kappa\in\mathbb{C}$. This geometry has an $\mathfrak{so}_8^3$ flavor algebra. Resolving the collision requires three exceptional divisors resulting in $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{Outlierso8.pdf} \end{array}\end{aligned}$$
The $\mathbb{Z}_3$ case allows the collision of even more components of the discriminant locus at the origin. The maximal singularity is achieved for the collision of five type $IV$ fibers, for $$\begin{aligned}
\label{eq:Z3IVtunings}
a_3 = \prod_i^5 (u-\kappa_i v)\,. \end{aligned}$$ The first blow-up yields an $\mathfrak{e}_6$ gauge factor. A smooth base requires five more blow-ups and the final configuration is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{Outlierdevil.pdf} \end{array}\end{aligned}$$ Alternatively, one can replace two of the $\mathfrak{su}_{3,IV}$ with one $\mathfrak{e}_6$ flavor algebra. Technically this is done by setting some of the parameters $\kappa_i$ in to zero. Two possibles configurations can be generated in this way $$\begin{aligned}
\begin{array}{c c}
\includegraphics[scale=0.5]{Outliergod.pdf} & \includegraphics[scale=0.5]{Outlierzilla.pdf} \end{array}\end{aligned}$$ where the flavor symmetry contains one and two $\mathfrak{e}_6$ factors, respectively.
Outlier Theories {#app:outlier}
================
### $\mathbb{Z}_2$ outlier theories {#mathbbz_2-outlier-theories .unnumbered}
A simple class of outlier theories can be constructed by tuning the $I_1$ locus such that it has a double- or triple-point singularity at the origin. This increases a regular collision to a singularity with vanishing order $\geq(4,6,12)$ in codimension two and hence enhances singularities that would have had perturbative matter to require a blow-up at the origin and hence lead to superconformal matter. This has already been encountered in in Section \[sssec:Z3SCM\] for $\mathbb{Z}_3$ torsion models. Here, we use the tuning $a_4 = u^n v^m$ and $a_2 =\kappa$, which results in $\mathfrak{su}_{2n}\times \mathfrak{su}_{2m}$ flavor algebras with perturbative bi-fundamental matter at the origin. Setting $a_2 = \kappa_1 u+\kappa_2 v$, the $\mathfrak{su}_{2n}$ and $\mathfrak{su}_{2m}$ factors reduce to $\mathfrak{sp}_{n} \times \mathfrak{sp}_m$ and their intersection is of $\mathfrak{so}$ type i.e. $(2,3,2(n+m+1))$; this still results in perturbative matter. However, setting $a_2$ to a generic quadratic polynomial, $a_2 = \kappa_1 u^2+\kappa_2 v^2+\kappa_3 u v$, leads back to an $\mathfrak{su}_n$ flavor group, but with a non-minimal collision. The tensor branch is given as $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SU_SU_Z2.pdf}\end{array}\end{aligned}$$ Notably, if $n+m=4$ the $(-1)$ curve is has no gauge algebra and we obtain a discrete holonomy instanton theory. Similarly, if we choose $a_2= \kappa_1 u^3+\kappa_2 v^3 + \kappa_3 u v^2 + \kappa_4 u^2 v$, the flavor factors become $\mathfrak{sp}_n$ factors again, but with a non-minimal collision giving rise to $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_Sp_Sp_Z2.pdf}\end{array}\end{aligned}$$ in cases with $n+m \geq7$. For the lower cases we find $$\begin{aligned}
\begin{array}{|c|c|}\hline
n+m & $gauge symmetry$ \\ \hline
\leq 3 & \text{no blowup}\\
4 & \emptyset \\
5 & \mathfrak{su}_{2,III} \\
6 & \mathfrak{so}_7 \\ \hline
\end{array}\end{aligned}$$ In the following we continue the systematic enhancement of E-string theories. They all feature an additional monodromy that folds the outer flavor group to a non-simply laced version. We start with E-string theory 8 of Table with flavor group $\text{Sp}(4)/\mathbb{Z}_2$ and enhance it by setting $a_2=u^3$ and $a_4 = v^n$. For $n\in\{4,5,6,7\}$, this leads to the following tensor branches: $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outliersp.pdf} \end{array}\end{aligned}$$ The first case is the discrete holonomy instanton theory with no gauge group. In the $\mathfrak{sp}_5$ case there are bi-fundamentals of the $\mathfrak{su}_2$. Also note that one would expect an additional $\mathfrak{sp}_2$ flavor factor [@Bertolini:2015bwa], which we do not see geometrically realized. There are two singular $I_1$ components thought that intersect the $(-1)$ curve. In the last case, we find bi-fundamental half-hypermultiplets in the $(\mathbf{12,14})$ representation, but in addition we expect three half-hypers in the spinor representation of $\mathfrak{so}_{12}$. These arise from an $\mathfrak{e}_7$ enhancement of multiplicity three at the codimension two locus $u=e_1=0$. Note that this is consistent if one assumes that the gauge group is Spin$(12)/\mathbb{Z}_2$ where the $\mathbb{Z}_2$ is identified with the $\mathbb{Z}_2$ factor in the $\mathbb{Z}_2\times\mathbb{Z}_2$ center of Spin$(12)$ that acts trivially on the spinor representation.
The chain above can in fact be increased up to $\mathfrak{sp}_{11}$ factors by increasing $n$ up to $11$. However, at this point, the chain starts introducing multiple gauge factors: $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outliersp8.pdf} \end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outliersp9.pdf} \end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outliersp10.pdf} \end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outliersp11.pdf} \end{array}\end{aligned}$$
We continue with the discrete holonomy instanton theory 9 with $[\text{SU}(2)_{III}\times \text{Sp}(3)]/\mathbb{Z}_2$ flavor group and continue enhancing the symplectic factor to an $\mathfrak{sp}_{3+n}$ by setting $a_2 = u^3$ and $a_4 = u v^{3+n}$. At the same time the $I_1$ component in the discriminant, which is of the form $I_1 = u^5+v^{3+2n}$, enhances as well. For $n>6$, one finds a non-crepantly resolvable singularity. Each chain is slightly different, given by the following tensor branches $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SU_Sp_Z2I.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SU_Sp_Z2II.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SU_Sp_Z2III.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SU_Sp_Z2IV.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SU_Sp_Z2V.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{Outlier_SU_Sp_Z2VI.pdf}\end{array}\end{aligned}$$
Next, we reconsider the discrete holonomy instanton theory 1 with $[$E$_7 \times $SU$(2)]/\mathbb{Z}_2$ flavor symmetry. For those theories we can enhance the $\mathfrak{su}_{2}$ flavor factor further to $\mathfrak{sp}_n$ by setting $a_2 = u ^3$ and $a_4= u^3 v^n$ with $n \leq 5$ to avoid non-minimal singularities, $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_E7_Sp_1.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_E7_Sp_2.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_E7_Sp_3.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_E7_Sp_4.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{Outlier_E7_Sp_5.pdf}\end{array}\end{aligned}$$
Let us next illustrate the importance of the exact form of the $I_1$ locus for the tensor branch. For that consider the family of models with tuning $a_2 = u ^{3+2n}$ and $a_4= u^3 v^3$ with $n=0,1,2$. Those three theories all have an $\mathfrak{e}_7$ collision with an $\mathfrak{sp}_3$ flavor brane, but all lead to different tensor branches: $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E7_Sp3_1_Z2.pdf} \end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E7_Sp3_2_Z2.pdf} \end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{E7_Sp3_3_Z2.pdf} \end{array}\end{aligned}$$
Finally, we consider the discrete holonomy instanton theory 10 with $[\text{Spin}(8) \times \text{Sp}(2)]/\mathbb{Z}_2$ flavor group, given by the factorization $a_2 = u^3$ and $a_4 = u^2 v^2$, and enhance the $\text{Sp}(2)$ side to $\text{Sp}(n)$ by setting $a_4 = u^2 v^{n}$. This chain is bounded by $n \leq 7$ in order to admit crepant resolutions. The flavor algebra further depends on whether $n$ is even or odd. For $n$ even, the flavor $\mathfrak{so}_7$ algebra enhances to an $\mathfrak{so}_8$ factor, leading to the chains $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SO8_Sp4.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SO8_Sp6.pdf}\end{array}\end{aligned}$$
For $n$ odd, we get the three theories $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SO7_Sp3.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_SO7_Sp5.pdf}\end{array}\end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.5]{Outlier_SO7_Sp7.pdf}\end{array}\end{aligned}$$
$\mathbb{Z}_3$ outlier theories {#mathbbz_3-outlier-theories .unnumbered}
-------------------------------
In this section we study the factorization of an $\mathfrak{su}_{3n}$ flavor algebra into three different pieces. This is enforced by setting $$\begin{aligned}
a_1 \rightarrow u\,,~a_3 \rightarrow (u+v)^k (u-v)^l (c_1 u+c_2 v)^m\,. \end{aligned}$$ The model is an $[\text{SU}(3k)\times \text{SU}(3l) \times \text{SU}(3m)] /\mathbb{Z}_3$ theory with all groups intersecting at the origin. After blowing up in the base, we get an $\mathfrak{su}_{3(k+l+m-3)}$ gauge symmetry over the $(-1)$ curve and bi-fundamental matter at the intersections with the three flavor branes. $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{SUOutlier.pdf} \end{array}\end{aligned}$$ In case that $k+l+m=2$, there is an $\mathfrak{su}_6$ gauge algebra found on the $(-1)$ curve that introduces an additional half-hyper in the triple anti-symmetric representation. If $k+l+m>3$, we need to perform additional blow-ups until we end on a $(-1)$ curve with an $\mathfrak{su}_{3 (k+l+m \text{ mod } 3)}$ factor: $$\begin{aligned}
\label{eq:SUOutlierZ3}
\begin{array}{c}
\includegraphics[scale=0.7]{SUOutlierZ3.pdf} \end{array}\end{aligned}$$ As before, the above theory has a similar tensor branch structure as e.g. and . Therefore, we might expect that theory can be understood as a deformation of these theories.
$\mathbb{Z}_4$ outlier theories {#mathbbz_4-outlier-theories .unnumbered}
-------------------------------
In this section we consider theories of type $[\text{Spin}(10+4n) \times \text{SU}(4)]/\mathbb{Z}_4$ and enhance the $\mathfrak{su}_4$ to $\mathfrak{su}_8$ via the factorization $a_1 \rightarrow u^{1+n}\,,~ a_2 \rightarrow u v^2 $. This modifies the tensor branch to $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{OutlierZ4.pdf} \end{array}\end{aligned}$$
Note that if $n>1$, one cannot enhance the $\mathfrak{su}_4$ higher than $\mathfrak{su}_8$, since one would encounters $(8,12,24)$ singularities in codimension two. The structure of the tensor branch includes the superconformal matter theories of $\mathfrak{so}_{10+4m}\times \mathfrak{so}_{10+4n}$ type, as well as the discrete holonomy instanton theory of type $[\text{Spin}(10)\times \text{SU}(4)]/\mathbb{Z}_4$ at the end. Note that the above theory can also be enhanced to $[\text{Spin}(10) \times \text{SU}(12) ]/\mathbb{Z}_4$ (but not further) by setting $a_1 = u^1$ and $a_2 = u v^3$, with the tensor branch $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{OutlierZ4mod.pdf} \end{array}\end{aligned}$$ Finally, we can turn the $\mathfrak{so}_{10}$ flavor algebra into $\mathfrak{sp}_2$ by setting $a_1 = u^2$ and $a_2=v^2$. The tensor branch then reads $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{OutlierZ4alt.pdf} \end{array}
\end{aligned}$$ Any further enhancement on either the $\mathfrak{sp}_2$ or $\mathfrak{su}_{12}$ side would lead to non-crepantly resolvable singularities.
$\mathbb{Z}_2 \times \mathbb{Z}_2 $ outlier theories {#mathbbz_2-times-mathbbz_2-outlier-theories .unnumbered}
----------------------------------------------------
We start by enhancing $\mathfrak{so}_{12}$ to $\mathfrak{so}_{16}$ in the discrete holonomy instanton theory 2 by setting $b_1 = u v$ and $c_1=v^3$. This leads to the tensor branch $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_1Z2Z2.pdf} \end{array} \end{aligned}$$ When one of the $\mathfrak{sp}$ flavor factor is further enhanced, we obtain $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_2Z2Z2.pdf} \end{array} \end{aligned}$$ as well as $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_3Z2Z2.pdf} \end{array} \end{aligned}$$ For tunings of the type $b_2 =u^n$ and $c_2 = v^m$ there are a couple of models that have not yet been discussed for values $n+m<9$ and $n>2$. In the following, we fix $n=3$. For $m=3$, we find $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_4Z2Z2.pdf} \end{array} \end{aligned}$$ The model has the special property that its $\mathfrak{sp}_1$ locus is of the form $z=u^3 - v^3$, i.e. it is reducible and can be reduced into three different pieces. Upon blow-up, all components intersect the $(-1)$ curve that hosts the $\mathfrak{so}_8$ at different points. This contributes the three spinor, vector and co-spinor representations required from anomaly cancellation. The flavor symmetry is the same as the one found in [@Bertolini:2015bwa].
For $m =4,5$ we obtain the following two theories $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_5Z2Z2.pdf} \end{array} \end{aligned}$$ and $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Outlier_6Z2Z2_mod.pdf} \end{array} \end{aligned}$$
$\mathbb{Z}_2 \times \mathbb{Z}_4$ outlier theories {#mathbbz_2-times-mathbbz_4-outlier-theories .unnumbered}
---------------------------------------------------
The outlier theories we consider here start from the $[\text{SU}(4)^2 \times \text{SU}(2)^2]/[\mathbb{Z}_2 \times \mathbb{Z}_4]$ theory. We enhance the flavor $\mathfrak{su}_{2n}$ algebra to $\mathfrak{su}_{2n} \times \mathfrak{su}_{2m}$ by setting $$\begin{aligned}
a_1 \rightarrow (u+ v)^n ( u-v)^m \, , \quad b_1 \rightarrow v \, .\end{aligned}$$ Hence, there are five flavor factors in total, all intersecting at the origin. The resulting tensor branch is $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{OutlierZ4Z2.pdf} \end{array} \end{aligned}$$ A similar factorization can be done for the $\mathfrak{su}_4$ part, leading to a similar chain but with jumps by multiples of four in the rank of the gauge group on the tensor branch. Again, we note that the above tuning looks very similar to the theories constructed from an $[\text{SU}(4)^2 \times \text{SU}(2) \times \text{SU}(2k)]/[\mathbb{Z}_2 \times \mathbb{Z}_4]$ theory with $k=n+m$ as shown in . Hence, its is conceivable that these theories arise from a deformation of the theory with $\mathfrak{su}_{2(n+m)}$ flavor algebra.
$\mathbb{Z}_3 \times \mathbb{Z}_3$ outlier theories {#mathbbz_3-times-mathbbz_3-outlier-theories .unnumbered}
---------------------------------------------------
The only simple outlier theories we can construct here come from further factorizing the components of the $\mathfrak{su_{3n}}$ loci as $$\begin{aligned}
a_1 = u \, , \quad b_1 = (u- v)^m (u- \omega v)^n (u- \omega^2 v)^k (u+ v)^l \,,\end{aligned}$$ which results in splitting the toric $\mathfrak{su}_{3(m+n+k+l)}$ locus into four individual blocks that all intersect at the origin. This is reflected in the tensor branch, where all four loci intersect the first resolution divisor, while the other three $\mathfrak{su}_3$ factors sit at the final $(-1)$ curve: $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{OutlierZ3Z3.pdf} \end{array} \end{aligned}$$ The picture is very similar to the cases we had e.g. in the $\mathbb{Z}_2 \times \mathbb{Z}_4$ case in the section before. Hence also in this case we might expect the four flavor factors on the right to arise as a deformation of an $\mathfrak{su}_{3(n+m+l+l)}$ theory.
Higher Order Torsion {#sec:ExoticTorsion}
====================
Throughout the paper we have worked with Weierstrass models whose torsion groups can appear in global models. However, it is also possible to construct higher order torsion models which are only consistent with non-compact 3-fold geometries. We construct one such quiver-like gauge theory with higher torsion that can flow to a non-trivial SCFT in the IR. For this we use a $\mathbb{Z}_7$ model over $\mathbb{F}_0 \equiv \mathbb{P}^1_s \times \mathbb{P}^1_t$ [@Hajouji:2019vxs], given by $$\begin{aligned}
y^2& + (s_0 s_1 t_0 t_1 + (s_0 t_0 - s_1 t_1) (2 s_1 t_1-s_0t_0)) xy + s_0 s_1^3 t_0 t_1^3 (s_0 t_0 - 2 s_1 t_1) (s_1 t_1 - s_0 t_0) y \, \nonumber \\
&= x^3 + s_1^2 t_1^2 (s_0 t_0 - 2 s_1 t_1) (s_1 t_1 - s_0 t_0)x^2 \, .\end{aligned}$$ The model has two $(8,12,24)$ singularities over the loci $s_0 =t_1 =0$ and $s_1 = t_0 =0$. However, we ignore these points for now, decompactify the two $\mathbb{P}^1$ factors to $\mathbb{C}^2$, and set the coordinates $s_1$ and $t_1$ to one. The discriminant of the Weierstrass model then becomes $$\begin{aligned}
\Delta=\tfrac16 s_0^7 t_0^7 ( s_0 t_0-1)^7 (1 - 8 s_0 t_0 + 5 s_0^2 t_0^2 + s_0^3 t_0^3) \, .\end{aligned}$$ Note that the $(8,12,24)$ singularities have been pushed off to infinity and the resulting model is crepantly resolvable. Moreover, since the local coordinates always appear in the combination $s_0 t_0$, we can add to this model a $\mathbb{Z}_n$ singularity at the origin via the action $$\begin{aligned}
(s_0,~t_0)~\to~(e^{2\pi i/n}s_0,~e^{-2\pi i/n}t_0)\,.\end{aligned}$$ Resolving the geometry with a chain of $n-1$ exceptional divisors $\{e_i =0\}$ leads to the discriminant $$\begin{aligned}
\Delta=\tfrac16 e^7 s_0^7 t_0^7 \big(s_0 t_0 e-1\big)^7 \big(1- 8 s_0 t_0 e + 5 s_0^2 t_0^2 e^2 + s_0^3 t_0^3 e^3\big) \, .\end{aligned}$$ with $e=\prod_i^{n-1}e_i$. The tensor branch is given by $$\begin{aligned}
\begin{array}{c}
\includegraphics[scale=0.7]{Z7_model.pdf}\end{array}\end{aligned}$$ with bi-fundamental matter between adjacent symmetry group factors.
[^1]: Note that F-theory in general only leads to a subgroup of the maximal flavor algebra, cf. [@Bertolini:2015bwa].
[^2]: Moreover, a conjecture was made in [@Klevers:2014bqa; @Oehlmann:2016wsb], that fibrations with torsion are related to genus-one fibrations via mirror duality in the fiber.
[^3]: This is also true for higher-dimensional Calabi-Yau manifolds, but elliptically fibered K3’s admit more freedom [@Hajouji:2019vxs].
[^4]: Note that the action of $T$ on the center of some of the factors in $\mathcal{G}^*$ might be trivial, see e.g. [@Baume:2017hxm].
[^5]: Hence the name superconformal matter as a non-perturbative generalization of bi-fundamental matter.
[^6]: Another alternative is a non-flat resolution of the fiber which encodes part of the tensor branch structure [@Buchmuller:2017wpe; @Dierigl:2018nlv; @Paul-KonstantinOehlmann:2019jgr] and can be used to explore 5D SCFTs via the M-theory duality [@Apruzzi:2018nre; @Apruzzi:2019vpe; @Apruzzi:2019opn; @Apruzzi:2019enx].
[^7]: To obtain the breaking pattern one can delete the root with Dynkin index $5$ from the affine Dynkin diagram of $\text{E}_8$. This extends to other breaking patterns with other discrete groups.
[^8]: The subscripts $F$ and $H$ are used to distinguish the two $\mathbb{P}^1$s, which we call [@Braun:2018ovc] F-theory $\mathbb{P}_F^1$ and heterotic $\mathbb{P}_H^1$.
[^9]: Note that this is different from the type II limit, where the elliptic fiber is not part of geometry, but the whole base is.
[^10]: The squares indicate flavor symmetries and the the circles with number $m$ encode compact curves with self-intersection $(-m)$ and mutual intersection according to the picture. If the fiber becomes singular over the compact curves, we further give the corresponding gauge algebra above the circle.
[^11]: This can be computed using $C\cdot (K + C) = 2 g - 2$ for the genus $g$ of a curve $C$.
[^12]: The starting configuration is a single curve with negative self-intersection. Several of these models, however, require further blow-ups leading to additional compact curves.
[^13]: In order to distinguish the $\mathfrak{su}_2$ algebras arising from type $I_2$ and type $III$, singularities, we add a subscript for the latter. We proceed similarly for the $\mathfrak{su}_{3}$ of type $I_3$ vs type $IV$.
[^14]: Notably, the Sp group on the tensor branch does not have a $\mathbb{Z}_4$ center. As we shall see later, often only a subgroup of the full torsion group is modded out.
[^15]: This can be seen from the decomposition: $\mathfrak{e}_7 \rightarrow \mathfrak{su}_3 \times \mathfrak{su}_6$, where the adjoint is decomposed as $\mathbf{133}\rightarrow (\mathbf{8},\mathbf{1}) + (\mathbf{1},\mathbf{35}) + (\boldsymbol{\overline{3}},\boldsymbol{15}) + (\boldsymbol{3},\boldsymbol{\overline{15}})$.
[^16]: Lines connecting the nodes are only used in order to unclutter the notation and do not indicate any special kind of matter.
[^17]: As mentioned in Section \[sec:rev\], the group structure is sometimes not uniquely specified by the indicated quotients. In these cases, the group action of $T$ can be deduced from the matter states that are present on the tensor branch of the theories.
[^18]: Indeed, by shifting the coordinates as $ \widetilde{u} = 2 (u - v), ~ \widetilde{v} = \frac12 (u + v)$, one obtains the same model from the $\mathbb{Z}_2 \times \mathbb{Z}_4$ torsion model, cf. .
[^19]: In contrast to the torsion models, progenitor theories have an $\text{E}_8$ flavor group factor with trivial center.
|
=0em =0em =0em =0em =0em @addto@macro
1em 0em 0.3em
|
---
abstract: 'Hundreds of brown dwarfs (BDs) have been discovered in the last few years in stellar clusters and among field stars. BDs are almost as numerous as hydrogen burning stars and so a theory of star formation should also explain their origin. The “mystery” of the origin of BDs is that their mass is two orders of magnitude smaller than the average Jeans’ mass in star–forming clouds, and yet they are so common. In this work we investigate the possibility that gravitationally unstable protostellar cores of BD mass are formed directly by the process of turbulent fragmentation. Supersonic turbulence in molecular clouds generates a complex density field with a very large density contrast. As a result, a fraction of BD mass cores formed by the turbulent flow are dense enough to be gravitationally unstable. We find that with density, temperature and rms Mach number typical of cluster–forming regions, turbulent fragmentation can account for the observed BD abundance.'
author:
- 'Paolo Padoan[^1], Åke Nordlund[^2]'
nocite: '[@Larson81; @Falgarone+92]'
title: 'The “Mysterious” Origin of Brown Dwarfs'
---
Introduction
============
A large number of Brown Dwarfs (BDs) have been discovered in the last few years, both within stellar clusters and among field stars [e.g. @Kirkpatrick+99; @Kirkpatrick+2000; @Chabrier2002]. It is now well established that BDs do not hide a significant amount of baryonic dark matter, at least in our galaxy [e.g. @Najita+2000; @Bejar+2001; @Chabrier2002]. The stellar initial mass function (IMF) is flat or decreasing toward sub–stellar masses.
Even if the total mass of BDs is not dynamically important, their abundance relative to hydrogen burning stars is so large that their existence cannot be overlooked in the context of a theory of star formation. According to cluster and field IMFs extended to sub–stellar masses, there are almost as many BDs as regular stars [e.g. @Bejar+2001; @Chabrier2002; @Chabrier2003]. BDs may be relevant also for understanding the formation of planets, because their mass is intermediate between that of hydrogen burning stars ($M>0.07$–$0.08$ M$_{\odot}$) and that of planets ($M<0.011$–$0.013$ M$_{\odot}$, using the deuterium burning limit to separate planets from BDs).
Surprisingly, very little theoretical research has addressed the problem of the formation of BDs. Most theoretical research on BDs is concerned with modeling the structure and evolution of sub–stellar objects and their atmospheres, in order to derive their observational properties [e.g. @Chabrier+2000; @Chabrier+Baraffe2000; @Baraffe+2002]
The classical Jeans’ mass [@Jeans02] at the mean density and temperature of a typical star–forming cloud is several solar masses. Based on the usual assumption that the Jeans’ mass is an approximate estimate of the lower limit to the stellar mass [@Larson92; @Elmegreen99], the formation of BDs is an unsolved problem.
Recent observational results have shown that the mass distribution of prestellar condensations is indistinguishable from the stellar IMF , both in the functional shape and in the range of masses, including BD mass cores [@Walsh+04]. The problem of the formation of BDs in molecular clouds with a Jeans’ mass much larger than the mass of BDs is thus unlikely to be solved by relying on a significant mass difference between a single collapsing core and its final star.
[@Elmegreen99] stressed the importance of BDs for testing star formation theories. He proposed that BDs are more abundant in ultra-cold regions in the inner disk of M31 or in spiral–arm dust lanes than in normal star–forming clouds. He assumed that the smallest stellar mass is of the order of the thermal Jeans’ mass and therefore the BD abundance should increase with decreasing gas temperature or increasing pressure. His argument is consistent with the present work, where we focus on more typical regions of star formation.
[@Reipurth+Clarke2001] proposed that BDs are the leftover of a prematurely interrupted accretion process. Their model assumes that stars are born as multiple systems of small “embryos” sharing a common reservoir of accreting gas. The accretion process of one of them is “aborted” when it is ejected by gravitational interaction with a pair of companions. One problem with this model is that protostellar disks are frequently found around BDs [@Natta+Testi2001; @Natta+2002; @Liu2003; @Jayawardhana+2003; @Jayawardhana2003a; @Klein+2003; @Martin+2004; @LopezMarti+2004; @Mohanty+2004]. Other problems posed by observational data are discussed in [@Briceno+2002].
[@Bate+2002] have interpreted the results of a numerical simulation as evidence in favor of the [@Reipurth+Clarke2001] model. They simulated the collapse of a 50 M$_{\odot}$ isothermal gas cloud, with an initial random velocity field and uniform density, using a smooth particle hydrodynamic (SPH) code. They found that BDs are formed mainly as members of multiple systems by the gravitational fragmentation of a common protostellar disk. In their simulation, the initial velocity field is not obtained as the solution of the fluid equations self–consistently with the density field. It is instead generated artificially by imposing a power spectrum consistent with Larson’s velocity-size correlation [@Larson79; @Larson81], while the initial density is uniform. As the turbulence is not externally driven, it decays while the cloud collapses and there is no time for the flow to rearrange as a realistic supersonic turbulent flow independent of the initial conditions. Another problem is the absence of a magnetic field, which affects significantly the fragmentation process as the magnetic field may modify the shock jump conditions and the angular momentum transport.
Given the numerical limitations in the simulation by [@Bate+2002], it is possible that turbulent fragmentation has been overlooked as the origin of collapsing cores of BD mass. In the present work we investigate the possibility that BDs are the direct consequence of turbulent fragmentation , in the sense that they are assembled by the turbulent flow as gravitationally unstable objects. We first compute the upper limit to the BD abundance in § 2, based on the probability density function (PDF) of gas density in supersonic turbulence. In § 3 we compute the actual BD abundance, according to our analytical model of the stellar IMF, and in § 4 we show that numerical simulations of supersonic magneto–hydrodynamic (MHD) turbulence provide support for our analytical model. Results are summarized and conclusions drawn in § 5.
An Upper Limit to the BD Abundance
==================================
The gas density and velocity fields in star–forming clouds are highly non–linear due to the presence of supersonic turbulence. The kinetic energy of turbulence is typically 100 times larger than the gas thermal energy on the scale of a few pc (the typical rms Mach number is of the order of 10) and the gas is roughly isothermal, so that very large compressions due to a complex network of interacting shocks cannot be avoided. Under such conditions the concept of gravitational instability, based on a comparison between gravitational and thermal energies alone in a system with mild perturbations, does not apply. Dense cores of any size can be formed in the turbulent flow, independent of the Jeans’ mass. Those cores that are massive and dense enough (larger than their own Jeans’ mass) collapse into protostars, while smaller subcritical ones re-expand into the turbulent flow. This is a process that we call [*turbulent fragmentation*]{}, to stress the point that stars and BDs, formed in supersonically turbulent clouds, are not primarily the result of gravitational fragmentation.
Nevertheless, the idea of a critical mass for gravitational collapse is often applied to star forming clouds. The critical mass is defined as the Jeans’ or the Bonnor–Ebert mass [@Bonnor56; @McCrea57] computed with the average density, temperature and pressure. The effect of the kinetic energy of turbulence is modeled as an external pressure, $P_t\sim\rho \sigma_{\rm v}^2$, where $\rho$ is the gas density and $\sigma_{\rm v}$ is the rms velocity of the turbulence. [@Elmegreen99], for example, has proposed that the minimum stellar mass is of the order of this Bonnor–Ebert mass. The minimum stellar mass in typical star–forming clouds is then several M$_{\odot}$, which contradicts the relatively large abundance of BDs.
Dense cores formed by the turbulent flow need to be larger than their thermal critical mass to collapse (neglecting the magnetic field). Therefore, a necessary condition for the formation of BDs by supersonic turbulence is the existence of a finite mass fraction, in the turbulent flow, with density at least as high as the critical one for the collapse of a BD mass core. As an estimate of the critical mass we use the Bonnor–Ebert mass, instead of the classical Jeans’ mass, because the cores assembled by the turbulent flow are non–linear density enhancements bounded by the shock ram pressure, rather than linear density perturbations as in Jeans’ assumption. The mass, $m_{\rm BE}$, of the critical Bonnor–Ebert isothermal sphere is [@Bonnor56]: $$m_{\rm BE}=3.3\, {\rm M}_{\odot}\left(\frac{T}{10\, K}\right)^{3/2}
\left(\frac{n}{10^3\, cm^{-3}}\right)^{-1/2},
\label{mj0}$$ We have verified numerically that this expression is a good estimate of the critical mass, both in simple geometries and in turbulence simulations including selfgravity.
We want to estimate the mass fraction with density at least as high as the critical one for the collapse of a BD mass core. Using the value $m_{\rm BD}=0.075$ M$_{\odot}$ for the largest BD mass, the condition $m_{\rm BE}<m_{\rm BD}=0.075$ M$_{\odot}$ corresponds to $n > n_{\rm BD}=1.8\times 10^6$ cm$^{-3}$, using equation (\[mj0\]). The fraction of the total mass of the system that can form BDs is given by the probability that $n > n_{BD}$. This is $\int_{n_{\rm BD}}^{\infty} p(n)\,dn$, where $p(n)$ is the probability density function (PDF) of the gas density $n$. This probability can be interpreted as a fractional volume and therefore the mass fraction available for the formation of BDs is: $$f_{\rm BD}={\int_{n_{\rm BD}}^{\infty} n\,p(n)\,dn \over \int_{0}^{\infty} n\,p(n)\,dn}
\label{fBD}$$ It is important to stress that $f_{\rm BD}$ is [*not*]{} the BD mass fraction, but only the mass fraction of the gas that [*could*]{} end up in BDs, or the upper limit for the BD mass fraction (assuming BDs are due only to turbulent fragmentation). Most likely the star formation process is rather inefficient for BDs as it is for hydrogen burning stars, and so the actual fractional mass in BDs may be significantly smaller than $f_{BD}$.
One of the most important universal properties of turbulent fragmentation is that the PDF of gas density is Log–Normal for an isothermal gas, so that $$p(n)dn=\frac{1/n}{(2\pi\sigma^{2})^{1/2}}exp\left[-\frac{1}{2}
\left(\frac{\ln n-\overline{\ln n}}{\sigma}\right)^{2}\right]dn,
\label{pdf1}$$ for the case of $\langle n\rangle=1$. The average value of the logarithm of density, $\overline{\ln n}$, is determined by the standard deviation $\sigma$ of the logarithm of density (a property of the Log–Normal, here used again for the case $\langle n\rangle=1$): $$\overline{\ln n}=-\frac{\sigma^{2}}{2} ,
\label{pdf3}$$ and the standard deviation of the logarithm of density, $\sigma$, is a function of the rms sonic Mach number of the flow, $M_{\rm S}$: $$\sigma^{2}=\ln(1+b^2 M_{\rm S}^2)
\label{pdf4}$$ or, equivalently, the standard deviation of the linear density is: $$\sigma_{\rho}=b M_{\rm S}
\label{pdf5}$$ where $b\approx0.5$ [@Nordlund+Padoan_Puebla98; @Ostriker+99]. This result is very useful because the rms Mach number of the turbulence in molecular clouds is easily estimated through the spectral line width of molecular transitions and by estimating the kinetic temperature. The three dimensional PDF of gas density is then fully determined by the value of the rms Mach number.
A contour plot of $f_{\rm BD}$, computed from (\[fBD\]) and with the Log–Normal PDF (\[pdf1\]), on the plane $\langle n \rangle$–$M_{\rm S}$, is shown in Figure \[fig1\]. The dotted line corresponds to values of rms Mach number and average gas density of typical Larson relations [@Larson81; @Brunt03; @Heyer+Brunt04]. Figure \[fig1\] shows that in regions following the average Larson relations, approximately 1% of the total mass is available for the formation of BDs (dotted line in Figure \[fig1\]). If most of the available 1% of the total mass were turned into BDs, the number of BDs would be comparable to the number of hydrogen burning stars even if as much as 10% of the total mass was turned into stars. However, it is possible that the formation of BDs has a typical efficiency of only a few percent, similar to that of hydrogen burning stars. In that case their relative abundance in molecular clouds following the average Larson relations is expected to be rather low.
For a given size and velocity dispersion, cluster–forming regions are a few times denser than clouds following the average density–size Larson relation. In cluster–forming regions, the mass available for the formation of BDs can be very large, $f_{\rm BD}\sim 0.1$. As an example, the stellar mass density in the central $5\times 5$ arcmin ($\approx 0.35\times0.35$ pc) of the young cluster IC 348 [@Luhman+2003] corresponds approximately to $2\times 10^4$ cm$^{-3}$. Because the star formation efficiency is likely to be less than unity, the initial gas density in that region must have been even larger. If we assume a gas density of $5\times 10^4$ cm$^{-3}$ (corresponding to a star formation efficiency of 40%) and a velocity dispersion taken from the average velocity–size Larson relation at a size equal to 0.35 pc, we get $f_{\rm BD}\approx 0.1$, as shown by the square in Figure \[fig1\]. Other cluster–forming regions, such as the central region of the Trapezium cluster [@Hillenbrand+Carpenter2000; @Luhman+2000], show similar stellar mass densities as IC 348. With such a large value of $f_{\rm BD}$, cluster–forming regions may in principle form as many BDs as hydrogen burning stars, even if the efficiency of BD formation (from the gas with density larger than $n_{\rm BD}$) is as low as the star formation efficiency for hydrogen burning stars.
The cross in the middle of Figure \[fig1\] shows the values of $\langle n \rangle$ and $M_{\rm S}$ corresponding to the initial conditions in the simulation by [@Bate+2002]. They simulate a cloud with a total mass of 50 M$_{\odot}$, a diameter of 0.375 pc, a mean molecular weight of the gas of 2.46, a temperature of 10 K and a kinetic energy of the turbulence equal to the cloud gravitational potential energy. From these initial conditions we obtain $\langle n \rangle=3.35\times 10^4$ cm$^{-3}$ and rms sonic Mach number of the turbulence $M_{\rm S}=6.5$. The contour plot indicates that 5% of the total mass is in this case available for the formation of BDs, sufficient to generate as many BDs as hydrogen burning stars, as discussed above. This suggests that under the conditions assumed in that simulation BDs could originate in large abundance as the result of the turbulent fragmentation.
The Brown Dwarf IMF from Turbulent Fragmentation
================================================
In order to provide a quantitative estimate of the BD abundance, a model for the structure of the density distribution is required. There has been significant progress in the analytical theory of supersonic turbulence in recent works [@Boldyrev2002; @Boldyrev+2002scaling; @Boldyrev+2002structure]. Some results regarding the scaling of structure functions of the density field are already available [@Boldyrev+2002structure; @Padoan+2002scaling] and could be used in the future for a rigorous analytical study of the process of turbulent fragmentation.
A simple model of the expected mass distribution of dense cores generated by supersonic turbulence has been proposed in [@Padoan+Nordlund2002IMF], on the basis of the two following assumptions: i) The power spectrum of the turbulence is a power law; ii) the typical size of a dense core scales as the thickness of the postshock gas. The first assumption is a basic result for turbulent flows and holds also in the supersonic regime [@Boldyrev+2002scaling]. The second assumption is suggested by the fact that postshock condensations are assembled by the turbulent flow in a dynamical time. Condensations of virtually any size can therefore be formed, independent of their Jeans’ mass.
With these assumptions, together with the jump conditions for MHD shocks (density contrast proportional to the Alfvénic Mach number of the shock), the mass distribution of dense cores can be related to the power spectrum of turbulent velocity, $E(k)\propto k^{-\beta}$. The result is the following expression for the core mass distribution: $$N(m)\,{\rm d}\ln m\propto m^{-3/(4-\beta)}{\rm d}\ln m ~.
\label{imf}$$ If the turbulence spectral index $\beta$ is taken from the analytical prediction [@Boldyrev+2002scaling], which is consistent with the observed velocity dispersion-size Larson relation [@Larson79; @Larson81] and with our numerical results [@Boldyrev+2002scaling], then $\beta \approx 1.74$ and the mass distribution is $$N(m)\,{\rm d}\ln m\propto m^{-1.33}{\rm d}\ln m ~,
\label{salpeter}$$ almost identical to the Salpeter stellar IMF [@Salpeter55]. The exponent of the mass distribution is rather well constrained, because the value of $\beta$ for supersonic turbulence cannot be smaller than the incompressible value, $\beta= 1.67$ (sligthly larger with intermittency corrections), and the Burgers case, $\beta=2.0$. As a result, the exponent of the mass distribution is predicted to be well within the range of values of 1.3 and 1.5. In the following we use $\beta=1.74$, corresponding to a core mass distribution $\propto m^{-1.36}$.
While massive cores are usually larger than their critical mass, $m_{\rm BE}$, the probability that small cores are dense enough to collapse is determined by the statistical distribution of core density. In order to compute this collapse probability for small cores, we assume i) the distribution of core density can be approximated by the Log–Normal PDF of gas density and ii) the core density and mass are statistically independent. Because of the intermittent nature of the Log-Normal PDF, even very small (sub–stellar) cores have a finite chance to be dense enough to collapse. Based on the first assumption, we can compute the distribution of the critical mass, $p(m_{\rm BE})\,d m_{\rm BE}$, from the Log–Normal PDF of gas density assuming constant temperature [@Padoan+97ext]. The fraction of cores of mass $m$ larger than their critical Bonnor–Ebert mass is given by the integral of $p(m_{\rm BE})$ from 0 to $m$. Using the second assumption of statistical independence of core density and mass, the mass distribution of collapsing cores is $$N(m)\, {\rm d}\ln m\propto m^{-3/(4-\beta)}\left[\int_0^m{p(m_{\rm BE}){\rm d}m_{\rm BE}}\right]\,{\rm d}\ln m ~.
\label{imfpdf}$$ The mass distribution is found to be a power law, determined by the power spectrum of turbulence, for masses larger than approximately 1 M$_{\odot}$ (using physical parameters typical of molecular clouds). At smaller masses the mass distribution flattens, reaches a maximum at a fraction of a solar mass, and then decreases with decreasing stellar mass.
The upper panel of Figure \[fig2\] shows five mass distributions computed from equation (\[imfpdf\]). Three of them (solid lines) are computed for $\langle n\rangle=10^4$ cm$^{-3}$, $T=10$ K and for three values of the sonic rms Mach number, $M_{\rm S}=5$, 10 and 20. An increase in the rms Mach number by a factor of two, from $M_{\rm S}=5$ to $M_{\rm S}=10$, results in a growth of the abundance of 0.07 M$_{\odot}$ stars by more than a factor of ten (relative to stars of approximately 1 M$_{\odot}$ or larger). From $M_{\rm S}=10$ to $M_{\rm S}=20$, the abundance of 0.07 M$_{\odot}$ stars increases by approximately a factor of three. The other two mass distributions (dotted lines) are computed for $M_{\rm S}=10$, $T=10$ K and density $\langle n\rangle=5\times 10^3$ cm$^{-3}$ (lower plot), and $\langle n\rangle=2\times 10^4$ cm$^{-3}$ (upper plot).
The IMF of the cluster IC 348 in Perseus, obtained by [@Luhman+2003], is plotted in the lower panel of Figure \[fig2\] (solid line histogram). The IMF of this cluster has been chosen for the comparison with the theoretical model because it is probably the most reliable observational IMF including both brown dwarfs and hydrogen burning stars. Spectroscopy has been obtained for every star and the sample is unbiased in mass and nearly complete down to 0.03 $M_{\odot}$. In the lower panel of Figure \[fig2\] we have also plotted the theoretical mass distribution computed for $\langle n\rangle=5\times 10^4$ cm$^{-3}$ , $T=10$ K and $M_{\rm S}=7$. As discussed above, these parameters are appropriate for the central $5\times 5$ arcmin of the cluster ($0.35\times 0.35$ pc), where the stellar density corresponds to approximately $2\times 10^4$ cm$^{-3}$. The figure shows that the theoretical distribution of collapsing cores, computed with parameters inferred from the observational data, is roughly consistent with the observed stellar IMF in the cluster IC 348.
Similar IMFs were obtained for the Trapezium cluster in Orion by [@Luhman+2000] and for the inner region of the Orion Nebula Cluster by [@Hillenbrand+Carpenter2000], using D’Antona and Mazzitelli’s 1997–evolutionary models. However, based on [@Baraffe+98] evolutionary models, these two IMFs contain a slightly larger abundance of brown dwarfs than found in IC 348 and predicted by the theoretical model (unless larger values of density or Mach number are assumed). A larger BD abundance is found in $\sigma$ Orionis by [@Bejar+2001], while the IMFs obtained by [@Najita+2000] for IC348 and the Pleiades’ IMF [@Bouvier+98] are consistent with the IMF in the Orion Nebula Cluster. Several other IMFs of young clusters, including both stellar and sub–stellar masses, have been recently obtained.
The present theoretical model may in some cases underestimate the BD abundance, if a significant fraction of BDs are formed as members of binary systems, because the process of binary formation is not taken into account. As an example, if most prestellar cores assembled by the turbulence were able to fragment into binary stars due to processes unrelated to turbulent fragmentation, the final BD abundance would be increased, while at larger masses the mass distribution would be indistinguishable from the one predicted by the model.
[@Luhman2000] found that the number of BDs in Taurus is 12.8 times lower than in the Trapezium cluster [@Luhman+2000]. This result was based on a single BD detection and on several low mass stars. The deficit of BDs in Taurus relative to the Trapezium cluster has been confirmed in a more recent work by [@Briceno+2002], although reduced to approximately a factor of two between the BD abundance of Orion and Taurus. The smaller relative abundance of BDs in Taurus may be explained by the analytical model as due to a decrease in the turbulent velocity dispersion (rms Mach number) or in the average gas density by less than a factor of two. This is consistent with the lower velocity dispersion and density in Taurus relative to Orion.
Numerical Results
=================
The mass distribution of prestellar condensations can be measured directly in numerical simulations of supersonic turbulence. With a mesh of 250$^3$ computational cells, and assuming a size of the simulated region of a few pc, it is not possible to follow numerically the gravitational collapse of individual protostellar condensations. However, dense cores at the verge of collapse can be selected in numerical simulations by an appropriate clumpfind algorithm. We use an algorithm that selects cores by scanning the full range of density levels. It eliminates large cores that are fragmented into smaller and denser ones. Cores are also excluded if their gravitational energy is not large enough to overcome thermal and magnetic support against the collapse, because only collapsing cores are selected.
A mass distribution of collapsing cores, derived from the density distribution in a numerical simulation is shown in Figure \[fig3\]. The mass distribution is computed from two snapshots of a 250$^3$ simulation with rms Mach number $M_{\rm S}\approx 10$. We have used a random external force on large scale and an isothermal equation of state (for details of the numerical method see [@Padoan+Nordlund99MHD] and references therein). The average gas density has been scaled to 500 cm$^{-3}$ and the size of the computational box to 10 pc. These values have been chosen to be able to select condensations in a range of masses from a sub–stellar mass to approximately 10 M$_{\odot}$. With this particular values of average gas density, size and resolution of the computational box, the smallest mass that can be achieved numerically is 0.057 M$_{\odot}$.
The analytical mass distribution, $N(m)$, computed with the same physical parameters used in the numerical simulation ($\langle n\rangle=500$ cm$^{-3}$, $T=10$ K and $M_{\rm S}=10$) is plotted in Figure \[fig3\] as a dashed line. There is no free parameter to adjust the shape of the analytical function and its mass scale, once the values of density, temperature, and rms Mach number have been specified to agree with those assumed in the numerical experiment. The agreement between the numerical and the analytical mass distributions provides strong support for our simple analytical model of the mass distribution of collapsing cores generated by supersonic turbulence.
Summary and Conclusions
=======================
In this work we have investigated the possibility that gravitationally unstable protostellar cores of BD mass are assembled by turbulent shocks. We have found that a fraction of BD mass cores formed by the turbulence are dense enough to collapse. The predicted BD abundance is consistent with the abundance observed in young stellar clusters if the theoretical IMF is computed with average density and rms sonic Mach number appropriate for dense cluster–forming regions inside molecular cloud complexes.
We have not studied the evolution of turbulent density fluctuations smaller than their critical mass. If subcritical fluctuations of BD mass are inside a larger collapsing core, they would be increasing their density as the background collapses. Additional fluctuations may also be created by turbulence during the collapse. Under appropriate conditions, a fraction of these fluctuations may be able to collapse into additional BDs or giant planets.
Future numerical simulations designed to study this process will require not only a very large dynamical range of scales, possibly achieved only by particle or adaptive mesh refinement codes, but also an accurate physical description of the supersonic turbulence including magnetic forces.
We are grateful to Kevin Luhman and Gilles Chabrier for valuable discussions on the stellar IMF in clusters and to Bo Reipurth for pointing out a numerical error in the definition of the critical mass. The work of [Å]{}N was supported by a grant from the Danish Natural Science Research Council. Computing resources were provided by the Danish Center for Scientific Computing.
[63]{} natexlab\#1[\#1]{}
, I., [Chabrier]{}, G., [Allard]{}, F., & [Hauschildt]{}, P. H. 1998, , 337, 403
—. 2002, , 382, 563
, M. R., [Bonnell]{}, I. A., & [Bromm]{}, V. 2002, MNRAS, 332, L65
, V. J. S., [Mart[' i]{}n]{}, E. L., [Zapatero Osorio]{}, M. R., [Rebolo]{}, R., [Barrado y Navascu[' e]{}s]{}, D., [Bailer-Jones]{}, C. A. L., [Mundt]{}, R., [Baraffe]{}, I., [Chabrier]{}, C., & [Allard]{}, F. 2001, , 556, 830
, S. 2002, ApJ, 569, 841
, S., [Nordlund]{}, [Å]{}., & [Padoan]{}, P. 2002, , 573, 678
—. 2002, Physical Review Letters, 89, 031102
, W. B. 1956, , 116, 351
, J., [Stauffer]{}, J. R., [Martin]{}, E. L., [Barrado y Navascues]{}, D., [Wallace]{}, B., & [Bejar]{}, V. J. S. 1998, , 336, 490
, C., [Luhman]{}, K. L., [Hartmann]{}, L., [Stauffer]{}, J. R., & [Kirkpatrick]{}, J. D. 2002, , 580, 317
, C. M. 2003, , 584, 293
, G. 2002, , 567, 304
—. 2003, , 115, 763
, G. & [Baraffe]{}, I. 2000, , 38, 337
, G., [Baraffe]{}, I., [Allard]{}, F., & [Hauschildt]{}, P. 2000, , 542, 464
, F., [Neuh[" a]{}user]{}, R., & [Kaas]{}, A. A. 2000, , 359, 269
, B. G. 1999, , 522, 915
Falgarone, E., Puget, J. L., & Pérault, M. 1992, A&A, 257, 715
, M. H. & [Brunt]{}, C. 2004, ApJ, submitted
, L. A. & [Carpenter]{}, J. M. 2000, , 540, 236
, R., [Ardila]{}, D. R., [Stelzer]{}, B., & [Haisch]{}, K. E. 2003, , 126, 1515
, R., [Mohanty]{}, S., & [Basri]{}, G. 2003, , 592, 282
Jeans, J. H. 1902, Phil. Trans. A, 199, 1
, D., [Fich]{}, M., [Mitchell]{}, G. F., & [Moriarty-Schieven]{}, G. 2001, , 559, 307
, D., [Wilson]{}, C. D., [Moriarty-Schieven]{}, G., [Giannakopoulou-Creighton]{}, J., & [Gregersen]{}, E. 2000, , 131, 505
, D., [Wilson]{}, C. D., [Moriarty-Schieven]{}, G., [Joncas]{}, G., [Smith]{}, G., [Gregersen]{}, E., & [Fich]{}, M. 2000, , 545, 327
, J. D., [Reid]{}, I. N., [Liebert]{}, J., [Cutri]{}, R. M., [Nelson]{}, B., [Beichman]{}, C. A., [Dahn]{}, C. C., [Monet]{}, D. G., [Gizis]{}, J. E., & [Skrutskie]{}, M. F. 1999, , 519, 802
, J. D., [Reid]{}, I. N., [Liebert]{}, J., [Gizis]{}, J. E., [Burgasser]{}, A. J., [Monet]{}, D. G., [Dahn]{}, C. C., [Nelson]{}, B., & [Williams]{}, R. J. 2000, , 120, 447
, R., [Apai]{}, D., [Pascucci]{}, I., [Henning]{}, T., & [Waters]{}, L. B. F. M. 2003, , 593, L57
, B., [Eisl[" o]{}ffel]{}, J., [Scholz]{}, A., & [Mundt]{}, R. 2004, , 416, 555
, R. B. 1979, , 186, 479
—. 1981, , 194, 809
Larson, R. B. 1992, MNRAS, 256, 641
, M. C. 2003, in Brown Dwarfs, IAU Symposium, Vol. 211, 2003, E. L. Martín, ed., (see also astro–ph/0207477)
, N., [Caux]{}, E., [Monin]{}, J.-L., & [Klotz]{}, A. 2002, , 383, L15
, K. L. 2000, , 544, 1044
, K. L., [Rieke]{}, G. H., [Young]{}, E. T., [Cotera]{}, A. S., [Chen]{}, H., [Rieke]{}, M. J., [Schneider]{}, G., & [Thompson]{}, R. I. 2000, , 540, 1016
, K. L., [Stauffer]{}, J. R., [Muench]{}, A. A., [Rieke]{}, G. H., [Lada]{}, E. A., [Bouvier]{}, J., & [Lada]{}, C. J. 2003, , 593, 1093
, E. L., [Basri]{}, G., [Zapatero-Osorio]{}, M. R., [Rebolo]{}, R., & [L[' o]{}pez]{}, R. J. G. . 1998, , 507, L41
, E. L., [Dougados]{}, C., [Magnier]{}, E., [M[' e]{}nard]{}, F., [Magazz[\` u]{}]{}, A., [Cuillandre]{}, J.-C., & [Delfosse]{}, X. 2001, , 561, L195
, E. L., [Delfosse]{}, X., & [Guieu]{}, S. 2004, , 127, 449
, W. H. 1957, , 117, 562
, S., [Jayawardhana]{}, R., [Natta]{}, A., [Fujiyoshi]{}, T., [Tamura]{}, M., & [Barrado y Navascu[' e]{}s]{}, D. 2004, , 609, L33
, E., [Bouvier]{}, J., & [Stauffer]{}, J. R. 2001, , 367, 211
, F., [Andr[' e]{}]{}, P., [Ward-Thompson]{}, D., & [Bontemps]{}, S. 2001, , 372, L41
, F., [André]{}, P., & [Neri]{}, R. 1998, A&A, 336, 150
, J. R., [Tiede]{}, G. P., & [Carr]{}, J. S. 2000, , 541, 977
, A. & [Testi]{}, L. 2001, , 376, L22
, A., [Testi]{}, L., [Comer[' o]{}n]{}, F., [Oliva]{}, E., [D’Antona]{}, F., [Baffa]{}, C., [Comoretto]{}, G., & [Gennari]{}, S. 2002, , 393, 597
Nordlund, [Å]{}. & Padoan, P. 1999, in Interstellar Turbulence, ed. J. Franco & A. Carrami[ñ]{}ana (Cambridge University Press), 218
Nordlund, A. & Paodan, P. 2002, in Simulations of magnetohydrodynamic turbulence in astrophysics: recent achievements and perspectives, ed. E. Falgarone & T. Passot, Lecture Notes in Physics (Springer), astro-ph/0209244
, T., [Mizuno]{}, A., [Kawamura]{}, A., & [Fukui]{}, Y. 1999, in Star Formation 1999, Proceedings of Star Formation 1999, held in Nagoya, Japan, June 21 - 25, 1999, Editor: T. Nakamoto, Nobeyama Radio Observatory, p. 153-158, 153–158
, T., [Mizuno]{}, A., [Kawamura]{}, A., [Tachihara]{}, K., & [Fukui]{}, Y. 2002, , 575, 950
, E. C., [Gammie]{}, C. F., & [Stone]{}, J. M. 1999, ApJ, 513, 259
, P., [Boldyrev]{}, S., [Langer]{}, W., & [Nordlund]{}, [Å]{}. 2002, in preparation
Padoan, P., Jones, B., & Nordlund, [Å]{}. 1997, ApJ, 474, 730
, P., [Juvela]{}, M., [Goodman]{}, A. A., & [Nordlund]{}, [Å]{}. 2001, ApJ, 553, 227
Padoan, P. & Nordlund, [Å]{}. 1999, ApJ, 526, 279
, P. & [Nordlund]{}, [Å]{}. 2002, , 576, 870
, B. & [Clarke]{}, C. 2001, , 122, 432
, E. E. 1955, , 121, 161
, L. & [Sargent]{}, A. I. 1998, ApJL, 508, L91
Walsh, A. J., Di Francesco, J., Myers, P. C., Bourke, T. L., & Wilner, D. J. 2004, “93 N2H+ Clumps in NGC 1333”, poster from “Cores, Disks, Jets and Outflows in Low and High Mass Star Forming Environments: Observations, Theory and Simulations”, Banff, Alberta, Canada, July 12-16, 2004
, M. R., [B[' e]{}jar]{}, V. J. S., [Mart[' i]{}n]{}, E. L., [Rebolo]{}, R., [Barrado y Navascu[' e]{}s]{}, D., [Bailer-Jones]{}, C. A. L., & [Mundt]{}, R. 2000, Science, 290, 103
[**Figure captions:**]{}\
[**Figure \[fig1\]:**]{} Contour plot of the fractional mass available for the formation of BDs, $f_{\rm BD}$, on the plane of gas density–rms Mach number, $\langle n \rangle$–$M_{\rm S}$. The dotted line corresponds to values of rms Mach number and average gas density of typical Larson relations. The cross corresponds to the initial conditions in the simulation by [@Bate+2002] and the square to the physical parameters appropriate for the central region of the young cluster IC 348 [@Luhman+2003].\
[**Figure \[fig2\]:**]{} Upper panel: Analytical mass distributions computed for $\langle n\rangle=10^4$ cm$^{-3}$, $T=10$ K and for three values of the sonic rms Mach number, $M_{\rm S}=5$, 10 and 20 (solid lines). The dotted lines show the mass distribution for $T=10$ K, $M_{\rm S}=10$ and $\langle n\rangle=5\times10^3$ cm$^{-3}$ (lower plot) and $\langle n\rangle=2\times10^4$ cm$^{-3}$ (upper plot). Lower panel: IMF of the cluster IC 348 in Perseus obtained by [@Luhman+2003] (solid line histogram) and theoretical IMF computed for $\langle n\rangle=5\times 10^4$ cm$^{-3}$, $T=10$ K and $M_{\rm S}=7$ (dashed line). The histogram of IC 348 mass function in [@Luhman+2003] is computed with 9 bins, while the histogram shown here is computed with 12 bins.\
[**Figure \[fig3\]:**]{} Solid line: Mass distribution of collapsing cores, derived from the density distribution of two snapshots of a 250$^3$ simulation with rms Mach number $M_{\rm S}\approx 10$, external random forcing on large scale and isothermal equation of state. The simulation is scaled to physical units assuming $\langle n\rangle=500$ cm$^{-3}$, $T=10$ K, and a mesh size of 10 pc. The fractional mass in collapsing cores is 5% of the total mass. Dashed line: Analytical mass distribution computed for $\langle n\rangle=500$ cm$^{-3}$, $T=10$ K and $M_{\rm S}=10$.\
[^1]: ppadoan@ucsd.edu
[^2]: aake@astro.ku.dk
|
---
abstract: |
Consider the high-order heat-type equation $\partial u/\partial
t=\pm\,\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$.
author:
- 'Valentina <span style="font-variant:small-caps;">Cammarota</span>[^1]$\,$ and Aimé <span style="font-variant:small-caps;">Lachal</span>[^2]'
title:
---
**Keywords:** pseudo-process, joint distribution of the process and its sojourn time, Spitzer’s identity.
**AMS 2000 Subject Classification:** Primary 60G20; Secondary 60J25, 60K35, 60J05.
Introduction
============
Let $N$ be an integer equal or greater than $2$ and $\kappa_{\!_{ N}}=(-1)^{1+N/2}$ if $N$ is even, $\kappa_{\!_{ N}}=\pm1$ if $N$ is odd. Consider the heat-type equation of order $N$: $$\frac{\partial u}{\partial t}=\kappa_{\!_{ N}}\,\frac{\partial^N u}{\partial x^N}.
\label{EDP}$$ For $N=2$, this equation is the classical normalized heat equation and its relationship with linear Brownian motion is of the most well-known. For $N>2$, it is known that no ordinary stochastic process can be associated with this equation. Nevertheless a Markov “pseudo-process” can be constructed by imitating the case $N=2$. This pseudo-process, $X=(X(t))_{t\ge 0}$ say, is driven by a signed measure as follows. Let $p(t;x)$ denote the elementary solution of Eq. (\[EDP\]), that is, $p$ solves (\[EDP\]) with the initial condition $p(0;x)=\delta(x)$. This solution is characterized by its Fourier transform (see, e.g., [@2007]) $$\int_{-\infty}^{+\infty} e^{i\mu x}\,p(t;x)\,\mathrm{d}x
=e^{\kappa_{\!_{ N}}t(-i\mu)^N}.$$ The function $p$ is real, not always positive and its total mass is equal to one: $$\int_{-\infty}^{+\infty} p(t;x)\,\mathrm{d}x=1.$$ Moreover, its total absolute value mass $\rho$ exceeds one: $$\rho=\int_{-\infty}^{+\infty} |p(t;x)|\,\mathrm{d}x>1.$$ In fact, if $N$ is even, $p$ is symmetric and $\rho<+\infty$, and if $N$ is odd, $\rho=+\infty$. The signed function $p$ is interpreted as the pseudo-probability for $X$ to lie at a certain location at a certain time. More precisely, for any time $t>0$ and any locations $x,y\in\mathbb{R}$, one defines $$\mathbb{P}\{X(t)\in \mathrm{d}y|X(0)=x\}/\mathrm{d}y=p(t;x-y).$$ Roughly speaking, the distribution of the pseudo-process $X$ is defined through its finite-dimensional distributions according to the Markov rule: for any $n>1$, any times $t_1,\dots,t_n$ such that $0<t_1<\dots<t_n$ and any locations $x,y_1,\dots,y_n\in\mathbb{R}$, $$\mathbb{P}\{X(t_1)\in \mathrm{d}y_1,\dots X(t_n)\in \mathrm{d}y_n|X(0)=x\}
/\mathrm{d}y_1\dots\mathrm{d}y_n=\prod_{i=1}^n p(t_i-t_{i-1};y_{i-1}-y_i)$$ where $t_0=0$ and $y_0=x$.
This pseudo-process has been studied by several authors: see the references [@bho] to [@bor] and the references [@hoch] to [@ors]. Now, we consider the sojourn time of $X$ in the interval $[0,+\infty)$ up to a fixed time $t$: $$T(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)} (X(s))\,\mathrm{d}s.$$ The computation of the pseudo-distribution of $T(t)$ has been done by Beghin, Hochberg, Nikitin, Orsingher and Ragozina in some particular cases (see [@bho; @bor; @ho; @no; @ors]), and by Krylov and the second author in more general cases (see [@kry; @2003]).
The method adopted therein is the use of the Feynman-Kac functional which leads to certain differential equations. We point out that the pseudo-distribution of $T(t)$ is actually a genuine probability distribution and in the case where $N$ is even, $T(t)$ obeys the famous Paul Lévy’s arcsine law, that is $$\mathbb{P}\{T(t)\in\mathrm{ds}\}/\mathrm{ds}=\frac{{1\hspace{-.27em}\mbox{\rm l}}_{(0,t)}(s)}{\pi\sqrt{s(t-s)}}.$$ We also mention that the sojourn time of $X$ in a small interval $(-\varepsilon,\varepsilon)$ is used in [@bo] to define a local time for $X$ at 0. The evaluation of the pseudo-distribution of the sojourn time $T(t)$ together with the up-to-date value of the pseudo-process, $X(t)$, has been tackled only in the particular cases $N=3$ and $N=4$ by Beghin, Hochberg, Orsingher and Ragozina (see [@bho; @bor]). Their results have been obtained by solving certain differential equations leading to some linear systems. In [@bho; @bor; @2003], the Laplace transform of the sojourn time serves as an intermediate tool for computing the distribution of the up-to-date maximum of $X$.
In this paper, our aim is to derive the joint pseudo-distribution of the couple $(T(t),X(t))$ for any integer $N$. Since the Feynman-Kac approach used in [@bho; @bor] leads to very cumbersome calculations, we employ an alternative method based on Spitzer’s identity. Since the pseudo-process $X$ is properly defined only in the case where $N$ is an even integer, the results we obtain are valid in this case. Throughout the paper, we shall then assume that $N$ is even. Nevertheless, we formally perform all computations also in the case where $N$ is odd, even if they are not justified.
The paper is organized as follows.
- In Section \[section-settings\], we write down the settings that will be used. Actually, the pseudo-process $X$ is not well defined on the whole half-line $[0,+\infty)$. It is properly defined on dyadic times $k/2^n$, $k,n\in\mathbb{N}$. So, we introduce ad-hoc definitions for $X(t)$ and $T(t)$ as well as for some related pseudo-expectations. For instance, we shall give a meaning to the quantity $$E(\lambda,\mu,\nu)=\mathbb{E}\!\left[\int_0^{\infty}
e^{-\lambda t+i \mu X(t)-\nu T(t)}\,\mathrm{d}t\right]$$ which is interpreted as the $3$-parameters Laplace-Fourier transform of $(T(t),X(t))$. We also recall in this part some algebraic known results.
- In Section \[section-evaluation\], we explicitly compute $E(\lambda,\mu,\nu)$ with the help of Spitzer’s identity. This is Theorem \[theorem\].
- Sections \[section-inverting-mu\], \[section-inverting-nu\] and \[section-inverting-lambda\] are devoted to successively inverting the Laplace-Fourier transform with respect to $\mu$, $\nu$ and $\lambda$ respectively. More precisely, in Section \[section-inverting-mu\], we perform the inversion with respect to $\mu$; this yields Theorem \[theorem-wrt.mu\]. Next, we perform the inversion with respect to $\nu$ which gives Theorems \[theorem-wrt.nu+1\] and \[theorem-wrt.nu+2\]. Finally, we carry out the inversion with respect to $\lambda$ and the main results of this paper are Theorems \[theorem-wrt.lambda+1\] and \[theorem-wrt.lambda+2\]. In each section, we examine the particular cases $N=2$ (case of rescaled Brownian motion), $N=3$ (case of an asymmetric pseudo-process) and $N=4$ (case of the biharmonic pseudo-process). Moreover, our results recover several known formulas concerning the marginal distribution of $T(t)$.
- The final appendix (Section \[section-appendix\]) contains a discussion on Spitzer’s identity as well as some technical computations.
Settings {#section-settings}
========
A first list of settings
------------------------
In this part, we introduce for each integer $n$ a step-process $X_n$ coinciding with the pseudo-process $X$ on the times $k/2^n$, $k\in\mathbb{N}$. Fix $n\in\mathbb{N}$. Set, for any $k\in\mathbb{N}$, $X_{k,n}=X(k/2^n)$ and for any $t\in [k/2^n,(k+1)/2^n),$ $X(t)=X_{k,n}$. We can write globally $$X_n(t)=\sum_{k=0}^{\infty} X_{k,n} {1\hspace{-.27em}\mbox{\rm l}}_{[k/2^n,(k+1)/2^n)}(t).$$ Now, we recall from [@2007] the definitions of tame functions, functions of discrete observations, and admissible functions associated with the pseudo-process $X$. They were introduced by Nishioka [@nish2] in the case $N=4$.
Fix $n\in\mathbb{N}$. A tame function for $X$ is a function of a finite number $k$ of observations of the pseudo-process $X$ at times $j/2^n$, $1\le j\le k$, that is a quantity of the form $F_{k,n}=F(X(1/2^n),\dots,X(k/2^n))$ for a certain $k$ and a certain bounded Borel function $F:\mathbb{R}^k\longrightarrow \mathbb{C}$. The “expectation” of $F_{k,n}$ is defined as $$\mathbb{E}(F_{k,n})=\int\dots\int_{\mathbb{R}^k} F(x_1,\dots,x_k)
\,p(1/2^n;x-x_1)\dots p(1/2^n;x_{k-1}-x_k)\,\mathrm{d}x_1\dots \mathrm{d}x_k.$$
\[def2\] Fix $n\in\mathbb{N}$. A function of the discrete observations of $X$ at times $k/2^n$, $k\ge 1$, is a convergent series of tame functions: $F_{X_n}=\sum_{k=1}^{\infty} F_{k,n}$ where $F_{k,n}$ is a tame function for all $k\ge 1$. Assuming the series $\sum_{k=1}^{\infty} |\mathbb{E}(F_{k,n})|$ convergent, the “expectation” of $F_{X_n}$ is defined as $$\mathbb{E}(F_{X_n})=\sum_{k=1}^{\infty} \mathbb{E}(F_{k,n}).$$
\[def3\] An admissible function is a functional $F_X$ of the pseudo-process $X$ which is the limit of a sequence $(F_{X_n})_{n\in\mathbb{N}}$ of functions of discrete observations of $X$: $F_X=\lim_{n\to\infty} F_{X_n},$ such that the sequence $(\mathbb{E}(F_{X_n}))_{n\in\mathbb{N}}$ is convergent. The “expectation” of $F_X$ is defined as $$\mathbb{E}(F_X)=\lim_{n\to\infty}\mathbb{E}(F_{X_n}).$$
In this paper, we are concerned with the sojourn time of $X$ in $[0,+\infty)$: $$T(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X(s))\,\mathrm{d}s.$$ In order to give a proper meaning to this quantity, we introduce the similar object related to $X_n$: $$T_n(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_n(s))\,\mathrm{d}s.$$ For determining the distribution of $T_n(t)$, we compute its $3$-parameters Laplace-Fourier transform: $$E_n(\lambda,\mu,\nu)=\mathbb{E}\!\left[\int_0^{\infty}
e^{-\lambda t+i\mu X_n(t)-\nu T_n(t)}\,\mathrm{d}t\right]\!.$$ In Section \[section-evaluation\], we prove that the sequence $(E_n(\lambda,\mu,\nu))_{n\in\mathbb{N}}$ is convergent and we compute its limit: $$\lim_{n\to\infty} E_n(\lambda,\mu,\nu)=E(\lambda,\mu,\nu).$$ Formally, $E(\lambda,\mu,\nu)$ is interpreted as $$E(\lambda,\mu,\nu)=\mathbb{E}\!\left[\int_0^{\infty}
e^{-\lambda t+i\mu X(t)-\nu T(t)}\,\mathrm{d}t\right]$$ where the quantity $\int_0^{\infty}e^{-\lambda t+i\mu X(t)-\nu T(t)}\,\mathrm{d}t$ is an admissible function of $X$. This computation is performed with the aid of Spitzer’s identity. This latter concerns the classical random walk. Nevertheless, since it hinges on combinatorial arguments, it can be applied to the context of pseudo-processes. We clarify this point in Section \[section-evaluation\].
A second list of settings
-------------------------
We introduce some algebraic settings. Let $\theta_i$, $1 \le i \le N$, be the $N^{\mathrm{th}}$ roots of $\kappa_{\!_{ N}}$ and $$J=\{i \in \{1,\dots, N\}:\,\Re{\theta_i}>0\},\qquad
K=\{i \in \{1,\dots, N\}:\,\Re{\theta_i}<0\}.$$ Of course, the cardinalities of $J$ and $K$ sum to $N$: $\#J+\#K=N$. We state several results related to the $\theta_i$’s which are proved in [@2003; @2007]. We have the elementary equalities $$\sum_{j \in J} \theta_j+\sum_{k \in K} \theta_k=\sum_{i=1}^N \theta_i=0,
\qquad \left(\vphantom{\prod_{k}}\right.\!\prod_{j\in J} \theta_j
\!\left.\vphantom{\prod_{k}}\right) \!\! \left(\prod_{k\in K} \theta_k\right)
=\prod_{i=1}^N \theta_i=(-1)^{N-1}\kappa_{\!_{ N}}
\label{set4}$$ and $$\prod_{i=1}^{N} (x-\theta_i)=\prod_{i=1}^{N} (x-\bar{\theta}_i)=x^N-\kappa_{\!_{ N}}.
\label{set6}$$ Moreover, from formula (5.10) in [@2007], $$\prod_{k\in K}(x-\theta_k)=\sum_{\ell=0}^{\# K} (-1)^{\ell}
\sigma_{\ell} \,x^{\# K-\ell},
\label{set10}$$ where $\sigma_{\ell}=\sum_{\substack{k_1<\dots<k_\ell\\ k_1,\dots, k_\ell\in K}}
\theta_{k_1} \dots \theta_{k_\ell}.$ We have by Lemma 11 in [@2003] $$\sum_{j \in J} \theta_j \prod_{i \in J\setminus \{j\}}
\frac{\theta_i x -\theta_j}{\theta_i-\theta_j}=\sum_{j \in J} \theta_j
=-\sum_{k \in K} \theta_k=
\begin{cases}
\displaystyle{\frac{1}{\sin \frac{\pi}{N}}}& \mbox{if } N \mbox{\,is\, even},
\\[2ex]
\displaystyle{\frac{1}{2 \sin \frac{\pi}{2N}}=\frac{\cos \frac{\pi}{2 N}}{\sin \frac{\pi}{N}}}
& \mbox{if } N \mbox{\,is\, odd}.
\end{cases}
\label{set7}$$
Set $A_j=\prod_{i \in J\setminus \{j\}} \frac{\theta_i}{\theta_i-\theta_j}$ for $j \in J$, and $B_k=\prod_{i \in K\setminus \{k\}}
\frac{\theta_i}{\theta_i-\theta_k}$ for $k \in K$. The $A_j$’s and $B_k$’s solve a Vandermonde system: we have $$\begin{gathered}
\sum_{j \in J} A_j=\sum_{k \in K} B_k=1
\nonumber\\[-3ex]
\label{set13}
\\[-1ex]
\sum_{j \in J} A_j\theta_j^m=0 \mbox{ for } 1\le m\le \# J-1,\qquad
\sum_{k \in K} B_k\theta_k^m=0 \mbox{ for } 1\le m\le \# K-1.
\nonumber\end{gathered}$$ Observing that $1/\theta_j=\bar{\theta}_j$ for $j\in J$, that $\{\theta_j,j\in J\}=\{\bar{\theta}_j,j\in J\}$ and similarly for the $\theta_k$’s, $k\in K$, formula (2.11) in [@2007] gives $$\begin{aligned}
\sum_{j \in J} \frac{A_j\theta_j}{\theta_j-x}
&=
\sum_{j \in J}\frac{A_j}{1-\bar{\theta}_j x}= \frac{1}{\prod_{j \in J}(1-\theta_j x)}
=-\kappa_{\!_{ N}}\,\frac{\prod_{k \in K} (1-\theta_k x)}{x^N-\kappa_{\!_{ N}}}
=-\kappa_{\!_{ N}}\,\frac{\prod_{k \in K} (1-\bar{\theta}_k x)}{x^N-\kappa_{\!_{ N}}},
\nonumber\\
\label{set5}
\\
\sum_{k \in K} \frac{B_k\theta_k}{\theta_k-x}
&=
\sum_{k \in K} \frac{B_k}{1-\bar{\theta}_k x}= \frac{1}{\prod_{k \in K}(1-\theta_k x)}
=-\kappa_{\!_{ N}}\,\frac{\prod_{j \in J} (1-\theta_j x)}{x^N-\kappa_{\!_{ N}}}
=-\kappa_{\!_{ N}}\,\frac{\prod_{j \in J} (1-\bar{\theta}_j x)}{x^N-\kappa_{\!_{ N}}}.
\nonumber\end{aligned}$$ In particular, $$\sum_{j \in J}\frac{A_j\theta_j}{\theta_j-\theta_k}=\frac{1}{N B_k},
\hspace{1cm} \sum_{k \in K}\frac{B_k\theta_k}{\theta_k-\theta_j}
=\frac{1}{N A_j}.
\label{set11}$$
Set, for any $m\in\mathbb{Z}$, $\alpha_{m}=\sum_{j \in J} A_j\theta_j^m$ and $\beta_m=\sum_{k \in K} B_k\theta_k^m$. We have, by formula (2.11) of [@2007], $\beta_{\#K}=(-1)^{\#K -1}\prod_{k \in K} \theta_k$. Moreover, $\beta_{\#K+1}=(-1)^{\#K -1}\left(\prod_{k \in K} \theta_k\right)\!\left(\sum_{k \in K} \theta_k\right)$. The proof of this claim is postponed to Lemma \[lemma-vdm\] in the appendix. We sum up this information and (\[set13\]) into $$\beta_m=\begin{cases}
1 & \mbox{if } m=0,
\\
0 & \mbox{if } 1 \le m \le \#K-1,
\\
(-1)^{\#K-1} \prod_{k \in K} \theta_k & \mbox{if } m=\# K ,
\\
(-1)^{\#K-1} \left(\prod_{k \in K} \theta_k\right)\!
\left(\sum_{k \in K} \theta_k\right) & \mbox{if } m=\# K+1,
\\
\kappa_{\!_{ N}} & \mbox{if } m=N.
\end{cases}
\label{set26}$$ We also have $$\alpha_{-m}=\sum_{j \in J} \frac{A_j}{\theta_j^m}
=\kappa_{\!_{ N}} \sum_{j \in J} A_j\theta_j^{N-m}=\kappa_{\!_{ N}} \alpha_{N-m}$$ and then $$\alpha_{-m}=\begin{cases}
1 & \mbox{if } m=0,
\\
\kappa_{\!_{ N}} (-1)^{\# J -1} \left(\vphantom{\prod_{\in}}\right.\!\prod_{j \in J}
\theta_j\left.\vphantom{\prod_{\in}}\!\right)\!
\left(\vphantom{\prod_{\in}}\right.\!\sum_{j \in J} \theta_j
\left.\vphantom{\prod_{\in}}\!\right) & \mbox{if } m=\# K-1,
\\
\kappa_{\!_{ N}} (-1)^{\#J -1} \prod_{j \in J} \theta_j & \mbox{if } m=\#K,
\\
0 & \mbox{if } \#K+1 \le m \le N-1,
\\
\kappa_{\!_{ N}} & \mbox{if } m=N.
\end{cases}
\label{set14}$$ In particular, by (\[set4\]), $$\alpha_0\beta_0=\alpha_{-N}\beta_N=1,\quad
\alpha_{-\#K}\beta_{\#K}=-1,\quad
\alpha_{-\#K}\beta_{\#K+1}=\sum_{j \in J} \theta_j,\quad
\alpha_{1-\#K}\beta_{\#K}=\sum_{k \in K} \theta_k.
\label{set15}$$ With $\sigma_0=1$, $\sigma_{\#K-1}=\left(\prod_{k \in K} \theta_k\right)\!
\left(\sum_{k \in K} \bar{\theta}_k\right)$ and $\sigma_{\#K}=\prod_{k \in K} \theta_k$, we also have $$\bar{\sigma}_0\beta_0=1,\quad
\bar{\sigma}_{\#K-1}\beta_{\#K}=\bar{\sigma}_{\#K}\beta_{\#K+1}
=(-1)^{\#K-1}\sum_{k \in K} \theta_k,\quad
\bar{\sigma}_{\#K}\beta_{\#K}=(-1)^{\#K-1}.
\label{set16}$$
Concerning the kernel $p$, we have from Proposition 1 in [@2003] $$p(t;0)=\begin{cases}
\displaystyle{\frac{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!\frac{1}{N}
\!\!\left.\vphantom{\frac aN}\right)}{N \pi t^{1/N}}}
& \mbox{if $N$ is even},
\\[2ex]
\displaystyle{\frac{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!\frac{1}{N}
\!\!\left.\vphantom{\frac aN}\right) \cos\!\left(\vphantom{\frac aN}\right.\!\!
\frac{\pi}{2 N}\!\!\left.\vphantom{\frac aN}\right)}{N \pi t^{1/N}}}
& \mbox{if $N$ is odd}.
\end{cases}
\label{p-at-zero}$$ Proposition 3 in [@2003] states $$\mathbb{P}\{X(t)\ge 0\}=\int_0^{\infty} p(t;-\xi) \,\mathrm{d} \xi=\frac{\# J}{N},
\qquad \mathbb{P}\{X(t)\le 0\}=\int_{-\infty}^0 p(t;-\xi) \,\mathrm{d} \xi=\frac{\# K}{N}
\label{set3}$$ and formulas (4.7) and (4.8) in [@2007] yield, for $\lambda>0$ and $\mu\in\mathbb{R}$, $$\begin{aligned}
\int_0^{\infty} \frac{e^{-\lambda t}}{t} \,\mathrm{d}t
\int_{-\infty}^0\left(e^{i\mu\xi}-1\right) p(t;-\xi) \,\mathrm{d}\xi
&=
\log \!\left(\,\prod_{k \in K} \frac{\!\sqrt[N]{\lambda}}{\!\sqrt[N]{\lambda}
-i\mu\theta_k}\right)\!,
\nonumber\\[-1ex]
\label{set1}\\[-1ex]
\int_0^{\infty} \frac{e^{-\lambda t}}{t} \,\mathrm{d}t
\int_0^{\infty}\left(e^{i\mu\xi}-1\right) p(t;-\xi) \,\mathrm{d}\xi
&=
\log \!\left(\vphantom{\prod_{\in}}\right.\!\prod_{j \in J} \frac{\!\sqrt[N]{\lambda}}{\!\sqrt[N]{\lambda}
-i\mu\theta_j}\!\left.\vphantom{\prod_{\in}}\right)\!.
\nonumber\end{aligned}$$
Let us introduce, for $m \le N-1$ and $x \ge 0$, $$I_{j,m}(\tau;x)=\frac{N i}{2 \pi} \left(e^{-i \frac{m}{N}\pi}\!
\int_0^{\infty} \xi^{N-m-1} e^{-\tau\xi^N-\theta_j e^{i \frac{\pi}{N}} x \xi}
\,\mathrm{d}\xi
-e^{i \frac{m}{N}\pi} \!\int_0^{\infty} \xi^{N-m-1} e^{-\tau\xi^N-\theta_j
e^{-i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi\right)\!.
\label{set18}$$ Formula (5.13) in [@2007] gives, for $0 \le m \le N-1$ and $x\ge 0$, $$\int_0^{\infty} e^{-\lambda \tau} I_{j,m}(\tau;x)\,\mathrm{d}\tau
=\lambda^{-\frac{m}{N}}e^{-\theta_j\!\!\sqrt[N]{\lambda}\,x}.
\label{set24}$$
\[example1\] Case $N=2$: we have $\kappa_2=+1$. This is the case of rescaled Brownian motion. The square roots of $\kappa_2$ are $\theta_1=1$, $\theta_2=-1$ and then $J=\{1\}$, K={2}, $A_1=B_2=1$, $\alpha_0=\alpha_{-1}=1$, $\beta_0=1$, $\beta_{-1}=-1$. Moreover, $$I_{1,0}(\tau;x)=\frac{i}{\pi} \left(\int_0^{\infty}\xi \,e^{-\tau \xi^2-ix \xi}
\,\mathrm{d}\xi-\int_0^{\infty}\xi \,e^{-\tau \xi^2+ix \xi}\,\mathrm{d}\xi\right)\!.$$ The function $I_{1,0}$ can be simplified. In fact, we have $$\begin{aligned}
I_{1,0}(\tau;x)
&=
\frac{i}{\pi} \int_{-\infty}^{\infty} \xi \,e^{-\tau \xi^2-ix \xi} \,\mathrm{d}\xi
=\frac{i}{\pi} \,e^{-\frac{x^2}{4 \tau}} \int_{-\infty}^{\infty} \xi\,
e^{-\tau(\xi+\frac{ix}{2 \tau})^2}\,\mathrm{d}\xi
\\
&=
\frac{i}{\pi} \,e^{-\frac{x^2}{4 \tau}} \int_{-\infty}^{\infty}
\left(\xi-\frac{i x}{2 \tau}\right) e^{-\tau\xi^2}\,\mathrm{d}\xi
=\frac{x \,e^{-\frac{x^2}{4 \tau}}}{2 \pi \tau} \int_{-\infty}^{\infty}e^{-\tau \xi^2} \,\mathrm{d}\xi
=\frac{x \,e^{-\frac{x^2}{4 \tau}}}{2 \pi \tau} \int_0^{\infty}\frac{e^{-\tau \xi}}{\sqrt{\xi}} \,\mathrm{d}\xi.\end{aligned}$$ Finally, $$I_{1,0}(\tau;x)=\frac{x \,e^{-\frac{x^2}{4 \tau}}}{2 \sqrt \pi \,\tau^{3/2}}.
\label{case2I}$$
\[example2\] Case $N=3$.
$\bullet$ For $\kappa_3=+1$, the third roots of $\kappa_3$ are $\theta_1=1$, $\theta_2=e^{i \frac{2\pi}{3}}$, $\theta_3=e^{-i \frac{2\pi}{3}}$, and the settings read $J=\{1\}$, $K=\{2,3\}$, $A_1=1$, $B_2=\frac{e^{-i\frac{\pi}{6}}}{\sqrt{3}}$, $B_3=\frac{e^{i\frac{\pi}{6}}}{\sqrt{3}}$, $\alpha_0=\alpha_{-1}=\alpha_{-2}=1$, $\beta_0=1$, $\beta_{-1}=-1$. Moreover, $$I_{1,0}(\tau;x)=\frac{3i}{2\pi} \left(
\int_0^{\infty} \xi^2\,e^{-\tau\xi^3-e^{i\frac{\pi}{3}}x\xi}\,\mathrm{d}\xi
-\int_0^{\infty} \xi^2\,e^{-\tau\xi^3-e^{-i\frac{\pi}{3}}x\xi}\,\mathrm{d}\xi\right)\!.$$
$\bullet$ For $\kappa_3=-1$, the third roots of $\kappa_3$ are $\theta_1=e^{i \frac{\pi}{3}}$, $\theta_2=e^{-i \frac{\pi}{3}}$, $\theta_3=-1$. The settings read $J=\{1,2\}$, $K=\{3\}$, $A_1=\frac{e^{i \frac{\pi}{6}}}{\sqrt{3}}$, $A_2=\frac{e^{-i\frac{\pi}{6}}}{\sqrt{3}}$, $B_3=1$, $\alpha_0=\alpha_{-1}=1$, $\beta_0=\beta_{-2}=1$, $\beta_{-1}=-1$. Moreover, $$\begin{aligned}
I_{1,1}(\tau;x)
&=
\frac{3i}{2\pi} \left(e^{-i\frac{\pi}{3}}
\int_0^{\infty} \xi\,e^{-\tau\xi^3-e^{i\frac{2\pi}{3}}x\xi}\,\mathrm{d}\xi
-e^{i\frac{\pi}{3}} \int_0^{\infty} \xi\,e^{-\tau\xi^3-x\xi}\,\mathrm{d}\xi\right)\!,
\\
I_{2,1}(\tau;x)
&=
\frac{3i}{2\pi} \left(e^{-i\frac{\pi}{3}}
\int_0^{\infty} \xi\,e^{-\tau\xi^3-x\xi}\,\mathrm{d}\xi
-e^{i\frac{\pi}{3}} \int_0^{\infty} \xi\,e^{-\tau\xi^3-e^{-i\frac{2\pi}{3}}x\xi}
\,\mathrm{d}\xi\right)\!.\end{aligned}$$ Actually, the three functions $I_{1,0}$, $I_{1,1}$ and $I_{2,1}$ can be expressed by mean of the Airy function $\mathrm{Hi}$ defined as $\mathrm{Hi}(z)=\frac{1}{\pi}\int_0^{\infty} e^{-\frac{\xi^3}{3}+z\xi}\,\mathrm{d}\xi$ (see, e.g., [@as Chap. 10.4]). Indeed, we easily have by a change of variables, differentiation and integration by parts, for $\tau>0$ and $z\in\mathbb{C}$, $$\begin{aligned}
\int_0^{\infty} e^{-\tau\xi^3+z\xi}\,\mathrm{d}\xi
&=
\frac{\pi}{(3\tau)^{4/3}}\,\mathrm{Hi}\!\left(\frac{z}{\sqrt[3]{3\tau}}\right)\!,
\\
\int_0^{\infty} \xi\,e^{-\tau\xi^3+z\xi}\,\mathrm{d}\xi
&=
\frac{\pi}{(3\tau)^{2/3}}\,\mathrm{Hi}'\!\left(\frac{z}{\sqrt[3]{3\tau}}\right)\!,
\\
\int_0^{\infty} \xi^2 \,e^{-\tau\xi^3+z\xi}\,\mathrm{d}\xi
&=
\frac{\pi z}{(3\tau)^{4/3}}\,\mathrm{Hi}\!\left(\frac{z}{\sqrt[3]{3\tau}}\right)+\frac{1}{3\tau}.\end{aligned}$$ Therefore, $$\begin{aligned}
I_{1,0}(\tau;x)
&=
\frac{x}{2\sqrt[3]{3} \,\tau^{4/3}} \left[\vphantom{\frac{z}{\sqrt{3}}}\right.\!
e^{i\frac{\pi}{6}} \mathrm{Hi}\!\left(\vphantom{\frac{z}{\sqrt{3}}}\right.\!\!
-\frac{e^{-i\frac{\pi}{3}}x}{\sqrt[3]{3\tau}} \!\left.\vphantom{\frac{z}{\sqrt{3}}} \right)
+e^{-i\frac{\pi}{6}} \mathrm{Hi}\!\left(\vphantom{\frac{z}{\sqrt{3}}}\right.\!\!
-\frac{e^{i\frac{\pi}{3}}x}{\sqrt[3]{3\tau}} \!\left.\vphantom{\frac{z}{\sqrt{3}}} \right)\!
\!\left.\vphantom{\frac{z}{\sqrt{3}}} \right]\!,
\label{case3I10}\\
I_{1,1}(\tau;x)
&=
\frac{\sqrt[3]3}{2\tau^{2/3}} \left[\vphantom{\frac{z}{\sqrt{3}}}\right.\!
e^{i\frac{\pi}{6}} \mathrm{Hi}'\!\left(\vphantom{\frac{z}{\sqrt{3}}}\right.\!\!
-\frac{e^{i\frac{2\pi}{3}}x}{\sqrt[3]{3\tau}} \!\left.\vphantom{\frac{z}{\sqrt{3}}} \right)
+e^{-i\frac{\pi}{6}} \mathrm{Hi}'\!\left(-\frac{x}{\sqrt[3]{3\tau}}\right)
\!\!\left.\vphantom{\frac{z}{\sqrt{3}}}\right]\!,
\label{case3I11}\\
I_{2,1}(\tau;x)
&=
\frac{\sqrt[3]3}{2\tau^{2/3}} \left[\vphantom{\frac{z}{\sqrt{3}}}\right.\!
e^{i\frac{\pi}{6}} \mathrm{Hi}'\!\left(-\frac{x}{\sqrt[3]{3\tau}}\right)
+e^{-i\frac{\pi}{6}} \mathrm{Hi}'\!\left(\vphantom{\frac{z}{\sqrt{3}}}\right.\!\!
-\frac{e^{-i\frac{2\pi}{3}}x}{\sqrt[3]{3\tau}}\!\left.\vphantom{\frac{z}{\sqrt{3}}}\right)
\!\!\left.\vphantom{\frac{z}{\sqrt{3}}} \right]\!.
\label{case3I21}\end{aligned}$$
\[example3\] Case $N=4$: we have $\kappa_4=-1$. This is the case of the biharmonic pseudo-process. The fourth roots of $\kappa_4$ are $\theta_1=e^{-i \frac{\pi}{4}}$, $\theta_2=e^{i \frac{\pi}{4}}$, $\theta_3=e^{i\frac{3\pi}{4}}$, $\theta_4=e^{-i\frac{3\pi}{4}}$ and the notations read in this case $J=\{1,2\}$, $K=\{3,4\}$, $A_1=B_3=\frac{e^{-i \frac{\pi}{4}}}{\sqrt{2}}$, $A_2=B_4=\frac{e^{i\frac{\pi}{4}}}{\sqrt{2}}$, $\alpha_0=\alpha_{-2}=1$, $\alpha_{-1}=\sqrt 2$, $\beta_0=\beta_{-2}=1$, $\beta_{-1}=-\sqrt 2$. Moreover, $$\begin{aligned}
I_{1,1}(\tau;x)
&=
\frac{2}{\pi} \left(e^{i\frac{\pi}{4}}
\int_0^{\infty} \xi^2\,e^{-\tau\xi^4-x\xi}\,\mathrm{d}\xi
+e^{-i\frac{\pi}{4}} \int_0^{\infty} \xi^2\,e^{-\tau\xi^4+ix\xi}\,\mathrm{d}\xi\right)\!,
\nonumber\\[-1ex]
\label{case4I}\\[-1ex]
I_{2,1}(\tau;x)
&=
\frac{2}{\pi} \left(e^{i\frac{\pi}{4}}
\int_0^{\infty} \xi^2\,e^{-\tau\xi^4-ix\xi}\,\mathrm{d}\xi
+e^{-i\frac{\pi}{4}} \int_0^{\infty} \xi^2\,e^{-\tau\xi^4-x\xi}
\,\mathrm{d}\xi\right)\!.
\nonumber\end{aligned}$$
Evaluation of $E(\lambda,\mu,\nu)$ {#section-evaluation}
==================================
The goal of this section is to evaluate the limit $E(\lambda,\mu,\nu)=\lim_{n \to \infty} E_n(\lambda,\mu,\nu)$. We write $E_n(\lambda,\mu,\nu)=\mathbb{E}[F_n(\lambda,\mu,\nu)]$ with $$F_n(\lambda,\mu,\nu)=\int_0^{\infty} e^{-\lambda t+i \mu X_n(t)-\nu T_n(t)} \,\mathrm{d}t.$$ Let us rewrite the sojourn time $T_n(t)$ as follows: $$\begin{aligned}
T_n(t)
&=
\sum_{j=0}^{[2^n t]}\int_{j/2^n}^{(j+1)/2^n} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_n(s)) \,\mathrm{d}s
-\int_t^{([2^n t]+1)/2^n} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_n(s)) \,\mathrm{d}s
\\
&=
\sum_{j=0}^{[2^n t]}\int_{j/2^n}^{(j+1)/2^n} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{j,n}) \,\mathrm{d}s
-\int_t^{([2^n t]+1)/2^n} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{[2^n t],n}) \,\mathrm{d}s
\\
&=
\frac{1}{2^n} \sum_{j=0}^{[2^n t]} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{j,n})
+\left(t-\frac{[2^n t]+1}{2^n}\right){1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{[2^n t],n}).\end{aligned}$$ Set $T_{0,n}=0$ and, for $k\ge 1$, $$T_{k,n}=\frac{1}{2^n} \sum_{j=1}^k {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{j,n}).$$ For $k\ge 0$ and $t\in[k/2^n,(k+1)/2^n)$, we see that $$T_n(t)=T_{k,n}+\left(t-\frac{k+1}{2^n}\right) {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})+\frac{1}{2^n}.$$ With this decomposition at hand, we can begin to compute $F_n(\lambda,\mu,\nu)$: $$\begin{aligned}
F_n(\lambda,\mu,\nu)
&=
\int_0^{\infty} e^{-\lambda t+i\mu X_n(t)-\nu T_n(t)}\,\mathrm{d}t
\\
&=
\sum_{k=0}^{\infty} \int_{k/2^n}^{(k+1)/2^n}
e^{-\lambda t+i\mu X_{k,n}-\nu T_{k,n}-\frac{\nu}{2^n}
+\nu(\frac{k+1}{2^n}-t){1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_{k,n})} \,\mathrm{d}t
\\
&=
e^{-\nu/2^n} \left(\sum_{k=0}^{\infty} \int_{k/2^n}^{(k+1)/2^n}
e^{-\lambda t+\nu(\frac{k+1}{2^n}-t){1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_{k,n})} \,\mathrm{d}t\right)
e^{i\mu X_{k,n}-\nu T_{k,n}}.\end{aligned}$$ The value of the above integral is $$\int_{k/2^n}^{(k+1)/2^n} e^{-\lambda t+\nu(\frac{k+1}{2^n}-t)
{1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_{k,n})} \,\mathrm{d}t
=e^{-\lambda(k+1)/2^n}\,\frac{e^{[\lambda
+\nu{1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_{k,n})]/2^n}-1}{\lambda+\nu{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})}.$$ Therefore, $$\begin{aligned}
F_n(\lambda,\mu,\nu)
&=
\frac{1-e^{-(\lambda+\nu)/2^n}}{\lambda+\nu} \sum_{k=0}^{\infty}
e^{-\lambda k/2^n+i\mu X_{k,n}-\nu T_{k,n}} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})
\\
&\hphantom{=\,}
+ e^{-\nu/2^n}\,\frac{1-e^{-\lambda/2^n}}{\lambda} \sum_{k=0}^{\infty}
e^{-\lambda k/2^n+i\mu X_{k,n}-\nu T_{k,n}} {1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_{k,n}).\end{aligned}$$ Before applying the expectation to this last expression, we have to check that it defines a function of discrete observations of the pseudo-process $X$ which satisfies the conditions of Definition \[def2\]. This fact is stated in the proposition below.
\[proposition\] Suppose $N$ even and fix an integer $n$. For any complex $\lambda$ such that $\Re(\lambda)>0$ and any $\nu>0$, the series $\sum_{k=0}^{\infty} e^{-\lambda k/2^n} \mathbb{E}\!\left[e^{i \mu X_{k,n}
-\nu T_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]$ and $\sum_{k=0}^{\infty} e^{-\lambda k/2^n} \mathbb{E}\!\left[e^{i \mu X_{k,n}
-\nu T_{k,n}}\right.$ $\left.{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_{k,n})\right]$ are absolutely convergent and their sums are given by $$\begin{aligned}
\sum_{k=0}^{\infty} e^{-\lambda k/2^n} \mathbb{E}\!\left[e^{i \mu X_{k,n}
-\nu T_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]
&=
\frac{e^{\nu/2^n}-S_n^+(\lambda,\mu,\nu)}{e^{\nu/2^n}-1},
\\
\sum_{k=0}^{\infty} e^{-\lambda k/2^n} \mathbb{E}\!\left[e^{i \mu X_{k,n}
-\nu T_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_{k,n})\right]
&=
\frac{e^{\nu/2^n}[S_n^-(\lambda,\mu,\nu)-1]}{e^{\nu/2^n}-1},\end{aligned}$$ where $$\begin{aligned}
S_n^+(\lambda,\mu,\nu)
&=
\exp\!\left(-\sum_{k=1}^{\infty} \left(1-e^{-\nu k/2^n}\right)
\frac{e^{-\lambda k/2^n}}{k}\,
\mathbb{E}\!\left[e^{i \mu X_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right)\!,
\\[1ex]
S_n^-(\lambda,\mu,\nu)
&=
\exp\!\left(\,\sum_{k=1}^{\infty} \left(1-e^{-\nu k/2^n}\right)
\frac{e^{-\lambda k/2^n}}{k}\,
\mathbb{E}\!\left[e^{i \mu X_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_{k,n})\right]\right)\!.\end{aligned}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}**$\bullet$ Step 1.** First, notice that for any $k\ge 1$, we have $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\left|\mathbb{E}\!\left[e^{i \mu X_{k,n}-\nu T_{k,n}}
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right|}$}}
&=
\left|\,\int \right.\!\!\dots\! \int_{\mathbb{R}^{k-1}\times [0,+\infty)}
e^{i \mu x_k-\frac{\nu}{2^n} \sum_{j=1}^k {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(x_j)}\,
\mathbb{P}\{X_{1,n}\in \mathrm{d}x_1,\dots,X_{k,n}\in \mathrm{d}x_k\}
\!\left.\vphantom{\int}\right|
\\
&=
\left|\,\int\right.\!\!\dots\! \int_{\mathbb{R}^{k-1}\times [0,+\infty)}
e^{i \mu x_k-\frac{\nu}{2^n}\sum_{j=1}^k {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(x_j)}\,
p\!\left(\frac{1}{2^n}; x_1\right) \prod_{j=1}^{k-1}
p\!\left(\frac{1}{2^n}; x_j-x_{j+1}\right) \mathrm{d}x_1 \dots \mathrm{d}x_k
\!\left.\vphantom{\int}\right|
\\
&\le
\int\!\dots\!\int_{\mathbb{R}^k} \left|\vphantom{\int}\right.\!
p\!\left(\frac{1}{2^n}; x_1\right)
\prod_{j=1}^{k-1} p\!\left(\frac{1}{2^n}; x_j-x_{j+1}\right)
\!\!\left.\vphantom{\int}\right| \mathrm{d}x_1 \dots \mathrm{d}x_k
\\
&=
\int\!\dots\!\int_{\mathbb{R}^k} \prod_{j=1}^k \left|\, p\!\left(\frac{1}{2^n};
y_j\right)\right| \mathrm{d}y_1 \dots \mathrm{d}y_k
=\prod_{j=1}^k \int_{-\infty}^{+\infty} \left|\, p\!\left(\frac{1}{2^n}; y_j\right)\right|\mathrm{d}y_j=\rho^k.\end{aligned}$$ Hence, we derive the following inequality: $$\sum_{k=1}^{\infty}\left| e^{-\lambda k/2^n} \mathbb{E}\!\left[e^{i \mu X_{k,n} -\nu T_{k,n}}
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right| \le \sum_{k=1}^{\infty} \rho^k \left| e^{-\lambda k/2^n}\right|
=\frac{1}{1-\rho e^{-\Re(\lambda)/2^n}}.$$ We can easily see that this bound holds true also when the factor ${1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})$ is replaced by ${1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_{k,n})$. This shows that the two series of Proposition \[proposition\] are finite for $\lambda \in \mathbb{C}$ such that $ \rho e^{-\Re(\lambda)/2^n} <1$, that is $\Re(\lambda)>2^n \log \rho$.\
**$\bullet$ Step 2.** For $\lambda \in \mathbb{C}$ such that $\Re(\lambda)>2^n \log \rho$, the Spitzer’s identity (\[spitzer-identitybis\]) (see Lemma \[lemma-spitzer\] in the appendix) gives for the first series of Proposition \[proposition\] $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\sum_{k=0}^{\infty} e^{-\lambda k/2^n} \mathbb{E}\!
\left[e^{i \mu X_{k,n} -\nu T_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]}$}}
&=
\frac{1}{e^{\nu/2^n}-1}\left[e^{\nu/2^n}-
\exp\!\left(-\sum_{k=1}^{\infty} \left(1-e^{-\nu k/2^n}\right)\frac{e^{-\lambda k/2^n}}{k}\,
\mathbb{E}\!\left[e^{i \mu X_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right)\right]\!.
\label{series}\end{aligned}$$ The right-hand side of (\[series\]) is an analytic continuation of the Dirichlet series lying in the left-hand side of (\[series\]), which is defined on the half-plane $\{\lambda \in \mathbb{C}: \Re(\lambda)>0\}$. Moreover, for any $\varepsilon>0$, this continuation is bounded over the half-plane $\{\lambda \in \mathbb{C}: \Re(\lambda)\ge \varepsilon\}$. Indeed, we have $$\left| \mathbb{E}\!\left[e^{i \mu X_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right|
= \left|\, \int_0^{+\infty} e^{i \mu \xi} p\!\left(\frac{k}{2^n};-\xi\right) \mathrm{d}\xi\right|
\le \int_0^{+\infty} \left|\, p\!\left(\frac{k}{2^n}; -\xi\right)\right| \mathrm{d}\xi <\rho$$ and then $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\left|\,\exp\!\left(-\sum_{k=1}^{\infty} \left(1-e^{-\nu k/2^n}\right)\frac{e^{-\lambda k/2^n}}{k}\,
\mathbb{E}\!\left[e^{i \mu X_{k,n}} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right)\right|}$}}
&\le
\exp\!\left(\rho \sum_{k=1}^{\infty} \frac{e^{-\Re(\lambda) k/2^n}}{k}\right)
=\exp\!\left(-\rho \log (1-e^{-\Re(\lambda)/2^n})\right)
=\frac{1}{(1-e^{-\Re(\lambda)/2^n})^{\rho}}.\end{aligned}$$ Therefore, if $\Re(\lambda)\ge\varepsilon$, $$\begin{aligned}
\left|\,\exp\!\left(-\sum_{k=1}^{\infty} \left(1-e^{-\nu k/2^n}\right)\frac{e^{-\lambda k/2^n}}{k}\,
\mathbb{E}\!\left[e^{i \mu X_{k,n}} {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]\right)\right|
&\le
\frac{1}{(1-e^{-\varepsilon/2^n})^{\rho}}.\end{aligned}$$ This proves that the left-hand side of this last inequality is bounded for $\Re(\lambda) \ge \varepsilon$. By a lemma of Bohr ([@bohr]), we deduce that the abscissas of convergence, absolute convergence and boundedness of the Dirichlet series $\sum_{k=0}^{\infty} e^{-\lambda k/2^n}
\mathbb{E}\!\left[e^{i\mu X_{k,n}-\nu T_{k,n}}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_{k,n})\right]$ are identical. So, this series converges absolutely on the half-plane $\{\lambda \in \mathbb{C}: \Re(\lambda)>0\}$ and (\[series\]) holds on this half-plane. A similar conclusion holds for the second series of Proposition \[proposition\]. The proof is finished. [ $\blacksquare$\
]{}
Thanks to Proposition \[proposition\], we see that the functional $F_n(\lambda,\mu,\nu)$ is a function of the discrete observations of $X$ and, by Definition \[def2\], its expectation can be computed as follows: $$\begin{aligned}
E_n(\lambda,\mu,\nu)
&=
\frac{1-e^{-(\lambda+\nu)/2^n}}{\lambda+\nu}\,\frac{e^{\nu/2^n}-S_n^+(\lambda,\mu,\nu)}{e^{\nu/2^n}-1}
+\frac{1-e^{-\lambda/2^n}}{\lambda}\,\frac{S_n^-(\lambda,\mu,\nu)-1}{e^{\nu/2^n}-1}
\nonumber\\
&=
\left(\frac{e^{\nu/2^n}(1-e^{-(\lambda+\nu)/2^n})}{(\lambda+\nu)(e^{\nu/2^n}-1)}
-\frac{1-e^{-\lambda/2^n}}{\lambda(e^{\nu/2^n}-1)}\right)
\nonumber\\
&\hphantom{=\,}
+\frac{1-e^{-\lambda/2^n}}{\lambda (e^{\nu/2^n}-1)}\,S_n^-(\lambda,\mu,\nu)
-\frac{1-e^{-(\lambda+\nu)/2^n}}{(\lambda+\nu) (e^{\nu/2^n}-1)}\,S_n^+(\lambda,\mu,\nu).
\label{En}\end{aligned}$$ Now, we have to evaluate the limit $E(\lambda,\mu,\nu)$ of $E_n(\lambda,\mu,\nu)$ as $n$ goes toward infinity. It is easy to see that this limit exists; see the proof of Theorem \[theorem\] below. Formally, we write $E(\lambda,\mu,\nu)=\mathbb{E}[F(\lambda,\mu,\nu)]$ with $$F(\lambda,\mu,\nu)=\int_0^{\infty} e^{-\lambda t+i \mu X(t)-\nu T(t)} \,\mathrm{d}t.$$ Then, we can say that the functional $F(\lambda,\mu,\nu)$ is an admissible function of $X$ in the sense of Definition \[def3\]. The value of its expectation $E(\lambda,\mu,\nu)$ is given in the following theorem.
\[theorem\] The $3$-parameters Laplace-Fourier transform of the couple $(T(t),X(t))$ is given by $$E(\lambda,\mu,\nu)=\frac{1}{\prod_{j \in J}(\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j )
\prod_{k \in K} (\!\sqrt[N]{\lambda}-i \mu \theta_k )}.
\label{expressionE}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}It is plain that the term lying within the biggest parentheses in the last equality of (\[En\]) tends to zero as $n$ goes towards infinity and that the coefficients lying before $S_n^+(\lambda,\mu,\nu)$ and $S_n^-(\lambda,\mu,\nu)$ tend to $1/\nu$. As a byproduct, we derive at the limit when $n\to \infty$, $$E(\lambda,\mu,\nu)=\frac{1}{\nu}\left[S^-(\lambda,\mu,\nu)-S^+(\lambda,\mu,\nu)\right]
\label{expressionEinter}$$ where we set $$\begin{aligned}
S^+(\lambda,\mu,\nu)
&=
\lim_{n\to\infty}S_n^+(\lambda,\mu,\nu)
=\exp\!\left(-\int_0^{\infty} \mathbb{E}\!\left[\vphantom{e^X}\right.\!\! e^{i \mu X(t)}
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X(t))\!\!\left.\vphantom{e^X}\right]
\!\!\left(1-e^{-\nu t}\right)\frac{e^{-\lambda t}}{t}\,\mathrm{d}t\right)\!,
\\
S^-(\lambda,\mu,\nu)
&=
\lim_{n\to\infty}S_n^-(\lambda,\mu,\nu)
=\exp\!\left(\,\int_0^{\infty} \mathbb{E}\!\left[\vphantom{e^X}\right.\!\! e^{i \mu X(t)}
{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X(t))\!\!\left.\vphantom{e^X}\right]\!\!
\left(1-e^{-\nu t}\right)\frac{e^{-\lambda t}}{t}\,\mathrm{d}t\right)\!.\end{aligned}$$ We have $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} \mathbb{E}\!\left[\vphantom{e^X}\right.\!\! e^{i \mu X(t)}
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X(t))\!\!\left.\vphantom{e^X}\right]
\!\!\left(1-e^{-\nu t}\right)\frac{e^{-\lambda t}}{t}\,\mathrm{d}t}$}}
&=
\int_0^{\infty} \mathbb{E}\!\left[\left(\!\!\vphantom{e^X}\right.\right.
e^{i \mu X(t)}-1\left.\left.\vphantom{e^X}\!\!\right)\!
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X(t))\right]\frac{e^{-\lambda t}}{t}\,\mathrm{d}t
-\int_0^{\infty} \mathbb{E}\!\left[\left(\!\!\vphantom{e^X}\right.\right.
e^{i \mu X(t)}-1\left.\left.\vphantom{e^X}\!\!\right)\!
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X(t))\right]\frac{e^{-(\lambda+\nu) t}}{t}\,\mathrm{d}t
\\
&\hphantom{=\,}
+\int_0^{\infty} \mathbb{P}\{X(t)\ge 0\} \,\frac{e^{-\lambda t}-e^{-(\lambda+\nu) t}}{t}\,\mathrm{d}t
\\
&=
\int_0^{\infty} \frac{e^{-\lambda t}}{t} \,\mathrm{d}t
\int_0^{\infty}\left(e^{i \mu \xi}-1\right) p(t;-\xi)\,\mathrm{d}\xi
-\int_0^{\infty} \frac{e^{-(\lambda+\nu)t}}{t} \,\mathrm{d}t
\int_0^{\infty}\left(e^{i \mu \xi}-1\right) p(t;-\xi)\,\mathrm{d}\xi
\\
&\hphantom{=\,}
+\mathbb{P}\{X(1)\ge 0\} \int_0^{\infty} \frac{e^{-\lambda t}-e^{-(\lambda+\nu) t}}{t}\,\mathrm{d}t.\end{aligned}$$ In view of (\[set3\]) and (\[set1\]) and using the elementary equality $\int_0^{\infty} \frac{e^{-\lambda t}-e^{-(\lambda+\nu) t}}{t} \,\mathrm{d}t
=\log \left(\frac{\lambda+\nu}{\lambda}\right)$, we have $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} \mathbb{E}\!\left[\vphantom{e^X}\right.\!\! e^{i \mu X(t)}
{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X(t))\!\!\left.\vphantom{e^X}\right]\!\!
\left(1-e^{-\nu t}\right)\frac{e^{-\lambda t}}{t}\,\mathrm{d}t}$}}
&=
\log\!\left(\!\vphantom{\prod_{\in}}\right.
\prod_{j \in J} \frac{\sqrt[N]{\lambda}}{\!\sqrt[N]{\lambda}-i\mu \theta_j}\left.\vphantom{\prod_{\in}}\!\right)\!
-\log\!\left(\!\vphantom{\prod_{\in}}\right.
\prod_{j \in J} \frac{\!\sqrt[N]{\lambda+\nu}}{\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j}
\left.\vphantom{\prod_{\in}}\!\right)\!
+\frac{\#J}{N} \log\!\left(\frac{\lambda+\nu}{\lambda}\right)\!
=\log\!\left(\vphantom{\prod_{k}}\right.\!
\prod_{j \in J} \frac{\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j}{\!\sqrt[N]{\lambda}-i\mu \theta_j}
\!\left.\vphantom{\prod_{k}}\right)\!.\end{aligned}$$ We then deduce the value of $S^+(\lambda,\mu,\nu)$. By (\[set6\]), $$\begin{aligned}
S^+(\lambda,\mu,\nu)
&=
\prod_{j \in J} \frac{\!\sqrt[N]{\lambda}-i\mu \theta_j}{\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j}
=\frac{\prod_{\ell=1}^N (\!\sqrt[N]{\lambda}-i\mu \theta_{\ell})}
{\prod_{j \in J} (\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j)
\prod_{k \in K} (\!\sqrt[N]{\lambda}-i\mu \theta_k)}
\nonumber\\
&=
\frac{\lambda-\kappa_{\!_{ N}}(i\mu)^N}{\prod_{j \in J} (\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j)
\prod_{k \in K} (\!\sqrt[N]{\lambda}-i\mu \theta_k)}.
\label{S+}\end{aligned}$$ Similarly, the value of $S^-(\lambda,\mu,\nu)$ is given by $$S^-(\lambda,\mu,\nu)=\prod_{k \in K} \frac{\!\sqrt[N]{\lambda+\nu}-i\mu \theta_k}{\!\sqrt[N]{\lambda}-i\mu \theta_k}
=\frac{\lambda+\nu-\kappa_{\!_{ N}}(i\mu)^N}{\prod_{j \in J} (\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j)
\prod_{k \in K} (\!\sqrt[N]{\lambda}-i\mu \theta_k)}.
\label{S-}$$ Finally, putting (\[S+\]) and (\[S-\]) into (\[expressionEinter\]) immediately leads to (\[expressionE\]). [ $\blacksquare$\
]{}
\[remark\] In the particular case $\mu=0$, we get the very simple result: $$E(\lambda, 0,\nu)=\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}
\!\left[\!\vphantom{e^t}\right. e^{-\nu T(t)}\!\left.\vphantom{e^t}\right] \mathrm{d}t
=\frac{1}{\lambda^{\frac{\# K}{N}} (\lambda+\nu)^{\frac{\# J}{N}}} .$$ This is formula (20) of [@2003]. On the other hand, we can rewrite (\[expressionE\]) as $$E(\lambda,\mu,\nu)=\frac{1}{\lambda^{\frac{\# K}{N}} (\lambda+\nu)^{\frac{\# J}{N}}}
\prod_{j \in J} \frac{\!\sqrt[N]{\lambda+\nu}}{\!\sqrt[N]{\lambda+\nu}-i\mu \theta_j}
\prod_{k \in K} \frac{\!\sqrt[N]{\lambda}}{\!\sqrt[N]{\lambda}-i \mu \theta_k}.
\label{expressionEbis}$$ Actually, this form is more suitable for the inversion of the Laplace-Fourier transform.
In the three next sections, we progressively invert the $3$-parameters Laplace-Fourier transform $E(\lambda,\mu,\nu)$.
Inverting with respect to $\mu$ {#section-inverting-mu}
===============================
In this part, we invert $E(\lambda,\mu,\nu)$ given by (\[expressionEbis\]) with respect to $\mu$.
\[theorem-wrt.mu\] We have, for $\lambda,\nu>0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
&=
\begin{cases}
\displaystyle{\frac{1}{\lambda^{\frac{\# K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{j \in J} A_j\theta_j \left(\,\sum_{k \in K}
\frac{B_k\theta_k}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}\right)
e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x}} & \mbox{if } x \ge 0,
\\[3ex]
\displaystyle{\frac{1}{\lambda^{\frac{\# K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{k \in K}B_k\theta_k \left(\vphantom{\prod_{k}}\right.\!\sum_{j \in J}
\frac{A_j\theta_j}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}
\!\left.\vphantom{\prod_{k}}\right)
e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}} & \mbox{if } x \le 0.
\end{cases}
\label{wrt.mu}\end{aligned}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}By (\[set5\]) applied to $x=i\mu/\!\sqrt[N]{\lambda+\nu}$ and $x=i\mu/\!\sqrt[N]{\lambda}$, we have
$$\begin{aligned}
\prod_{j \in J} \frac{\!\sqrt[N]{\lambda+\nu}}{ \sqrt[N]{\lambda+\nu}-i\mu \theta_j}
\prod_{k \in K} \frac{\!\sqrt[N]{\lambda}}{ \sqrt[N]{\lambda} -i \mu \theta_k}
&=
\prod_{j \in J} \frac{1}{1-\frac{i\mu}{\!\sqrt[N]{\lambda+\nu}}\,\theta_j}
\prod_{k \in K} \frac{1}{1-\frac{i\mu}{\!\sqrt[N]{\lambda}}\,\theta_k}
\\
&=
\sum_{j \in J} \frac{A_j\theta_j}{\theta_j-\frac{i\mu}{\!\sqrt[N]{\lambda+\nu}}}
\sum_{k \in K} \frac{B_k\theta_k}{\theta_k-\frac{i\mu}{\!\sqrt[N]{\lambda}}}
\\
&=
\sqrt[N]{\lambda (\lambda+\nu)} \sum_{\substack{j \in J \\k \in K}}
\frac{A_j B_k\theta_j \theta_k}{(\theta_j\!\!\sqrt[N]{\lambda+\nu}-i\mu ) (\theta_k\!\!\sqrt[N]{\lambda}-i\mu )}.\end{aligned}$$
Let us write that $$\begin{aligned}
\frac{1}{(\theta_j\!\!\sqrt[N]{\lambda+\nu}-i\mu ) (\theta_k\!\!\sqrt[N]{\lambda}-i\mu )}
&=
\frac{1}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\mu}}
\left(\frac{1}{\theta_j\!\!\sqrt[N]{\lambda+\nu}-i \mu}
-\frac{1}{\theta_k\!\!\sqrt[N]{\lambda} -i \mu}\right)
\\
&=
\frac{1}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\mu}}
\left(\int_0^{\infty} e^{(i \mu -\theta_j\!\!\sqrt[N]{\lambda+\mu}) x} \,\mathrm{d}x
+\int_{-\infty}^0 e^{(i \mu -\theta_k\!\!\sqrt[N]{\lambda}) x} \,\mathrm{d}x\right)\!.\end{aligned}$$ Therefore, we can rewrite $E(\lambda,\mu,\nu)$ as $$\begin{aligned}
E(\lambda,\mu,\nu)
&=
\frac{1}{\lambda^{\frac{\# K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\\
&\hphantom{=\,}
\times\sum_{\substack{j \in J \\k \in K}}
\frac{A_j B_k\theta_j \theta_k}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}
\int_{-\infty}^{\infty} e^{i \mu x} \left(e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}
{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0]}(x)+ e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x} {1\hspace{-.27em}\mbox{\rm l}}_{[0,\infty)}(x)\right) \mathrm{d}x\end{aligned}$$ which is nothing but the Fourier transform with respect to $\mu$ of the right-hand side of (\[wrt.mu\]). [ $\blacksquare$\
]{}
$\bullet$ By integrating (\[wrt.mu\]) on $(-\infty,0]$, we obtain $$\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)},\, X(t) \le 0 \left.\!\vphantom{e^t}\right) \mathrm{d}t
=-\frac{1}{\lambda^{\frac{\# K}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{k \in K} B_k \left(\!\vphantom{\prod_{k}}\right.
\sum_{j \in J}\frac{A_j\theta_j}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}
\!\left.\vphantom{\prod_{k}}\right)\!.$$ Using (\[set5\]) applied to $x=\theta_k\!\!\sqrt[N] \lambda/ \!\sqrt[N]{\lambda+\nu}$ and (\[set10\]), we see that $$\begin{aligned}
\sum_{j \in J} \frac{A_j\theta_j}{\theta_j-\theta_k\!\!\sqrt[N]{\frac{\lambda}{\lambda+\nu}}}
&=
\frac{\lambda+\nu}{\nu} \prod_{i\in K} \left(1-\bar{\theta}_i\theta_k
\!\!\sqrt[N]{\frac{\lambda}{\lambda+\nu}}\,\right)
\nonumber\\
&=
\theta_k^{\#K} \frac{\lambda+\nu}{\nu} \left(\frac{\lambda}{\lambda+\nu}\right)^{\!\frac{\#K}{N}}
\prod_{i\in K} \left(\bar{\theta}_k\!\!\sqrt[N]{\frac{\lambda+\nu}{\lambda}}-\bar{\theta}_i\right)
\nonumber\\
&=
\frac{1}{\nu}\,\theta_k^{\#K} \lambda^{\frac{\#K}{N}}(\lambda+\nu)^{\frac{\#J}{N}}\,
\sum_{\ell=0}^{\#K} (-1)^{\ell} \bar{\sigma}_{\ell}\,
\bar{\theta}_k^{\#K-\ell} \left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\#K-\ell}{N}}
\nonumber\\
&=
\frac{1}{\nu}\, \lambda^{\frac{\#K}{N}}(\lambda+\nu)^{\frac{\#J}{N}}\,
\sum_{\ell=0}^{\#K} (-1)^{\ell} \bar{\sigma}_{\ell}\,
\theta_k^{\ell} \left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\#K-\ell}{N}}.
\label{sum-inter}\end{aligned}$$ This entails that $$\begin{aligned}
\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)},\, X(t) \le 0 \left.\!\vphantom{e^t}\right) \mathrm{d}t
&=
\frac{1}{\nu} \,\sum_{k \in K} B_k \sum_{\ell=0}^{\# K} (-1)^{\ell}
\bar{\sigma}_{\ell} \,\theta_k^{\ell}
\left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\#K-\ell}{N}}
\\
&=
\frac{1}{\nu} \,\sum_{\ell=0}^{\# K} (-1)^{\ell} \bar{\sigma}_{\ell}\,
\beta_{\ell} \left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\#K-\ell}{N}}.\end{aligned}$$ By (\[set26\]), we know that all the $\beta_\ell$, $1\le \ell\le \#K-1$ vanish and it remains, with (\[set16\]), $$\begin{aligned}
\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)},\, X(t) \le 0 \left.\!\vphantom{e^t}\right) \mathrm{d}t
&=
\frac{1}{\nu} \left[\bar{\sigma}_0\beta_0\left(\frac{\lambda+\nu}{\lambda}
\right)^{\!\frac{\# K}{N}}+(-1)^{\#K}\bar{\sigma}_{\#K}\beta_{\# K}\right]
\nonumber\\
&=
\frac{1}{\nu} \left[\left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\# K}{N}} -1\right]\!.
\label{le2}\end{aligned}$$ We retrieve (30) of [@2003].
$\bullet$ Likewise, we have $$\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)},\, X(t) \le 0 \left.\!\vphantom{e^t}\right) \mathrm{d}t
=\frac{1}{\nu} \left[1-\left(\frac{\lambda}{\lambda+\nu}\right)^{\!\frac{\# J}{N}}\right]\!,
\label{ge2}$$ which coincides with (29) of [@2003].
$\bullet$ Adding formulas (\[le2\]) and (\[ge2\]) we obtain $$\begin{aligned}
\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)} \left.\!\vphantom{e^t}\right) \mathrm{d}t
&=
\frac{1}{\nu} \left[\left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\# K}{N}}
-\left(\frac{\lambda}{\lambda+\nu}\right)^{\!\frac{\# J}{N}}\right]
=\frac{(\lambda+\nu)^{\frac{\#J+\#K}{N}}-\lambda^{\frac{\# J+\#K}{N}}}{\nu
\lambda^{\frac{\# K}{N}}(\lambda+\nu)^{\frac{\# J}{N}}}
\\
&=
\frac{1}{\lambda^{\frac{\#K}{N}}(\lambda+\nu)^{\frac{\# J}{N}}}.\end{aligned}$$ This is formula (10) of [@2003] which has already been pointed out in Remark \[remark\]. Another way of checking this formula consists of integrating (\[wrt.mu\]) with respect to $x$ directly on $\mathbb{R}$. Indeed, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)} \left.\!\vphantom{e^t}\right) \mathrm{d}t}$}}
\\[-5ex]
&=
\frac{1}{\lambda^{\frac{\#K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\left(\frac{1}{\!\sqrt[N]{\lambda+\nu}} \sum_{{\substack{j\in J \\ k \in K}}}
\frac{A_j B_k\theta_k}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}
-\frac{1}{\!\sqrt[N]{\lambda}} \sum_{{\substack{j\in J \\ k \in K}}}
\frac{A_j B_k\theta_j}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}\right)
\\
&=
\frac{1}{\lambda^{\frac{\#K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{\substack{j\in J \\ k \in K}} \left(\frac{A_j B_k\theta_k}{\!\sqrt[N]{\lambda+\nu}\,
(\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu})}
-\frac{A_j B_k\theta_j}{\!\sqrt[N]{\lambda}\,(\theta_k\!\!\sqrt[N]{\lambda}
-\theta_j\!\!\sqrt[N]{\lambda+\nu})}\right)
\\
&=
\frac{1}{\lambda^{\frac{\#K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{\substack{j\in J \\ k \in K}} \frac{A_j B_k}{\!\sqrt[N]{\lambda(\lambda+\nu)}}.\end{aligned}$$ By (\[set13\]), we have $\sum_{\substack{j\in J \\ k \in K}} A_j B_k
=\sum_{j\in J} A_j \sum_{k\in K} B_k=1$ and then $$\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)} \left.\!\vphantom{e^t}\right) \mathrm{d}t
=\frac{1}{\lambda^{\frac{\#K}{N}}(\lambda+\nu)^{\frac{\# J}{N}}}.$$
\[remark-x0\] By replacing $x$ by $0$ into (\[wrt.mu\]) and by using (\[sum-inter\]), we get $$\begin{aligned}
\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)},\, X(t)\in\mathrm{d}x \left.\!\vphantom{e^t}\right)\!/\mathrm{d}x\,\Big|_{x=0} \,\mathrm{d}t
&=
\frac{1}{\lambda^{\frac{\# K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{k \in K} B_k\theta_k \left(\!\vphantom{\prod_{k}}\right.\sum_{j \in J}
\frac{A_j\theta_j}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}
\left.\!\vphantom{\prod_{k}}\right)\!.
\\
&=
-\frac{\!\sqrt[N]{\lambda}}{\nu} \,\sum_{k \in K} B_k\theta_k
\sum_{\ell=0}^{\# K} (-1)^{\ell} \bar{\sigma}_{\ell} \,\theta_k^{\ell}
\left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\#K-\ell}{N}}
\\
&=
-\frac{\!\sqrt[N]{\lambda}}{\nu}\,\sum_{\ell=0}^{\# K} (-1)^{\ell}
\bar{\sigma}_{\ell} \,\beta_{\ell+1}
\left(\frac{\lambda+\nu}{\lambda}\right)^{\!\frac{\#K-\ell}{N}}.\end{aligned}$$ In this sum, all the $\beta_{\ell+1}$, $0\le\ell\le \#K-2$, vanish and it remains, with (\[set16\]), $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \,\mathbb{E}\!\left(\!\vphantom{e^t}\right.
e^{-\nu T(t)},\, X(t)\in\mathrm{d}x \left.\!\vphantom{e^t}\right)\!
/\mathrm{d}x\,\Big|_{x=0} \,\mathrm{d}t}$}}
&=
-\frac{\!\sqrt[N]{\lambda}}{\nu}\left(
(-1)^{\#K-1}\bar{\sigma}_{\#K-1}\beta_{\# K}\!\sqrt[N]{\frac{\lambda+\nu}{\lambda}}
+(-1)^{\#K}\bar{\sigma}_{\#K}\beta_{\# K+1}\right)
\\
&=
-\frac{\!\sqrt[N]{\lambda}}{\nu} \,\sum_{k \in K} \theta_k\!
\left(\!\sqrt[N]{\frac{\lambda+\nu}{\lambda}}-1\right)
=\left(\!\vphantom{\prod_{k}}\right.\sum_{j \in J}\theta_j
\left.\!\!\vphantom{\prod_{k}}\right) \frac{\!\sqrt[N]{\lambda+\nu}-\sqrt[N]\lambda}{\nu}.\end{aligned}$$ We retrieve formula (26) of [@2003].
For $\nu=0$, formula (\[wrt.mu\]) yields with (\[set11\]) the $\lambda$-potential of $X$: $$\begin{aligned}
\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{X(t) \in \mathrm{d}x\}/\mathrm{d}x\right] \mathrm{d}t
&=
\begin{cases}
\displaystyle{\frac{1}{\lambda^{1-\frac{1}{N}}} \sum_{j \in J} A_j\theta_j \sum_{k \in K}
\frac{B_k\theta_k}{\theta_k-\theta_j} \,e^{-\theta_j\!\!\sqrt[N]{\lambda} \,x}}& \mbox{if } x \ge 0,
\\
\displaystyle{\frac{1}{\lambda^{1-\frac{1}{N}}} \sum_{k \in K} B_k\theta_k \sum_{j \in J}
\frac{A_j\theta_j}{\theta_k-\theta_j} \,e^{-\theta_k\!\!\sqrt[N]{\lambda} \,x}}& \mbox{if } x \le 0,
\end{cases}
\\[2ex]
&=
\begin{cases}
\displaystyle{\frac{1}{N \lambda^{1-\frac{1}{N}}} \sum_{j \in J} \theta_j\,
e^{-\theta_j\!\!\sqrt[N]{\lambda} \,x}}& \mbox{if } x \ge 0,
\\
\displaystyle{-\frac{1}{N \lambda^{1-\frac{1}{N}}}\sum_{k \in K} \theta_k
e^{-\theta_k\!\!\sqrt[N]{\lambda} \,x}} & \mbox{if } x \le 0.
\end{cases}\end{aligned}$$ We retrieve (12) of [@2003].
For $N=2$, formula (\[wrt.mu\]) gives, with the numerical values of Example \[example1\], $$\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t
=\begin{cases}
\displaystyle{\frac{1}{\sqrt{\lambda}+\sqrt{\lambda+\nu}}
\,e^{-\sqrt{\lambda+\nu}\,x}} & \mbox{if } x \ge 0,
\\[3ex]
\displaystyle{ \frac{1}{\sqrt{\lambda}+\sqrt{\lambda+\nu}}
\,e^{\sqrt{\lambda}\,x}}& \mbox{if } x \le 0.
\end{cases}$$ This is formula 1.4.5, p. 129, of [@bs].
For $N=3$, we have two cases to consider. Formula (\[wrt.mu\]) yields, with the numerical values of Example \[example2\], in the case $\kappa_3=1$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
=\begin{cases}
\displaystyle\frac{e^{-\sqrt[3]{\lambda +\nu}\,x}}{\lambda^{2/3}
+\sqrt[3]{\lambda(\lambda+\nu)}+ (\lambda+\nu)^{2/3}} & \mbox{if } x\ge 0,
\\[3ex]
\displaystyle
\frac{e^{\frac{\sqrt[3]{\lambda}}{2}\,x}}{\sqrt3\,\sqrt[3] \lambda}
\frac{\sqrt3 \,\sqrt[3] \lambda \, \cos \!\left(\!\vphantom{\frac12}\right.
\frac{\sqrt3\sqrt[3]{\lambda}}{2}\,x \left.\vphantom{\frac12}\!\right)
-(2\sqrt[3]{\lambda+\nu}+\sqrt[3]\lambda\,) \sin\!\left(\!\vphantom{\frac12}\right.
\frac{\sqrt3\sqrt[3]{\lambda}}{2}\,x \left.\vphantom{\frac12}\!\right)}{
\lambda^{2/3} +\sqrt[3]{\lambda(\lambda+\nu)}+ (\lambda+\nu)^{2/3}} & \mbox{if } x\le 0,
\end{cases}\end{aligned}$$ and in the case $\kappa_3=-1$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
=\begin{cases}
\displaystyle\frac{e^{-\frac{\sqrt[3]{\lambda+\nu}}{2}\,x}}{ \sqrt 3 \sqrt[3]{ \lambda+\nu}}\,
\frac{\sqrt 3 \,\sqrt[3]{\lambda+\nu} \,\cos\!\left(\!\vphantom{\frac12}\right.
\frac{\sqrt3\sqrt[3]{\lambda+\nu}}{2}\,x \left.\vphantom{\frac12}\!\right)
+(\sqrt[3]{\lambda+\nu}+2 \sqrt[3] \lambda\,) \sin\!\left(\!\vphantom{\frac12}\right.
\frac{\sqrt3\sqrt[3]{\lambda+\nu}}{2}\,x \left.\vphantom{\frac12}\!\right)}{\lambda^{2/3}
+\sqrt[3]{\lambda(\lambda+\nu)}+ (\lambda+\nu)^{2/3}} & \mbox{if } x\ge 0,
\\[2ex]
\displaystyle
\frac{e^{\sqrt[3]{\lambda}\,x}}{\lambda^{2/3} +\sqrt[3]{\lambda(\lambda+\nu)}+ (\lambda+\nu)^{2/3}}
& \mbox{if } x\le 0.
\end{cases}\end{aligned}$$
For $N=4$, formula (\[wrt.mu\]) supplies, with the numerical values of Example \[example3\], $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
=\begin{cases}
\displaystyle\frac{\sqrt2\,e^{-\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}\,x}}{\sqrt[4]{\lambda+\nu}\,(\sqrt \lambda
+\sqrt{\lambda+\nu})(\sqrt[4] \lambda +\sqrt[4]{\lambda+\nu})}
\left[\sqrt[4]{\lambda+\nu} \,\cos \!\left(\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}\,x\right)\!
+ \sqrt[4]{\lambda} \,\sin \!\left(\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}\,x\right)\right]& \mbox{if } x\ge 0,
\\[3ex]
\displaystyle
\frac{\sqrt2\,e^{\frac{\sqrt[4]{\lambda}}{\sqrt{2}}\,x}}{\sqrt[4]{\lambda} \,(\sqrt \lambda
+\sqrt{\lambda+\nu})(\sqrt[4] \lambda +\sqrt[4]{\lambda+\nu})}
\left[\vphantom{\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}}\!\!\right.
\sqrt[4]{\lambda} \,\cos \!\left(\vphantom{\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}}\right.\!\!
\frac{\sqrt[4]{\lambda}}{\sqrt{2}}\,x \left.\vphantom{\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}}\!\!\right)\!
-\sqrt[4]{\lambda+\nu} \,\sin \!\left(\vphantom{\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}}\right.\!\!
\frac{\sqrt[4]{\lambda}}{\sqrt{2}}\,x \left.\vphantom{\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}}\!\!\right)
\left.\vphantom{\frac{\sqrt[4]{\lambda+\nu}}{\sqrt{2}}}\!\!\right]
& \mbox{if } x\le 0.
\end{cases}\end{aligned}$$
Using quite analogous computations to those of Remark \[remark-x0\], we could derive another expression for formula (\[wrt.mu\]). Actually it will not be used for the inversion. $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
&=
\begin{cases}
\displaystyle{\frac{\!\sqrt[N]{\lambda+\nu}-\sqrt[N]{\lambda}}{ \nu} \sum_{j \in J} \theta_j
\left(\prod_{i \in J \setminus \{j\}} \frac{\theta_i\!\!\sqrt[N]{\frac{\lambda}{\lambda+\nu}}
-\theta_j}{\theta_i-\theta_j}\right) e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x}} & \mbox{if } x \ge 0,
\\[4ex]
\displaystyle{-\frac{\!\sqrt[N]{\lambda+\nu}-\sqrt[N]{\lambda}}{ \nu} \sum_{k \in K}
\theta_k \left(\prod_{i \in K \setminus \{k\}} \frac{\theta_i\!\!\sqrt[N]{\frac{\lambda+\nu}{\lambda}}
-\theta_k}{\theta_i-\theta_k}\right)e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}} & \mbox{if } x \le 0.
\end{cases}
\label{wrt.mu2}\end{aligned}$$ Formula (\[wrt.mu2\]) looks like formula (24) in [@2003]. Nevertheless, (\[wrt.mu2\]) involves the distribution of $(T(t), X(t))$ when the pseudo-process starts at zero while (24) of [@2003] involves the density of $(T(t), X(t))$ evaluated at the extremity $X(t)=0$ when the starting point is $x$. Actually, both formulas are identical by invoking the duality upon changing $x$ into $-x$, but they were obtained through different approaches.
Inverting with respect to $\nu$ {#section-inverting-nu}
===============================
In this section, we carry out the inversion with respect to the parameter $\nu$. The cases $x\le 0$ and $x\ge 0$ lead to results which are not quite analogous. This is due to the asymmetry of our problem. So, we split our analysis into two subsections related to the cases $x\le 0$ and $x\ge 0$.
The case $x\le 0$
-----------------
\[theorem-wrt.nu+1\] The Laplace transform with respect to $t$ of the density of the couple $(T(t),X(t))$ is given, when $x\le 0$, by $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t)\in \mathrm{d}s,
X(t) \in \mathrm{d}x\} /(\mathrm{d}s \,\mathrm{d}x)\right]\mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K -1}{N}} s^{\frac{\# K}{N}}}
\sum_{m=0}^{\#K} \alpha_{-m}\,(\lambda s)^{\frac mN} E_{1,\frac{m+\# J}{N}}(\lambda s)
\sum_{k \in K} B_k\theta_k^{m+1} e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}.\end{aligned}$$ \[wrt.nu\]
[<span style="font-variant:small-caps;">Proof\
</span>]{}Recall (\[wrt.mu\]) in the case $x \le 0$: $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
&=
\frac{1}{\lambda^{\frac{\# K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{k \in K} B_k\theta_k \left(\!\vphantom{\prod_{k}}\right.
\sum_{j \in J} \frac{A_j\theta_j}{\theta_k\!\!\sqrt[N]{\lambda}
-\theta_j\!\!\sqrt[N]{\lambda+\nu}}\left.\!\vphantom{\prod_{k}}\right)e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}.\end{aligned}$$ We have to invert with respect to $\nu$ the quantity $$\frac{1}{(\lambda+\nu)^{\frac{\# J-1}{N}}} \sum_{j \in J} \frac{A_j\theta_j}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}
=-\sum_{j \in J} \frac{A_j}{(\lambda+\nu)^{\frac{\# J-1}{N}} (\!\sqrt[N]{\lambda+\nu}
-\frac{\theta_k}{\theta_j} \sqrt[N]{\lambda})}.$$ By using the following elementary equality, which is valid for $\alpha>0$, $$\frac{1}{(\lambda+\nu)^{\alpha}}=\frac{1}{\Gamma(\alpha)} \int_0^{\infty}
e^{-(\lambda+\nu)s}s^{\alpha-1}\,\mathrm{d}s=\int_0^{\infty} e^{-\nu s}
\left(\frac{s^{\alpha-1} e^{-\lambda s}}{\Gamma(\alpha)}\right) \mathrm{d}s,$$ we obtain, for $|\beta|<\sqrt[N]{\lambda+\nu}$, $$\begin{aligned}
\frac{1}{\!\sqrt[N]{\lambda+\nu}-\beta}
&=
\frac{1}{\!\sqrt[N]{\lambda+\nu}} \,\frac{1}{1-\frac{\beta}{\!\sqrt[N]{\lambda+\nu}}}
=\sum_{r=0}^{\infty} \frac{\beta^r}{(\lambda+\nu)^{\frac{r+1}{N}}}
= \sum_{r=0}^{\infty} \frac{\beta^r}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\frac{r+1}{N} \!\!\left.\vphantom{\frac aN}\right)}\int_0^{\infty}e^{-(\lambda+\nu)s}
s^{\frac{r+1}{N}-1} \,\mathrm{d}s
\\
&=
\int_0^{\infty} e^{-\nu s} \left(s^{\frac{1}{N}-1}e^{-\lambda s} \sum_{r=0}^{\infty}
\frac{(\beta \!\sqrt[N]{s}\,)^r}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\frac{r+1}{N}\!\!\left.\vphantom{\frac aN}\right)}\right) \mathrm{d}s.\end{aligned}$$ The sum lying in the last displayed equality can be expressed by means of the Mittag-Leffler function (see [@erdelyi Chap. <span style="font-variant:small-caps;">xviii</span>]): $E_{a,b}(\xi)=\sum_{r=0}^{\infty} \frac{\xi^r}{\Gamma(ar+b)}$. Then, $$\frac{1}{\!\sqrt[N]{\lambda+\nu}-\beta}=\int_0^{\infty} e^{-\nu s}
\left(s^{\frac{1}{N}-1}e^{-\lambda s} E_{\frac{1}{N},\frac{1}{N}}(\beta \sqrt[N]{s}\,)\right) \mathrm{d}s.
\label{mittag}$$ Next, we write $$\sum_{j \in J} \frac{A_j}{ \sqrt[N]{\lambda+\nu}-\frac{\theta_k}{\theta_j}
\sqrt[N]{\lambda}}=\int_0^{\infty} e^{-\nu s} \left[\vphantom{\sum_\in}\right.\!
s^{\frac{1}{N}-1}e^{-\lambda s}
\sum_{j \in J} A_j \,E_{\frac{1}{N},\frac{1}{N}} \!\left(\frac{\theta_k}{\theta_j}\!
\sqrt[N]{\lambda s}\right)\!\!\left.\vphantom{\sum_\in}\right] \mathrm{d}s,
\label{a_sum}$$ where $$\sum_{j \in J} A_j \,E_{\frac{1}{N},\frac{1}{N}} \!\left(\frac{\theta_k}{\theta_j}\!
\sqrt[N]{\lambda s}\right)=\sum_{j \in J} A_j \sum_{r=0}^{\infty}
\left(\frac{\theta_k}{\theta_j}\right)^{\!r} \frac{(\lambda s)^{\frac{r}{N}}}{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!\frac{r+1}{N}\!\!\left.\vphantom{\frac aN}\right)}
=\sum_{r=0}^{\infty} \left(\!\vphantom{\prod_{k}}\right.
\theta_k^r \,\sum_{j \in J} \frac{A_j}{\theta_j^r}\!\left.\vphantom{\prod_{k}}\right)
\frac{(\lambda s)^{\frac{r}{N}}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\frac{r+1}{N}\!\!\left.\vphantom{\frac aN}\right)}.$$ When performing the euclidian division of $r$ by $N$, we can write $r$ as $r=\ell N+m$ with $\ell\ge 0$ and $0 \le m \le N-1$. With this, we have $\theta_j^{-r}=(\theta_j^N)^{-\ell} \,\theta_j^{-m}=\kappa_{\!_{ N}}^{\ell} \,\theta_j^{-m}$ and $\theta_k^r=\kappa_{\!_{ N}}^{\ell} \,\theta_k^m$. Then, $$\theta_k^r \,\sum_{j \in J} \frac{A_j}{\theta_j^r}
=\theta_k^m \,\sum_{j \in J} \frac{A_j}{\theta_j^m}=\theta_k^m \alpha_{-m}.$$ Hence, since by (\[set14\]) the $\alpha_{-m}$, $\#K+1\le m\le N$, vanish, $$\sum_{j \in J} A_j \,E_{\frac{1}{N},\frac{1}{N}}\!\left(\frac{\theta_k}{\theta_j}\!\sqrt[N]{\lambda s}\right)
=\sum_{\ell=0}^{\infty} \sum_{m=0}^{\# K} \alpha_{-m} \theta_k^m\,
\frac{(\lambda s)^{\ell+\frac{m}{N}}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{m+1}{N}\!\!\left.\vphantom{\frac aN}\right)}
=\sum_{m=0}^{\# K} \alpha_{-m} \,\theta_k^m (\lambda s)^{\frac{m}{N}} E_{1,\frac{m+1}{N}}(\lambda s)$$ and (\[a\_sum\]) becomes $$\sum_{j \in J} \frac{A_j}{\!\sqrt[N]{\lambda+\nu}-\frac{\theta_k}{\theta_j}\!\sqrt[N]{\lambda}}
=\int_0^{\infty}e^{-\nu s} \left(s^{\frac{1}{N}-1}e^{-\lambda s}
\sum_{m=0}^{\# K} \alpha_{-m} \,\theta_k^m (\lambda s)^{\frac{m}{N}}
E_{1,\frac{m+1}{N}}(\lambda s)\right) \mathrm{d}s.$$ As a result, by introducing a convolution product, we obtain $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
&=
-\frac{1}{\lambda^{\frac{\# K -1}{N}}} \sum_{k \in K} B_k\theta_k e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}
\\
&\hphantom{=\,}
\times \int_0^{\infty} e^{-\nu s}\left(\int_0^s \frac{\sigma^{\frac{\# J-1}{N}-1}e^{-\lambda \sigma}}{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!\frac{\#J-1}{N}\!\!\left.\vphantom{\frac aN}\right)}
\times e^{-\lambda (s-\sigma)} \sum_{m=0}^{\# K} \alpha_{-m} \theta_k^m \lambda^{\frac m N}
(s-\sigma)^{\frac{m+1}{N}-1} E_{1,\frac{m+1}{N}} (\lambda(s-\sigma)) \,\mathrm{d}\sigma
\right)\mathrm{d}s.\end{aligned}$$ By removing the Laplace transforms with respect to the parameter $\nu$ of each member of the foregoing equality, we extract $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{T(t)\in \mathrm{d}s,
\, X(t) \in \mathrm{d}x\} /(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K -1}{N}}} \sum_{m=0}^{\#K} \alpha_{-m}\lambda^{\frac{m}{N}}
\left(\,\sum_{k \in K} B_k\theta_k^{m+1} e^{-\theta_k\!\!\sqrt[N]{\lambda}\, x}\right)
\int_0^s \frac{\sigma^{\frac{\# J-1}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\frac{\# J-1}{N}\!\!\left.\vphantom{\frac aN}\right)}
\,(s-\sigma)^{\frac{m+1}{N}-1} E_{1,\frac{m+1}{N}} (\lambda(s-\sigma))\,\mathrm{d}\sigma.\end{aligned}$$ The integral lying on the right-hand side of the previous equality can be evaluated as follows: $$\begin{aligned}
\int_0^s \frac{\sigma^{\frac{\# J-1}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\frac{\# J-1}{N}\!\!\left.\vphantom{\frac aN}\right)}\,
(s-\sigma)^{\frac{m+1}{N}-1} E_{1,\frac{m+1}{N}} (\lambda(s-\sigma))\,\mathrm{d}\sigma
&=
\int_0^s \frac{\sigma^{\frac{\# J-1}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\frac{\# J-1}{N}\!\!\left.\vphantom{\frac aN}\right)}
(s-\sigma)^{\frac{m+1}{N}-1} \sum_{\ell=0}^{\infty}
\frac{\lambda^{\ell} (s-\sigma)^{\ell}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+1}{N}\!\!\left.\vphantom{\frac aN}\right)}
\,\mathrm{d}\sigma
\\
&=
\sum_{\ell=0}^{\infty} \lambda^{\ell}
\int_0^s \frac{\sigma^{\frac{\# J-1}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\! \frac{\# J-1}{N} \!\!\left.\vphantom{\frac aN}\right)}\,
\frac{(s-\sigma)^{\ell+\frac{m+1}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+1}{N}\!\!\left.\vphantom{\frac aN}\right)}
\,\mathrm{d}\sigma
\\
&=
\sum_{\ell=0}^{\infty} \frac{\lambda^{\ell} s^{\ell+\frac{m+\# J}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+\#J}{N}
\!\!\left.\vphantom{\frac aN}\right)}
=s^{\frac{m+\# J}{N}-1}E_{1,\frac{m+\#J}{N}} (\lambda s)\end{aligned}$$ from which we deduce (\[wrt.nu\]). [ $\blacksquare$\
]{}
Let us integrate formula (\[wrt.nu\]) with respect to $x$ on $(-\infty,0]$. This gives $$\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P} \{T(t) \in \mathrm{d}s,\,
X(t)\le 0\}/\mathrm{d}s\right] \mathrm{d}t
=\frac{e^{-\lambda s}}{(\lambda s)^{\frac{\# K}{N}}} \sum_{m=0}^{\# K} \alpha_{-m}\beta_m
(\lambda s)^{\frac{m}{N}} E_{1,\frac{m+\# J}{N}}(\lambda s).$$ In view of (\[set14\]) and (\[set15\]), since $E_{1,1}(\lambda s)=e^{\lambda s}$, $$\begin{aligned}
\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P} \{T(t) \in \mathrm{d}s,\,
X(t) \le 0\}/\mathrm{d}s\right] \mathrm{d}t
&=
\frac{e^{-\lambda s}}{(\lambda s)^{\frac{\# K}{N}}} \left(\alpha_0\beta_0
E_{1,\frac{\# J}{N}}(\lambda s)+\alpha_{-\#K}\beta_{\#K}
(\lambda s)^{\frac{\# K}{N}} E_{1,1}(\lambda s)\right)
\nonumber\\
&=
\frac{e^{-\lambda s}}{(\lambda s)^{\frac{\# K}{N}}} E_{1,\frac{\# J}{N}}(\lambda s)-1.
\label{star}\end{aligned}$$ We can rewrite $E_{1,\frac{\# J}{N}}(\lambda s)$ as an integral by using Lemma \[8.2\]. We obtain that $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P} \{T(t)\in \mathrm{d}s,
\, X(t) \le 0\}/\mathrm{d}s\right] \mathrm{d}t}$}}
&=
\frac{e^{-\lambda s}}{(\lambda s)^{\frac{\# K}{N}}}\, \frac{1}{(\lambda s)^{\frac{\# J}{N}-1}}
\left(e^{\lambda s}+\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right.
\frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi} \int_0^{\infty}
\frac{t^{1-\frac{\# J}{N}}}{t+1} \,e^{-\lambda s t}\,\mathrm{d}t\right)-1
\\
&=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi} e^{-\lambda s} \int_0^{\infty}
\frac{t^{\frac{\# K}{N}}}{t+1} \,e^{-\lambda s t}\,\mathrm{d}t
=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi s^{\frac{\# K}{N}}} e^{-\lambda s}
\int_0^{\infty} \frac{t^{\frac{\# K}{N}}}{t+s} \,e^{-\lambda t}\,\mathrm{d}t
\\
&=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi s^{\frac{\# K}{N}}}
\int_s^{\infty} \frac{(t-s)^{\frac{\# K}{N}}}{t} \,e^{-\lambda t}\,\mathrm{d}t.\end{aligned}$$ From this, we extract, for $0 < s <t$, $$\mathbb{P}\{T(t) \in \mathrm{d}s,\, X(t) \le 0\}/\mathrm{d}s
=\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi t} \left(\frac{t-s}{s}\right)^{\!\frac{\# K}{N}}.
\label{f1}$$ We retrieve Theorem 14 of [@2003]. By integrating (\[f1\]) with respect to $s$, we get $$\mathbb{P} \{ X(t) \le 0\} =\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right.
\frac{\# K}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi t}
\int_0^t \left(\frac{t-s}{s}\right)^{\!\frac{\# K}{N}}\mathrm{d}s
=\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi}
B\!\left(\frac{\# K}{N}+1,1-\frac{\# K}{N}\right)
=\frac{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}+1\!\!
\left.\vphantom{\frac aN}\right)}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right.
\frac{\# K}{N}\!\!\left.\vphantom{\frac aN}\right)}$$ which simplifies to $\mathbb{P} \{ X(t) \le 0\} =\# K/N.$ We retrieve (11) of [@2003].
An alternative expression for formula (\[wrt.nu\]) is for $x\le 0$ $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t)\in \mathrm{d}s, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
=-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K-1}{N}}s^{\frac{\# K}{N}}}
\sum_{\substack{j \in J \\k \in K}} A_j B_k\theta_k \,E_{\frac{1}{N},\frac{\# J}{N}}\!
\left(\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda s}\right) e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}.
\label{wrt.nu2}\end{aligned}$$ In effect, by (\[wrt.nu\]), $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t)\in \mathrm{d}s,
X(t) \in \mathrm{d}x\} /(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K-1}{N}}s^{\frac{\# K}{N}}} \sum_{\ell=0}^{\infty}
\sum_{m=0}^{\# K} \sum_{k\in K} \alpha_{-m} B_k\theta_k^{m+1}
\frac{(\lambda s)^{\ell+\frac{m}{N}}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\,e^{-\theta_k\!\!\sqrt[N]{\lambda}\, x}
\\
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K-1}{N}}s^{\frac{\# K}{N}}}
\sum_{\ell=0}^{\infty} \sum_{m=0}^{N-1} \sum_{\substack{j \in J \\k \in K}} A_j B_k\theta_k
\left(\frac{\theta_k}{\theta_j}\right)^{\!m} \frac{(\lambda s)^{\ell+\frac{m}{N}}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\,e^{-\theta_k\!\!\sqrt[N]{\lambda}\, x}.\end{aligned}$$ In the last displayed equality, we have extended the sum with respect to $m$ to the range $0\le m\le N-1$ because, by (\[set14\]), the $\alpha_{-m}$, $\# K+1 \le m \le N-1$, vanish. Let us introduce the index $r=\ell N+m$. Since $\left(\frac{\theta_k}{\theta_j}\right)^{\!m}=\left(\frac{\theta_k}{\theta_j}\right)^{\!r}$, we have $$\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t)\in \mathrm{d}s,
X(t) \in \mathrm{d}x\} /(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t
=-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K-1}{N}}s^{\frac{\# K}{N}}}
\sum_{\substack{j \in J \\k \in K}} A_j B_k\theta_k \sum_{r=0}^{\infty}
\frac{\left(\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda s}\right)^{\!r}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\frac{r+\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\,e^{-\theta_k\!\!\sqrt[N]{\lambda}\, x}$$ which coincide with (\[wrt.nu2\]).
Case $N=2$. Suppose $x\le 0$. The first expression (\[wrt.nu\]) reads $$\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s, X(t)\in \mathrm{d}x\}
/ (\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t
=\frac{e^{-\lambda s}}{\sqrt s} \left(E_{1 ,\frac12} (\lambda s) -\sqrt{\lambda s}\,
E_{1,1}(\lambda s)\right) e^{\sqrt \lambda \, x}$$ while the second expression (\[wrt.nu2\]) reads $$\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{T(t) \in \mathrm{d}s, X(t)\in \mathrm{d}x\} /
(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t
=\frac{e^{-\lambda s}}{\sqrt s} \,
E_{\frac12,\frac12} \!\left(\vphantom{\sqrt a}\right.\!\!
-\sqrt{\lambda s}\!\left.\vphantom{\sqrt a}\right) e^{\sqrt \lambda \, x}.$$ From Lemma \[2half\], we have $$E_{\frac12,\frac12} \!\left(\vphantom{\sqrt a}\right.\!\!
-\sqrt{\lambda s}\!\left.\vphantom{\sqrt a}\right)\!
=E_{1,\frac12}(\lambda s)-\sqrt{\lambda s} \,e^{\lambda s}
\label{kub}$$ which proves the coincidence of both formulas. Moreover, from Lemma \[2half\], for $x \le 0$, $$\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s,
X(t)\in \mathrm{d}x\} /(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t
=\left(\frac{e^{-\lambda s}}{\sqrt{\pi s}}-\sqrt{\lambda}
\,\mathrm{Erfc} (\sqrt{\lambda s})\right)
e^{\sqrt \lambda \,x}.$$ We retrieve formula 1.4.6, page 129, of [@bs].
Case $N=3$. We have for $x\le 0$, when $\kappa_3=-1$: $$\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t=
e^{\sqrt[3]{\lambda} \,x} \left(\frac{e^{-\lambda s}}{\sqrt[3]{s}}\,
E_{1,\frac23}(\lambda s)-\sqrt[3] \lambda\,\right)$$ and when $\kappa_3=1$: $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right]\mathrm{d}t}$}}
&=
\frac{e^{-\lambda s+\frac{\sqrt[3]{\lambda}}{2}x}}{\sqrt 3\,\sqrt[3]{\lambda s^2}}
\left[\vphantom{\frac tt}\right.\!\!\sqrt 3 \,\cos\!\left(\vphantom{\frac tt}\right.\!\!
\frac{\sqrt 3 \,\sqrt[3] \lambda \,x}{2}\!\!\left.\vphantom{\frac tt}\right)\!\!
\left(\sqrt[3]{\lambda s}\,
E_{1,\frac23}(\lambda s)-(\lambda s)^{2/3} e^{\lambda s}\right)
\\
&\hphantom{=\,}
+\sin\!\left(\vphantom{\frac tt}\right.\!\!
\frac{\sqrt 3 \,\sqrt[3] \lambda \,x}{2}\!\!\left.\vphantom{\frac tt}\right)\!\!
\left(\sqrt[3]{\lambda s}\,
E_{1,\frac23}(\lambda s)+(\lambda s)^{2/3} e^{\lambda s}-2E_{1,\frac13} (\lambda s)\right)
\!\!\left.\vphantom{\frac tt}\right]\!.\end{aligned}$$
Case $N=4$. We have, for $x\ge 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
\frac{\sqrt 2\, e^{-\lambda s+\frac{\sqrt[4]\lambda}{\sqrt 2}x}}{\sqrt[4]\lambda \,\sqrt s}
\left[\vphantom{\frac tt}\right.\!\!\cos\!\left(\vphantom{\frac tt}\right.\!\!
\frac{\sqrt[4]\lambda \,x}{\sqrt 2}\!\!\left.\vphantom{\frac tt}\right)\!\!
\left(\sqrt[4]{\lambda s}\,E_{1,\frac34}(\lambda s) -\sqrt{\lambda s} \,e^{\lambda s}\right)
+\sin\!\left(\vphantom{\frac tt}\right.\!\!
\frac{\sqrt[4]\lambda \,x}{\sqrt 2}\!\!\left.\vphantom{\frac tt}\right)\!\!
\left(\sqrt[4]{\lambda s}\,E_{1,\frac34}(\lambda s) -E_{1,\frac12}(\lambda s)\right)
\!\!\left.\vphantom{\frac tt}\right]\!.\end{aligned}$$
The case $x\ge 0$
-----------------
\[theorem-wrt.nu+2\] The Laplace transform with respect to $t$ of the density of the couple $(T(t),X(t))$ is given, when $x\ge 0$, by $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} [\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)] \,\mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K-1}{N}}} \sum_{\substack{j \in J \\k \in K}}
A_j B_k\theta_k \int_0^s \sigma^{\frac 1N-1} E_{\frac 1N,\frac 1N}\!
\left(\frac{\theta_k}{\theta_j} \sqrt[N]{\lambda \sigma}\right)
I_{j,\# J-1}(s-\sigma;x)\,\mathrm{d}\sigma
\label{wrt.nu+1}\end{aligned}$$ where the function $I_{j,\# J-1}$ is defined by (\[set18\]).
[<span style="font-variant:small-caps;">Proof\
</span>]{}Recall (\[wrt.mu\]) in the case $x \ge 0$: $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{E}\!\left(\!\!\vphantom{e^t}\right.\right.
e^{-\nu T(t)},\, X(t) \in \mathrm{d}x \!\!\left.\left.\vphantom{e^t}\right)\!/\mathrm{d}x\right]\mathrm{d}t}$}}
&=
\frac{1}{\lambda^{\frac{\# K-1}{N}}(\lambda+\nu)^{\frac{\# J-1}{N}}}
\sum_{j \in J} A_j \theta_j \left(\,\sum_{k \in K}
\frac{B_k\theta_k}{\theta_k\!\!\sqrt[N]{\lambda}-\theta_j\!\!\sqrt[N]{\lambda+\nu}}\right)
e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x}.\end{aligned}$$ We have to invert the quantity $\frac{e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x}}{(\lambda+\nu)^{\frac{\# J-1}{N}}
\left(\vphantom{\sqrt t}\right.\!\!\!\sqrt[N]{\lambda+\nu}
-\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda}\left.\vphantom{\sqrt t}\right)}$ with respect to $\nu$. Recalling (\[mittag\]) and (\[set24\]), $$\begin{aligned}
\frac{1}{\!\sqrt[N]{\lambda +\nu}-\beta}
&=
\int_0^{\infty} e^{-\nu s}\left(s^{\frac{1}{N}-1}
e^{-\lambda s} E_{\frac 1N,\frac 1N} \!\left(\beta \sqrt[N]{s}\,\right)\right) \mathrm{d}s,
\\
\frac{e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x}}{(\lambda+\nu)^{\frac{\# J-1}{N}}}
&=
\int_0^{\infty} e^{-\nu s}\left(e^{-\lambda s} I_{j,\# J-1} (s;x)\right)\mathrm{d}s,\end{aligned}$$ we get by convolution $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\frac{e^{-\theta_j\!\!\sqrt[N]{\lambda+\nu}\,x}}{(\lambda+\nu)^{\frac{\# J-1}{N}}
\left(\vphantom{\sqrt t}\right.\!\!\!\sqrt[N]{\lambda+\nu}-\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda}
\left.\vphantom{\sqrt t}\right)}}$}}
&=
\int_0^{\infty} e^{-\nu s} \left(\int_0^s \sigma^{\frac 1N-1} e^{-\lambda \sigma}
E_{\frac 1N,\frac1N} \!\left(\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda \sigma}\right)
\times e^{-\lambda (s-\sigma)} I_{j,\# J-1} (s-\sigma;x)\,\mathrm{d}\sigma\right)\mathrm{d}s
\\
{\noalign{\noindent $\displaystyle{}$}}
&=
\int_0^{\infty} e^{-\nu s} \left(e^{-\lambda s} \int_0^s \sigma^{\frac 1N-1}
E_{\frac 1N,\frac1N} \!\left(\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda \sigma}\right)
I_{j,\# J-1} (s-\sigma;x)\,\mathrm{d}\sigma\right)\mathrm{d}s.\end{aligned}$$ This immediately yields (\[wrt.nu+1\]). [ $\blacksquare$\
]{}
Noticing that $$E_{\frac 1N,\frac 1N} \!\left(\frac{\theta_k}{\theta_j} \!\sqrt[N]{\lambda \sigma}\right)
=\sum_{r=0}^{\infty} \frac{\theta_k^r}{\theta_j^r} \,
\frac{(\lambda \sigma)^{\frac{r}{N}}}{\Gamma\!\left(\frac{r+1}{N}\right)}
=\sum_{\ell=0}^{\infty} \sum_{m=0}^{N-1} \frac{\theta_k^m}{\theta_j^m}\,
\frac{(\lambda \sigma)^{\ell+\frac{m}{N}}}{\Gamma\!\left(\ell+\frac{m+1}{N}\right)}
=\sum_{m=0}^{N-1} \frac{\theta_k^m}{\theta_j^m} \,(\lambda \sigma)^{\frac{m}{N}}
E_{1,\frac{m+1}{N}}(\lambda \sigma)$$ and reminding that, from (\[set26\]), the $\beta_m$, $1\le m \le \# K-1$, vanish, we can rewrite (\[wrt.nu+1\]) in the following form. For $x\ge 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
-e^{-\lambda s} \sum_{m=\# K -1}^{N-1} \left(\,\sum_{k \in K} B_k\theta_k^{m+1}\right)
\lambda^{\frac{m-\# K +1}{N}} \int_0^s \sigma^{\frac{m+1}{N}-1}
E_{1,\frac{m+1}{N}} (\lambda \sigma) \left(\!\vphantom{\sum_{\in}}\right.\sum_{j \in J}
\frac{A_j}{\theta_j^m}I_{j,\#J-1}(s-\sigma;x) \left.\vphantom{\sum_{\in}}\!\right)\mathrm{d}\sigma
\nonumber \\
&=
-e^{-\lambda s} \sum_{m=\# K}^{N} \beta_m\, \lambda^{\frac{m-\# K}{N}} \int_0^s
\sigma^{\frac{m}{N}-1} E_{1,\frac{m}{N}} (\lambda \sigma) \,\Phi_m(s-\sigma;x) \,\mathrm{d}\sigma
\label{wrt.nu+2}\end{aligned}$$ with $\Phi_m(\tau;x)= \sum_{j \in J} \frac{A_j}{\theta_j^{m-1}} \,I_{j,\#J-1}(\tau;x)$.
Let us integrate (\[wrt.nu+2\]) with respect to $x$ on $[0,\infty)$. We first compute $$\begin{aligned}
\int_0^{\infty} \Phi_m(\tau;x) \,\mathrm{d}x
&=
\frac{N i}{2 \pi}\left(\!\vphantom{\sum_{\in}}\right.\sum_{j \in J}
\frac{A_j}{\theta_j^m}\left.\!\vphantom{\sum_{\in}}\right)\!\!
\left(e^{-i\frac{\#J}{N}\pi}-e^{i\frac{\#J}{N}\pi}\right)
\int_0^{\infty} \xi^{\# K-1} \,e^{-\tau \xi^N} \,\mathrm{d}\xi
\\
&=
\frac{\Gamma\!\left(\!\!\vphantom{\frac aN}\right.\frac{\# K}{N}\!\!
\left.\vphantom{\frac aN}\right)\sin\!\left(\!\!\vphantom{\frac aN}\right.
\frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi \tau^{\frac{\# K}{N}}}\,\alpha_{-m}
=\frac{\alpha_{-m}}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right) \tau^{\frac{\# K}{N}}}.\end{aligned}$$ Then, with the aid of (\[set26\]) and (\[set15\]), we get $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t)\ge 0\}
/ \mathrm{d}s\right] \mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right.\frac{\#J}{N}\!\!
\left.\vphantom{\frac aN}\right)}\left(\alpha_{-\# K}\,
\beta_{\#K} \int_0^s\frac{\sigma^{\frac{\#K}{N}-1}}{(s-\sigma)^{\frac{\# K}{N}}}
E_{1,\frac{\# K}{N}}(\lambda \sigma)\,\mathrm{d}\sigma + \alpha_{-N}\,\beta_{N} \,
\lambda^{\frac{\#J}{N}} \int_0^s\frac{E_{1,1}(\lambda \sigma)}{(s-\sigma)^{\frac{\# K}{N}}}
\,\mathrm{d}\sigma\right)
\\
&=
\frac{e^{-\lambda s}}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}\left(\int_0^s
\frac{E_{1,\frac{\# K}{N}}(\lambda \sigma)}{\sigma^{\frac{\#J}{N}} (s-\sigma)^{\frac{\#K}{N}}}
\,\mathrm{d}\sigma-\lambda^{\frac{\#J}{N}}
\int_0^s\frac{e^{\lambda \sigma}}{(s-\sigma)^{\frac{\# K}{N}}}\,\mathrm{d}\sigma\right)
\\
&=
\frac{e^{-\lambda s}}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right)} \sum_{\ell=0}^{\infty}
\frac{\lambda^{\ell}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{\# K}{N}\!\!\left.\vphantom{\frac aN}\right)}
\int_0^s \frac{\sigma^{\ell-\frac{\#J}{N}}}{(s-\sigma)^{\frac{\#K}{N}}} \,\mathrm{d}\sigma
-\frac{\lambda^{\frac{\#J}{N}}}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\int_0^s \frac{e^{-\lambda( s-\sigma)}}{(s-\sigma)^{\frac{\# K}{N}}} \,\mathrm{d}\sigma
\\
&=
e^{-\lambda s} \sum_{\ell=0}^{\infty} \frac{(\lambda s)^{\ell}}{\ell!}
-\frac{1}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\int_0^{\lambda s} \frac{e^{-\sigma}}{\sigma^{\frac{\# K}{N}}} \,\mathrm{d}\sigma
=
1-\frac{1}{\Gamma\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\int_0^{\lambda s} \sigma^{\frac{\# J}{N}-1} e^{-\sigma} \,\mathrm{d}\sigma.\end{aligned}$$ By Lemma \[1alpha\], this simplifies into $$\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t)\ge 0\}/
\mathrm{d}s\right] \mathrm{d}t
=1-(\lambda s)^{\frac{\# J}{N}} e^{-\lambda s} E_{1,\frac{\# J}{N}+1}(\lambda s).
\label{TLXpositive}$$ Now, using Lemma \[8.2\], we derive another representation for the foregoing Laplace transform: $$\begin{aligned}
\int_0^{\infty}e^{-\lambda t}\left[\mathbb{P}\{T(t)\in \mathrm{d}s, X(t) \ge 0\}/
\mathrm{d}s\right] \mathrm{d}t
&=
1-e^{-\lambda s} \left(e^{\lambda s}-\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi}
\int_0^{\infty} \frac{t^{-\frac{\# J}{N}}}{t+1} \,e^{-\lambda s t}\,\mathrm{d}t\right)
\\
&=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi} \,s^{\frac{\# J}{N}}
\int_0^{\infty} \frac{e^{-\lambda(t+s)}}{(t+s) \,t^{\frac{\# J}{N}}} \,\mathrm{d}t
\\
&=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi} \,s^{\frac{\# J}{N}}
\int_{s}^{\infty} \frac{e^{-\lambda t}}{t (t-s)^{\frac{\# J}{N}}} \,\mathrm{d}t.\end{aligned}$$ As a result, we derive $$\mathbb{P}\{T(t)\in \mathrm{d}s, X(t) \ge 0\}/ \mathrm{d}s
=\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi t}
\left(\frac{s}{t-s}\right)^{\!\frac{\# J}{N}},\qquad 0<s<t.
\label{f2}$$ This is formula (11) of [@2003].
If we add (\[star\]) and (\[TLXpositive\]), we find $$\int_0^{\infty}e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s\} /\mathrm{d}s\right] \mathrm{d}t
=e^{-\lambda s} \left(\sum_{\ell=0}^{\infty}
\frac{(\lambda s)^{\ell+\frac{\# J}{N}-1}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
-\sum_{\ell=0}^{\infty} \frac{(\lambda s)^{\ell+\frac{\# J}{N}}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\! \ell+\frac{\# J}{N}+1 \!\!
\left.\vphantom{\frac aN}\right)}\right)
=\frac{e^{-\lambda s}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\frac{\# J}{N}\!\!\left.\vphantom{\frac aN}\right) \!(\lambda s)^{\frac{\# K}{N}}}.$$ It is easy to invert this Laplace transform. Indeed, $$\begin{aligned}
\frac{e^{-\lambda s}}{\lambda^{\frac{\#K}{N}}}=\frac{e^{-\lambda s}}{\Gamma\!
\left(\vphantom{\frac aN}\right.\!\!\frac{\# K}{N} \!\!\left.\vphantom{\frac aN}\right)}
\int_0^{\infty} t^{\frac{\#K}{N}-1} e^{-\lambda t}\,\mathrm{d}t
&=
\frac{1}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\frac{\# K}{N}
\!\left.\vphantom{\frac aN}\right)} \int_{s}^{\infty} (t-s)^{\frac{\# K}{N}-1} e^{-\lambda t}\,\mathrm{d}t
\\
&=
\frac{1}{\pi}\,\Gamma\!\left(\frac{\# J}{N}\right) \,\sin\!\left(\frac{\# J}{N}\pi\right)
\int_{s}^{\infty} \frac{e^{-\lambda t}}{(t-s)^{\frac{\# J}{N}}} \,\mathrm{d}t.\end{aligned}$$ This implies that $$\mathbb{P}\{T(t) \in \mathrm{d}s\} /\mathrm{d}s
=\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!\left.\vphantom{\frac aN}\right)}{\pi}
\frac{{1\hspace{-.27em}\mbox{\rm l}}_{(0,t)}(s)}{ s^{\frac{\# K}{N}}(t-s)^{\frac{\# J}{N}}}$$ which can be also obtained by adding directly (\[f1\]) and (\[f2\]). Thus, we retrieve the famous counterpart to the Paul Lévy’s arc-sine law stated in [@2003] (Corollary 9).
For $x=0$, using formula (\[wrt.nu\]) which is valid for $x\le0$, we get, by (\[set26\]), (\[set14\]) and (\[set15\]), $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty}e^{-\lambda t} \,\mathbb{P}\{T(t) \in \mathrm{d}s,\, X(t)\in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x) \Big|_{x=0}\,\mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\#K-1}{N}} s^{\frac{\# K}{N}}}
\sum_{m=0}^{\# K} \alpha_{-m}\,\beta_{m+1}\, (\lambda s)^{\frac mN} E_{1,\frac{m+\# J}{N}} (\lambda s)
\nonumber\\
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\#K-1}{N}} s^{\frac{\# K}{N}}}
\left(\alpha_{1-\#K}\,\beta_{\#K}\, (\lambda s)^{\frac{\#K-1}{N}} E_{1,1-\frac1N}(\lambda s)
+\alpha_{-\#K}\,\beta_{\#K+1}\, (\lambda s)^{\frac{\#K}{N}} E_{1,1}(\lambda s)\right)
\nonumber
\\
&=
\frac{e^{-\lambda s}}{\lambda^{\frac{\#K-1}{N}} s^{\frac{\# K}{N}}}
\left(\!\vphantom{\sum_\in}\right.\sum_{j \in J} \theta_j (\lambda s)^{\frac{\# K-1}{N}}
E_{1,1-\frac1N}(\lambda s)+\sum_{k \in K} \theta_k (\lambda s)^{\frac{\#K}{N}}
e^{\lambda s}\left.\!\!\vphantom{\sum_\in}\right)
\nonumber\\
&=
\left(\!\vphantom{\sum_{\in}}\right.\sum_{j \in J} \theta_j
\left.\!\vphantom{\sum_{\in}}\right) \frac{e^{-\lambda s}}{\!\!\sqrt[N]{s}}
\left(E_{1,1-\frac 1N}(\lambda s)-\sqrt[N]{\lambda s}\,e^{\lambda s}\right)\!.
\label{LTX-zero1}\end{aligned}$$ On the other hand, with formula (\[wrt.nu+2\]) which is valid for $x \ge 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty}e^{-\lambda t} \,\mathbb{P}\{T(t) \in \mathrm{d}s,
\,X(t)\in \mathrm{d}x\}/(\mathrm{d}s\,\mathrm{d}x )\Big|_{x=0} \,\mathrm{d}t}$}}
&=
-e^{-\lambda s} \sum_{m=\# K}^{N} \beta_m\,\lambda^{\frac{m-\# K}{N}}
\int_0^s \sigma^{\frac{m}{N}-1} E_{1,\frac mN} (\lambda \sigma) \,\Phi_m(s-\sigma;0) \,\mathrm{d}\sigma
\label{LTX-zero2}\end{aligned}$$ with $$\begin{aligned}
\Phi_m(\tau;0)
&=
\frac{Ni}{2 \pi} \left(\!\vphantom{\sum_{\in}}\right.
\sum_{j \in J} \frac{A_j}{\theta_j^{m-1}} \left.\!\vphantom{\sum_{\in}}\right)\!\!
\left(e^{-i \frac{\# J-1}{N}\pi}-e^{i \frac{\#J-1}{N}\pi}\right)
\int_0^{\infty} \xi^{\# K} e^{-\tau \xi^N} \,\mathrm{d}\xi
\\
&=
\frac{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\frac{\# K+1}{N} \!\!\left.\vphantom{\frac aN}\right)
\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J-1}{N}\pi
\!\!\left.\vphantom{\frac aN}\right)}{\pi\,\tau^{\frac{\#K+1}{N}}}\,\alpha_{1-m}
= \frac{\alpha_{1-m}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!\frac{\#J-1}{N}
\!\!\left.\vphantom{\frac aN}\right) \tau^{\frac{\#K+1}{N}}}.\end{aligned}$$ In view of (\[set26\]), (\[set14\]) and (\[set15\]), we have $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \,\mathbb{P}\{T(t) \in \mathrm{d}s,
\, X(t)\in \mathrm{d}x\}/(\mathrm{d}s\,\mathrm{d}x ) \Big|_{x=0} \,\mathrm{d}t}$}}
&=
\frac{e^{-\lambda s}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!\frac{\# J-1}{N}
\!\!\left.\vphantom{\frac aN}\right)} \left[\left(\!\vphantom{\sum_{\in}}\right.
\sum_{j \in J} \theta_j \!\left.\vphantom{\sum_{\in}}\right)
\int_0^s \frac{\sigma^{\frac{\#K}{N}-1}}{(s-\sigma)^{\frac{\#K+1}{N}}}\,
E_{1,\frac{\#K}{N}} (\lambda \sigma)\,\mathrm{d}\sigma\right.
\\
&\hphantom{=\,}
\left.+ \left(\,\sum_{k \in K} \theta_k\right) \!\!\sqrt[N]{\lambda}
\int_0^s \frac{\sigma^{\frac{\#K+1}{N}-1}}{(s-\sigma)^{\frac{\#K+1}{N}}}\,
E_{1,\frac{\#K+1}{N}} (\lambda \sigma) \,\mathrm{d}\sigma\right]
\\
&=
\left(\!\vphantom{\sum_{\in}}\right.\sum_{j \in J} \theta_j\!\left.\vphantom{\sum_{\in}}\right)
\frac{e^{-\lambda s}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\frac{\#J-1}{N} \!\!\left.\vphantom{\frac aN}\right)} \,e^{-\lambda s}
\left(\,\sum_{\ell=0}^{\infty} \frac{B\!\left(\!\!\vphantom{\frac aN}\right.
\ell+\frac{\# K}{N}, 1-\frac{\# K+1}{N}\!\!\left.\vphantom{\frac aN}\right)}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{\# K}{N}\!\!\left.\vphantom{\frac aN}\right)}\,
\lambda^{\ell}s^{\ell-\frac 1N}\right.
\\
&\hphantom{=\,}
\left.-\!\sqrt[N]\lambda \,\sum_{\ell=0}^{\infty} \frac{B\!\left(\!\!\vphantom{\frac aN}\right.
\ell+\frac{\# K+1}{N}, 1-\frac{\# K+1}{N}\!\!\left.\vphantom{\frac aN}\right)}{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{\# K+1}{N}\!\!\left.\vphantom{\frac aN}\right)}\,
(\lambda s)^{\ell}\right)
\\
&=
\left(\!\vphantom{\sum_{\in}}\right.\sum_{j \in J} \theta_j
\left.\!\vphantom{\sum_{\in}}\right) \frac{e^{-\lambda s}}{\!\!\sqrt[N]s}
\left(E_{1, 1-\frac 1N}(\lambda s)-\sqrt[N]{\lambda s} \,e^{\lambda s}\right)\!.\end{aligned}$$ Thus, we have checked that the two different formulas (\[LTX-zero1\]) and (\[LTX-zero2\]) lead to the same result.
Case $N=2$. Suppose $x\ge 0$. Formula (\[wrt.nu+1\]) reads, with the numerical values of Example \[example1\], $$\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{T(t) \in \mathrm{d}s, X(t)\in \mathrm{d}x\}
/(\mathrm{d}s\,\mathrm{d}x)\right] \mathrm{d}t
=e^{-\lambda s} \int_0^s \frac{1}{\sqrt \sigma}\,
E_{\frac12,\frac12}\!\left(\vphantom{\sqrt a}\right.\!\!-\sqrt{\lambda \sigma}
\!\left.\vphantom{\sqrt a}\right) I_{1,0}(s-\sigma;x)\,\mathrm{d}\sigma$$ while formula (\[wrt.nu+2\]) gives, because of $\Phi_1=\Phi_2=I_{1,0}$ and (\[kub\]), $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s, \,X(t)\in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
e^{-\lambda s} \left(\int_0^s \frac{1}{\sqrt{\sigma}}\,
E_{1,\frac12} (\lambda \sigma) \,\Phi_1(s-\sigma;x) \,\mathrm{d}\sigma -\sqrt \lambda
\int_0^s E_{1,1}(\lambda \sigma) \,\Phi_2(s-\sigma;x) \,\mathrm{d}\sigma\right)
\\
&=
e^{-\lambda s} \int_0^s \frac{1}{\sqrt{\sigma}} \left(E_{1,\frac12}(\lambda \sigma)
-\sqrt{\lambda \sigma}\,e^{\lambda \sigma}\right) I_{1,0} (s-\sigma;x)\,\mathrm{d}\sigma
\\
&=
e^{-\lambda s} \int_0^s \frac{1}{\sqrt{\sigma}} \,
E_{\frac12,\frac12}\!\left(\vphantom{\sqrt a}\right.\!\!-\sqrt{\lambda \sigma}
\!\left.\vphantom{\sqrt a}\right) I_{1,0} (s-\sigma;x)\,\mathrm{d}\sigma.\end{aligned}$$ We have checked that the two different representations (\[wrt.nu+1\]) and (\[wrt.nu+2\]) lead to the same result. Let us pursue the computations. In view of (\[case2I\]) and Lemma \[2half\], we get $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s,
\, X(t)\in \mathrm{d}x\} /(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
\frac{x e^{-\lambda s}}{2 \sqrt{\pi}} \int_0^s \frac{e^{-\frac{x^2}{4(s-\sigma)}}}{\sqrt \sigma\,
(s-\sigma)^{3/2}} \,E_{\frac12,\frac12}\!\left(\vphantom{\sqrt a}\right.
\!\!-\sqrt{\lambda \sigma}\!\left.\vphantom{\sqrt a}\right) \mathrm{d}\sigma
\\
&=
\frac{x e^{-\lambda s}}{2 \sqrt{\pi}} \int_0^s \frac{e^{-\frac{x^2}{4\sigma}}}{
\sigma^{3/2}\sqrt{s-\sigma}} \,E_{\frac12,\frac12}\!\left(\vphantom{\sqrt a}\right.
\!\!-\sqrt{\lambda(s-\sigma)}\!\left.\vphantom{\sqrt a}\right) \mathrm{d}\sigma
\\
&=
\frac{xe^{-\lambda s}}{2 \sqrt \pi} \int_0^s \frac{e^{-\frac{x^2}{4\sigma}}}{
\sigma^{3/2}\sqrt{s-\sigma}} \left(\frac{1}{\sqrt \pi} -\sqrt{\lambda (s-\sigma)}
\,e^{\lambda(s-\sigma)}\,
\mathrm{Erfc}\!\left(\vphantom{\sqrt a}\right.\!\!\sqrt{\lambda (s-\sigma)}
\!\left.\vphantom{\sqrt a}\right)\right)\mathrm{d}\sigma
\\
&=
\frac{xe^{-\lambda s}}{2 \sqrt \pi} \left[\frac{1}{\sqrt \pi}
\int_0^s \frac{e^{-\frac{x^2}{4\sigma}}}{\sigma^{3/2}\sqrt{s-\sigma}} \,\mathrm{d}\sigma
-\frac{2 \sqrt{\lambda}}{\sqrt \pi} \int_0^s \frac{e^{\lambda(s-\sigma)-\frac{x^2}{4\sigma}}}{\sigma^{3/2}}
\left(\int_{\sqrt{\lambda (s-\sigma)}}^{\infty} e^{-\xi^2}\,\mathrm{d}\xi\right) \mathrm{d}\sigma\right]\!.\end{aligned}$$ The first integral in the last displayed equality writes, with the change of variable $\sigma=s^2/\tau$, $$\int_0^s \frac{e^{-\frac{x^2}{4\sigma}}}{ \sigma^{3/2} \sqrt{s-\sigma}} \,\mathrm{d}\sigma
=\frac{1}{s^{3/2}}\int_{s}^{\infty} \frac{e^{-\frac{x^2}{4 s^2}\tau}}{\sqrt{\tau-s}} \,\mathrm{d}\tau
=\frac{e^{-\frac{x^2}{4 s}}}{s^{3/2}} \int_0^{\infty}
\frac{e^{-\frac{x^2}{4 s^2}\tau}}{\sqrt \tau} \,\mathrm{d}\tau
=\frac{2 \sqrt \pi}{\sqrt s \,x} \,e^{-\frac{x^2}{4 s}},$$ and then $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s, X(t)\in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right]\mathrm{d}t}$}}
&=
\frac{1}{\sqrt{\pi s}} \, e^{-\lambda s-\frac{x^2}{4s}}
-\frac{x \,\sqrt \lambda}{\pi} \int_0^s \frac{e^{-\lambda\sigma-\frac{x^2}{4\sigma}}}{\sigma^{3/2}}
\left(\int_{\sqrt{\lambda(s-\sigma)}}^{\infty} e^{-\xi^2} \,\mathrm{d}\xi\right)\mathrm{d}\sigma.\end{aligned}$$ The computation of the integral lying on the right-hand side of the foregoing equality being cumbersome is postponed to Lemma \[secondint\] in the appendix. The final result is, for $x\ge 0$, $$\int_0^{\infty} e^{-\lambda t}\left[\mathbb{P}\{T(t) \in \mathrm{d}s, X(t)\in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t=\frac{e^{-\lambda s-\frac{x^2}{4s}}}{\sqrt{\pi s}}
-\sqrt \lambda \,e^{\sqrt{\lambda} \,x}\, \mathrm{Erfc}\!\left(\frac{x}{2 \sqrt s} +\sqrt{\lambda s}\right)\!.$$ This is formula 1.4.6, page 129, of [@bs].
Case $N=3$. For $x \ge 0$, (\[wrt.nu+1\]) supplies with the numerical values of Example \[example2\], when $\kappa_3=-1$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,
\, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
\frac{e^{-\lambda s}}{\sqrt 3} \left(e^{\frac{i \pi}{6}} \int_0^s \sigma^{-2/3}
E_{\frac13,\frac13}\!\left(\vphantom{\sqrt a}\right.\!\!
-e^{-i\frac{\pi}{3}} \sqrt[3]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{1,1}(s-\sigma;x) \,\mathrm{d}\sigma\right.
\\
&\hphantom{=\,}
\left. +\, e^{-\frac{i \pi}{6}} \int_0^s \sigma^{-2/3}
E_{\frac13,\frac13}\!\left(\vphantom{\sqrt a}\right.\!\!
-e^{i\frac{\pi}{3}} \sqrt[3]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{2,1}(s-\sigma;x) \,\mathrm{d}\sigma\right)\end{aligned}$$ and when $\kappa_3=1$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,
\, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
\frac{i\,e^{-\lambda s}}{\sqrt 3 \,\sqrt[3]\lambda} \left(\int_0^s \sigma^{-2/3}
E_{\frac13,\frac13}\!\left(\vphantom{\sqrt a}\right.\!\!
e^{-i\frac{2\pi}{3}} \sqrt[3]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{1,0}(s-\sigma;x) \,\mathrm{d}\sigma\right.
\\
&\hphantom{=\,}
\left. - \int_0^s \sigma^{-2/3} E_{\frac13,\frac13}\!\left(\vphantom{\sqrt a}\right.\!\!
e^{i\frac{2\pi}{3}} \sqrt[3]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{1,0}(s-\sigma;x) \,\mathrm{d}\sigma\right)\!.\end{aligned}$$ The functions $I_{1,0}$, $I_{1,1}$ and $I_{2,1}$ above are respectively given by (\[case3I10\]), (\[case3I11\]) and (\[case3I21\]).
Case $N=4$. For $x \ge 0$, (\[wrt.nu+1\]) supplies, with the numerical values of Example \[example3\], $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \left[\mathbb{P}\{ T(t)\in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}
/(\mathrm{d}s \,\mathrm{d}x)\right] \mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{2 \sqrt[4] \lambda}\left(e^{i\frac{\pi}{4}}
\int_0^s \sigma^{-3/4} E_{\frac14,\frac14}\!\left(\vphantom{\sqrt a}\right.\!\!
-\!\sqrt[4]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{1,1}(s-\sigma;x)\,\mathrm{d}\sigma\right.
\\
&\hphantom{=\,}
\left. +\,e^{-i\frac{3\pi}{4}} \int_0^s \sigma^{-3/4}
E_{\frac14,\frac14}\!\left(\vphantom{\sqrt a}\right.\!\!
-i\sqrt[4]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{1,1}(s-\sigma;x)\,\mathrm{d}\sigma\right.
\\
&\hphantom{=\,}
\left. +\,e^{i\frac{3\pi}{4}} \int_0^s \sigma^{-3/4}
E_{\frac14,\frac14}\!\left(\vphantom{\sqrt a}\right.\!\!
i\sqrt[4]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{2,1}(s-\sigma;x)\,\mathrm{d}\sigma\right.
\\
&\hphantom{=\,}
\left.+\,e^{-i\frac{\pi}{4}} \int_0^s \sigma^{-3/4}
E_{\frac14,\frac14}\!\left(\vphantom{\sqrt a}\right.\!\!
-\!\sqrt[4]{\lambda\sigma}\!\left.\vphantom{\sqrt a}\right)
I_{2,1}(s-\sigma;x)\,\mathrm{d}\sigma\right)\!.\end{aligned}$$ The functions $I_{1,1}$ and $I_{2,1}$ above are given by (\[case4I\]).
Inverting with respect to $\lambda$ {#section-inverting-lambda}
===================================
In this section, we perform the last inversion in $F(\lambda,\mu,\nu)$ in order to derive the distribution of the couple $(T(t),X(t))$. As in the previous section, we treat separately the two cases $x\le 0$ and $x\ge 0$.
The case $x\le 0$
-----------------
The distribution of the couple $(T(t),X(t))$ is given, for $x \le 0$, by $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/ \mathrm{d}s \,\mathrm{d}x}$}}
&=
-\frac{Ni}{2 \pi}\sum_{m=0}^{\# K} \alpha_{-m} s^{\frac{m-\#K}{N}}
\int_0^{\infty} \xi^{m+\# J} e^{-(t-s)\xi^N}
\mathcal{K}_m(x \xi) \,E_{1,\frac{m+\# J}{N}}(-s \xi^N)\,\mathrm{d}\xi
\label{wrt.lambda}\end{aligned}$$ where $$\mathcal{K}_m(z)=e^{-i \frac{\# K-m-1}{N}\pi} \sum_{k \in K} B_k\theta_k^{m+1}
e^{-\theta_k e^{i\frac{\pi}{N}}z}-e^{i \frac{\# K -m-1}{N}\pi} \sum_{k \in K} B_k\theta_k^{m+1}
e^{-\theta_k e^{-i\frac{\pi}{N}}z}.$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Assume $x \le 0$. Recalling (\[wrt.nu\]), we have $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} e^{-\lambda t} \,[\mathbb{P}\{ T(t)\in \mathrm{d}s,
\, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)] \mathrm{d}t}$}}
&=
-\frac{e^{-\lambda s}}{\lambda^{\frac{\# K -1}{N}} s^{\frac{\# K}{N}}}
\sum_{m=0}^{\#K}\alpha_{-m}(\lambda s)^{\frac{m}{N}} E_{1,\frac{m+\# J}{N}}(\lambda s)
\sum_{k \in K} B_k\theta_k^{m+1} e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}
\nonumber \\
&=
-\sqrt[N]{\lambda}\,e^{-\lambda s} \sum_{m=0}^{\#K}\alpha_{-m} \sum_{\ell=0}^{\infty}
\frac{(\lambda s)^{\ell+\frac{m-\#K}{N}}}{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\sum_{k \in K} B_k\theta_k^{m+1} e^{-\theta_k\!\!\sqrt[N]{\lambda}\,x}
\nonumber \\
&=
-\sum_{\ell=0}^{\infty} \sum_{m=0}^{\#K} \alpha_{-m}
\frac{s^{\ell+\frac{m-\#K}{N}}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{m+\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\sum_{k \in K} B_k\theta_k^{m+1} \,\lambda^{\ell+\frac{m-\# K+1}{N}}
e^{-\lambda s-\theta_k\!\!\sqrt[N]{\lambda}\,x}.
\label{wrt.lambda.inter}\end{aligned}$$ We need to invert the quantity $\lambda^{\ell+\frac{m-\# K+1}{N}}
e^{-\lambda s-\theta_k\!\!\sqrt[N]{\lambda}\,x}$ for $\ell \ge 0$ and $0 \le m \le \#K$ with respect to $\lambda$. We intend to use (\[set24\]) which is valid for $0\le m\le N-1$. Actually (\[set24\]) holds true also for $m \le 0$; the proof of this claim is postponed to Lemma \[brom\] in the appendix. As a byproduct, for any $\ell \ge 0$ and $0 \le m \le \# K$, $$\begin{aligned}
\lambda^{\ell+\frac{m-\# K+1}{N}} e^{-\lambda s-\theta_k\!\!\sqrt[N]\lambda\,x}
&=
e^{-\lambda s} \int_0^{\infty} e^{-\lambda u}I_{k,\#K-\ell N-m-1}(u;x)\,\mathrm{d}u
\nonumber \\
&=
\int_s^{\infty} e^{-\lambda t} I_{k,\# K-\ell N-m-1}(t-s;x)\,\mathrm{d}t.
\label{wrt.lambda.inter2}\end{aligned}$$ Then, by putting (\[wrt.lambda.inter2\]) into (\[wrt.lambda.inter\]) and next by eliminating the Laplace transform with respect to $\lambda$, we extract $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(s)\in \mathrm{d}s,\,X(t)\in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
&=
-\sum_{\ell=0}^{\infty} \sum_{m=0}^{\# K} \alpha_{-m}
\frac{s^{\ell+\frac{m-\# K}{N}}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{m+\# J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\sum_{k \in K} B_k\theta_k^{m+1} \,I_{k,\#K-\ell N-m-1}(t-s;x)
\\
&=
-\frac{Ni}{2 \pi} \sum_{\ell=0}^{\infty} \sum_{m=0}^{\# K} \alpha_{-m}
\frac{s^{\ell+\frac{m-\# K}{N}}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!\ell+\frac{m+\# J}{N}
\!\!\left.\vphantom{\frac aN}\right)}
\\
&\hphantom{=\,}
\times\sum_{k \in K} B_k\theta_k^{m+1} \left(e^{-i \frac{\# K -\ell N-m-1}{N}\pi}
\int_0^{\infty} \xi^{N-\#K+\ell N+m} \,e^{-(t-s) \xi^N
-\theta_k e^{i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left.-\,e^{i \frac{\#K-\ell N-m-1}{N}\pi} \int_0^{\infty} \xi^{N-\#K+\ell N+m}
\,e^{-(t-s) \xi^N-\theta_k e^{-i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi\right)
\\
&=
-\frac{Ni}{2 \pi} \sum_{m=0}^{\# K} \alpha_{-m} s^{\frac{m-\# K}{N}}
\sum_{k \in K} B_k\theta_k^{m+1}
\\
&\hphantom{=\,}
\times\left(e^{-i \frac{\# K -m-1}{N}\pi} \int_0^{\infty} \left(\sum_{\ell=0}^{\infty}
\frac{\left(-s \xi^N\right)^{\!\ell}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{m+\#J}{N}\!\!\left.\vphantom{\frac aN}\right)} \right)
\xi^{m+\#J} \,e^{-(t-s) \xi^N-\theta_k
e^{i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left.-\,e^{i \frac{\#K-m-1}{N}\pi} \int_0^{\infty} \left(\sum_{\ell=0}^{\infty}
\frac{\left(-s \xi^N\right)^{\!\ell}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{m+\#J}{N}\!\!\left.\vphantom{\frac aN}\right)}\right)
\xi^{m+\#J} \,e^{-(t-s) \xi^N-\theta_k
e^{-i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi\right)
\\
{\noalign{\noindent $\displaystyle{}$}}
&=
-\frac{Ni}{2 \pi} \sum_{m=0}^{\# K} \alpha_{-m} s^{\frac{m-\#K}{N}}
\sum_{k \in K} B_k\theta_k^{m+1}
\\
&\hphantom{=\,}
\times\left(e^{-i \frac{\# K -m-1}{N}\pi}
\int_0^{\infty} \xi^{m+\# J} \,e^{-(t-s) \,\xi^N-\theta_k e^{i \frac{\pi}{N}} x \xi} \,
E_{1,\frac{m+\# J}{N}} \!\left(-s \xi^N\right)\mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left.-\,e^{i \frac{\#K-m-1}{N}\pi} \int_0^{\infty}
\xi^{m+\#J}\,e^{-(t-s) \,\xi^N-\theta_k e^{-i \frac{\pi}{N}} x \xi} \,
E_{1,\frac{m+\# J}{N}}\!\left(-s \xi^N\right) \mathrm{d}\xi\right)\!.\end{aligned}$$ The proof of (\[wrt.lambda\]) is established. [ $\blacksquare$\
]{}
\[remark-integ\] Let us integrate (\[wrt.lambda\]) with respect to $x$ on $(-\infty,0]$. We first compute, by using (\[set26\]), $$\begin{aligned}
\int_{-\infty}^0 \mathcal{K}_m (x \xi) \,\mathrm{d}x
&=
-\frac{1}{\xi} \left(\,\sum_{k \in K} B_k\theta_k^m\right)\!\!
\left(e^{-i \frac{\# K-m}{N}\pi}-e^{i\frac{\#K-m}{N}\pi}\right)
\\
&=
\frac{2i}{\xi} \sin\!\left(\frac{\#K-m}{N}\pi\right) \beta_m
=\begin{cases}
0 & \mbox{if } 1 \le m \le \#K ,\\[1ex]
\displaystyle{\frac{2i}{\xi} \,\sin\!\left(\frac{\#K}{N}\pi\right)} & \mbox{if } m=0.
\end{cases}\end{aligned}$$ We then obtain $$\begin{aligned}
\mathbb{P}\{T(t)\in \mathrm{d}s, X(t)\le 0\}/\mathrm{d}s
&=
\frac{N\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi s^{\frac{\# K}{N}}}
\int_0^{\infty} \xi^{\#J-1}\, e^{-(t-s) \xi^N} \,
E_{1,\frac{\# J}{N}}\!\left(-s \xi^N\right)\mathrm{d}\xi
\\
&=
\frac{N\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi s^{\frac{\# K}{N}}}
\sum_{\ell=0}^{\infty} \frac{(-s)^{\ell}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{\#J}{N}\!\!\left.\vphantom{\frac aN}\right)}
\int_0^{\infty}\xi^{\ell N+\#J-1} e^{-(t-s) \xi^N} \,\mathrm{d}\xi
\\
&=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi s^{\frac{\# K}{N}} (t-s)^{\frac{\#J}{N}}}
\sum_{\ell=0}^{\infty} \left(-\frac{s}{t-s}\right)^{\!\ell}
=\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# K}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi t} \left(\frac{t-s}{s}\right)^{\!\frac{\#K}{N}}.\end{aligned}$$ We retrieve (\[f1\]).
Let us evaluate $\mathbb{P}\{T(t)\in \mathrm{d}s, X(t)\in \mathrm{d}x\}/(\mathrm{d}s\,\mathrm{d}x)$ at $x=0$. For $0 \le m \le \# K$, $$\mathcal{K}_m(0)=e^{-i \frac{\# K-m-1}{N}\pi}
\sum_{k \in K} B_k\theta_k^{m+1} -e^{i \frac{\# K-m-1}{N}\pi} \sum_{k \in K} B_k\theta_k^{m+1}
=-2i \sin\!\left(\frac{\# K-m-1}{N}\,\pi\right) \beta_{m+1}.$$ Observing that $\sin\!\left(\!\!\vphantom{\frac aN}\right.
\frac{\# K-m-1}{N}\pi\!\!\left.\vphantom{\frac aN}\right)=0$ if $m=\# K-1$, in view of (\[set26\]), (\[set14\]) and (\[set15\]), we get $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}/ \mathrm{d}s \Big|_{x=0}}$}}
&=
\frac{N}{\pi} \sin\!\left(\frac \pi N\right)\alpha_{-\#K} \beta_{\#K+1}
\int_0^{\infty} \xi^N \,e^{-(t-s) \xi^N} E_{1,1}\!\left(-s \xi^N\right) \mathrm{d}\xi
\\
&=
\frac{N}{\pi} \sin\!\left(\frac \pi N\right) \!\!\left(\!\vphantom{\sum_\in}\right.
\sum_{j \in J} \theta_j\!\!\left.\vphantom{\sum_in}\right) \int_0^{\infty} \xi^N \,e^{-t \xi^N} \mathrm{d}\xi
=\frac{\sin\!\left(\frac \pi N\right) \!\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\frac{1}{N}\!\!\left.\vphantom{\frac aN}\right)}{N \pi \,t^{1+\frac 1N}}\sum_{j \in J}\theta_j.\end{aligned}$$ Thanks to (\[set7\]) and (\[p-at-zero\]), we see that $$\mathbb{P}\{T(t) \in \mathrm{d}s,\, X(t) \in \mathrm{d}x\}/ \mathrm{d}s \Big|_{x=0}=\frac{1}{t} \,p(t;0)$$ and we deduce $$\mathbb{P}\{ T(t) \in \mathrm{d}s| X(t)=0\}/\mathrm{d}s=\frac{{1\hspace{-.23em}\mathrm{l}}_{(0,t)}(s)}{t},$$ that is, $(T(t)| X(t)=0)$ has the uniform law on $(0,t)$. This is Theorem 2.13 of [@2003].
The case $x\ge 0$
-----------------
The case $x \ge 0$ can be related to the case $x \le 0$ by using the duality. Let us introduce the dual process $(X^*_t)_{t \ge 0}$ of $(X_t)_{t \ge 0}$ defined as $X^*_t=-X_t$ for any $t\ge 0$. It is known that (see [@2003]):
- If $N$ is even, the processes $X$ and $X^*$ are identical in distribution (because of the symmetry of the heat kernel $p$): $X^*\stackrel{d}{=}X$;
- If $N$ is odd, we have the equalities in distribution $(X^+)^*\stackrel{d}{=}X^-$ and $(X^-)^*\stackrel{d}{=}X^+$ where $X^+$ is the pseudo-process associated with $\kappa_{\!_{ N}}=+1$ and $X^-$ the one associated with $\kappa_{\!_{ N}}=-1$.
When $N$ is even, we have $\{-\theta_j, j\in J\}=\{\theta_k, k \in K\}$. In this case, for any $j \in J$, there exists a unique $k \in K$ such that $\theta_j=-\theta_k$ and then $$A_j=\prod_{i \in J \setminus \{j\}} \frac{\theta_i}{\theta_i-\theta_j}
=\prod_{i \in K \setminus \{k\}} \frac{-\theta_i}{-\theta_i+\theta_k}
=\prod_{i \in K \setminus \{k\}} \frac{\theta_i}{\theta_i-\theta_k}=B_k$$ and $$\alpha_{m}=\sum_{j \in J} A_j \theta_j^m=\sum_{k \in K} B_k (-\theta_k)^m=(-1)^m \beta_m.$$ When $N$ is odd, we distinguish the roots of $\kappa_{\!_{ N}}$ in the cases $\kappa_{\!_{ N}}=+1$ and $\kappa_{\!_{ N}}=-1$:
- For $\kappa_{\!_{ N}}=+1$, let $\theta^+_i$, $1 \le i \le N$, denote the roots of $1$ and set $J^+=\{i\in\{1,\dots,N\}: \Re(\theta^+_i) >0\}$ and $K^+=\{i\in\{1,\dots,N\}: \Re(\theta_i^+)<0\}$;
- For $\kappa_{\!_{ N}}=-1$, let $\theta^-_i$, $1 \le i \le N$, denote the roots of $-1$ and set $J^-=\{i\in\{1,\dots,N\}: \Re(\theta^-_i) >0\}$ and $K^-=\{i\in\{1,\dots,N\}: \Re(\theta_i^-)<0\}$.
We have $\{\theta^-_j, i \in J^-\}=\{-\theta_k^+, k \in K^+\}$ and $\{\theta_k^-, k\in K^-\}=\{-\theta^+_j, j \in J^+\}$. In this case, for any $j \in J^-$, there exists a unique $k \in K^+$ such that $\theta_j^-=-\theta_k^+$ and then $$A_j^-=\prod_{i \in J^-\setminus \{j\}} \frac{\theta^-_i}{\theta^-_i-\theta^-_j}
=\prod_{i \in K^+ \setminus \{k\}} \frac{-\theta^+_i}{-\theta^+_i-\theta^+_k}
=\prod_{i \in K^+ \setminus \{k\}} \frac{\theta^+_i}{\theta^+_i-\theta^+_k}=B^+_k$$ and similarly $A_j^+=B_k^-$. Moreover, we have $$\alpha_{m}^-=\sum_{j \in J^-}A_j^-(\theta_j^-)^m=\sum_{k \in K^+} B_k^+(-\theta_k^+)^m
=(-1)^m \sum_{k \in K^+} B_k^+ (\theta_k^+)^m=(-1)^m \beta_m^+$$ and similarly $\alpha_m^+=(-1)^m \beta_m^-$.\
Now, concerning the connection between sojourn time and duality, we have the following fact. Set $$\tilde{T}(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{(0,+\infty)}(X(u))\,\mathrm{d}u
\quad \mbox{and} \quad T^*(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X^*(u))\,\mathrm{d}u.$$ Since Spitzer’s identity holds true interchanging the closed interval $[0,+\infty)$ and the open interval $(0,+\infty)$, it is easy to see that $T(t)$ and $\tilde{T}(t)$ have the same distribution. On the other hand, we have $$\tilde{T}(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{(0,+\infty)}(X(u))\,\mathrm{d}u
=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X^*(u))\,\mathrm{d}u
=\int_0^t [1-{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X^*(u))]\,\mathrm{d}u=t-T^*(t).$$ We then deduce that $T(t)$ and $t-T^*(t)$ have the same distribution. Consequently, we can state the lemma below.
\[identity-duality\] The following identity holds: $$\mathbb{P}\{T(t)\in \mathrm{d}s, X(t)\in \mathrm{d}x\}/ (\mathrm{d}s \,\mathrm{d}x)
=\mathbb{P}\{T^*(t) \in \mathrm{d}(t-s), X^*(t)\in \mathrm{d}(-x)\}/(\mathrm{d}s \,\mathrm{d}x).$$
As a result, the following result ensues.
\[theorem-wrt.lambda+1\] Assume $N$ is even. The distribution of $(T(t),X(t))$ is given, for $x \ge 0$, by $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t)\in \mathrm{d}s, X(t)\in \mathrm{d}x\}/ (\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{Ni}{2 \pi} \sum_{m=0}^{\# J} \beta_{-m}\, (t-s)^{\frac{m-\#J}{N}}
\int_0^{\infty} \xi^{m+\# K} \,e^{-s \xi^N} \,\mathcal{J}_m(x \xi)\,
E_{1,\frac{m+\# K}{N}} \!\left(-(t-s)\xi^N\right) \mathrm{d}\xi
\label{wrt.lambda+1}\end{aligned}$$ where $$\mathcal{J}_m(z)=e^{-i \frac{\# J-m-1}{N}\pi} \sum_{j \in J} A_j \theta_j^{m+1}
e^{-\theta_j e^{i\frac{\pi}{N}}z}-e^{i \frac{\# J-m-1}{N}\pi}
\sum_{j \in J} A_j \theta_j^{m+1} e^{-\theta_j e^{-i\frac{\pi}{N}}z}.$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}When $N$ is even, we know that $X^*$ is identical in distribution to $X$ and $(T^*(t),X^*(t))$ is then distributed like $(T(t),X(t))$. Thus, by (\[wrt.lambda\]) and Lemma \[identity-duality\], for $x\ge 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t)\in \mathrm{d}s,\, X(t)\in \mathrm{d}x\}/ (\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\mathbb{P}\{T(t) \in \mathrm{d} (t-s),\, X(t) \in \mathrm{d}(-x)\}/ (\mathrm{d}s \,\mathrm{d}x)
\\
&=
-\frac{Ni}{2 \pi} \sum_{m=0}^{\# K} \alpha_{-m} (t-s)^{\frac{m-\#K}{N}}
\int_0^{\infty} \xi^{m+\# J} e^{-s \xi^N}\,\mathcal{K}_m(-x \xi) \,
E_{1,\frac{m+\# J}{N}} (-(t-s) \xi^N)\,\mathrm{d}\xi.\end{aligned}$$ The discussion preceding Lemma \[identity-duality\] shows that $$\mathcal{K}_m(z)
=e^{-i \frac{\# J-m-1}{N}\pi} \sum_{j \in J} A_j (-\theta_j)^{m+1} e^{\theta_j e^{i\frac{\pi}{N}} z}
-e^{i \frac{\# J -m-1}{N}\pi} \sum_{j \in J} A_j (-\theta_j)^{m+1} e^{\theta_j e^{-i\frac{\pi}{N}}z}.$$ We see that $\mathcal{K}_m(z)=(-1)^{m+1}\mathcal{J}_m(-z)$ where the function $\mathcal{J}_m$ is written in Theorem \[theorem-wrt.lambda+1\]. Finally, by replacing $\alpha_{-m}$ by $(-1)^m \beta_{-m}$ and $\#J$, $\# K$ by $\# K$, $\# J$ respectively (which actually coincide since $N$ is even), (\[wrt.lambda+1\]) ensues. [ $\blacksquare$\
]{}
If $N$ is odd, although the results are not justified, similar formulas can be stated. We find it interesting to produce them here. We set $T^{\pm}(t)=\int_0^t {1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X^{\pm} (u))\,\mathrm{d}u$.
\[theorem-wrt.lambda+2\] Suppose that $N$ is odd. The distribution of $(T^+(t),X^+(t))$ is given, for $x \ge 0$, by $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T^+(t)\in \mathrm{d}s, \, X^+(t)\in \mathrm{d}x\}/ (\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{Ni}{2 \pi} \sum_{m=0}^{\# J^+} \beta^+_{-m}(t-s)^{\frac{m-\#J^+}{N}}
\int_0^{\infty} \xi^{m+\# K^+} \,e^{-s \xi^N} \,\mathcal{J}^+_m(x \xi)
\,E_{1,\frac{m+\# K^+}{N}} \!\left(-(t-s)\xi^N\right) \mathrm{d}\xi
\label{wrt.lambda+2}\end{aligned}$$ where $$\mathcal{J}^+_m(z)=e^{-i \frac{\# J^+-m-1}{N}\pi} \sum_{j \in J^+} A_j^+(\theta_j^+)^{m+1}
e^{-\theta_j^+ e^{i\frac{\pi}{N}}z}-e^{i \frac{\# J^+-m-1}{N}\pi}
\sum_{j \in J^+} A_j^+ (\theta_j^+)^{m+1} e^{-\theta_j^+ e^{-i\frac{\pi}{N}}z}.$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}When $N$ is odd, we know that $(X^+)^*\stackrel{d}{=}X^-$ and then $((T^+)^*(t),(X^+)^*(t))\stackrel{d}{=}(T^-(t),X^-(t))$. Thus, by (\[wrt.lambda\]) and Lemma \[identity-duality\], for $x\ge0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T^+(t)\in \mathrm{d}s, \,X^+(t)\in \mathrm{d}x\}/ (\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\mathbb{P}\{T^-(t) \in \mathrm{d} (t-s), \,X^-(t) \in \mathrm{d}(-x)\}/ (\mathrm{d}s \,\mathrm{d}x)
\\
&=
-\frac{Ni}{2 \pi} \sum_{m=0}^{\# K^-} \alpha^-_{-m} (t-s)^{\frac{m-\#K^-}{N}}
\int_0^{\infty} \xi^{m+\# J^-} \,e^{-s \xi^N} \,\mathcal{K}^-_m(-x \xi)
\,E_{1,\frac{m+\# J^-}{N}} \!\left(-(t-s)\xi^N\right)\,\mathrm{d}\xi\end{aligned}$$ where $$\mathcal{K}^-_m(z)=e^{-i \frac{\# K^--m-1}{N}\pi} \sum_{k \in K^-} B^-_k(\theta_k^-)^{m+1}
e^{-{\theta}_k^-e^{i\frac{\pi}{N}}z}-e^{i \frac{\# K^--m-1}{N}\pi}
\sum_{k \in K^-} B^-_k(\theta^-_k)^{m+1} e^{-\theta_k^-e^{-i\frac{\pi}{N}}z}.$$ As in the proof of Theorem \[theorem-wrt.lambda+1\], we can write $\mathcal{K}^-_m(z)=(-1)^{m+1}\mathcal{J}^+_m(-z)$ where the function $\mathcal{J}^+_m$ is defined in Theorem \[theorem-wrt.lambda+2\]. Finally, by replacing $\alpha^-_m$ by $(-1)^m \beta^+_m$ and $\#J^-$, $\# K^-$ by $\# K^+$, $\# J^+$ respectively, (\[wrt.lambda+2\]) ensues. [ $\blacksquare$\
]{}
Formula (\[wrt.lambda+2\]) involves only quantities with associated ‘$+$’ signs. We have a similar formula for $X^-$ by changing all ‘$+$’ into ‘$-$’. So, we can remove these signs in order to get a unified formula (this is (\[wrt.lambda+1\])) which is valid for even $N$ and, at least formally, for odd $N$ without sign.
Let us integrate (\[wrt.lambda+1\]) with respect to $x$ on $[0,\infty)$. We first calculate, recalling that $\mathcal{J}_m(z)=(-1)^{m+1}\mathcal{K}_m(-z)$ and referring to Remark \[remark-integ\], $$\int_0^{\infty} \mathcal{J}_m (x \xi) \,\mathrm{d}x
=(-1)^{m+1}\int_{-\infty}^0 \mathcal{K}_m (x \xi) \,\mathrm{d}x
=\begin{cases}
\displaystyle{0} & \mbox{if } 1 \le m \le \#J ,
\\[1ex]
\displaystyle{-\frac{2i}{\xi} \sin\!\left(\frac{\#J}{N}\pi\right)} &\mbox{if } m=0.
\end{cases}$$ Then, $$\begin{aligned}
\mathbb{P}\{T(t)\in \mathrm{d}s, X(t)\ge 0\}/\mathrm{d}s
&=
\frac{N\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi(t-s)^{\frac{\# J}{N}}}
\int_0^{\infty} \xi^{\#K-1} \, e^{-s \xi^N}
\,E_{1,\frac{\# K}{N}}\!\left(-(t-s) \xi^N\right) \mathrm{d}\xi
\\
&=
\frac{N\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi(t-s)^{\frac{\# J}{N}}}
\sum_{\ell=0}^{\infty} \frac{(-(t-s))^{\ell}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{\#K}{N}\!\!\left.\vphantom{\frac aN}\right)}
\int_0^{\infty}\xi^{N\ell+\#K-1} e^{-s \xi^N} \,\mathrm{d}\xi
\\
&=
\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi s^{\frac{\# K}{N}} (t-s)^{\frac{\#J}{N}}}
\sum_{\ell=0}^{\infty} \left(-\frac{t-s}{s}\right)^{\!\ell}
=\frac{\sin\!\left(\!\!\vphantom{\frac aN}\right. \frac{\# J}{N}\pi\!\!
\left.\vphantom{\frac aN}\right)}{\pi t} \left(\frac{s}{t-s}\right)^{\!\frac{\#J}{N}}.\end{aligned}$$ We retrieve (\[f2\]).
Examples
--------
In this part, we write out the distribution of the couple $(T(t),X(t))$ in the cases $N=2$, $N=3$ and $N=4$.
Case $N=2$. Using the numerical values of Example \[example1\], formula (\[wrt.lambda\]) yields, for $x \le 0$, $$\begin{aligned}
\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)
&=
\frac{1}{\pi} \left(\frac{1}{\sqrt s} \int_0^{\infty}
\xi \,e^{-(t-s) \xi^2} \,\tilde{\mathcal{K}}_0(x \xi)
\,E_{1,\frac12}\!\left(-s \xi^2\right) \mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left. +\int_0^{\infty} \xi^2 \,e^{-(t-s)\xi^2} \,\tilde{\mathcal{K}}_1(x \xi)
\,E_{1,1}\!\left(-s \xi^2\right) \mathrm{d}\xi\right)\end{aligned}$$ with $$E_{1,1} \!\left(-s \xi^2\right)\!=e^{-s \xi^2},
\quad\tilde{\mathcal{K}}_0(z)=-i\,\mathcal{K}_0(z)=i(e^{iz}-e^{-iz}),
\quad\tilde{\mathcal{K}}_1(z)=-i\,\mathcal{K}_1(z)=e^{iz}+e^{-iz}.$$ On the other hand, we learn from Lemma \[1alpha\] that, for $\xi\ge 0$, $$E_{1,\frac12}(-s \xi^2)=\frac{1}{\sqrt \pi} \left(\!\!\vphantom{\int}\right.
1-2 \xi e^{-s \xi^2} \int_0^{s \xi}
e^{\frac{u^2}{s}} \,\mathrm{d}u \!\!\left.\vphantom{\int}\right)\!.$$ Therefore, $$\begin{aligned}
\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)
&=
\frac{i}{ \pi^{3/2} \sqrt s} \int_0^{\infty} \xi \left(
e^{i x \xi -(t-s) \xi^2}-e^{-ix \xi-(t-s) \xi^2}\right) \mathrm{d}\xi
\\
&\hphantom{=\,}
-\frac{2i}{\pi^{3/2} \sqrt s} \int_0^{\infty} \xi^2 \left(e^{i x \xi -t \xi^2}
-e^{-i x \xi-t \xi^2}\right) \!\!\left(\!\!\vphantom{\int}\right.
\int_0^{s \xi} e^{\frac{u^2}{s}} \,\mathrm{d}u\!\!\left.\vphantom{\int}\right) \mathrm{d}\xi
\\
&\hphantom{=\,}
+\frac{1}{\pi} \int_0^{\infty} \xi^2 \left(e^{i x \xi -t \xi^2} +e^{-i x \xi-t \xi^2}\right) \mathrm{d}\xi
\\
&=
\frac{i}{ \pi^{3/2} \sqrt s} \int_{-\infty}^{\infty} \xi \,e^{ix\xi-(t-s)\xi^2} \mathrm{d}\xi
+\frac{1}{\pi} \int_{-\infty}^{\infty} \xi^2 e^{ix\xi-t\xi^2} \mathrm{d}\xi
\\
&\hphantom{=\,}
+\frac{4}{\pi^{3/2} \sqrt s} \int_0^{\infty} \xi^2 \sin(x \xi) \,e^{-t\xi^2}
\left(\!\!\vphantom{\int}\right.\int_0^{s \xi} e^{\frac{u^2}{s}}
\,\mathrm{d}u\!\!\left.\vphantom{\int}\right) \mathrm{d}\xi.\end{aligned}$$ Let us compute two intermediate integrals: $$\begin{aligned}
\int_{-\infty}^{\infty} \xi \, e^{ix\xi-(t-s) \xi^2} \,\mathrm{d}\xi
&=
e^{-\frac{x^2}{4(t-s)}} \int_{-\infty}^{\infty} \xi \,e^{-(t-s)
\left(\xi-\frac{ix}{2(t-s)}\right)^{\!2}}\mathrm{d}\xi=e^{-\frac{x^2}{4(t-s)}}
\int_{-\infty}^{\infty}\left(\xi+\frac{ix}{2(t-s)}\right)e^{-(t-s) \xi^2} \,\mathrm{d}\xi
\\
&=
\frac{i \sqrt \pi \,x}{2 (t-s)^{3/2}} e^{-\frac{x^2}{4(t-s)}},\end{aligned}$$ $$\begin{aligned}
\int_{-\infty}^{\infty} \xi^2 e^{i x \xi -t \xi^2} \,\mathrm{d}\xi
&=
e^{-\frac{x^2}{4t}} \int_{-\infty}^{\infty}
\left(\xi + \frac{i x}{2t}\right)^{\!2} e^{-t \xi^2} \,\mathrm{d}\xi
=e^{-\frac{x^2}{4t}} \left(\int_{-\infty}^{\infty} \xi^2 e^{-t \xi^2} \,\mathrm{d}\xi
-\frac{x^2}{4 t^2} \int_{-\infty}^{\infty} e^{-t \xi^2} \,\mathrm{d}\xi\right)
\\
&=
\frac{\sqrt \pi}{2 t^{3/2}} \, e^{-\frac{x^2}{4t}} \left(1-\frac{x^2}{2t}\right)\!.\end{aligned}$$ We deduce the following representation: for $x\le 0$, $$\begin{aligned}
\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)
&=
\frac{4\sqrt s}{\pi^{3/2}} \left[\!\vphantom{\int}\right.
\int_0^{\infty} \xi^2 \sin(x \xi) \, e^{-t \xi^2}
\left(\!\!\vphantom{\int}\right.\int_0^{\xi} e^{su^2} \,\mathrm{d}u\!\!
\left.\vphantom{\int}\right)\mathrm{d}\xi -\frac{\sqrt \pi}{8}
\frac{xe^{-\frac{x^2}{4(t-s)}}}{s(t-s)^{3/2}}\!\!\left.\vphantom{\int}\right]
\\
&\hphantom{=\,}
+\frac{1}{2 \sqrt \pi t^{3/2}} \left(1-\frac{x^2}{2t}\right) e^{-\frac{x^2}{4t}}.\end{aligned}$$
For $x \ge 0$, (\[wrt.lambda+1\]) gives $$\begin{aligned}
\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)
&=
\frac{1}{\pi} \left(\frac{1}{\sqrt{t-s}} \int_0^{\infty}
\xi \,e^{-s\xi^2} \,\tilde{\mathcal{J}}_0(x \xi)
\,E_{1,\frac12}\!\left(-(t-s)\xi^2\right) \mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left. +\int_0^{\infty} \xi^2 \,e^{-s\xi^2} \,\tilde{\mathcal{J}}_1(x \xi)
\,E_{1,1}\!\left(-(t-s)\xi^2\right) \mathrm{d}\xi\right)\end{aligned}$$ with $$\tilde{\mathcal{J}}_0(z)=i\,\mathcal{J}_0(z)=-i(e^{iz}-e^{-iz}),
\quad\tilde{\mathcal{J}}_1(z)=-i\,\mathcal{J}_1(z)=e^{iz}+e^{-iz}.$$ As previously, $$\begin{aligned}
\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)
&=
-\frac{i}{ \pi^{3/2} \sqrt{t-s}} \int_{-\infty}^{\infty} \xi \,e^{ix\xi-s\xi^2} \mathrm{d}\xi
+\frac{1}{\pi} \int_{-\infty}^{\infty} \xi^2 e^{ix\xi-t\xi^2} \mathrm{d}\xi
\\
&\hphantom{=\,}
-\frac{4}{\pi^{3/2} \sqrt{t-s}} \int_0^{\infty} \xi^2 \sin(x \xi) \,e^{-t\xi^2}
\left(\!\!\vphantom{\int}\right.\int_0^{(t-s) \xi} e^{\frac{u^2}{t-s}}
\,\mathrm{d}u\!\!\left.\vphantom{\int}\right) \mathrm{d}\xi
\\
&=
\frac{4\sqrt{t-s}}{\pi^{3/2}} \left[
\frac{\sqrt \pi}{8}\frac{xe^{-\frac{x^2}{4s}}}{s^{3/2}(t-s)}-\!\!\left.\vphantom{\int}
\!\vphantom{\int}\right.\int_0^{\infty} \xi^2 \sin(x \xi) \,e^{-t \xi^2}
\left(\!\!\vphantom{\int}\right.\int_0^{\xi} e^{su^2} \,\mathrm{d}u\!\!
\left.\vphantom{\int}\right)\mathrm{d}\xi
\right]
\\
&\hphantom{=\,}
+\frac{1}{2 \sqrt \pi t^{3/2}} \left(1-\frac{x^2}{2t}\right) e^{-\frac{x^2}{4t}}.\end{aligned}$$
Actually, the density of $(T(t),X(t))$ related to (rescaled) Brownian motion is well-known under another form. For instance, in [@bs] pages 129–131, we find that $$\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)=\begin{cases}
\displaystyle{\int_0^{\infty} \frac{y(y-x)}{4 \pi s^{3/2}(t-s)^{3/2}}\,
e^{-\frac{y^2}{4 s}-\frac{(y-x)^2}{4(t-s)}} \,\mathrm{d}y} & \mbox{if } x\le 0,
\\[2ex]
\displaystyle{\int_0^{\infty} \frac{y(y+x)}{4 \pi s^{3/2}(t-s)^{3/2}}\,
e^{-\frac{(y+x)^2}{4 s}-\frac{y^2}{4(t-s)}} \,\mathrm{d}y} & \mbox{if } x\ge 0.
\end{cases}$$ The coincidence of our representation and that of [@bs] can be checked by using Lemma \[s\] and Lemma \[ss\] in the appendix.
Case $N=3$.
$\bullet$ Suppose $\kappa_3=1$. Using $E_{1,1}\!\left(-s \xi^3\right)=e^{-s \xi^3}$ and the values of Example \[example2\], (\[wrt.lambda\]) writes, for $x \le 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{\sqrt 3}{2\pi} \left(s^{-2/3} \int_0^{\infty} \xi \,e^{-(t-s) \xi^3}\,
\tilde{\mathcal{K}}_0(x \xi)\,E_{1,\frac13} \!\left(-s \xi^3\right) \mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left. +\, s^{-1/3} \int_0^{\infty} \xi^2 \,e^{-(t-s) \xi^3}\,\tilde{\mathcal{K}}_1(x \xi)\,
E_{1,\frac23}\!\left(-s \xi^3\right) \mathrm{d}\xi
+ \int_0^{\infty} \xi^3\, e^{-t\xi^3}\,\tilde{\mathcal{K}}_2(x \xi) \,\mathrm{d}\xi\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{K}}_0(z)
&=
-i\sqrt 3\,\mathcal{K}_0(z)= e^z-e^{-z/2}
\left(\!\!\vphantom{\frac{\sqrt a}{a}}\right.\cos\frac{\sqrt 3\,z}{2}
+\sqrt 3 \,\sin\frac{\sqrt 3\,z}{2}\left.\!\!\vphantom{\frac{\sqrt a}{a}}\right)\!,
\\
\tilde{\mathcal{K}}_1(z)
&=
-i\sqrt 3\,\mathcal{K}_1(z)= -e^z+e^{-z/2}
\left(\!\!\vphantom{\frac{\sqrt a}{a}}\right.
\cos\frac{\sqrt 3\,z}{2}-\sqrt 3 \,\sin\frac{\sqrt 3\,z}{2}
\left.\!\!\vphantom{\frac{\sqrt a}{a}}\right)\!,
\\
\tilde{\mathcal{K}}_2(z)
&=
-i\sqrt 3\,\mathcal{K}_2(z)= e^z+2e^{-z/2} \cos\frac{\sqrt 3\,z}{2}.\end{aligned}$$ For $x \ge 0$, (\[wrt.lambda+2\]) gives $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{3}{2\pi} \left(\frac{1}{\!\sqrt[3]{t-s}} \int_0^{\infty} \xi^2 \,e^{-s \xi^3}\,
\tilde{\mathcal{J}}_0(x\xi)\,E_{1,\frac23}\!\left(-(t-s) \xi^3\right) \mathrm{d}\xi
+\int_0^{\infty} \xi^3\,e^{-t \xi^3}\,\tilde{\mathcal{J}}_1(x\xi) \,\mathrm{d}\xi\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{J}}_0(z)
&=
i\,\mathcal{J}_0(z)=2 \,e^{-z/2} \,\sin\frac{\sqrt 3\,z}{2},
\\
\tilde{\mathcal{J}}_1(z)
&=
-i\,\mathcal{J}_1(z)= \,e^{-z/2} \left(\!\!\vphantom{\frac{\sqrt a}{a}}\right.
\sqrt 3\,\cos\frac{\sqrt 3\,z}{2}-\sin \frac{\sqrt 3\,z}{2}\left.\!\!\vphantom{\frac{\sqrt a}{a}}\right)\!.\end{aligned}$$
$\bullet$ Suppose $\kappa_3=-1$. Likewise, for $x \le 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t)\in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{3}{2\pi} \left(\frac{1}{\!\sqrt[3]{s}} \int_0^{\infty} \xi^2\,e^{-(t-s) \xi^3}\,
\tilde{\mathcal{K}}_0(x\xi)\,E_{1,\frac2 3}\!\left(-s \xi^3\right) \mathrm{d}\xi\right.
+ \left. \int_0^{\infty} \xi^3\,e^{-t\xi^3} \tilde{\mathcal{K}}_1(x\xi) \,\mathrm{d}\xi\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{K}}_0(z)
&=
-i\,\mathcal{K}_0(z)=-2\,e^{z/2} \,\sin \frac{\sqrt 3\,z}{2},
\\
\tilde{\mathcal{K}}_1(z)
&=
-i\,\mathcal{K}_1(z)=e^{z/2}
\left(\!\!\vphantom{\frac{\sqrt a}{a}}\right.\sqrt 3 \,\cos\frac{\sqrt 3\,z}{2}
+\sin\frac{\sqrt 3\,z}{2}\left.\!\!\vphantom{\frac{\sqrt a}{a}}\right)\!.\end{aligned}$$ For $x \ge 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
\\[-3ex]
&=
\frac{\sqrt3}{2\pi} \left((t-s)^{-2/3}
\int_0^{\infty} \xi \,e^{-s \xi^3}\,\tilde{\mathcal{J}}_0(x \xi)\,
E_{1,\frac13}\!\left(-(t-s) \xi^3\right) \mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left.+\,(t-s)^{-1/3} \int_0^{\infty} \xi^2 \,e^{-s \xi^3}\,
\tilde{\mathcal{J}}_1(x \xi)\,E_{1,\frac23}\!\left(-(t-s) \xi^3\right) \mathrm{d}\xi
+\int_0^{\infty} \xi^3\,e^{-t\xi^3}\,\tilde{\mathcal{J}}_2(x \xi) \,\mathrm{d}\xi\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{J}}_0(z)
&=
i\sqrt 3\,\mathcal{J}_0(z)=e^{-z}-e^{z/2}
\left(\!\!\vphantom{\frac{\sqrt a}{a}}\right.\cos\frac{\sqrt 3 \,z}{2}
-\sqrt 3 \,\sin\frac{\sqrt 3 \,z}{2}\left.\!\!\vphantom{\frac{\sqrt a}{a}}\right)\!,
\\
\tilde{\mathcal{J}}_1(z)
&=
-i\sqrt 3\,\mathcal{J}_1(z)=-e^{-z}+e^{z/2} \left(\!\!\vphantom{\frac{\sqrt a}{a}}\right.
\cos \frac{\sqrt 3 \,z}{2}+\sqrt 3 \,\sin\frac{\sqrt 3 \,z}{2}\left.\!\!\vphantom{\frac{\sqrt a}{a}}\right)\!,
\\
\tilde{\mathcal{J}}_2(z)
&=
i\sqrt 3\,\mathcal{J}_2(z)=e^{-z}+2\,e^{z/2} \cos\frac{\sqrt 3 \,z}{2}.\end{aligned}$$
Case $N=4$. Referring to Example \[example3\], formula (\[wrt.lambda\]) writes, for $x \le 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{2}{ \pi} \left(\frac{1}{\sqrt{s}} \int_0^{\infty} \xi^2 \,e^{-(t-s) \xi^4}\,
\tilde{\mathcal{K}}_0(x \xi) \,E_{1,\frac12}\!\left(-s \xi^4\right) \mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left. +\,\frac{\sqrt 2}{\sqrt[4]{s}} \int_0^{\infty} \xi^3\,e^{-(t-s) \xi^4}\,
\tilde{\mathcal{K}}_1(x \xi) \,E_{1,\frac34}\!\left(-s \xi^4\right) \mathrm{d}\xi
+\int_0^{\infty} \xi^4 \,e^{-t\xi^4}\, \tilde{\mathcal{K}}_2(x \xi) \,\mathrm{d}\xi\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{K}}_0(z)
&=
-i\,\mathcal{K}_0(z)= e^z-\cos z-\sin z,
\\
\tilde{\mathcal{K}}_1(z)
&=
-i\,\mathcal{K}_1(z)= -e^z+\cos z-\sin z,
\\
\tilde{\mathcal{K}}_2(z)
&=
-i\,\mathcal{K}_2(z)= e^z+\cos z+\sin z.\end{aligned}$$ For $x \ge 0$, (\[wrt.lambda+1\]) reads $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\mathbb{P}\{T(t) \in \mathrm{d}s, X(t) \in \mathrm{d}x\}/(\mathrm{d}s \,\mathrm{d}x)}$}}
&=
\frac{2}{\pi} \left(\frac{1}{\sqrt{t-s}} \int_0^{\infty} \xi^2 \,e^{-s \xi^4}\,
\tilde{\mathcal{J}}_0(x \xi)\,E_{1,\frac12}\!\left(-(t-s) \xi^4\right) \mathrm{d}\xi\right.
\\
&\hphantom{=\,}
\left.+\,\frac{\sqrt 2}{\!\sqrt[4]{t-s}} \int_0^{\infty} \xi^3 \,e^{-s \xi^4}\,
\tilde{\mathcal{J}}_1(x \xi)\,E_{1,\frac34}\!\left(-(t-s) \xi^4\right) \mathrm{d}\xi
+ \int_0^{\infty} \xi^4 \,e^{-t\xi^4}\,\tilde{\mathcal{J}}_2(x \xi) \,\mathrm{d}\xi\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{J}}_0(z)
&=
i\,\mathcal{J}_0(z)=e^{-z}-\cos z+\sin z,
\\
\tilde{\mathcal{J}}_1(z)
&=
-i\,\mathcal{J}_1(z)=-e^{-z}+\cos z+\sin z,
\\
\tilde{\mathcal{J}}_2(z)
&=
i\,\mathcal{J}_2(z)=e^{-z}+\cos z-\sin z.\end{aligned}$$
Appendix {#section-appendix}
========
\[lemma-spitzer\] Let $(\xi_k)_{k \ge 1}$ be a sequence of independent identically distributed random variables and set $X_0=0$ and $T_0=0$ and, for any $k\ge 1$, $$X_k=\xi_1+\dots+\xi_k,\qquad T_k=\sum_{j=1}^{k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k).$$ Then, for $\mu\in\mathbb{R}$, $\nu>0$ and $|z|<1$, $$\begin{aligned}
\sum_{k=0}^{\infty}\mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}\right] z^k
&=
\exp\!\left(\,\sum_{k=1}^{\infty} \mathbb{E}\!
\left[e^{i \mu X_k-\nu k {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_k)}\right]\frac{z^k}{k}\right)\!,
\label{spitzer-identity}\\
\sum_{k=0}^{\infty}\mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)\right] z^k
&=
\frac{1}{e^{\nu}-1}\left[e^{\nu}\!\!-
\exp\!\left(\!-\sum_{k=1}^{\infty} \left(1-e^{-\nu k}\right) \mathbb{E}\!
\left[e^{i \mu X_k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)\right]\!\frac{z^k}{k}\right)\right]\!,
\label{spitzer-identitybis}
\\
\sum_{k=0}^{\infty}\mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_k)\right] z^k
&=
\frac{e^{\nu}}{e^{\nu}-1}\left[
\exp\!\left(\,\sum_{k=1}^{\infty} \left(1-e^{-\nu k}\right) \mathbb{E}\!
\left[e^{i \mu X_k}{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_k)\right]\frac{z^k}{k}\right)-1\right]\!.
\label{spitzer-identityter}\end{aligned}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Formula (\[spitzer-identity\]) is stated in [@spitzer] without proof. So, we produce a proof below which is rather similar to one lying in [@spitzer] related to the maximum functional of the $X_k$’s.\
$\bullet$ **Step 1.** Set, for any $(x_1,\dots, x_n)\in \mathbb{R}^n$ and $\sigma \in \mathfrak{S}_n$ ($\mathfrak{S}_n$ being the set of the permutations of $1,2,\dots,n$), $$U(x_1,\dots, x_n)=\sum_{k=1}^{n}{1\hspace{-.27em}\mbox{\rm l}}_{[0,\infty)} \!\left(\vphantom{\sum_{\in}}\right.\!
\sum_{j=1}^{k}x_j \!\!\left.\vphantom{\sum_{\in}}\right)$$ and $$V(\sigma; x_1,\dots, x_n)=\sum_{k=1}^{n_{\sigma}}\# c_k(\sigma)
{1\hspace{-.27em}\mbox{\rm l}}_{[0,\infty)}\!\left(\vphantom{\sum_{\in}}\right.\!
\sum_{j \in c_k(\sigma)}x_j \!\!\left.\vphantom{\sum_{\in}}\right)\!.$$ In the definition of $V$ above, the permutation $\sigma$ is decomposed into $n_{\sigma}$ cycles: $\sigma=(c_1(\sigma))(c_2(\sigma))\dots (c_{n_\sigma}(\sigma))$.
In view of Theorem 2.3 in [@spitzer], we have the equality between the two following sets: $$\{U(\sigma(x_1),\dots,\sigma(x_n)),\sigma \in \mathfrak{S}_n\}
=\{V(\sigma; x_1,\dots, x_n),\sigma \in \mathfrak{S}_n\}.$$ We then deduce, for any bounded Borel functions $\phi$ and $F$, $$\begin{aligned}
\mathbb{E}\!\left[\phi(X_n) F(U(\xi_1,\dots,\xi_n))\right]
&=
\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \mathbb{E}
\!\left[\vphantom{\sum_{\in}}\right.\!\phi\!\left(\vphantom{\sum_{\in}}\right.\!\sum_{j=1}^n
\xi_{\sigma(j)}\!\!\left.\vphantom{\sum_{\in}}\right)\! F(V(\sigma; \xi_1,\dots,\xi_n))
\!\left.\vphantom{\sum_{\in}}\right]\!.\end{aligned}$$ In particular, for $\phi(x)=e^{i\mu x}$ and $F(x)=e^{-\nu x}$ (where $\mu\in\mathbb{R}$ and $\nu>0$ are fixed), $$\begin{aligned}
\mathbb{E}\!\left[e^{i \mu X_n-\nu \sum_{k=1}^{n}
{1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(\sum_{j=1}^k \xi_j)}\right]
&=
\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \mathbb{E}
\!\left[\vphantom{\sum_{\in}}\right.\!\exp
\!\left(\vphantom{\sum_{\in}}\right.\!
i \mu \sum_{k=1}^{n_\sigma}\sum_{j \in c_k(\sigma)}\xi_j -\nu \sum_{k=1}^{n_\sigma} \# c_k(\sigma)
{1\hspace{-.27em}\mbox{\rm l}}_{[0,\infty)} \!\left(\vphantom{\sum_{\in}}\right.\!
\sum_{j \in c_k(\sigma)}\xi_j \!\left.\vphantom{\sum_{\in}}\right)\!\!
\!\left.\vphantom{\sum_{\in}}\right)\!\!\left.\vphantom{\sum_{\in}}\right]\!
\\
&=
\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \prod_{k=1}^{n_\sigma}
\mathbb{E} \!\left[\vphantom{\sum_{\in}}\right.\!
\exp\!\left(\vphantom{\sum_{\in}}\right.\!
i \mu \sum_{j \in c_k(\sigma)}\xi_j -\nu \,(\# c_k(\sigma))
{1\hspace{-.27em}\mbox{\rm l}}_{[0,\infty)}
\!\left(\vphantom{\sum_{\in}}\right.\!
\sum_{j \in c_k(\sigma)}\xi_j
\!\left.\vphantom{\sum_{\in}}\right)\!\!\!\left.\vphantom{\sum_{\in}}\right)
\!\!\left.\vphantom{\sum_{\in}}\right]\!
\\
&=
\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \prod_{k=1}^{n_\sigma}
\mathbb{E} \!\left[\vphantom{\sum_{\in}}\right.\!
\exp\!\left(\vphantom{\sum_{\in}}\right.\! i \mu \sum_{j=1}^{\# c_k(\sigma)}\xi_j -\nu \,(\# c_k(\sigma))
{1\hspace{-.27em}\mbox{\rm l}}_{[0,\infty)}
\!\left(\vphantom{\sum_{\in}}\right.\!
\sum_{j =1}^{\# c_k(\sigma)}\xi_j
\!\left.\vphantom{\sum_{\in}}\right)\!\!\!\left.\vphantom{\sum_{\in}}\right)\!\!\left.\vphantom{\sum_{\in}}\right]\!.\end{aligned}$$ Denote by $r_\ell(\sigma)$ the number of cycles of length $\ell$ in $\sigma$ for any $\ell\in\{1,\dots,n\}$. We have $r_1(\sigma)+2r_2(\sigma)+\dots+nr_n(\sigma)=n$. Then, $$\begin{aligned}
\mathbb{E}\!\left[e^{i \mu X_n-\nu T_n}\right]
&=
\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \prod_{\ell=1}^{n} \left(\mathbb{E}\!\left[e^{i \mu X_l-\nu\ell\,
{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right]\right)^{\!r_\ell(\sigma)}
\\
&=
\frac{1}{n!} \sum_{\substack{k_1,\dots, k_n\ge0: \\k_1+2k_2+\dots+nk_n=n}}
N_{k_1,\dots,k_n} \prod_{\ell=1}^{n} \left(\mathbb{E}\!\left[e^{i \mu X_\ell-\nu \ell\,
{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right]\right)^{\!k_\ell}\end{aligned}$$ where $N_{k_1,\dots,k_n}$ is the number of the permutations $\sigma$ of $n$ objects satisfying $r_1(\sigma)=k_1,\dots,r_n(\sigma)=k_n$; this number is equal to $$N_{k_1,\dots,k_n}=\frac{n!}{(k_1! 1^{k_1}) (k_2! 2^{k_2})\dots (k_n! n^{k_n})}.$$ Then, $$\begin{aligned}
\mathbb{E}\!\left[e^{i \mu X_n-\nu T_n}\right]
&=
\sum_{\substack{k_1,\dots, k_n\ge0: \\ k_1+2k_2+\dots+nk_n=n}}
\prod_{\ell=1}^{n}\frac{1}{k_\ell! \ell^{k_\ell}}
\left(\mathbb{E} \left[e^{i \mu X_\ell-\nu \ell \,
{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right]\right)^{\!k_\ell}.\end{aligned}$$\
$\bullet$ **Step 2.** Therefore, the identity between the generating functions follows: for $|z|<1$, $$\begin{aligned}
\sum_{n=0}^{\infty} \mathbb{E}\!\left[e^{i \mu X_n-\nu T_n}\right] z^n
&=
\sum_{\substack{n\ge 0,k_1,\dots, k_n\ge0: \\k_1+2k_2+\dots+nk_n=n}}
\prod_{\ell=1}^{n} \frac{1}{k_\ell!}\left(\mathbb{E}\!\left[e^{i \mu X_\ell-\nu \ell
\,{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right] \frac{z^{\ell}}{\ell}\right)^{\!k_\ell}
\\
&=
\sum_{k_1,k_2,\dots\ge 0} \prod_{\ell=1}^{\infty} \frac{1}{k_\ell!} \left(\mathbb{E}
\left[e^{i \mu X_\ell-\nu \ell \,{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right]
\frac{z^{\ell}}{\ell}\right)^{\!k_\ell}
\\
&=
\prod_{\ell=1}^{\infty} \Bigg[\sum_{k=1}^{\infty} \frac{1}{k!}
\left(\mathbb{E} \left[e^{i \mu X_\ell-\nu \ell \,{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right]
\frac{z^{\ell}}{\ell}\right)^{\!k}\Bigg]
\\
&=
\prod_{\ell=1}^{\infty} \exp\!\left(\mathbb{E}
\!\left[e^{i \mu X_\ell-\nu \ell \,{1\hspace{-.23em}\mathrm{l}}_{[0,\infty)}(X_\ell)}\right]
\frac{z^{\ell}}{\ell}\right)
\\
{\noalign{\noindent $\displaystyle{}$}}
&=
\exp\!\left(\sum_{n=1}^{\infty}
\mathbb{E}\!\left[e^{i \mu X_n-\nu n {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_n)}\right]
\frac{z^n}{n}\right)\!.\end{aligned}$$ The proof of (\[spitzer-identity\]) is finished.\
$\bullet$ **Step 3.**
Using the elementary identity $e^{a {1\hspace{-.23em}\mathrm{l}}_A(x)}-1=(e^a-1){1\hspace{-.27em}\mbox{\rm l}}_A(x)$ and noticing that $T_k=T_{k-1}+{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)$, we get for any $k\ge 1$, $$\mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)\right]
=\mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}\,\frac{e^{\nu{1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_k)}-1}{e^{\nu}-1}\right]
=\frac{1}{e^{\nu}-1}\left[\mathbb{E}\!\left(e^{i \mu X_k-\nu T_{k-1}}\right)\!
-\mathbb{E}\!\left(e^{i \mu X_k-\nu T_k}\right)\right]\!.$$ Now, since $X_k=X_{k-1}+\xi_k$ where $X_{k-1}$ and $\xi_k$ are independent and $\xi_k$ have the same distribution as $\xi_1$, we have, for $k\ge 1$, $$\mathbb{E}\!\left(e^{i\mu X_k-\nu T_{k-1}}\right)=
\mathbb{E}\!\left(e^{i\mu\xi_1}\right)
\mathbb{E}\!\left(e^{i \mu X_{k-1}-\nu T_{k-1}}\right)\!.$$ Therefore, $$\begin{aligned}
\sum_{k=1}^{\infty} \mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)\right]z^k
&=
\frac{1}{e^{\nu}-1}\sum_{k=1}^{\infty} \left(
\mathbb{E}\!\left[e^{i \mu X_k-\nu T_{k-1}}\right]\!
-\mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}\right]\right)z^k
\nonumber\\
&=
\frac{1}{e^{\nu}-1}\left(\mathbb{E}\!\left(e^{i\mu\xi_1}\right)
\sum_{k=1}^{\infty} \mathbb{E}\!\left[e^{i \mu X_{k-1}-\nu T_{k-1}}\right]z^k
-\sum_{k=1}^{\infty} \mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}\right]z^k\right)
\nonumber\\
&=
\frac{1}{e^{\nu}-1}\left(\left(z\,\mathbb{E}\!\left(e^{i\mu\xi_1}\right)-1\right)
\sum_{k=0}^{\infty} \mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}\right]z^k+1\right)\!.
\label{spitzer-inter}\end{aligned}$$ By putting (\[spitzer-identity\]) into (\[spitzer-inter\]), we extract $$\sum_{k=0}^{\infty} \mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)\right]z^k
=\frac{1}{e^{\nu}-1}\left[e^{\nu}-\left(1-z\,\mathbb{E}\!\left(e^{i\mu\xi_1}\right)\right)
S(\mu,\nu,z)\right]
\label{spitzerbis-inter}$$ where we set $$S(\mu,\nu,z)=\exp\!\left(\,\sum_{k=1}^{\infty} \mathbb{E}\!
\left[e^{i \mu X_k-\nu k {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_k)}\right]\frac{z^k}{k}\right)\!.$$ Next, using the elementary identity $1-\zeta=\exp[\log(1-\zeta)]
=\exp\!\left[-\sum_{k=1}^{\infty} \zeta^k/k\right]$ valid for $|\zeta|<1$, $$1-z\,\mathbb{E}\!\left(e^{i\mu\xi_1}\right)
=\exp\!\left(-\sum_{k=1}^{\infty} \left[\mathbb{E}\!\left(e^{i\mu\xi_1}\right)\right]^k \frac{z^k}{k}\right)
=\exp\!\left(-\sum_{k=1}^{\infty} \mathbb{E}\!\left(e^{i\mu X_k}\right) \frac{z^k}{k}\right)$$ and then $$\begin{aligned}
\left(1-z\,\mathbb{E}\!\left(e^{i\mu\xi_1}\right)\right) S(\mu,\nu,z)
&=
\exp\!\left(\,\sum_{k=1}^{\infty} \mathbb{E}\!
\left[e^{i \mu X_k-\nu k {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_k)}-e^{i \mu X_k}\right]\frac{z^k}{k}\right)
\nonumber\\
&=
\exp\!\left(-\sum_{k=1}^{\infty} \left(1-e^{-\nu k}\right) \mathbb{E}\!
\left[e^{i \mu X_k}{1\hspace{-.27em}\mbox{\rm l}}_{[0,+\infty)}(X_k)\right]\frac{z^k}{k}\right)\!.
\label{spitzerbis-interbis}\end{aligned}$$ Hence, by putting (\[spitzerbis-interbis\]) into (\[spitzerbis-inter\]), formula (\[spitzer-identitybis\]) entails.
By subtracting (\[spitzerbis-inter\]) from (\[spitzer-identity\]), we obtain the intermediate representation $$\sum_{k=0}^{\infty} \mathbb{E}\!\left[e^{i \mu X_k-\nu T_k}{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_k)\right]z^k
=\frac{1}{e^{\nu}-1}\left[\left(e^{\nu}-z\,\mathbb{E}\!
\left(e^{i\mu\xi_1}\right)\right)S(\mu,\nu,z)-e^{\nu}\right]\!.$$ By writing, as previously, $$e^{\nu}-z\,\mathbb{E}\!\left(e^{i\mu\xi_1}\right)
=e^{\nu}\exp\!\left(-\sum_{k=1}^{\infty} \mathbb{E}\!\left(e^{i\mu X_k}\right) \frac{e^{-\nu k}z^k}{k}\right),$$ we find $$\begin{aligned}
\left(e^{\nu}-z\,\mathbb{E}\!\left(e^{i\mu\xi_1}\right)\right) S(\mu,\nu,z)
&=
e^{\nu}\exp\!\left(\,\sum_{k=1}^{\infty} \mathbb{E}\!
\left[e^{i \mu X_k-\nu k {1\hspace{-.23em}\mathrm{l}}_{[0,+\infty)}(X_k)}-e^{i \mu X_k-\nu k}\right]\frac{z^k}{k}\right)
\\
&=
e^{\nu}\exp\!\left(\,\sum_{k=1}^{\infty} \left(1-e^{-\nu k}\right) \mathbb{E}\!
\left[e^{i \mu X_k}{1\hspace{-.27em}\mbox{\rm l}}_{(-\infty,0)}(X_k)\right]\frac{z^k}{k}\right)\!.\end{aligned}$$ Finally, (\[spitzer-identityter\]) ensues. [ $\blacksquare$\
]{}
\[lemma-vdm\] The following identities hold: $$\beta_{\#K}=(-1)^{\#K -1}\prod_{k \in K} \theta_k,\qquad
\beta_{\#K+1}=(-1)^{\#K -1}\left(\prod_{k \in K} \theta_k\right)\!\!\left(\sum_{k \in K} \theta_k\right)\!.$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}We label the set $K$ as $\{1,2,3,\dots,\#K\}$. By (\[set13\]), we know that the $B_k$’s solve a Vandermonde system. Then, by Cramer’s formulas, we can write them as fractions of some determinants: $B_k=V_k/V$ where $$V=\begin{vmatrix}
1 & \dots & 1 \\
\theta_1 & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots \\[.5ex]
\theta_1^{\#K-1} & \dots & \theta_{\#K}^{\#K-1}
\end{vmatrix}
\quad\mbox{and}\quad
V_k=\begin{vmatrix}
1 & \dots & 1 & 1 & 1 & \dots & 1 \\
\theta_1 & \dots & \theta_{k-1} & 0 & \theta_{k+1} & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{k-1}^2 & 0 & \theta_{k+1}^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots & \vdots & \vdots & & \vdots \\[.5ex]
\theta_1^{\#K-1} & \dots & \theta_{k-1}^{\#K-1} & 0 & \theta_{k+1}^{\#K-1} & \dots & \theta_{\#K}^{\#K-1}
\end{vmatrix}.$$ By expanding the determinant $V_k$ with respect to its $k^{\mathrm{th}}$ column and next factorizing it suitably, we easily see that $$\begin{aligned}
V_k
&=
(-1)^{k+1}\begin{vmatrix}
\theta_1 & \dots & \theta_{k-1} & \theta_{k+1} & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{k-1}^2 & \theta_{k+1}^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots & \vdots & & \vdots \\[.5ex]
\theta_1^{\#K-1} & \dots & \theta_{k-1}^{\#K-1} & \theta_{k+1}^{\#K-1} & \dots & \theta_{\#K}^{\#K-1}
\end{vmatrix}
\\
&=
(-1)^{k+1}\frac{\prod_{i\in K} \theta_i}{\theta_k}
\begin{vmatrix}
1 & \dots & 1 & 1 & \dots & 1 \\
\theta_1 & \dots & \theta_{k-1} & \theta_{k+1} & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{k-1}^2 & \theta_{k+1}^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots & \vdots & & \vdots \\[.5ex]
\theta_1^{\#K-2} & \dots & \theta_{k-1}^{\#K-2} & \theta_{k+1}^{\#K-2} & \dots & \theta_{\#K}^{\#K-2}
\end{vmatrix}.\end{aligned}$$ With this at hands, we have $$\beta_{\#K}=\sum_{k\in K} B_k\theta_k^{\#K}
=\frac{\prod_{k\in K} \theta_k}{V} \sum_{k\in K}(-1)^{k+1}\theta_k^{\#K-1}
\begin{vmatrix}
1 & \dots & 1 & 1 & \dots & 1 \\
\theta_1 & \dots & \theta_{k-1} & \theta_{k+1} & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{k-1}^2 & \theta_{k+1}^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots & \vdots & & \vdots \\[.5ex]
\theta_1^{\#K-2} & \dots & \theta_{k-1}^{\#K-2} & \theta_{k+1}^{\#K-2} & \dots & \theta_{\#K}^{\#K-2}
\end{vmatrix}.$$ We can observe that the sum lying on the above right-hand side is nothing but the expansion of the determinant $V$ with respect to its last row multiplied by the sign $(-1)^{\#K -1}$. This immediately ensues that $\beta_{\#K}=(-1)^{\#K -1}\prod_{k \in K} \theta_k$. Similarly, $$\beta_{\#K+1}=\sum_{k\in K} B_k\theta_k^{\#K+1}
=\frac{\prod_{k\in K} \theta_k}{V} \sum_{k\in K}(-1)^{k+1}\theta_k^{\#K}
\begin{vmatrix}
1 & \dots & 1 & 1 & \dots & 1 \\
\theta_1 & \dots & \theta_{k-1} & \theta_{k+1} & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{k-1}^2 & \theta_{k+1}^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots & \vdots & & \vdots \\[.5ex]
\theta_1^{\#K-2} & \dots & \theta_{k-1}^{\#K-2} & \theta_{k+1}^{\#K-2} & \dots & \theta_{\#K}^{\#K-2}
\end{vmatrix}.$$ The above sum is the expansion with respect to its last row, multiplied by the sign $(-1)^{\#K -1}$, of the determinant $V'$ defined as $$V'=\begin{vmatrix}
1 & \dots & 1 \\
\theta_1 & \dots & \theta_{\#K} \\
\theta_1^2 & \dots & \theta_{\#K}^2 \\
\vdots & & \vdots \\[.5ex]
\theta_1^{\#K-2} & \dots & \theta_{\#K}^{\#K-2}\\
\theta_1^{\#K} & \dots & \theta_{\#K}^{\#K}
\end{vmatrix}.$$ Let $R_0,R_1,R_2,\dots,R_{\#K-2},R_{\#K-1}$ denote the rows of $V'$. We perform the substitution $R_{\#K-1}\leftarrow R_{\#K-1}+\sum_{\ell=2}^{\#K}
(-1)^{\ell}\sigma_{\ell} R_{\#K-\ell}$ where the $\sigma_{\ell}$’s are defined by (\[set10\]). This substitution does not affect the value of $V'$ and it transforms, e.g., the first term of the last row into $$\begin{aligned}
\theta_1^{\#K}+\sum_{\ell=2}^{\#K} (-1)^{\ell}\sigma_{\ell} \,\theta_1^{\#K-\ell}.\end{aligned}$$ Recall that $\sigma_{\ell}=\sum_{1\le k_1<\dots<k_\ell\le\#K}
\theta_{k_1} \dots \theta_{k_\ell}$. We decompose $\sigma_{\ell}$, by isolating the terms involving $\theta_1$, into $$\theta_1 \sum_{2\le k_2<\dots<k_\ell\le \#K} \theta_{k_2} \dots \theta_{k_\ell}
+\sum_{2\le k_1<k_2<\dots<k_\ell\le \#K} \theta_{k_1}\theta_{k_2} \dots \theta_{k_\ell}
=\theta_1\,\sigma'_{\ell-1}+\sigma'_{\ell}$$ where we set $\sigma'_{\#K}=0$ and $\sigma'_{\ell}=\sum_{2\le k_1<k_2<\dots<k_\ell\le \#K}
\theta_{k_1}\theta_{k_2} \dots \theta_{k_\ell}.$ Therefore, we have $$\begin{aligned}
\theta_1^{\#K}+\sum_{\ell=2}^{\#K} (-1)^{\ell}\sigma_{\ell} \,\theta_1^{\#K-\ell}
&=
\theta_1^{\#K}+\sum_{\ell=2}^{\#K} (-1)^{\ell}\sigma'_{\ell-1} \,\theta_1^{\#K-\ell+1}
+\sum_{\ell=2}^{\#K} (-1)^{\ell}\sigma'_{\ell} \,\theta_1^{\#K-\ell}
\\
&=
\theta_1^{\#K}+\sigma'_1\,\theta_1^{\#K-1}=\theta_1^{\#K-1}(\theta_1+\sigma'_1)
=\theta_1^{\#K-1}\left(\sum_{k \in K} \theta_k\right)\!.\end{aligned}$$ The foregoing manipulation works similarly for each term of the last row of $V'$. So, we deduce that $V'=\left(\sum_{k \in K} \theta_k\right)V$ and finally $\beta_{\#K+1}=(-1)^{\#K -1}\left(\prod_{k \in K} \theta_k\right)
\!\!\left(\sum_{k \in K} \theta_k\right)$. [ $\blacksquare$\
]{}
\[1alpha\] For $\alpha>0$, the Mittag-Leffler functions $E_{1,\alpha}$ and $E_{1,\alpha+1}$ admit the following integral representations: $$\begin{aligned}
E_{1,\alpha}(x)
&=
\begin{cases}
\displaystyle{\frac{1}{\Gamma(\alpha)} \left(1+ x^{1-\alpha} e^{x}
\int_0^x u^{\alpha-1} e^{-u} \,\mathrm{d}u\right)} & \mbox{if } x>0,
\\[3ex]
\displaystyle{\frac{1}{\Gamma(\alpha)} \left(1-|x|^{1-\alpha} e^{x}
\int_0^{|x|} u^{\alpha-1} e^u \,\mathrm{d}u\right)} & \mbox{if } x<0,
\end{cases}
\label{mittag1}\\[1ex]
E_{1,\alpha+1}(x)
&=
\begin{cases}
\displaystyle \frac{e^x}{\Gamma(\alpha) \,x^{\alpha}}
\int_0^x u^{\alpha-1} e^{-u} \,\mathrm{d}u & \mbox{if } x>0,
\\[2ex]
\displaystyle \frac{e^x}{\Gamma(\alpha) \,|x|^{\alpha}}
\int_0^{|x|} u^{\alpha-1} e^u \,\mathrm{d}u & \mbox{if } x<0.
\end{cases}
\label{mittag2}\end{aligned}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Using the series expansion of $E_{1,\alpha}$, we obtain $$\begin{aligned}
x E_{1,\alpha}'(x)
&=
\sum_{r=1}^{\infty} \frac{r x^r}{\Gamma(r+\alpha)}
=\sum_{r=1}^{\infty} \frac{(r +\alpha-1)x^r}{\Gamma(r+\alpha)}+
(1-\alpha)\sum_{r=1}^{\infty} \frac{x^r}{\Gamma(r+\alpha)}
\\
&=
x \sum_{r=1}^{\infty} \frac{x^{r-1}}{\Gamma(r+\alpha-1)}
+(1-\alpha) \left(E_{1,\alpha}(x)-\frac{1}{\Gamma(\alpha)}\right)
=\left(x+1-\alpha\right)E_{1,\alpha}(x)+\frac{\alpha-1}{\Gamma(\alpha)}.\end{aligned}$$ Hence, $E_{1,\alpha}$ solves the differential equation $x E_{1,\alpha}'(x)=(x+1-\alpha)E_{1,\alpha}(x)+\frac{\alpha-1}{\Gamma(\alpha)}$. In view of this equation, we know that $E_{1,\alpha}(x)$ is of the form $$E_{1,\alpha}(x)=\lambda(x)\,e^{\int \frac{x+1-\alpha}{x}\,
\mathrm{d}x}=\lambda(x) \, |x|^{1-\alpha} \, e^x$$ where the unknown function $\lambda$ solves $$\lambda'(x)=\frac{\alpha-1}{\Gamma(\alpha)x} |x|^{\alpha-1} e^{-x}
=\begin{cases}
\displaystyle{\frac{\alpha-1}{\Gamma(\alpha)} \,x^{\alpha-2}} \, e^{-x} &\mbox{if } x>0,
\\[2ex]
\displaystyle{-\frac{\alpha-1}{\Gamma(\alpha)} \,|x|^{\alpha-2}}\, e^{-x} &\mbox{if } x<0.
\end{cases}$$ This implies that, for a certain $x_0>0$ and $\lambda_0 \in \mathbb{R}$ ($\lambda_0$ could be different for $x>0$ and $x <0$), we have, for $x >0$, $$\lambda(x)=\frac{\alpha-1}{\Gamma(\alpha)}
\int_{x_0}^{x} u^{\alpha-2}\,e^{-u}\,\mathrm{d}u+\lambda_0
=\frac{1}{\Gamma(\alpha)}\left(x^{\alpha-1} \,e^{-x}-x_0^{\alpha-1} \,e^{-x_0}
+\int_{x_0}^{x} u^{\alpha-1}\,e^{-u} \,\mathrm{d}u\right)+\lambda_0,$$ and, for $x<0$, $$\lambda(x)=\frac{\alpha-1}{\Gamma(\alpha)}
\int_{x_0}^{|x|} u^{\alpha-2}e^u\,\mathrm{d}u +\lambda_0
=\frac{1}{\Gamma(\alpha)}\left(|x|^{\alpha-1}\,e^{|x|}
-x_0^{\alpha-1}\,e^{x_0} - \int_{x_0}^{|x|}
u^{\alpha-1}\,e^{u} \,\mathrm{d}u\right)+\lambda_0.$$ Then $$E_{1,\alpha}(x)=\begin{cases}
\displaystyle{\frac{1}{\Gamma(\alpha)}+\left(\lambda_0
-\frac{e^{-x_0}x_0^{\alpha-1}}{\Gamma(\alpha)}\right)
x^{1-\alpha} e^{x}+\frac{x^{1-\alpha}e^{x}}{\Gamma(\alpha)}
\int_{x_0}^{x} u^{\alpha-1} e^{-u} \,\mathrm{d}u} & \mbox{if } x>0,
\\[3ex]
\displaystyle{\frac{1}{\Gamma(\alpha)}+\left(\lambda_0
-\frac{e^{x_0}x_0^{\alpha-1}}{\Gamma(\alpha)}\right)
|x|^{1-\alpha} e^{x}-\frac{|x|^{1-\alpha}e^{x}}{\Gamma(\alpha)}
\int_{x_0}^{|x|} u^{\alpha-1} e^u \,\mathrm{d}u} & \mbox{if } x<0.
\end{cases}
\label{mittag-inter}$$ Because of the singularity of the differential equation at zero, the initial value at zero does not determine $x_0$ and $\lambda_0$. Nevertheless, we know that $E_{1,\alpha}$ is $C^1$ at zero. So, we need to compute $E_{1,\alpha}'(x)$ for $x\neq 0$: $$E_{1,\alpha}'(x)=\begin{cases}
\displaystyle{\left(\lambda_0-\frac{x_0^{\alpha-1}e^{-x_0}}{\Gamma(\alpha)}
+ \frac{1}{\Gamma(\alpha)} \int_{x_0}^{x} u^{\alpha-1} e^{-u} \,\mathrm{d}u\right)\!
\left(x^{1-\alpha} e^{x}+(1-\alpha)x^{-\alpha} e^{x}\right)+\frac 1{\Gamma(\alpha)}}
& \mbox{if } x>0,
\\[3ex]
\displaystyle{\left(\vphantom{\int_{x_0}^x}\right.\!\!
\lambda_0-\frac{x_0^{\alpha-1}e^{x_0}}{\Gamma(\alpha)}
-\frac{1}{\Gamma(\alpha)} \int_{x_0}^{|x|} u^{\alpha-1} e^u \,\mathrm{d}u
\!\!\left.\vphantom{\int_{x_0}^x}\right)\!
\left(|x|^{1-\alpha} e^{x}-(1-\alpha)|x|^{-\alpha} e^{x}\right)+\frac 1{\Gamma(\alpha)}}
& \mbox{if } x<0.
\end{cases}$$ In order to have a $C^1$-function at $0$, we must have $$\lambda_0-\frac{x_0^{\alpha-1}e^{-x_0}}{\Gamma(\alpha)}
= \frac{1}{\Gamma(\alpha)} \int_0^{x_0} u^{\alpha-1} e^{-u} \,\mathrm{d}u
\quad\mbox{ and }\quad\lambda_0-\frac{x_0^{\alpha-1}e^{x_0}}{\Gamma(\alpha)}
=-\frac{1}{\Gamma(\alpha)} \int_0^{x_0} u^{\alpha-1} e^u \,\mathrm{d}u.
\label{conditions}$$ Putting (\[conditions\]) into (\[mittag-inter\]) yields (\[mittag1\]). Next, formula (\[mittag2\]) can be deduced from (\[mittag1\]) by simply observing that, e.g., for $x>0$, $$E_{1,\alpha+1}(x)=\frac{x^{-\alpha}e^x}{\Gamma(\alpha+1)} \left(x^{\alpha}e^{-x}+
\int_0^x u^{\alpha} e^{-u} \mathrm{d}u\right)=\frac{x^{-\alpha} e^x}{\Gamma(\alpha+1)}
\left(\alpha \int_0^x u^{\alpha-1} e^{-u} \,\mathrm{d}u\right)\!.$$ [ $\blacksquare$\
]{}
\[8.2\] For $\alpha \in (0,2)$, the function $E_{1,\alpha}$ admits the following representation. For $x > 0$, $$E_{1,\alpha}(x)=x^{1-\alpha} \left(e^x +\frac{\sin (\alpha \pi)}{\pi}
\int_0^{\infty} \frac{\xi^{1-\alpha}}{\xi+1} \,e^{-x \xi} \,\mathrm{d}\xi\right)\!.$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Writing $\frac{e^{-\xi x}}{\xi+1}=e^x \int_x^{\infty} e^{-( \xi +1)u}\,\mathrm{d}u$, we obtain $$\begin{aligned}
\int_0^{\infty} \frac{\xi^{1-\alpha}}{\xi+1} \,e^{-x \xi} \,\mathrm{d}\xi
&=
e^{x} \int_0^{\infty} \xi^{1-\alpha} \,\mathrm{d}\xi \int_x^{\infty} e^{-(\xi+1)u} \,\mathrm{d}u
=\Gamma(2-\alpha) e^x \int_x^{\infty} u^{\alpha-2}e^{-u}\,\mathrm{d}u
\\
&=
\frac{\Gamma(2-\alpha)}{1-\alpha}\,e^x\!\left(x^{\alpha-1} e^{-x}-
\int_{x}^{\infty} u^{\alpha-1} e^{-u}\,\mathrm{d}u\right)
\\
&=
\Gamma(1-\alpha)\,e^x \!\left(x^{\alpha-1} e^{-x}-\Gamma(\alpha)
+\int_0^x u^{\alpha-1} e^{-u}\,\mathrm{d}u\right)\!.\end{aligned}$$ From this, it entails that $$\int_0^x u^{\alpha-1} e^{-u}\,\mathrm{d}u=\frac{e^{-x}}{\Gamma(1-\alpha)}
\int_0^{\infty} \frac{\xi^{1-\alpha}}{\xi+1} \,e^{-x \xi} \,\mathrm{d}\xi
+\Gamma(\alpha)-x^{\alpha-1} e^{-x}$$ which, by (\[mittag1\]), proves Lemma \[8.2\]. [ $\blacksquare$\
]{}
\[2half\] The Mittag-Leffler functions $E_{1,\frac12}$ and $E_{\frac12,\frac12}$ are related to the error function according to $$\begin{aligned}
E_{1,\frac12}(x)
&=
\begin{cases}
\displaystyle{\frac{1}{\sqrt \pi} +\sqrt x \,e^x \mathrm{Erf} (\sqrt x\,)} &\mbox{for } x\ge0,
\\[2ex]
\displaystyle{\frac{1}{\sqrt \pi} -\sqrt{|x|}\,e^x \mathrm{Erf} (\sqrt{|x|}\,)} &\mbox{for } x\le0,
\end{cases}
\\
E_{\frac12,\frac12} (x)
&=
\frac{1}{\sqrt \pi} + x e^{x^2}+|x| e^{x^2}
\mathrm{Erf} (|x|) \quad \mbox{ for } x \in \mathbb{R}.\end{aligned}$$ In particular, $$E_{\frac12 ,\frac12} (x)=\frac{1}{\sqrt \pi}+x e^{x^2}
\mathrm{Erfc}(|x|)\quad \mbox{ for } x \le 0.$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Recall that $\mathrm{Erf} (x)=\frac{2}{\sqrt \pi} \int_0^x e^{-t^2}\,\mathrm{d}t=\frac{1}{\sqrt \pi}
\int_0^{x^2} \frac{e^{-u}}{\sqrt u} \,\mathrm{d}u$ and $\mathrm{Erfc}(x)=1-\mathrm{Erf}(x)$. By (\[mittag1\]), we have, e.g., for $x \ge 0$, that $$E_{1,\frac12}(x)= \frac{1}{\sqrt \pi} \left(1+\sqrt x \,e^x
\int_0^x \frac{e^{-u}}{\sqrt u} \,\mathrm{d}u\right)=\frac{1}{\sqrt \pi}
+\sqrt x \,e^x \mathrm{Erf} (\sqrt x).$$ On the other hand, we have, for $x \le 0$, $$E_{\frac12,\frac12} (x)=\sum_{r=0}^{\infty} \frac{x^r}{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!\frac{r+1}{2}\!\!\left.\vphantom{\frac aN}\right)}
=\sum_{\ell=0}^{\infty} \frac{x^{2\ell}}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
\ell+\frac{1}{2}\!\!\left.\vphantom{\frac aN}\right)}
+\sum_{\ell=0}^{\infty} \frac{x^{2\ell+1}}{\Gamma(\ell+1)}
=E_{1,\frac12}(x^2)+x e^{x^2}$$ from which we immediately extract the aforementioned representation. [ $\blacksquare$\
]{}
\[secondintlem\] For $\alpha\neq 0$, the following equality holds: $$\int_0^{\infty} \frac{e^{-w^2}}{w^2+\alpha^2}\,\mathrm{d}w
=\frac{\pi}{2 \alpha} \,e^{\alpha^2} \mathrm{Erfc}(\alpha).$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Put, for $\alpha\neq 0$, $$F(\alpha)=\alpha\int_0^{\infty} \frac{e^{-w^2}}{w^2+\alpha^2}\,\mathrm{d}w=
\int_0^{\infty} \frac{e^{-\alpha^2 w^2}}{w^2+1}\,\mathrm{d}w.$$ We plainly have $$\begin{aligned}
F'(\alpha)
&=
-2\alpha \int_0^{\infty} \frac{w^2}{w^2+1}\,e^{-\alpha^2 w^2} \,\mathrm{d}w
=-2\alpha \int_0^{\infty} \left(1-\frac{1}{w^2+1}\right)e^{-\alpha^2 w^2} \,\mathrm{d}w
\\
&=
-2\alpha \int_0^{\infty}e^{-\alpha^2 w^2} \,\mathrm{d}w+2\alpha \,F(\alpha).\end{aligned}$$ So, the function $F$ solves the differential equation $F'(\alpha)=2 \alpha F(\alpha)-\sqrt{\pi}$ with initial value . We deduce that $F(\alpha)$ has the form $F(\alpha)=G(\alpha)\,e^{\alpha^2}$ where the unknown function $G$ satisfies and $G(0)=\frac{\pi}{2}$. This implies that $G(\alpha)=\sqrt{\pi} \int_{\alpha}^{\infty}e^{-\xi^2}\,\mathrm{d}\xi
=\frac \pi 2 \,\mathrm{Erfc}(\alpha)$ and then $F(\alpha)=\frac{\pi}{2} \,e^{\alpha^2} \mathrm{Erfc}(\alpha)$. The proof of Lemma \[secondintlem\] is established. [ $\blacksquare$\
]{}
\[secondint\] The following identity holds: for $\lambda>0$, $s>0$ and $x>0$, $$\int_0^s \frac{e^{-\lambda\sigma -\frac{x^2}{4\sigma}}}{\sigma^{3/2}}
\left(\int_{\sqrt{\lambda (s-\sigma)}}^{\infty} e^{-\xi^2} \,\mathrm{d}\xi\right)\mathrm{d}\sigma
=\frac{\pi}{x} \,e^{\sqrt \lambda \,x}\,
\mathrm{Erfc}\!\left(\frac{x}{2 \sqrt s}+\sqrt{\lambda s}\right)$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Put $I=\int_0^s \frac{e^{-\lambda\sigma-\frac{x^2}{4\sigma}}}{\sigma^{3/2}}
\left(\int_{\sqrt{\lambda (s-\sigma)}}^{\infty} e^{-\xi^2} \,\mathrm{d}\xi\right)\mathrm{d}\sigma$. With the change of variables $\xi=\sqrt{\lambda s z}$ and $v=z+\sigma/s$, we have $$I=\frac{\sqrt{\lambda s}}{2} \int_0^s \frac{e^{-\lambda\sigma
-\frac{x^2}{4\sigma}}}{\sigma^{3/2}} \left( \int_{(s-\sigma)/s}^{\infty}
\frac{e^{-\lambda s z}}{\sqrt{z}} \,\mathrm{d}z\right)\mathrm{d}\sigma
=\frac{\sqrt{\lambda s}}{2} \int_0^s \frac{e^{-\frac{x^2}{4\sigma}}}{\sigma^{3/2}}
\left(\int_{1}^{\infty} \frac{e^{-\lambda s v}}{\sqrt{v-\frac{\sigma}{s}}} \,\mathrm{d}v
\right)\mathrm{d}\sigma.$$ With the new change of variable $u=s/\sigma$, it becomes $$\begin{aligned}
I
&=
\frac{\sqrt{\lambda s}}{2} \int_{1}^{\infty} \left(\frac{u}{s}\right)^{\!3/2}
\frac{s}{u^2} \,e^{-\frac{x^2}{4s}u} \left(\int_{1}^{\infty}
\frac{e^{-\lambda s v}}{\sqrt{v-\frac 1u}} \,\mathrm{d}v\right)\mathrm{d}u =\frac{\sqrt \lambda}{2}
\int_{1}^{\infty} \!\!\!\int_{1}^{\infty} \frac{e^{-\left(\frac{x^2}{4s}\right)u
-\lambda s v}}{\sqrt{uv-1}} \,\mathrm{d}u \,\mathrm{d}v
\\
&=
\frac{\sqrt \lambda}{2} \int_{1}^{\infty} \!\!\!\int_{1}^{\infty}
\frac{e^{-\frac a 2 u-\frac b 2 v}}{\sqrt{uv-1}} \,\mathrm{d}u \,\mathrm{d}v\end{aligned}$$ where we set $a=\frac{x^2}{2s}$ and $b=2 \lambda s$. With the change of variable $(v,w)=(v,\frac a2 u+ \frac b2 v)$, this gives $$\begin{aligned}
I
&=
\frac{\sqrt{\lambda}}{\sqrt{ab}} \int_{w=\frac{a+b}{2}}^{w=\infty} \int_{v=1}^{v=\frac{2w-a}{b}}
\frac{e^{-w}}{\sqrt{\frac 2 b \,v w-v^2-\frac a b}} \,\mathrm{d}v \,\mathrm{d}w=\frac{1}{x}
\int_{\frac{a+b}{2}}^{\infty} e^{-w} \left(\int_{1}^{\frac{2w-a}{b}}
\frac{\mathrm{d}v}{\sqrt{\frac{w^2-ab}{b^2}-(v-\frac{w}{b})^2}}\right) \mathrm{d}w
\\
&=
\frac{1}{x}\int_{\frac{a+b}{2}}^{\infty} \left(\arcsin \frac{w-a}{\sqrt{w^2-ab}}
+ \arcsin \frac{w-b}{\sqrt{w^2-ab}}\right) e^{-w} \,\mathrm{d}w.\end{aligned}$$ Let us integrate by parts this last integral. First, we have $$\frac{\mathrm{d}}{\mathrm{d}w}\left(\arcsin \frac{w-a}{\sqrt{w^2-ab}}\right)
=\frac{\sqrt{a}\,(w-b)}{(w^2-ab)\sqrt{2w-a-b}},$$ and then, $$\begin{aligned}
I
&=
\frac{1}{x} \left( \left[-\left(\arcsin \frac{w-a}{\sqrt{w^2-ab}}
+ \arcsin \frac{w-b}{\sqrt{w^2-ab}}\right)e^{-w}\right]_{\frac{a+b}{2}}^{\infty}
+\int_{\frac{a+b}{2}}^{\infty} \frac{\sqrt{a} \,(w-b)
+\sqrt b \,(w-a)}{(w^2-ab)\sqrt{2w-a-b}}\,e^{-w}\,\mathrm{d}w\right)
\\
&=
\frac{\sqrt{a}+\sqrt{b}}{x} \int_{\frac{a+b}{2}}^{\infty}
\frac{e^{-w}}{(w+\sqrt{ab})\sqrt{2w-(a+b)}}\,\mathrm{d}w=\frac{\sqrt{a}+\sqrt{b}}{x \sqrt 2}\,
e^{-\frac{a+b}{2}}\int_0^{\infty} \frac{e^{-w}}{(w+\frac{a+b}{2}+\sqrt{ab})\sqrt{w}}\,\mathrm{d}w
\\
&=
\sqrt 2 \,\frac{\sqrt{a}+\sqrt{b}}{x}\,e^{-\frac{a+b}{2}}
\int_0^{\infty} \frac{e^{-w^2}}{w^2+\left(\frac{\sqrt a+\sqrt b}{\sqrt 2}\right)^{\!2}} \,\mathrm{d}w.\end{aligned}$$ As a byproduct, in view of Lemma \[secondintlem\], we have $$I=\sqrt 2 \,\frac{\sqrt{a}+\sqrt{b}}{x}
\,\frac{\pi e^{-\frac{a+b}{2}}}{\sqrt 2 (\sqrt a+\sqrt b)}\,
e^{\frac{a+b}{2}+\sqrt{ab}} \,\mathrm{Erfc}\!\left(\frac{\sqrt a+\sqrt b}{\sqrt 2}\right)
=\frac{\pi}{x}\,e^{\sqrt \lambda \,x} \,\mathrm{Erfc}\!\left(\frac{x}{2 \sqrt s}+\sqrt{\lambda s}\right)\!.$$ Lemma \[secondint\] is proved. [ $\blacksquare$\
]{}
\[brom\] For any integer $m\le N-1$ and any $x\ge 0$, $$\int_0^{\infty} e^{-\lambda u} I_{j,m}(u;x)\,\mathrm{d}u
=\lambda^{-\frac{m}{N}}e^{-\theta_j\!\!\sqrt[N]{\lambda}\,x}.
\eqno{(\ref{set24})}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}This formula is proved in [@2007] for $0\le m\le N-1$. To prove that it holds true also for negative $m$, we directly compute the Laplace transform of $I_{j,m}(u;x)$. We have $$\int_0^{\infty} e^{-\lambda u} I_{j,m}(u;x) \,\mathrm{d}u
=\frac{N i}{2 \pi} \left(e^{-i \frac{m}{N}\pi } \int_0^{\infty} \frac{\xi^{N-m-1}}{\xi^N+\lambda}\,
e^{-\theta_j e^{i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi -e^{i \frac{m}{N}\pi } \int_0^{\infty}
\frac{\xi^{N-m-1}}{\xi^N+\lambda}\, e^{-\theta_j e^{-i \frac{\pi}{N}} x \xi} \,\mathrm{d}\xi\right)\!.$$ Let us integrate the function $H:z \to \frac{z^{M-1}}{z^N+ \lambda} \,e^{-a z}$ for fixed $a$ and $M$ such that $\Re(a)>0$ and $M>0$ on the contour $\Gamma_R=\{\rho e^{i \varphi}\in\mathbb{C}: \varphi=0, \rho \in [0,R]\}
\cup \{\rho e^{i \varphi}\in\mathbb{C}: \varphi \in (0,-\frac{2 \pi} N), \rho=R\}
\cup\{\rho e^{i \varphi}\in\mathbb{C}: \varphi=-\frac{2 \pi}N,$ $\rho\in(0,R]\}$. We get, by residues theorem, $$\begin{aligned}
-\int_0^{\infty} \frac{z^{M-1}}{z^N+\lambda} \,e^{-a z} \,\mathrm{d}z
+e^{-2i \frac{M}{N}\pi} \int_0^{\infty} \frac{z^{M-1}}{z^N+\lambda}\,
e^{-a e^{-i\frac{2\pi}{N}}z} \,\mathrm{d}z
&=
2i \pi \,\mathrm{Residue}\!\left(H,\!\sqrt[N]{\lambda} \,e^{-i \frac{\pi}{N}}\right)
\\
&=
\frac{2i\pi}{N} \left(\!\!\!\vphantom{\sqrt a}\right.\sqrt[N] \lambda
\!\left.\vphantom{\sqrt a}\right)^{\!M-N} e^{-i \frac{M-N}{N}\pi}
e^{-a \!\sqrt[N] \lambda \,e^{-i\frac{\pi}{N}}}
\\
&=
\frac{2 \pi}{Ni} \,\lambda^{\frac{M}{N}-1} e^{-i\frac{M}{N}\pi}
e^{-a\!\sqrt[N] \lambda \,e^{-i\frac{\pi}{N}}}.\end{aligned}$$ For $M=N-m$ and $a=\theta_j\,e^{i \frac{\pi}{N}}x$, this yields $$\int_0^{\infty} e^{-\lambda u} I_{j;m} (u;x) \,\mathrm{d}u=
-e^{-i\frac{m}{N}\pi} \lambda^{-\frac{m}{N}} \times (-e^{i\frac{m}{N}\pi})
e^{-\theta_j\!\!\sqrt[N] \lambda \,x}=\lambda^{-\frac{m}{N}}e^{-\theta_j\!\!\sqrt[N] \lambda \,x}.$$ Hence, (\[set24\]) is valid for $m\le N-1$. [ $\blacksquare$\
]{}
\[s\] The following identity holds: for any $x\in\mathbb{R}$ and $0<s<t$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} \xi^2 \sin(x \xi) e^{-t \xi^2} \!\left(\vphantom{\int}\right.\!
\int_0^{\xi} e^{s u^2} \,\mathrm{d}u\!\!\left.\vphantom{\int}\right) \mathrm{d}\xi}$}}
&=
\frac{\sqrt \pi}{8} \frac{2t-s}{t^2 (t-s)^{3/2}} \,x e^{-\frac{x^2}{4(t-s)}}
-\frac{\pi}{8 \sqrt s \,t^{3/2}} \left(1-\frac{x^2}{2t}\right) e^{-\frac{x^2}{4t}}\,
\mathrm{Erf}\!\left(-\frac12 \sqrt{\frac{s}{t(t-s)}} \,x\right)\!.
\label{identity-s}\end{aligned}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}Set $$\begin{aligned}
A
&=
\int_0^{\infty} \xi^2 \sin(x \xi) e^{-t \xi^2} \!\left(\vphantom{\int}\right.\!
\int_0^{\xi} e^{s u^2} \,\mathrm{d}u\!\!\left.\vphantom{\int}\right) \mathrm{d}\xi,
\\
B
&=
\frac{\sqrt \pi}{8} \frac{2t-s}{t^2 (t-s)^{3/2}} \,x e^{-\frac{x^2}{4(t-s)}}
-\frac{\pi}{8\sqrt s \,t^{3/2}} \left(1-\frac{x^2}{2t}\right) e^{-\frac{x^2}{4t}}
\mathrm{Erf}\!\left(-\frac{1}{2} \sqrt{\frac{s}{t(t-s)}} \,x\right)\!.\end{aligned}$$ Using the expansion of the sine function, we get $$A=\sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!} \int_0^{\infty} \left(\xi^{2n+3}
e^{-t\xi^2}\right)\!\!\left(\vphantom{\int}\right.\!
\int_0^{\xi} e^{s u^2} \,\mathrm{d}u\!\!\left.\vphantom{\int}\right)\mathrm{d}\xi.$$ In order to integrate by parts this last integral, we search for a primitive of $\xi^{2n+3} e^{-t \xi^2}$. With the change of variable $\zeta=t\xi^2$, we have $$\int \xi^{2n+3} e^{-t \xi^2}\,\mathrm{d}\xi
=\frac{1}{2t^{n+2}} \int \zeta^{n+1}e^{-\zeta} \,\mathrm{d}\zeta
=-\frac{(n+1)!}{2t^{n+2}} \left(\,\sum_{k=0}^{n+1} \frac{\zeta^k}{k!}\right) e^{-\zeta}
=-\frac{(n+1)!}{2t^{n+2}} \sum_{k=0}^{n+1} \frac{t^k \xi^{2 k}}{k!} \, e^{-t\xi^2}.$$ Then, $$\begin{aligned}
A
&=
\frac12 \sum_{n=0}^{\infty} (-1)^n \frac{(n+1)!}{(2n+1)! \,t^{n+2}}
\left[\,\sum_{k=0}^{n+1} \frac{t^k}{k!}
\int_0^{\infty} \xi^{2 k} e^{-t \xi^2} \,\frac{\mathrm{d}}{\mathrm{d}\xi}
\left(\vphantom{\int}\right.\!\int_0^{\xi} e^{s u^2} \,\mathrm{d}u\!\!
\left.\vphantom{\int}\right)\mathrm{d}\xi\right] x^{2n+1}
\nonumber\\
&=
\frac{x}{4 t^2 \sqrt{t-s}} \sum_{n=0}^{\infty} (-1)^n \frac{(n+1)!}{(2n+1)! \,t^n}
\left[\,\sum_{k=0}^{n+1} \frac{\Gamma\!\left(\vphantom{\frac aN}\right.\!\! k+\frac12
\!\!\left.\vphantom{\frac aN}\right)}{k!} \left(\frac{t}{t-s}\right)^{\!k}\right]x^{2n}.
\label{expression1}\end{aligned}$$ On the other hand, by expanding $\mathrm{Erf}(\xi)=\frac{2}{\sqrt \pi} \int_0^{\xi} \sum_{n=0}^{\infty}
(-1)^n \frac{u^2}{n!} \,\mathrm{d}u=\frac{2}{\sqrt \pi}
\sum_{n=0}^{\infty} (-1)^n \frac{\xi^{2n+1}}{(2n+1)n!}$, we get $$\begin{aligned}
B
&=
\frac{\sqrt \pi}{8} \frac{2t-s}{t^2 (t-s)^{3/2}} \,x e^{-\frac{x^2}{4(t-s)}}
\nonumber\\
&\hphantom{=\,}
-\frac{\pi}{8\sqrt s \,t^{3/2}} \left(\,\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n! (4t)^n}
-2\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+2}}{n! (4t)^{n+1}}\right) \frac{2 x}{\sqrt \pi}
\sum_{n=0}^{\infty} \left(\frac12 \sqrt{\frac{s}{t(t-s)}}\,\right)^{\!2n+1}
\frac{(-1)^{n+1}x^{2n}}{(2n+1)n!}
\nonumber\\
{\noalign{\noindent $\displaystyle{}$}}
&=
\frac{\sqrt \pi}{8t^2} \frac{x}{\sqrt{t-s}} \left[\frac{2t-s}{t-s}
\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n! (4(t-s))^n}\right.
\nonumber\\
&\hphantom{=\,}
\left.+\left(\,\sum_{p=0}^{\infty}(-1)^p \frac{2p+1}{p!} \frac{x^{2p}}{(4t)^p}\right) \!\!
\left(\,\sum_{q=0}^{\infty}\frac{(-1)^q}{(2q+1)q!}
\left(\frac{s}{4t(t-s)}\right)^{\!q} x^{2q}\right)\right]\!.
\label{somme1}\end{aligned}$$ The computation of the above product of series can be carried out as follows: $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\left(\,\sum_{p=0}^{\infty}(-1)^p \frac{2p+1}{p!} \frac{x^{2p}}{(4t)^p}\right) \!\!
\left(\,\sum_{q=0}^{\infty}\frac{(-1)^q}{(2q+1)q!} \left(\frac{s}{4t(t-s)}\right)^{\!q} x^{2q}\right)}$}}
&=
\sum_{n=0}^{\infty} \left[\vphantom{\sum_p}\right.
\sum_{\substack{p,q\ge 0:\\ p+q=n}} \frac{2p+1}{(2q+1)p!q!}
\left(\frac{s}{t-s}\right)^{\!q} \!\left.\vphantom{\sum_p}\right]\frac{(-1)^nx^{2n}}{(4t)^n}
\nonumber\\
&=
\sum_{n=0}^{\infty} \left[\,\sum_{q=0}^n \frac{2n-2q+1}{2q+1} \binom{n}{q}\!\!
\left(\frac{s}{t-s}\right)^{\!q} \right]\frac{(-1)^nx^{2n}}{n!(4t)^n}
\nonumber\\
&=
\sum_{n=0}^{\infty} \left[(2n+2)\sum_{q=0}^n \frac{\binom{n}{q}}{2q+1}
\left(\frac{s}{t-s}\right)^{\!q} -\sum_{q=0}^n \binom{n}{q}\!\!
\left(\frac{s}{t-s}\right)^{\!q} \right]\frac{(-1)^nx^{2n}}{n!(4t)^n}
\nonumber\\
&=
\sum_{n=0}^{\infty} \left[(2n+2)\sum_{q=0}^n \frac{\binom{n}{q}}{2q+1}
\left(\frac{s}{t-s}\right)^{\!q} \right]\frac{(-1)^nx^{2n}}{n!(4t)^n}
-\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n}}{n!(4(t-s))^n}.
\label{somme2}\end{aligned}$$ By inserting (\[somme2\]) into (\[somme1\]), we derive $$B=\frac{\sqrt \pi}{8 t^2} \frac{x}{\sqrt{t-s}} \sum_{n=0}^{\infty}
\left[\left(\frac{t}{t-s}\right)^{\!n+1}\! + \sum_{q=0}^n \frac{\binom{n}{q}}{2q+1}
\left(\frac{s}{t-s}\right)^{\!q}\right] \frac{(-1)^nx^{2n}}{n!(4t)^n}.
\label{somme3}$$ Writing now $(\frac{s}{t-s})^q=(\frac{t}{t-s}-1)^q
=\sum_{k=0}^q (-1)^{q-k}\binom{q}{k} (\frac{t}{t-s})^k$, we have $$\sum_{q=0}^n \frac{\binom{n}{q}}{2q+1}\left(\frac{s}{t-s}\right)^{\!q}
=\sum_{k=0}^n (-1)^k \left(\vphantom{\frac{\binom{q}{k}}{q}}\right.\!
\sum_{q=k}^n (-1)^q \left.\frac{\binom{q}{k} \binom{n}{q}}{2q+1}\right)
\!\!\left(\frac{t}{t-s}\right)^{\!k}.
\label{somme3bis}$$ The sum with respect to $q$ can be calculated as follows: $$\begin{aligned}
\sum_{q=k}^{n} (-1)^q \frac{\binom{q}{k} \binom{n}{q}}{2q+1}
&=
\binom{n}{k} \sum_{q=k}^{n} (-1)^q \frac{\binom{n-k}{q-k}}{2q+1}
=(-1)^k\binom{n}{k} \sum_{q=0}^{n-k} (-1)^q \frac{\binom{n-k}{q}}{2q+2k+1}
\nonumber\\
&=
(-1)^k\binom{n}{k} \int_0^1 \sum_{q=0}^{n-k} (-1)^q \binom{n-k}{q} \xi^{2q+2k}\,\mathrm{d}\xi
\nonumber\\
&=
(-1)^k\binom{n}{k} \int_0^1 (1-\xi^2)^{n-k}\xi^{2k}\,\mathrm{d}\xi
=(-1)^k\frac12\binom{n}{k} \int_0^1 (1-\xi)^{n-k}\xi^{k-1/2}\,\mathrm{d}\xi
\nonumber\\
&=
(-1)^k\frac{1}{2} \binom{n}{k} B\!\left(n-k+1,k+\frac12\right)
=(-1)^k\frac{n!\,\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
k+\frac12 \!\!\left.\vphantom{\frac aN}\right)}{2k!\,\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!n+\frac32 \!\!\left.\vphantom{\frac aN}\right)}.
\label{somme4}\end{aligned}$$ Plugging (\[somme4\]) into (\[somme3bis\]), and next (\[somme3bis\]) into (\[somme3\]), we obtain $$\begin{aligned}
B
&=
\frac{\sqrt \pi}{8 t^2} \frac{x}{\sqrt{t-s}} \sum_{n=0}^{\infty}
\frac{(-1)^n}{n! (4t)^n} \left[\left(\frac{t}{t-s}\right)^{\!n+1}+\frac{(n+1)!}{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!n+\frac32\!\!\left.\vphantom{\frac aN}\right)}
\sum_{k=0}^n \frac{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!k+\frac12
\!\!\left.\vphantom{\frac aN}\right)}{k!} \left(\frac{t}{t-s}\right)^{\!k}\right] x^{2n}
\nonumber\\
&=
\frac{\sqrt \pi}{8 t^2} \frac{x}{\sqrt{t-s}} \sum_{n=0}^{\infty} \frac{(-1)^n}{n! (4t)^n}
\frac{(n+1)!}{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!n+\frac32
\!\!\left.\vphantom{\frac aN}\right)} \left[\,\sum_{k=0}^{n+1} \frac{\Gamma
\!\left(\vphantom{\frac aN}\right.\!\!k+\frac12\!\!\left.\vphantom{\frac aN}\right)}{k!}
\left(\frac{t}{t-s}\right)^{\!k}\right] x^{2n}
\nonumber\\
&=
\frac{x}{4 t^2 \sqrt{t-s}} \sum_{n=0}^{\infty} (-1)^n \frac{(n+1)!}{(2n+1)! t^n}
\left[\,\sum_{k=0}^{n+1} \frac{\Gamma\!\left(\vphantom{\frac aN}\right.\!\!
k+\frac12 \!\!\left.\vphantom{\frac aN}\right)}{k!} \left(\frac{t}{t-s}\right)^{\!k}\right]x^{2n}
\label{expression2}\end{aligned}$$ where we used in the last equality $\Gamma\!\left(\vphantom{\frac aN}\right.\!\!n+\frac32\!\!\left.\vphantom{\frac aN}\right)\!
=\frac{\sqrt \pi}{2\cdot 4^n} \frac{(2n+1)!}{n!}$. In view of (\[expression1\]) and (\[expression2\]), we see that both members of (\[identity-s\]) are equal: $A=B$. [ $\blacksquare$\
]{}
\[ss\] We have, for $x \le 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} y(y-x) \,e^{-\frac{y^2}{4 s}-\frac{(y-x)^2}{4(t-s)}} \,\mathrm{d}y}$}}
&=
-2\,\frac{s(t-s)^2}{t^2} \,x e^{-\frac{x^2}{4(t-s)}}+2\sqrt \pi\,\frac{s^{3/2} (t-s)^{3/2}}{t^{3/2}}
\left(1-\frac{x^2}{2t}\right) e^{-\frac{x^2}{4t}}\,
\mathrm{Erfc}\!\left(-\frac12 \,\sqrt{\frac{s}{t(t-s)}}\,x\right)\!,\end{aligned}$$ and, for $x \ge 0$, $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} y(y+x) \,e^{-\frac{(y+x)^2}{4 s}-\frac{y^2}{4(t-s)}} \,\mathrm{d}y}$}}
&=
2\,\frac{s^2(t-s)}{t^2} \,x e^{-\frac{x^2}{4s}}+2\sqrt \pi\,
\frac{s^{3/2} (t-s)^{3/2}}{t^{3/2}}\left(1-\frac{x^2}{2t}\right) e^{-\frac{x^2}{4t}}\,
\mathrm{Erfc}\!\left(\frac12 \,\sqrt{\frac{t-s}{s\,t}}\,x\right)\!.\end{aligned}$$
[<span style="font-variant:small-caps;">Proof\
</span>]{}We only produce the proof for $x\le 0$ since the case $x\ge 0$ is quite similar. We have $$\begin{aligned}
{\noalign{\noindent $\displaystyle{\int_0^{\infty} y(y-x) \,e^{-\frac{y^2}{4 s}-\frac{(y-x)^2}{4(t-s)}} \,\mathrm{d}y}$}}
&=
e^{-\frac{x^2}{4t}} \int_0^{\infty} y(y-x) \,e^{-\frac{t}{4s(t-s)} (y-\frac{sx}{t})^2} \,\mathrm{d}y
\\
&=
e^{-\frac{x^2}{4t}} \int_{-sx/t}^{\infty} \left(y+\frac{sx}{t}\right)\!\!
\left(y-\frac{(t-s)x}{t}\right) e^{-\frac{t}{4s(t-s)}y^2}\,\mathrm{d}y
\\
&=
e^{-\frac{x^2}{4t}} \int_{-sx/t}^{\infty} \left(y^2-\frac{s(t-s)}{t^2}\,x^2\right)
e^{-\frac{t}{4s(t-s)}y^2} \,\mathrm{d}y+\frac{2s-t}{t}\,x e^{-\frac{x^2}{4t}}
\int_{-sx/t}^{\infty} y e^{-\frac{t}{4s(t-s)}y^2} \,\mathrm{d}y
\\
&=
e^{-\frac{x^2}{4t}} \int_{-sx/t}^{\infty} y^2 e^{-\frac{t}{4s(t-s)}y^2} \,\mathrm{d}y
-\frac{s(t-s)}{t^2} \,x^2 e^{-\frac{x^2}{4t}}
\int_{-sx/t}^{\infty} e^{-\frac{t}{4s(t-s)}y^2}\,\mathrm{d}y
+ 2\,\frac{s(t-s)(2s-t)}{t^2} \, x e^{-\frac{x^2}{4(t-s)}}.\end{aligned}$$ Integrating by parts, we observe that $$\int_{-sx/t}^{\infty} y^2 e^{-\frac{t}{4s(t-s)}y^2} \,\mathrm{d}y
=2\,\frac{s^2(t-s)}{t^2}\,xe^{-\frac{sx^2}{4t(t-s)}}
+2\,\frac{s(t-s)}{t} \int_{-sx/t}^{\infty} e^{-\frac{t}{4s(t-s)}y^2}\,\mathrm{d}y,$$ we finally get the result. [ $\blacksquare$\
]{}
[3]{}
Abramowitz, M. and Stegun, I.-A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, 1992.
Beghin, L., Hochberg, K.-J. and Orsingher, E.: Conditional maximal distributions of processes related to higher-order heat-type equations. Stochastic Process. Appl. 85 (2000), no. 2, 209–223.
Beghin, L. and Orsingher, E.: The distribution of the local time for “pseudoprocesses” and its connection with fractional diffusion equations. Stochastic Process. Appl. 115 (2005), 1017–1040.
Beghin, L., Orsingher, O. and Ragozina, T.: Joint distributions of the maximum and the process for higher-order diffusions. Stochastic Process. Appl. 94 (2001), no. 1, 71–93.
Bohr, H.: Über die gleichmässige Konvergenz Dirichletscher Reihen. J. für Math. 143 (1913), 203–211.
Borodin, A.-N. and Salminen, P.: Handbook of Brownian motion—facts and formulae. First edition. Probability and its Applications. Birkhäuser Verlag, 1996.
Erdélyi, A., Ed.: Higher transcendental functions, vol. III, Robert Krieger Publishing Company, Malabar, Florida, 1981.
Hochberg, K.-J.: A signed measure on path space related to Wiener measure. Ann. Probab. 6 (1978), no. 3, 433–458.
Hochberg, K.-J. and Orsingher, E.: The arc-sine law and its analogs for processes governed by signed and complex measures. Stochastic Process. Appl. 52 (1994), no. 2, 273–292.
Krylov, V. Yu.: Some properties of the distribution corresponding to the equation $\frac{\partial u}{\partial t}=(-1)^{q+1}\frac{\partial^{2q} u}{\partial^{2q} x}$. Soviet Math. Dokl. 1 (1960), 760–763.
Lachal, A.: Distribution of sojourn time, maximum and minimum for pseudo-processes governed by higer-order heat-typer equations. Electron. J. Probab. 8 (2003), paper no. 20, 1–53.
Lachal, A.: Joint law of the process and its maximum, first hitting time and place of a half-line for the pseudo-process driven by the equation $\frac{\partial}{\partial t}=\pm\frac{\partial^N}{\partial x^N}$. C. R. Acad. Sci. Paris, Sér. I 343 (2006), no. 8, 525–530.
Lachal, A.: First hitting time and place, monopoles and multipoles for pseudo-procsses driven by the equation $\frac{\partial}{\partial t}=\pm \frac{\partial^N}{\partial x^N}$. Electron. J. Probab. 12 (2007), 300–353.
Lachal, A.: First hitting time and place for the pseudo-process driven by the equation $\frac{\partial}{\partial t}=\pm\frac{\partial^n}{\partial x^n}$ subject to a linear drift. Stoch. Proc. Appl. 118 (2008), 1–27.
Nakajima, T. and Sato, S.: On the joint distribution of the first hitting time and the first hitting place to the space-time wedge domain of a biharmonic pseudo process. Tokyo J. Math. 22 (1999), no. 2, 399–413.
Nikitin, Ya. Yu. and Orsingher, E.: On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 (2000), no. 4, 997–1012.
Nishioka, K.: Monopole and dipole of a biharmonic pseudo process. Proc. Japan Acad. Ser. A 72 (1996), 47–50.
Nishioka, K.: The first hitting time and place of a half-line by a biharmonic pseudo process. Japan J. Math. 23 (1997), 235–280.
Nishioka, K.: Boundary conditions for one-dimensional biharmonic pseudo process. Electronic Journal of Probability 6 (2001), paper no. 13, 1–27.
Orsingher, E.: Processes governed by signed measures connected with third-order “heat-type” equations. Lithuanian Math. J. 31 (1991), no. 2, 220–231.
Spitzer, F.: A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 (1956), 323–339.
[^1]: Dipartimento di Statistica, Probabilità e Statistiche Applicate, <span style="font-variant:small-caps;">University of Rome ‘La Sapienza’</span>, P.le A. Moro 5, 00185 Rome, <span style="font-variant:small-caps;">Italy</span>. *E-mail address*: `valentina.cammarota@uniroma1.it`
[^2]: Pôle de Mathématiques/Institut Camille Jordan/CNRS UMR5208, Bât. L. de Vinci, <span style="font-variant:small-caps;">Institut National des Sciences Appliquées de Lyon</span>, 20 av. A. Einstein, 69621 Villeurbanne Cedex, <span style="font-variant:small-caps;">France</span>. *E-mail address:* `aime.lachal@insa-lyon.fr`, *Web page:* `http://maths.insa-lyon.fr/\mbox{}^{\sim}lachal`
|
Introduction
============
The solar neutrino experiments - Homestake (HM), Kamiokande (K), GALLEX, SAGE, and Super-Kamiokande (SK) - have shown the existence of robust quantitative differences between the experiments and the combined predictions of minimal standard electroweak theory and stellar evolution theory. On the other hand, the latter is nowadays in significant agreement with the constraints posed by helioseismology and thus we can consider the possibility that the neutrinos have properties other than those included in the standard electroweak model. The Mikheyev-Smirnov-Wolfenstein [@msw] (MSW) matter-enhanced oscillation and vacuum (“just-so”) oscillation [@vo] (VO) provide an explanation of the neutrino deficit, although it is not yet clear which mechanism produces the required suppression. Due to the increasing accuracy of the results of the present and future neutrino experiments, it is interesting to investigate the effects of the uncertainty in solar physics parameters on the solar neutrino oscillation scenarios.
We study how the allowed regions in the parameter space of the two-flavour oscillations are modified when $S_{11}$ and $S_{17}$, the astrophysical zero energy S-factors of the reactions $^1{\rm H}(p,e^+\nu_{\rm e})^2{\rm H}$ and $^7{\rm Be}(p,\gamma)^8{\rm
B}$, are changed within the ranges derived from the nuclear physics calculations and experiments, using up-to-date solar models.
The efficiency of the first reaction determines, through $S_{11}$, the evolution of the chemical composition in the Sun and its hydrostatic structure. Since the meteoritic age of the Sun is fairly well known [@BP95], any modification of $S_{11}$ changes the present central abundance of hydrogen and hence the behaviour of the adiabatic sound speed which can also be determined by helioseismic p-mode data inversion. Most of the astrophysical S-factors of the relevant nuclear reactions in the Sun are determined from measurements in the laboratory at higher energies, extrapolated down to zero energy. However, due to the very rare event rate of $^1{\rm
H}(p,e^+\nu_{\rm e})^2{\rm H}$ at high energies (1 reaction in $\sim
10^6$ years at 1 MeV for a proton beam of 1 mA [@PR91]) this procedure is not applicable to $S_{11}$ and its estimation must be obtained from standard weak-interaction theory [@Kam94]. The latest suggested value [@Adel98] is $S_{11}=4.00\;10^{22}~
{\rm keV\,barn}$ with an uncertainty of $\simeq\pm\,2.5\,\%$ at $1\,\sigma$. This is of the same order as the uncertainty of the free neutron decay time, which is linked to the ratio of the axial-vector to the Fermi weak-coupling constants.
The reaction $^7{\rm Be}(p,\gamma)^8{\rm B}$ produces the dominant signal in the HM, SK and Sudbury Neutrino Observatory (SNO) neutrino experiments. Unfortunately $S_{\rm 17}$ is one of the most poorly known quantity of the entire nucleosynthesis chain which leads to the ${\rm^8B}$ formation. The reaction cross-section is measured down to $134~{\rm keV}$ with large statistical and systematic errors which dominate the uncertainty in the determination of the astrophysical factor at low energies [@Tur93]. [@Adel98] recently quoted $S_{17}=19^{+4}_{-2}~{\rm eV\,barn}$ at $1\sigma$, suggesting a conservative value of $S_{17}$ in the range $15~{\rm eV\,barn}$ to $27~{\rm eV\, barn}$ with an error of $\simeq\pm 30\%$ at $3\sigma$. Any change in $S_{\rm 17}$ affects only the $^8{\rm B}$ neutrino flux, $\phi_{\nu}(^8{\rm B})$, and leaves all the other relevant quantities of the solar model, such as the sound speed profile and the neutrino fluxes produced in the other reactions, unaltered [@Pat97].
The value of $S_{11}$ influences indirectly the total $\phi_{\nu}(^8{\rm B})$ which is quite sensitive to the central temperature of the Sun $T_{\rm c}$ ($\phi_\nu(^8 {\rm B})\propto S_{17} {T}_{\rm c}^{24}$ [@newb].). In fact, a change in $S_{11}$ determines a change in both the total $pp$-neutrino flux, $\phi_{\nu}(pp)$, and $T_{\rm c}$, being $\Delta T_{\rm c}/T_{\rm c}\simeq -0.15\, \Delta S_{11}/S_{11}$ and $\phi_\nu(pp)\propto T^{-1}_{\rm c}$. We therefore constrain $S_{11}$ with the help of helioseismology, in order to reduce its influence on the total $\phi_{\nu}(^8{\rm B})$ uncertainty. Nonetheless, the greatest uncertainty in this flux still remains the measurement of $S_{17}$.
Since the efficiency of $^1{\rm H}(p,e^+\nu_{\rm e})^2{\rm H}$ influences mainly the structure of the solar model and the neutrino rates, whereas the situation is opposite for the strength of $^7{\rm Be}(p,\gamma)^8{\rm B}$, we have considered the following cases: 1) $S_{17}$ standard and $S_{11}$ varied; 2) $S_{11}$ standard and $S_{17}$ varied. Here by [*standard*]{} we denote the most favoured values for $S_{17}$ and $S_{11}$ suggested by [@Adel98] and by [*varied*]{} we mean a conservative range of variations allowed at $\sim 99\%$ confidence level. All the other reaction rates, such as $S_{33}$ and $S_{34}$, are left unaltered to their standard values as given in [@Adel98]. In Section II we investigate case 1) by using helioseismic data in order to obtain more stringent limits on the “unsuppressed” total $\phi_{\nu}(pp)$.
------------- ------------------------------------------------------- -------------
Experiment Data $\pm$ (stat)$\pm$(syst.) theor. err.
HM $2.56\pm 0.16 \pm 0.15~{\rm SNU}$ 13.1%
SAGE $69.9^{+8.0\,+3.9}_{-7.7\, -4.1}~{\rm SNU}$ 5.8%
GALLEX $76.4\pm6.3^{+4.5}_{-4.9}~{\rm SNU}$ 5.8%
SK $2.44\pm0.05^{+0.09}_{-0.07}~{\rm 10^6cm^{-2}s^{-1}}$ 14%
GALLEX+SAGE $72.4\pm6.6~{\rm SNU}$ 5.8%
------------- ------------------------------------------------------- -------------
: \[rates\] Solar neutrino event rates with $1\sigma$ errors. In the theoretical errors the $S_{\rm 17}$-uncertainty is removed.
\
The behaviour of the neutrino mixing parameters $\Delta m^2$ and $\sin^22\theta$ as a function of $S_{17}$ is presented in Section III. The mixing parameters are obtained through $\chi^2$-fits by using the recent results of HM, GALLEX, SAGE and SK experiments as shown in Table \[rates\]. We consider MSW and VO transitions into active (non-sterile) and sterile neutrinos as well. Previous analyses in this direction have been carried out by other authors which used different approaches and considered an arbitrary $\phi_{\nu}(^8{\rm B})$ [@lan] as an additional free parameter. In our calculations the Earth regeneration effect is included and the exact evolution equation for the neutrino mixing is solved numerically without resorting to analytical approximations.
In Section IV, the first and second moments of the recoil electron spectra in SK are calculated for the best-fit values of ${\sin}^{2}2\theta$ and $\Delta {\rm m}^{2}$ obtained in the previous section, and we discuss the possibility of considering $S_{17}$ as a free parameter in the analysis of the forthcoming data from both SNO and Borexino experiments. It is shown that a determination of the lower and upper limits on the $S_{17}$ values can be derived from the measurement of the charged current to neutral current relative ratio (CC/NC) in SNO. Section V is devoted to the conclusions.
solar models
============
The solar models were computed by using the latest version of the GArching SOlar MOdel (GARSOM) code, which originates from the Kippenhahn stellar evolution program [@KWH67]. Its numerical and physical features are described in more details in [@Sch97]. In particular, it uses the latest OPAL-opacities [@OPOPAC] and equation of state [@OPEOS] and it takes into account microscopic diffusion of hydrogen, helium and heavier elements (e.g. C, N, O). The diffusion constants are calculated by solving Burgers’ equation for a multicomponent fluid via the routine described in [@Thoul]. The standard values of the reaction rates are taken from [@Adel98]. In the present version the equations for nuclear network and diffusion are solved simultaneously. We follow the evolution of the models from ZAMS to an age of 4.6 Gyr. The metal abundances are taken from [@Gre93]. The convection is described by the mixing length theory [@MLT]. Unlike previous work [@Sch97] where models of solar atmosphere were used, here we consider an Eddington atmosphere for the outer boundary conditions since we focus our attention on processes occurring in the deep interior, where the exact stratification of the atmosphere has almost no influence.
A comparison of the present model with other up-to-date standard solar models is given in [@Sch99]. In Fig. \[cspeed\] we show the behavior of sound speed in our standard solar model as compared with the seismic model derived by S. Basu and J. Christensen-Dalsgaard by inverting the GOLF+MDI data [@silv1]. The values of some basic quantities of our models are summarized in Table \[models\]. These values refer to the computation of different solar models with $S_{11}$ varied within the extreme cases of $3.89\;10^{-22}~{\rm keV\,barn}$ and $4.20\;10^{-22}~{\rm keV\,barn}$, and $S_{17}$ kept at the standard value (case 1). Each model with a given value of $S_{11}$ has been obtained by following the whole evolution and adjusting the mixing length, initial helium abundance, and chemical composition to fit solar luminosity, effective temperature and the surface value of $Z/X=0.0245$ [@Gre93]. All other input physics, like opacity or the equation of state, is the same for all the models. In particular, the luminosity and effective temperatures of the models differ from solar luminosity and effective temperature less than $10^{-4}$ for all the models considered.
to
---------- ----------------- ------------- ------------------ ------------------------- -------- -------
$s_{11}$ ${\rm T_{c}}$ ${\mu_{c}}$ $R_{cz}/R_\odot$ $\phi(^8{\rm B})$ GALLEX HM
$(10^7{\rm K})$ $({\rm (SNU) (SNU)
cm}^{-2} {\rm s}^{-1})$
3.89 1.578 0.860 0.715 5.54 $10^6$ 131.8 8.2
4.00 1.574 0.859 0.713 5.16 $10^6$ 129.5 7.7
4.10 1.567 0.858 0.712 4.85 $10^6$ 127.5 7.3
4.20 1.563 0.857 0.711 4.56 $10^6$ 125.6 6.9
---------- ----------------- ------------- ------------------ ------------------------- -------- -------
: \[models\] Solar models with different values of $s_{11}$= $S_{11}/(10^{-22}~{\rm keV\,barn})$, the *standard* model has $s_{11}$=4.00.
The production region of the $pp$ neutrinos extends up to $r<0.3\,R_\odot$. This region is within the reach of the low order $p$-modes. It can be of interest to verify to which extent the uncertainty in the theoretical calculations of $S_{11}$ can be constrained by helioseismic data. In order to investigate this possibility we have compared the sound speed profile of solar models with different $S_{11}$ with the sound speed profile derived from helioseismic data inversion in [@silv1]. The result is shown in Fig. \[cspeed\] where it appears that a model with the highest $S_{11}$ better reproduces the internal stratification.
A different method is the forward approach where small differences in frequencies of low order modes are compared. The small spacing differences $\delta \nu_{n,l} =\nu_{n,l} -\nu_{n-1,l+2}$, for $l=0$ and $l=1$, are in fact highly sensitive to the sound speed gradient in the very central region of the Sun. For this purpose we have then used a weighted average of the first 144 days of MDI and of 8 months GOLF data [@golf] for $l=0,1,2,3$ and $n$ from $10$ up to $26$. We have thus calculated $\delta\nu_{n ,l}$ for $l=0$ and $l=1$ relative to solar models with different $S_{11}$.
to
From an ispection of Fig. \[sms\] it appears that, for $l=1$, the model with the highest $S_{11}$ seems to approach more closely the real Sun (a similar conclusion is obtained for $l=0$). This is consistent with the results of secondary inversions for the temperature profile where it has been estimated that $S_{11}=(4.15\pm 0.25)\;10^{-22}~{\rm
keV \, barn}$ [@Antia98]. Since both the inverse and forward helioseismic approach indicate that higher values of $S_{11}$ seem more favoured, we are allowed to conclude that the total $\phi_{\nu}(pp)
\propto S_{11}^{0.14}S_{33}^{0.03}S_{34}^{-0.06}$ can be considered as bounded from below at the value $$5.93\;10^{10} {\rm cm^{-2} s^{-1}}\leq \phi(pp)$$ from helioseismic data.
The greatest uncertainty in the neutrino flux predicted by solar models comes from the poorly known $^8{\rm B}$-neutrinos, whose flux is mainly determined by the reaction rate of $^7{\rm Be}(p,\gamma)^8{\rm B}$, the first reaction of the $pp$III-subcycle. This subcycle contributes by only 0.01% to the total energy production of the $pp$-cycle though it is responsible for the emission of the most energetic neutrinos produced in this subcycle. Its contribution has practically no influence on the solar structure, thus excluding any possibility of producing signatures in the helioseismic frequencies. For the computations that follow we keep $S_{11}$ fixed at its standard value, and we vary $S_{17}$ within the allowed “conservative” range.
results for neutrino oscillation parameters
===========================================
In this section we present the results obtained from the total rates in the GALLEX/SAGE, HM and SK detectors (Table \[rates\]) for our modified solar model introduced in the previous section. We have calculated the allowed parameter space ($\Delta m^2, {\sin ^2
2\theta}$) for neutrino oscillations in the two-flavour case, taking the theoretical errors from [@Bah98]. As we study the influence of $S_{17}$ on the oscillation parameters we remove its contribution from the total theoretical uncertainty.
to
For the calculation of the MSW-effect we piecewise linearize the density profile of the respective solar models, and the evolution equations for neutrino oscillations are then integrated by using the exact solution on each linear part. We also include the average earth-regeneration effect [@Earth]. Since the models with different values of $S_{17}$ predict a different $^8$B-neutrino flux, the expected event rate changes for SK and also for GALLEX/SAGE and HM. Thus, different conversion probabilities are needed for each value of $S_{17}$ in order to explain the measured rates in these experiments. This leads to different confidence regions in the $\sin^22\theta-\Delta m^2$-plane (see Fig. \[plane\]).
to
The general trend in the small mixing angle (SMA) solution shows that an increase of $S_{17}$ shifts the mixing towards larger angles, while keeping the mass difference almost constant. Similar trend can also be noted for the VO case (Fig. \[sno2\]). In the large mixing angle (LMA) solution both the mass difference and the mixing angle decrease with increasing $S_{17}$.
The results shown in Fig. \[plane\] and Fig. \[sno2\] indicate that if on one hand there are always three possible well separated solutions of VO, SMA and LMA, on the other hand it is difficult to disentangle additional effects in each of the solutions for the present experimental status, since $\chi^2$ has rather shallow minima (Fig. \[sno\_chi\]).
---------- ---------------- ------------------- ---------------- ------------------- ----------------- ------------------- --
$S_{17}$ $\Delta m^2$ $\sin^2(2\theta)$ $\Delta m^2$ $\sin^2(2\theta)$ $\Delta m^2$ $\sin^2(2\theta)$
15 $5.2\;10^{-6}$ $4.2\;10^{-3}$ $2.7\;10^{-4}$ 0.88 $1.1\;10^{-10}$ 0.88
17 $5.2\;10^{-6}$ $6.1\;10^{-3}$ $8.5\;10^{-5}$ 0.88 $1.1\;10^{-10}$ 0.93
19 $5.3\;10^{-6}$ $6.5\;10^{-3}$ $7.4\;10^{-5}$ 0.82 $9.1\;10^{-11}$ 0.78
23 $5.2\;10^{-6}$ $8.8\;10^{-3}$ $2.1\;10^{-5}$ 0.69 $6.6\;10^{-11}$ 0.85
27 $5.3\;10^{-6}$ $1.0\;10^{-2}$ $1.6\;10^{-5}$ 0.57 $8.7\;10^{-11}$ 0.95
---------- ---------------- ------------------- ---------------- ------------------- ----------------- ------------------- --
: \[matter\] Best-fit solutions for the total event rates in Table I. The first two column refers to SMA solution, the second to LMA and the last ones to VO. $S_{17}$ is given in ${\rm eV\,barn}$
A constraint on $S_{17}$ at $1\sigma$ ($\chi^2$$-$$\chi^2_{\rm min}$=1 in Fig. \[sno\_chi\]) can be obtained from the $\chi^2$ analysis of the total neutrino rate in the case of SMA solution which gives $9\leq S_{17}\leq 25$ and it leads to the following constraint on the $\phi_{\nu}(^8{\rm B})$ $$0.6 \leq f_{\nu_n}(^8{\rm B}) \leq 1.8 ~~~~~ (2\sigma),$$ where $f_{\nu_n}(^8{\rm B})$ is the normalized neutrino flux $\phi_{\nu}(^8{\rm B})/\phi_{\nu}(^8{\rm B})|_{standard}$.
to
We have also analysed the case of a non-standard $S_{11}$ concluding that, if the range of variation is limited by both helioseismology and nuclear physics uncertainties, the differences in the best-fit solutions are not very significant. Unfortunately, at the present time it is not clear which kind of oscillation mechanism is responsible for the neutrino suppression. Additional information should be available from the future data of SK, Borexino and SNO experiments.
Future data and experiments
===========================
In the following sections the expected forthcoming data for SK, Borexino and SNO are summarized. We focus (i) on the ability of these experiments to identify the oscillation mechanism (LMA, SMA or VO)) and (ii) on what is expected to be measured in these detectors using solar models with different values of $S_{17}$ and taking into account the present data of GALLEX/SAGE, HM and SK.
Super-Kamiokande
----------------
Recently, the SK-collaboration published first data about the zenith angle dependence [@SKzenit] of neutrino flux and electron recoil energy spectrum [@SKele] which seem to disfavour any of the above investigated solutions. However the present statistics and detector threshold at 6.5 MeV is not yet sufficient to exclude them. More precise conclusions can be reached in the future with the improvement of statistic and lowering of the threshold to 5 MeV.
We determined the values of $\Delta m^2$ and $\sin^2 2\theta$ needed in order to reproduce the present event rates in HM, GALLEX/SAGE and SK by using models with different values of $S_{17}$ (c.f. Fig \[plane\]). For the best fit LMA, SMA and VO-solutions (depending on $S_{17}$) we calculated the electron recoil spectrum by convolving the neutrino spectrum with the calculated survival probability, the neutrino-electron scattering cross section and the energy resolution function. Apparently, the spectrum in SK does not allow us to discriminate among different values of $S_{17}$ (ii), but it can provide important information to distinguish the different types of solutions (i). We thus calculated the first and second electron moments of the recoil electron energy distribution assuming a threshold of $5~\rm{MeV}$ and a energy scale uncertainty $\delta=\pm100$ keV as in [@bkr]. Further information can in fact be extracted from the relative deviations of the above two moments from the corresponding moments in the case of non-oscillating neutrinos $(\langle
E\rangle-\langle E\rangle_0)/\langle E\rangle_0$ and $(\langle
\sigma^2\rangle -\langle \sigma^2\rangle_0)/\langle
\sigma^2\rangle_0$ (the subscript “0” refers to the no-oscillation case.). As it is shown in Table \[mome\], different solutions lead to different relative deviations of the first two spectral moments.
We note in particular that in the SMA case an increase of $S_{17}$ leads to an increase of the relative deviation of both first and second moments, while in the LMA case, one finds the opposite behavior with a weaker relative variation. A trend that is qualitatively very similar to this one can also be observed for the sterile case.
[cccccccc]{}\
$S_{17}$ & $\Delta E\;[\%]$ & $\Delta \sigma^2\;[\%]$ & $\Delta E\;[\%]$ & $\Delta\sigma^2\;[\%]$ &$\Delta E\;[\%]$ & $\Delta\sigma^2\;[\%]$\
14 & 0.98 & 3.38 & -0.37 & -1.51 & 5.90 & 6.88\
19 & 1.41 & 4.98 & -0.49 & -1.58 & 3.32 & -1.64\
23 & 1.56 & 5.61 & -0.12 & -0.32 & 0.77 & -9.80\
\
$S_{17}$ & $\Delta E\;[\%]$ & $\Delta
\sigma^2\;[\%]$ & $\Delta E\;[\%]$ & $\Delta\sigma^2\;[\%]$&$\Delta
E\;[\%]$ & $\Delta\sigma^2\;[\%]$\
14 & 1.31 & 2.04 & -0.15 & -0.36 & 3.06 & -19.7\
19 & 2.17 & 2.91 & -0.55 & -0.72 & -0.21 & -21.3\
23 & 2.62 & 3.53 & 0.02 & 0.31 & -3.10 & -24.2\
Borexino
--------
The Borexino-experiment will measure mainly the ${\rm ^7Be}$-neutrinos via neutrino-electron-scattering, therefore no significant information can be obtained from this experiment about the value of $S_{17}$ (ii), as the expected counting rate is independent of $S_{17}$ (Fig. \[borexino\]a).
However, it is interesting to note that although the 1$\sigma$-regions of the SMA and LMA solution are well separated, at 2$\sigma$ level there is some overlap. In this case it may also be possible that the measurement of the event rate will not be sufficient to discriminate these solutions unless the value of $S_{17}$ is quite low.
The expected recoil electron spectra are shown in Fig. \[borexino\]b for the different types of solution. The SMA solution shows a rise in the signal at low energies, thus it is crucial to have good statistical data just above the detector threshold of ${\rm 0.25~MeV}$. The
to
behaviour of the SMA solution is described by the typical shape of the survival probability of electron neutrinos with varying energy (“valley” at intermediate energies). This leads to an almost full conversion of the $^7\rm{Be}$-neutrinos into $\nu_\tau, \nu_\mu$ or $\nu_{\rm s}$, partial conversion of the ${\rm ^8B}$-neutrino and almost no change of the $pp$-neutrinos. In the case of the LMA-solution the survival probability of $\nu_{\rm e}$ is almost constant for all the energies. In the light of the present solar neutrino experiments results (total rates) the MSW-SMA solution seems to be the most viable one for explaining the lack of $^7{\rm Be}$ and a reduction by a factor 2 of the $^8{\rm B}$ neutrinos.
In the VO case ($10^{-11}$$\le$$\Delta m^2$$\le$$10^{-9}$) the eccentric orbit of the earth leads to seasonal variations in the neutrino flux due to the long oscillation length $l_{\rm V} \approx
2.48 E/\Delta m^2$ ($l_{\rm V}$ in m, $E$ in MeV, $\Delta m^2$ in $eV^2$). Since 90% of the $^7{\rm Be}$-neutrinos are emitted in a monoenergetic line, this effect is more pronounced for these neutrinos than for $pp$ and $^8{\rm B}$-neutrinos, which are emitted in a continuous range of energies. In the SMA and LMA solutions no seasonal variation appears, thus Borexino should be able to discriminate between these cases and the VO solution (i).
SNO
---
The SNO experiment will measures the recoil electron spectrum of the reaction $$\nu_e+d \rightarrow p+p+e^-$$ and the ratio of the charged to neutral current events (CC/NC). Using the neutrino fluxes of a solar model with a fixed value of $S_{17}$, the expected (CC/NC)-ratio is determined by letting $\Delta m^2$ and $\sin^2 2\theta$ vary within the 68.4% and 95.4% C.L.-region of the LMA, SMA or VAC-solution. As shown in Fig. \[plane\], varying $S_{17}$ change the oscillation parameters which are able to reproduce the present results of GALLEX/SAGE, SK and HM. These changes alter the expected (CC/NC)-ratio, and thus provide an indirect dependence of the (CC/NC)-ratio on $S_{17}$ (ii). The found relations are shown in Fig. \[sno\]. For the calulation of the (CC/NC)-ratio in SNO we have used the energy resolution corresponding to a typical statistics of 5000 CC events.
to
In the case of VO solution the $1\sigma$ level of uncertainty is significantly larger than in the MSW solution case (Fig. \[sno\]). In the SMA scenario it is possible to determine an effective constraint on $S_{17}$ from the $1\sigma$-level strip of the (CC/NC) ratio. For instance, from Fig. \[sno\] it can be inferred that a measurement of (CC/NC) of $\simeq 0.8$ would imply $$S_{17}=19.0^{+2.0}_{-3.0}~{\rm keV\,barn},$$ if SMA turns out to be the solution of the solar neutrino puzzle. In the VO case the limits are not very stringent but they nevertheless provide independent constraints on the allowed value of $S_{17}$. However, this procedure is not very useful for sterile neutrinos, because no sensible variation of the (CC/NC) ratio occurs when $S_{17}$ is varied.
The recoil electron spectrum provides additional information about the type of the solution (i). In particular we have employed a Gaussian energy resolution function of width $\sigma_{10}=
1~{\rm MeV}$ at the electron energy $E_{\rm e}=10~{\rm MeV}$ as adopted in [@bkr]. For the best fit SMA and LMA solutions obtained from solar models with different values of $S_{17}$, the expected electron energy spectrum in SNO is shown in Fig. \[snospec\] for the case of active neutrinos.
to
The separation in the recoil electron spectra of both solutions is not very pronounced, therefore these data alone may not be sufficient to discriminate between LMA and SMA solution. We remark that the overall behaviour of the SMA and LMA solutions in SNO is very similar to the one in SK, namely that the average energy of the recoil electrons is higher in the SMA than in the LMA case for every value of $S_{17}$ (see also Table \[mome\]).
In the sterile case the differences among various cases with altered $S_{17}$ are much smaller (ii), and it is even more unlikely that any significant variation in the spectra will be visible, neither in SNO nor in SK.
Conclusions
===========
We have investigated the influence of $S_{11}$ on the sound speed and the small spacing frequency differences by comparing the model predictions with helioseismic data using up-to-date solar models. Moreover we discussed the change in the allowed parameter space for SMA, LMA and VO solutions with varying $S_{17}$. As shown in Section II the latest results from helioseismology suggest that the value of $S_{11}$ is slightly greater than the theoretically calculated one. However, since the statistical significance is weak, we conclude that the limits inferred from helioseismology and those derived from the theory are consistent. The influence of the value of $S_{11}$ on the solar neutrino flux is too small to alter the resulting neutrino mixing parameters significantly. However, the proposed LENS detector [@rrr] can observe in principle a suppression of the $pp$-neutrino flux and therefore it is reasonable to expect relevant differences in the signal as function of the $S_{11}$ value.
The present experiments GALLEX/SAGE, HM and SK favour neutrino oscillations as the solution to the solar neutrino deficit. Improved statistics in SK and future experiments like Borexino and SNO will provide powerful tools to support this solution. We have calculated the expected rates, electron moments, electron spectra or (CC/NC)-ratios of the above experiments for the SMA, LMA and VO solution provided by the present data. We expect that the combined data of the recoil electron spectra in SK, SNO and Borexino enable us to discriminate among these solutions.
Since the $^7{\rm Be}(p,\gamma)^8{\rm B}$ reaction has no influence on the solar structure, it is impossible to get information about its strength from helioseismology. Moreover the exact value for $S_{17}$ is crucial to calculate the flux of the most energetic solar neutrinos, which are measured in the SK and SNO experiments. The (CC/NC)-ratio expected in SNO is sensitive to the $\phi_{\nu}(^8{\rm B})$ which is directly related to the strength of $S_{17}$.
We conclude that the combination of SK, SNO and Borexino will be useful to test the consistency of the value of $S_{17}$ found by direct nuclear physics measurements with the combined analysis of theoretical models and neutrino experiments as described in Sections III and IV. Of course, the whole analysis was done under the assumption of neutrino-oscillations (either MSW or “just so”) as solution to the solar neutrino puzzle. In the case of oscillations into sterile neutrinos the strength of $^7{\rm Be}({\rm p},\gamma)^8{\rm B}$ does not leave any signature in the future experiments. However, this solution can be at least discriminated from the oscillations into active neutrinos by means of the behaviour of the (CC/NC) ratio.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to S. Turck-Chièze for useful discussions and for allowing us to use a set of the GOLF data, and to J. Christensen-Dalsgaard and S. Basu for providing us with the inverted sound speed profile derived from the GOLF+MDI data. We would also like to express our thanks to A. Weiss, H. M. Antia, J. N. Bahcall and S. Degli’Innocenti for useful comments and advices. The work of H. S. was partly supported by the “Sonderforschungbereich 375-95 für Astrophysik” der Deutschen Forschungsgemeinschaft. Furthermore A. B. acknowledges the INFN, Sezione di Catania and the MPA for financial support, and thanks the scientists at the MPA for their warm hospitality.
[10]{}
S. P. Mikheyev and A. Y. Smirnov, Sov. J. Nucl. Phys. [**42**]{}, 913 (1985); L. Wolfenstein, Phys. Rev. D [**17**]{}, 2369, (1978).
B. Pontecorvo, Sov. Phys. JETP [**26**]{}, 984 (1968); J. N. Bahcall and S. C. Frautschi, Phys. Lett. [**29B**]{}, 623 (1969); S. M. Bilenky and B. M. Pontecorvo, Phys. Rep. [**41**]{}, 225 (1978).
J. N. Bahcall, M. H. Pinsonneault and G. J. Wasserburg, Rev. Mod. Phys. [**67**]{}, 885, (1995).
P. D. Parker and C. Rolfs, in [*Solar Interior and Atmosphere*]{}, edited by A. N. Cox, W. C. Livingston, and M. S. Matthews (The University of Arizona Press, 1991), p. 31.
M. Kamionkowski and J. N. Bahcall, Astrophys. J. [**420**]{}, 884 (1994).
E. G. Adelberger [*et al.*]{}, Rev. Mod. Phys. [**70**]{}, 1265 (1998).
S. Turck-Chièze [*et al.*]{}, Phys. Rep. [**230**]{}, 57 (1993).
L. Paternò, in [*Fouth International Neutrino Confernce*]{}, edited by W. Hampel (Max-Planck-Institut für Kernphysik, Heidelberg, 1997), p. 54.
J. N. Bahcall and A. Ulmer, Phys. Rev. D [**53**]{} 4202, (1996);
N. Hata and P. Langaker, Phys. Rev. D [**52**]{}, 420, (1995); P. Krastev and A. Smirnov, Phys. Lett. B[**338**]{}, 282 (1994); P. I. Krastev and S. T. Petcov, Phys. Lett. B[**395**]{}, 69, (1997).
R. Kippenhahn, A. Weigert, and E. Hofmeister, in [*Methods for Calculating Stellar Evolution*]{}, Vol. 7 of [*Methods in Computational Physics*]{} (New York: Academic Press, 1967), p. 129.
H. Schlattl, A. Weiss, and H.-G. Ludwig, Astron. Astrophys. [**322**]{}, 64 (1997).
C. A. Iglesias and F. J. Rogers, Astrophys. J. [**464**]{}, 943 (1996).
F. J. Rogers, F. J. Swenson, and C. A. Iglesias, Astrophys. J. [**456**]{}, 902 (1996).
A. A. Thoul, J. N. Bahcall, and A. Loeb, Astrophys. J. [**421**]{}, 828 (1994).
N. Grevesse and A. Noels, Phys. Scripta [**T47**]{}, 133 (1993).
E. Vitense, Z. Astrophys. [**32**]{}, 135 (1953).
H. Schlattl and A. Weiss, submitted to Astron. Astrophys. (1999).
S. Turck-Chièze [*et al.*]{}, [*Sensitivity of the Sound Speed to the Physical Processes Included in the Standard Solar Model*]{}, 1998, ed. by S. Korzennik and A. Wilson, ESA-SP 418, 555, (ESA Publication Division, Noordwijk , Netherlands).
M. Lazrek [*et al.*]{}, Sol. Phys. [**175**]{}, 227 (1997).
H. M. Antia and S. M. Chitre, Astron. Astrophys. [**339**]{}, 239 (1998).
J. N. Bahcall, S. Basu, and M. H. Pinsonneault, Phys. Lett. B [**433**]{}, 1 (1998).
J. Bouchez [*et al.*]{}, Z. Phys. C [**32**]{}, 499 (1986). M. Cribier, W. Hampel, J. Rich and D. Vignaud, Phys. Lett. B [**182**]{}, 89 (1986). A. J. Baltz and J. Wesener, Phys. Rev. D [**35**]{}, 528 (1987); ibid. [**37**]{}, 3364 (1988). A. Dar, A. Mann, Y. Melina and D. Zajfman, Phys. Rev. D [**35**]{}, 3607 (1987). A. Dar and A. Mann, Nature (London) [**325**]{}, 790 (1987). S. P. Mikeyev and A. Yu. Smirnov, in [*Moriond ’87*]{}, Proceedings of the 7th Moriond Workshop on New and Exotic Phenomena, Les Arcs, 1987, edited by O. Fackler and J. Trân Thanh Vân (Frontières, Paris, 1987), p. 405. M. L. Cherry and K. Lande, Phys. Rev. D [**36**]{}, 3571 (1987). S. Hiroi, H. Sakuma, T. Yanagida and M. Yoshimura, Phys. Lett. B [**198**]{}, 403 (1987); Prog. Theor. Phys. [**78**]{}, 1428 (1987). M. Spiro and D. Vignaud, Phys. Lett. B [**242**]{}, 279 (1990).
The Super-Kamiokande Collaboration, Y. Fukuda [*et al.*]{}, hep-ex/9812009 (1998).
The Super-Kamiokande Collaboration, Y. Fukuda [*et al.*]{}, hep-ex/9812011 (1998).
J. N. Bahcall, P. I. Krastev and E. Lisi, Phys. Rev. C [**55**]{}, 494, (1997).
R. S. Raghavan, Phys. Rev. Lett. [**19**]{}, 3618, (1997).
|
---
author:
- |
for the CMS Collaboration\
University of Antwerp, Belgium\
E-mail:
title: 'Measurement of the inelastic proton-proton cross section at $\sqrt{S}$ = 13 TeV'
---
Introduction
============
The hadronic cross section is a fundamental observable in high energy particle physics, and has been measured in many experiments, covering several orders of magnitude in centre-of-mass energy. It can be decomposed in an elastic and inelastic part, where the latter can again be decomposed into diffractive and non-diffractive contributions that can be described by non-perturbative, phenomenological models. However, these models have large uncertainties when one wants to extrapolate existing measurements towards higher centre-of-mass energies. Therefore precise measurements of the hadronic cross sections are needed to provide input to phenomenological models and for the tuning of Monte Carlo event generators. In addition, the value of the inelastic proton-proton cross section ($\sigma_{\text{inel}}$) is used to estimate the average pile-up in the data, i.e. the number of simultaneous proton-proton interactions occurring in the same bunch crossings of the accelerator beams. This can be an important background to new physics searches and the measurement of $\sigma_{\text{inel}}$ is thus important to control this pile-up contribution in proton-proton interactions measured at the LHC.
Experiments at the LHC measured $\sigma_{\text{inel}}$ at a centre-of-mass energy of $\sqrt{s}$ = 7 TeV, reporting values ranging between 66.9 mb and 73.2 mb [@Abelev:2012sea; @Aad:2014dca; @CMS:2011xpa; @Chatrchyan:2012nj; @Aaij:2014vfa; @Antchev:2013iaa]. This large spread is mainly due to the different methods used to extrapolate the measured value to the total inelastic phase space, and indicates that a precise determination of $\sigma_{\text{inel}}$ is very challenging. The most precise values of $\sigma_{\text{inel}}$ are currently given by the ATLAS Collaboration [@Aad:2014dca] : $\sigma_{\text{inel}} = 71.34 \pm 0.9$ mb; and by the TOTEM Collaboration [@Antchev:2013iaa]: $\sigma_{\text{inel}} = 72.9 \pm 1.5$ mb. In addition, the TOTEM Collaboration also performed a measurement of $\sigma_{\text{inel}}$ at a centre-of-mass energy of $\sqrt{s}$ = 8 TeV and reported a value of $\sigma_{\text{inel}} = 74.7 \pm 1.7$ mb [@Antchev:2013paa].
Recently the ATLAS Collaboration presented a preliminary result on the measurement of $\sigma_{\text{inel}}$ at a centre-of-mass energy of $\sqrt{s}$ = 13 TeV [@ATLASXS13TeV], and reported a measured value of $65.2 \pm 0.8 \ \text{(exp.)} \pm 5.9 \ \text{(lum.)}$ mb within an acceptance of $\xi > 10^{-6}$ (corresponding to $M$ > 13 GeV) with $\xi = M^{2}/s$ and $M$ the mass of the largest diffractive dissociation system. This value was extrapolated to the total inelastic phase space, resulting in $\sigma_{\text{inel}} = 73.1 \pm 0.9 \ \text{(exp.)} \pm 6.6 \ \text{(lum.)} \pm 3.8 \ \text{(ext.)}$ mb.
In these proceedings we present a measurement of the inelastic proton-proton cross section at $\sqrt{s}$ = 13 TeV with data collected by the CMS detector at the LHC [@CMS:2016ael]. The analysis method is based on the usage of extensive forward calorimetry available in CMS, and enables us to measure the inelastic cross section in two different detector acceptances. The first measurement only requires a signal in the Hadronic Forward (HF) calorimeters of CMS. They are placed at both sides of the interaction point and consist out of iron absorbers and quartz fibers for read out. The energy scale is known to an uncertainty of 10%, and the used acceptance in this analysis is $3.152 < \left| \eta \right| < 5.205$. The second measurement includes the very forward CASTOR calorimeter of CMS. It consists out of tungsten absorbers and quarts plates for read out, and is only located at the minus side of the interaction point. The acceptance is $-6.6 < \eta < -5.2$, the energy scale uncertainty is 15%, and the alignment of the detector is known to a $\pm 2$ mm accuracy [@CASTORDPS2016]. A detailed description of the CMS detector is available in [@Chatrchyan:2008aa]. The use of the very forward CASTOR calorimeter allows us to measure the inelastic cross section in the largest acceptance possible, and thus to reduce the extrapolation factor, and its uncertainty, towards the full inelastic phase space.
Event selection and reconstruction
==================================
The measurement uses LHC Run2 low pile-up proton-proton data with a centre-of-mass energy of $\sqrt{s}$ = 13 TeV with the solenoid of CMS both at B = 0 T and B = 3.8 T values. The CMS data acquisition was triggered by the presence of both beams in the interaction point, and this defines our total event sample (ZeroBias). Two different event selections are further applied offline to count the interactions in different detector acceptances:
- **HF OR**: require an energy deposit above 5 GeV in any of the two HF calorimeters.
- **HF OR CASTOR**: require either an energy deposit above 5 GeV in any of the two HF calorimeters or an energy deposit above 5 GeV in the CASTOR calorimeter.
This first set of selected inelastic events is then corrected for remaining noise contributions: $$N_{cor} = N_{ZB}\left[ \left( F_{ZB} - F_{EB} \right) + F_{EB} \left( F_{ZB} - F_{EB} \right) \right],$$ where $N_{ZB}$ represents the number of ZeroBias triggered events; $F_{ZB}$ is the fraction of ZeroBias triggered events that are selected offline; and $F_{EB}$ is the fraction of no-beam triggered events that are selected offline. The selected number of interactions is further corrected for pile-up effects. The observed number of proton-proton collisions ($n$) per bunch crossing follows a Poisson distribution $P(n,\lambda)$ with average value $\lambda$. The probability to have no interaction in a ZeroBias sample is then given by $P(0,\lambda) \equiv \text{exp}(-\lambda) = 1 - N_{cor} / N_{ZB}$, which allows one to determine the mean number of inelastic collisions per bunch crossing $\lambda = -\text{ln}(1 - N_{cor} / N_{ZB})$. In the data used this value ranges from 0.05 to 0.5. It is then possible to correct the inelastic event count with the following factor: $$f_{\text{pu}} = \frac{\Sigma^{\infty}_{n=0} \ n P \left( n, \lambda \right) }{\Sigma^{\infty}_{n=1} \ P \left( n, \lambda \right)} = \frac{\lambda}{1 - P \left( 0, \lambda \right)}.$$ This correction is applied bunch by bunch, and the total reconstructed number of interactions, corrected for contributions of noise and pile-up, is then given by: $$N_{int} = \sum_{\text{bunches}} N^{b}_{cor} \times f^{b}_{\text{pu}},$$ where $N_{cor}^{b}$ is the number of noise-corrected events per bunch, and $f_{\text{pu}}^{b}$ the pile-up correction factor per bunch.
Measurement of the visible inelastic cross section
==================================================
The measured number of inelastic interactions is corrected for various detector effects, such as the event selection efficiency and the resolution on the energy measurement, in order to compare to theoretical predictions. The acceptance on stable-particle level is defined as a function of $\xi$, which can be determined by first splitting the final state into systems X (negative side) and Y (positive side) separated by the largest rapidity gap, to then calculate their invariant masses $M_{X(Y)}$ and finally take the ratio with respect to the available centre-of-mass energy: $$\xi_{X} = \frac{M^{2}_{X}}{s}, \quad \xi_{Y} = \frac{M^{2}_{Y}}{s}, \quad \xi = \text{max}(\xi_{X},\xi_{Y}).$$ Although the selection criteria on detector level are chosen to have the largest possible acceptance of inelastic events, low mass diffractive dissociation events will escape and not be included in the selected inelastic event sample. Hence only interactions with a value of $\xi_{X}$ or $\xi_{Y}$ above a certain threshold will be detected. These acceptance limits are obtained from a dedicated study using fully simulated events from various Monte Carlo event generators. The relation between the stable-particle level phase space definition and the offline detector level selection is quantified by efficiency and contamination factors. The efficiency ($\epsilon_{\xi}$) is defined as the fraction of selected stable-particle level events that also pass the offline detector-level selection, while the contamination ($b_{\xi}$) is defined as the fraction of offline detector-level selected events that are not part of the considered stable-particle level phase space domain.
The optimal acceptances on stable-particle level are found to be $\xi > 10^{-6}$ for the HF OR detector-level selection, and the asymmetric (due to CASTOR being only present at the minus side) $\xi_{X} > 10^{-7} \text{or} \ \xi_{Y} > 10^{-6}$ for the HF OR CASTOR detector-level selection. Combining all ingredients, the stable-particle level inelastic cross section value for a given acceptance in $\xi$ is thus given by: $$\sigma = \frac{N_{int} \left( 1 - b_{\xi} \right)}{\epsilon_{\xi} \int \mathcal{L} dt}.$$ The systematic uncertainties are summarised in table \[tab:syst\] and include the model dependence of the correction factors, the HF and CASTOR energy scale uncertainties, the alignment of the CASTOR calorimeter, and a run-to-run variation. Finally the uncertainty due to the luminosity measurement is given, which is known to a 2.7 % (2.9 %) precision for B = 3.8 T (0 T) data following a dedicated analysis of Van der Meer scans performed in August 2015 [@CMS:2016eto].
$\sigma\left(\xi > 10^{-6}\right)$ (mb) $\sigma\left(\xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6} \right)$ (mb)
--------------------------------- ----------------------------------------- --------------------------------------------------------------------------------
Model dependence 0.66 0.38
HF energy scale uncertainty 0.34 0.13
CASTOR energy scale uncertainty - 0.04
CASTOR alignment - 0.03
Run-to-run variation 0.15 0.14
Total 0.76 0.44
Luminosity 1.78 1.96
: Summary of systematic uncertainties from all sources.[]{data-label="tab:syst"}
Results
=======
The fully corrected inelastic cross section measured with the HF calorimeters only is: $$\sigma \left( \xi > 10^{-6} \right) = 65.77 \pm 0.03 \ \text{(stat.)} \pm 0.76 \ \text{(sys.)} \pm 1.78 \ \text{(lum.) mb} ,$$ and including the CASTOR calorimeter into the event selection to extend the acceptance yields: $$\sigma \left( \xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6} \right) = 66.85 \pm 0.06 \ \text{(stat.)} \pm 0.44 \ \text{(sys.)} \pm 1.96 \ \text{(lum.) mb.}$$ Figure \[fig:FinalXSPlots\_relative\_bin1\] shows the ratio of the two measured cross sections in the left bin, indicating that most models are able to describe the relative increase from $\xi > 10^{-6}$ to $\xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6}$. The right bin represents the model dependent extrapolation factors to go from the measured phase space to the full inelastic phase space domain. Using the average value of all models we obtain the following total inelastic cross section: $$\sigma_{\text{inel}} = 71.26 \pm 0.06 \ \text{(stat.)} \pm 0.47 \ \text{(sys.)} \pm 2.09 \ \text{(lum.)} \pm 2.72 \ \text{(ext.) mb.}$$ The maximal difference between the model extrapolation factors is taken as uncertainty (ext.). Figure \[fig:FinalXSPlots\_absolute\] then shows the absolute values of all results, compared to various model predictions, and the preliminary ATLAS result [@ATLASXS13TeV]. While it is clear that the results of ATLAS and CMS are compatible within systematic uncertainties, all models predict an absolute cross section that is too high.
![Ratio of the HF OR CASTOR ($\xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6}$) measured cross section to the HF OR ($\xi > 10^{-6}$) acceptance (left bin); and the ratio of $\sigma_{\text{inel}}$ to the HF OR CASTOR ($\xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6}$) cross section (right bin). The latter represents the extrapolation factors used to calculate $\sigma_{\text{inel}}$. [@CMS:2016ael][]{data-label="fig:FinalXSPlots_relative_bin1"}](./FinalXSPlots_relative_bin1.pdf){width="80.00000%"}
![Fully corrected inelastic cross sections measured in various phase space regions, compared to different models and the preliminary results of the ATLAS experiment. The data point for $\sigma_{\text{inel}}$ is calculated using a model-dependent extrapolation of the measured cross section for $\xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6}$. [@CMS:2016ael][]{data-label="fig:FinalXSPlots_absolute"}](./FinalXSPlots_absolute.pdf){width="80.00000%"}
Summary
=======
A measurement of the inelastic proton-proton cross section at $\sqrt{s}$ = 13 TeV obtained with the CMS detector at the LHC has been presented. Visible cross sections in two acceptances are obtained: $\sigma \left( \xi > 10^{-6} \right) = 65.8 \pm 0.8 \ \text{(exp.)} \pm 1.8 \ \text{(lum.)}$ mb; and $\sigma \left( \xi_{X} > 10^{-7} \ \text{or} \ \xi_{Y} > 10^{-6} \right) = 66.9 \pm 0.4 \ \text{(exp.)} \pm 2.0 \ \text{(lum.)}$ mb. The latter visible cross section is extrapolated to the full inelastic phase space domain, yielding $71.3 \pm 0.5 \ \text{(exp.)} \pm 2.1 \ \text{(lum.)} \pm 2.7 \ \text{(ext.)}$ mb. The measured cross section is lower than predicted by models for hadronic scattering.
[99]{}
B. Abelev [*et al.*]{} \[ALICE Collaboration\], Eur. Phys. J. C [**73**]{} (2013) no.6, 2456 \[arXiv:1208.4968\]. G. Aad [*et al.*]{} \[ATLAS Collaboration\], Nucl. Phys. B [**889**]{} (2014) 486 \[arXiv:1408.5778\]. CMS Collaboration, “Inelastic $pp$ cross section at 7 TeV,” CMS-PAS-FWD-11-001. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**722**]{} (2013) 5 \[arXiv:1210.6718\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], JHEP [**1502**]{} (2015) 129 \[arXiv:1412.2500\]. G. Antchev [*et al.*]{} \[TOTEM Collaboration\], Europhys. Lett. [**101**]{} (2013) 21004. G. Antchev [*et al.*]{} \[TOTEM Collaboration\], Phys. Rev. Lett. [**111**]{} (2013) no.1, 012001. The ATLAS collaboration, “Measurement of the Inelastic Proton-Proton Cross Section at $\sqrt{s} = 13$ TeV with the ATLAS Detector at the LHC,” ATLAS-CONF-2015-038. CMS Collaboration, “Measurement of the inelastic proton-proton cross section at $\sqrt{s}=13~\mathrm{TeV}$,” CMS-PAS-FSQ-15-005. CMS Collaboration, “Results on CASTOR performance during LHC Run 2,” CMS-DP-2016-006
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], JINST [**3**]{} (2008) S08004.
CMS Collaboration, “CMS Luminosity Measurement for the 2015 Data Taking Period,” CMS-PAS-LUM-15-001.
|
---
abstract: |
The anomalous magnetic moment of the negative muon has been measured to a precision of 0.7 parts per million (ppm) at the Brookhaven Alternating Gradient Synchrotron. This result is based on data collected in 2001, and is over an order of magnitude more precise than the previous measurement of the negative muon. The result $a_{\mu^-} = 11\,659\,214 (8)(3) \times 10^{-10}$ (0.7ppm), where the first uncertainty is statistical and the second is systematic, is consistent with previous measurements of the anomaly for the positive and negative muon. The average for the muon anomaly is $a_\mu(\text{exp}) = 11\,659\,208 (6) \times
10^{-10}$ (0.5ppm).
author:
- |
G.W. Bennett$^{2}$, B. Bousquet$^{9}$, H.N. Brown$^2$, G. Bunce$^2$, R.M. Carey$^1$, P. Cushman$^{9}$, G.T. Danby$^2$, P.T. Debevec$^7$, M. Deile$^{11}$, H. Deng$^{11}$, S.K. Dhawan$^{11}$, V.P. Druzhinin$^3$, L. Duong$^{9}$, F.J.M. Farley$^{11}$, G.V. Fedotovich$^3$, F.E. Gray$^7$, D. Grigoriev$^3$, M. Grosse-Perdekamp$^{11}$, A. Grossmann$^6$, M.F. Hare$^1$, D.W. Hertzog$^7$, X. Huang$^1$, V.W. Hughes$^{11}$, M. Iwasaki$^{10}$, K. Jungmann$^5$, D. Kawall$^{11}$, B.I. Khazin$^3$, F. Krienen$^1$, I. Kronkvist$^{9}$, A. Lam$^1$, R. Larsen$^2$, Y.Y. Lee$^2$, I. Logashenko$^{1,3}$, R. McNabb$^{9}$, W. Meng$^2$, J.P. Miller$^1$, W.M. Morse$^2$, D. Nikas$^2$, C.J.G. Onderwater$^{7}$, Y. Orlov$^4$, C.S. Özben$^{2,7}$, J.M. Paley$^1$, Q. Peng$^1$, C.C. Polly$^7$, J. Pretz$^{11}$, R. Prigl$^{2}$, G. zu Putlitz$^6$, T. Qian$^{9}$, S.I. Redin$^{3,11}$, O. Rind$^1$, B.L. Roberts$^1$, N. Ryskulov$^3$, Y.K. Semertzidis$^2$, P. Shagin$^9$, Yu.M. Shatunov$^3$, E.P. Sichtermann$^{11}$, E. Solodov$^3$, M. Sossong$^7$, L.R. Sulak$^{1}$, A. Trofimov$^1$, P. von Walter$^6$, and A. Yamamoto$^8$.\
(Muon $(g-2)$ Collaboration)
title: 'Measurement of the Negative Muon Anomalous Magnetic Moment to 0.7 ppm'
---
The anomalous magnetic moments of the muon and the electron have played an important role in the development of the standard model. Compared to the electron, the muon anomaly has a relative sensitivity to heavier mass scales which typically is proportional to $(m_{\mu} /m_e)^2$. At the present level of accuracy the muon anomaly gives an experimental sensitivity to virtual $W$ and $Z$ gauge bosons as well as a potential sensitivity to other, as yet unobserved, particles in the few hundred GeV/$c^2$ mass range [@kh].
We report our result for the negative muon anomalous magnetic moment $a_{\mu^-}=(g-2)/2$ from data collected in early 2001. The measurement is based on muon spin precession in a magnetic storage ring with electrostatic focusing. The same experimental technique was used as in our most recent measurements of $a_{\mu^+}$ [@muplus; @g2_2000], and a similar precision of 0.7 ppm was achieved. Detailed descriptions of the apparatus may be found elsewhere [@nimpapers; @fei; @nimpapers2; @kicker; @quads].
For polarized muons moving in a uniform magnetic field $\vec{B}$ perpendicular to the muon spin and to the plane of the orbit and in an electric quadrupole field $\vec{E}$, which is used for vertical focusing [@quads], the angular frequency difference, $\omega_a$ between the spin precession frequency and the cyclotron frequency, is given by $$\vec{\omega}_a= {e \over m \, c} \left[ a_\mu \vec{B} - \left( a_\mu - {1 \over \gamma^2 - 1} \right) \vec{\beta} \times \vec{E} \right].
\label{eq:amu}$$ The dependence of $\omega_a$ on the electric field is eliminated by storing muons with the “magic” $\gamma=29.3$ [@cern], which corresponds to a muon momentum $p=3.09$ GeV/$c$. Hence measurement of $\omega_a$ and of $B$, in terms of the free proton NMR frequency $\omega_p$ and the ratio of muon to proton magnetic moments $\lambda$, determines $a_\mu$. At the magic $\gamma$, the muon lifetime is approximately $ 64.4 \, {\rm \mu s}$ and the $(g-2)$ precession period is $4.37\, {\rm \mu s}$. With a field of 1.45 T in our storage ring [@nimpapers], the central orbit radius is 7.11 m. The difference frequency $\omega_a$ was determined by counting the number $N(t)$ of decay electrons above an energy threshold. The time spectrum of decay electrons is then given by $$N(t) = N_0 e^{-t/(\gamma\tau)} \left[ 1 + A \sin(\omega_a t + \phi_a) \right].
\label{eq:spectrum}$$ The normalization $N_0$, asymmetry $A$, and phase $\phi_a$ vary with the chosen energy threshold. The measurement of the magnetic field frequency $\omega_p$ is based on proton NMR in water. A trolley with 17 NMR probes was moved typically every three days throughout the entire muon storage region. About 150 fixed NMR probes distributed around the ring in the top and bottom walls of the vacuum chamber were used to interpolate the field between trolley measurements. The system was calibrated with respect to a standard probe with a spherical sample [@fei]. The homogeneity of the field in 2001 (Figure \[fig:field\]) was similar to that achieved for the opposite polarity field in 2000 [@g2_2000].
![A two-dimensional multipole expansion of the 2001 field averaged over azimuth from one out of 20 trolley measurements. Half ppm contours with respect to a central azimuthal average field are shown. The multipole amplitudes relative to $B_0$ are given at the beam aperture, which had a radius of 4.5cm and is indicated by the circle.\[fig:field\]](multi2001_4){width="45.00000%"}
The field $\langle B \rangle$ weighted with the analyzed event sample was obtained from two largely independent analyses, whose results were found to agree to within 0.05ppm. Its final value is expressed in terms of the free proton resonance frequency and is given by Table \[table:field\] lists the uncertainties. The improved 2001 uncertainties resulted from refinements in the calibration measurements, and from an upgraded system to determine the azimuthal trolley position in the storage ring.
Source of uncertainty Size \[ppm\]
---------------------------------------- --------------
Absolute calibration of standard probe 0.05
Calibration of trolley probe 0.09
Trolley measurements of $B_0$ 0.05
Interpolation with fixed probes 0.07
Uncertainty from muon distribution 0.03
Others$^\dagger$ 0.10
Total systematic error on $\omega_p$ 0.17
: Systematic uncertainties for the $\omega_p$ analysis. \[table:field\]
The 2001 $\omega_a$ data taking was similar to that in 2000. However, the hardware energy threshold of the detectors was kept lower and equal for all counters at 0.9GeV compared to 1.0-1.4 GeV in 2000. This was made possible by reducing the intensity of the injected beam, which in turn reduced the light flash in the detectors [@muplus; @g2_2000]. These factors allowed all the detectors to be turned on and be stable by 32$\mu$s after beam injection, as opposed to 50$\mu$s in 2000. As a result of the reduced rates, the fraction of overlapping signals (pileup) after 32$\mu$s in 2001 was comparable to the pileup fraction after 50$\mu$s in 2000. In 2000 the field focusing index $n$, which is proportional to the electric field gradient, was $n=0.137$, corresponding to a horizontal coherent betatron oscillation frequency (CBO) of 466 kHz [@g2_2000]. This frequency was close to twice the $(g-2)$ frequency of 229 kHz, which resulted in a sizable uncertainty in the fitted $\omega_a$ value [@g2_2000]. In 2001 we used two different $n$-values, $n = 0.122$ and $n= 0.142$, which resulted in CBO frequencies, 419 kHz and 491 kHz that are further from twice the $(g-2)$ frequency (see Figure \[fig:FT\_high\_low\]). Consequently, the uncertainty caused by CBO is smaller. Furthermore it also reduced the correlation between the CBO and detector gain effects in the fits to the time spectrum.
![The Fourier spectrum of the residuals of a fit to the five free parameters given in Eq. \[eq:spectrum\] for the high (top) and low (bottom) $n$-value data. The corresponding CBO frequencies, located at 491 kHz (top), and 419 kHz (bottom) as well as their $(g-2)$ sidebands are clearly visible. Dashed lines indicate the $(g-2)$ frequency. \[fig:FT\_high\_low\]](FT_high_low3){width="40.00000%"}
Two independent implementations of the algorithm to reconstruct the electron times and energies from the calorimeter signals were used. The frequency $\omega_a$ was determined by fitting the time distribution of decay electrons. Five independent analyses were performed in order to probe the systematic uncertainties and, of course, to protect against mistakes. All five results agreed within the expected statistical deviations due to different data selection and weightings. These analyses are described below.
Two of the analyses used slightly different parametrizations [@muplus; @g2_2000] that included CBO modulations and fitted the combined electron spectrum in the energy range 1.8-3.4 GeV. In the third analysis, the counts were weighted with the experimentally determined energy-dependent modulation asymmetry, which optimized the statistical power of the data. This method permitted the analyzed energy range to be extended. We used an energy range of 1.5 to 3.4 GeV, which together with the asymmetry weighting resulted in a 10% improvement of the statistical uncertainty. As in the first two analyses, the resulting spectrum of [*weighted*]{} counts was fitted to a function that parametrized all known and statistically significant perturbations.
The remaining analyses fit the ratio [@muplus; @g2_2000] formed by randomly assigning the data to four statistically independent subsets $n_1$ to $n_4$. The subsets were rejoined in $u(t)=n_1(t) + n_2(t)$ and $v(t)=n_3(t-\tau_a / 2) + n_4(t + \tau_a /2)$, where $\tau_a$ is an estimate of the $(g-2)$ period, and then combined to form the time spectrum $ r(t) = [{u(t)-v(t)}]/[{u(t)+v(t)}]$. The $(g-2)$ rate modulation of $v$ is 180$^\circ$ degrees out of phase compared to that of $u$, and to sufficient precision $r(t)$ can be described by $A\sin(\omega_a
t + \phi_a)$. The ratio $r(t)$ is largely insensitive to changes of observed counts on time scales larger than $\tau_a = 2\pi/\omega_a \sim
4\,\mu\mathrm{s}$. In one of the ratio analyses, the sensitivity to CBO was reduced by combining the data from both $n$-values and all detectors prior to fitting. The data were fitted from 32$\mu$s after injection when all detectors were on. In the second ratio analysis the data were fitted separately for each calorimeter and $n-$value. The fits began between 24$\mu$s and 32$\mu$s, and required the parametrization of the CBO effects in the fit function.
Changes in the radial and vertical muon distributions with time were quantified, and were found to have negligible effect on $\omega_a$. A small reduction in the pulsed electrostatic quadrupole voltages [@quads] during the measurement period could change the vertical muon distribution. Analysis of the data from scintillator counter hodoscopes placed in front of the calorimeters combined with a beam tracking calculation and a GEANT based simulation set a systematic error limit of 0.03 ppm. The muon radial distribution is determined by the magnetic field and the momentum distribution [@muplus; @yuri]. The magnetic field does not change with time after injection, except due to the field from eddy currents induced by the fast kicker [@kicker]. This was measured, and found to have a negligible effect on the muon radial distribution. Muons of lower momenta decay earlier in the laboratory frame than muons of higher momenta. The momentum distribution of the stored beam thus changes during the $~600\, \rm \mu s$ measurement period. The effect on $\omega_a$ due to this change was studied in simulation, and was found to be 0.03 ppm.
![Comparison of the $\omega_a$ values from the five analyses for the low-$n$ (filled) and high-$n$ (open) data sets. Analysis 4 used only the combined low and high-$n$ data (square). The divisions on the vertical axis are separated by 1 ppm, and the indicated uncertainties are statistical. The systematic uncertainties are considerably smaller. []{data-label="fig:all_results"}](all_analysesv2){width="45.00000%"}
The results for $\omega_a$ for the two $n$-values are consistent, see Figure \[fig:all\_results\], and were combined for each of the analyses. The values for $\omega_a$ from the five analyses are in agreement to within variations expected from the differences in the analyzed event samples and the treatment of the data. The analysis techniques are expected to have somewhat different sensitivities to different systematic effects. Detailed comparisons of the results, using all analyzed data as well as only the data in overlap, showed no evidence for unaccounted systematic differences. The five resulting values for $\omega_a$ were combined in a simple arithmetic mean to obtain a single value for $\omega_a$.
The resulting frequency value is $\omega_a/(2\pi) = 229\,073.59(15)(5)\,\mathrm{Hz}$ (0.7ppm), which includes a correction of $+0.77(6)$ ppm for contributions to Eq. \[eq:amu\] caused by vertical oscillations (0.30ppm) and for the effect of the horizontal electric fields on muons with (0.47ppm). The stated uncertainties account for strong correlations among the individual results, both statistical and systematic. Table \[table:omegaa\] lists the systematic uncertainties in the combined result with these correlations taken into account.
Source of errors Size \[ppm\]
-------------------------------------- --------------
Coherent betatron oscillations 0.07
Pileup 0.08
Gain changes 0.12
Lost muons 0.09
Others$^\dagger$ 0.11
Total systematic error on $\omega_a$ 0.21
: Systematic uncertainties for the combined $\omega_a$ analysis.
\[table:omegaa\]
After the $\omega_p$ and $\omega_a$ analyses were finalized separately and independently, $a_\mu$ was evaluated. The result is $$a_{\mu^-} = \frac{R}{\lambda - R} = 11\,659\,214(8)(3)\ \times\ 10^{-10}~~\mbox{(0.7\,ppm)},
\label{eq:result}$$ where $R_{\mu^-} \equiv \omega_a/\omega_p = 0.003 \, 707 \, 208 \, 3 (2\,6) $ and $\lambda = \mu_\mu/\mu_p = 3.183\,345\,39(10)$ [@liu]. This new result is in good agreement with the average of $R_{\mu^+} = 0.003\, 707\, 204\, 8 (2\,5) $ [@muplus] as predicted by the CPT theorem. The difference $\Delta R = R_{\mu^-} - R_{\mu^+} = (3.5 \pm 3.4)\times 10^{-9}$. The new average is $R_{\mu} = 0.003\,707\,206\,3 (2\,0) $ and $$a_\mu(\mathrm{exp}) = 11\,659\,208(6) \times 10^{-10}~~\mbox{(0.5\,ppm)},$$ in which the total uncertainty consists of $5 \times 10^{-10}$ (0.4ppm) statistical uncertainty and $4 \times 10^{-10}$ (0.3ppm) systematic uncertainty. The correlation of systematic uncertainties between the data sets has been taken into account. The combined result for the positive muon [@g2_2000], $a_{\mu^+}(\mathrm{exp}) = 11\, 659\, 203(8) \times 10^{-10}$ (0.7 ppm) has a statistical uncertainty of $6 \times 10^{-10}$ (0.6 ppm) and a systematic uncertainty of $5 \times 10^{-10}$ (0.4 ppm). It is shown in Figure \[fig:results\] together with the new result for the negative muon and their average.
![Measurements of $a_\mu$ by E821 with the SM predictions (see text for discussion). Uncertainties indicated on the measurements are total uncertainties. []{data-label="fig:results"}](g2results2){width="45.00000%"}
The standard model prediction for $a_{\mu}$ consists of QED, hadronic and weak contributions. The uncertainty on the standard model value is dominated by the uncertainty on the lowest-order hadronic vacuum polarization. This contribution can be determined directly from the annihilation of $e^+e^-$ to hadrons through a dispersion integral [@disp]. The indirect determination using data from hadronic $\tau$ decays, the conserved vector current hypothesis, plus the appropriate isospin corrections, could in principle improve the precision of $a_{\mu}({\rm had})$. However, discrepancies between the $\tau$ and the $e^+e^-$ results exist [@dehz03; @jeg]. The two data sets do not give consistent results for the pion form factor. Using $e^+e^-$ annihilation data the corresponding theoretical value is $a_\mu(\mathrm{SM}) = 11\,659\,181(8)\,\times\,10^{-10} \mathrm{\ (0.7\,ppm)}$. The value deduced from $\tau$ decay is larger by $15 \times 10^{-10}$ and has a stated uncertainty of $7\,\times\,10^{-10} \mathrm{\ (0.6\,ppm)}$. The difference between the experimental determination of $a_\mu$ and the standard model theory using the $e^+e^-$ or $\tau$ data for the calculation of the hadronic vacuum polarization is 2.7 $\sigma$ and 1.4 $\sigma$, respectively.
This is the final analysis of the anomalous magnetic moment from experiment E821 at the Brookhaven Alternating Gradient Synchrotron. We aim to substantially improve our result in a new measurement and look forward to continued efforts to improve the theoretical evaluation.
We thank T. Kirk, D.I. Lowenstein, P. Pile, and the staff of the BNL AGS for the strong support they have given this experiment. This work was supported in part by the U.S. Department of Energy, the U.S. National Science Foundation, the U.S. National Computational Science Alliance, the German Bundesminister für Bildung und Forschung, the Russian Ministry of Science, and the U.S.-Japan Agreement in High Energy Physics.
[99]{}
V.W. Hughes and T. Kinoshita, Comments. Nucl. Part. Phys. [**14**]{}, 341 (1985).
H.N. Brown et al., (Muon $(g-2)$ Collaboration), Phys. Rev. [**D62**]{}, 091101 (2000); H.N. Brown, et al., (Muon $(g-2)$ Collaboration), Phys. Rev. Lett. [**86**]{} 2227 (2001).
G.W. Bennett et al., (Muon $(g-2)$ Collaboration), Phys. Rev. Lett. [**89**]{}, 101804 (2002).
A. Yamamoto et al., Nucl. Instrum. Methods Phys. Res. [**A491**]{} 23 (2002); G.T. Danby et al., Nucl. Instrum. Methods Phys. Res. [**A 457**]{}, 151 (2001); S.I. Redin et al., Nucl. Instrum. Methods Phys. Res. [**A473**]{}, 260 (2001); R. Prigl et al., Nucl. Instrum. Methods Phys. Res. [**A374**]{} 118 (1996).
X. Fei, V.W. Hughes and R. Prigl, Nucl. Instrum. Methods Phys. Res. [**A394**]{}, 349 (1997).
J. Ouyang et al., Nucl. Instrum. Methods Phys. Res. [**A374**]{}, 215 (1996); S.A. Sedykh et al., Nucl. Instrum. Methods Phys. Res. [**A455**]{} 346 (2000).
E. Efstathiadis et al., Nucl. Instrum. Methods Phys. Res. [**A496**]{}, 8 (2003).
Y.K. Semertzidis et al., Nucl. Instrum. Methods Phys. Res. [**A503**]{} 458 (2003).
J. Bailey et al., Nucl. Phys. [**B150**]{}, 1 (1979).
Y. Orlov, C.S. Özben and Y.K. Semertzidis, Nucl. Instrum. Meth. [**A482**]{}, 767 (2002).
W. Liu et al., [Phys. Rev. Lett.]{} [**82**]{}, 711 (1999), and K. Hagiwara et al., (Particle Data Group) Phys. Rev. [**D66**]{}, 010001 (2002).
B.E. Lautrup, A. Peterman and E. de Rafael, Phys. Rep. [**C 3**]{}, 193 (1972).
M. Davier, S. Eidelman, A. Höcker, Z. Zhang Aug 2003, Eur. Phys. J. [**C 31**]{}, 503 (2003).
S. Ghozzi and F. Jegerlehner, hep-ph/0310181, Phys. Lett. [**B**]{}, in Press, (2004).
|
---
abstract: 'Quantum measurements of physical quantities are usually described as ideal measurements. However, only a few measurements fulfil the conditions of ideal measurements. The aim of the present work is to describe real position measurements with detectors that are able to detect single particles. For this purpose, a detector model has been developed that can describe the time dependence of interactions between nonrelativistic particles and a detector. At the beginning of a position measurement, the detector behaves as a target consisting of a large number of quantum mechanical systems. The incident object interacts with a single atom, electron or nucleus, but not with the whole detector. This reaction is a quantum mechanical process. At the end of the measurement, the detector can be considered as a classical apparatus. A detector is neither a quantum mechanical system nor a classical apparatus. The detector model explains why one obtains a well-defined result for each individual measurement. It further explains that, in general, it is impossible to predict the outcome of the next measurement. The main advantage is that it describes real rather than only ideal measurements.'
author:
- |
Klaus Wick\
\
title: |
On the quantum mechanical description of\
the interaction between particle and detector
---
[^1]
*Keywords:* Quantum measurement process; Position measurement; State reduction
Introduction
============
A measurement device may consist of various components (magnets, cavities, crystals, detectors and so on). In the literature, such a complex device is often referred to as a detector. In the following, the word detector will only be employed for the components of the measurement device that can provide an output signal. Examples include ionisation chambers, semiconducting detectors, photomultiplier tubes, scintillation counters, and cloud and bubble chambers. The output signal delivered by a detector indicates that a quantum object (particle or photon) has been detected. The detection of an object is also a position measurement, because at the moment of the measurement each detector has a well-defined position. The uncertainty of the measured position is defined by the size of the detector. A measurement is complete when the detector has produced the output signal and the result has been registered. Detectors are highly important for experimental performance, because measurement results for single particles or photons are usually determined by means of detectors.
One problem is that quantum theory describes the interactions between quantum objects and detectors in a highly simplified manner. The Copenhagen interpretation of quantum mechanics assumes that a detector is a classical apparatus. However, it is not clear how a quantum mechanical system can interact with a classical one. On the other hand, John von Neumann [@Ne32] considers a detector as a quantum mechanical system. An ideal measurement then describes an interaction between two quantum mechanical systems: a microscopic one (the quantum object) and macroscopic one (the detector[^2]). Unfortunately, the time evolution of the wave functions of both systems during the measurement process cannot be described by the time-dependent Schrödinger equation.[^3]
Following an ideal measurement, the incident object will be in a well-defined quantum mechanical state. Hence, an ideal measurement is repeatable, and the same object can interact a second time with an identical measurement device. If one repeats the same measurement *immediately*, then one must obtain the same result as for the first measurement. According to R. Omnès [@Om94], this statement can be considered as the definition of an ideal measurement.
Most real measurements do not fulfil the conditions of ideal measurements. Either they cannot be repeated, or if they are repeatable one will not obtain the same outcome as in the first measurement. Neutrons and photons are often absorbed when they strike a detector. If one wants to measure the energy of a charged particle, this particle has to deposit its full energy in the detector. Hence, these measurements are not repeatable.
The first aim of the present work is to describe real position measurements. A necessary prerequisite for achieving this goal is a better understanding of the interaction between a quantum object and detector. As a first step, the time dependence of this interaction will be studied in the next section.
The interaction\
between a quantum object and detector {#ch_qo-det}
=====================================
![Detection of a single particle or photon. The voltage pulse $U(t)$ generated by the detector is transformed by the discriminator into a logical output signal $U^{\rm L}$ if $U(t)$ exceeds the threshold $U^{\rm thr}$. The output signal $U^{\rm L}$ increases the content of the counter memory by one.[]{data-label="rAbb_pos-meas"}](pos-meas-1.pdf){width="11.5cm"}
Figure \[rAbb\_pos-meas\] shows a simple electronic set-up that can detect a single quantum object. An incident charged particle deposits a certain amount of energy in the detector. This energy is utilised to provide a voltage pulse $U(t)$ (where $t$ denotes time). To suppress thermal noise, the experimenter will set a fixed threshold $U^{\rm thr}$. The discriminator in Fig. \[rAbb\_pos-meas\] will only deliver a logical output signal $U^{\rm L}$ if $U(t)$ exceeds the threshold. In most detectors, this means that an output signal $U^{\rm L}$ can only be generated if the energy $E^{\rm dep}$ deposited in the detector is greater than a certain threshold energy $E^{\rm thr}$: $$\label{r_E-thr}
E^{\rm dep} > E^{\rm thr} \;.$$
An incident charged particle (a) with sufficient energy can ionise and excite many atoms (X) during its passage through matter. $$\begin{aligned}
\mbox{a + X} & \to & \mbox{a + X}^* \;, \label{r_Anrg}\\
\mbox{a + X} & \to & \mbox{a + X}^+ + \mbox{e}^- \;. \label{r_Ionis}\end{aligned}$$ In a semiconducting detector, a large number of electrons are transferred from the valence band to the conduction band. In a scintillator, many photons are produced. These examples show that the numbers of charged objects and photons are vastly increased along the path of a charged particle. Thus, a single particle state becomes a many particle state. In the following, this effect will be called amplification. Such amplification processes are employed in various detectors to produce a detectable output signal $U^{\rm L}$.
Neutrons and photons do not ionise or excite atoms during their passage through matter. Hence, one utilises special detectors to detect such objects. In these detectors, reactions in which one or two charged particles are released can occur. Examples include nuclear reactions for neutrons, and pair production and the photoelectric and Compton effects for photons. In the following, these reactions will be called start reactions, because the charged particles ionise and excite many atoms and initiate the amplification process mentioned above. If condition (\[r\_E-thr\]) is fulfilled, then a logical output signal $U^{\rm L}$ will be produced.
In the next step, we consider the time dependence of the interaction between a quantum object and detector. Successful measurements can often be divided into three phases:
In *phase 1*, the start reaction occurs. The incident object interacts with a microscopic part of the detector (atom, molecule, electron or nucleus), rather than with the whole detector.
*Phase 2* is the amplification phase. An avalanche of secondary objects (charged particles and/or photons) is released.
*Phase 3* is the readout phase. A fully operable detector will produce a detectable output signal.
As a first example, we consider the detection of single photons of visible light with a photomultiplier tube. The photomultiplier tube consists of three parts: a photocathode, dynode system, and anode. In phase 1, the incident photon interacts with a single atom in the photocathode. This is the start reaction. The photon will be absorbed, and a photoelectron will be ejected. In this case, the photoelectron will not fulfil condition (\[r\_E-thr\]). Hence, it will be accelerated to the first dynode in an electric field. Here, it will eject several secondary electrons, which are accelerated to the second dynode. In phase 2, many secondary electrons are released in the dynode system. In phase 3, a voltage pulse $U(t)$ is readout at the anode, and the discriminator in Fig. \[rAbb\_pos-meas\] will produce a detectable output signal $U^{\rm L}$.
Thermal neutrons can be detected in a proportional chamber filled with boron trifluoride (BF$_3$) gas enriched to 96% $^{10}$B. Here, the reaction $$\label{r_Bor-Fluo}
\mbox{n} + \mbox{$^{10}$B} \to \mbox{$^{11}$B}^* \to \alpha + \mbox{$^7$Li}$$ is the start reaction. The charged particles $\alpha$ and $^7$Li ionise many molecules in phase 2. Electrons and ions move to the electrodes of the chamber in an external electric field and produce the output signal. Both examples show that the incident object (photon or neutron) interacts with an atom or nucleus in phase 1, but not with the whole detector.
Many reactions occur in phases 1 and 2 of a position measurement. Although each reaction is a quantum mechanical process, the sum effect of these reactions (the production of incoherent light or an electric current in phase 3) can be described in the language of classical physics. A scintillation counter collects the light emitted by many excited atoms at the photocathode of a photomultiplier tube. In a semiconducting detector, electrons and holes are accelerated in an external electric field and produce a current pulse.
A start reaction initiates the amplification process in a detector, and with certainty produces a logical output signal $U^{\rm L}$. Neutrons and photons can produce various reactions that do not start an amplification process in the detector. Hence, we define the start reaction as the first interaction occurring in a detector that fulfils the following two conditions:
\(I) A start reaction is an interaction between two quantum objects.
\(II) In the exit channel of a start reaction, there must be at least one or two charged particles that are able to start the amplification process and provide the logical output signal $U^{\rm L}$.
When charged particles interact with a detector that is placed under vacuum conditions, we define its first interaction with an atom of the detector as the start reaction. The first interaction may be an elastic or inelastic process. This definition fulfils conditions (I) and (II).
An experimenter who wants to detect particles will first choose an appropriate detector, and then bring it into an operable state. In some cases, there only exists one start reaction. Reaction (\[r\_Bor-Fluo\]) is an example of this. On the other side, an incident photon can initiate various reactions that fulfil the conditions (I) and (II) (pair production, and the photoelectric and Compton effects). In the following, we define the ‘start reaction’ as consisting of all reactions fulfilling both conditions (I) and (II). A particle that produces a start reaction in a detector will produce an output signal with certainty.
Detector model for position measurements {#ch_det-mod}
========================================
Interactions between quantum objects and different types of detectors can be considered as sequences consisting of three phases (start reaction, amplification, and readout). It will be shown in [@Wick] that this statement holds for neutrons, photons, atoms, and nonrelativistic charged particles. This observation will be utilised to develop a detector model that can describe individual measurements of quantum objects. In the following, this model will be called the three-phase model.
Only detectors that fulfil the following requirements will be considered:
\(A) The detector should be able to detect single particles or photons.
\(B) The above-mentioned amplification process should be initiated by reactions of type (\[r\_Anrg\]) or (\[r\_Ionis\]).
These requirements are fulfilled when photons, neutrons or nonrelativistic charged particles interact with different types of detectors. Examples include ionisation chambers, semiconducting detectors, photomultiplier tubes, and scintillation counters.[^4] In all cases, the quantum mechanical state of the incident object is destroyed when it is detected in a detector.
If the incident object performs a start reaction in a fully operable detector, then the same detector will provide a logical output signal $U^{\rm L}$ with certainty. For detectors that fulfil the requirements (A) and (B), the converse is also true. If the detector delivers a logical output signal, then an amplification process must have occurred that was initiated by a start reaction in the same detector. We conclude that there is a one-to-one correlation between a quantum mechanical event (a start reaction occurs in phase 1) and a classical event (the detector and discriminator produce a logical output signal $U^{\rm L}$ in phase 3).
The reactions occurring in phase 2 appear to have no influence. However, this is not true. They are important if one measures the energies of charged particles or if one investigates tracks of particles in a bubble chamber. However, in a position measurement one is only interested in the question of whether a logical output signal has been produced. This yes/no decision depends only on the question of whether a start reaction has been performed. In the following, only position measurements will be discussed.
So far, we have only studied the interactions between quantum objects and a single detector. In the following example, we will consider interactions between charged particles and an array of several ($Z$) small ionisation chambers. This may be useful if one wants to measure the angular distribution of reaction products of a certain reaction. From the classical viewpoint, an ionisation chamber is a capacitor filled with argon gas. From the quantum mechanical viewpoint, the detector is a collection of atoms or molecules. In phase 1, the state of the projectile is described by a probability wave that can interact with all atoms of all detectors. Each incident particle views the different detectors as one big target consisting of all the atoms of all the detectors. The spatial order of the detectors does not play an essential role in phase 1 of the measurement.
The wave function of the projectile collapses at the moment at which it interacts with a single atom of a detector. Let us assume that the start reaction randomly occurs in detector D$_m$ (with $1 \le m \le Z$). Because the mean free path length of a charged particle is considerably smaller than the size of a typical detector, all the reactions following in phase 2 will occur in the same detector D$_m$. At this point, the spatial order of the detectors is important. In phase 3, the ionisation chamber D$_m$ can be considered as a capacitor in which clouds of positively and negatively charged particles move. Under the influence of an external electric field, a large number of electrons and ions move towards the capacitor plates. The uncorrelated movement of several tens of thousands of charged particles can be described as a classical electric current. This current generates a short voltage pulse in detector D$_m$. Before measurement, the outputs of all discriminators were in the ground state ($U_n^{\rm L}=0$ V for $n=1~- Z$). Immediately after the detection of the particle, for a short time one detector (D$_m$) will be in an ‘excited’ state with a definite output voltage ($U_m^{\rm L}=1$ V). $$U_n^{\rm L} = \left\{ \begin{array}{rl} 1\, \mbox{V for} & n=m \;, \\
0\, \mbox{V for} & n \ne m \;. \end{array} \right\}$$ Only detector D$_m$ will produce a logical output signal. This indicates that the particle has been detected in detector D$_m$. A well-defined outcome will be obtained for each successful measurement. However, one cannot predict the detector in which the incident particle will be detected, because it is impossible to predict the atom with which it will produce a start reaction.
Probability distribution\
of measurement results {#ch_prob-dis}
=========================
In the present work, we are only interested in measurements of physical quantities for which quantum theory only provides statistical predictions. Again, we consider position measurements with $Z$ small detectors D$_n$. The aim of the investigation is to determine the probability distributions of measurement results. These distributions are determined in quite a different manner in experiment and theory. For the experimenter, the detector and discriminator are classical apparatuses, which produce a yes/no decision for each object that leaves the source. The discriminator will either generate a logical output signal or deliver no signal. This yes/no decision is described in the language of classical physics.
Let us assume that the source has emitted $N^{\rm s}$ particles during an experiment, whose state is described by the same wave function, and $N({\rm D}_n)$ particles have been detected by detector D$_n$. For each detector, the experimenter determines the probability $\Delta P({\rm D}_n)$ that detector D$_n$ and the corresponding discriminator (see Fig. \[rAbb\_pos-meas\]) generate a logical output signal $U_n^{\rm L}$. In the limit $N^{\rm s} \to \infty$, the probabilities $\Delta P({\rm D}_n)$ are defined as the ratios $N({\rm D}_n)/N^{\rm s}$ (with $n = 1 - Z$).
The (classical) probability $\Delta P({\rm D}_n)$ cannot be calculated in the framework of quantum mechanics. However, the probability $\Delta P(\mbox{\boldmath$R$}_n)$ that a nonrelativistic charged particle has passed the entrance window of detector D$_n$ (see Fig. \[rAbb\_pos-meas\]) can be determined from the wave function of the particle. The centre of this entrance window is located at position $\mbox{\boldmath$R$}_n$. For single events, one can of course obtain only statistical predictions. On average, $N_n = N^{\rm s} \Delta P(\mbox{\boldmath$R$}_n)$ particles will pass this entrance window. If these charged particles fulfil condition (\[r\_E-thr\]), then they will initiate a start reaction in detector D$_n$ with certainty, and produce a detectable output signal $U_n^{\rm L}$. Hence, the number $N_n$ must be equal to the number of detected particles $N({\rm D}_n)$. For incident charged particles, one can conclude that the probabilities $\Delta P(\mbox{\boldmath$R$}_n)$ and $\Delta P({\rm D}_n)$ are equal: $$\label{r_pos-m-Dn}
\Delta P(\mbox{\boldmath$R$}_n) = \frac{N_n}{N^{\rm s}}
= \frac{N({\rm D}_n)}{N^{\rm s}} = \Delta P({\rm D}_n)
\quad \mbox{for} \quad n = 1, \, 2,\dots,Z \;.$$ It should be noted that $\Delta P({\rm D}_n)$ is determined by the experimenter using methods of classical physics, while $\Delta P(\mbox{\boldmath$R$}_n)$ is calculated in the framework of quantum mechanics. From (\[r\_pos-m-Dn\]), one can conclude that the measured probability distribution $\Delta P({\rm D}_n)$ can be reproduced using the methods of quantum mechanics.
The last statement also holds for neutrons, as will be shown in [@Wick]. In this case, the probability $\Delta P_n^{\rm St}$ that the incident object performs a start reaction in detector D$_n$ can be calculated using the methods of quantum physics. One can show that $\Delta P_n^{\rm St}$ is equal to the measured probability $\Delta P({\rm D}_n)$. Here, we use the statement of the three-phase model (see Section \[ch\_det-mod\]) that there is a one-to-one correlation between the start reaction (one of the possible start reactions) in detector D$_n$ and the production of a logical output signal $U_n^{\rm L}$.
Summary and outlook
===================
In the preceding sections, we have mainly studied interactions between quantum objects and a small number of detectors. The incident object is in a quantum mechanical state. Conversely, the detector and discriminator produce a yes/no decision, and provide classical information.
For various types of detectors (such as ionisation chambers, semiconducting detectors, photomultiplier tubes, and scintillation and proportional counters) the measurement process can be considered as a sequence of three phases (the start reaction, amplification, and readout). This observation has been utilised to develop the three-phase model, which can describe individual measurements for nonrelativistic particles.
Detectors behave quite differently during the three phases. Let us assume that charged particles strike an array of $Z$ ionisation chambers. In phase 1, the $Z$ detectors form one big target with a large number of quantum mechanical systems. In the start reaction, the incident particle interacts with one of these systems (atom, molecule, nucleus, or electron), but not with the whole detector. If the start reaction occurs in detector D$_1$, then all reactions following in phase 2 will occur in the same detector. All reactions are quantum mechanical processes. In phases 1 and 2, the incident particle ionises a large number of atoms and produces a cloud of charged particles in detector D$_1$. In phase 3, this cloud determines the properties of the output signal, while the incident particle does not play a role. At the beginning of a position measurement, the detector can be considered as a collection of many quantum mechanical systems (atoms), and at the end the detector behaves like a classical apparatus. We conclude that the detector is neither a quantum mechanical system nor a classical apparatus. The transition from the quantum mechanical to the classical description occurs in the detector.
Probability distributions are determined quite differently in experimental and theoretical physics. The experimenter employs methods of classical physics to determine the probability that detector D$_n$ generates an output signal. This probability cannot be calculated in the framework of quantum physics. However, one can calculate the probability that the incident object performs a start reaction in detector D$_n$. According to the three-phase model, both probabilities are equal. Here, the one-to-one correlation between the start reaction and logical output signal $U_n^{\rm L}$ is utilised.
For position measurements with detectors that fulfil the requirements (A) and (B) (in Section \[ch\_det-mod\]), the essential properties of the three-phase model can be summarised as follows:
- The model explains how nonrelativistic particles can interact with a detector in the framework of quantum physics.
- It explains why it is impossible to predict the outcome of the next measurement. The outcome will be determined at random, because one cannot predict the detector in which the incident particle will perform a start reaction.
- If the start reaction occurs in detector D$_n$, the same detector will provide the classical information comprising the logical output signal $U_n^{\rm L}$. This signal indicates the measurement result, comprising the position of the detected particle.
Position measurements play an important role in many other measurements. This is because of the fact that position coordinates can be measured directly, unlike practically all other physical quantities. As an example, let us consider the measurement of the spin component $S_z$ in the Stern–Gerlach experiment. Each measurement is a two-step process. In the first step, the incident particle (atom) interacts with an inhomogeneous magnetic field, and in the second step a position measurement is performed. A particle that has passed the magnetic field is in an entangled state. The spin state and the centre-of-mass motion of the particle are correlated (see [@Pa58] and [@GoYa]). The spin component $S_z$ of the particle cannot be measured directly, but it can be determined by a position measurement with a correctly calibrated Stern–Gerlach apparatus. The detection of the particle in one of the two detectors leads to a state reduction. At the moment of the measurement, the particle is in a state with a well-defined spin component $S_z$.
The three-phase model is extended in such a manner that it can describe the interaction between the incident particle and the whole measuring device. In phase 1, the particle interacts with the magnetic field and performs a start reaction in one of the two detectors. Both processes can be described in the framework of quantum physics. The amplification process in phase 2 and the readout in phase 3 are similar to the corresponding processes in a position measurement.
A more general discussion of quantum measurements will be provided in [@Wick]. In addition, the question of whether one can find a solution for the measurement problem of quantum mechanics will be discussed.\
Acknowledgments {#acknowledgments .unnumbered}
===============
For helpful discussions, I am grateful to Prof. Dr. J. Bartels, Prof. Dr. K. Fredenhagen, Prof. Dr. R. Klanner and Prof. Dr. E. Lohrmann of the University of Hamburg.
[9999]{}
Von Neumann, J., Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin, 1932. Second edition: Springer-Verlag, 1996 Zurek, W. H., Decoherence and the transition from quantum to classical, Physics Today, Oct. 1991, 36 Schlosshauer, M., Decoherence and the quantum-to-classical transition, Springer-Verlag, Berlin, Heidelberg, 2008 Omnès, R., The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, New Jersey, 1994 Wick, K., Atomphysik und Quantenmessung, to be published (in German) Pauli, W., Die allgemeinen Prinzipien der Wellenmechanik. In: Flügge, S. (Ed.), Handbuch der Physik, Band V, Teil 1, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1958 Gottfried, K., Yan, T.-W., Quantum Mechanics: Fundamentals, Springer-Verlag, New York, 2003
[^1]: e-mail: klaus.wick@desy.de
[^2]: W. H. Zurek [@Zu91] discusses the question of whether macroscopic objects can be treated as quantum mechanical systems.
[^3]: A more detailed description of the so-called measurement problem of quantum mechanics is provided by M. Schlosshauer [@Schl].
[^4]: The photographic process does not fulfil requirement (A). Detectors that use the Cherenkov effect do not fulfil requirement (B).
|
---
abstract: 'In sentence classification tasks, additional contexts, such as the neighboring sentences, may improve the accuracy of the classifier. However, such contexts are [**domain-dependent**]{} and thus cannot be used for another classification task with an inappropriate domain. In contrast, we propose the use of translated sentences as [**domain-free**]{} context that is always available regardless of the domain. We find that naive feature expansion of translations gains only marginal improvements and may decrease the performance of the classifier, due to possible inaccurate translations thus producing noisy sentence vectors. To this end, we present multiple context fixing attachment (MCFA), a series of modules attached to multiple sentence vectors to fix the noise in the vectors using the other sentence vectors as context. We show that our method performs competitively compared to previous models, achieving best classification performance on multiple data sets. We are the first to use translations as domain-free contexts for sentence classification.'
author:
- 'Reinald Kim Amplayo$^\dag$[,]{} Kyungjae Lee$^\dag$[,]{} Jinyeong Yeo$^\ddag$'
- |
Seung-won Hwang$^\dag$\
$^\dag$Yonsei University, Seoul, South Korea\
$^\ddag$Pohang University of Science and Technology, Pohang, South Korea\
[{rktamplayo, lkj0509, seungwonh}@yonsei.ac.kr jinyeo@postech.edu]{}\
bibliography:
- 'ijcai18.bib'
title: Translations as Additional Contexts for Sentence Classification
---
Introduction {#sec:intro}
============
One of the primary tasks in natural language processing (NLP) is sentence classification, where given a sentence (e.g. a sentence of a review) as input, we are tasked to classify it into one of multiple classes (e.g. into positive or negative). This task is important as it is widely used in almost all subareas of NLP such as sentiment classification for sentiment analysis [@Pang2007OpinionMA] and question type classification for question answering [@li2002learning], to name a few. While past methods require feature engineering, recent methods enjoy neural-based methods to automatically encode the sentences into low-dimensional dense vectors [@Kim2014ConvolutionalNN; @Joulin2017BagOT]. Despite the success of these methods, the major challenge in this task is that extracting features from a single sentence limits the performance.
To overcome this limitation, recent works attempted to augment different kinds of features to the sentence, such as the neighboring sentences [@lin2015hierarchical] and the topics of the sentences [@zhao2017topic]. However, these methods used [**domain-dependent**]{} contexts that are only effective when the domain of the task is appropriate. For one thing, neighboring sentences may not be available in some tasks such as question type classification. Moreover, topics inferred using topic models may produce less useful topics when the data set is domain-specific such as movie review sentiment classification [@mimno2011optimizing].
![PCA visualizations of unaltered sentence vectors on TREC data set, where each language is effective for a specific class, highlighted using a yellow circle.[]{data-label="fig:trecvecs"}](example_trec){width="47.00000%"}
In this paper, we propose the usage of translations as compelling and effective **domain-free** contexts, or contexts that are always available no matter what the task domain is. We observe two opportunities when using translations.
[UTF8]{} First, each language has its own linguistic and cultural characteristics that may contain different signals to effectively classify a specific class. Figure \[fig:trecvecs\] contrasts the sentence vectors of the original English sentences and their Arabic-translated sentences in the question type classification task. A yellow circle signifies a clear separation of a class. For example, the green class, or the numeric question type, is circled in the Arabic space as it is clearly separated from other classes, while such separation cannot be observed in English. Meanwhile, location type questions (in orange) are better classified in English.
Second, the original sentences may include language-specific ambiguity, which may be resolved when presented with its translations. Consider the example English sentence “*The movie is terribly amazing*” for the sentiment classification task. In this case, *terribly* can be used in both positive and negative sense, thus introduces ambiguity in the sentence. When translated to Korean, it becomes “*영화는 대단히 훌륭합니다*” which means “*The movie is greatly magnificent*”, removing the ambiguity.
The above two observations hold only when translations are supported for (nearly) arbitrary language pairs with sufficiently high quality. Thankfully, translation services (e.g. Google Translate) Moreover, recent research on neural machine translation (NMT) [@bahdanau2014neural] improved the efficiency and even enabled zero-shot translation [@johnson2016google] of models for languages with no parallel data. This provides an opportunity to leverage on as many languages as possible to any domain, providing a much wider context compared to the limited contexts provided by past studies.
![PCA visualizations of unaltered sentence vectors (left) and the corresponding MCFA-altered vectors (right) on the MR data set. $d$ is the Mahalanobis distance between the two class clusters. []{data-label="fig:mrvecs"}](mrvecs4){width="47.00000%"}
However, despite the maturity of translation, naively concatenating their vectors to the original sentence vector may introduce more noise than signals. The unaltered translation space on the left of Figure \[fig:mrvecs\] shows an example where translation noises make the two classes indistinguishable. In this paper, we propose a method to mitigate the possible problems when using translated sentences as context based on the following observations. Suppose there are two translated sentences $a$ and $b$ with slight errors. We posit that $a$ can be used to fix $b$ when $a$ is used as a context of $b$, and vice versa[^1]. Revisiting the example above, to fix the vector of the English sentence “*The movie is terribly amazing*”, we use the Korean translation to move the vector towards the location where the vector “*The movie is greatly magnificent*” is. Based on these observations, we present a neural attention-based multiple context fixing attachment (MCFA). MCFA is a series of modules that uses all the sentence vectors (e.g. Arabic, English, Korean, etc.) as context to fix a sentence vector (e.g. Korean). Fixing the vectors is done by selectively moving the vectors to a location in the same vector space that better separates the class, as shown in Figure \[fig:mrvecs\]. Noises from translation may cause adverse effects to the vector **itself** (e.g. when a noisy vector is directly used for the task) and **relatively** to other vectors (e.g. when a noisy vector is used to fix another noisy vector). MCFA computes two sentence usability metrics to control the noise when fixing vectors: (a) **self usability** $\rho_i(a)$ weighs the confidence of using sentence $a$ in solving the task. (b) **relative usability** $\rho_r(a, b)$ weighs the confidence of using sentence $a$ in fixing sentence $b$.
Listed below are the three main strengths of the MCFA attachment. (1) MCFA is attached after encoding the sentence, which makes it widely adaptable to other models. (2) MCFA is extensible and improves the accuracy as the number of translated sentences increases. (3) MCFA moves the vectors inside the same space, thus preserves the meaning of vector dimensions. Results show that a convolutional neural network (CNN) attached with MCFA significantly improves the classification performance of CNN, achieving state of the art performance over multiple data sets.
Preliminaries
=============
[0.45]{} {width="\textwidth"}
[0.45]{} {width="\textwidth"}
Problem: Translated Sentences as Context
----------------------------------------
In this paper, the ultimate task that we solve is the sentence classification task where given a sentence and a list of classes, one is task to classify which class (e.g. positive or negative sentiment) among the list of classes does the sentence belong. However, the main challenge that we tackle is the task on how to utilize translated sentences as additional context in order to improve the performance of the classifier. Specifically, the problem states: given the original sentence $s$, the goal is to use $t_1, t_2, ..., t_n$, or sentences in other languages which are translated from $s$, as additional context.
#### Base Model: Convolutional Neural Network.
The base model used is the convolutional neural network (CNN) for sentences [@Kim2014ConvolutionalNN]. It is a simple variation of the original CNN for texts [@collobert2011natural] to be used on sentences. Let $\mathbf{x}_i \in \mathbb{R}^d$ be the $d$-dimensional word vector of the $i$-th word in a sentence of length $n$. A convolution operation involves applying a filter matrix $\mathbf{W} \in \mathbb{R}^{h \times d}$ to a window of $h$ words and producing a new feature vector $c_i$ using the equation $c_i = f([\mathbf{x}_i; ...; \mathbf{x}_{i+h-1}]^\top \mathbf{W}+b)$, where $b$ is a bias vector and $f(.)$ is a non-linear function. By doing this on all possible windows of words we produce a feature map $\mathbf{c} = [c_1, c_2, ...]$. We then apply a max-over-time pooling operation [@collobert2011natural] over the feature map and take the maximum value as the feature vector of the filter. We do this on all feature vectors and concatenate all the feature vectors to obtain the final feature vector $\mathbf{v}$. We can then use this vector as input features to train a classifier such as logistic regression. We use CNN to create sentence vectors for all sentences $s, t_1, t_2, ..., t_n$. From here on, we refer to these vectors as $\mathbf{v}_s, \mathbf{v}_{t_1}, \mathbf{v}_{t_2}, ..., \mathbf{v}_{t_n}$, respectively. We refer to them collectively as $\mathbb{V}$.
#### Baseline 1: Naive Concatenation.
A simple method in order to use the translated sentences as additional context is to naively concatenate their vectors with the vector of the original sentence. That is, we create a wide vector $\mathbf{\hat{v}} = [\mathbf{v}_s; \mathbf{v}_{t_1}; ...; \mathbf{v}_{t_n}]$, and use this as the input feature vector of the sentence to the classifier. This method works fine if the translated sentences are translated properly. However, sentences translated using machine translation models usually contain incorrect translation. In effect, this method will have adverse effects on the overall performance of the classifier. This will especially be very evident if the number of additional sentences increases.
#### Baseline 2: L2 Regularization.
In order to alleviate the problems above, we can use L2 regularization to automatically select useful features by weakening the appropriate weights. The main problem of this method occurs when almost all of the weights coming from the vectors of the translated sentence are weakened. This leads to making the additional context vectors useless and to having a similar performance when there are no additional context. Ultimately, this method does not make use of the full potential of the additional context.
Model
=====
To solve the problems of the baselines discussed above, we introduce an attention-based neural multiple context fixing attachment (MCFA)[^2], a series of modules attached to the sentence vectors $\mathbb{V}$. MCFA attachment is used to fix the sentence vectors, by slightly modifying the per-dimension values of the vector, before concatenating them into the final feature vector. The sentence vectors are altered using other sentence vectors as context (e.g. $\mathbf{v}_{t_1}$ is altered using $\mathbf{v}_s, \mathbf{v}_{t_2}, ..., \mathbf{v}_{t_n}$). This results to moving the vectors in the same vector space. The full architecture is shown in Figure \[fig:arch\].
Self Usability Module
---------------------
To fix a source sentence vector[^3], we use the other sentence vectors as guide to know which dimensions to fix and to what extent do we need to fix them. However, other vectors might also contain errors which may reflect to the fixing of the source sentence vector. In order to cope with this, we introduce self usability modules. A self usability module contains the **self usability** of the vector $\rho_i(a)$, which measures how confident sentence $a$ is for the task at hand. For example, an ambiguous sentence (e.g. “*The movie is terribly amazing*”) may receive a low self usability, while a clear and definite sentence (e.g. “*The movie is very good*”) may receive a high self usability. Mathematically, we calculate the self usability of the vector $\mathbf{v}_i$ of sentence $i$, denoted as $\rho_{i}(\mathbf{v}_i)$, using the equation $\rho_i(\mathbf{v}_i) = \sigma(\mathbf{v}_i^\top \mathbf{T}_i)$, where $\mathbf{T}_i \in \mathbb{R}^{d \times 1}$ is a matrix to be learned. The produced value is a single real number from 0 to 1. We pre-calculate the self usability of all sentence vectors $\mathbf{v}_i \in \mathbb{V}$. These are used in the next module, the relative usability module.
Relative Usability Module
-------------------------
**Relative usability** $\rho_r(a, b)$ measures how useful $a$ can be when fixing $b$, relative to other sentences. There are two main differences between $\rho_i(a)$ and $\rho_r(a, b)$. First, $\rho_i(a)$ is calculated before $a$ knows about $b$ while $\rho_r(a, b)$ is calculated when $a$ knows about $b$. Second, $\rho_r(a, b)$ can be low even though $\rho_i(a)$ is not. This means that $a$ is not able to help in fixing the wrong information in $b$. Here, we extend the additive attention module [@bahdanau2014neural] and use it as a method to calculate the relative usability of two sentences of different languages. To better visualize the original attention mechanism, we present the equations below. $$\begin{aligned}
\label{eq:addattscore}
e_i &= u^\top tanh( s^\top \mathbf{W}+t_i^\top \mathbf{U} ) \\
\alpha_i &= \frac{exp(e_i)}{\sum_{j \in T} exp(e_j)}\end{aligned}$$ One major challenge in using the attention mechanism in our problem is that the sentence vectors do not belong to the same vector space. Moreover, one characteristic of our problem is that the sentence vectors can be both a source and a context vector (e.g. $\mathbf{v}_s$ can be both $s$ and $t_i$ in Equation \[eq:addattscore\]). Because of these, we cannot directly use the additive attention module. We extend the module such that (1) each sentence vector $\mathbf{v}_k$ has its own projection matrix $\mathbf{X}_k \in \mathbb{R}^{d \times d}$, and (2) each projection matrix $\mathbf{X}_k$ can be used as projection matrix of both the source (e.g. when sentence $k$ is the current source) and the context vectors. Finally, we incorporate the self usability function $\rho_i(\mathbf{v}_k)$ to reflect the self usability of a sentence. Ultimately, the relative usability denoted as $\rho_r(\mathbf{v}_i, \mathbf{v}_j)$ is calculated using the equations below, where $\times$ is the multiplication of a vector and a scalar through broadcasting. $$\begin{aligned}
e(\mathbf{v}_i, \mathbf{v}_j) &= x^\top tanh( \mathbf{v}_i^\top \mathbf{X}_i+\mathbf{v}_j^\top \mathbf{X}_j \times \rho_i(\mathbf{v}_j) ) \\
\rho_r(\mathbf{v}_i, \mathbf{v}_j) &= \frac{exp(e(\mathbf{v}_i, \mathbf{v}_j))}{\sum_{\mathbf{v}_k \in \mathbb{V}} exp(e(\mathbf{v}_i, \mathbf{v}_k))}\end{aligned}$$
Vector Fixing Module
--------------------
The vector fixing module applies the attention weights to the sentence vectors and creates an integrated context vector. We then use this vector alongside with the source sentence vector to create a weighted gate vector. The weighted gate vector is used to determine to what extent should a dimension of the source sentence vector be altered.
The common way to apply the attention weights to the context vectors and create an integrated context vector $c_i$ is to directly do weighted sum of all the context vectors. However, this is not possible because the context vectors are not on the same space. Thus, we use a projection matrix $\mathbf{U}_k \in \mathbb{R}^{d \times d}$ to linearly project the sentence vector $\mathbf{v}_k$ to transform the sentence vectors into a common vector space. The integrated context vector $c_i$ is then calculated as $c_i = \sum_{\mathbf{v}_k \in \mathbb{V}} \rho_r(\mathbf{v}_i, \mathbf{v}_k) \mathbf{v}_k^\top \mathbf{U}_k$.
Finally, we construct a weighted gate vector $w_k$ and use it to fix the source sentence vectors using the equations below, where $\mathbf{V}_k \in \mathbb{R}^{2d \times d}$ is a trainable parameter and $\otimes$ is the element-wise multiplication procedure. The weighted gate vector is a vector of real numbers between 0 and 1 to modify the intensity of per-dimension values of the sentence vector. This causes the vector to move in the same vector space towards the correct direction. $$w_k = \sigma([\mathbf{v_k}; c_k]^\top \mathbf{V}_k)$$ $$\hat{\mathbf{v}}_k = \mathbf{v}_k \otimes w_k$$ An alternative approach to do vector correction is using a residual-style correction, where instead of multiplying a gate vector, a residual vector [@he2016deep] is added to the original vector. However, this approach makes the correction not interpretable; it is hard to explain what does adding a value to a specific dimension mean. One major advantage of MCFA is that the corrections in the vectors are interpretable; the weights in the gate vector correspond to the importance of the per-dimension features of the vector. The altered vectors $\hat{\mathbf{v_s}}, ..., \hat{\mathbf{v_{t_n}}}$ are then concatenated and fed directly as an input vector to the logistic regression classifier for training.
Experiments
===========
Experimental Setting
--------------------
We test our model on four different data sets as listed below and summarized in Table \[tab:data\]. (a) **MR**[^4] [@pang2005seeing]: Movie reviews data where the task is to classify whether the review sentence has positive or negative sentiment. (b) **SUBJ** [@pang2004sentimental]: Subjectivity data where the task is to classify whether the sentence is subjective or objective. (c) **CR**[^5] [@hu2004mining]: Customer reviews where The task is to classify whether the review sentence is positive or negative. (d) **TREC**[^6] [@li2002learning]: TREC question data set the task is to classify the type of question.
**Data set** $c$ $|w|$ $M$ *Test*
-------------- ----- ------- ------- --------
MR 2 20 10662 CV
SUBJ 2 19 10000 CV
CR 2 23 3775 CV
TREC 6 10 5952 500
: Statistics of the four data sets used in this paper. $c$: number of target classes. $|w|$: average number of words. $M$: number of data instances. *Test*: size of the test data, if available. If not, we use 10-fold cross validation (marked as CV) with random split.[]{data-label="tab:data"}
All our data sets are in English. For the additional contexts, we use ten other languages, selected based on their diversity and their performance on prior experiments: Arabic, Finnish, French, Italian, Korean, Mongolian, Norwegian, Polish, Russian, and Ukranian. We translate the data sets using Google Translate. Tokenization is done using the polyglot library[^7]. We experiment on using only one additional context ($N=1$) and using all ten languages at once ($N=10$). For $N=1$, we only show the accuracy of the best classifier for conciseness.
For our CNN, we use rectified linear units and three filters with different window sizes $h=3,4,5$ with $100$ feature maps each, following [@Kim2014ConvolutionalNN]. For the final sentence vector, we concatenate the feature maps to get a 300-dimension vector. We use dropout [@srivastava2014dropout] on all non-linear connections with a dropout rate of 0.5. We also use an $l_2$ constraint of 3, following [@Kim2014ConvolutionalNN] for accurate comparisons. We use FastText pre-trained vectors[^8] [@bojanowski2016enriching] for all our data sets and their corresponding additional context. During training, we use mini-batch size of 50. Training is done via stochastic gradient descent over shuffled mini-batches with the Adadelta update rule. We perform early stopping using a random $10\%$ of the training set as the development set.
We present several competing models, listed below to compare the performance of our model. (A) Aside from the base model (**CNN**) [@Kim2014ConvolutionalNN], we use Dependency-based CNN (**Dep-CNN**) [@ma2015dependency], which parses the sentences first and does convolution on ancestor paths and Dependency-sensitivity CNN (**DSCNN**) [@zhang2016dependency], which uses LSTM to capture dependency information within each sentence; (B) **AdaSent** [@zhao2015self] adopts a hierarchical structure, where consecutive levels are connected through gated recursive composition of adjacent segments, and feeds the hierarchy as a multi-scale representation through a gating network; (C) Topic-aware Convolutional Neural Network (**TopCNN**) [@zhao2017topic] uses topics as additional contexts and changes the CNN architecture. TopCNN uses two types of topics: word-specific topic and sentence-specific topic; and (D) **CNN+B1** and **CNN+B2** are the two baselines presented in this paper. We do not show results from RNN models because they were shown to be less effective in sentence classification in our prior experiments. For models with additional context, we further use an ensemble classification model using a commonly used method by averaging the class probability scores generated by the multiple variants (in our model’s case, $N=1$ and $N=10$ models), following [@zhao2017topic].
**Model**
--------------- -- -- ------ -- -- ------ -- -- ------ -- -- ------
CNN
Dep-CNN
DSCNN
AdaSent
**C = Topic**
TopCNN ****
**C = Trans**
CNN+B1
CNN+B2
CNN+MCFA **** **** **** ****
Results and Discussion
----------------------
We report the classification accuracy of the competing models in Table \[tab:result\]. We show that CNN+MCFA achieves state of the art performance on three of the four data sets and performs competitively on one data set. When $N=1$, MCFA increases the performance of a normal CNN from $85.0$ to $87.6$, beating the current state of the art on the CR data set. When $N=10$, MCFA additionally beats the state of the art on the TREC data set. Finally, our ensemble classifier additionally outperforms all competing models on the MR data set. We emphasize that we only use the basic CNN as our sentence encoder for our experiments, yet still achieve state of the art performance on most data sets. Hence, MCFA is successful in effectively using translations as additional context to improve the performance of the classifier.
We compare our model (CNN+MCFA) and the baselines discussed above (CNN+B1, CNN+B2). On all settings, our model outperforms the baselines. When $N=10$, the performance of our model increases over the performance when $N=1$, however the performance of CNN+B1 decreases when compared to the performance when $N=1$. We also show the accuracies of the worst classifiers when $N=1$ in Table \[tab:minresult\]. On all data sets except SUBJ, the accuracy of CNN+B1 decreases from the base CNN accuracy, while the accuracy of our model always improves from the base CNN accuracy. This is resolved by CNN+B2 by applying L2 regularization, however the increase in performance is marginal.
**Model** [**MR**]{} **SUBJ** **CR** [**TREC**]{}
----------- ------------ ---------- -------- --------------
CNN 81.5 93.4 85.0 93.6
CNN+B1 94.2
CNN+B2 81.7 94.2
CNN+MCFA 81.8 94.4 85.8 94.2
: Accuracies of the worst CNN+translation classifiers when $N=1$. Accuracies less than CNN accuracies are highlighted in .[]{data-label="tab:minresult"}
We also compare two different kinds of additional context: topics (TopCNN) and translations (CNN+B1, CNN+B2, CNN+MCFA). Overall, we conclude that translations are better additional contexts than topics. When using a single context (i.e. TopCNN$_\text{word}$, TopCNN$_\text{sent}$, and our models when $N=1$), translations always outperform topics even when using the baseline methods. Using topics as additional context also decreases the performance of the CNN classifier on most data sets, giving an adverse effect to the CNN classifier.
Model Interpretation
====================
[0.4]{} ![Attention weights of example Korean sentences from the MR data set. The red color fill represents the attention weights given to each sentence. The darker the fill, the larger the attention weight.[]{data-label="fig:attention"}](attention1 "fig:"){width="85.00000%"}
[0.4]{} ![Attention weights of example Korean sentences from the MR data set. The red color fill represents the attention weights given to each sentence. The darker the fill, the larger the attention weight.[]{data-label="fig:attention"}](attention2 "fig:"){width="85.00000%"}
[UTF8]{}
[0.47]{}
[X]{} Original sentence:\
*skip this turd and pick your nose instead because you’re sure to get more out of the latter experience .*\
Korean translation:\
*이 웅덩이를 건너 뛰고 .*\
Human re-translation:\
*In order to get more from the latter experience , you need to skip this puddle and choose your nose .*\
**Self Usability: 0.3958**\
[0.47]{}
[X]{} Original sentence:\
*michael moore’s latest documentary about america’s thirst for violence is his best film yet . . .*\
Korean translation:\
*마이클 무어 ( Michael Moore ) 의 최근 미국 다큐멘터리 은 그의 최고의 영화 다 . . .*\
Human re-translation:\
*Michael Moore’s latest American documentary “ Violent Scene ” is his best film yet . . .*\
**Self Usability: 1.0000**\
We first provide examples shown in Table \[tab:usability\] on how the self usability module determines the score of sentences. In the first example, it is hard to classify whether the translated sentence is positive or negative, thus it is given a low self usability score. In the second example, although the sentence contains mistranslations, these are minimal and may actually help the classifier by telling it that *thirst for violence* is not a negative phrase. Thus, it is given a high self usability score.
![PCA visualization of unaltered (left) and altered (right) vectors of the MR data set. $d$ is the Mahalanobis distance between two class clusters.[]{data-label="fig:mrvecsbefaf"}](comb_mr4){width="45.00000%"}
[|m[0.06]{}|m[0.36]{}|]{} **Sentence** & may take its sweet time to get wherever it’s going, but if you have the patience for it, you won’t feel like it’s wasted yours.\
**NN (Unaltered)** & you know that ten bucks you’d spend on a ticket? just send it to cranky. we don’t get paid enough to sit through crap like this.\
**NN (altered)** & what might have been readily dismissed as the tiresome rant of an aging filmmaker still thumbing his nose at convention takes a surprising, subtle turn at the midway point.\
[|m[0.06]{}|m[0.36]{}|]{} **Sentence** & every nanosecond of the new guy reminds that you could be doing something else more pleasurable. like scrubbing the toilet. emptying rat traps. or doing last year’s taxes with your ex-wife.\
**NN (Unaltered)** & in the new release of cinema paradiso, the tale has turned from sweet to bittersweet, and when the tears come during that final, beautiful scene, they finally feel absolutely earned.\
**NN (altered)** & after scenes of nonsense, you’ll be wistful for the testosterone-charged wizardry of jerry bruckheimer productions, especially because half past dead is like the rock on walmart budget.\
Figure \[fig:attention\] shows two data instance examples where we show the attention weights given to the other contexts when fixing a Korean sentence. The larger the attention weight is, the more the context is used to fix the Korean sentence. In the first example, the Korean sentence contains translation errors; especially, the words *bore* and *climactic setpiece* were not translated and were only spelled using the Korean alphabet. In this example, the English attention weight is larger than the Korean attention weight. In the second example, the Korean sentence correctly translates all parts of the English sentence, except for the phrase *as it does in trouble*. However, this phrase is not necessary to classify the sentence correctly, and may induce possible vagueness because of the word *trouble*. Thus, the Korean attention weight is larger.
Figure \[fig:mrvecsbefaf\] shows the PCA visualization of the unaltered and the altered vectors of four different languages. In the first example, the unaltered sentence vectors are mostly in the middle of the vector space, making it hard to draw a boundary between the two examples. After the fixing, the boundary is much clearer.
We also show the English sentence vectors in the second example. Even without fixing the unaltered English sentence vectors, it is easy to distinguish both classes. After the fix, the sentence vectors in the middle of the space are moved, making the distinction more obvious and clearer. We also provide quantitative evidence by showing that the Mahalanobis distance between the two classes in the altered vectors are significantly farther than that of the unaltered vectors.
We also show two examples sentences from English and Korean vector spaces and their corresponding nearest neighbors on both the unaltered and altered vector spaces in Table \[tab:vecfixexample\]. In the first example, the unaltered vector focuses on the meaning of *“wasted yours”* in the sentence, which puts it near sentences regarding wasted time or money. After fixing, the sentence vector focuses its meaning on the slow yet worth-the-wait pace of the movie, thus moving it closer to the correct vectors. In the second example, all three sentences have highly descriptive tones, however, the nearest neighbor on the altered space is hyperbolically negative, comparing the movie to a description unrelated to the movie itself.
Related Work
============
One way to improve the performance of a sentence classifier is to introduce new context. Common and obvious kinds of context are the neighboring sentences of the sentence [@lin2015hierarchical], and the document where the sentence belongs [@huang2012improving]. Topics of the words in the sentence induced by a topic model were also used as contexts [@zhao2017topic]. In this paper, we introduce yet another type of additional context, sentence translations, which to the best of our knowledge have not been used previously.
Sentence encoders trained from neural machine translation (NMT) systems were also used for transfer learning [@hill2016learning]. [@hill2017representational] demonstrated that altered-length sentence vectors from NMT encoders outperform sentence vectors from monolingual encoders on semantic similarity tasks. Recent work used representation of each word in the sentence to create a sentence representation suitable for multiple NLP tasks [@mccann2017learned]. Our work shares the commonality of using NMT for another task, but instead of using NMT to encode our sentences, we use it to translate the sentences into new contexts. Increasing the number of data instances of the training set has also been explored to improve the performance of a classifier. Recent methods include the usage of thesaurus [@zhang2015character], paraphrases [@fu2014improving], among others. These simple variation techniques are preferred because they are found to be very effective despite their simplicity. Our work similarly augments training data, not by adding data instances (vertical augmentation), but rather by adding more context (horizontal augmentation). Though the paraphrase of $p$ can be alternatively used as an augmented context, this could not leverage the added semantics coming from another language, as discussed in Section \[sec:intro\].
Conclusion
==========
This paper investigates the use of translations as better additional contexts for sentence classification. To answer the problem on mistranslations, we propose multiple context fixing attachment (MCFA) to fix the context vectors using other context vectors. We show that our method improves the classification performance and achieves state-of-the-art performance on multiple data sets. In our future work, we plan to use and extend our model to other complex NLP tasks.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by Microsoft Research Asia and the ICT R&D program of MSIT/IITP. \[2017-0-01778, Development of Explainable Human-level Deep Machine Learning Inference Framework\]
[^1]: Hereon, we mean to “fix” as to “correct, repair, or alter.”
[^2]: The code we use in this paper is publicly shared: <https://github.com/rktamplayo/MCFA>
[^3]: Hereon, we say that $\mathbf{v}_k$ is a *source sentence vector* if $\mathbf{v}_k$ is the current vector to be altered.
[^4]: <https://www.cs.cornell.edu/people/pabo/movie-review-data/>
[^5]: <http://www.cs.uic.edu/~liub/FBS/sentiment-analysis.html>
[^6]: <http://cogcomp.cs.illinois.edu/Data/QA/QC/>
[^7]: <https://pypi.python.org/pypi/polyglot>
[^8]: <https://github.com/facebookresearch/fastText/blob/master/pretrained-vectors.md>
|
---
abstract: 'The time-dependent fluorescence Stokes shifts monitors the relaxation of the polarization of a polar solvent in the surroundings of a photoexcited solute molecule, but also the structural variation of the solute following photoexcitation and the subsequent molecular charge redistribution. Here, we formulate a simple nonequilibrium quantum theory of solvation for an explicitly time-dependent continuous solvent. The time-dependent solvent induces nonequilibrium fluctuations on the solvent dynamics which are directly reflected in different time components in the time-dependent Stokes shift. We illustrate the structural dynamics in the presence of an explicitly time-dependent solvent by the example of a dynamically shrinking solute which leads to a bi-modal Stokes shift. Interestingly, both contributions are mutually coupled. Furthermore, we can explain a prominent long-tail decay of the Stokes shift associated to slow structural dynamical variations.'
author:
- Henning Kirchberg
- Michael Thorwart
title: 'Time-Resolved Probing of the Nonequilibrium Structural Solvation Dynamics by the Time-Dependent Stokes Shift'
---
Introduction {#StokesShift}
============
A photoionized solvated molecule with its suddenly rearranged electronic charge distribution is no longer in equilibrium with its surrounding solvent. The subsequent electronic relaxation can be monitored spectroscopically by measuring the time-dependent Stokes shift. During the relaxation, the molecular structure is dynamically reorganized and multiple time scales may be involved in this structural dynamics of the solute-solvent system [@lag2017]. One time scale is set by the correlation time of the fluctuating electric field which is induced by the surrounding polar solvent molecules. Commonly, these fluctuations are described by thermal equilibrium variations about the local mean, which modify, e.g., the free energies of reactants, products and transition states and, thus, affect the energy of activation or even the course of the chemical process [@NitzanBook]. A further time scale may be associated to the intrinsic structural dynamics of the solute itself which may influence its electronic relaxation properties, in concert with the dynamical rearrangement of the solvent which can occur far from thermal equilibrium. Such a dynamical solvent induces then nonequilibrium fluctuations of the electric field acting on the solute’s charge configuration and their influence on the relaxation dynamics is more difficult to be included.
A large body of literature exits for the relaxational dynamics in solution due to equilibrium fluctuations of the solvent molecules. For instance, Fleming and co-workers [@jim1994; @ros1991] have uncovered that the solvation dynamics at very short times is associated to inertial collective relaxation of solvent molecules mostly driven by the large force constant of solvent free energy surfaces [@ros1991; @lag2017]. The latter are also employed in the famous Marcus theory to describe solvent-assisted charge transfer reactions [@mar1964; @mar1965]. Other ultrafast processes in water are the intramolecular O-H stretching (10 fs) and bending vibrations (20 fs), which occur on the time scale of a few tens of femtoseconds[@lag2017].
To probe molecular motion at the interfaces between the solute and the solvent on different time scales, time-dependent fluorescence spectroscopy is a common technique. By determining the Stokes shift, which is the shift of the maximum of the fluorescence emission spectrum a time span $t$ after initial excitation, solvent relaxation in the close proximity of the solute can be recorded for varying times [@hei2019; @nil2005; @bag1984]. The time-dependent Stokes shift is commonly discussed in the context when an electronic excitation occurs in the framework of the Franck-Condon principle [@fra1926; @con1926]. According to this, the configuration of the polar solvent immediately after photoionization of the solute is still that of the solvent with the unexcited solute. For the total solute-solvent system, this is an energetically unfavorable configuration, and in the course of time, the solvent adopts dynamically (different parts of it on different timescales) to the altered charge distribution of the solute. It reaches energetically more favorable configurations, so that the energy of the fluorescence emission decreases over time.
The time evolution of the Stokes shift in polar media at equilibrium is described by the well-established Bagchi-Oxtoby-Fleming theory in terms of a continuum Onsager model of the solvent [@bag1984]. It may be understood as a time-dependent description of the Ooshika-Lippert-Mataga equation of the average shift in frequency of the absorption and fluorescence transition in solution due to polar interactions [@bag1984; @oos1954; @mat1956; @lip1957]. Using the theory of linear response, it is possible to link the fluctuations of the solute-solvent interaction to the dipole equilibrium correlation function of the solute alone, which depends on solvent time constants and dielectric properties only [@bag1984]. For the simplest case of a Debye-type dielectric medium [@deb1913], the model predicts an exponential relaxation of the solvation energy with the time constant being given by the solvent’s longitudinal relaxation time $\tau_L=[(2\epsilon_\infty+1)/(2\epsilon_s+1)]\tau_D$. In general, $\tau_L$ is strongly reduced in comparison to the Debye relaxation time $\tau_D$ for a solvent with a large static dielectric constant $\epsilon_s$ like water due to collective polarization dynamics. This becomes observable by the time-dependent Stokes shift [@bag1984].
The approach in terms of a continuum model used for time-resolved fluorescence spectroscopy is a powerful description until today because the fluorescence response is mostly insensitive to the motion of individual water molecules but only to their collective motion [@hei2019; @nil2005]. Spectroscopic measurements confirm relaxation dynamics in accordance with $\tau_L$ which is much faster than the process of reorientation of individual solvent molecules. This reflects the fact that the solvent response involves the concerted motion of many molecules. The prediction of $\tau_L$ forms a cornerstone for the comparison of experimental data of the time-dependent Stokes shift to theoretical models [@mar1988].
However, experimental data sometimes deviate from the predictions of a homogeneous continuum model. This may be understood from the microscopic picture. The solvent in close proximity to the solute contains an insufficient number of molecules to attain the full cooperativeness described by the $\tau_L$ response, but the solvent further away may look like a continuum fluid where the $\tau_L$ response pertains. This fact is known as the “Onsager inverse snowball picture”, which associates the shorter time scales with solvent layers further away from the solute [@ons1977]. Therefore, relaxation may occur on several different time scales, ranging from single particle relaxation processes up to the collective polarization dynamics characterized by $\tau_L$. Even ultrafast relaxation contributions characterized by a non-exponential decay may arise [@mar1988; @bag1992]. However, Bagchi and Chandra have shown that when translational motions of the solvent become important, which are neglected in Onsager’s approach, relaxation is generated by different solvent shells such that the fastest dynamics is associated with the closest shells in the vicinity of the solute [@bag1988; @bag1989; @bag2012].
An alternative to a continuum description are molecular dynamics (MD) simulations or combined quantum mechanics/ molecular mechanics (QM/MM) simulations. They help to mimic at least the most prominent solvent reorientational and relaxational time scales to explain experimental time-dependent fluorescence studies. Neira and Nitzan, for example, have used MD simulations to confirm a slow component of the dynamical solvent associated with salt ions in electrolyte solutions. Their signature becomes visible by a slower time component in the Stokes shift in comparison to the conventional bulk water relaxation [@ner1993].
Recent experimental time-dependent fluorescence studies of aqueous proteins provide evidence of different contributions to relaxation associated with distinct motions of such complex systems like proteins [@cer2019; @hei2019]. Heid and Braun, for example, decompose the fluorescence Stokes shift into a water and a protein component by performing MD simulations at nine different sites of the protein in water [@hei2019]. The water component dominates the static Stokes shift at short times, but its magnitude decays rapidly. In turn, the self-motion of the protein becomes visible after a few picoseconds. The resulting Stokes shift therefore leads to a bi- or even multimodal decay [@hei2019; @nil2005]. Despite the computational accessibility to describe different dynamical contributions of the solvation process, MD simulations are usually limited to a numerically tractable number of solvent molecules. Furthermore, simulations often use linear solvent response for a varity of pertubations such as changes of the solute’s electronic state. By this, the determination of computationally expensive forces using excited-state interaction potentials can be avoided, since the impact of the perturbation can be calculated from the fluctuations of the equilibrium solute-solvent interaction. However, nonequilibrium simulations clearly identify “mechanical” relaxation due to changes in the spatial extension of the solute which are key to the breakdown of linear response. [@ahe2000]
In this work, we combine the self-motion of the solute with a continuum description of a time-dependent solvent and determine the resulting time-dependent Stokes shift. We formulate an effective nonequilibrium theory in which an explicit time-dependent motion of the continuum solvent beyond thermal fluctuations enters. By this, we rejuvenate and generalize the accostable Bagchi-Oxtoby-Fleming continuum theory by including an explicitly time-dependent Onsager model introduced in Ref. [@kir2018]. To be specific, we focus on the case of a dynamically shrinking radius of an Onsager sphere and determine fully analytically the time-dependent Stokes shift. This model combines nonpolar and dielectric relaxation dynamics and is an archetypal example for the solvation dynamics beyond Onsager’s regression hypothesis which normally associates dynamical spectral signatures of the solute to spontaneous equilibrium fluctuations of the solvent. [@cha1987] We find a bimodal decay in the time-dependent Stokes shift in which the shrinking rate of the solute and the correlation time of the solvent both appear. Throughout the work, we concentrate on water as a solvent, but the theory applies to any other polar solvent.
Model {#StokesModel}
=====
We consider the time-dependent Stokes shift characterized by the function [@NitzanBook] $$\begin{aligned}
\label{eq3.3.1}
S(t)=\frac{\Delta E_{solv}(t)-\Delta E_{solv}(\infty)}{\Delta E_{solv}(0)-\Delta E_{solv}(\infty)} \end{aligned}$$ for a photoexcited molecular complex with explicit molecular motions at the interface to a dipolar solvent. Here, $\Delta E_{solv}(t)$ is the resulting solvation energy difference between the excited and the ground state molecule at a given time $t$. It results from the electrostatic interaction between the charge distribution of the solute and the surrounding polar solvent. If there is only little internal vibrational excitation of the solvent during the initial photoexcitation, the time-dependent Stokes shift mainly results from the time-dependent solvation energy [@hsu1997].
To illustrate the mechanism of the structural dynamics of a coupled solute-solvent system, we assume that the self-motion of the solute is given by the model of a dynamically shrinking Onsager cavity [@ons1936; @kir2018] in the center of which the central molecular dipole moment suddenly changes from an initial ground state dipole moment $\boldsymbol{\mu}_g$ to $\boldsymbol{\mu}_e$ upon photoexcitation which initiates the geometrical ‘shrinking’. The shrinking cavity radius may mimic generically motional changes at the solute-solvent interface such as the observed self-motion of dissolved proteins [@nil2005; @hei2019]. Another example is the photodetachment of an electron from a sodium anion leaving behind a smaller neutral atom that drives solvent molecules into that locations which were occupied before by the volume of the solute. [@bed2003]
The electrostatic interaction energy of the dipole moment with the homogeneous and isotropic electric field provided by the polar solvent is given by $V_I=-\boldsymbol{\mu}(t)\cdot \textbf{R}(t)\equiv E_{solv}(t)$. We disregard solute-solvent coupling of higher multipole contributions because the change upon excitation in the shape and size of the solute is usually small. The reaction field for a not to rapidly varying Onsager radius $a(t)$ according to Ref. [@kir2018] can be calculated to be $$\begin{aligned}
\label{eq1}
\textbf{R}(t)&=\frac{1}{a(t)^3}\int_{-\infty}^t dt' \frac{2(\epsilon_s-1)}{3\tau_D}\exp[-\omega_D (t-t')] \boldsymbol{\mu}(t')\\ \notag &=\frac{1}{a(t)^3}\int_{-\infty}^t dt' \chi(t-t') \boldsymbol{\mu}(t'),\end{aligned}$$ where $\omega_D=(2\epsilon_s+1)/(3\tau_D)=\tau_L^{-1}$. The static dielectric constant of water at 20$^{\circ}$C is $\epsilon_s=78.3$ and the Debye relaxation time $\tau_D=8.2$ ps [@mck2005]. The reaction field portrays the time-dependent back action of the polarization of the surrounding solvent due to the dipole on itself. We neglect in our model a possible change of the solute’s polarizability upon excitation which may be included by additionally filling the Onsager cavity with a dielectric. A thorough discussion on this can be found in the work of Bagchi, Oxtoby and Fleming [@bag1984]. Here, we focus on structural changes of the solute at the solute-solvent interface which enter via a time-dependent Onsager radius $a(t)$. For an exponentially shrinking Onsager radius by a not too large amount from an initial value $a_0+a_1$ to a final value $a_0$, we use the expansion $1/a^3(t)=1/(a_0+a_1\exp[-\alpha t]\Theta(t))^{3}\approx 1/a_0^{3}-3a_1\exp[-\alpha t]\Theta(t)/a_0^4$ up to first order in $a_1/a_0$. Here, $\alpha$ is the phenomenological shrinking rate. By this, we can split the reaction field of Eq. (\[eq1\]) into two terms according to $$\begin{aligned}
\label{eq3.3.2}
\textbf{R}(t)=&\frac{1}{a_0^3}\int_{-\infty}^t d t' \bigg[\chi(t-t')-\frac{3a_1}{a_0}\exp[-\alpha t]\Theta(t)\chi(t-t') \bigg] \boldsymbol{\mu}(t'),\end{aligned}$$ where the shrinking begins upon photoexcitation at time $t=0$. We assume the optical excitation of the solute to occur instantaneously such that the dipole moment changes from $\boldsymbol{\mu}_g$ to $\boldsymbol{\mu}_e$ at $t=0$ which coincides with the beginning of the radial shrinking. Moreover, we assume that the dipole moment does not change its direction but its magnitude according to $$\begin{aligned}
\label{eq3.3.3}
\boldsymbol{\mu}(t)=\mu_g \hat{\textbf{e}}_z+ \Theta(t)(\mu_e-\mu_g)\hat{\textbf{e}}_z,\end{aligned}$$ where $\Theta(t)$ is the unit Heaviside function. Then, the reaction field in Eq. (\[eq3.3.2\]) becomes $$\begin{aligned}
\label{eq3.3.4}
\textbf{R}(t)&=\frac{1}{a_0^3}\chi_{s} \mu_g \hat{\textbf{e}}_z + \frac{1}{a_0^3} \int_{-\infty}^t d t' \bigg[\chi(t-t')-\frac{3a_1}{a_0}\exp[-\alpha t]\chi(t-t')\bigg] \Theta(t')\Delta\mu \hat{\textbf{e}}_z,\end{aligned}$$ where $\chi_{s}=\int_{-\infty}^t dt' \chi(t-t') = \frac{2(\epsilon_s-1)}{2\epsilon_s+1}$ and $\Delta\mu=\mu_e-\mu_g$. The first term in Eq. (\[eq3.3.4\]) describes the static reaction field before excitation when the solvent is in equilibrium with the ground state dipole moment $\boldsymbol{\mu}_g$. The second term is the change of the reaction field after the sudden change of the dipole moment to $\boldsymbol{\mu}_e$ which the solvent has to readjust to. In addition, due to the excitation, the molecular radius begins to shrink from its initial value $a_0+a_1$ to $a_0$, which gives rise to an additional explicit time-dependent contribution given by the third term.
At time $t$, the solute suddenly fluoresces and reaches again its ground state such that also its dipole moment goes back to $\boldsymbol{\mu}_g$. The solvent immediately reacts in the continuum’s approach with the fast (or optical) contribution $\epsilon_\infty$. As we set this value to $\epsilon_\infty =1$, there is no further contribution to the reaction field coming from the sudden dipole change arising from $\chi_\infty=2a(t)^{-3}(\epsilon_\infty-1)(2\epsilon_\infty+1)^{-1}$.
The difference of the solvation energies between the excited state and the ground state of the molecule at time $t$ thus readily follows as $$\begin{aligned}
\label{eq3.3.5}
\Delta E_{solv}(t)=&-\boldsymbol{\mu}_e\textbf{R}(t)+\boldsymbol{\mu}_g\textbf{R}(t) \\ \notag
=& -\frac{1}{a_0^3}\mu_g \chi_s \Delta \mu - \frac{\Delta\mu^2}{a_0^3} \int_{-\infty}^t d t' \bigg[\chi(t-t')-\frac{3a_1}{a_0}\exp[-\alpha t]\chi(t-t')\bigg] \Theta(t') \\ \notag
=& -\frac{1}{a_0^3}\mu_g \chi_s \Delta \mu + \Delta E (t),\end{aligned}$$ where only the second term is time-dependent such that the first term cancels out in determining $S(t)$ of Eq. (\[eq3.3.1\]). Next, we use the complex Fourier transform $f(z)=\int_{-\infty}^\infty dt \, e^{-izt} f(t)=\mathcal{F}[f(t)]$, where $f(z)$ is analytic for $\rm{Im}(z)<0$ and $f(t\to \infty) < \infty$. Applied to $\Delta E(t)$ and by using the convolution theorem and the Fourier transform of the Heaviside function $\mathcal{F}[\Theta(t)]=1/iz$, we find $$\begin{aligned}
\label{eq3.3.6}
\Delta E(z)=-\frac{\Delta\mu^2}{a_0^3} \bigg[\frac{\chi(z)}{iz}-\frac{3a_1}{a_0}\frac{\chi(z-i\alpha)}{i(z-i\alpha)}\bigg],\end{aligned}$$ where $z=\omega-i\eta$ with $\omega$ real and $\eta$ representing a small positive number [@hsu1997]. The inverse transform of Eq. (\[eq3.3.6\]) for $t\geq0$ leads to $$\begin{aligned}
\label{eq3.3.7}
\Delta E(t)=&-\frac{\Delta\mu^2}{2\pi a_0^3} \int_C dz e^{izt}\bigg[\frac{\chi(z)}{iz}-\frac{3a_1}{a_0}\frac{\chi(z-i\alpha)}{i(z-i\alpha)}\bigg] \\ \label{eq3.3.8}
=&-\frac{4\Delta\mu^2}{\pi a_0^3} \int_0^{\infty} d\omega \frac{\cos[\omega t]}{\omega}\Im\bigg[\frac{\epsilon(\omega)-1}{2\epsilon(\omega)+1}\bigg]-\frac{\Delta\mu^2 \chi_s}{a_0^3}\bigg[1-3\frac{a_1}{a_0}e^{-\alpha t}\bigg] \\
\label{eq3.3.9}
=&\frac{\Delta \mu^2}{a_0^3}\chi_s\bigg\{ e^{-\omega_D t}-\bigg[ 1-3\frac{a_1}{a_0}e^{-\alpha t} \bigg]\bigg \}. \end{aligned}$$ The contour $C$ of integration in Eq. (\[eq3.3.7\]) is a path parallel to but slightly below the real axis in the complex plane. Further details for calculating the first term in Eq. (\[eq3.3.7\]) are given in the Supporting Information and in Ref. [@hsu1997] . Moreover, we apply the theorem of residues to the second term for the singularity at $z=i\alpha$. We use the Debye form of the dielectric function $\epsilon(\omega)=1+\frac{\epsilon_s-1}{1+i\omega \tau_D}$ to evaluate the integral in Eq. (\[eq3.3.8\]). The resulting form $\omega_D=(2\epsilon_s+1)/(3\tau_D)=\tau_L^{-1}$ in Eq. (\[eq3.3.9\]) is the inverse dipolar longitudinal relaxation time [@bag1984].
By this, we obtain the final result of the time-dependent fluorescence Stokes shift according to Eq. (\[eq3.3.1\]) in the form $$\begin{aligned}
\label{eqStokes}
S(t)&=\frac{1}{1+Q} e^{-\omega_D t} + \frac{Q}{1+Q} e^{-\alpha t} \\
& \simeq (1-Q )e^{-\omega_D t} + Q e^{-\alpha t} \, , \notag\end{aligned}$$ where $Q=3\frac{a_1}{a_0}$.
Results {#StokesResults}
=======
![\[figStokes\] Nonequilibrium time-dependent fluorescence Stokes shift due to a suddenly changing dipole moment in an shrinking Onsager cavity for different shrinking rates $\alpha$ for $a_1=0.01a_0$. The dashed red line shows the time-dependent Stokes shift in a static Onsager cavity of constant radius $a_0$.](Fig1.eps){width="\textwidth"}
The nonequilibrium time-dependent Stokes shift of Eq. (\[eqStokes\]) reveals a bi-exponential decay where the first term reflects the collective bulk water relaxation while the second term describes the motional changes at the solute-solvent interface associated with a possible self-motion of the solute and the corresponding shrinking rate $\alpha$. Interestingly, also the first term related to the bulk water relaxation carries information about the time-dependence of the solute via the prefactor proportional to $Q$. Fig. \[figStokes\] shows the nonequilibrium time-dependent Stokes shift for different shrinking rates, which are equal to (black-star line), larger (black-diamond line) or smaller (red-diamond line) than the inverse longitudinal relaxation time. For $\alpha > \omega_D$, the bulk contribution dominates and the rapidly decaying contribution of the solute motion is almost negligible. When $\alpha < \omega_D$, the long-term tail of $S(t)$ is dominated by the solute’s motion and the bulk contribution has decayed rapidly. For the special case $\alpha = \omega_D$, the two contributions cannot be separated in the Stokes shift.
![\[figStokes2\] (a) Average relaxation time $\langle \tau \rangle$ for the time-dependent fluorescence Stokes shift in dependence of the relative radial change $a_1/a_0$ for different shrinking rates $\alpha$. (b) Average relaxation time $\langle \tau \rangle$ in dependence of the shrinking rate $\alpha$ for different relative radial changes $a_1/a_0$.](Fig2.eps){width="\textwidth"}
Next, we determine the average relaxation time $$\begin{aligned}
\label{eqavtime}
\langle \tau \rangle & =\frac{1}{1+Q}\tau_L+\frac{Q}{1+Q} \alpha^{-1} \\
& \simeq (1-Q )\tau_L + Q \alpha^{-1} \, , \notag\end{aligned}$$ for the time-dependent Stokes shift of Eq. (\[eqStokes\]), where the prefactors sum up to $1$. Fig. \[figStokes2\] (a) shows different modes of behavior for the average relaxation time with relative changes of the Onsager radius $a_1/a_0$. For $\alpha<\omega_D$, the shrinking is slower than the longitudinal relaxation which highly increases the average relaxation time in comparison to bulk water relaxation determined by $\tau_L$. In this regime, the shrinking exceeds the conventional water relaxation process and becomes directly measurable via the slow long-time decay of the time-dependent fluorescence spectrum. With a larger absolute change of the Onsager radius $a_0+a_1\to a_0$, the average relaxation time grows further because the shrinking process is noticeable for longer times. When $\alpha=\omega_D$ the average relaxation time shows the bulk water property $\tau_L$ although a shrinking occurs. In this special case, the interfacial solute-solvent motion remains undisclosed. A faster shrinking for $\alpha>\omega_D$ leads to a smaller average relaxation time in comparison to bulk water. Now, the fast Onsager radial motion reduces the overall relaxation which decays further with relative radial change $a_1/a_0$. Interestingly enough, the relative impact of a slowly shrinking Onsager cavity on the average relaxation time is more pronounced than for a rapidly shrinking sphere. Fig. \[figStokes2\] (b) shows the diverging increase of $\langle \tau \rangle$ for small shrinking rates $\alpha\ll \omega_D$. Such a behavior can in principle be recorded by a long-tail decay in the associated fluorescence spectrum which ranges up to several hundred ps [@pal2004]. Thus, the slow solute-solvent interfacial motion by, e.g., self-motion of a protein can become directly detectable. For the fast shrinking $\alpha >\omega_D$, the average relaxation time approaches quickly its steady-state value $\langle \tau \rangle\simeq (1-Q) \tau_L$. In this regime, the faster shrinking has no further impact on the relaxation time and one can consider the fluctuating solvent to be “quasi-instantaneously” close to the solute. It is evident, that this slightly reduces the average relaxation time, see also Refs. [@nal2014; @kir2018].
Discussion
==========
An example for a bimodal experimental fit to the measured time-dependent fluorescent Stokes shift is given by the dye Hoechst 33258 in solution with DNA . The data show clear evidence of different motional contributions to relaxation. The experimental data for the Stokes shift confirm that relaxation due to the fluctuations in the bulk water solvating the dye bound to DNA is slowed down, but contributes to the fast relaxation times ($0.2$ ps and $1.2$ ps). On the other hand, the DNA self-motion, which occurs on a time scale of $\sim$20 ps, modifies the long-time components ($1.4$ ps and $19$ ps) of the solvation response [@fur2008; @fur2010].
We have shown that the standard Onsager continuum model for the time-dependent Stokes shift can be generalized to a generic model which includes an explicit time-dependent motion in the interfacial solute-solvent region. By this, we are able to generalize the Bagchi-Oxtoby-Fleming theory towards nonequilibrium. The change in the solute radius upon photoexcitation drives nearby solvent molecules into locations that are never occupied at equilibrium. Clearly, the proposed dielectric model is still phenomenological, but provides an intuitive picture of the involved contributions to the relaxation dynamics. Our theory can, in principle, be extended to cases where the solute has different geometrical shapes such as spheroids or where the solvent is a mixture of polar solvent molecules by adapting dielectric constants. Furthermore, the dielectric solvent incorporates important solute-solvent interactions which are commonly neglected in molecular modeling [@bag2010]. The explicit inclusion of electronic and collective polarization in MD simulations, for example, is numerically expensive, especially from the point of view of the actual accurate force field parametrization [@kirb2019]. The used continuum model incorporates long-range electrostatic interactions as well as polarization and, in addition, gives a configurationally sampled solvent effect avoiding delicate statistical sampling averages. The frequency-dependent dielectric function entering in fluorescence Stokes shift in Eq. (\[eqStokes\]) automatically accounts for time averages [@men2010]. In addition, microscopic models where the motion of individual molecules is incorporated may be more accurate, but only for the specific situation under investigation. Yet, molecular dynamics simulations have successfully examined the origin of the Stokes shift and have revealed numerically that the bulk water component dominates at short times but rapidly decays. The long-time behavior is dominated by the solute dynamics, i.e., mostly the self-motion of a dissolved protein [@nil2005], such as DNA solvation dynamics [@muk2018]. The proposed nonequilibrium continuum model clearly reveals this mechanism in a simple and transparent manner and agrees with this observation. It also confirms experimental measurements of fluorescence Stokes shifts which show two (or more) components of decaying exponentials with different time scales involved.
On the whole, different spectroscopic decay times may be assigned to specific motions in a more complex system. However, the complexity of the relaxation channels and their interplay in a macromolecule in solution make it often hard to interpret the various contributions to the time-dependent Stokes shift. In the study of the hydration dynamics, for example, one can attribute the dynamic exchange of ’bound’ and ’free’ water molecules between the hydration shell and the bulk water to a slow relaxational component, but also to self motion of the protein or its side chains. Also, a coupled motion of both dynamical effects could occur [@bag2010]. Our proposed time-dependent continuum model explicitly confirms the coupling of different motional contributions related by the prefactor $Q$ where the time-dependent configurational changes of the solute enters. In total, we suggest a combination of exact microscopic modeling of distinct solute-solvent motions and explicit dynamical dielectric continuum models to team the strengths of both. Then, solvation dynamics and the resulting solvent effects in physical and chemical processes in liquids can be further elucidated even in highly complex biological systems like proteins.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the DFG-Sonderforschungsbereich 925 “Light-induced dynamics and control of correlated quantum systems” (A4, project number 170620586).
Supporting Information {#supporting-information .unnumbered}
======================
The Supporting Information includes the detailed calculation of the first term in Eq. (\[eq3.3.7\]).
Laage, D.; Elsaesser, T.; Hynes, J. T. Water Dynamics in the Hydration Shells of Biomolecules.*Chem. Rev.* [**2017**]{}, *117*, 10649-10725.
Nitzan, A. *Chemical Dynamics in Condensed Phases*; Oxford University Press: Oxford, U.K.; 2006.
Jimenez, R.; Fleming, G. R.; Kumar, P.; Maroncelli, M. Femtosecond Solvation Dynamics of Water. *Nature* [**1994**]{}, *369*, 471-473.
Rosenthal, S. J.; Xie, X.; Du, M.; Fleming, G. R. Femtosecond solvation dynamics in acetonitrile. Observation of the Inertial contribution to Solvent Response. *J. Chem. Phys.* [**1991**]{}, *95*, 4715-4718.
Marcus, R. A. Chemical and Electrochemical Electron-Transfer Theory. *Annu. Rev. Phys. Chem.* **1964**, *15*, 155-196.
Marcus, R. A. On the Theory of Electron-Transfer Reactions. VI. Unified Treatment for Homogeneous and Electrode Reactions. *J. Chem. Phys.* **1965**, *43*, 679-701.
Bagchi, B.; Oxtoby, D. W.; Fleming, G. R. Theory of the Time Development of the Stokes Shift in Polar Media. *Chem. Phys.* [**1984**]{}, *86*, 257-267.
Heid, E.; Braun, D. Fundamental Limitations of the Time-Dependent Stokes Shift for Investigating Protein Hydration dynamics. *Phys. Chem. Chem. Phys.* [**2019**]{}, *21*, 4435-4443.
Cerveny, S.; Swenson, J. Water Dynamics in the Hydration Shells of Biological and Non-Biological Polymers. *J. Chem. Phys.* **2019**, *150*, 234904.
Nilsoon, L.; Halle, B. Molecular Origin of Time-Dependent Fluorescence Shifts in Proteins. *Proc. Nat. Ac. Sc.* **2005** *39*, 13867-13872.
Franck, J. Elementary Processes of Photochemical Reactions. *Faraday Soc.* **1926**, *21*, 536-542.
Condon, E. U. A Theory of Intensity Distribution in Band Systems. *Phys. Rev.* **1926**, *28*, 1182-1201.
Oshika, Y. Absorption Spectra of Dyes in Solution. *J. Phys. Soc. Jpn.* **1954**, *9*, 594-602.
Mataga, N.; Kaifu, Y.; Koizumi, M. Solvent Effects upon Fluorescence Spectra and the Dipolemoments of Excited Molecules. *Bull. Chem. Soc. Jpn.* **1956**, *29*, 465-470.
Lippert, E. Spektroskopische Bestimmung des Dipolmomentes aromatischer Verbindungen im ersten angeregten Singulettzustand. *Z. Electrochem.* **1957**, *61*, 962-975.
Debye, P. Zur Theorie der annomalen Dispersion im Gebiet der langwelligen elektrischen Strahlung. *Verh. d. Deutsch. Phys. Ges.* **1913**, *15*, 777-793.
Maroncelli, M.; Fleming, G. R. Computer Simulation of the Dynamics of Aqueous Solvation. *J. Chem. Phys.* **1988**, *89*, 5044-5069.
Onsager, L. after Jortner, J.; Gathoon, A. Effects of Phase Density on Ionization Processes and Electron Localization in Fluids. *Can. J. Chem.* **1977**, *55*, 1801-1819.
Bagchi, B.; Chandra, A. Ultrafast Solvation Dynamics: Molecular Explanation of Computer Simulation Results in a Simple Dipolar Solvent. *J. Chem. Phys.* **1992**, *97*, 5126-5131.
Bagchi, B.; Chandra, A. Polarization Relaxation, Dielectric Dispersion, and Solvation Dynamics in Dense Dipolar Liquid. *J. Chem. Phys.* **1989**, *90*, 7338-7345.
Bagchi, B.; Chandra, A. The Role of Translational Diffusion in the Polarization Relaxation in Dense Polar Liquids. *Chem. Phys. Lett.* **1988**, *151*, 47-53.
Bagchi, B. *Molecular Relaxation in Liquids*; Oxford University Press: New York, 2012.
Neria, E.; Nitzan, A. Semiclassical Evaluation of Nonadiabatic Rates in Condensed Phases. *J. Chem. Phys.* [**1993**]{}, *99*, 1109-1123.
Onsager, L. Electric Moments of Molecules in Liquids. *J. Am. Chem. Soc.* **1936**, *58*, 1486-1493.
Aherne, D.; Tran, V.: Nonlinear, Nonpolar Solvation Dynamics in Water: The Roles of Electrostriction and Solvent Translation in the Breakdown of Linear Response. *J. Phys. Chem. B* **2000**, *104*, 5382-5394.
Kirchberg, H.; Nalbach, P.; Thorwart, M. Nonequilibrium Quantum Solvation with a Time-Dependent Onsager Cavity. *J. Chem. Phys.* **2018**, *148*, 164301.
Chandler, D. *Introduction to Modern Statistical Mechanics*; Oxford University Press: New York, 1987.
Gilmore, J.; R. H. McKenzie, R. H. Spin Boson Models for Quantum Decoherence of Electronic Excitations of Biomolecules and Quantum Dots in a Solvent. *J. Phys.: Condens. Matter* **2005**, *17*, 1735-1746.
Nalbach, P.; Achner, A. J. A.; Frey, M.; Grosser, M.; Bressler, C., Thorwart, M. Hydration Shell Effects in the Relaxation Dynamics of Photoexcited Fe-II Complexes in Water. *J. Chem. Phys.* **2014**, *141*, 044304.
Hsu, C.-P.; Song, X.; Marcus, R. A. Time-Dependent Stokes Shift and Its Calculation from Solvent Dielectric Dispersion Data. *J. Phys. Chem. B* **1997**, *101*, 2546-2551.
Bedard-Hearn, M.; Larsen, R. E.; Schwartz, B. J. Understanding Nonequilibrium Solute and Solvent Motions through Molecular Projections: Computer Simulations of Solvation Dynamics in Liquid Tetrahydrofuran (THF). *J. Phys. Chem. B* **2003**, *107*, 14464-14475.
Mennucci, B. Continuum Solvation Models: What Else Can We Learn from Them? *J. Phys. Chem. Lett.* **2010**, *1*, 1666-1674.
Pal, S. K.; Zewail, A. H. Dynamics of Water in Biological Recognition. *Chem. Rev.* **2004**, *104*, 2099-2124.
Bagchi, B.; Jana, B. Solvation Dynamics in Dipolar Liquids. *Chem. Soc. Rev.* **2010**, *39*, 1936-1954.
Kirby, B. J.; Jungwirth, P. Charge Scaling Manifesto: A Way of reconciling the Inherently Macroscopic and Microscopic Natures of Molecular Simulations. *J. Phys. Chem. Lett.* **2019**, *10*, 7531-7536.
Furse, K. E.; Corcelli, S. A. The Dynamics of Water at DNA Interfaces: Computational Studies of Hoechst 33258 Bound to DNA. *J. Am. Chem. Soc.* **2008**, *130*, 13103-13109.
Furse, K. E.; Corcelli, S. A. Effects of an Unnatural Base Pair Replacement on the Structure and Dynamics of DNA and Neighboring Water and Ions. *J. Phys. Chem. B* **2010**, *114*, 9934-9945.
Mukherjee, S.; Modal, S.; Archarya, S.; Bagchi, B. DNA Solvation Dynamics. *J. Phys. Chem. B* **2018**, *122*, 11743-11761.
|
---
abstract: 'Recent discoveries of strongly misaligned transiting exoplanets pose a challenge to the established planet formation theory which assumes planetary systems to form and evolve in isolation. However, the fact that the majority of stars actually do form in star clusters raises the question how isolated forming planetary systems really are. Besides radiative and tidal forces the presence of dense gas aggregates in star-forming regions are potential sources for perturbations to protoplanetary discs or systems. Here we show that subsequent capture of gas from large extended accretion envelopes onto a passing star with a typical circumstellar disc can tilt the disc plane to retrograde orientation, naturally explaining the formation of strongly inclined planetary systems. Furthermore, the inner disc regions may become denser, and thus more prone to speedy coagulation and planet formation. Pre-existing planetary systems are compressed by gas inflows leading to a natural occurrence of close-in misaligned hot Jupiters and short-period eccentric planets. The likelihood of such events mainly depends on the gas content of the cluster and is thus expected to be highest in the youngest star clusters.'
author:
- |
I. Thies$^{1}$[^1], P. Kroupa$^{1}$[^2], S. P. Goodwin$^{2}$, D. Stamatellos$^{3}$, A. P. Whitworth$^{3}$\
$^{1}$Argelander-Institut für Astronomie (Sternwarte), Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany\
$^{2}$Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK\
$^{3}$School of Physics & Astronomy, Cardiff University, Cardiff CF24 3AA, UK
title: 'A natural formation scenario for misaligned and short-period eccentric extrasolar planets'
---
hydrodynamics — planets and satellites: formation — planet–disc interactions — planet–star interactions — protoplanetary disks — open clusters and associations: general
Introduction {#sec:intro}
============
The discoveries of retrograde or strongly misaligned transiting exoplanets like WASP-17b and others [@Heetal08; @Gillon09; @Joetal09; @Naritaetal09; @Pontetal09; @Pontetal10; @Winetal09a; @Winetal09b; @wasp17b_01; @Triaudetal2010] pose a challenge for established planet formation theories. These classically assume the birth of planetary systems out of collapsing protostellar cloud fragments. While contracting from sub-parsec to milliparsec (or hundreds of AU) scale the conservation of its initial angular momentum causes the cloud to spin and flatten, while part of its mass settles towards the center, eventually igniting the hydrogen burning in the stellar core. Given such a protostar with a circumstellar disc that contains a certain amount of dust, solid particles collide and stick together, forming larger particles. This coagulation leads to bodies (cores) of sufficient mass to accrete surrounding material (dust and even gas) by gravity until no more disc material is available. This mechanism is probably the dominant process of the formation of the Solar System, including the Earth [@2008ASPC..398..235M]. As an alternative, gravitational instability as a forming mechanism for giant planets and brown dwarfs is being discussed [@2004ApJ...610..456B], and at least for brown dwarfs it has actually been shown to work [@StHuWi07; @Thiesetal2010]. More recent variations of these models assume gravitational instability of the solid phase only or dust trapping in vortices as a speed-up for coagulation.
Being isolated from any external perturbation, everything inside this protostar-disc system spins in the same direction, thus the forming planets orbit their host star the same way. This simple model has been severely questioned by the findings mentioned above. The transiting exoplanet WASP-17b [@wasp17b_01], for example, has been found to have a sky-projected inclination of the orbital plane normal against the stellar spin axis of $148.5{{}^{+5.1}_{-4.2}}$ degrees. A number of other mis-aligned transiting exoplanets have been discovered [@Heetal08; @Gillon09; @Joetal09; @Naritaetal09; @Pontetal09; @Pontetal10; @Winetal09a; @Winetal09b; @wasp17b_01]. These discoveries suggest that misalignment between planetary orbits and the spin of their host star (hereafter called spin-orbit misalignment) are quite common, at least among close-in transiting planets for which the spin-orbit alignment can be measured by using the Rossiter-McLaughlin Effect [@Rossiter1924; @McLaughlin1924]. The nearby $\upsilon$ And planetary system exhibits even mutually inclined planetary orbits [@2010ApJ...715.1203M].
Such planetary orbits cannot be explained by the standard planet formation model according to which the conservation of angular momentum forces any coplanar system to remain coplanar forever. There have been recent suggestions that may shed some light into the possible mechanism. [@FabTre07; @Naozetal2011] deduce a formation scenario for misaligned “hot jupiters” through long-term mutual perturbations of two planets (or an inner planet and an outer brown dwarf; for simplicity, we use the term “planet” for all substellar companions in this paper) orbiting the same star. Given an initial mutual inclination between 40 and 140 the Kozai-Lidov mechanism leads to secular oscillations of eccentricity vs. inclination. Tidal friction circularises the orbit of the inner planet if its periastron falls below a few stellar radii, eventually leaving the planet on a close-in orbit with near-random inclination with respect to the stellar spin. This scenario, however, requires a sufficient initial mutual inclination of both planets.
Another possible mechanism is tilting the spin axis of the star. Direct stellar encounters within a stellar radius (about 0.01 AU) can be effectively ruled out, because they are improbable, even within stellar clusters [@TKT05; @TK07] and would destroy the disc. Secular transfer of angular momentum between the protostellar spin and the protoplanetary disc via magnetic fields have been investigated by @Laietal11.
Multi-stage or episodic accretion of circumstellar material may provide another viable mechanism for misaligned planetary systems. The underlying idea is that accretion from different sources (i.e. gas filaments, accretion envelopes) from different directions may sever the classically assumed spin-orbit correlation. The star itself may get its angular momentum from a different accretion event than the bulk of circumstellar material. @BLP10 have analysed turbulent accretion events using data from @BBB03. They deduce a high frequency of misaligned planets as a consequence of multi-directional accretion. However, due to the limited simulation data available, they were forced to make simplified assumptions to what happens within the close vicinity of a star. Our work aims to take a deeper insight into such scenarios by self-consistent calculations of close interactions of a circumstellar disc with encountered new material, and by also treating the effects on pre-existing planets.
As @PTS08 [@OPE10] have shown, encounters between stars and pre-/protostellar objects do occur frequently in dense stellar environments and may lead to significant accretion bursts. Although the precise mechanisms proposed [@PTS08] differ from ours, their calculations emphasise the importance of encounter events to accretion processes.
In this paper the effects of gas capture onto a pre-existing circumstellar disc are studied. In Section \[sec:model\] we depict the physical scenario while Section \[sec:methods\] briefly describes the numerical methods used in our computations. The results are presented in Section \[sec:results\].
Model {#sec:model}
=====
In our model, both the initial mutual inclination as well as the inclination with respect to the stellar equator can be naturally explained by multi-stage accretion during the formation process of the planetary system. The scenario can be described as follows: First, a star with a protoplanetary disc (PPD) is forming the classical way, i.e. out of a collapsing spinning cloud fragment. Soon after this the protostar passes another region in the star cluster that contains dense gas, e.g. the protostellar cloud fragment or accretion envelope of another star, hereafter called the target. Accordingly, the star which encounters the target and its disc are referred to as the bullet star and the bullet disc, respectively. In our calculations, a massive circumstellar disc around another star is used as the target. While grazing the outer regions of this target the star captures material from it at an angle that depends on the encounter orbit and the arbitrarily chosen orientation of the target and the bullet disc. Since there is no correlation between the orientations of the bullet disc, the target and the encounter orbit, this configuration reflects an uncorrelated or random in-flow of material. The subsequent evolution mainly depends on the amount and infall angle of this additional material, and also on the protoplanetary evolution stage during the capture.
In our calculations we focus on the case of an encounter before any planet has formed in the passing star’s disc. It is thus a purely gaseous interaction computed via smoothed particle hydrodynamics (SPH). In our model, the star is a sun-type star with a low-mass PPD (${M_\mathrm{d}}\le0.1\,{M_{\sun}}$, radius $<100$ AU), passing through the target disc (0.5 ).
The initial conditions for the PPD are taken from @StaWhi08 [@StaWhi09a]. The disc model has power-law profiles for temperature, $T$, and surface density, $\Sigma$, from @StaWhi08, both as a function of the distance, $R$, from the central star $$\label{eq:sigma}
\Sigma(R)=\Sigma_0\,\left(\frac{R}{\mathrm{AU}}\right)^{-{q_{\Sigma}}}\,,$$ $$\label{eq:temp}
T(R)=\left[T_0^2\,\left(\frac{R}{\mathrm{AU}}\right)^{-2{q_{T}}}+T_\infty^2\right]^{1/2}\,,$$ where $1.5\le{q_{\Sigma}}\le1.75$ and $0.5\le{q_{T}}\le1.0$. $T_\infty=10$ K is a background temperature to account for background radiation from other stars within the host cluster. The radiation temperature of the stars are estimated assuming a mass-to-luminosity relation of $L\propto M^4$ for low-mass stars.
The disc is initially populated between 4 and 40 AU. Preceding the actual encounter the disc is allowed to “settle” during the approach time of about 5000 years (equivalent to 20 orbits at the outer rim), erasing artefacts from the initial distribution function. During this settling the disc smears out at the inner and the outer border extending its radial range from a few AU up to $\gtrsim 60$ AU. The actual surface density after the settling and thus the mass content within a given radius is therefore slightly smaller than in the initial setup.
Similarly, the target is set up as a massive extended disc with mass of 0.48 and a radius of $\approx400$ AU, as well as a 0.75 star to keep the gas gravitationally bound. The setup is therefore equivalent to those used in @Thiesetal2010, where triggered fragmentation inside this envelope upon tidal perturbations have been studied, and is indeed realistic as long as encounters with large circumstellar discs are considered. However, future models will also use different setups (protostellar cores, dense gas filaments etc.) as the target.
Both systems are initially set on a hyperbolic orbit around their centre of mass with the discs being inclined mutually as well as with respect to the orbital plane. The periapsis is set to 500 AU initially. The eccentricity is set to 1.1, corresponding to a pre-encounter relative velocity of about 0.7. The periapsis relative velocity is 2.9. The hyperbolic orbit is inclined 15 degrees against the target disc plane using Eulerian $z$-$x$-$z$ rotation angles 0, 15, 0, while the plane of the bullet disc (initially in the $x$-$y$ plane) is tilted by 0, 135, 60.
The eccentricity value is in agreement with the likelihood estimation of encounter parameters by @OPE10. In addition, encounters at distances of the order of a few hundred AU are likely to happen in clusters comparable to the Orion Nebula Cluster [@TKT05; @OPE10], thus our scenario is rather likely than exceptional. The actual periapsis and eccentricity may differ slightly due to tidal angular momentum transfer, but these effects on the orbit are negligible at that time. After the passage, when both stars have reached a sufficient distance ($\sim1000$ AU) again, the gas particles around the bullet star (i.e. the remains of its own disc and any captured gas) are separated from the system and analysed in the centre-of-mass system of the bullet star.
Numerical method {#sec:methods}
================
All computations were performed by using the well-tested smoothed-particle hydrodynamics (SPH) code [dragon]{} by @Goetal04 including the radiative heat transfer extension by @Staetal07. Most of the numerical parameter settings have been adopted from @StHuWi07 [@StaWhi09a]. The radiative-transfer algorithm is not strictly required in our model, but due to its high numerical efficiency it does not significantly slow down the calculations, so we left it in to keep it consistent with previous calculations by @Thiesetal2010. All gas particles have the same mass and equation of state. The artificial viscosity parameters are $\alpha=0.1$ and $\beta=0.2$, i.e. lower then the usually used values (1 and 2, respectively, @monaghan1992), to keep the disc dispersion low. Stars and planets are treated by sink particles which are set in the initial conditions, but may also form if the local volume density exceeds $10^{-9}{\mbox{$\,\mathrm{g\,cm^{-3}}$}}$ (not expected in the current model, however). The sink radius is chosen as 0.5 AU while the sink masses are the masses of the stars involved, i.e. 1.0 for the bullet star and 0.75 for the host star of the target disc. Any gas particle that becomes gravitationally bound to a sink within less than the sink radius is treated as being accreted by the sink. The code currently does not treat magnetic field nor dust.
The numerical model used here is the same as used in @Thiesetal2010, using 250 000 SPH particles for the target while the PPD of the passing star has been set up by 25 000 to 50 000 particles, for disc masses between 0.05 to 0.1 , respectively.
Results {#sec:results}
=======
![\[x100i\] Snapshot of a bullet star with a 0.1 circumstellar disc passing through a massive extended (0.5 , 400 AU) target disc on an inclined orbit. While the original disc is partially truncated, partially condensed to the center, and somewhat tilted, the bullet star accretes additional material from the target disc, forming an inclined annulus of gas around it (frames b to d). The initially strongly inclined structures progressively align each other within 30 000 years after the encounter (frames e, f). The time stamp in each frame refers to the time of the encounter.](fig1.eps){width="45.00000%"}
![\[twoalign\] The angular momentum unit vectors of the particles of the material around the bullet star in spherical coordinate angles, $\vartheta$ (latitude) and $\varphi$ (longitude), i.e. projected on the unit sphere. The reference frame is adjusted so that the normal vector of the bullet disc points at $(\varphi, \vartheta)=(0, 0)$. The topmost two frames show the angular momenta initially and at the time of the closest approach. While the original bullet star disc is altered only slightly there is clearly another population of particles settling in an annulus which is tilted against the initial disc plane. After formation, both structures progressively align to each other, and the angular difference between both degrades (subsequent frames). The total angular momentum vector of each disc is marked by a cross, while the open circle indicates the initial angular momentum vector of the bullet star disc before the encounter (being in fact nearly identical to that at the moment of the encounter, $t=0$). This also denotes the direction of the bullet’s star spin vector. The time stamp in each frame refers to the time of the encounter.](fig2.eps){width="45.00000%"}
General effects {#ssec:general}
---------------
Figure \[x100i\] shows six snapshots of the encounter and subsequent gas capture. The view is centred on the passing star (the “bullet star”) at face-on direction towards the unperturbed disc. The time stamp refers to the moment of the periastron passage. Frame (a) shows the unperturbed disc. Its core has a radius of about 40 AU while the low-density outer regions extent to about 100 AU. Shortly after the encounter, as shown in frame (b–d), the captured material starts to form an annulus around the star while, at the same time, the original bullet disc is partly truncated, with the inner region being condensed. The persistent quadrupole forces continue to alter the disc’s orientation and eventually tilt it into edge-on orientation (frames e and f). At the same time, the orientation of the annulus is only slightly changed due to its larger radius and thus higher moment of inertia. As clearly visible, both the captured annulus and the disc are progressively aligning each other, eventually forming a combined disc. The dynamical evolution of the mutual orientation and the total tilt of the original bullet disc is depicted in Figures \[twoalign\] and \[twoaligncurve\]. Here, the SPH particles have been separated into groups sharing a similar angular momentum unit vector and thus belonging to the same rotational structure. Particles which deviate more than 15 degrees from the bulk angular momentum as well as unbound or loosely bound particles beyond 100 AU from the bullet star are neglected. The mutual inclination of both structures is declining from initially 130 degrees to near zero degrees within 35 kyr. On the other hand, the original disc is tilted by nearly 100 degrees, i.e., [*if being initially aligned to the stellar equator it is now retrograde.*]{}
During the encounter, typically around 10 to 30 of gas is captured by the bullet star (Figure \[x100i\], frames b–d; Section \[sec:accrete\]). In the cases where a protoplanetary disc (hereafter called the original disc) is already present, two general effects can be distinguished: (i) During the capture process, the original disc is compressed to about one-half or one-third of the initial radius, accordingly increasing the surface density. (ii) The captured material settles in an outer annulus the inclination of which depends on the orbital orientation and the orientation of the target disc relative to the encounter orbit.
The orientation of this annulus, shown in Figure \[twoalign\] in spherical coordinates, is not constant, but changes over time. Within about a few thousand years after formation the annulus and the bullet disc have considerably lowered their mutual inclination (third and fourth frame), as shown in Figure \[twoaligncurve\]. A few ten thousand years after the encounter, they have almost aligned to each other, eventually forming a single circumstellar disc. Consequently, the orientation of this disc differs considerably from its initial orientation and therefore from the stellar equatorial plane. It should be noted that this tilt may be much lower if a steady and homogeneous gas flow is assumed [@MoeThr09].
![\[twoaligncurve\]The mutual angular tilt between the pre-existing disc of the bullet star and the annulus of captured gas, as well as the net angular tilt of the pre-existing disc as a function of time. While being almost retrograde wrt. each other before the encounter, the two structures progressively align with each other over time. 25 kyr after the encounter the mutual tilt has degraded to about 35 degrees. At the same time, the pre-existing disc has been tilted wrt. its initial orientation by about 90 degrees. It has become retrograde wrt. the bullet star’s spin after about 27 000 yr.](fig3.eps){width="45.00000%"}
Captured mass {#sec:accrete}
-------------
![\[accrete\]The mass of the pre-existing bullet star disc (solid curve) and of the captured annulus (dashed curve; see Section \[ssec:general\]) as a function of time. The peaks in the solid curve at about the moment of the closest encounter are due to the perturbations of the pre-existing disc caused by the impact of in-flowing gas and tidal forces by the target mass. The capture process starts shortly before the encounter and continues to drag material from the tidal arm (Figure \[x100i\]) even several kyr later. The slight decrease of mass of both structures is mainly caused by dynamical scattering and, in particular for the inner disc, accretion onto the star, but also by a small amount due the numerical diffusion that is unavoidable in SPH calculations. In a real disc the final masses would thus be somewhat larger.](fig4.eps){width="45.00000%"}
The amount of captured mass is an important quantity to judge its impact on planet formation. In our calculations, typically between 0.01 and 0.02 is captured from the target disc, i.e. about one third of the initial bullet star disc mass. Figure \[accrete\] depicts the time evolution of the masses of the bullet star disc and the captured annulus. As shown here, the capture of material from the target disc begins quickly about the closest encounter, and is almost finished within about 3000 yr. At the same time, the pre-existing disc suffers a radial compression (see frames b–d in Figure \[x100i\]). The distinction of these two structures has been done by the angular momentum grouping method described in Section \[ssec:general\]. It has to be noted that the curves in Figure \[accrete\] show a representative fraction of each gas body defined by this method. Since part of the gas particles are also subject to scattering into large separations ($>100$ AU) and/or inclinations ($>15$ degrees) these are omitted by this selection process, thus leading to a decreasing mass of each disc. These losses are in part due to dynamical scattering and frictional heating of the gas, in part due to accretion by the bullet star. To a certain degree, they are also due to numerical diffusion, which is unavoidable in SPH, especially for relatively low resolution, and may thus be reduced by using larger particle numbers in future calculations. However, the mass-loss does not affect the central conclusion of this paper in any way.
Pre-existing planets {#sec:planets}
--------------------
-------- ----------------------------------------- -------------------------------------- -------------------- ----------------------------------------- -------------------------------------- --------------------
Object $\frac{m_{\text{ini}}}{{M_\mathrm{J}}}$ $\frac{a_{\text{ini}}}{\text{[AU]}}$ ${e}_{\text{ini}}$ $\frac{m_{\text{fin}}}{{M_\mathrm{J}}}$ $\frac{a_{\text{fin}}}{\text{[AU]}}$ ${e}_{\text{fin}}$
1 1.0 5.2 0.05 2.8 1.6 0.67
2 0.30 9.5 0.05 3.1 5.1 0.20
3 0.046 19.2 0.05 3.2 6.5 0.70
4 0.054 30.1 0.01 2.2 2.2 0.89
-------- ----------------------------------------- -------------------------------------- -------------------- ----------------------------------------- -------------------------------------- --------------------
: \[x200aei\]Masses and orbital elements of four model planets before and after a flyby and capture event equivalent to the scenario in Fig. \[x100i\]. The planets, which replace the bullet star disc of the first scenario, are initially equivalent to Jupiter, Saturn, Uranus and Neptune. 20 kyr after the flyby, when the capture and accretion process is essentially over, the orbits are shrunk to a fraction of their initial values while being highly eccentric now. Furthermore, the orbital positions of planets 2–4 have been altered. The system has apparently become unstable.
We also studied the effects of such a gas capture event on a pre-existing planetary system. Here, the circumstellar disc of the bullet star has been replaced by a model planetary system equivalent Jupiter, Saturn, Uranus, and Neptune. Like the stars, the planets are represented by sink particles with a sink radius of 0.5 AU. This value is a compromise between spatial resolution and mass resolution. Gas particles passing through the sphere defined by the sink radius are considered as captured if they are gravitationally bound to the sink, otherwise they simply pass through it. The current results, summarised in Table \[x200aei\], show that a planetary system equivalent to the Sun orbited by Jupiter, Saturn, Uranus, and Neptune (with initial orbital radii of 5, 10, 20, and 30 AU, respectively) is severely affected by gas-planet interaction. During the capture some 0.01 of gas flow towards the bullet star, crossing the planetary orbits. In response, the planets migrate inward to semimajor axes of about one tenth their initial separations, i.e. between about 0.5 and 4 AU. In addition, some orbits get strongly eccentric (up to $e=0.8$, where $e=0$ means a circular orbit and $e\ge1.0$ means ejection). Such a system is dynamically unstable, i.e. some planets are probably ejected over time, typically leaving one to three planets, with the most massive planet on a close-in orbit. Systems like these are expected to undergo significant subsequent evolution. Even after ejection of the most unstable planets the remaining ones may still be subject to Kozai resonance, altering their eccentricities and inclinations over many orbital times. But it is clear that these results suggest that misaligned planets and hot Jupiters as well as short-period planets on eccentric orbits may both be an outcome of gas flow onto a forming or existing system.
Stellar spin
------------
The current work assumes that the stellar spin is not largely changed since the assembly of its circumstellar disc, and thus its angular momentum vector points into approximately the same direction as the angular momentum vector of the disc before the encounter. This assumption, however, may not be true in general. As @Laietal11 have found, the stellar spin may, in some cases, be tilted through magnetic interaction with the disc. Furthermore, during the early phases, when the protostar has still a considerable diameter (tenths of an AU), it may still re-align to the disc after an encounter event of the kind proposed in our paper. However, a large fraction of stars will already be compact enough to be effectively decoupled from angular momentum transfer with their disc, given the absence of strong magnetic torque. Since encounter events and gas capture are a natural consequence of star formation in dense environments, we conclude that they indeed do play an important role in forming mis-aligned planetary systems.
Resolution issues {#ssec:resolution}
-----------------
The required resolution strongly depends on the kind of scenario to be modelled. For example, for studying fragmentation the local Jeans mass has to be sufficiently resolved, meaning by at least about $~50$ SPH particles. The study of the global behaviour of gas masses without focus on fragmentation, on the other hand, requires much less resolution. In our model, the mass of each gas particle is $1.92{{}\cdot10^{-6}\,}\,{M_{\sun}}$ or 0.64 ${M_{\oplus}}$. The well-resolved mass therefore is about 32 ${M_{\oplus}}$ or twice the mass of Neptune. The transferred mass in both models (encounter of disc with target and planetary-system vs. target, respectively) is of the order of 0.01 or 5000 SPH particles, so mass resolution is not an issue here. For calculating the impact on dust components or to study vortex formation [@BaSo1995; @KlahrBodenheimer2004] this resolution is likely to be insufficient. These issues will be addressed in future work.
One could argue that the masses of Uranus and Neptune equivalents (planets 3 and 4, see Section \[sec:planets\]), which are roughly equal to the interacting mass required for efficient momentum transfer, are less-optimally resolved by only 20 to 25 SPH particles. The statistical scatter in terms of the number of gas particles interacting with these minor giant planets is about $\sqrt{1/25}=20\pct$ and therefore does not significantly influence the outcome of drag-induced migration. Furthermore, this behaviour is fully consistent with the migration of the Jupiter and Saturn equivalents (planets 1 and 2). A word of caution has, however, to be stated at this point: Apart from mere resolution issues the proper treatment of accretion onto sinks shows a number of pitfalls. For example, the void of material in the sink may lead to artificial suction effects and thus speed-up accretion. The time-evolution of the planetary orbits has therefore to be treated with great caution. The long-term outcome, however, largely depends on basic physical principles like conservation of angular momentum and can therefore be expected to reflect the real consequences on a pre-existing planetary system.
Discussion and conclusions {#sec:conclusions}
==========================
We have analysed the general effects of inclined gas capture onto a pre-existing circumstellar disc upon passage through a dense gaseous reservoir. The secondary gas capture does not destroy a pre-existing disc even if the new material inflows at a highly inclined angle. Rather, the inner regions of the disc (inside about 30 AU) become denser, while some of the outer disc material is scattered away. The major result is the formation of an inclined combined circumstellar disc from captured material and the pre-existing disc. After capturing a few 0.01 into an initially inclined annulus both structures tend to align with each other within a few 10 000 years after the encounter. An annulus with an initial inclination of $135{\mbox{\textdegree}}$ aligns to about $35{\mbox{\textdegree}}$ within 20 000 years. Due to conservation of angular momentum this results in a net shift of more than $90{\mbox{\textdegree}}$ in our model, and even more for lower-mass pre-existing discs or disc-less stars. If planets form from the resulting inclined disc one naturally arrives at misaligned planets.
The resulting disc may have two chemically distinct radial regions, the inner region which is associated with the star, and the outer region composed from the captured material.
In addition, the basic effect of capture onto a pre-existing planetary system have been estimated. Orbits typically shrink and become strongly eccentric in response to the contact with captured gas and subsequent planet-planet interaction, while their orbital planes can be dramatically altered thus naturally leading to (even mutually) misaligned short-period planets with eccentric orbits.
Important constraints are given by recent observations of apparent inclinations between circumstellar discs and their host stars. In a most recent analysis @Watsonetal2011 have found no significant misalignment between the normal vectors of debris discs and the spin axis of low-mass to solar-type stars. In particular, the projected inclinations between the star and its disc typically differ by 5–10 degrees, and 15–45 degrees in a few cases. Their results do not necessarily contradict our scenario since even moderate disc tilts and warps may lead to a Kozai resonance in a planetary system born out of such a disc. Following the mechanism described by @FabTre07 and @Naozetal2011 near-random misalignment of a close-in planet and the spin of its host star may result.
Future studies will also consider protoplanetary discs with embedded planets and systems with a brown dwarf on a wide orbit (about 100–200 AU). In addition, the consequences for dust content and mixing of differently composed material will be analysed.
The mechanism described in this paper provides a natural scenario for the formation of misaligned planetary systems in gas-rich dense star-forming regions. While other models have been proposed our scenario requires rather simple assumptions, and may even provide the initially misaligned planetary orbits required by the recent model of @Naozetal2011.
Acknowledgements {#acknowledgements .unnumbered}
================
This project has been funded by DFG grant KR1635/25 as part of the SPP1385.
[^1]: E-mail: ithies@astro.uni-bonn.de
[^2]: pavel@astro.uni-bonn.de
|
---
author:
- 'Duarte Fontes,'
- 'Jorge C. Romão,'
- 'Rui Santos,'
- 'and João P. Silva'
title: Large pseudoscalar Yukawa couplings in the complex 2HDM
---
Introduction {#sec:intro}
============
The discovery of the Higgs boson at the Large Hadron Collider (LHC) by the ATLAS [@ATLASHiggs] and CMS [@CMSHiggs] collaborations has ignited a very large number of studies in the context of multi-Higgs models. It is now clear that some features of the Higgs couplings to fermions and gauge bosons have to be well within the Standard Model (SM) predictions. Also, even if other heavy scalars are far from being experimentally excluded, there is still no hint of scalar particles other than the 125 GeV one. However, even if no large deviations from the SM were found, many of its extension are still in agreement with all experiment data. Many models provide interesting scenarios that can be probed at the next LHC run while contributing to solve some of the outstanding problems in particle physics. Such is the case of the complex two-Higgs double model (C2HDM). The 2HDM was first proposed by T. D. Lee [@Lee:1973iz] as an attempt to understand the matter-antimatter asymmetry of the universe (the 2HDM is described in detail in [@hhg; @ourreview]).
The 2HDM is a simple extension of the SM where the potential is still invariant under $SU(2) \times U(1)$ but is now built with two complex scalar doublets. The complex two-Higgs doublet model is the version of the model that allows for CP-violation in the potential, providing therefore an extra source of CP-violation to the theory. The existing experimental data and in particular the one recently analysed at the LHC has been used in several studies with the goal of constraining the parameter space of the C2HDM [@Barroso:2012wz; @Inoue:2014nva; @Cheung:2014oaa; @Fontes:2014xva] or just the Yukawa couplings [@Brod:2013cka].
The main purpose of this work is to analyse scenarios in the C2HDM that deviate from the SM predictions, while being in agreement with all available experimental and theoretical constraints. These are scenarios where the scalar component of the Higgs coupling to leptons or to b-quarks vanishes. The respective pseudoscalar component has to be non-zero which does not necessarily imply a very large CP-violating parameter. Even if the scalar component is not exactly zero, there are still Yukawa couplings where the pseudoscalar component can be much larger than the corresponding scalar component.
We will start by discussing the status of the C2HDM. Presently the processes $pp \to h \to WW (ZZ)$, $pp \to h \to \gamma \gamma$ and $pp \to h \to \tau^+ \tau^-$ are measured with an accuracy of about $20$%. On the other hand $pp \to V (h \to b \bar b)$ has been measured at the Tevatron and at the LHC with an accuracy of about $50$% [@cms:bb; @Tuchming:2014fza] while for $pp \to h \to Z \gamma$ an upper bound of the order of ten times the SM expectation at the $95$% confidence level was found [@atlas:Zph; @cms:Zph]. In order to understand how the model will perform at the end of the next LHC run we use the expected precisions on the signal strengths of different Higgs decay modes by the ATLAS [@ATLASpred] and CMS [@CMSpred] collaborations (see also [@Dawson:2013bba]) for $\sqrt{s}=14$ TeV and for 300 and 3000 $fb^{-1}$ of integrated luminosities. As previously shown in [@Fontes:2014xva], the final states $VV$, $\gamma \gamma$ and $\tau^+ \tau^-$ are enough to reproduce quantitatively the effect of all possible final states in the Higgs decay. Therefore, taking into account the predicted precision for the signal strength, we will consider the situations where, at $13$ TeV, the rates are measured within either $10$% or $5$% of the SM prediction. We should note that no difference can be seen in the plots when the energy is changed from $13$ to $14$ TeV as discussed in [@Fontes:2014xva].
This paper is organized as follows. In Section \[sec:model\], we describe the complex 2HDM and the constraints imposed by theoretical and phenomenological considerations including the most recent LHC data. In Section \[status\] we discuss the present status of the model and in Section \[zero\] we discuss the scenarios where the pure scalar component of the Yukawa coupling is allowed to vanish. Our conclusions are presented in Section \[sec:conc\].
The complex 2HDM {#sec:model}
================
The complex 2HDM was recently reviewed in great detail in [@Fontes:2014xva] (see also [@Ginzburg:2002wt; @Khater:2003wq; @ElKaffas:2007rq; @ElKaffas:2006nt; @Grzadkowski:2009iz; @Arhrib:2010ju; @Barroso:2012wz]). Therefore, in this section we will just briefly describe the main features of the complex two two-Higgs doublet with a softly broken $Z_2$ symmetry $\phi_1 \ra \phi_1, \phi_2 \ra -\phi_2$ whose scalar potential we write as [@ourreview] V\_H &=& m\_[11]{}\^2 |\_1|\^2 + m\_[22]{}\^2 |\_2|\^2 - m\_[12]{}\^2 \_1\^\_2 - (m\_[12]{}\^2)\^ \_2\^\_1\
&& + |\_1|\^4 + |\_2|\^4 + \_3 |\_1|\^2 |\_2|\^2 + \_4 (\_1\^\_2) (\_2\^\_1)\
&& + (\_1\^\_2)\^2 + (\_2\^\_1)\^2. \[VH\] All couplings except $m_{12}^2$ and $\lambda_5$ are real due to the hermiticity of the potential. The complex 2HDM model as first defined in [@Ginzburg:2002wt], is obtained by forcing $\textrm{arg}(\lambda_5) \neq 2\, \textrm{arg}(m_{12}^2)$ in which case the two phases cannot be removed simultaneously. From now on we will refer to this model as C2HDM. We choose a basis where the vacuum expectation values (vevs) are real. Whenever we refer to the CP-conserving 2HDM, not only the vevs, but also $m_{12}^2$ and $\lambda_5$ are taken real. Therefore, 2HDM refers to a softly broken $Z_2$ symmetric model where all parameters of the potential and the vevs are real. Writing the scalar doublets as \_1 = (
[c]{} \_1\^+\
(v\_1 + \_1 + i \_1)
), \_2 = (
[c]{} \_2\^+\
(v\_2 + \_2 + i \_2)
), with $v = \sqrt{v_1^2 + v_2^2} = (\sqrt{2} G_\mu)^{-1/2} = 246$ GeV, they can be transformed into the Higgs basis by [@LS; @BS] (
[c]{} H\_1\
H\_2
) = (
[cc]{} c\_ & s\_\
- s\_ & c\_
) (
[c]{} \_1\
\_2
), where $\tan{\beta} = v_2/v_1$, $c_\beta = \cos{\beta}$, and $s_\beta = \sin{\beta}$. In the Higgs basis the second doublet does not get a vev and the Goldstone bosons are in the first doublet.
Defining $\eta_3$ as the neutral imaginary component of the $H_2$ doublet, the mass eigenstates are obtained from the three neutral states via the rotation matrix $R$ (
[c]{} h\_1\
h\_2\
h\_3
) = R (
[c]{} \_1\
\_2\
\_3
) \[h\_as\_eta\] which will diagonalize the mass matrix of the neutral states via R [M]{}\^2 R\^T = (m\_1\^2, m\_2\^2, m\_3\^2 ), and $m_1 \leq m_2 \leq m_3$ are the masses of the neutral Higgs particles. We parametrize the mixing matrix $R$ as [@ElKaffas:2007rq] R = (
[ccc]{} c\_1 c\_2 & s\_1 c\_2 & s\_2\
-(c\_1 s\_2 s\_3 + s\_1 c\_3) & c\_1 c\_3 - s\_1 s\_2 s\_3 & c\_2 s\_3\
- c\_1 s\_2 c\_3 + s\_1 s\_3 & -(c\_1 s\_3 + s\_1 s\_2 c\_3) & c\_2 c\_3
) \[matrixR\] with $s_i = \sin{\alpha_i}$ and $c_i = \cos{\alpha_i}$ ($i = 1, 2, 3$) and - /2 < \_1 /2, - /2 < \_2 /2, - /2 \_3 /2. \[range\_alpha\] The potential of the C2HDM has 9 independent parameters and we choose as input parameters $v$, $\tan \beta$, $m_{H^\pm}$, $\alpha_1$, $\alpha_2$, $\alpha_3$, $m_1$, $m_2$, and $\textrm{Re}(m_{12}^2)$. The mass of the heavier neutral scalar is then given by m\_3\^2 = . \[m3\_derived\] The parameter space will be constrained by the condition $m_3 > m_2$. In order to perform a study on the light Higgs bosons we need the Higgs coupling to gauge bosons that can be written as [@Barroso:2012wz] C = c\_R\_[11]{} + s\_R\_[12]{} = , \[C\] and the Higgs couplings to a pair of charged Higgs bosons [@Barroso:2012wz] - = c\_R\_[11]{} + s\_R\_[12]{} + s\_c\_ (\_5) R\_[13]{}, \[lambda\_hHpHm\] where $\lambda_{145} = \lambda_1 - \lambda_4 - \textrm{Re}(\lambda_5)$ and $\lambda_{245} = \lambda_2 - \lambda_4 - \textrm{Re}(\lambda_5)$. Finally we also need the Yukawa couplings. In order to avoid flavour changing neutral currents (FCNC) we extend the $Z_2$ symmetry to the Yukawa Lagrangian [@GWP]. The up-type quarks couple to $\phi_2$ and the usual four models are obtained by coupling down-type quarks and charged leptons to $\phi_2$ (Type I) or to $\phi_1$ (Type II); or by coupling the down-type quarks to $\phi_1$ and the charged leptons to $\phi_2$ (Flipped) or finally by coupling the down-type quarks to $\phi_2$ and the charged leptons to $\phi_1$ (Lepton Specific). The Yukawa couplings can then be written, relative to the SM ones, as $a + i b \gamma_5$ with the coefficients presented in table \[tab:1\].
------------------- -- ---------------------------------------------------------------------------- -- ---------------------------------------------------------------------------- -- ---------------------------------------------------------------------------- -- ----------------------------------------------------------------------------
Type I Type II Lepton Flipped
Specific
Up $\tfrac{R_{12}}{s_{\beta}} - i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$ $\tfrac{R_{12}}{s_{\beta}} - i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$ $\tfrac{R_{12}}{s_{\beta}} - i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$ $\tfrac{R_{12}}{s_{\beta}} - i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$
\*\[2mm\] Down $\tfrac{R_{12}}{s_{\beta}} + i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$ $\tfrac{R_{11}}{c_{\beta}} - i s_\beta \tfrac{R_{13}}{c_{\beta}} \gamma_5$ $\tfrac{R_{12}}{s_{\beta}} + i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$ $\tfrac{R_{11}}{c_{\beta}} - i s_\beta \tfrac{R_{13}}{c_{\beta}} \gamma_5$
\*\[2mm\] Leptons $\tfrac{R_{12}}{s_{\beta}} + i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$ $\tfrac{R_{11}}{c_{\beta}} - i s_\beta \tfrac{R_{13}}{c_{\beta}} \gamma_5$ $\tfrac{R_{11}}{c_{\beta}} - i s_\beta \tfrac{R_{13}}{c_{\beta}} \gamma_5$ $\tfrac{R_{12}}{s_{\beta}} + i c_\beta \tfrac{R_{13}}{s_{\beta}} \gamma_5$
\*\[2mm\]
------------------- -- ---------------------------------------------------------------------------- -- ---------------------------------------------------------------------------- -- ---------------------------------------------------------------------------- -- ----------------------------------------------------------------------------
: \[tab:1\] Yukawa couplings of the lightest scalar, $h_1$, in the form $a + i b\gamma_5$.
From the form of the rotation matrix $R$ , it is clear that when $s_2=0$, the pseudoscalar $\eta_3$ does not contribute to the mass eigenstate $h_1$. It is also obvious that when $s_2=0$ the pseudoscalar components of all Yukawa couplings vanish. Therefore, we can state |s\_2| = 0 & & h\_1 , \[pure\_scalar\]\
|s\_2| = 1 & & h\_1 . \[pure\_pseudoscalar\]
There are however other interesting scenarios that could be in principle allowed. We could ask ourselves if a situation where the scalar couplings $a_{F} \approx 0$ ($F=U,D,L$) is still allowed after the 8 TeV run. As $a_U$ is fixed (the same for all Yukawa types) and given by $a_U = R_{12}/s_{\beta} = s_1 c_2/s_{\beta}$, it can only be small if $s_1 \approx 0$. If instead $c_2 \approx 0$ the $h_1 V V$ coupling $C$ in eq. would vanish which is already disallowed by experiment. There is one other coupling that could also vanish, which is $R_{11}/c_{\beta} = c_1 c_2/c_{\beta}$ (this is for example the expression for $a_D$ in Type II). Again this scalar part could vanish if $c_1 \approx 0$. In either case, $s_1 \approx 0$ or $c_1 \approx 0$, the important point to note is that the pseudoscalar component of the 125 GeV Higgs is not constrained by the choice of $\alpha_1$ because it depends only on $s_2$. We will discuss these scenarios in detail in section \[zero\].
Present status of the C2HDM {#status}
===========================
We start by briefly reviewing the status of the C2HDM after the 8 TeV run. We will generate points in parameter space with the following conditions: the lightest neutral scalar is $m_1 = 125$ GeV [^1], the angles $\alpha_{1,2,3}$ all vary in the interval $[-\pi/2, \, \pi/2]$, $1 \leq \tan{\beta} \leq 30$, $ m_1 \leq m_2 \leq 900\, \textrm{GeV}$ and $-(900\, \textrm{GeV})^2 \leq Re(m_{12}^2) \leq (900\, \textrm{GeV})^2$. Finally, we consider different ranges for the charged Higgs mass because the constraints from B-physics, and in particular the ones from $b \ra s \gamma$, affect differently Type II/F and Type I/LS. In Type II and F the range for the charged Higgs mass is $340\, \textrm{GeV} \leq m_{H^\pm} \leq 900\, \textrm{GeV}$ due to $b \ra s \gamma$ which forces $\mhpm \gtrsim 340$ GeV almost independently of $\tan \beta$ [@BB]. In Type I and LS the range is $100\, \textrm{GeV} \leq m_{H^\pm} \leq 900\, \textrm{GeV}$ because the constraint from $b \ra s \gamma$ is not as strong. The remaining constraints from B-physics [@Deschamps:2009rh; @gfitter1] and from the $R_b\equiv\Gamma(Z\to b\bar{b})/\Gamma(Z\to{\rm hadrons})$ [@Ztobb] measurement have a similar effect on all models forcing $\tan{\beta} \gtrsim 1$. The choice of the lower bound of 100 GeV is due to LEP searches on $e^+ e^- \to H^+ H^-$ [@Abbiendi:2013hk] and the latest LHC results on $pp \to \bar t \, t (\to H^+ \bar b )$ [@ATLASICHEP; @CMSICHEP]. Very light neutral scalars are also constrained by LEP results [@lepewwg].
All points comply to the following theoretical constraints: the potential is bounded from below [@Deshpande:1977rw], perturbative unitarity is enforced [@Kanemura:1993hm; @Akeroyd:2000wc; @Ginzburg:2003fe] and all allowed points conform to the oblique radiative parameters [@Peskin:1991sw; @Grimus:2008nb; @Baak:2012kk]. The signal strength is defined as $$\mu^{h_i}_f \, = \, \frac{\sigma \, {\rm BR} (h_i \to
f)}{\sigma^{\scriptscriptstyle {\rm SM}} \, {\rm BR^{\scriptscriptstyle{\rm SM}}} (h_i \to f)}
\label{eg-rg}$$ where $\sigma$ is the Higgs boson production cross section and ${\rm BR} (h_i \to f)$ is the branching ratio of the $h_i$ decay into the final state $f$; $\sigma^{\scriptscriptstyle {\rm {SM}}}$ and ${\rm BR^{\scriptscriptstyle {\rm SM}}}(h \to f)$ are the respective quantities calculated in the SM. The gluon fusion cross section is calculated at NNLO using HIGLU [@Spira:1995mt] together with the corresponding expressions for the CP-violating model in [@Fontes:2014xva]. SusHi [@Harlander:2012pb] at NNLO is used for calculating $b \bar{b} \ra h$, while $Vh$ (associated production), $t \bar{t} h$ and $VV \ra h$ (vector boson fusion) can be found in [@LHCCrossSections]. As previously discussed we will consider the rates for the processes $\mu_{VV}$, $\mu_{\gamma \gamma}$ and $\mu_{\tau \tau}$ to be within $20$% of the expected SM value, which at present roughly matches the average precision at $1\sigma$. It was shown in [@Fontes:2014xva] that taking into account other processes with the present attained precision has no significant impact in the results.
![$\alpha_2$ vs. $\alpha_1$ for Type I (left) and Type II (right). The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F1"}](plot_200_T1.pdf "fig:"){width="0.46\linewidth"} ![$\alpha_2$ vs. $\alpha_1$ for Type I (left) and Type II (right). The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F1"}](plot_200_T2.pdf "fig:"){width="0.46\linewidth"}
We start by examining the parameter space for a center of mass energy of $\sqrt{s} = 8$ TeV corresponding to the end of the first LHC run. The rates are taken at $20$% and the effect of considering each of the rates at a time is shown by superimposing the colours, cyan/light-grey ($\mu_{VV}$), blue/black ($\mu_{\tau \tau}$) and finally red/dark-grey ($\mu_{\gamma \gamma}$). In figure \[fig:F1\] we present the allowed space for the angles $\alpha_2$ vs. $\alpha_1$ for Type I (left) and Type II (right) with all theoretical and collider constraints taken into account. The corresponding plots for the Flipped (Lepton specific) are very similar to the one for Type II (I) and are not shown. It was expected that $\alpha_2$ would be centred around zero where the pseudoscalar component vanishes. Also $\alpha_1$ plays the role of $\alpha + \pi/2$, where $\alpha$ is the rotation angle in the CP-conserving case [^2]. In previous works for the CP-conserving model [@Ferreira:2012nv; @Fontes:2014tga] we have made estimates for the allowed parameters based on the assumption that the production is dominated by gluon fusion and that $\Gamma(h_1 \to b \bar b)$ is to a good approximation the Higgs total width. Under a similar approximation, we can write for Type I and large $\tan \beta$ (when $\tan \beta \gg 1$, $b_i \ll 1$ and we recover the CP-conserving Yukawa couplings) $$\mu_{VV}^{I} \approx \cos^2 \alpha_2 \, \cos^2 (\beta - \alpha_1) \, .
\label{eq:muI}$$ Since we are considering a $20$% accuracy, it is clear that neither $\cos \alpha_2$ nor $\cos (\beta - \alpha_1)$ can be close to zero. In fact, a measurement of $\mu_{VV}$ with a $20$% ($5$%) accuracy and centred at the SM expected value implies $\cos^2 \alpha_2 \gtrsim 0.8 (0.95)$ and consequently $|\sin \alpha_2| \lesssim 0.45 (0.22)$ and $|\alpha_2| \lesssim 27 \degree (13 \degree)$. Although the approximations captures the features, the plot does not reproduce the exact value of the limit, which for a $20$% accuracy is slightly below $50 \degree$.
![Top: $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ for Type I. Bottom: same but for Type II. The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F2"}](plot_430_T1.pdf "fig:"){width="0.33\linewidth"} ![Top: $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ for Type I. Bottom: same but for Type II. The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F2"}](plot_440_T1.pdf "fig:"){width="0.33\linewidth"} ![Top: $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ for Type I. Bottom: same but for Type II. The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F2"}](plot_450_T1.pdf "fig:"){width="0.33\linewidth"}\
![Top: $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ for Type I. Bottom: same but for Type II. The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F2"}](plot_430_T2.pdf "fig:"){width="0.33\linewidth"} ![Top: $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ for Type I. Bottom: same but for Type II. The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F2"}](plot_440_T2.pdf "fig:"){width="0.33\linewidth"} ![Top: $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ for Type I. Bottom: same but for Type II. The rates are taken to be within 20$\%$ of the SM predictions. The colours are superimposed with cyan/light-grey for $\mu_{VV}$, blue/black for $\mu_{\tau \tau}$ and finally red/dark-grey for $\mu_{\gamma \gamma}$ with a center of mass energy of 8 TeV.[]{data-label="fig:F2"}](plot_450_T2.pdf "fig:"){width="0.33\linewidth"}
In figure \[fig:F2\] we show $\tan \beta$ as a function of $R_{11}$ (left), $R_{12}$ (middle) and $R_{13}$ (right). The upper plots are for Type I and the lower plots for Type II. Again the differences of Type II (I) relative to F (LS) are small and we do not show the corresponding plots. We start with $R_{13}$ which is just $\sin \alpha_2$, thus measuring the amount of CP-violation for the 125 GeV Higgs, that is, the magnitude of its pseudoscalar component. The allowed points are centred around zero where we recover a SM-like Higgs Yukawa coupling for the lightest scalar state. The differences between the models only occur for large $\tan \beta$, reflecting the different angle dependence of the couplings in the various models. We now discuss $R_{12}=\sin \alpha_1 \cos \alpha_2$. Using the same approximation for $\mu_{VV}$ as in eq.(\[eq:muI\]) we can write for large $\tan \beta$ $$\mu_{VV}^{I} \approx R_{12}^2 \, ,
\label{eq:muI2}$$ which means that if we take $|R_{12}^2| > 0.8$ then $R_{12}>0.89$ or $R_{12}< - 0.89$. These are exactly the bounds we see in the plots for Type I. Therefore, as already happened for the CP-conserving case it is mainly $\mu_{VV}$ that constrains $|R_{12}|$ to be close to $1$ especially for large $\tan \beta$. Finally $R_{11}=\cos \alpha_1 \cos \alpha_2$ is only indirectly constrained by the bounds on $\alpha_1$ and $\alpha_2$. Since the pure scalar part of the coupling relative to the SM is proportional to $R_{11}^2 \, (1+\tan^2 \beta)$ it is natural that when $R_{11}$ increases, $\tan \beta$ decreases. However, the most important point to note is that $R_{11} = 0$ is allowed. Although $R_{11}$ is never part of the Yukawa couplings in Type I, it appears in pure scalar couplings for down-type quarks or/and charged leptons in the remaining types. This in turn implies that scenarios where $a_D =0$ and/or $a_L =0$ are not excluded. Models Type II, F and LS can therefore have a pure pseudoscalar component for some of its Yukawa couplings. This scenario will be discussed in detail in the next section.
![$\tan \beta$ as a function of $\sin (\alpha_1-\pi/2)$ with all rates at $20$% for Type I (left), Type II (middle) and LS (right). All angles are free to vary in their allowed range (cyan/light-grey) and we impose the constraint $s_2 < 0.1$ (blue/black) and $s_2 < 0.05$ (red/dark-grey).[]{data-label="fig:F3"}](plot_104_T1.pdf "fig:"){width="0.33\linewidth"} ![$\tan \beta$ as a function of $\sin (\alpha_1-\pi/2)$ with all rates at $20$% for Type I (left), Type II (middle) and LS (right). All angles are free to vary in their allowed range (cyan/light-grey) and we impose the constraint $s_2 < 0.1$ (blue/black) and $s_2 < 0.05$ (red/dark-grey).[]{data-label="fig:F3"}](plot_104_T2.pdf "fig:"){width="0.33\linewidth"} ![$\tan \beta$ as a function of $\sin (\alpha_1-\pi/2)$ with all rates at $20$% for Type I (left), Type II (middle) and LS (right). All angles are free to vary in their allowed range (cyan/light-grey) and we impose the constraint $s_2 < 0.1$ (blue/black) and $s_2 < 0.05$ (red/dark-grey).[]{data-label="fig:F3"}](plot_104_LS.pdf "fig:"){width="0.33\linewidth"}
In figure \[fig:F3\] we present $\tan \beta$ as a function of $\sin (\alpha_1-\pi/2)$ with all rates at $20$% for Type I (left), Type II (middle) and LS (right). All angles are free to vary in their allowed range and we present scenarios for which $s_2 < 0.1$ and $s_2 < 0.05$. We plot $\alpha_1-\pi/2$ instead of $\alpha_1$ to match the usual definition for the CP-conserving model. Since we recover the CP-conserving $h_1$ couplings when $s_2 =0$, the red/dark-grey outer layer for Type II and LS has to match the bounds for the angle $\alpha$ in the CP conserving case which is indeed the case [@ourreview]. If we identify $\alpha_1$ with $\alpha+\pi/2$, where $\alpha$ is the rotation angle for the CP-conserving scenario, we can write the coupling to gauge bosons as g\_[hVV]{}\^ = g\_[hVV]{}\^ . \[C1\] Hence, for Type I $\mu_{VV}$ will either give the same bound as in the CP-conserving case or worse as $\cos{(\alpha_2)}$ decreases. However, for Type II, the same approximation that lead to eq. (\[eq:muI\]) for Type I results for Type II in $$\mu_{VV}^{II} \approx \frac{ \cos^2 \alpha_2 \, \cos^2 (\beta - \alpha_1)}{\tan^2 \beta} \, \, \,
\frac{\sin^2 \alpha_1 \, \cos^2 \alpha_2 + \sin^2 \alpha_2 \, \cos^2 \beta}
{\cos^2 \alpha_1 \, \cos^2 \alpha_2 + \sin^2 \alpha_2 \, \sin^2 \beta} \, .
\label{approxtype2}$$ Again, if $s_2=0$ we recover the CP-conserving expression. However, it can be shown that larger values of $s_2$ together with smaller values of $\tan \beta$ still fulfil the constraints on the rates. We conclude that in Type I, the allowed parameter space is the same as in the CP-conserving case while, for the remaining types and for a given $\alpha_1$, the upper bound on $\tan \beta$ is the same as in the CP-conserving case. But, now, there is no lower bound on $\tan \beta$.
The zero scalar components scenarios and the LHC run 2 {#zero}
======================================================
In the previous section we have shown that $R_{11}=0$ is still allowed, which implies that the pure scalar components of the Yukawa couplings can be zero in some scenarios. This possibility arises in Type II, F and LS. In particular for Type II we can have either $a_D = 0$ or $a_L = 0$ while in F (LS) only $a_D = 0$ ($a_L = 0$) is possible. For definiteness let us now analyse the case where $a_D = 0$ in Type II. Since $a_D = R_{11}/c_{\beta} = c_1 c_2/c_{\beta}$ we could in principle have $c_1=0$ or $c_2=0$. However, $c_2=0$ would mean that the gauge bosons would not couple at tree level to the Higgs, a scenario that is ruled out by experiment as shown in the previous section. Setting $c_1=0$ we get, in Type II, $a_D=a_L =0$ and $$a_U^2=c_2^2/s_\beta^2, \quad b_U^2=s_2^2/t_\beta^2, \quad b_D^2=b_L^2=t_{\beta}^2 s_2^2, \quad C^2=s_\beta^2 c_2^2 \,.$$
![Left: sgn$(C)$ $b_D$ $=$ sgn$(C)$ $b_L$ as a function of sgn$(C)$ $a_D$ $=$ sgn$(C)$ $a_L$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black), $5$% (red/dark-grey), and $1$% (cyan/light-grey). Right: same, but for sgn$(C)$ $b_U$ as a function of sgn$(C)$ $a_U$.[]{data-label="fig:F4"}](TypeII-aDbD-1000.pdf "fig:"){width="0.49\linewidth"} ![Left: sgn$(C)$ $b_D$ $=$ sgn$(C)$ $b_L$ as a function of sgn$(C)$ $a_D$ $=$ sgn$(C)$ $a_L$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black), $5$% (red/dark-grey), and $1$% (cyan/light-grey). Right: same, but for sgn$(C)$ $b_U$ as a function of sgn$(C)$ $a_U$.[]{data-label="fig:F4"}](TypeII-aUbU-1010.pdf "fig:"){width="0.49\linewidth"}
In the left panel of figure \[fig:F4\] we show $b_D=b_L$ as a function of $a_D=a_L$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black), $5$% (red/dark-grey), and $1$% (cyan/light-grey) (in order to avoid the dependence on the phase conventions in choosing the range for the angles $\alpha_i$, we plot sgn$(C) \,a_i$ (sgn$(C) \,b_i$) instead of $a_i$ ($b_i$) with $i=U,D,L$). It is quite interesting to note that this scenario is still possible with the rates at $5$% of the SM value at the LHC at 13 TeV. We have checked that this is still true at $2$% and only when the accuracy reaches $1$% are we able to exclude the scenario. So far we have discussed $a_D=0$. Another interesting point is that when $|a_D| \to 0$, $|b_D| \to 1$. The requirement that $|b_D| \approx 1$ implies that the couplings of the up-type quarks to the lightest Higgs take the form $$a_U^2=( 1 -s_2^4)=(1-1/t_\beta^4), \quad b_U^2=s_2^4=1/t^4_\beta,$$ while the coupling to massive gauge bosons is now $$C^2 = (t^2_\beta-1)/(t^2_\beta+1) = (1 - s_2^2)/(1 + s_2^2) \, .$$ In the right panel of figure \[fig:F4\] we now show $b_U$ as a function of $a_U$ for Type II with the same colour code. We conclude from the plot that the constraint on the values of $(a_U,\, b_U)$ are already quite strong and will be much stronger in the future just taking into account the measurement of the rates. We would like to understand why $a_U \sim 1$ and $b_U \sim 0$, while the bounds are much looser for $a_D$. We start by noting that the couplings impose different constraints on the up and down sectors. Indeed, from table \[tab:1\] and eq. , $$R_{11} =\frac{C- s_\beta^2 \, a_U}{c_\beta}; \quad R_{12} = s_\beta \, a_U; \quad R_{13} = - \tan{\beta} \, b_U,
\label{RsUP}$$ for the up sector, while $$R_{11} = c_\beta \, a_D; \quad R_{12} = \frac{C- c_\beta^2 \, a_D}{s_\beta}; \quad R_{13} = - \frac{c_\beta}{s_\beta}\, b_D,
\label{RsDOWN}$$ for the down sector. In the first case, $R_{12}^2 + R_{13}^2 < 1$ leads to a\_U\^2 + < . \[eq:elliUP1\] Noting that the $\tan{\beta} > 1$ constraint forces $c_\beta < 1/\sqrt{2}$ and $s_\beta > 1/\sqrt{2}$, we find $b_U < 1$, while $a_U < \sqrt{2}$. This is what we see in the right panel of figure \[fig:NEW\], where in cyan we show points which are subject only to the theoretical constraints.
![Left: Simulation points on the sgn$(C)\, a_D$ versus sgn$(C)\, b_D$ plane. In cyan/light-grey (orange/dark-grey; blue/black) we show the points which pass all theoretical constraints (pass, in addition, the restriction from $\mu_{VV}$ at 10%; pass, in addition, the restriction from all $\mu$’s at 10%). Right: Same constraints on the sgn$(C)\, a_U$ versus sgn$(C)\, b_U$ plane.[]{data-label="fig:NEW"}](TypeII-aDbD-Theory-VV10-All10.pdf "fig:"){width="0.49\linewidth"} ![Left: Simulation points on the sgn$(C)\, a_D$ versus sgn$(C)\, b_D$ plane. In cyan/light-grey (orange/dark-grey; blue/black) we show the points which pass all theoretical constraints (pass, in addition, the restriction from $\mu_{VV}$ at 10%; pass, in addition, the restriction from all $\mu$’s at 10%). Right: Same constraints on the sgn$(C)\, a_U$ versus sgn$(C)\, b_U$ plane.[]{data-label="fig:NEW"}](TypeII-aUbU-Theory-VV10-All10.pdf "fig:"){width="0.49\linewidth"}
We see that all points lye inside the ellipse in eq. . The constraint from the $\mu_{VV}$ bound (orange/dark-grey points in the right panel of figure \[fig:NEW\]) then places the points on a section of that ellipse close to $(a_U, b_U) \sim (0,1)$.
The situation is completely different for the down sector. Indeed, a similar analysis starting from eqs. and $R_{11}^2 + R_{13}^2 < 1$, would lead to a\_D\^2 + < . \[eq:elliDOWN1\] Since $c_\beta$ can be very small, this entails no constraint at all, agreeing with the fact that the cyan/light-grey points in the left panel of figure \[fig:NEW\] have no restriction. In contrast, it is the bound on $\mu_{VV}$ which constrains the parameter space to the orange/dark-grey circle centered at $(0,0)$. But now, the whole circle is allowed.
The constraints from $\mu_{VV}$ can be understood with simple arguments as follows. It was shown in [@Fontes:2014tga], in the real 2HDM, that the limits on $\mu_{VV}$ impose rather non trivial constraints on the coupling to fermions which, however, can be understood from simple trigonometry. Following the spirit of that article, we assume that the production is mainly due to $gg \rightarrow h_1$ with an intermediate top in the triangle loop, and that the scalar decay width is dominated by the decay into $b \bar{b}$. As a result, \_[VV]{} \~(a\_U\^2 + 1.5 b\_U\^2) , where the approximate factor of 1.5 is what one would obtain either from a naive one-loop calculation[^3], or from a full HIGLU simulation [@Spira:1995mt]. Applying this formula, we obtain figure \[fig:JR1\], where we have taken $\mu_{VV}$ within $20\%$ (orange/light-grey) or $10\%$ (blue/black) of the SM, letting the angles vary freely within their theoretically allowed ranges.
![Left: sgn$(C)$ $b_D$ $=$ sgn$(C)$ $b_L$ as a function of sgn$(C)$ $a_D$ $=$ sgn$(C)$ $a_L$ for Type II and a center of mass energy of $13$ TeV with $\mu_{VV}$ within $20\%$ (orange/light-grey) and $10\%$ (blue/black) of the SM. []{data-label="fig:JR1"}](Type2-aDbD-Trig.pdf "fig:"){width="0.49\linewidth"} ![Left: sgn$(C)$ $b_D$ $=$ sgn$(C)$ $b_L$ as a function of sgn$(C)$ $a_D$ $=$ sgn$(C)$ $a_L$ for Type II and a center of mass energy of $13$ TeV with $\mu_{VV}$ within $20\%$ (orange/light-grey) and $10\%$ (blue/black) of the SM. []{data-label="fig:JR1"}](Type2-aUbU-Trig.pdf "fig:"){width="0.49\linewidth"}
The similarity between the left (right) panes of figures \[fig:NEW\] (a full model simulation) and figure \[fig:JR1\] (a simple trigonometric exercise) is uncanny.
Further constraints are brought about by a second simple geometrical argument. They place all solutions close to $(a,b) \sim (1,0)$ when $C$ is close to unity. We use eqs. to derive $$1=R_{11}^2 + R_{12}^2 + R_{13}^2=
\frac{(C - s_\beta^2 \, a_U)^2}{c_\beta^2}
+ s_\beta^2 \, a_U^2 + \tan^2{\beta} \, b_U^2,$$ leading to $$\left( a_U - C \right)^2 + b_U^2
=
\frac{1}{\tan^2{\beta}}\, [1-C^2].
\label{circleUP}$$ This is a circle centered at $(C,0)$, which excludes most cyan/light-grey points on the right panel of figure \[fig:NEW\]. Since $C$ is close to unity, and appears divided by $\tan{\beta}$ (which must be larger than one), the radius is almost zero, forcing $a_U$ to lie close to $C \sim 1$, and $b_U$ close to $0$. Including all channels at 10% restricts the region even further, as seen in the blue/black points on the right panel of figure \[fig:NEW\]. It is true that an equation similar to eq. can be found for the down sector: $$\left( a_D - C \right)^2 + b_D^2
=
\tan^2{\beta}\, [1-C^2].
\label{circleDOWN}$$ However, the different placement of $\tan{\beta}$ is crucial. For intermediate to large $\tan{\beta}$, the $\tan^2{\beta}$ factor in eq. enhances the radius with respect to that allowed by the $\cot^2{\beta}$ factor in eq. . This explains the difference between the red/dark-grey points on the two panels in figure \[fig:F4\].
We now turn to the constraints on the $\sin \alpha_2$-$\tan{\beta}$ plane.
![Left: $\tan \beta$ as a function of $\sin \alpha_2$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black). In red/dark-grey we show the points with $|a_D| < 0.1$ and $||b_D|-1| < 0.1$ and in green $|b_D| < 0.05$ and $||a_D|-1| < 0.05$ Right: same, with $\tan \beta$ replaced by $\cos \alpha_1$.[]{data-label="fig:F4b"}](TypeII-s2tb-1020.pdf "fig:"){width="0.49\linewidth"} ![Left: $\tan \beta$ as a function of $\sin \alpha_2$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black). In red/dark-grey we show the points with $|a_D| < 0.1$ and $||b_D|-1| < 0.1$ and in green $|b_D| < 0.05$ and $||a_D|-1| < 0.05$ Right: same, with $\tan \beta$ replaced by $\cos \alpha_1$.[]{data-label="fig:F4b"}](TypeII-s2c1-1030.pdf "fig:"){width="0.49\linewidth"}
When we choose $\mu_{VV} > 0.9$ in the exact limit $(|a_D|,|b_D|)=(0,1)$, we obtain, using the approximation in eq. (\[approxtype2\]) $\tan \beta > 4.4$. Because we are not in the exact limit, the bound we present in the left plot of figure \[fig:F4b\] for $\tan \beta$ is closer to $3$. The left panel of figure \[fig:F4b\] shows $\tan \beta$ as a function of $\sin \alpha_2$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black). In red/dark-grey we show the points with $|a_D| < 0.1$ and $||b_D| - 1| < 0.1$ and in green $|b_D| < 0.05$ and $||a_D| - 1| < 0.05$. In the right panel, $\tan \beta$ is replaced by $\cos \alpha_1$. These two plots allow us to distinguish the main features of the SM-like scenario, where $(|a_D|,|b_D|) \approx (1,0)$ from the pseudoscalar scenario where $(|a_D|,|b_D|) \approx (0,1)$. In the SM-like scenario $\sin \alpha_2 \approx 0$, $\tan \beta$ is not constrained and the allowed values of $\sin \alpha_2$ grow with increasing $\cos \alpha_1$. In the pseudoscalar scenario $\cos \alpha_1 \approx 0$, $\sin \alpha_2$ and $\tan \beta$ are strongly correlated and $\tan \beta$ has to be above $\approx 3$. Clearly, all values of $a_D$ and $b_D$ are allowed provided $a_D^2 + b_D^2 \approx 1$.
![Left: sgn$(C)$ $b_U$ as a function of sgn$(C)$ $a_U$ for Type I and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black) and $5$% (red/dark-grey). Right: sgn$(C)$ $b_L$ as a function of sgn$(C)$ $a_L$ for LS and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black) and $5$% (red/dark-grey).[]{data-label="fig:F5"}](TypeI-aUbU-2000.pdf "fig:"){width="0.49\linewidth"} ![Left: sgn$(C)$ $b_U$ as a function of sgn$(C)$ $a_U$ for Type I and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black) and $5$% (red/dark-grey). Right: sgn$(C)$ $b_L$ as a function of sgn$(C)$ $a_L$ for LS and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black) and $5$% (red/dark-grey).[]{data-label="fig:F5"}](LS-aLbL-1000.pdf "fig:"){width="0.49\linewidth"}
In the left panel of figure \[fig:F5\] we show $b_U$ as a function of $a_U$ for Type I and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black) and $5$% (red/dark-grey). In Type I this plot is valid for all Yukawa couplings, because $a_U=a_D=a_L$ and $b_U=b_D=b_L$. It is interesting that even at $10$% there are points close to $(a,b)=(0.5,0.6)$ still allowed and no dramatic changes happen when we move to $5$%. In the right plot we show $b_L$ as a function of $a_L$ for LS with the same colour code. Here again the $(a_L,b_L)=(0,1)$ scenario is still allowed both with $10$% and $5$% accuracy. However, as was previously shown, the wrong sign limit is not allowed for the LS model [@Fontes:2014tga; @Ferreira:2014dya]. Nevertheless, in the C2HDM, the scalar component sgn$(C) \, a_L$ can reach values close to $-0.8$. Finally, for the up-type and down-type quarks, the plots are very similar to the one in the right panel of figure \[fig:F4\] for Type II.
Direct measurements of the CP-violating angle
---------------------------------------------
Although precision measurements already constrain both the scalar and pseudoscalar components of the Yukawa couplings in the C2HDM, there is always the need for a direct (and thus, more model independent) measurement of the relative size of pseudoscalar to scalar components of the Yukawa couplings. The angle that measures this relative strength, $\phi_i$, defined as $$\tan \phi_i = b_i/a_i \qquad i=U,\, D, \, L \, ,$$ could in principle be measured for all Yukawa couplings. The experimental collaborations at CERN will certainly tackle this problem when the high luminosity stage is reached, through any variables able to measure the ratio of the pseudoscalar to scalar component of the Yukawa couplings. There are several proposals for a direct measurement of this ratio, which focus mainly on the $tth$ and on the $\tau^+ \tau^- h$ couplings. Measurement of $b_U/a_U$ were first proposed for $pp \to t \bar t h$ in [@Gunion:1996xu] and more recently reviewed in [@Ellis:2013yxa; @He:2014xla; @Boudjema:2015nda]. A proposal to probe the same vertex through the process $pp \to hjj$ [@DelDuca:2001fn] was put forward in [@Field:2002gt] and again more recently in [@Dolan:2014upa]. In reference [@Dolan:2014upa] an exclusion of $\phi_t > 40 \degree$ ($\phi_t > 25 \degree$) for a luminosity of 50 fb$^{-1}$ (300 fb$^{-1}$) was obtained for 14 TeV and assuming $\phi_t=0$ as the null hypothesis. A study of the $\tau^+ \tau^- h$ vertex was proposed in [@Berge:2008wi] and a detailed study taking into account the main backgrounds [@Berge:2014sra] lead to an estimate in the precision of $\Delta \phi_\tau$ of $27 \degree$ ($14.3 \degree$) for a luminosity of 150 fb$^{-1}$ (500 fb$^{-1}$) and a center of mass energy of 14 Tev.
Since in the C2HDM the couplings are not universal, one would need in principle three independents measurements, one for up-type quarks, one for down-type quarks and one for leptons. The number of independent measurements is of course model dependent. For Type I one such measurement is enough because the Yukawa couplings are universal. For all other Yukawa types we need two independent measurements. It is interesting to note that for model F, since the leptons and up-type quarks coupling to the Higgs are the same, a direct measurement of the $hbb$ vertex is needed to probe the model. On the other hand, and again using model F as an example, a different result for $\phi_t$ and $\phi_\tau$ would exclude model F (and also Type I).
![Top: $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I with rates at $20$% (green) and $5$% (red/dark-grey). Bottom: same but for Type II.[]{data-label="fig:F6"}](TypeI-atanaUbU-tanb-2090.pdf "fig:"){width="0.32\linewidth"} ![Top: $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I with rates at $20$% (green) and $5$% (red/dark-grey). Bottom: same but for Type II.[]{data-label="fig:F6"}](TypeI-atanaUbU-s2-2110.pdf "fig:"){width="0.32\linewidth"} ![Top: $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I with rates at $20$% (green) and $5$% (red/dark-grey). Bottom: same but for Type II.[]{data-label="fig:F6"}](TypeI-atanaUbU-c1-2100.pdf "fig:"){width="0.32\linewidth"}\
![Top: $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I with rates at $20$% (green) and $5$% (red/dark-grey). Bottom: same but for Type II.[]{data-label="fig:F6"}](TypeII-atanbUaU-tanb-1090.pdf "fig:"){width="0.32\linewidth"} ![Top: $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I with rates at $20$% (green) and $5$% (red/dark-grey). Bottom: same but for Type II.[]{data-label="fig:F6"}](TypeII-atanbUaU-s2-1110.pdf "fig:"){width="0.32\linewidth"} ![Top: $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I with rates at $20$% (green) and $5$% (red/dark-grey). Bottom: same but for Type II.[]{data-label="fig:F6"}](TypeII-atanbUaU-c1-1100.pdf "fig:"){width="0.32\linewidth"}
Let us first discuss what we can already say about the allowed range for the $\phi_U \equiv \phi_t$ angle and what to expect by the end of the LHC’s run 2 using only the rates’ measurements. In figure \[fig:F6\] we show on the top row $\tan \beta$ (left), $\sin \alpha_2$ (middle) and $\cos \alpha_1$ (right) as a function of $\phi_U=\tan^{-1} (b_U/a_U)$ for Type I, with rates within $20$% (green) and $5$% (red/dark-grey) of the SM prediction. In the bottom row we present the same plots but for Type II. The green points are a good approximation for the allowed region after run 1, while the red/dark-grey points are a good prediction for the allowed space with the run 2 high luminosity results. The most striking features of the plots are the following. For Type I the angle $\phi_U=\phi_D=\phi_L$ is between $-75 \degree$ and $75 \degree$ and this interval will be reduced to roughly $-45 \degree$ and $45 \degree$ provided the measured rates are in agreement with the SM predictions. For Type II only $\phi_U$ is constrained; we get $|\phi_U| < 30 \degree$ and the prediction of roughly $|\phi_U| < 15 \degree$ when rates are within 5$\%$ of the SM predictions. Since the Higgs couplings to top quarks are the same for all models, the angle that relates scalar and pseudoscalar components for this vertex is related to the lightest Higgs CP-violating angle $\alpha_2$ by $$\tan \phi_t = - c_\beta/s_1 \, \tan \alpha_2 \qquad \Rightarrow \qquad \tan \alpha_2 = - s_1/c_\beta \, \tan \phi_t \, .
\label{eq:phi1}$$ The parameter space is restricted in such a way that high $\tan \beta$ implies low $\alpha_2$. Since $s_1$ cannot be too small, it is clear from equation (\[eq:phi1\]) that large $\tan \beta$ necessarily implies a small $\phi_t$. This is clearly seen in the left top and bottom plots of figure \[fig:F6\] where for large $\tan \beta$ the pseudoscalar component of the up-type quarks Yukawa coupling is very close to zero. Interestingly, for both Type I and Type II the values of $\tan \beta \thicksim$ O$(1)$ are the ones for which the angle $\phi_t$ is less constrained. These are exactly the values for which the the coupling $tth$ has a maximum value (already considering the remaining constraints that disallow values of $\tan \beta$ below 1). Therefore, a direct measurement of $\phi_t$ could still be competitive with the rates measurement in Type I.
![Left: $\cos \alpha_1$ as a function of $\tan^{-1} (b_D/a_D)$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black). In red/dark-grey we show the points with $|a_D| < 0.1$ and $||b_D|-1| < 0.1$ and in green $|b_D| < 0.05$ and $||a_D|-1| < 0.05$. Right: same, with $\cos \alpha_1$ replaced by sgn$(C) \, a_D$.[]{data-label="fig:F7"}](TypeII-atanbDaDc1-1040.pdf "fig:"){width="0.49\linewidth"} ![Left: $\cos \alpha_1$ as a function of $\tan^{-1} (b_D/a_D)$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black). In red/dark-grey we show the points with $|a_D| < 0.1$ and $||b_D|-1| < 0.1$ and in green $|b_D| < 0.05$ and $||a_D|-1| < 0.05$. Right: same, with $\cos \alpha_1$ replaced by sgn$(C) \, a_D$.[]{data-label="fig:F7"}](TypeII-atanbDaD-aD-1055.pdf "fig:"){width="0.49\linewidth"}
Let us now move to the Yukawa versions that can have a zero scalar component not only at the end of run 1, but also at the end of run 2, if only the rates are considered. For definiteness we focus on Type II. As previously discussed, a direct measurement involving the vertex $h \tau^+ \tau^-$ [@Berge:2008wi; @Berge:2014sra] could lead to a precision in the measurement of $\phi_\tau$, $\Delta \phi_\tau$, of $27 \degree$ ($14.3 \degree$) for a luminosity of 150 fb$^{-1}$ (500 fb$^{-1}$) and a center of mass energy of 14 Tev. In figure \[fig:F7\] (left) we show $\cos \alpha_1$ as a function of $\tan^{-1} (b_D/a_D)$ for Type II and a center of mass energy of $13$ TeV with all rates at $10$% (blue/black). In red/dark-grey we show the points with $|a_D| < 0.1$ and $||b_D|-1| < 0.1$ and in green $|b_D| < 0.05$ and $||a_D|-1| < 0.05$. In the right panel $\cos \alpha_1$ is replaced by sgn$(C) \, a_D$. It is clear that the SM-like scenario sgn$(C)$ $(a_D, \, b_D ) = (1,0)$ is easily distinguishable from the $(0,1)$ scenario. In fact, a measurement of $\phi_\tau$ even if not very precise would easily exclude one of the scenarios. Obviously, all other scenarios in between these two will need more precision (and other measurements) to find the values of scalar and pseudoscalar components. The $\tau^+ \tau^- h$ angle is related to $\alpha_2$ as $$\tan \phi_\tau = - s_\beta/c_1 \, \tan \alpha_2 \qquad \Rightarrow \qquad
\tan \alpha_2 = - c_1/s_\beta \, \tan \phi_\tau$$ and therefore a measurement of the angle $\phi_\tau$ does not directly constrain the angle $\alpha_2$. In fact, the measurement gives a relation between the three angles. A measurement of $\phi_t$ and $\phi_\tau$ would give us two independent relations to determine the three angles.
Constraints from EDM
--------------------
Models with CP violation are constrained by bounds on the electric dipole moments (EDMs) of neutrons, atoms and molecules. Recently the ACME Collaboration [@Baron:2013eja] improved the bounds on the electron EDM by looking at the EDM of the ThO molecule. This prompted several groups to look again at the subject. For what concerns us here, the complex 2HDM, several analyses have been performed recently [@Buras:2010zm; @Cline:2011mm; @Jung:2013hka; @Shu:2013uua; @Inoue:2014nva; @Brod:2013cka]. In ref. [@Inoue:2014nva] it was found that the most stringent limits are obtained from the ThO experiment, except in cases where there are cancellations among the neutral scalars. These cancellations were pointed out in [@Jung:2013hka; @Shu:2013uua] and arise due to orthogonality of the $R$ matrix in the case of almost degenerate scalars [@Fontes:2014xva]. So far, there is no complete scan of EDM in the C2HDM; only some benchmark points have been considered, making it difficult to see when these cancellations are present.
What can be learned from these studies is that the EDMs are very important and their effect in the C2HDM has to be taken in account in a systematic way, in the sense that, for each point in the scan, the EDMs have to be calculated and compared with the experimental bounds. However, for the purpose of the studies in this work and for the present experimental sensitivity, this is not required. This is because we are looking at scenarios where the couplings of the up-type sector (top quark) are very close to the SM and the differences, still allowed by the LHC data, are in the couplings of the down-type sector; the tau lepton and bottom quark. As was shown in ref. [@Brod:2013cka], while the pseudoscalar coupling of the top quark is very much constrained (in our notation $|b_U| \leq 0.01$), the corresponding couplings for the b quark and tau lepton are less constrained by the EDMs than by the LHC data. Since we are taking in account the collider data, our scenarios are in agreement with the present experimental data. But one should keep in mind that, as pointed out in refs. [@Brod:2013cka; @Dekens:2014jka], the future bounds from the EDMs can alter this situation. In the future, the interplay between the EDM bounds and the data from the LHC Run 2 will pose relevant new constraints in the complex 2HDM in general, and in particular for the scenarios presented in this work.
Conclusions {#sec:conc}
===========
We discuss the present status of the allowed parameter space of the complex two-Higgs doublet model where we have considered all pre-LHC plus the theoretical constraints on the model. We have also taken into account the bounds arising from assuming that the lightest scalar of the model is 125 GeV Higgs boson discovered at the LHC. We have shown that the parameter space is already quite constrained and recovered all the limits on the couplings of a CP-conserving 125 GeV Higgs. The allowed space for some variables, as for example for the $\tan \beta$ parameter, is now increased as a natural consequence of having a larger number of variables to fit the data as compared to the CP-conserving case.
The core of the work is the discussion of scenarios where the scalar component of the Yukawa couplings of the lightest Higgs to down-type quarks and/or to leptons can vanish. In these scenarios, that can occur for Type II, F and LS, the pseudoscalar component plays the role of the scalar component in assuring the measured rates at the LHC. A direct measurement of the angle that gauges the ratio of pseudoscalar to scalar components in the $tth$ vertex, $\phi_t$, will probably help to further constrain this ratio. However, it is the measurement of $\phi_\tau$, the angle for the $\tau^+ \tau^- h$ vertex, that will allow to rule out the scenario of a vanishing scalar even with a poor accuracy. We have also noted that for the F model only a direct measurement of $\phi_D$ in a process involving the $bbh$ vertex would be able to probe the vanishing scalar scenario. Finally a future linear collider [@Ono:2012ah; @Asner:2013psa] will certainly help to further probe the vanishing scalar scenarios.
R.S. is supported in part by the Portuguese *Fundação para a Ciência e Tecnologia* (FCT) under contract PTDC/FIS/117951/2010. D.F., J.C.R. and J.P.S. are also supported by the Portuguese Agency FCT under contracts CERN/FP/123580/2011, EXPL/FIS-NUC/0460/2013 and PEst-OE/FIS/UI0777/2013.
[99]{}
G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Lett. B [**716**]{}, 1 (2012) \[arXiv:1207.7214 \[hep-ex\]\]. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**716**]{}, 30 (2012) \[arXiv:1207.7235 \[hep-ex\]\]. T.D. Lee, Phys. Rev. D [**8**]{} (1973) 1226.
J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, *The Higgs Hunter’s Guide* . G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, *Theory and phenomenology of two-Higgs-doublet models*, *Phys. Rept. * [**516**]{} (2012) 1 \[arXiv:1106.0034 \[hep-ph\]\].
A. Barroso, P. M. Ferreira, R. Santos and J. P. Silva, Phys. Rev. D [**86**]{} (2012) 015022 \[arXiv:1205.4247 \[hep-ph\]\]. S. Inoue, M. J. Ramsey-Musolf and Y. Zhang, Phys. Rev. D [**89**]{} (2014) 115023 \[arXiv:1403.4257 \[hep-ph\]\]. K. Cheung, J. S. Lee, E. Senaha and P. Y. Tseng, JHEP [**1406**]{} (2014) 149 \[arXiv:1403.4775 \[hep-ph\]\]. D. Fontes, J. C. Romão and J. P. Silva, JHEP [**1412**]{}, 043 (2014) \[arXiv:1408.2534 \[hep-ph\]\]. J. Brod, U. Haisch and J. Zupan, JHEP [**1311**]{} (2013) 180 \[arXiv:1310.1385 \[hep-ph\], arXiv:1310.1385\].
S. Chatrchyan *et al.* \[CMS Collaboration\], *Search for the standard model Higgs boson produced in association with a W or a Z boson and decaying to bottom quarks*, *Phys. Rev. D* **89** (2014) 012003 \[arXiv:1310.3687 \[hep-ex\]\].
B. Tuchming, (for the CDF and D0 Collaborations) *Higgs boson production and properties at the Tevatron*, arXiv:1405.5058 \[hep-ex\].
G. Aad [*et al.*]{} \[ATLAS Collaboration\], *Search for Higgs boson decays to a photon and a Z boson in pp collisions at $\sqrt{s}$=7 and 8 TeV with the ATLAS detector*, *Phys. Lett. B* **732** (2014) 8 \[arXiv:1402.3051 \[hep-ex\]\].
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], *Search for a Higgs boson decaying into a Z and a photon in pp collisions at $\sqrt{s}$ = 7 and 8 TeV*, *Phys. Lett. B* **726** (2013) 587 \[arXiv:1307.5515 \[hep-ex\]\].
ATLAS Collaboration, Physics at a High-Luminosity LHC with ATLAS, (2013), arXiv:1307.7292 \[hep-ex\], SNOW13-00078; ATLAS Collaboration, Projections for measurements of Higgs boson cross sections, branching ratios and coupling parameters with the ATLAS detector at a HL-LHC, ATLAS-PHYS-PUB-2013-014 (2013).
CMS Collaboration, Projected Performance of an Upgraded CMS Detector at the LHC and HL-LHC: Contribution to the Snowmass Process, (2013), arXiv:1307.7135 \[hep-ex\], SNOW13-00086.
S. Dawson, A. Gritsan, H. Logan, J. Qian, C. Tully, R. Van Kooten, A. Ajaib and A. Anastassov [*et al.*]{}, arXiv:1310.8361 \[hep-ex\]. I. F. Ginzburg, M. Krawczyk and P. Osland, *Two Higgs doublet models with CP violation*, hep-ph/0211371.
W. Khater and P. Osland, *CP violation in top quark production at the LHC and two Higgs doublet models*, *Nucl. Phys. B* **661** (2003) 209 \[hep-ph/0302004\].
A. W. El Kaffas, P. Osland and O. M. Ogreid, *CP violation, stability and unitarity of the two Higgs doublet model*, *Nonlin. Phenom. Complex Syst.* **10**(2007) 347 \[hep-ph/0702097\].
A. W. El Kaffas, W. Khater, O. M. Ogreid, and P. Osland, *Consistency of the two Higgs doublet model and CP violation in top production at the LHC*, *Nucl. Phys. B* **775** (2007) 45 \[hep-ph/0605142\].
B. Grzadkowski and P. Osland, *Tempered Two-Higgs-Doublet Model*, *Phys. Rev. D* [**82**]{} (2010) 125026 \[arXiv:0910.4068 \[hep-ph\]\].
A. Arhrib, E. Christova, H. Eberl and E. Ginina, *CP violation in charged Higgs production and decays in the Complex Two Higgs Doublet Model*, *JHEP* [**1104**]{} (2011) 089 \[arXiv:1011.6560 \[hep-ph\]\].
L. Lavoura, J. P. Silva, *Fundamental CP violating quantities in a $SU(2) x U(1)$ model with many Higgs doublets*, *Phys. Rev. D* **50** (1994) 4619 \[hep-ph/9404276\].
F. J. Botella and J. P. Silva, *Jarlskog - like invariants for theories with scalars and fermions*, *Phys. Rev. D* **51** (1995) 3870 \[hep-ph/9411288\].
S.L. Glashow and S. Weinberg, Phys. Rev. D [**15**]{}, 1958 (1977); E.A. Paschos, Phys. Rev. D [**15**]{}, 1966 (1977).
G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Rev. D [**90**]{} (2014) 5, 052004 \[arXiv:1406.3827 \[hep-ex\]\]. V. Khachatryan [*et al.*]{} \[CMS Collaboration\], arXiv:1412.8662 \[hep-ex\]. T. Hermann, M. Misiak and M. Steinhauser, JHEP [**1211**]{} (2012) 036 \[arXiv:1208.2788 \[hep-ph\]\]; F. Mahmoudi and O. Stal, Phys. Rev. D [**81**]{}, 035016 (2010) \[arXiv:0907.1791 \[hep-ph\]\].
O. Deschamps, S. Descotes-Genon, S. Monteil, V. Niess, S. T’Jampens and V. Tisserand, Phys. Rev. D [**82**]{}, 073012 (2010) \[arXiv:0907.5135 \[hep-ph\]\]. M. Baak, M. Goebel, J. Haller, A. Hoecker, D. Ludwig, K. Moenig, M. Schott and J. Stelzer, *Updated Status of the Global Electroweak Fit and Constraints on New Physics*, *Eur. Phys. J. C* **72** (2012) 2003 \[arXiv:1107.0975 \[hep-ph\]\]
A. Denner, R.J. Guth, W. Hollik and J.H. Kuhn, Z. Phys. C [**51**]{}, 695 (1991); H.E. Haber and H.E. Logan, Phys. Rev. D [**62**]{}, 015011 (2000) \[hep-ph/9909335\]; A. Freitas and Y.-C. Huang, JHEP [**1208**]{}, 050 (2012) \[arXiv:1205.0299 \[hep-ph\]\]. G. Abbiendi [*et al.*]{} \[ALEPH and DELPHI and L3 and OPAL and LEP Collaborations\], Eur. Phys. J. C [**73**]{} (2013) 2463 \[arXiv:1301.6065 \[hep-ex\]\]. ATLAS collaboration, ATLAS-CONF-2013-090; G. Aad [*et al.*]{} \[ATLAS Collaboration\], JHEP [**1206**]{} (2012) 039 \[arXiv:1204.2760 \[hep-ex\]\]; G. Aad [*et al.*]{} \[ATLAS Collaboration\], arXiv:1412.6663 \[hep-ex\].
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], JHEP [**1207**]{} (2012) 143 \[arXiv:1205.5736 \[hep-ex\]\]; CMS Note, CMS-PAS-HIG-14-020. The ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the Tevatron Electroweak Working Group, and the SLD electroweak and heavy flavour Groups, arXiv:1012.2367 \[hep-ex\]. N. G. Deshpande and E. Ma, *Pattern of Symmetry Breaking with Two Higgs Doublets*, *Phys. Rev. D* **18** (1978) 2574 .
S. Kanemura, T. Kubota and E. Takasugi, *Lee-Quigg-Thacker bounds for Higgs boson masses in a two doublet model*, *Phys. Lett. B* **313** (1993) 155 \[hep-ph/9303263\].
A. G. Akeroyd, A. Arhrib and E. -M. Naimi, *Note on tree level unitarity in the general two Higgs doublet model*, *Phys. Lett. B* **490**(2000) 119 \[hep-ph/0006035\].
I. F. Ginzburg and I. P. Ivanov, *Tree level unitarity constraints in the 2HDM with CP violation*, hep-ph/0312374.
M.E. Peskin and T. Takeuchi, Phys. Rev. D [**46**]{}, 381 (1992). W. Grimus, L. Lavoura, O. M. Ogreid and P. Osland, *The Oblique parameters in multi-Higgs-doublet models* *Nucl. Phys. B* **801** (2008) 81 \[arXiv:0802.4353 \[hep-ph\]\].
M. Baak, M. Goebel, J. Haller, A. Hoecker, D. Kennedy, R. Kogler, K. Moenig and M. Schott et al., *The Electroweak Fit of the Standard Model after the Discovery of a New Boson at the LHC*, *Eur. Phys. J. C* **72** (2012) 2205 \[arXiv:1209.2716 \[hep-ph\]\].
M. Spira, *HIGLU: A program for the calculation of the total Higgs production cross-section at hadron colliders via gluon fusion including QCD corrections*, hep-ph/9510347.
R. V. Harlander, S. Liebler and H. Mantler, *SusHi: A program for the calculation of Higgs production in gluon fusion and bottom-quark annihilation in the Standard Model and the MSSM*, *Comput. Phys. Commun.* **184** (2013) 1605 \[arXiv:1212.3249 \[hep-ph\]\].
https://twiki.cern.ch/twiki/bin/view/LHCPhysics/CrossSectionsFigures . P. M. Ferreira, R. Santos, H. E. Haber and J. P. Silva, Phys. Rev. D [**87**]{}, no. 5, 055009 (2013) \[arXiv:1211.3131 \[hep-ph\]\].
D. Fontes, J. C. Romão and J. P. Silva, Phys. Rev. D [**90**]{}, no. 1, 015021 (2014) \[arXiv:1406.6080 \[hep-ph\]\]. P. M. Ferreira, R. Guedes, M. O. P. Sampaio and R. Santos, JHEP [**1412**]{} (2014) 067 \[arXiv:1409.6723 \[hep-ph\]\]. J. F. Gunion and X. G. He, Phys. Rev. Lett. [**76**]{} (1996) 4468 \[hep-ph/9602226\]. J. Ellis, D. S. Hwang, K. Sakurai and M. Takeuchi, JHEP [**1404**]{} (2014) 004 \[arXiv:1312.5736 \[hep-ph\]\]. X. G. He, G. N. Li and Y. J. Zheng, arXiv:1501.00012 \[hep-ph\]. F. Boudjema, R. M. Godbole, D. Guadagnoli and K. A. Mohan, arXiv:1501.03157 \[hep-ph\]. V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt and D. Zeppenfeld, Nucl. Phys. B [**616**]{} (2001) 367 \[hep-ph/0108030\]. B. Field, Phys. Rev. D [**66**]{} (2002) 114007 \[hep-ph/0208262\]. M. J. Dolan, P. Harris, M. Jankowiak and M. Spannowsky, Phys. Rev. D [**90**]{} (2014) 7, 073008 \[arXiv:1406.3322 \[hep-ph\]\].
S. Berge, W. Bernreuther and J. Ziethe, Phys. Rev. Lett. [**100**]{} (2008) 171605 \[arXiv:0801.2297 \[hep-ph\]\]. S. Berge, W. Bernreuther and S. Kirchner, Eur. Phys. J. C [**74**]{} (2014) 11, 3164 \[arXiv:1408.0798 \[hep-ph\]\]. J. Baron [*et al.*]{} \[ACME Collaboration\], Science [**343**]{}, 269 (2014) \[arXiv:1310.7534 \[physics.atom-ph\]\]. A. J. Buras, G. Isidori and P. Paradisi, Phys. Lett. B [**694**]{}, 402 (2011) \[arXiv:1007.5291 \[hep-ph\]\]. J. M. Cline, K. Kainulainen and M. Trott, JHEP [**1111**]{}, 089 (2011) \[arXiv:1107.3559 \[hep-ph\]\]. M. Jung and A. Pich, JHEP [**1404**]{}, 076 (2014) \[arXiv:1308.6283 \[hep-ph\]\]. J. Shu and Y. Zhang, Phys. Rev. Lett. [**111**]{}, no. 9, 091801 (2013) \[arXiv:1304.0773 \[hep-ph\]\]. W. Dekens, J. de Vries, J. Bsaisou, W. Bernreuther, C. Hanhart, U. G. Mei[ß]{}ner, A. Nogga and A. Wirzba, JHEP [**1407**]{} (2014) 069 \[arXiv:1404.6082 \[hep-ph\]\].
H. Ono and A. Miyamoto, Eur. Phys. J. C [**73**]{} (2013) 2343. D.M. Asner, T. Barklow, C. Calancha, K. Fujii, N. Graf, H.E. Haber, A. Ishikawa, S. Kanemura [*et al.*]{}, arXiv:1310.0763 \[hep-ph\].
[^1]: The latest results on the measurement of the Higgs mass are $125.36 \pm 0.37$ GeV from ATLAS [@Aad:2014aba] and $125.02 +0.26 -0.27$ (stat) $+0.14 -0.15$ (syst) GeV from CMS [@Khachatryan:2014jba].
[^2]: We can choose a parametrization where the angle $\alpha_1$ is exactly $\alpha$ in that limit. See for example the definition of the rotation matrix in [@Inoue:2014nva] as compared to our equation \[matrixR\].
[^3]: See, for instances, equations (A7)-(A9) and (A14) in [@Barroso:2012wz].
|
---
abstract: 'Understanding the formation of earliest supermassive black holes is a question of prime astrophysical interest. In this chapter, we focus on the formation of massive black holes via gas dynamical processes. The necessary requirement for this mechanism are large inflow rates of about 0.1 solar mass per year. We discuss how to obtain such inflow rates via an isothermal collapse in the presence of atomic hydrogen cooling, and the outcome of such a collapse from three dimensional cosmological simulations in subsection 2.2. Alternatives to an isothermal direct collapse are discussed in subsection 3 which include trace amounts of metals and/or molecular hydrogen. In the end, we briefly discuss future perspectives and potential detection of massive black hole seeds via upcoming missions.'
address: |
Department of Physics, United Arab Emirates University,\
Po Box, 15551, Al-Ain, UAE,\
latifne@gmail.com
author:
- 'Muhammad A. Latif'
bibliography:
- 'ref.bib'
title: Formation of the First Black Holes
---
[.5ex]{}[ ]{} [.5ex]{}[ ]{}
[.5ex]{}[ ]{} [.5ex]{}[ ]{}
Black hole formation via gas-dynamical processes$^1$ {#dcbh}
=====================================================
Introduction
------------
Understanding the formation of supermassive black holes remains an open and fascinating issue, as outlined in detailed reviews on this topic by @Volonteri10 [@Volonteri2012; @Haiman13; @LatifFerrara2016]. The three main pathways for the formation of supermassive black holes are: (1) stellar remnants, (2) seed black holes forming in dense stellar clusters via dynamical processes, (3) monolithic collapse of proto-galactic gas cloud into a massive black hole so-called direct collapse scenario. In case of stellar remnants, the most promising pathway is likely in the context of massive Population III stars, as introduced in the previous chapter 4. Such a mechanism, if it was to produce the observed supermassive black holes, would certainly require a mechanism of super-Eddington accretion, as described in chapter 11, which may however also be present in other scenarios. Black hole formation via collisions in stellar clusters will be discussed in chapter 7, and the observed masses and constraints on the supermassive black hole population at high redshift are reported in chapter 12. In this chapter, the main focus will be on black hole formation via a monolithic collapse.
The idea for the formation of a massive black hole directly via the gas dynamical processes was conceived in the pioneering work of Martin Rees [@Rees1984]. The expectation is that gas in the low spin halos collapses on the viscous time scale, forms a rotationally supported compact disk which later may lead to the formation of a massive black hole (BH) [@Loeb1994; @Eisenstein1995]. Similarly, models proposed in the cosmological context suggest that conditions for the formation of a massive black hole are ideal in high redshift protogalaxies with the lowest angular momentum gas [@Koushiappas2004; @Volonteri2005; @Lodato2006]. It is proposed that gas can rapidly loose angular momentum via ’bars within bars’ instabilities and may lead to rapid formation of a self-gravitating core supported by gas pressure. Such a core catastrophically cools by thermal neutrino emission and contracts to potentially form a massive BH [@Begelman2006]. The first smoothed particle hydrodynamical simulations starting from idealised initial conditions showed that a low spin metal free halo collapses into a single clump while higher spin halo formed a binary [@Bromm03].
The key requirement for this scenario is that gas should rapidly collapse by efficiently transporting angular momentum and avoiding fragmentation. The goal is to bring large inflows ($\rm \geq 0.1~M_{\odot}/yr$, see discussion below) of gas into the halo centre within a short time scale of about $\sim$1 Myr and rapidly build up a massive object of $\rm 10^4-10^6~M_{\odot}$. Such large inflow rates can be obtained thermodynamically by keeping the gas warm as the inflow rate is $\propto T^{3/2}$ and also via dynamical processes such as ’bars within bars’ instabilities [@Shlosman1989; @Begelman2006] or in the aftermath of galaxy mergers [@Mayer2010]. Depending on the time evolution of mass inflow rates, the central object may form a supermassive star/quasi-star (details are mentioned below) or directly collapse into a massive black hole.
Black hole formation in primordial atomic gas
---------------------------------------------
The mass inflow rate ($\dot{M}$) of collapsing gas is related to its thermodynamical properties, as $\rm \dot{M} \sim c_s^3/G \sim 0.1 ~M_{\odot}/yr \left( \frac{T}{8000 ~K} \right)^{3/2} $ where $c_s$ is the sound speed and T is the gas temperature. The higher the sound speed the larger the mass inflow rate. Therefore, the thermodynamical requirement for getting large inflows is that gas should not cool down to lower temperatures, otherwise it will fragment and form ordinary stars. The cooling ability of the gas strongly depends on its chemical composition. In the presence of a trace amount of dust/metals, the gas cools down to a few tens of Kelvin by radiating away its thermal energy and forming stars. Even in primordial gas, molecular hydrogen cooling can bring the gas temperature down to $\rm \sim 200$ K and induces star formation. In the absence of molecular hydrogen, primordial gas remains in the atomic phase, cools mainly via atomic line radiation and the gas temperature remains around 8000 K. In atomic primordial gas collapse is expected to proceed isothermally with $\rm T \sim 8000~K$ and large mass inflow rates of the order of $0.1 ~M_{\odot}/yr$ can be achieved easily. Therefore, the conditions for forming a massive object are ideal in massive primordial halos with $T_{vir} \geq 10^4$ K cooled only via atomic lines. Such halos have masses of a few times $\rm \geq 10^7~M_{\odot}$, formed at $z>10$ and their gravitational potentials are sufficiently deep to allow the rapid collapse. Therefore, massive primordial halos deprived of $\rm H_2$ cooling are the potential cradles for the formation of massive black holes.
The prerequisites for the formation of massive black holes in atomic cooling halos are that they should be metal-free and the formation of molecular hydrogen remains suppressed. In the next sub-section, we discuss in detail how to quench the molecular hydrogen formation which could be detrimental for forming massive black holes via isothermal direct collapse (DC).
### Conditions to keep the gas primordial and atomic
Our understanding of structures in the cosmos is based on the hierarchical paradigm of structure formation according to which minihalos of $\rm 10^5-10^6~M_{\odot}$ were formed first at $z \sim 30-40$ which later merged to form larger halos. According to the Big Bang theory, the gas in the Universe initially has a primordial composition of predominantly hydrogen and helium. The first stars, so-called Population III stars, will then form out of a primordial gas in these minihalos. As structure formation proceeds, depending on the their mass, these stars end their lives as supernovae and pollute the ambient medium with metals. At earlier cosmic times, a few hundred Myrs after the Big Bang, the metal enrichment is expected to be patchy and also found from the large scale numerical simulations [@Trenti2009; @Maio2011; @Ritter2014; @Pallottini2014; @Habouzit2016a]. Therefore, some of the halos may remain pristine until their masses reach the atomic cooling regime (above $10^7~M_{\odot}$). Indeed, there is observational evidence that pockets of metal free gas can exist down to z=7 [@Simcoe2012] and even at z=3 [@Fumagalli2011]. The estimates about the fraction of metal free halos computed from cosmological simulations including self-consistently star formation and supernova feedback suggest that about 40 % of halos remain metal free down to z=10 [@Latif2016D; @Habouzit2016b], see figure \[figure1\].
![ Fraction of halos with metallicity below the given value in the figure legged and masses between $\rm 2 \times ~10^7-10^8~M_{\odot} $. Adopted from @Latif2016D, AAS. Reproduced with permission. []{data-label="figure1"}](fraction_halos.pdf)
As we discuss below, due to the requirement of a strong UV flux, a DC halo may form in the surroundings of an intensely star forming galaxy. This may affect the abundance of DC halos by metal pollution from supernova winds from a nearby star forming halo [@Dijksta2014; @Habouzit2016a]. Even such pollution can be avoided in a synchronised pair of halos where the halo which is the radiation source forms first while a rapid collapse in the DC halo helps in avoiding the metal pollution [@Visbal2014]. Moreover, cosmological hydrodynamical simulations starting from first principles show that metal ejection preferentially occurs in the low density regime [@Ritter2014; @Pallottini2014] and neighboring halos may remain metal free. In a nutshell, metal pollution is not a bottleneck in the formation of DCBHs.
The second main constraint for DCBHs is that the formation of $\rm H_2$ should remain suppressed in DC halos. Trace amount of $\rm H_2$ can be formed via gas phase reactions where a residual fraction of electrons from the recombination epoch acts as a catalyst. As discussed in the previous chapter on the chemistry of the early universe, the main pathway for the formation of $\rm H_2$ is the following: $$\mathrm{H + e^{-} \rightarrow H^{-} +} \gamma$$ $$\mathrm{ H + H^{-} \rightarrow H_{2} + e^-.}\\
\label{h21}$$
The $\rm H_2$ can be dissociated either directly or indirectly by UV radiation depending on on the stellar spectra. The low energy photons with $\rm 0.75$ eV can photo-detach $\rm H^-$ which is the main channel for the formation of $\rm H_2$. The photons with energy between 11.2-13.6 eV, so-called Lyman Werner (LW) photons directly photo-dissociate molecular hydrogen via the Solomon process. The dissociation processes are described by the following reactions: $$\mathrm{H_{2}} + h\nu \mathrm{\rightarrow H + H}
\label{h20}$$ $$\mathrm{H^{-}} + h\nu \mathrm{\rightarrow H + e^{-}}\\
\label{h2}$$
The competition between the formation and dissociation timescales defines the critical value of UV flux (hereafter $J_{21}^{crit}$) above which the formation of $\rm H_2$ remains quenched. Pop. III stars with $T_{rad}=10^5$ K produce more high energy photons and are very effective in directly dissociating $\rm H_2$ while normal stars with $\rm T_{rad}=10^4$ K photo-detach $\rm H^-$. The previous studies employed idealized spectra to compute the $J_{21}^{crit}$ and found that it varies from 30-300 for $\rm T_{rad}=10^4$ K and 1000 for $\rm T_{rad}=10^5$ K [@Omukai2001; @Shang2010; @Latif2014; @Johnson2014]. However, recent estimates of $J_{21}^{crit}$ for a realistic spectra of the first galaxies suggest that it has no single value but it depends on the stellar age, metallicity and mode of star formation [@Sugimura2014; @Agarwal2015; @Agarwal2016]. Moreover, such spectra can be mimicked with $\rm T_{rad}=2 \times 10^4-10^5$ K. The values of $J_{21}^{crit}$ from cosmological simulations for a realistic spectra of first galaxies found that it further depends on the properties of the host halos and varies between 20,000-50,000 considering a uniform background UV flux [@Latif2015] or anisotropic flux [@Regan2014B; @Regan15b]. X-rays catalyse $\rm H_2$ formation by boosting electron fraction and consequently the value of $J_{21}^{crit}$ may further get enhanced [@Latif2015; @Inayoshi2015; @Regan2016], see figure \[figure2\]. The accurate treatment of $\rm H_2$ self-shielding is also necessary to calculate $J_{21}^{crit}$ [@WolcottGreen2011; @Hartwig2015].
![Estimates of the critical value of the UV flux ($J_{21}^{\rm crit}$) both from one zone models and 3D simulations including variations from halo to halo, dependence on the radiation spectra and the impact of X-ray ionization. Adopted from @Latif2015, reproduced by permission of Oxford University Press / on behalf of the RAS. []{data-label="figure2"}](Sum1.pdf)
Estimate of $J_{21}^{crit}$ and the fraction of metal free halos are crucial to estimate the the number density of direct collapse black holes and to assess the feasibility of this scenario. A detailed assessment of its dependence on the stellar spectra reflecting our present understanding is therefore provided in chapter 6.
### Outcome of an isothermal collapse
In this section, we discuss the outcome of an isothermal collapse in massive primordial halos with $T_{vir} \geq 10^4$ K illuminated by a strong LW flux above the critical strength. Numerical cosmological simulations performed under these conditions confirm that the gas collapses isothermally in dark matter potentials with $\rm T \sim 8000$ K in a self-similar way and the density profile follows an $R^{-2}$ behaviour [@Bromm03; @Wise2008; @Regan09; @Latif2011; @Latif2013c; @Inayoshi2014; @Bcerra2014; @Latif2016]. The gas continues to cool by Lyman alpha radiation and keeps collapsing. At densities of $\rm 10^{8}~cm^{-3}$, cooling due to $\rm H^-$ comes into play and brings the gas temperature down to 5000 K [@VanBorm2014; @Latif2016]. Above $\rm 10^{16}~cm^{-3}$, the gas cloud becomes optically thick to both Lyman alpha and $\rm H^-$ cooling, and the temperature starts to rise as shown in figure \[fig3\]. The collisional ionization cooling becomes important and maintains the gas temperature close to $\rm 10^4$ K. Eventually, at densities higher than $\rm 10^{20}~cm^{-3}$, the gas cloud becomes completely opaque and collapses adiabatically.
![Spherically averaged and radially binned profiles of density, temperature, mass and mass inflow rates are depicted here. The simulated halos are metal free and illuminated by a strong LW flux above the $J_{21}^{crit}$. Consequently, $\rm H_2$ formation remains suppressed and cooling proceeds via atomic lines. Adopted from @Latif2016, reproduced by permission of Oxford University Press / on behalf of the RAS. []{data-label="fig3"}](Temperature1.pdf)
Under isothermal conditions, the gas is expected to collapse monolithically without fragmentation. Previous low resolution studies employing a fixed Jeans resolution of four cells confirmed this hypothesis [@Regan09; @Latif2011]. However, recent work employing a detailed chemical model, resolving the collapse to unprecedentedly high densities of $\rm 10^{21}~cm^{-3}$ with a Jeans resolution of 32 cells shows that fragmentation occasionally occurs depending on the properties of host halos, but does not prevent the formation of a massive central object [@Bcerra2014; @Latif2016]. The clumps forming due to the fragmentation on small scales quickly migrate inwards and merge with the central object [@Inayoshi2014b; @Latif2015Disk]. Analytical models of primordial disks around the central object suggest that in the presence of large inflows and rapid rotation, viscous heating stabilises the disk and helps in the formation of a massive central object [@Latif2015Disk2; @Schleicher2016]. Similarly, magnetic fields amplified via the so-called small scale dynamo reach the equipartition field strength within a dynamical timescale and provide a support against gravity. Such strong fields further help in suppressing the fragmentation in atomically cooled halos [@Schobera; @Schliecher2010dyn; @Latif2013a; @LatifMag2014]. On larger scales, the angular momentum in these halos gets transferred due to the gravitational torques exerted by the triaxility of DM haloes via ’bars within bars’ instabilities [@Choi2013; @Choi2015].
![Averaged density along the line of sight in the central 20 AU for two halos (top and bottom). The simulated halos are metal free and illuminated by a strong LW flux above the $J_{21}^{crit}$. Therefore, cooling proceeds via atomic lines and an isothermal collapse occurs. Adopted from @Latif2016, reproduced by permission of Oxford University Press / on behalf of the RAS. []{data-label="fig4"}](Density_Projection.pdf)
One of the salient features of forming DCBHs through isothermal collapse is that large inflow rates of $\rm 0.1-1~M_{\odot}/yr$ are easily available, see bottom right panel of figure \[fig3\]. If these inflow rates can be sustained for about one Myr then the formation of a massive object of $\rm \sim 10^4 - 10^5 ~M_{\odot}$ is feasible. In fact, it has been confirmed from cosmological numerical simulations that a massive central object of $\rm 10^5~M_{\odot}$ can be formed within a Myr after the initial collapse [@Latif2013d; @Shlosman2016], though these simulations did not employ the feedback from a supermassive proto-star. In fact, stellar evolution calculations show that for large mass accretion rates $\dot{m}$) of $\geq 0.1~M_{\odot}/yr$, the radius of the star monotonically increases with mass because of the short accretion time compared to the Kelvin Helmholtz contraction timescale [@Hosokawa12; @Schleicher13]. These stars have low surface temperatures of about 5000 K like supergaints and do not produce strong UV flux. Due to the short accretion time in comparison with the nuclear burning time, the core of such a supermassive star may collapse into a BH provided that sufficiently large accretion rates can be retained [@Begelman2008; @Begelman2010]. Such stars with a BH at their centre are called quasi-stars and it has been found that $\dot{m} >0.14~M_{\odot}/yr$ is required for their formation [@Schleicher13]. These supermassive/quasi-stars may collapse into a BH via general relativistic instabilities [@Volonteri2010b; @Ball2011; @Johnson2014; @Ferrara14] and are the potential embryos of massive black holes forming via direct collapse. A more detailed picture on the evolution of such supermassive stars is provided in chapter 8.
Most of the previous work studying the evolution of a supermassive star used constant mass accretion rates. However, recent studies employing time dependent accretion rates show that if the time interval between to consecutive episodes of accretion ($\Delta t_{acc}$) is longer than 1000 years then star can sufficiently contract and produces strong UV flux which may halt further accretion [@Sakurai2015]. It is expected that accretion onto the supermassive protostar will be episodic due to the possible fragmentation of proto-stellar disk and clumps inward migration. Under isothermal conditions, $\Delta t_{acc}$ is expected to be much shorter in 1000 years [@Sakurai2016; @Latif2015Disk2; @Latif2016]. Therefore, intermittent accretion does not halt the formation of a potential supermassive star. As mentioned earlier, due to the requirement of a strong LW flux, the host halos of DCBHs are expected to form in the close vicinity of actively star forming galaxy and some of these DC halos get tidally disrupted by the nearby massive halos and can not host a DCBH [@chon16]. Recently, radiation hydrodynamical simulations investigating the impact of ionising radiation on the formation of DCBHs found that rich structures such as filaments and clumps between DC halo and star forming galaxy shield it from ionising photons [@Chon2017]. So, the formation of DCBHs is expected to continue under these conditions.
### Outcome of collapse under less idealized conditions
Various alternatives to isothermal direct collapse mediated by a strong LW flux have been proposed. For large columns ($\rm \geq 10^{20}~cm^{-2}$) of neutral hydrogen, the escape time for Lyman alpha photons becomes longer than the cloud collapse time scale. Consequently, Lyman alpha photons get trapped inside the cloud, the equation of state gets stiffened, the temperature of the gas cloud starts to increase and eventually collapse proceeds adiabatically [@Spaans2006]. However, later studies found that cooling still proceeds via 2s-1s transition and collapse becomes isothermal identical to the Lyman alpha cooling case [@Schleicher10; @Latif2011b]. Similarly, large baryonic streaming motions (3 $\sigma$ fluctuations) produced prior to the epoch of recombination naturally increase the critical mass for $\rm H_2$ cooling until it reaches the atomic cooling limit. Such motions may also collisionaly dissociate $\rm H_2$ molecules and avoid metal enrichment by suppressing in-situ star formation [@Tanaka2013; @Tanaka2014]. However, it was found that streaming motions are not effective in quenching $\rm H_2$ formation and may require the ubiquity of a strong LW flux [@Latif2014Stream].
The large inflow rates of the gas required to assemble massive back holes can be obtained dynamically via ’bars within bars’ instabilities, and fragmentation even in metal rich gas may be suppressed via supersonic turbulence [@Begelman2009]. However, studies of contemporary star formation show that supersonic turbulence locally compresses the gas and induces star formation in molecular clouds [@Federrath2010; @Federrath11]. During galaxy mergers, gravitational torques drive large inflows of about $10^3-10^4~M_{\odot}/yr$ to the halo centre and form a compact circumnuclear disk which later may coalesce into a massive BH [@Mayer2010]. The cooling timescale of such disk is very short, of the order of 100 years, and hence the resulting mass scale is still controversial [@Ferrara13]. In a recent study, improving on their previous work of galaxy mergers, @Mayer2015 found that the gas becomes optically thick to cooling radiation and the central stable core may directly collapse into a massive BH via general relativistic instabilities. In this scenario, the central core collapse must avoid the stellar phase otherwise it will blow away most of the mass via stellar winds as the mass loss from star is directly proportional to the metallicity of the gas.
So far, we have mainly focused on the isothermal direct collapse with emphasis on keeping the halo metal free and devoid of $\rm H_2$. The studies of primordial star formation suggest that although the protostellar disk forming in natal primordial clouds fragments, most of the clumps migrate inward, leading to intermittent accretion and merging with the central protostar. The typical accretion rates in primordial minihalos are $\rm 10^{-2}-10^{-4}~M_{\odot}/yr$ which is about 3-4 orders of magnitudes larger than the present day star formation in molecular clouds. Moreover, the accretion rate in the bursty mode exceeds $\rm 10^{-2}~M_{\odot}/yr$, which keeps the stellar envelope bloated up and consequently the supermassive star produces weak UV feedback. Thus, even in the presence of $\rm H_2$, massive stars up to $\rm 1000 ~M_{\odot}$ or even higher may form [@Hirano2014; @Latif2015Disk; @Hosokawa2016].
The numerical experiments exploring collapse in massive primordial halos with $\rm T_{vir} \geq 10^4$ K irradiated by moderate strength of LW flux found that a trace amount of molecular hydrogen forms in the halo centre surrounded by warm gas with $T \sim 8000$ K. It was found that fragmentation occasionally occurs but clumps migrate inward and merge with the central star. Large inflow rates of $\rm \sim 0.1~M_{\odot}/yr$ are generated in halos which have recently gone through a major merger [@Latif2014ApJ; @LatifVolonteri15]. Under these conditions, the central star may reach $\rm 10^4~M_{\odot}$ within 1 Myr even if in situ star formation occurs in these halos. Estimates about the mass of the central object for various strengths of LW flux below $J_{21}^{crit}$ are shown in figure \[fig5\]. These findings suggest that a massive central object/star of $\rm 10^3-10^4~M_{\odot}$ can be formed within 1 Myr after the initial collapse for moderate LW flux. Even if in situ star formation occurs, the gas flow to the central star may still continue until stars reach the main-sequence and below away the gas. This study was performed assuming that the spectra of Pop. II stars is soft with $\rm T_{rad}= 10^4~K$. However, similar results are expected for a realistic spectra of Pop. II stars.
![Expected stellar masses for different strengths of background LW flux. The dashed vertical shaded region represents the range of the critical values. The blue and red spheres two representative halos. Adopted from @Latif2014ApJ, AAS. Reproduced with permission. []{data-label="fig5"}](MassUV.pdf)
In the presence of a strong LW flux ,star formation within the halo remains suppressed but the halo may be polluted by supernova winds from a nearby star forming galaxy. It has been found that for a trace amount of metals/dust as low as $Z = 10^{-6}Z_{\odot}$, where Z is the metallicity of the halo and $Z_{\odot}$ is the solar metallicity, dust cooling becomes important at densities above $\rm 10^8-10^{12}~cm^{-3}$ and lowers the gas temperature to a few hundred K [@Omukai2008; @Latif2016D]. For very low metallicities $\rm \leq 10^{-5}$ and a strong LW flux, cooling remains confined to the central 10 AU and large inflows of gas are available. The density structure for two halos illuminated by a strong LW flux and polluted by a trace amount of metals are shown in figure \[fig6\]. It has been found that for $\rm Z/Z_{\odot}= 10^{-4}$, due the efficient dust cooling, a filamentary structure emerges and gravitationally bound clumps form. While for $\rm Z/Z_{\odot}= 10^{-5}$, the density structure remains spherical and gravitationally unbound sub-solar clumps are observed. The clumps for the cases with $\rm Z/Z_{\odot} \leq 10^{-5}$ cases may migrate inwards and merge with the central object. In the case of efficient fragmentation, a dense stellar cluster is expected to form as fragmentation occurs in the very inner part of the halo. It is expected that the run-away collisions in such a dense cluster may lead to the formation of a very massive central star which may later collapse into a massive BH of about a thousand solar masses. So, the conditions for growing a massive object in very metal poor halos with $\rm Z \leq 10^{-5}~Z_{\odot}$ and illuminated by a strong UV flux are still favourable [@Latif2016D].
![ Average gas density along x-axis for metallicities of $\rm Z/Z_{\odot}=10^{-6}, 10^{-5}, 10^{-4}$ and is shown for the central 4000 AU of a halo. Each row represents a halo (halo 1 on top and halo 2 on bottom) and each column represents metallicity (increasing from left to right). Adopted from @Latif2016D, AAS. Reproduced with permission.[]{data-label="fig6"}](Density_Zoom_x.pdf)
Future outlook
--------------
Despite the tremendous progress made regarding our understanding of massive black hole formation during the past decade, there are still many open questions. It is yet not clear what is the final outcome of an isothermal collapse, if it is always a supermassive star or a quasi-star or a direct massive black hole. The final result depends on the long term sustainability of the mass accretion rates and the amount of angular momentum retained during the end stages of gravitational collapse. Both of these quantities remain uncertain and are not fully comprehended due to the numerical constraints. In the future, numerical simulations starting from ab initio initial conditions should be performed to assess the feasibility of both parameters and to determine the ultimate fate of direct collapse. Similarly, stellar evolution calculations of supermassive proto-star formation ignored the role of rotation which may impede the collapse and may shut further mass accretion.
Some studies suggest that rapid accretion may launch strong winds leading to large mass loss [@Dotan2011; @Fiacconi2016]. Also, preliminary work exploring the role of rotation during the quasi-stellar phase indicates that the quasi-stellar phase may be skipped for objects massive than $\rm 10^5~M_{\odot}$ [@Fiacconi2017]. However, detailed three-dimensional radiation hydrodynamical computations are necessary to support/reject this finding. The dynamical ways to obtain large inflow rates look promising, but it is not completely clear whether such inflows can be maintained down to AU scales. Also, whether metal-enriched gas remains optically thick or radiates away thermal energy to form stars remains to be determined. Detailed investigations exploring this channel will thus be necessary.
One of the biggest uncertainty in assessing the feasibility of direct collapse black holes is their number density which sensitively depends on the value of $\rm J_{21}^{crit}$. The variation in $J_{21}^{crit}$ by a factor of a few changes the abundance of direct collapse black holes by an order of magnitude or even larger. Our current understanding suggests that values of $J_{21}^{crit}$ vary by orders of magnitude and may also change with the age of stars and metallicity. In the future, more realistic estimates of $\rm J_{21}^{crit}$ are required from 3D radiation hydrodynamical simulations by properly modelling the SED of the source galaxy for different modes of star formation, as also discussed in chapter 6.
Observational evidence is necessary to constrain these models of black hole formation. The first observations of CR7, the brightest Lyman alpha emitter at z=6.6, revealed that it shows strong Lyman alpha and He-1640$~\AA$ emission and no signatures of metal lines were observed [@Sobral]. The drivers of strong Lyman alpha and He-1640$\AA$ can be either a cluster of primordial stars with $\rm 10^7~M_{\odot}$ or a direct collapse black hole of $\rm 10^6-10^7~M_{\odot}$. However, due to the metal enrichment forming such a young massive cluster of primordial stars at redshifts as low as z= 6 seems infeasible and a black hole of $\rm 10^6~M_{\odot}$ possibly forming via DC is more likely source of CR7 [@Pallottini2015; @Hartwig16; @Agarwal16; @Dijkstra2016; @Smidt2016; @Smith2016]. The recent deep observations of CR7 show doubly ionized oxygen (OIII) emission [@Bowler17] but the current photometry is still consistent with a mild amount of pollution DCBH site with metals [@Hartwig16; @Pacucci2017; @Agarwal2017]. It is expected that JWST[^1] will be able to directly probe the observational signatures of DCBHs in high redshift galaxies.
The future X-ray space observatory ATHENA[^2] is expected to detect a few hundred low luminosity active galactic nuclei (AGN) in the early universe with X-ray luminosities of $\rm L_{X} \geq 10^{43} erg/s$. These observations will provide direct constraints on the masses and luminosity functions of the first AGN. Also future 21 cm experiments such as SKA[^3] and LOFAR[^4] will probe the proximity zones of high redshift quasars and help in better understanding their formation and growth mechanisms [@Whalen2017]. In the local universe the deep observation of dwarf galaxies may help in tracing the seed BHs forming at high redshift [@Reines2016]. The recent detections of BH-BH mergers with LIGO[^5] have opened a new window of gravitational waves astronomy and the upcoming European space mission eLISA[^6] will be able to detect the merging of massive black holes and help in understanding their formation mechanisms.
In the next chapters, we will clarify uncertainties in the value of $\rm J_{21}^{crit}$ (chapter 6), which is crucial to quantify the expected number of black holes. Black hole formation through mergers in stellar clusters will then be explored in chapter 7, and the evolution of supermassive stars is described in chapter 8. The growth of seed black holes, potentially forming from the first stars, will be discussed in chapters 10 and 11. Statistical predictions will be provided in chapter 9. A comparison with current observations of high-redshift quasars is given in chapter 12, while predictions on gravitational wave emission are provided in chapter 13, and expectations for future observations are outlined in chapter 14.
[^1]: https://jwst.nasa.gov
[^2]: http://www.the-athena-x-ray-observatory.eu
[^3]: http://skatelescope.org
[^4]: http://www.lofar.org/
[^5]: https://www.ligo.caltech.edu
[^6]: https://www.elisascience.org
|
---
abstract: |
We report long-slit spectroscopic observations of the dust-lane polar-ring galaxy obtained with the Southern African Large Telescope (SALT) during its performance-verification phase. The observations target the spectral region of the H$\alpha$, \[\] and \[\] emission-lines, but show also deep [NaI]{} stellar absorption lines that we interpret as produced by stars in the galaxy. We derive rotation curves along the major axis of the galaxy that extend out to about 8 kpc from the center for both the gaseous and the stellar components, using the emission and absorption lines. We derive similar rotation curves along the major axis of the polar ring and point out differences between these and the ones of the main galaxy.
We identify a small diffuse object visible only in H$\alpha$ emission and with a low velocity dispersion as a dwarf galaxy and argue that it is probably metal-poor. Its velocity indicates that it is a fourth member of the galaxy group in which belongs.
We discuss the observations in the context of the proposal that the object is the result of a major merger and point out some observational discrepancies from this explanation. We argue that an alternative scenario that could better fit the observations may be the slow accretion of cold intergalactic gas, focused by a dense filament of galaxies in which this object is embedded.
Given the pattern of rotation we found, with the asymptotic rotation of the gas in the ring being slower than that in the disk while both components have approximately the same extent, we point out that may be a galaxy in which a dark matter halo is flattened along the galactic disk and the first object in which this predicted behaviour of polar ring galaxies in dark matter haloes is fulfilled.
author:
- |
Noah Brosch,$^{1,2}$[^1] Alexei Y. Kniazev,$^{2,3}$ David Buckley,$^{2}$ Darragh O’Donoghue,$^{2}$ Yas Hashimoto,$^{2}$ Nicola Loaring,$^{2}$ Encarni Romero,$^{2}$ Martin Still,$^{2}$ Petri Vaisanen,$^{2}$ Eric B. Burgh,$^{4}$ Kenneth Nordsieck$^{4}$\
$^{1}$The Wise Observatory and the School of Physics and Astronomy, the Raymond and Beverly Sackler Faculty of Exactâ Sciences,\
Tel Aviv University, Tel Aviv 69978, Israel\
$^{2}$South African Astronomical Observatory, Observatory Road, Cape Town, South Africa\
$^{3}$Special Astrophysical Observatory, Nizhnij Arkhyz, Karachai-Circassia, 369167, Russia\
$^{4}$Space Astronomy Laboratory, University of Wisconsin, Madison, WI 53706, USA
date: 'Accepted 2007 April ??. Received 2007 March ??; in original form 2007 March ??'
title: 'The polar ring galaxy revisited [^2] '
---
\[firstpage\]
galaxies: ring galaxies — galaxies: evolution — galaxies: individual: — galaxies: dark matter — galaxies: galaxy haloes
Introduction {#introduction .unnumbered}
============
Ring galaxies posed significant astronomical interest since @LT76 modelled the Cartwheel galaxy as the result of a small galaxy passing through a larger one. While such events probably happen and produce some of the ring galaxies, in other instances different mechanisms might be at work. A particularly interesting kind of ring galaxy is the polar ring galaxy (PRG) where a flattened disk galaxy exhibits an outer ring of stars and interstellar matter that rotate in a plane approximately perpendicular to the central disk. An extensive catalog of PRGs was produced by @Whietal90.
The issue of PRGs was reviewed by @Co06. She reviewed a number of formation mechanisms for PRGs: minor or major mergers, tidal accretion events, or direct cold gas accretion from filaments of the cosmic web. @Co06 proposed that these objects can be used to probe the three-dimensional shape of dark matter (DM) haloes, provided the PRG is in equilibrium in the gravitational potential.
The well-known Spindle Galaxy (NGC 2685), an archetypal PRG, exhibits two sets of rings: an outer one visible only on HI maps and which might be in the plane of the galaxy, and an inner one that is helix-shaped, is perpendicular to the main axis of the galaxy, is optically bright, shows embedded present-day star formation, and is associated with prominent dust lanes. Shane (1980) explained the system as consisting of a lenticular galaxy that recently accreted an HI gas cloud that formed the inner ring, while the outer gas ring might be a remnant of the formation of the galaxy. Hagen-Thorn (2005) found that the stellar population of the inner system of dust and gas, arranged in a spiral around the ”spindle” but really in a disk, is 1.4$\times10^9$ years old.
In a different ring galaxy, NGC 660, Karataeva (2004) detected red and blue supergiants belonging to the ring system. They showed that the age of the youngest stars there is only $\sim$7 Myr; thus star formation is currently taking place. N660 is special in that both the disk and the polar ring contain stars, gas and dust. @Resh04, who analyzed three other ring galaxies, showed that their rings result from ongoing interactions or mergers where the main galaxy is a spiral and the rings are currently forming stars.
Other claims of interactions being at the origin of the rings and of the star formation taking place therein have been put forward by Mayya & Korchagin (2001, revised 2006). On the other hand, others claimed that rings are formed as a dynamical event in a larger disk galaxy (e.g., Mazzuca 2001). It is clear that more studies of ring galaxies, and in particular such investigations that can time the ring and star formation events, can help understand the particular instances when a galaxy-galaxy interaction took place, when a ring is formed, and when the event does trigger the SF process. There is also the possibility that careful tracing of the polar ring and of the galaxy itself, and their kinematic properties, might reveal the DM halo shape and properties, as advocated by @Co06. This singles out PRGs as valuable targets for DM studies.
In this paper we analyze new observations of the polar-ring galaxy , a PRG with an optical redshift of 11649$\pm$10 km sec$^{-1}$ located at l=341.02, b=-28.73, also identified as PRC B-18 in Whitmore (1990). The object was recently studied by @Resh06, who showed that this is a giant galaxy in a compact triplet, together with PGC 400092 (classified Sd/Irr:) and PGC 399718 (classified SBc:) at approximately the same redshift. The authors used the 1.6-meter telescope of the Pico dos Dias Observatory in Brazil for imaging in BVRI, the CTIO 1.5-meter telescope to collect spectral observations, and included data from IRAS and 21-cm line observations. However, most of their conclusions about the nature of the object rely on the morphological appearance of the galaxy.
@Resh06 modelled using an N-body code that includes gas dynamics using sticky particles and star formation. They concluded that the best-fitting model is of a major merger, whereby a gas-rich galaxy transferred a sizable amount of matter to during a parabolic encounter. The matter subsequently relaxed and now forms a complete ring of stars, gas, and dust around whereas the donor galaxy is one of the two other galaxies in the same group.
The reason to revisit this object was the availability of high-quality spectra obtained with the effectively 8-meter diameter Southern African Large Telescope (SALT) telescope. We derive, for the first time, rotation curves for the ionized gas and for the stellar components of both the main galaxy and the polar ring. Since PRGs might make good test cases for the properties of dark matter haloes in and around galaxies, as argued by @Co06, the more observational data collected on these objects and with higher quality, the better.
Very few PRG observations obtained with large telescopes have been published. A noticeable one is by Swaters & Rubin (2003), with the Baade 6.5-meter telescope on Las Campanas, tracing the dynamics of the stellar component of the prototype PRG NGC 4650A where they showed that the polar ring is actually a polar disk, an extended feature rather than a narrow gas disk. They favour a scenario by which the ring/disk was formed by the polar merger to two similar disks, as previously suggested by Iodice (2002). Iodice (2006) observed the gaseous component in the ring of N4650A with ESO’s FORS2 on UT4 and concluded that a scenario by which it could be formed was through slow gas accretion from the cosmic web filaments. We propose that the same situation could be taking place for .
This paper is organized as follows: § \[txt:Obs\_and\_Red\] gives a description of all the observations and data reduction. In § \[txt:results\] we present our results, analyze them in § \[txt:disc\], and present our interpretation in § \[txt:interp\]. The conclusions drawn from this study are summarized in § \[txt:summ\].
------------ -------------- ------------- ------ ------------------ ---------
Date Exp.time Spec. Range Slit [1]{}[c]{}[PA]{} Disp.
(sec) (Å) () ($^\circ$) (Å/pix)
16.07.2006 2$\times$600 3650–6740 1.5 140 0.98
16.07.2006 1$\times$600 3650–6740 1.5 35 0.98
20.09.2006 2$\times$900 6050–7315 1.5 140 0.40
20.09.2006 1$\times$750 6050–7315 1.5 27 0.40
21.09.2006 3$\times$900 6050–7315 1.5 27 0.40
------------ -------------- ------------- ------ ------------------ ---------
: Details of the RSS observations[]{data-label="t:Obs"}
[ ]{}
[ ]{}
[ ]{}
[ ]{}
Observations and data reduction {#txt:Obs_and_Red}
===============================
SALT was described by Buckley (2006) and by O’Donoghue (2006), its Robert Stobie Spectrograph (RSS) was described by Burgh (2003) and Kobulnicky et al. (2003), and the first scientific papers based on its observations were published by Woudt (2006) and by O’Donoghue (2006). We used the SALT and RSS to observe . The observations of were obtained during the Performance Verification (PV) phase of the SALT telescope with the RSS spectrograph and are described in Table \[t:Obs\].
The July 2006 spectra (see Table \[t:Obs\]) were obtained during unstable weather conditions (high humidity, seeing worse than 5), without fully stacking the SALT mirrors. They cover the range 3650–6740 Å with a spectral resolution of $\sim$1.0 Å pixel$^{-1}$ or a FWHM of 6–7 Å. These spectra do not show strong and extended emission lines but were used to measure equivalent widths (EWs) of absorption lines in that spectral range following observations.
The spectra obtained on the nights of September 2006 were taken during stable weather conditions with seeing $\sim$15. They cover the range from $\sim$6050Å to $\sim$7300Å with a spectral resolution of 0.4 Å pixel$^{-1}$ or 2.4 Å FWHM. All data were taken with a 15 wide slit and a final scale along the slit of 0258 pixel$^{-1}$ (after binning the CCDs by a factor of two). Each exposure was broken up into 2–3 sub-exposures to allow the removal of cosmic rays. Spectra of a Cu–Ar comparison lamp were obtained after the science exposures to calibrate the wavelength scale.
The September 2006 data include two spectra obtained at position angle 140$^{\circ}$ centered on extending about four arcmin along the galaxy’s major axis and at a shallow angle to the dust lane, where the northern part passes also through the “northwest companion” PGC 400092 [@Resh06], and three spectra centered on the same position but obtained at position angle 27$^{\circ}$, along the major axis of the ”polar ring” described by @Resh06. We emphasize that the sampling of the major axis spectra was at PA=140$^{\circ}$, not at 130$^{\circ}$ as done by @Resh06, since 140$^{\circ}$ is closer to the position angle of the disk as given by Reshetnikov (148$^{\circ}$) and allows for a moderate degree of disk warping.
Although the observations discussed here are mostly spectroscopic, one image of the galaxy was obtained with a two-sec exposure in the V filter with the SALTICAM camera (O’Donoghue 2006) prior to the spectrometer observations in order to adjust the slit orientation, and is shown here as Figure \[fig:AM\_direct\]. The $\sim$15 seeing during the observations, and the problematic image quality SALT exhibited at that time, which can be evaluated from the stellar images on Figure \[fig:AM\_direct\_masked\] (see below), caused the images far from the good-quality $\sim$3 arcmin region to assume complicated shapes. The full SALTICAM image is $\sim$10 arcmin across with 0.28 arcsec/pixel (after binning on-chip by a factor of two).
The data for each RSS chip were bias and overscan subtracted, gain corrected, trimmed and cross-talk corrected, sky-subtracted and mosaiced. All the primary reduction was done using the IRAF[^3] package [*salt*]{}[^4] developed for the primary reduction of SALT data. Cosmic ray removal was done with the FILTER/COSMIC task in MIDAS.[^5] We used the IRAF software tasks in the [*twodspec*]{} package to perform the wavelength calibration and to correct each frame for distortion and tilt. One-dimensional (1D) spectra were then extracted using the IRAF APALL task.
Figures \[fig:AM\_2D\_130\] and \[fig:AM\_2D\_27\] show parts of fully reduced and combined spectral images for PA=140$^{\circ}$ and PA=27$^{\circ}$, respectively. Figure \[fig:AM\_1D\_130\] shows the spectrum of the central part of . The $\sim$40Å missing sections at $\sim \lambda \lambda$ 6500 and 6930Å are produced by small gaps between the three CCDs of the RSS. The noisy region of the RSS images shown in Figs. \[fig:AM\_2D\_130\] and \[fig:AM\_2D\_27\] near $\sim$6685Å is a subtraction artifact of laser light scattered into the RSS from SALT’s interferometric auto-collimating system. Figure \[fig:AM\_1D\_130\] shows the 1D spectra of the central part of extracted from the 2D spectra. Figure \[fig:PGC40092\_1D\] shows the 1D spectrum of the galaxy PGC 400092 extracted from the 2D spectrum observed at PA=140.
The derived internal errors for the 2D wavelength calibrations were small and did not exceed 0.04 Å for a resolution of 0.4 Å pixel$^{-1}$, or $<$2 km s$^{-1}$ at the wavelength of redshifted H$\alpha$ line. To exclude systematic shifts originating from known RSS flexure, we calculated line-of-sight velocity distributions along the slit for both emission and absorbtion lines using a suite of MIDAS programs described in detail in @Zasov00. These programs allow the use of additional correction factors derived from tracing nearby night-sky lines whose accurate wavelengths are very well known to correct the observed wavelengths of the [NaI]{}D, H$\alpha$ \[\] $\lambda$6583 and \[\] $\lambda$6716 emission lines. After implementing the night-sky line corrections, the line-of-sight velocity distributions are accurate to $\sim$1.5 km s$^{-1}$. Most of the calculated velocity distributions are shown in Figures \[fig:AM\_rot\_130a\]–\[fig:AM\_rot\_27\]. All velocities derived with this procedure are heliocentric.
All emission lines were measured with the MIDAS programs described in detail in @SHOC [@Sextans]. These programs determine the location of the continuum, perform a robust noise estimation, and fit separate lines with single Gaussian components superposed on the continuum-subtracted spectrum. Nearby lines, such as the H$\alpha$ and \[\] $\lambda\lambda$6548, 6583 lines on the one hand, the \[\] $\lambda\lambda$6716, 6731 lines on the other, and [NaI]{}D $\lambda\lambda$5890, 5896 absorption doublet were fitted simultaneously as blends of two or more Gaussian features.
[ ]{}
[ ]{}
Results {#txt:results}
=======
Spectra of and PGC 400092
--------------------------
A cursory inspection of the spectra obtained at PA=140$^{\circ}$ (see Figure \[fig:AM\_2D\_130\]) shows rotation detectable in the same amount and behaviour exhibited by the H$\alpha$, \[\] $\lambda\lambda$6548,6583 and \[\] $\lambda\lambda$6716,6731 emission lines, and rotation as almost a solid body exhibited by the [NaI]{} $\lambda\lambda$5890,5896 doublet lines. The NE extension of the spectrum, away from and crossing the companion galaxy PGC 400092, shows that the same emission lines seen in are produced by the NE companion; the rotation there is much slower and the [NaI]{} doublet is not visible, even though the continuum there is visible. In addition, the spectrum of PGC 400092 shows also weak \[\] $\lambda$6300 and HeI $\lambda$5876 in emission, while the spectrum of shows \[\] $\lambda$6300 emission only in the central part.
------------------- ------------------
Absorption Line Equivalent Width
[1]{}[c]{}[(Å)]{} (Å)
CaII H 8.9$\pm$1.5
CaII K 10.3$\pm$1.8
H$\delta$ 6.5$\pm$2.1
H$\gamma$ 5.8$\pm$2.4
H$\beta$ 6.4$\pm$2.5
[MgI]{}b 3.5$\pm$0.8
[NaI]{}D 5.8$\pm$0.7
------------------- ------------------
: EWs of absorption lines in spectra of []{data-label="t:EW_abs"}
The short-wavelength spectra obtained in June 2006 (top panel of Fig. \[fig:AM\_1D\_130\]) show the blend of the [NaI]{} doublet as a single line (due to the lower resolution of this setup), and the H$\beta$, H$\gamma$ and H$\delta$ lines in absorption. The CaII H and K doublet is seen in absorption at the blue end of the spectrum. The spectra also show very weak \[\] $\lambda\lambda$4959, 5007 emission lines. In this figure and in the following plots we describe as ”intensity” the raw counts extracted from the spectra. Since our data have not been spectrophotometrically calibrated, this is in reality ”relative intensity”. The equivalent widths of the main absorption lines were measured for the central part of the galaxy and are shown in Table \[t:EW\_abs\]. Measurements of lines detected in more than one spectrum were averaged.
[ ]{}
[ ]{}
[ ]{}
[ ]{}
[ ]{}
The rotation curve of along the major axis, derived from the two-spectra combination shown in Figure \[fig:AM\_2D\_130\], is shown in Figures \[fig:AM\_rot\_130a\] and \[fig:AM\_rot\_130b\]. Figure \[fig:AM\_rot\_130a\] show the velocity-position plot and Figure \[fig:AM\_rot\_130b\] shows the galacto-centric velocity-distance plot. In general, the emission-line rotation curve derived here corresponds with that shown in Figure 5 of @Resh06, except that ours is better sampled, has a higher signal-to-noise, and the rotation curves derived from the different emission lines practically coincide, as can be estimated from the formal 1$\sigma$ error bars plotted in the figures and from the scatter of the individual points. Figure \[fig:AM\_rot\_130b\] shows also a comparison of our measurements with those of @Resh06.
Deriving the rotation curves shown in Figures \[fig:AM\_rot\_130a\] and \[fig:AM\_rot\_130b\] we found that the systemic radial velocity of is 11663$\pm$3 km sec$^{-1}$, formally higher by some 14 km sec$^{-1}$ than the value given by @Resh06 in their Table 3 but consistent with their value within the quoted uncertainties. This offset might be the result of a slightly different definition of the systemic velocity; we chose the value for which the NW branch of the rotation curve matched best that for the SE branch and by this procedure also found the rotation center of the galaxy. Independently, we found that this location on the velocity curve is also the central point for the linear fitting of all the measurements for the [NaI]{}D lines seen in absorption, as shown in Figure \[fig:AM\_rot\_130a\]. We obtained a best-fit line following the relation: $${\rm V_r = (11663 \pm 2) + (15.2 \pm 0.4) \times {\rm R} } $$ where $\rm R$ is the distance in arcsec from the point where the radial velocity of , defined using the emission lines, equals 11663 km sec$^{-1}$ and we adopt this location as the kinematic centre of the galaxy. The different symbols indicate the H$\alpha$ velocity (black squares), the \[\] $\lambda$6583 line velocity (red squares), and the \[\] $\lambda$6716 line velocity (blue triangles). The stellar rotation along the same slit position on the major axis, as derived from an average of the two [NaI]{}absorption lines, is depicted as filled black circles.
We detected a discrepant systemic velocity 11680$\pm$10 km sec$^{-1}$ for the NW companion PGC 400092 as well, where our value is significantly lower than the 11735$\pm$6 km sec$^{-1}$ given in @Resh06. Since the velocity discrepancies for and for PGC400092 are in opposite directions, we can probably rule out a systematic shift between our velocity scale and the one of @Resh06. This is confirmed also by the plot in Figure \[fig:AM\_rot\_130b\] where their derived velocity curve points are plotted over our results. The shift between our data for PGC400092 and that from @Resh06 could be the result of the slit position for $\rm PA = 140\degr$ used here that did not cross exactly the physical center of that galaxy.
We could also derive the velocity dispersion of the H$\alpha$ line along the slit for $\rm PA = 140\degr$; this is shown in the bottom panel of Figure \[fig:AM\_rot\_130b\]. The dispersion is shown as the FWHM of the line after correcting for the intrinsic spectrometer line width. The corrected H$\alpha$ line FWHM=5–7 Å found for the central part ($\pm$3 arcsec) of indicates internal motions of 200–300 km s$^{-1}$. The corrected FWHM=$<$1 Å measured for the H$\alpha$ line of PGC400092 indicates internal motions slower than 45 km s$^{-1}$.
The rotation curve along the polar ring axis, at $\rm PA =
27\degr$, is shown in Figure \[fig:AM\_rot\_27\] as a velocity-position plot. This, as already mentioned, relies mostly on the emission lines since the [NaI]{} absorptions are visible only in the central part of the spectrum, and is therefore more limited in extent. The spectra for $\rm PA = 27\degr$ show a linearly increasing rotation for $\sim$7 SW of the galaxy centre outwards, where the center position is that derived for the major axis. Since the NE and SW branches of the ring’s major axis show very different behaviour from that observed along the galaxy’s major axis, the method used previously to find the rotation center by matching the two branches could not be used in this case, thus we do not show a folded and combined velocity curve for the major axis of the ring.
The NE branch shows an approximately flat rotation from $\sim$2 away from the centre, as derived from the emission lines, with some oscillations from the center to the periphery at 10 arcsec from the center. These oscillations are evident in both H$\alpha$ and \[\] $\lambda$6583; they may be caused by the overlap of the emission lines from the ring with those from the main body of the galaxy. The plot in the top panel of Fig. \[fig:AM\_rot\_27\] shows that the strongest H$\alpha$ emission is encountered close to the location of the most intense continuum contribution (compare the solid and the dashed lines).
Our spectra along $\rm PA = 27\degr$ show a completely different kinematic behaviour than the one described by @Resh06. Their Fig. 7 shows a $\sim$50 km sec$^{-1}$ difference between the velocity of the \[\] $\lambda$6583 and H$\alpha$ at the galaxy centre that increases to $\sim$100 km sec$^{-1}$ at the SW end of the ring. We, on the other hand, see no difference between the velocities of these two lines. Moreover, the \[\] lines in our observed spectrum also show the same behavior as the \[\] $\lambda$6583 and H$\alpha$ lines. We also note that the extent to which the rotation is defined and measurable for this position angle and using the emission lines is practically the same as for the major axis of , some 8 kpc from the center (at 167 Mpc).
Similar to the case of the major axis, $\rm PA = 140\degr$, we see here also a straight-line behaviour with galacto-centric distance of the [NaI]{} absorption lines. We find a formal linear fit of the form $${\rm V_r = (11662 \pm 2) + (14.9 \pm 0.8) \times {\rm R}} $$ The [NaI]{} rotation curve is linear from 15 SW of the centre to $\sim$5" NE of the kinematic centre. Note that the value found for the slope at this position angle is virtually identical with that for the major axis in equation (1).
A comparison of the two panels of Fig. \[fig:AM\_rot\_27\], the lower one which is a velocity-position plot for $\rm PA = 27\degr$ and the upper one which is a plot of the line intensity vs. position along the slit, shows that the region where most of the line emission is produced is about 4 to the NE of the kinematic center of and that the emission is practically only along the NE part of the ring.
As for $\rm PA = 140\degr$, we derive the velocity dispersion for this position angle as the FWHM of the H$\alpha$ line vs. galacto-centric distance. This is shown in the bottom panel of Fig. \[fig:AM\_rot\_27\_all\] after correction for the intrinsic width of the lines using the night sky spectrum. The corrected FWHM=7 Å for the redshifted H$\alpha$ indicates internal motions of $\sim$300 km s$^{-1}$.
Although not spectrophotometrically calibrated, our spectra allow the derivation of a few physical parameters of the gas using line ratios. The good signal-to-noise of the spectra allows the derivation of these ratios along the slit, as shown in Figs. \[fig:AM\_140\_cond\] and \[fig:AM\_27\_cond\]. The ratios plotted in Fig. \[fig:AM\_140\_cond\] allow a derivation along the galaxy major axis and for its NW companion. Since these ratios are based on the very closely located emission lines, they practically do not depend on whether the spectral data were corrected for sensitivity or not. For the red spectral range, using the sensitivity curve cannot change these ratios by more that a few percent; this is less than the displayed errors.
Creating these ratios we took into account the possible stellar absorption in the H$\alpha$ line. Checking Table \[t:EW\_abs\], and considering the Balmer spectra of @Rosa99 we suggest that EW$_{abs}$(H$\alpha$)=6 Å with a constant value along the slit. Since EW(H$\alpha$)$\approx$15Å for the emission line at the center of , decreasing to the galaxy’s edges, this correction is very important and without it the ratios of \[\]/H$\alpha$ and \[\]/H$\alpha$ would increase from the centre to the edges. That could be interpreted as an increase in of metallicity with galacto-centric distance, which is not correct. With a measured line ratio for the central part of ($\pm$2 arcsec) $\rm ([\ionn{N}{II}]
\lambda6583/H\alpha)$=0.54$\pm$0.02, the metallicity in the center of is 12+log(O/H)=8.92$\pm$0.06 dex [@Den02] and drops down to 8.81$\pm$0.07 in the outer parts of the galaxy. The figure indicates that along the major axis of the galaxy n$_e\simeq$50 cm$^{-3}$. The measurements for detected part of PGC 400092 give 12+log(O/H) = 8.45$\pm$0.12 dex and n$_e \simeq$ 500 cm$^{-3}$.
In a similar way, we derive the gas properties along the major axis of the ring (see Fig. \[fig:AM\_27\_cond\]). With the line ratios measured in the central part of ($\pm$2 central arcsec) $\rm ([\ionn{N}{II}] \lambda6583/H\alpha)$=0.51$\pm$0.04, the metallicity in the center of is 12+log(O/H)=8.91$\pm$0.06 [@Den02], essentially the same value found from the major axis measurement. From the measured \[\] lines ratio we obtain the same value found previously: n$_e\simeq$ 50 cm$^{-3}$.
Newly detected H$\alpha$ emission knot
--------------------------------------
An isolated H$\alpha$ emission knot was detected at ; , some 78 away from the main body of the galaxy to the NE and on the extension of the ring’s major axis. This knot is real and was detected on all spectra observed at $\rm PA = 27\degr$ taken on 2006 September 20 and 21. The velocity distribution with distance is shown in the top panel of Figure \[fig:AM\_rot\_27\_all\]. It is evident that the line emitting knot is fairly isolated and is very distant from the galaxy, yet its radial velocity is close to that of the systemic velocity. The measured velocity for the knot is 11645$\pm$5 km sec$^{-1}$; this is more than three standard deviations away from the systemic velocity of and very many standard deviations away from the recession velocity measured for H$\alpha$ at the NW tip of the galaxy. It is also very different from the velocity of PGC 400092, the NW companion of , or from that of PGC 399718, the other companion in the triplet.
Our observations do not show a significant velocity dispersion of the H$\alpha$ line observed from the knot, as shown in the bottom panel of Fig. \[fig:AM\_rot\_27\_all\]; a formal measurement indicates that this H$\alpha$ line has the same FWHM ($\sim$2.4 Å) as the reference night-sky line. The corrected FWHM$\leq$1 Å for the redshifted H$\alpha$ from the knot indicates internal motions slower than 40 km s$^{-1}$. The size of the line-emitting region is only $\sim$5 arcsec; small, but well-resolved by our observations. The very weak continuum is detected; this allows a measurement of EW(H$\alpha$) = 120$\pm$15 Å. No additional emission lines are visible in the spectrum.
Analysis {#txt:disc}
========
The image of the field displayed in Figure \[fig:AM\_direct\] shows not only but also its two companion galaxies. Fig. \[fig:AM\_direct\] is a V-band image of the field obtained with SALTICAM in the same night as the spectroscopic observations on September 21. The image of the three galaxies allows one to note that (a) the region around the target contains many diffuse, low surface brightness (LSB) images that might be parts of galaxies or LSB dwarfs at the same redshift, or distant objects in the background, and (b) the appearance of the companion galaxy PGC 400092 to the NW is that of a Sd galaxy with a similar overall size to that of . The LSB objects are also visible on Digitized Sky Survey images of the region.
We performed unsharp masking of Figure \[fig:AM\_direct\] to emphasize the dust lane; this is shown in Figure \[fig:AM\_direct\_masked\] and, contrary to the claim of @Resh06 that the dust lane is split and embraces the galaxy nucleus from SE and NW, indicates that the lane is fairly straight, passes south and west of the brightest part of the galaxy, and is probably not split at all. The stars in Fig. \[fig:AM\_direct\_masked\] have the shapes of crescent moons. This arises from telescope optical problems which are being ironed out during the Performance Verification process, and have been emphasized by the unsharp masking.
The measured ratio of emission lines to corrected H$\alpha$, and the possibly very weak \[\] $\lambda$5007 emission detected in our July 2006 short-wavelength spectra, puts this object at the border between starburst nuclei (SBN) and LINERs. Norris et al. (1990) found no compact radio core in this galaxy and for this reason it should be classified as SBN; this is in agreement with the previous conclusions of @Al91.
The curves shown in Fig. \[fig:AM\_rot\_130b\] indicate that the gas rotation along the major axis has its maximum at $\sim$240 km sec$^{-1}$ and not at 195 km sec$^{-1}$ as given by @Resh06, and that this maximum is reached asymptotically for the NE part of the galaxy. Figire \[fig:AM\_rot\_130a\] shows that our measurements are compatible with those of @Resh06 for the regions of overlap. The last points of the rotation curve branch of the SE part of the galaxy, from galacto-centric distance of 6 to 10, drop from 200$\pm$7 km sec$^{-1}$ to 150$\pm$7 km sec$^{-1}$ in both H$\alpha$ and \[\] $\lambda$6583 lines. This drop is gradual from 6 to 8 but shows a step-like drop at this location, followed by a recovery with a similar distance-velocity gradient as for the central part of the galaxy.
A comparison of the major axis rotation curves shown in Fig. \[fig:AM\_rot\_130b\] shows clearly the difference between the kinematic behaviour of the two [NaI]{}D absorption lines and the H$\alpha$, \[\] $\lambda$6583 and \[\] $\lambda$6716 emission lines. At this point it is worth discussing the origin of the [NaI]{} absorption lines. These could be produced by stellar photospheres, or by diffuse gas in the interstellar medium of . For the case of dwarf starburst galaxies, Schwartz & Martin (2004) used giant and supergiant stars to show that the EW of the [MgI]{} triplet near 5180Å should be twice that of the [NaI]{} lines. If this would be the case for then our blue spectrum where the [MgI]{} triplet is barely visible would rule out a major [NaI]{} absorption contribution from stars.
[ ]{}
However, in giant galaxies such as the stellar populations are better represented by main sequence stars. These have stronger photospheric [NaI]{} than [MgI]{} (e.g., a M0V star from the same library as used by Schwartz & Martin (2004) has EW([MgI]{})=20Å and EW([NaI]{})=12Å. While it is not possible to separate the stellar [NaI]{} from the interstellar absorption, we can accept that at the least a fraction, and perhaps all of the observed absorption represents the stars in the galaxy. For example, in M82 Saito et al. (1984) detected [NaI]{} absorption that they attributed to stars and interpreted as solid-body rotation.
Assuming that most of the [NaI]{} absorption is photospheric, this would indicate that, while the gaseous component follows a “normal” galactic rotation law, the stellar component rotates almost like a solid body for $\sim$10 away from the centre. The maximal rotation velocity exhibited by the stellar component is only $\sim$150 km sec$^{-1}$ at 10 from the centre for both ends of the major axis.
The extent over which the emission is observed for the “polar ring” is almost the same as for the major axis, some 18overall as shown in Fig. \[fig:AM\_rot\_27\], but the derived rotation curve is completely different. The rotation curve indicates solid-body like rotation for 15 to the NE (one resolution element away from the centre, given the seeing) and for about 5 to the SW. The velocity difference between the outermost points on the slit where the absorption lines are measured is only 90 km sec$^{-1}$. The velocity gradients shown by the stellar components along the major axis of the galaxy and along the axis of the PR, in regions where a linear rotation curve can be defined, are very similar as equations (1) and (2) show. In both cases the gradients are $\sim$19 km sec$^{-1}$ kpc$^{-1}$, where we converted the observational gradients from equations (1) and (2) to physical units.
Interpretation {#txt:interp}
==============
At a distance to the object of 167 Mpc (H$_0$=70 km sec$^{-1}$ Mpc$^{-1}$) the radius of the galaxy to the outermost point where emission lines are visible is $\sim$8 kpc. We found the stellar component of a 16 kpc wide galaxy rotating as a solid body, while its gaseous component measured at the same slit position shows a smoothly increasing rotation curve which then flattens out. A ring or disk feature with an extent similar to that of the galaxy is observed at an inclination of $\sim$60$^{\circ}$ to the major axis of the galaxy. The stellar component observed with the spectrometer slit oriented along the major axis of the ring is also rotating as a solid body and with a similar velocity-distance gradient to that observed for the main body of the galaxy.
@Resh06 concluded from their photometry and spectroscopy, coupled with results of N-body modelling, that is a PRG. Their models indicate that the system might be the result of a major interaction between a ”donor” galaxy with a 17 kpc stellar disk and a 42 kpc gaseous disk, with a total mass of 3.6$\times10^{11}$ M$_{\odot}$, which encountered a 2$\times10^{11}$ M$_{\odot}$ and 14 kpc wide ”receptor” galaxy some 1.6 Gyrs ago with an impact parameter of 130 kpc and a relative velocity of 145 km sec$^{-1}$. This encounter transferred a large quantity of matter (stars, gas, and dust) from the donor to the receptor galaxy resulting in the formation of the polar ring which is inclined with respect to the galaxy disk and is warped. @Resh06 suggested that the donor galaxy survived and is PGC 399718, the southern companion in the triplet, and argued that their suggestion is supported by the reddish (B-V) colour of the galaxy and by its somewhat disturbed appearance.
In selecting this scenario in preference to those of minor mergers calculated by them, or of other possible models for the formation of ring galaxies, @Resh06 relied primarily on the morphological appearance of the galaxy. In particular, the minor merger models rejected by @Resh06 produced only partially-open rings that were not closed, whereas the preferred major merger model produced a “closed and regular ring” a few 10$^8$ years following the interaction.
Since the acceptance of the @Resh06 scenario as the explanation for the appearance of this system relies on their interpretation that the ring is closed and regular, it is worth examining whether the observations presented here support this assertion.
The specific items resulting from our observations that require understanding are:
1. Solid-body rotation is observed for stars vs. a ”regular” rotation for the gas at the same (projected) locations. No differential rotation, as expected from a stellar disk, is observed. This is true for the main body of the galaxy as well as for the ring, though with the gas showing a different distance-velocity gradient than the stars.
2. The ring is very faint and there is no evidence that it contains a considerable number of stars, as would be expected from the major merger claimed by @Resh06. Our observations of the intensity distribution along the slit at PA=27$^{\circ}$ show that the stars producing the continuum are located mostly where the HII is, namely some 2-5 NE of the centre.
3. The ring dynamics are different at its SW end, where the line and continuum emissions are very weak and the ring is more extended [@Resh06], in comparison with the other end of the ring.
4. The gas dynamics for the ring are very different from those of the gas in the galaxy. Specifically, at similar extents from the dynamical centre the gas in the ring spins much slower than the gas in the galaxy. This, while the stellar components have similar kinematic behaviours as evaluated from the velocity-distance gradients.
Apparent solid-body rotation of a galaxy could be produced, for example, by dust extinction. Baes (2003) modelled the light propagation through a dusty galactic disk and showed that, unless the disk is perfectly edge-on, no effects in the kinematics would be observable. The more the disk is edge on, and the stronger the extinction caused by the dust in the disk is, the more would the rotation curve resemble that of a solid body. Perusal of the DSS images of the object, of the image shown in Fig. 1 of @Resh06, and of our Figs. \[fig:AM\_direct\] and \[fig:AM\_direct\_masked\], shows that is not a purely edge-on galaxy and that, since the disk deviation from edge-on is definitely more than “a few degrees” but rather $\sim$25$^{\circ}$, as explained below, we should not expect to see a solid-body rotation just because of dust obscuration and light scattering. We can, therefore, reject the possibility that the solid-body rotation is an effect of dust obscuration.
Stars vs. gas in the disk
-------------------------
The key observation reported here is the difference in rotation curves between the emission lines produced by the gas and the stars as represented by the absorption lines. Such cases of different kinematic behaviour of the gas and the stars are known in the literature, Bettoni (1990), where NGC 2217 was shown to exhibit “counter-rotation” in that the gas motions in the inner parts of the galaxy indicated motions opposite those of the stars. This was interpreted there as a consequence of a warp in the disk coupled with the presence of a bar; this situation may exist for as well.
Macci[ò]{} (2006) tried to explain the origin of PRGs by accretion of cold intergalactic gas. They provide in their Fig. 4 plots of simulated velocity-position diagrams for gas and stars; the upper one, where the slit is aligned with the major axis of the galaxy, can be compared with our Figs. \[fig:AM\_rot\_130a\] and \[fig:AM\_rot\_130b\]. It seems that the presence of a stellar bar in could be producing the linearly-rising stellar rotation curve, whereas the rotation curve for the gas fits the simulation quite well.
Since none of our observations are of photometric-quality, we rely on parameters derived by @Resh06 to characterize the galaxy. In particular, we adopt their photometric disk parameters: a disk exponential scale length h(B)=5“.1$\pm$0”.3=3.8 kpc and their scaling to other bandpasses: h(B)/h(V)=1.18$\pm$0.11 and h(B)/h(R)=1.25$\pm$0.12. The R-band disk scale length is, therefore, 4.8$\pm$0.5 kpc. This is useful when comparing with properties of other galaxies or of model galaxies.
To compare with the rotational properties of other galaxies, we use the observations of edge-on galaxy disks from Kregel (2004) for the stellar kinematics and from Kregel & van der Kruit (2004) for the gas kinematics. Fig. 6 in Kregel shows that the stellar rotation curve can be almost linear with galacto-centric distance for about 1.5 disk scale lengths and this for galaxies earlier than Sbc. Note that this galaxy sample does not include barred galaxies, though Kregel mention that some do show boxy or peanut-shaped bulges. The gas in none of their galaxies (Kregel & van der Kruit 2004) rotates with as small a gradient with distance from the center as observed in .
It is also possible to compare both the imaged galaxy and its stellar kinematics with the diagnostic plots calculated by Bureau & Athanassoula (2005). Inspection of their Figs. 1 and 4 indicates that a good fit with could be obtained for an intermediate or strong bar viewed at least at 45$^{\circ}$ to the bar or even edge-on, and at a disk inclination of at least 80$^{\circ}$ to the line of sight. The conclusion is that does probably have a fairly strong bar that is almost side-on to our line of sight, and its disk is seen almost edge-on.
Another comparison for our rotation curve is with the collection of template rotation curves of Catinella (2006) who, however, studied normal galaxies, not PRGs. They normalize the rotation curves between 2 and 3 disk radii; applying this to , with the peak rotation derived from the curve, indicates that the galaxy should have an absolute I-band magnitude brighter than –23 mag. Indeed, using the photometry from @Resh06, with a measured M$_B \simeq$–21 mag and a color index (B-I)=2.06, the absolute I magnitude of is –23.06 mag. This confirms the assumption that, in analyzing the gaseous rotation curve along the major axis, it is a valid assumption to adopt the rotation pattern of a regular galaxy, not that of a PRG, since the presence of the polar ring does not affect significantly the kinematics of the galaxy.
HI vs. other kinematic indicators
---------------------------------
The HI in a number of PRGs, including , was studied by van Driel (2002) with the Parkes radio telescope. This observation produced a puzzling and troublesome result for ; van Driel reported the HI line at a heliocentric velocity of 11282$\pm$24 km sec$^{-1}$ with a full-width at half-maximum of the two-horned profile of 193 km sec$^{-1}$. Note that their data were taken with the Parkes multibeam system, which implies a beam width of 144 FWHM. The 12500 km sec$^{-1}$ bandwidth was centered at 10000 km sec$^{-1}$ and the channel separation was 6.6 km sec$^{-1}$.
If the HI would have been associated with , we would expect to find the neutral hydrogen line at a similar systemic velocity to that measured here, that in @Resh01, or that measured by @Resh06. We would also expect a much wider HI profile than quoted by van Driel (2002), since the H$\alpha$ kinematics indicate a width of $\sim$450 km sec$^{-1}$ along the major axis, as befitting a major galaxy given its bright absolute magnitude of M$_B$=–21.1 measured by @Resh06. The very wide Parkes beam implies that all three objects were included in the measurement, and probably many outlying HI clouds that may exist in this neighbourhood as well, but does not explain the velocity discrepancy since all three galaxies should have appeared on the red shoulder of the HI profile shown by van Driel
Another indication that something is wrong with the HI measurement comes from applying the Tully-Fisher relation to . @Co06 gives a Tully-Fisher diagram for PRGs in Fig. 2 of her paper; these galaxies seem to follow the T-F relation for spirals and S0 galaxies and it is worthwhile to check where fits in this diagram. Adopting the HI width given in van Driel (2002) indicates that should have an M$_B\simeq$–18 mag, completely different from the magnitude measured by @Resh06. Adopting a velocity width as measured by us albeit from the emission lines and not from the HI profile, namely 450 km sec$^{-1}$, yields the proper value of M$_B\simeq$–21 mag.
Irrespective of the explanation regarding the HI redshift discrepancy, it is possible that extended HI is present in the system. The possibility that such HI clouds or other gas-rich galaxies might be present is supported by our discovery of the H$\alpha$ knot (see below), and by the presence of a few low surface brightness (LSB) extended objects in the immediate vicinity. These resemble LSBs the nearby Universe that are often found to be very gas-rich. In addition, there are a few very blue star-like objects that stand out in comparisons of the Second Digitized Sky Survey images in different bands.
We do not have redshifts for these LSB objects but the fact that they are of similar sizes to the main galaxies in the group hints that they might be group members; such companions are seen in other groups as well (Grossi 2007) and could have interacted with in the past. We predict that once HI synthesis observations will be obtained for and its neighbours, for example with the ATNF, at least some of these candidates and in particular the H$\alpha$ knot discovered by us will prove to be actually gas-rich members of this group.
The H$\alpha$ knot
------------------
The H$\alpha$ knot reported above, which is $\sim$78 arcsec away to the NE from the galaxy center but almost at the same velocity, is in reality $\sim$630 kpc away in projected distance. Its detectable H$\alpha$ emission, combined with a lack of \[\], \[\] and only weak continuum emissions, argue that this is probably a metal-poor dwarf galaxy that belongs to the same group as . Such objects are known as ”HII galaxies” (Sargent & Searle 1970) since they show an HII region spectrum with negligible continuum and have considerable redshifts.
Our fitting procedure to the emission lines, used for the galaxies and for the ring, allows the derivation of an upper limit for the \[\] $\lambda$6583 flux that can be used to obtain an upper limit to the metal abundance. With a measured upper limit line ratio of $\rm log([\ionn{N}{ii}] \lambda6583/H\alpha) =
-1.46$ the metallicity upper limit is 12+log(O/H) $<$ 8.05 [@Den02]. The knot appears to be somewhat metal-poor, though we cannot set a definite upper limit on its metal abundance, and we have shown that it also shows a very low internal velocity dispersion as befits a dwarf galaxy.
Brosch et al. (2006) identified a considerable number of H$\alpha$-emitting knots in the neighbourhoods of a few star-forming galaxies qualified as “dwarfs” (M$_B\geq$–18) and located in some very under-dense regions of the nearby Universe. The study revealed these prospective neighbour galaxies through the presence of H$\alpha$ emission at or near the central galaxy redshift. It is possible that the knot found here is a similar type of object.
Differences between the two slit positions
------------------------------------------
Spectroscopy of the ring in NGC 4650A has been reported by Swaters & Rubin (2003). They found a ring rotation curve that seems to flatten out from the center to the North, but which is steadily increasing from the center to 20 arcsec on the South side and then flattens out. This is more pronounced for the stellar component of the ring than for its gaseous component. The galaxy itself, an S0 as most PRGs are, shows solid-body-like stellar rotation from the center to $\sim$15 arcsec out while the emission lines show a different pattern of constant velocity. This they interpret as due to the galaxy being devoid of gas while the line emission is produced only in the ring. Comparisons of the rotational properties of polar rings and of galaxy disks are valuable in understanding PRGs.
We return now to the appearance of the stellar rotation curves observed at $\rm PA = 140\degr$ and at $\rm PA = 27\degr$. These curves, derived from the [NaI]{} absorption lines, are very similar. They appear linear for a considerable distance and their velocity-distance gradients are $\sim$19 km sec$^{-1}$ kpc$^{-1}$.
We point the reader back to Fig. \[fig:AM\_2D\_27\] where the extent of the continuum that allows the detection and measurement of the [NaI]{}D lines is considerably narrower than that for $\rm PA
= 140\degr$. The discrepancy could be resolved by assuming that the absorption lines, and therefore most of the continuum, would not be produced by stars in the ring, as implicitly assumed in the previous sections, but by stars in the main galaxy, perhaps in a stellar disk or in a strong bar. The inclination can be derived from the axial ratio of the galaxy given in @Resh06: i$\simeq 63
^{\circ}$.
In this case, the angle difference between the two slit positions, $\rm 67\degr$, would explain the difference in the extent of the linear rotation curves at the two position angles as a combination of foreshortening and obscuration by the dust lane. The dust lane produces about one magnitude of extinction, as the intensity profiles along the slit in Fig. 4c of @Resh06 show. The sudden disappearance of the absorption lines only 15 SW of the dynamical center could be explained by the crossing of the dark lane by the slit at this position angle. The weak intensity of the underlying continuum of the H$\alpha$ line, plotted with a short-dashed line in the top panel of Fig. \[fig:AM\_rot\_27\], supports this interpretation.
The ring would, in this case, be composed mostly of gas, would be located between us and the disk with its dark lane, and would necessarily be much less massive than assumed by @Resh06. The emission lines measured within $\pm$5 of the kinematic centre (see e.g., Fig \[fig:AM\_rot\_27\]) would then be produced primarily in the disk, while those for $\rm PA = 27\degr$ but measured at a galacto-centric distance of more than 65 would originate in the ring.
The arguments presented above indicate that the model proposed by @Resh06 to explain as a major merger, with the donor galaxy being PGC 399718, might not fit the observations. We therefore propose another alternative, that the unsettled disk or ring around the galaxy was formed by accretion of cold gas from a cosmic filament, one of the possibilities accounting for ring galaxy formation put forward by @Co06. The presence of anomalous HI redshifts in the region, our discovery of an apparent dwarf HII galaxy in the group, and the circumstantial detection of large but low surface brightness galaxies in the immediate vicinity of , albeit lacking redshifts at present, argue in favour of this interpretation.
Perusing the large-scale structures identified in this region by Radburn-Smith (2006), specifically those in Panel 6 of their Figure 4, indicates that the location of , at $\rm l \simeq 341.02$, $\rm b\simeq -28.73$, corresponds to the tip of a galaxy filament extending out of the Zone of Avoidance. This might be a distant structure related to the Centaurus wall and the Norma and Pavo II clusters of galaxies at lower redshifts, through which intergalactic matter is accreted by the galaxy and forms the ring.
Models of cold gas accretion from cosmic filaments by Macci[ò]{} (2006) show how a ring galaxy, such as NGC 4650A or for that matter could be formed by such a process. Their simulations show that the accreted gas is not completely cold but rather at 15,000K due to its collapse within the gravitational potential of the filamentary structure. Moreover, they mention that some of the gas might also be shock-heated by the halo potential.
A similar process could take place in . There is no clear-cut evidence that the ring is closed or relaxed, or that it has a substantial stellar component. Its disturbed appearance at its SW end is more similar to that of an assemblage of diffuse gas clouds, not of a coherent and relaxed structure. The NE part is smaller and sharper; it is possible that accreted gas collides there with itself, becomes compressed and shocked, and reaches higher temperatures that produce the enhanced line emission. At this location the accreted gas could perhaps enter a circular or quasi-circular orbit.
An alternative could be that in the case we are indeed witnessing a merger with a gas-rich galaxy, which takes place in a polar configuration. This is, in a way, similar to the major merger scenario of @Resh06 with the exception that the ”donor” galaxy would now be the ring itself. The argument reducing the likelihood of this explanation is the lack of a significant stellar continuum from the ring, indicating its low mass.
The shape of the dark matter halo of
--------------------------------------
Considering the two gas rotation curves, the one along the galaxy’s major axis and the other along the ring’s major axis, one observation is in order. The two rotation curves derived from the emission lines extend a similar distance from the galaxy’s kinematic centre, are presumably in the same dark matter potential well if is indeed a PRG, yet show a completely different full amplitude. While the galaxy major axis rotation curve has a full end-to-end amplitude of $\sim$450 km sec$^{-1}$, that for the ring has a full amplitude of only $\sim$240 km sec$^{-1}$. The asymptotic rotation of the ring is slower than the asymptotic rotation of the galaxy.
The formation of PRGs has been studied by Bournaud & Combes (2003) via N-body simulations. They discussed, in particular, cases when both the galaxy disk and the ring contain gas. Their argument was that in such cases the polar ring must, by necessity, be wider than the galaxy. If this is not the case, the gas in the ring would interact with the gas in the disk and one of the components would join the other. Two orthogonal, or almost orthogonal gas rings, can coexist in the same galaxy only if they have different radii and do not cross each other. Such crossing presumably occurs in NGC 660, where both the disk and the ring contain gas; the N660 system is unstable and according to Bournaud & Combes did not have sufficient time to dissolve the ring since its formation.
The specific question of the DM halo shape in PRGs was studied by Iodice (2003). They explained that the ring material would move slower than the gas in the disk if the gravitational potential would be oblate, like the flattened disk galaxy. In this case the ring would be elliptical and would show a lower observed velocity than the disk at its outermost locations (see their Fig. 3). In the case of , since the ring and the galaxy appear to have the same size but the ring must be wider in order to avoid crossing the disk, a possible conclusion would be that the ring is elliptical with its major axis close to our line of sight to the object and its minor axis seen almost perpendicular to the disk. This way, the ring could indeed be larger than the galaxy, the gas in the ring and that in the galaxy would not cross, and the velocities at the apo-galactic ring locations would be slower than in the galaxy disk. In this case the outermost visible ring segments would correspond to locations near the ends of the minor axis and ring material should show there its highest orbital speed, larger than that of the galactic disk. As this is not observed, we conclude that the disk and the ring in are of similar sizes, their contents do cross, and the system is unstable.
With the additional kinematic information now available, could be considered a test case for DM gravitational potential tracing. The discussion of PRGs by @Co06 was based on the hope that PRGs would prove to be useful probes of the DM potential in which a galaxy and its polar ring find themselves. @Co06 found that the rings in observed PRGs show faster rotation than the maximal velocity observed in the host galaxy. The theoretical prediction is in the opposite direction to the observations, namely rings in PRGs devoid of DM halos or with spherical halos should be rotating slower than their galaxies. According to @Co06, this effect should be accentuated for flattened or oblate DM halos.
We find that fulfills the theoretical predictions for non-spherical oblate haloes in that the polar ring does rotate slower than its host galaxy. Any DM halo of , if it exists at all, would have to be flattened along the barred disk, but this configuration could not be stable on the long run because the ring would cross the disk. Resolving this possibility and deriving more constraints on the existence and shape of a possible DM halo for would require detailed modelling and further observations that are not within the scope of this paper.
Conclusions {#txt:summ}
===========
We presented observations obtained with SALT and RSS during their performance verification phase that emphasize the long-slit capabilities of the RSS for galaxy observations. We traced the stellar and gaseous rotation curves for the major axis of the galaxy and for the major axis of a polar ring-like feature almost perpendicular to the disk of the galaxy. We showed that, while the gas rotates regularly when sampled along the galaxy major axis, the stellar component shows rotation like a solid body, supporting an interpretation that this is an object with a strong bar viewed almost side-on.
The ionized gas rotation along the major axis of the ring was found to be much less regular than along the major axis of the galaxy and shows a somewhat shallower gradient with galacto-centric distance. The [NaI]{} stellar rotation from the $\sim$6.5 kpc ring segment where the lines are measurable shows a similar distance dependence to that seen along the galaxy’s major axis. The systemic velocity derived by us for differs from previously published values. We propose that the discrepancy of rotation curves along the two position angles can be resolved by recognizing that the absorption lines are probably produced only by the main galaxy or its bar, and not by the ring where only emission lines are produced.
We discovered a small H$\alpha$ knot at a projected distance of about 700 kpc from but at a similar velocity, which we interpret as a fourth member of this compact group of galaxies, presumably a metal-poor dwarf galaxy. The lack of continuum emission for this object while only the H$\alpha$ line is detected indicates that it might be forming stars for the first time. The low velocity dispersion measured from the knot indicates its low mass.
We argue that a more plausible explanation to the major merger scenario proposed by @Resh06 to explain could be the slow accretion of cold cosmic gas along a galaxy filament directed to the region. In the cold gas accretion case the flow is probably towards the galaxy from the South-West and becomes more compressed at the NE end of the polar ring feature. We point out that the kinematic properties we measured follow the theoretical predictions for PRGs in a dark matter halo that is not spherical, but is flattened along the plane of the galaxy.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper was written while NB was a sabbatical visitor at the South African Astronomical Observatory in Cape Town; NB is grateful for this opportunity offered by the SAAO management. We are grateful for the generous allocation of SALT observing time during the PV phase to complete this project. We acknowledge a private communication from Vladimir P. Reshetnikov concerning this galaxy. We acknowledge the use of products of the second Digitized Sky Survey produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images are based on photographic data obtained using the UK Schmidt Telescope. The UK Schmidt Telescope was operated by the Royal Observatory Edinburgh, with funding from the UK Science and Engineering Research Council (later the UK Particle Physics and Astronomy Research Council), until 1988 June, and thereafter by the Anglo-Australian Observatory. The blue plates of the southern Sky Atlas and its Equatorial Extension (together known as the SERC-J), as well as the Equatorial Red (ER), and the Second Epoch \[red\] Survey (SES) were all taken with the UK Schmidt. An anonymous referee provided some insightful comments that improved the clarity of the presentation.
[99]{}
Allen at al. 1991, [MNRAS]{}, 248, 528
Baes M., et al., 2003, MNRAS, 343, 1081
Bekki K., 1998, ApJ, 499, 635
Bettoni D., Fasano G., Galletta G., 1990, AJ, 99, 1789
Bournaud F., Combes F., 2003, [A&A]{}, 401, 817
Brosch, N., Bar-Or, C., and Malka, D. 2006, [MNRAS]{}, 368, 864
Brosch, N. 1985, A&A, 153, 199
Brosch, N. 1987, Mercury, 16, 174
Buckley D. A. H., Swart G. P., Meiring J. G., 2006, SPIE, 6267,
Bureau M., Athanassoula E., 2005, ApJ, 626, 159
Burgh E. B., Nordsieck K. H., Kobulnicky H. A., Williams T. B., O’Donoghue D., Smith M. P., Percival J. W., 2003, SPIE, 4841, 1463
Combes, F. 2006, EAS, 20, 97
Catinella B., Giovanelli R., Haynes M. P., 2006, ApJ, 640, 751
Denicoló, G., Terlevich, R., & Terlevich, E. 2002, [MNRAS]{}, 330, 69
González-Delgado, R.M., Leitherer, C., & Heckman, T.M. 1999, [ApJS]{}, 125, 489
Grossi M., Disney M. J., Pritzl B. J., Knezek P. M., Gallagher J. S., Minchin R. F., Freeman K. C., 2007, MNRAS, 374, 107
Kregel M., van der Kruit P. C., 2004, MNRAS, 352, 787
Kregel M., van der Kruit P. C., Freeman K. C., 2004, MNRAS, 351, 1247
Hagen-Thorn, V.A., Shalyapina, L.V., Karataeva, G.M., Yakovleva, V.A., Moiseev, A.V., and Burenkov, A.N. 2005, Astron. Reports, 49, 958
Iodice E., Arnaboldi M., Bournaud F., Combes F., Sparke L. S., van Driel W., Capaccioli M., 2003, ApJ, 585, 730
Karataeva, G.M., Tikhonov, N.A., Galazutdinova, O.A., Hagen-Thorn, V.A., and Yakovleva, V.A. 2004, A&A, 421, 833
Kobulnicky H. A., Nordsieck K. H., Burgh E. B., Smith M. P., Percival J. W., Williams T. B., O’Donoghue D., 2003, SPIE, 4841, 1634
Kniazev, A.Y., Pustilnik, S.A., Grebel, E.K., Lee, H., & Pramskij, A.G. 2004, ApJS, 153, 429
Kniazev, A.Y., Grebel, E.K., Pustilnik, S.A., Pramskij, A.G., & Zucker, D. 2005, [AJ]{}, 130, 1558
Iodice E., Arnaboldi M., De Lucia G., Gallagher J. S., III, Sparke L. S., Freeman K. C., 2002, AJ, 123, 195
Iodice E., et al., 2006, ApJ, 643, 200
Leitherer, C. 1999, ApJS, 123, 3
Lynds, R. & Toomre, A. 1976, ApJ, 209, 382
Macci[ò]{} A. V., Moore B., Stadel J., 2006, ApJ, 636, L25
Mayya, Y.D. & Korchagin, V. 2001, ApSSS, 277, 339 (on-line revision in 2006)
Mazucca, L.M., Knapen, J.H., Regan, M.W. & Böker, T. 2001, in [*The Central kiloparsec of Starbursts and AGN*]{}, J.H. Knapen , eds. San Francisco: ASP, 573
Norris R. P., Kesteven M. J., Troup E. R., Allen D. A., Sramek R. A., 1990, ApJ, 359, 291
O’Donoghue D., et al., 2006, MNRAS, 372, 151
Persic M., Salucci P., Stel F., 1996, MNRAS, 281, 27 (erratum 1996, MNRAS, 283, 1102)
Radburn-Smith D. J., Lucey J. R., Woudt P. A., Kraan-Korteweg R. C., Watson F. G., 2006, MNRAS, 369, 1131
Reshetnikov V. P., Fa[ú]{}ndez-Abans M., de Oliveira-Abans M., 2001, [MNRAS]{}, 322, 689
Reshetnikov, V.P. 2004, A&A, 416, 889
Reshetnikov, V., Bournaud, F., Combes, F., Fa[ú]{}ndez-Abans, M., and de Oliveira-Abans, M.: 2006, [*Astron. Astroph.*]{} 446, 447.
Sargent W. L. W., Searle L., 1970, ApJ, 162, L155
Schweizer, F., Ford, W.K. Jr., Jederzejewsky, R. & Giovanelli, R. 1987, ApJ, 320, 454
Shane, W.W. 1980, A&A 82, 314
Swaters R. A., Rubin V. C., 2003, ApJ, 587, L23
van Driel W., Combes F., Arnaboldi M., Sparke L. S., 2002, A&A, 386, 140
Whitmore B. C., Lucas R. A., McElroy D. B., Steiman-Cameron T. Y., Sackett P. D., Olling R. P., 1990, AJ, 100, 1489
Zasov, A., Kniazev, A., Pustilnik, S., et al. 2000, A&AS, 144, 429
Woudt P. A., et al., 2006, MNRAS, 371, 1497
\[lastpage\]
[^1]: E-mail: noah@wise.tau.ac.il (NB); akniazev@saao.ac.za (AYK); dibnob@saao.ac.za (DB); dod@saao.ac.za (DOD); hashimot@saao.ac.za (YH); nsl@saao.ac.za (NL); erc@saao.ac.za (ER); still@saao.ac.za (MS); petri@saao.ac.za (PV); ebb@sal.wisc.edu (EBB); khn@sal.wisc.edu (KN)
[^2]: Based on observations obtained with the Southern African Large Telescope (SALT).
[^3]: IRAF: the Image Reduction and Analysis Facility is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, In. (AURA) under cooperative agreement with the National Science Foundation (NSF).
[^4]: See http://www.salt.ac.za/partners-login/partners/data-analysis-software for more information.
[^5]: MIDAS is an acronym for the European Southern Observatory package – Munich Image Data Analysis System.
|
---
abstract: |
We introduce inner amenability for discrete p.m.p. groupoids and investigate its basic properties, examples, and connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid.
Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations which either are stable or have a non-trivial central sequence in their full group.
address:
- 'Graduate School of Mathematical Sciences, the University of Tokyo, Komaba, Tokyo 153-8914, Japan'
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843, USA'
author:
- 'Yoshikata Kida and Robin Tucker-Drob'
date: 'October 25, 2018'
title: Inner amenable groupoids and central sequences
---
[^1]
Introduction
============
A discrete countable group $G$ is called *inner amenable* if there exists a sequence $(\xi_n)_n$ of non-negative unit vectors in $\ell^1(G)$ such that $\Vert \xi_n^g-\xi_n\Vert_1\to 0$ and $\xi_n(g)\to 0$ for any $g\in G$, where the function $\xi_n^g$ is defined by $\xi_n^g(h)=\xi_n(ghg^{-1})$ for $h\in G$. This notion was introduced by Effros [@effros] who first observed its connection with property Gamma of the group von Neumann algebra. This connection has since become a common theme: inner amenability of a group $G$ can often be deduced from the existence of certain central sequences, either in the von Neumann algebra associated with $G$, or in the full group of a probability-measure-preserving (p.m.p.) action of $G$ (e.g., [@choda] and [@js]).
In this paper, we introduce inner amenability for discrete p.m.p. groupoids. We investigate its basic properties and examine its connection with central sequences, both in the full group of the groupoid and in the von Neumann algebra associated with the groupoid, highlighting many examples along the way. We expect results in this paper to accelerate the understanding of free ergodic p.m.p. actions of inner amenable groups and their orbit equivalence relations. We refer to [@dv], [@is], [@kida-inn]–[@kida-sce], [@mar], [@pv], [@td] and [@vaes] for recent progress on related topics.
We briefly outline some results in this paper. Inner amenability of a discrete p.m.p. groupoid ${\mathcal{G}}$ is defined in §\[sec-iag\] (Definition \[def:groupoid\]), generalizing the definition given above for groups; if the groupoid ${\mathcal{G}}$ is ergodic, then inner amenability is equivalent to the existence of a sequence $(\xi_n)_n$ of non-negative unit vectors in $L^1({\mathcal{G}})$ which is asymptotically conjugation-invariant and diffuse (see Theorem \[thm:equiv\] for various equivalent characterizations). One motivating example of an inner amenable groupoid is an ergodic discrete p.m.p. equivalence relation ${\mathcal{R}}$ which is *Schmidt*, i.e., admits a non-trivial central sequence in its full group $[{\mathcal{R}}]$. However there also exist inner amenable equivalence relations which are not Schmidt. We will obtain examples of equivalence relations ${\mathcal{R}}$ which are either not inner amenable, or not Schmidt, by imposing various spectral gap and mixing properties on actions which generate ${\mathcal{R}}$. For example, the Bernoulli shift action of any non-amenable group gives rise to an orbit equivalence relation that is not inner amenable. In fact, the product of such a Bernoulli shift action with any other p.m.p. action also gives rise to a non inner amenable orbit equivalence relation.
We observe permanence of inner amenability under several groupoid constructions (e.g., inflations, restrictions, ergodic decompositions and inverse limits). Of particular interest is that we establish permanence of inner amenability under compact extensions. This implies that inner amenability passes to finite-index Borel subgroupoids, and that any compact free p.m.p. action of an inner amenable group gives rise to an orbit equivalence relation which is inner amenable.
We also show that both the Schmidt property and stability of a discrete p.m.p. equivalence relation are preserved under compact extension, where a discrete p.m.p. equivalence relation is called *stable* if it absorbs the ergodic hyperfinite p.m.p. equivalence relation on a non-atomic probability space, under direct product. Those two properties therefore pass to a finite-index subrelation. However for a finite-index inclusion ${\mathcal{S}}<{\mathcal{R}}$ of ergodic discrete p.m.p. equivalence relations, while inner amenability of ${\mathcal{S}}$ implies that of ${\mathcal{R}}$, the Schmidt property and stability of ${\mathcal{S}}$ does not necessarily imply those of ${\mathcal{R}}$. We show this by giving examples, and also give a sufficient condition for this implication to hold, in terms of the algebra of asymptotically invariant sequences for ${\mathcal{S}}$ and the action of $F$ on it when ${\mathcal{R}}$ is written as ${\mathcal{R}}={\mathcal{S}}\rtimes F$ for some finite group $F$ acting on ${\mathcal{S}}$ by automorphisms. These results on finite-index inclusions should be compared with Pimsner-Popa’s result [@pp Proposition 1.11] on property Gamma and the McDuff stability for inclusions of $\text{II}_1$ factors.
In [@sch-prob], Schmidt asked whether every inner amenable group admits a free ergodic p.m.p. action whose orbit equivalence relation is Schmidt. Let us say that a group has the *Schmidt property* if it admits such an action. Schmidt’s question remains open and is one of the questions motivating the present work. We call a group *orbitally inner amenable* if it admits a free ergodic p.m.p. action whose orbit equivalence relation is inner amenable. As we shall see, every group with the Schmidt property is orbitally inner amenable, and every group which is orbitally inner amenable is inner amenable. While we do not know whether every inner amenable group is orbitally inner amenable, it follows from our aforementioned result on compact actions that this implication holds under the additional assumption that the group in question is residually finite. It also follows from [@td], using different methods, that every inner amenable linear group has the Schmidt property, and hence is orbitally inner amenable.
We say that a countable group is *stable* if it admits a free ergodic p.m.p. action whose orbit equivalence relation is stable. Since, as we mentioned above, stability passes to finite-index subrelations, it follows that stability of a countable group passes to finite-index subgroups as well. Combining this with the first author’s result [@kida-sce], we obtain the corollary that stability of a countable group is invariant under virtual isomorphism. Although we can also ask the same question for the Schmidt property and orbital inner amenability of a countable group, it remains unsolved. More precisely, we do not know if those two properties are invariant under central group-extension with finite central group.
The paper is organized as follows: In Section \[sec-pre\], we fix the notation and terminology on discrete p.m.p. groupoids. In Section \[sec-iag\], we define inner amenability of discrete p.m.p. groupoids and state several conditions equivalent to it. We also discuss basic facts around inner amenability and some permanence properties, and discuss its relationship with property Gamma of the groupoid von Neumann algebra in the case when the groupoid is principal. In Section \[sec-cpt-ia\], we show that inner amenability is preserved under compact extension. In Section \[sec-cpt-cs\], we show that stability and the Schmidt property are also preserved under compact extension. In Section \[sec-sg\], we use spectral gap properties to obtain constraints, either on sequences witnessing inner amenability, or on central sequences in full groups. This allows us to provide many examples in which such sequences do not exist. In Section \[sec-finite-index\], for a finite-index inclusion ${\mathcal{S}}<{\mathcal{R}}$ of ergodic discrete p.m.p. equivalence relations, we address the question of whether stability or the Schmidt property of ${\mathcal{S}}$ implies the corresponding property of ${\mathcal{R}}$. In Section \[sec-ex\], we collect miscellaneous examples of orbitally inner amenable groups and free ergodic p.m.p. actions such that central sequences in the full group are well controlled.
Throughout the paper, unless otherwise mentioned, all relations among Borel sets and maps are understood to hold up to sets of measure zero.
We thank Adrian Ioana for his valuable remarks on the first author’s earlier note and for kindly allowing us to incorporate them into Lemma \[lem-p\] and Corollary \[cor-c\] (ii) of the present paper. The second author would like to thank Andrew Marks for a suggestion which helped to simplify the proof of Theorem \[thm:equiv\].
Groupoid preliminaries and notation {#sec-pre}
===================================
For a groupoid ${\mathcal{G}}$, we denote the unit space of $\mathcal{G}$ by $\mathcal{G}^0$, and denote the source and range maps of $\mathcal{G}$ by $s$ and $r$, respectively. For a subset $D\subset \mathcal{G}$ and $x, y\in \mathcal{G}^0$, we set $D_x = D\cap s^{-1}(x)$, $D^y = D\cap r^{-1}(y)$ and $D_x^y=D_x\cap D^y$, and we say that $D$ is *bounded* if there is some $N\in {\mathbb{N}}$ with $|D_x|\leq N$ and $|D^x|\leq N$ for all $x\in \mathcal{G}^0$. For subsets $A, B\subset \mathcal{G}^0$, we set $\mathcal{G}_A = r^{-1}(A)\cap s^{-1}(A)$ and $\mathcal{G}_{A,B} = r^{-1}(A)\cap s^{-1}(B)$. The set ${\mathcal{G}}_A$ is then a groupoid with unit space $A$, with respect to the product inherited from ${\mathcal{G}}$. For $x\in {\mathcal{G}}^0$, we set ${\mathcal{G}}(x)={\mathcal{G}}_x^x$ and call it the *isotropy group* of $\mathcal{G}$ at $x$.
A *discrete Borel groupoid* is a groupoid $\mathcal{G}$ such that $\mathcal{G}$ is a standard Borel space, $\mathcal{G}^0$ is a Borel subset of $\mathcal{G}$, the maps $s,r\colon \mathcal{G}\rightarrow \mathcal{G}^0$ are Borel and countable to one, and the multiplication map $\{ \, (\gamma ,\delta ) \in \mathcal{G}\times \mathcal{G}\mid s(\gamma )=r(\delta )\, \} \rightarrow \mathcal{G}$, $(\gamma ,\delta )\mapsto \gamma \delta$ and the inverse map $\gamma \mapsto \gamma ^{-1}$ are both Borel maps. A *cocycle* $\alpha \colon {\mathcal{G}}\to L$ into a standard Borel group $L$ is a Borel homomorphism, i.e., a Borel map satisfying $\alpha (\gamma \delta )= \alpha (\gamma )\alpha (\delta )$ whenever $s(\gamma )=r(\delta )$.
A *discrete p.m.p. groupoid* is a pair $(\mathcal{G},\mu )$, where $\mathcal{G}$ is a discrete Borel groupoid, and $\mu$ is a Borel probability measure on $\mathcal{G}^0$ satisfying $\int _{\mathcal{G}^0}c^s_x \, d\mu(x) = \int _{\mathcal{G}^0} c^r_x\, d\mu(x)$, where $c^s_x$ and $c^r_x$ denote the counting measures on ${\mathcal{G}}_x$ and ${\mathcal{G}}^x$, respectively. We will write $\mu ^1$ for this common measure: $\mu ^1 = \int _{\mathcal{G}^0}c^s_x \, d\mu(x) = \int _{\mathcal{G}^0} c^r_x\, d\mu(x)$.
Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid. We say that $(\mathcal{G},\mu )$ is *aperiodic* if ${\mathcal{G}}_x$ is infinite for $\mu$-almost every $x\in \mathcal{G}^0$. A Borel subset $A\subset {\mathcal{G}}^0$ is called *${\mathcal{G}}$-invariant* if $r({\mathcal{G}}_x)\subset A$ for $\mu$-almost every $x\in A$. We say that $({\mathcal{G}}, \mu)$ is *ergodic* if for any ${\mathcal{G}}$-invariant Borel subset $A\subset {\mathcal{G}}^0$, we have $\mu(A)=0$ or $1$. A *local section* of $\mathcal{G}$ is a Borel map $\phi \colon \mathrm{dom}(\phi ) \rightarrow \mathcal{G}$, where $\mathrm{dom}(\phi )\subset \mathcal{G}^0$ is a Borel subset, such that
- $\phi (x)\in {\mathcal{G}}_x$ for every $x\in \mathrm{dom}(\phi )$, and
- the associated map $\phi ^0:\mathrm{dom}(\phi )\rightarrow \mathcal{G}^0$, defined by $\phi ^0(x)=r(\phi (x))$, is injective.
We identify two local sections if their domains and values agree up to a $\mu$-null set. The *composition* of the local section $\psi \colon B \rightarrow \mathcal{G}$ with the local section $\phi \colon A\rightarrow \mathcal{G}$ is the local section $\psi \circ \phi \colon (\phi ^0)^{-1}(\phi ^0(A)\cap B) \rightarrow \mathcal{G}$ defined by $(\psi \circ \phi )(x)= \psi (\phi ^0(x))\phi (x)$. The *inverse* of the local section $\phi \colon A\rightarrow \mathcal{G}$ is the local section $\phi ^{-1}\colon \phi ^0(A) \rightarrow \mathcal{G}$ defined by $\phi ^{-1}(x) = \phi ((\phi ^0)^{-1}(x))^{-1}$. Let $[\mathcal{G}]$ denote the group of all local sections $\phi$ of $\mathcal{G}$ with $\mathrm{dom}(\phi )=\mathcal{G}^0$ and call $[{\mathcal{G}}]$ the *full group* of ${\mathcal{G}}$. There is a natural complete separable metric $d$ on $[\mathcal{G}]$ given by $d(\phi , \psi ) = \mu ( \{ \, x\in \mathcal{G}^0 \mid \phi (x) \neq \psi (x) \, \} )$.
Let $\phi$ be a local section of $\mathcal{G}$, and set $\mathrm{ran}(\phi^0)=\phi^0(\mathrm{dom}(\phi))$. For $\gamma \in \mathcal{G}_{\mathrm{ran} (\phi ^0)}$, we set $$\gamma ^{\phi }=\phi ^{-1}(r(\gamma ))\gamma \phi ^{-1}(s(\gamma ))^{-1} \in \mathcal{G}_{\mathrm{dom} (\phi )}.$$ For a subset $D\subset {\mathcal{G}}$, we set $$D^\phi =\{ \, \gamma^\phi \mid \gamma \in D\cap {\mathcal{G}}_{\mathrm{ran}(\phi^0)}\, \}.$$ For a function $f\colon \mathcal{G} \to {\mathbb{C}}$, we define $f^{\phi }\colon \mathcal{G}\rightarrow {\mathbb{C}}$ by $$f^\phi (\gamma ) =
\begin{cases}
f(\gamma ^{(\phi ^{-1})} )=f(\phi(r(\gamma))\gamma \phi(s(\gamma))^{-1}) &\text{if }\gamma\in \mathcal{G}_{\mathrm{dom}(\phi )}, \\
0 &\text{otherwise}.
\end{cases}$$ If $\psi$ is another local section of ${\mathcal{G}}$, then $(\gamma^{\phi})^\psi =\gamma^{\phi \circ \psi}$ and $(f^\phi)^\psi =f^{\phi \circ \psi}$.
A *discrete p.m.p. equivalence relation* is a discrete p.m.p. groupoid which is *principal*, i.e., for which the map $\gamma \mapsto (r(\gamma ), s(\gamma ))$ is injective. Let $\mathcal{R}$ be a p.m.p. countable Borel equivalence relation on a standard probability space $(X,\mu )$. Then the pair $(\mathcal{R},\mu )$ is naturally a principal discrete p.m.p. groupoid with unit space $\mathcal{R}^0 {\coloneqq}\{ \, (x,x) \mid x\in X \, \}$, which we will simply identify with $X$ itself when there is no cause for confusion. The source and range maps are given by $s(y,x)=x$ and $r(y,x)=y$, respectively, and the multiplication and inverse operations are given by $(z,y)(y,x)=(z,x)$ and $(y,x)^{-1}=(x,y)$, respectively. In this setting, each local section $\phi$ of $\mathcal{R}$ is completely determined by its associated unit map $\phi ^0$, and we will abuse notation and identify $\phi$ with $\phi ^0$ when there is no cause for confusion. Under this identification, the group $[\mathcal{R}]$ is also called the *full group* of $\mathcal{R}$. For each $(y,x)\in \mathcal{R}$ and $\phi \in [\mathcal{R}]$, we then have $(y,x)^{\phi} = (\phi ^{-1}(y),\phi ^{-1}(x) )$.
The *translation groupoid* associated to a p.m.p. action $G\curvearrowright (X,\mu )$ of a countable group $G$ is the groupoid $G\ltimes (X,\mu ) = (\mathcal{G},\mu )$ defined as follows. The set of groupoid elements is $\mathcal{G} {\coloneqq}G\times X$ with unit space $\mathcal{G}^0{\coloneqq}\{ 1_G \} \times X$, which we will once again identify with $X$ itself when there is no cause for confusion. The source and range maps $s,r\colon \mathcal{G}\rightarrow \mathcal{G}^0$ are given by $s(g,x)=x$ and $r(g,x)=gx$, respectively, and the multiplication and inverse operations are given by $(g,hx)(h,x)= (gh,x)$ and $(g,x)^{-1}= (g^{-1}, gx )$, respectively. Let $p\colon G\times X \rightarrow G$ be the projection onto $G$. In this setting, each local section $\phi$ of $\mathcal{G}$ is completely determined by the associated map $p\circ \phi \colon \mathrm{dom}(\phi )\rightarrow G$, and we will abuse notation and identify $\phi$ with $p\circ \phi$ when there is no cause for confusion. The group $G$ embeds into $[\mathcal{G}]$ via the map $g\mapsto \phi _g$, where $\phi _g \colon X\rightarrow G$ is the constant map $\phi _g(x)=g$. Then for any $(h,x)\in G\times X$ and $g\in G$, we have $(h,x)^{\phi _g}= (g^{-1}hg,g^{-1}x)$. If the action $G{\curvearrowright}(X, \mu)$ is essentially free, i.e., the stabilizer of almost every point of $X$ is trivial, then the groupoid $G\ltimes (X,\mu )$ is isomorphic to the orbit equivalence relation $$\mathcal{R}(G\curvearrowright (X,\mu )){\coloneqq}\{ \, (gx,x) \mid g\in G,\, x \in X\, \}$$ associated with the action.
Inner amenable groupoids {#sec-iag}
========================
Definition and equivalent conditions
------------------------------------
We define inner amenability for discrete p.m.p. groupoids and state several conditions equivalent to it. The proof of their equivalence is postponed to subsection \[subsec-proof\], following the preliminary subsections \[subsec-conj\] and \[subsec-ame\].
\[def:groupoid\] A discrete p.m.p. groupoid $(\mathcal{G},\mu )$ is *inner amenable* if there exists a sequence $(\xi _i )_{i\in {\mathbb{N}}}$ of non-negative unit vectors in $L^1(\mathcal{G} , \mu ^1 )$ such that
1. $\| 1_{\mathcal{G}_A} \xi _i \| _1 \to \mu (A)$ for every Borel subset $A\subset \mathcal{G}^0$;
2. $\| \xi _i ^\phi - \xi _i \| _1 \to 0$ for every $\phi \in [\mathcal{G}]$;
3. $\| 1_{D}\xi _i \| _1 \to 0$ for every Borel subset $D\subset \mathcal{G}$ with $\mu ^1 (D)<\infty$; and
4. $\sum _{\gamma \in {\mathcal{G}}^x}\xi _i (\gamma ) = 1 = \sum _{\gamma \in {\mathcal{G}}_x}\xi _i (\gamma )$ for almost every $x\in \mathcal{G}^0$ and every $i$.
Such a sequence $(\xi _i ) _{i\in {\mathbb{N}}}$ is called an *inner amenability sequence* for $(\mathcal{G},\mu )$.
A discrete countable group $G$ is inner amenable in the above sense if there exists a sequence $(\xi_i)_{i\in {\mathbb{N}}}$ of non-negative unit vectors in $\ell^1(G)$ such that for every $g\in G$, we have $\Vert \xi_i^g-\xi_i\Vert_1\to 0$ and $\xi_i(g)\to 0$, where the function $\xi_i^g$ on $G$ is given by $\xi_i^g(h)=\xi_i(ghg^{-1})$.
We will see in Lemma \[lem:bal\] that if $(\mathcal{G} ,\mu )$ is ergodic and non-amenable, then any sequence $(\xi_i)$ of non-negative unit vectors in $L^1({\mathcal{G}}, \mu^1)$ satisfying condition (ii) automatically satisfies condition (i). Sequences $(\xi_i)$ satisfying condition (i) are said to be *balanced*.
A *mean* on a discrete p.m.p. groupoid $(\mathcal{G},\mu )$ is a finitely additive, probability measure $\bm{m}$ on $\mathcal{G}$ which is defined on the algebra of all $\mu ^1$-measurable subsets of $\mathcal{G}$ and is absolutely continuous with respect to $\mu ^1$. Equivalently, a mean on $\mathcal{G}$ is a state on $L^\infty (\mathcal{G},\mu ^1 )$. A mean $\bm{m}$ on $(\mathcal{G},\mu )$ is called
- *balanced* if $\bm{m}(\mathcal{G}_A ) = \mu (A)$ for every Borel subset $A\subset {\mathcal{G}}^0$;
- *conjugation-invariant* if $\bm{m}(D^{\phi}) = \bm{m}(D)$ for every $\phi \in [\mathcal{G}]$ and every Borel subset $D\subset \mathcal{G}$;
- *diffuse* if $\bm{m}(D)=0$ for every Borel subset $D\subset \mathcal{G}$ with $\mu ^1 (D)<\infty$; and
- *symmetric* if $\bm{m}(D) = \bm{m}(D^{-1})$ for every Borel subset $D\subset \mathcal{G}$.
\[rem-balanced\] Let $\bm{m}$ be a balanced mean on a discrete p.m.p. groupoid $({\mathcal{G}}, \mu)$. Then for any Borel subset $A\subset {\mathcal{G}}^0$ and any countable Borel partition $A=\bigsqcup_nA_n$ of $A$, we have $\bm{m}({\mathcal{G}}_A)=\mu(A)=\sum_n\mu(A_n)=\sum_n\bm{m}({\mathcal{G}}_{A_n})$. This implies that given Borel subsets $D_n\subset {\mathcal{G}}_{A_n}$, for each $n$, we have $\bm{m}(\bigsqcup_n D_n)=\sum_n\bm{m}(D_n)$.
\[thm:equiv\] Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid. If $(\mathcal{G},\mu )$ is ergodic, then the following conditions (1)–(6) are equivalent:
1. The groupoid $(\mathcal{G}, \mu )$ is inner amenable.
2. There exists a net $(\xi _i )$ (as opposed to a sequence) of non-negative unit vectors in $L^1(\mathcal{G},\mu ^1 )$ satisfying conditions (i)–(iv) of Definition \[def:groupoid\].
3. There exists a net $(\xi _i )$ of non-negative unit vectors in $L^1(\mathcal{G},\mu ^1 )$ satisfying conditions (ii) and (iii) of Definition \[def:groupoid\].
4. There exists a diffuse, conjugation-invariant mean on $(\mathcal{G},\mu )$.
5. There exists a diffuse, conjugation-invariant mean on $(\mathcal{G},\mu )$ which is symmetric and balanced.
6. There exists a positive linear map $P \colon L^\infty (\mathcal{G}, \mu ^1 ) \rightarrow L^\infty (\mathcal{G}^0 ,\mu )$ such that
- $P(1_{\mathcal{G}_A})=1_A$ for every Borel subset $A\subset \mathcal{G}^0$;
- $P(F)=P(F^{-1})$ and $P(F^{\phi})= P(F)\circ \phi ^0$ for every $F\in L^\infty (\mathcal{G},\mu ^1 )$ and every $\phi \in [\mathcal{G}]$, where the function $F^{-1}$ is defined by $F^{-1}(\gamma)=F(\gamma^{-1})$ for $\gamma \in {\mathcal{G}}$; and
- $P(F)=0$ for every $F \in L^1(\mathcal{G},\mu ^1 )\cap L^{\infty}(\mathcal{G},\mu ^1 )$.
In general, even without assuming that $(\mathcal{G},\mu )$ is ergodic, conditions (1), (2), (5) and (6) are all equivalent.
\[rem:nonerg\] In general, in the absence of ergodicity, condition (4) does not imply condition (5), since any groupoid $(\mathcal{G},\mu ){\coloneqq}(\mathcal{G}_0\sqcup \mathcal{G}_1 , \mu _0/2 + \mu _1/2)$, with $(\mathcal{G}_0, \mu _0)$ ergodic and inner amenable, and $(\mathcal{G}_1,\mu _1 )$ ergodic and not inner amenable, satisfies condition (4) but not condition (5).
\[rem:locsec\] Condition (ii) of Definition \[def:groupoid\] immediately implies its own strengthening that $\| \xi _i ^{\phi} - 1_{\mathcal{G}_{\mathrm{dom}(\phi )}}\xi _i \| _1 \to 0$ for every local section $\phi$ of $\mathcal{G}$ since every local section $\phi$ of $\mathcal{G}$ can be extended to a local section with conull domain. Likewise, any conjugation-invariant mean $\bm{m}$ on $(\mathcal{G},\mu )$ satisfies $\bm{m}(D^\phi ) = \bm{m}(D)$ for every local section $\phi$ of $\mathcal{G}$ and every Borel subset $D\subset \mathcal{G}_{\mathrm{ran} (\phi ^0 )}$. It follows that if a discrete p.m.p. groupoid $(\mathcal{G},\mu )$ is inner amenable, then so is $(\mathcal{G}_A,\mu _A )$ for every Borel subset $A\subset \mathcal{G}^0$ with positive measure, where $\mu_A$ is the normalized restriction of $\mu$ to $A$. For the converse, see Proposition \[prop:inflate\].
\[rem:suffice\] Suppose that $(\mathcal{G},\mu ) = G\ltimes (X,\mu )$ is the translation groupoid associated to a p.m.p. action $G\curvearrowright (X,\mu )$ of a countable group $G$. For $g\in G$, let $\phi _g \in [\mathcal{G}]$ be the local section defined by $\phi _g (x) = g$. Suppose that $\bm{m}$ is a balanced mean on $(\mathcal{G},\mu )$. Suppose in addition that $\bm{m}$ is invariant under conjugation by $\{ \phi _g \} _{g\in G}$, i.e., $\bm{m}(D^{\phi _g}) = \bm{m}(D)$ for every Borel subset $D\subset \mathcal{G}$ and every $g\in G$. Then $\bm{m}$ is in fact invariant under conjugation by every element of $[\mathcal{G}]$. Indeed, for any $\phi \in [\mathcal{G}]$ and any Borel subset $D\subset \mathcal{G}$, if for $g\in G$, we set $A_g = \{ \, x\in X \mid \phi ^{-1}(x) =g^{-1} \, \}$, then $X=\bigsqcup_{g\in G}A_g=\bigsqcup_{g\in G}g^{-1}A_g$, and by Remark \[rem-balanced\], we have $$\begin{aligned}
\bm{m}(D^{\phi} ) &= \bm{m}\Biggl(\bigsqcup _{g\in G} D^\phi \cap \mathcal{G}_{g^{-1}A_g} \Biggr)= \sum_{g\in G}\bm{m}( D^\phi \cap \mathcal{G}_{g^{-1}A_g}) = \sum _{g\in G}\bm{m}((D\cap \mathcal{G}_{A_g})^{\phi _g}) \\
&= \sum _{g\in G}\bm{m}(D\cap \mathcal{G}_{A_g}) = \bm{m}(D).\end{aligned}$$
Conjugation-invariant means {#subsec-conj}
---------------------------
Before proving Theorem \[thm:equiv\], we prepare several lemmas saying that under mild assumption, any conjugation-invariant mean is automatically balanced or diffuse. We will use the following characterization of amenability of an ergodic discrete p.m.p. groupoid.
\[lem:amengroupoid\] An ergodic discrete p.m.p. groupoid $(\mathcal{G},\mu )$ is amenable if and only if there exists a mean $\bm{m}$ on $(\mathcal{G}, \mu )$ which is right-invariant, i.e., satisfies $\bm{m}(R_{\phi}f)=\bm{m}(f)$ for every $\phi \in [\mathcal{G}]$ and $f\in L^{\infty}(\mathcal{G},\mu ^1 )$, where $R_{\phi} \colon L^{\infty}(\mathcal{G},\mu ^1 ) \rightarrow L^{\infty}(\mathcal{G},\mu ^1 )$ is the right translation map defined by $(R_{\phi}f)(\gamma ) = f(\gamma \phi (s(\gamma )) ^{-1})$.
See [@kerr-li Remark 4.67] for the case of principal groupoids. The extension to the general case is routine.
\[lem:nonamenbal\] Let $(\mathcal{G},\mu )$ be an ergodic discrete p.m.p. groupoid which is non-amenable, and let $\bm{m}$ be a conjugation-invariant mean on $(\mathcal{G},\mu )$. If $A\subset \mathcal{G}^0$ is a Borel subset with positive measure, then $\bm{m}(\mathcal{G}_{{\mathcal{G}^0}\setminus A, A}) =0$.
Suppose toward a contradiction that $\bm{m}(\mathcal{G}_{{\mathcal{G}^0}\setminus A, A}) >0$, and let $\bm{m}_0$ denote the normalized restriction of $\bm{m}$ to $\mathcal{G}_{{\mathcal{G}^0}\setminus A, A}$. For $\phi \in [\mathcal{G}_A]$, let $\tilde{\phi}\in [\mathcal{G}]$ denote the extension of $\phi$ defined by $\tilde{\phi} (y)= y$ for $y\in {\mathcal{G}^0}\setminus A$ (and $\tilde{\phi} (x) = \phi (x)$ for $x\in A$). Then for any $\phi \in [\mathcal{G}_A]$, both left and right translation by $\tilde{\phi}$ fixes the set $\mathcal{G}_{{\mathcal{G}^0}\setminus A, A}$, and for any $f\in L^{\infty}(\mathcal{G},\mu ^1 )$ that vanishes outside $\mathcal{G}_{{\mathcal{G}^0}\setminus A, A}$, we have $f^{\phi} = R_{\tilde{\phi}}f$ and hence $$\begin{aligned}
\label{eqn:fvan}
\bm{m}_0(R_{\tilde{\phi}}f) &= \bm{m}_0(f) .\end{aligned}$$ Since $(\mathcal{G},\mu )$ is ergodic, we can find a Borel map $x\mapsto T(x)\in {\mathcal{G}}_x$ with $r(T(x))\in A$ for almost every $x\in {\mathcal{G}^0}$. For $f\in L^{\infty}(\mathcal{G}_A,\mu ^1 _A )$, we define $L_Tf\in L^{\infty}(\mathcal{G},\mu ^1 )$ by $$(L_Tf)(\gamma ) =
\begin{cases}
f(T(r(\gamma ))\gamma ) &\text{if }\gamma \in \mathcal{G}_{{\mathcal{G}^0}\setminus A, A},\\
0 &\text{otherwise}.
\end{cases}$$ Then $L_TR_{\phi}f=R_{\tilde{\phi}}L_Tf$ for every $\phi \in [\mathcal{G}_A]$ and every $f\in L^{\infty}(\mathcal{G}_A, \mu ^1 _A )$. Define a mean $\bm{m}_1$ on $(\mathcal{G}_A, \mu _A)$ by $\bm{m}_1(f) = \bm{m}_0 (L_Tf)$. Then for any $\phi \in [\mathcal{G}_A]$ and any $f\in L^{\infty}(\mathcal{G}_A,\mu ^1 _A )$, by equation , we have $$\bm{m}_1(R_{\phi}f)= \bm{m}_0 (L_TR_{\phi}f)=\bm{m}_0(R_{\tilde{\phi}}L_Tf) = \bm{m}_0(L_Tf)= \bm{m}_1(f) .$$ Thus $\bm{m}_1$ is a right-invariant mean on $(\mathcal{G}_A, \mu _A )$, and hence $(\mathcal{G}_A, \mu _A)$ is amenable by Lemma \[lem:amengroupoid\]. Since $A$ has positive measure and $(\mathcal{G},\mu )$ is ergodic, this implies $(\mathcal{G},\mu )$ is amenable, a contradiction.
In Lemma \[lem:nonamenbal\], the condition that $(\mathcal{G},\mu )$ is non-amenable is necessary: Let the direct sum $G=\bigoplus_{\mathbb{N}}{\mathbb{Z}}/2{\mathbb{Z}}$ act on the compact group $X=\prod_{\mathbb{N}}{\mathbb{Z}}/2{\mathbb{Z}}$ by translation, and let ${\mathcal{R}}$ denote the associated orbit equivalence relation, where $X$ is equipped with the normalized Haar measure. For $n\in {\mathbb{N}}$, define the subgroup $F_n{\coloneqq}\bigoplus_{k=1}^n{\mathbb{Z}}/2{\mathbb{Z}}$ of $G$ and let ${\mathcal{R}}_n$ be the subrelation of ${\mathcal{R}}$ generated by $F_n$. Define the non-negative unit vector $\xi_n{\coloneqq}1_{{\mathcal{R}}_n}/2^n\in L^1({\mathcal{R}}, \mu^1)$, and let $\bm{m}$ be a weak${}^*$-cluster point of the sequence $(\xi_n)$ in $L^\infty({\mathcal{R}}, \mu^1)^*$. The mean $\bm{m}$ is then left and right-invariant and hence conjugation-invariant. However, if $A{\coloneqq}\{ \, (x_k)_k\in X \mid x_1=0\, \}$, then $\int_{{\mathcal{R}}}\xi_n1_{{\mathcal{R}}_{A, X\setminus A}}\, d\mu^1=\mu(A)(2^{n-1}/2^n)=1/4$ for every $n$, and therefore $\bm{m}({\mathcal{R}}_{A, X\setminus A})=1/4\neq 0$.
\[lem:bal\] Let $(\mathcal{G},\mu )$ be an ergodic discrete p.m.p. groupoid which is non-amenable. Then every conjugation-invariant mean on $(\mathcal{G},\mu )$ is balanced.
It follows that if $(\xi _i )$ is any net of non-negative unit vectors in $L^1(\mathcal{G},\mu ^1)$ satisfying $\| \xi _i ^{\phi} - \xi _i \| _1\rightarrow 0$ for all $\phi \in [\mathcal{G}]$, then $(\xi _i )$ is balanced.
Let $\bm{m}$ be a conjugation-invariant mean on $(\mathcal{G},\mu )$. Since $(\mathcal{G},\mu )$ is ergodic, if $A,B\subset \mathcal{G}^0$ are Borel subsets with $\mu (A) = \mu (B)$, then we can find some $\phi \in [\mathcal{G}]$ with $\phi ^0(A)= B$ and hence $$\bm{m}(\mathcal{G}_{A}) = \bm{m}(\mathcal{G}_{(\phi ^0) ^{-1} (B)}) = \bm{m} ((\mathcal{G}_B)^{\phi})= \bm{m}(\mathcal{G}_B) .$$ Therefore there is some function $\theta \colon [0,1]\rightarrow [0,1]$ such that $\bm{m}(\mathcal{G}_A)= \theta (\mu (A))$ for every Borel subset $A\subset \mathcal{G}^0$. Given $n\in {\mathbb{N}}$, we can find a Borel partition $A_1,\dots , A_n$ of $\mathcal{G}^0$ with $\mu (A_1)=\cdots = \mu (A_n)= 1/n$, and Lemma \[lem:nonamenbal\] implies that $1=\sum _{i=1}^n \bm{m}(\mathcal{G}_{A_i}) = n\theta (1/n)$, so that $\theta (1/n ) = 1/n$ and $$\theta (k/n) = \bm{m}(\mathcal{G}_{\bigsqcup _{i=1}^k A_i} ) = \sum _{i=1}^k \bm{m}(\mathcal{G}_{A_i}) = k\theta (1/n ) = k/n$$ for any $k\in \{ 1,\ldots, n\}$. Therefore $\theta (q)=q$ for every rational $q\in [0,1]$. Since $\theta$ is monotone increasing, this implies that $\theta (r)=r$ for every $r\in [ 0, 1]$, and hence $\bm{m}(\mathcal{G}_A)=\mu (A)$ for every Borel subset $A\subset \mathcal{G}^0$, i.e., $\bm{m}$ is balanced.
\[lem:diffuse\] Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid and set $$\mathcal{G}_{\mathrm{isot}} = \{ \, \gamma \in \mathcal{G} \mid s(\gamma ) =r(\gamma ) \, \}.$$ Suppose that $\bm{m}$ is a balanced mean on $(\mathcal{G},\mu )$ satisfying $\bm{m}(D)=0$ for every bounded Borel subset $D$ of $\mathcal{G}_{\mathrm{isot}}$. Then $\bm{m}$ is diffuse.
It follows that if $\mathcal{R}$ is a discrete p.m.p. equivalence relation on $(X,\mu )$, and if $\bm{m}$ is a balanced mean on $(\mathcal{R},\mu )$ satisfying $\bm{m} (\{ \, (x,x) \mid x\in X \, \} ) =0$, then $\bm{m}$ is diffuse.
Suppose toward a contradiction that $\bm{m}$ is not diffuse. Then there is some Borel $D\subset \mathcal{G}$ with $\bm{m}(D)>0$ and $\mu ^1(D)<\infty$. Since $\mu ^1(D)<\infty$, we have both $|D^x|<\infty$ and $|D_x| < \infty$ for $\mu$-almost every $x\in \mathcal{G}^0$. Then the sets $$A_{n,m} {\coloneqq}\{ \, x\in \mathcal{G}^0\mid |D_x| = n \text{ and } |D^x|=m \, \}\ \ \text{with}\ \ n,m\in {\mathbb{N}}\cup \{ 0\},$$ partition $\mathcal{G}^0$. Since $\bm{m}$ is balanced, we have $0<\bm{m}(D)=\sum _{n,m\in {\mathbb{N}}\cup \{ 0\}} \bm{m} (\mathcal{G}_{A_{n,m}}\cap D)$. Thus, after replacing $D$ by one of the sets $\mathcal{G}_{A_{n,m}}\cap D$ if necessary, we may assume without loss of generality that $D$ is bounded. Then, by hypothesis, $\bm{m}(D\cap \mathcal{G}_{\mathrm{isot}})=0$, so we may also assume without loss of generality that $D\cap \mathcal{G}_{\mathrm{isot}}=\emptyset$.
Since $D$ is bounded, it can be covered by the images of finitely many local sections of $\mathcal{G}$, and hence we can find a local section $\phi \colon A \rightarrow \phi (A)\subset D$ of $\mathcal{G}$ with $\bm{m}(\phi (A) ) > 0$. Since $D\cap \mathcal{G}_{\mathrm{isot}}=\emptyset$, we have $\phi ^0(x)\neq x$ for each $x\in A$. We can then partition $A$ into three Borel subsets, $A_0$, $A_1$ and $A_2$, such that $\phi ^0 (A_i)\cap A_i =\emptyset$ (modulo $\mu$) for each $i\in \{ 0, 1, 2\}$. Indeed, take $A_0$ to be an arbitrary maximal (modulo $\mu$) Borel subset of $A$ with $\phi^0(A_0)\cap A_0 =\emptyset$, and set $A_1= \phi ^0(A_0)\cap A$ and $A_2 = A\setminus (A_0\cup A_1)$. This works since then $\phi ^0(A_1)\cap A_1 \subset \phi^0(A\setminus A_0)\cap \phi^0(A_0)=\emptyset$, and $\mu (\phi ^0(A_2 )\cap A_2 )=0$ by maximality of $A_0$. For each $i\in \{ 0,1,2\}$, we have $\phi (A_i)\subset \mathcal{G}_{\mathcal{G}^0\setminus A_i , A_i }$, and hence $\bm{m}(\phi (A_i))= 0$ since $\bm{m}$ is balanced. Therefore $\bm{m}(\phi (A))=\bm{m}(\phi (A_0))+ \bm{m}(\phi (A_1))+ \bm{m}(\phi (A_2)) = 0$, a contradiction.
Amenability and inner amenability {#subsec-ame}
---------------------------------
Any discrete, countably infinite, amenable group is inner amenable. We extend this to a discrete p.m.p. groupoid.
\[prop:extend\] Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid. Suppose that $({\mathcal{G}}_{\mathrm{isot}}, \mu)$ is inner amenable, where ${\mathcal{G}}_{\mathrm{isot}}{\coloneqq}\{ \, \gamma \in {\mathcal{G}}\mid s(\gamma)=r(\gamma)\, \}$, and that the equivalence relation $$\mathcal{R}_{\mathcal{G}} {\coloneqq}\{ \, (r(\gamma ), s(\gamma ))\in \mathcal{G}^0\times \mathcal{G}^0 \mid \gamma \in \mathcal{G} \, \}$$ associated to $\mathcal{G}$ is hyperfinite. Then $(\mathcal{G},\mu )$ is inner amenable, and moreover there exists an inner amenability sequence $(\xi _n)_n$ for $\mathcal{G}$ which vanishes outside $\mathcal{G}_{\mathrm{isot}}$.
We note that by Proposition \[prop:ergdec\] shown below, the groupoid $({\mathcal{G}}_{\mathrm{isot}}, \mu)$ being inner amenable is equivalent to the the isotropy group ${\mathcal{G}}(x)$, of ${\mathcal{G}}$ at $x$, being inner amenable for $\mu$-almost every $x\in {\mathcal{G}}^0$.
By hypothesis, we can write $\mathcal{R}_{\mathcal{G}}=\bigcup _n \mathcal{R}_n$, where $\mathcal{R}_0\subset \mathcal{R}_1\subset \cdots$ is an increasing sequence of Borel equivalence relations on $\mathcal{G}^0$ with finite classes. Fix a countable collection $\{ \phi _n \} _{n\in {\mathbb{N}}}\subset [\mathcal{G}]$ which is dense in $[\mathcal{G}]$ and with $(x, \phi _i ^0(x)) \in \mathcal{R}_n$ for any $i\leq n$ and any $x\in \mathcal{G}^0$. For each $n$, we can find a Borel transversal $X_n\subset \mathcal{G}^0$ for $\mathcal{R}_n$, i.e., a Borel subset of ${\mathcal{G}}^0$ which meets each $\mathcal{R}_n$-equivalence class in exactly one point. Fix a Borel map $T_n \colon \mathcal{G}^0 \rightarrow \mathcal{G}$ with $(r(T_n(y)), s(T_n(y))) \in \mathcal{R}_n$, $s(T_n(y))\in X_n$, and $r(T_n(y))=y$ for any $y\in {\mathcal{G}}^0$. For $x\in {\mathcal{G}}^0$, let ${\mathcal{G}}(x)$ denote the isotropy group of ${\mathcal{G}}$ at $x$, and let $[x]_{{\mathcal{R}}_n}$ denote the ${\mathcal{R}}_n$-equivalence class of $x$. Since $({\mathcal{G}}_{\mathrm{isot}}, \mu)$ is inner amenable, for each $n\in {\mathbb{N}}$, we can find a Borel family $(\eta _n^ x)_{x\in X_n}$ of non-negative unit vectors $\eta ^x_n \in \ell ^1(\mathcal{G}(x))$ such that $\int_{X_n}\sum_{\gamma \in {\mathcal{G}}(x)}\eta_n^x(\gamma)1_D(\gamma)\, d\mu(x)\to 0$ for any bounded Borel subset $D\subset {\mathcal{G}}_{\mathrm{isot}}$ and such that $\| (\eta ^x_n )^{\delta} - \eta ^x_n \| _1 < 1/n$ for any $x\in X_n$ and any $\delta \in \mathcal{G}(x)$ belonging to the finite set $$F^n_x: = \{ \, T_n(\phi _i ^0(y))^{-1}\phi_i(y)T_n(y)\in {\mathcal{G}}(x) \mid y\in [x]_{\mathcal{R}_n} \text{ and }i\leq n \, \}.$$ Then define $\xi _n \colon \mathcal{G} \rightarrow [0,1]$ by $\xi _n (\gamma ) = 0$ for $\gamma \not\in \mathcal{G}_{\mathrm{isot}}$, and $$\xi _n (\gamma ) = \eta_n^{s(T_n(y))}(T_n(y)^{-1}\gamma T_n(y))$$ for $\gamma \in \mathcal{G}(y)$ with $y\in \mathcal{G}^0$. It is clear that $(\xi _n)$ satisfies conditions (i), (iii) and (iv) in Definition \[def:groupoid\]. To verify condition (ii), it is enough to show that $\| \xi _n ^{\phi _i}- \xi _n \| _1 \to 0$ for each $i\in {\mathbb{N}}$. For any $i\leq n$, $x\in X_n$, $y\in [x]_{\mathcal{R}_n}$ and $\gamma \in \mathcal{G}(y)$, we have $\phi _i (y)\gamma \phi _i (y)^{-1}\in {\mathcal{G}}(\phi _i ^0(y))$ and hence, setting $\delta = T_n(\phi _i ^0(y))^{-1}\phi_i(y)T_n(y)\in F^n_x$, we have $$\begin{aligned}
\xi _n^{\phi _i}(\gamma )&= \xi _n (\phi _i (y)\gamma \phi _i (y)^{-1}) = \eta _n^x (T_n(\phi _i^0(y))^{-1}\phi _i (y)\gamma \phi _i (y)^{-1}T_n(\phi _i^0(y)) )\\
&=(\eta _n ^x )^{\delta} (T_n(y)^{-1}\gamma T_n(y)),\end{aligned}$$ and thus $$\sum _{\gamma \in \mathcal{G}(y)} |\xi _n^{\phi _i}(\gamma ) - \xi _n (\gamma ) | = \sum _{\gamma \in \mathcal{G}(x)} |(\eta _n ^x )^{\delta}(\gamma ) - (\eta _n ^x )(\gamma )| < 1/n .$$ It follows that $\| \xi _n ^{\phi _i}- \xi _n \| _1 < 1/n \rightarrow 0$.
\[prop:aperamen\] Let $(\mathcal{G},\mu )$ be an amenable discrete p.m.p. groupoid which is aperiodic. Then $(\mathcal{G},\mu )$ is inner amenable.
By [@ar Corollary 5.3.33], the groupoid $({\mathcal{G}}_{\mathrm{isot}}, \mu)$ is amenable, and the equivalence relation $\mathcal{R}_{\mathcal{G}} {\coloneqq}\{ \, (r(\gamma ), s(\gamma )) \in \mathcal{G}^0\times \mathcal{G}^0 \mid \gamma \in \mathcal{G}\, \}$ is amenable and hence is hyperfinite by [@cfw]. By restricting ${\mathcal{G}}$ to a ${\mathcal{G}}$-invariant Borel subset of ${\mathcal{G}}^0$, we may assume that there exists an $M\in {\mathbb{N}}\cup \{ \infty \}$ such that $|\mathcal{G}(x)|=M$ for almost every $x\in \mathcal{G}^0$. If $M=\infty$ then we are done by Proposition \[prop:extend\] because infinite amenable groups are inner amenable.
Suppose now that $M\in {\mathbb{N}}$. Since $\mathcal{G}$ is aperiodic and almost every $\mathcal{G}(x)$ is finite, the equivalence relation $\mathcal{R}_{\mathcal{G}}$ is aperiodic and hyperfinite. Hence $[\mathcal{R}_{\mathcal{G}}]$ admits a central sequence $(T_n)_{n\in{\mathbb{N}}}$ with $T_nx\neq x$ for any $x\in \mathcal{G}^0$ and any $n\in {\mathbb{N}}$. For each $n\in {\mathbb{N}}$, let $\eta _n\in L^1({\mathcal{R}}_{\mathcal{G}}, \mu^1)$ be the indicator function of the graph $\{ \, (T_nx,x)\in {\mathcal{R}}_{{\mathcal{G}}} \mid x\in \mathcal{G}^0 \, \}$. Then $(\eta _n )$ is an inner amenability sequence for $(\mathcal{R}_{\mathcal{G}},\mu )$. Define $\xi _n \colon \mathcal{G}\rightarrow [0,1]$ by $\xi _n (\gamma )= \eta _n (r(\gamma ) , s(\gamma ))/M$. Then $(\xi _n)$ is an inner amenability sequence for $(\mathcal{G},\mu )$.
Proof of Theorem \[thm:equiv\] {#subsec-proof}
------------------------------
The only place where we use ergodicity is in the proof of the implication (4)$\Rightarrow$(5): Assume that $(\mathcal{G},\mu )$ is ergodic and that condition (4) holds, and let $\bm{m}$ be a diffuse, conjugation-invariant mean on $(\mathcal{G},\mu )$. Let $\check{\bm{m}}$ be the mean defined by $\check{\bm{m}}(D)=\bm{m}(D^{-1})$. After replacing $\bm{m}$ by $(\bm{m} + \check{\bm{m}})/2$ if necessary, we may assume without loss of generality that $\bm{m}$ is symmetric. Since $\bm{m}$ is diffuse, $(\mathcal{G},\mu )$ must be aperiodic. Thus if $(\mathcal{G},\mu )$ is amenable, then condition (1) holds by Proposition \[prop:aperamen\] and hence condition (5) holds, where the implication (1)$\Rightarrow$(5) will be proved for a general $({\mathcal{G}}, \mu)$ in the next paragraph. If $(\mathcal{G},\mu )$ is non-amenable, then $\bm{m}$ is balanced by Lemma \[lem:bal\], and hence condition (5) holds in this case as well.
For the rest of the proof, we no longer assume that $(\mathcal{G},\mu )$ is ergodic. The implications (1)$\Rightarrow$(2)$\Rightarrow$(3) are clear. The implication (2)$\Rightarrow$(1) follows from separability of $[\mathcal{G}]$ and of $L^1(\mathcal{G}^0,\mu )$. The implication (3)$\Rightarrow$(4) follows from weak${}^*$-compactness of the set of means on $\mathcal{G}$, by identifying both $L^1(\mathcal{G},\mu ^1 )$ and the set of means on $(\mathcal{G},\mu ^1 )$ with subsets of $L^{\infty}(\mathcal{G},\mu ^1 )^*$: If $(\xi _i )$ is a net as in condition (3), then any weak${}^*$-cluster point of $(\xi _i )$ in $L^{\infty}(\mathcal{G},\mu ^1 )^*$ is a diffuse, conjugation-invariant mean on $(\mathcal{G},\mu )$. The implication (2)$\Rightarrow$(5) is analogous: If $(\xi _i )$ is a net as in condition (2), then after replacing $\xi _i$ by $\xi _i '{\coloneqq}(\xi _i + \check{\xi} _i)/2$, where $\check{\xi}_i(\gamma ) {\coloneqq}\xi _i(\gamma ^{-1})$, we can assume that each $\xi _i$ is symmetric, and hence any weak${}^*$-cluster point of $(\xi _i)$ is a mean on $(\mathcal{G},\mu )$ satisfying condition (5).
(5)$\Leftrightarrow$(6): If $\bm{m}$ is a mean on $(\mathcal{G}, \mu )$ as in condition (5), then for any $F\in L^\infty (\mathcal{G},\mu ^1 )$, we can define a (countably additive!) finite, complex Borel measure $\mu _F$ on $\mathcal{G}^0$ by $\mu _F(A)=\int 1_{\mathcal{G}_A}F\, d\bm{m}$, which is absolutely continuous with respect to $\mu$. Countable additivity of $\mu _F$ follows from $\bm{m}$ being balanced. Then the map $P \colon L^\infty (\mathcal{G}, \mu ^1 ) \rightarrow L^\infty (\mathcal{G}^0,\mu )$ defined by $P(F) = d\mu _F/d\mu$ verifies condition (6). Conversely, if $P$ is as in condition (6), then $\bm{m}(D)=\int _{\mathcal{G}^0} P(1_D)\, d\mu$ defines a mean on $(\mathcal{G},\mu )$ as in condition (5).
It remains to prove the implication (5)$\Rightarrow$(2). For each $\eta \in L^1(\mathcal{G} ,\mu ^1 )$, we define $\check{\eta}$ by $\check{\eta}(\gamma ) = \eta (\gamma ^{-1})$, and call $\eta$ symmetric if $\check{\eta}=\eta$. In what follows, we denote by $L^1(\mathcal{G},\mu ^1 )_{+,1}$ the set of all non-negative unit vectors in $L^1(\mathcal{G},\mu ^1 )$. The next lemma (with $D=\emptyset$) will complete the proof.
\[lem:weakstar\] Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid and let $\bm{m}$ be a diffuse, conjugation-invariant mean on $(\mathcal{G},\mu )$ which is symmetric and balanced. Let $D\subset \mathcal{G}$ be a Borel subset with $\bm{m}(D)=0$. Then $\bm{m}$ is the weak${}^*$-limit of a net $(\xi _i)$ of symmetric vectors in $L^1(\mathcal{G},\mu ^1 )_{+, 1}$ which vanish on $D$ and satisfy conditions (i)–(iv) in Definition \[def:groupoid\].
Since $\bm{m}$ is symmetric, by replacing $D$ by $D\cup D^{-1}$, we may assume that $D$ is symmetric as well. For each $\eta \in L^1(\mathcal{G} ,\mu ^1 )$, define $r_{\eta}, s_{\eta} \in L^1(\mathcal{G}^0,\mu )$ by $r_{\eta} (x) = \sum _{\gamma \in {\mathcal{G}}^x}\eta (\gamma )$ and $s_{\eta} (x) = \sum _{\gamma \in {\mathcal{G}}_x}\eta (\gamma )$. If $\eta$ is a vector in $L^1(\mathcal{G},\mu ^1 )_{+, 1}$, then $r_{\eta}$ and $s_{\eta}$ are non-negative unit vectors in $L^1(\mathcal{G}^0,\mu )$, and if $\eta$ is symmetric, then $r_{\eta}=s_{\eta}$.
\[claim:transfin\] Let $\eta \in L^1(\mathcal{G},\mu ^1 )_{+, 1}$ be a symmetric vector which vanishes on $D$. Then there exists a symmetric vector $\xi \in L ^1 (\mathcal{G},\mu ^1 )_{+, 1}$ which vanishes on $D$ and satisfies $s_{\xi} = 1_{\mathcal{G}^0}$ and $\| \eta - \xi \| _1 = \| s_{\eta}- 1_{\mathcal{G}^0} \| _1$.
Let $\eta _0 = \eta$. We proceed by transfinite induction on countable ordinals $\alpha$ to define a symmetric vector $\eta _{\alpha}\in L^1 (\mathcal{G},\mu ^1 )_{+, 1}$ vanishing on $D$ and satisfying, for any $\beta < \alpha$:
1. $\| \eta _{\beta} - \eta _{\alpha} \| _1 = \| s_{\eta _{\beta }} -s_{\eta _{\alpha }} \| _1$.
2. For almost every $x\in \mathcal{G}^0$, if $s_{\eta _{\beta}}(x) \leq 1$, then $s_{\eta _{\beta}}(x)\leq s_{\eta _{\alpha}}(x)\leq 1$.
3. For almost every $x\in \mathcal{G}^0$, if $s_{\eta _{\beta}}(x) \geq 1$, then $s_{\eta _{\beta}}(x)\geq s_{\eta _{\alpha}}(x)\geq 1$.
4. If $\| s_{\eta _{\beta}}- 1_{\mathcal{G}^0}\| _1 >0$, then $\| s_{\eta _{\alpha}} - 1_{\mathcal{G}^0}\| _1 < \| s_{\eta _{\beta}}- 1_{\mathcal{G}^0} \| _1$, and if $s_{\eta _{\beta}} = 1_{\mathcal{G}^0}$, then $\eta _{\alpha}=\eta _{\beta}$.
Assume that $\{ \eta _{\beta } \} _{\beta <\alpha}$ has already been defined, and we show how to define $\eta _{\alpha}$. If $\alpha$ is a limit ordinal, say $\alpha = \sup _{n\in {\mathbb{N}}}\beta _n$, where $\beta _1<\beta _2<\cdots$, then, by the induction hypothesis (namely, properties (ii) and (iii)), the sequence $(s_{\eta _{\beta _n}})_{n\in {\mathbb{N}}}$ is Cauchy in $L^1(\mathcal{G}^0,\mu )$. Hence property (i) implies that the sequence $(\eta _{\beta _n} ) _{n\in {\mathbb{N}}}$ is Cauchy in $L^1(\mathcal{G},\mu ^1 )$, so we define $\eta _{\alpha}$ to be its limit in $L^1(\mathcal{G},\mu ^1)$. If $\alpha$ is a successor, then we define $\eta _{\alpha}$ from $\eta _{\alpha -1}$ as follows: If $s_{\eta _{\alpha - 1}} = 1_{\mathcal{G}^0}$, then we put $\eta _{\alpha} = \eta _{\alpha -1}$. Otherwise, i.e., if $s_{\eta _{\alpha - 1}} \neq 1_{\mathcal{G}^0}$, then for some ${\varepsilon}>0$, both the sets $$A _0 {\coloneqq}\{ \, x\in {\mathcal{G}^0} \mid s_{\eta _{\alpha -1}}(x)< 1-{\varepsilon}\, \} \quad \text{and} \quad A_1 {\coloneqq}\{ \, x\in {\mathcal{G}^0} \mid s_{\eta _{\alpha -1}}(x)>1+{\varepsilon}\, \}$$ have positive measure. Since $\bm{m}(\mathcal{G}_{A_i}\setminus D) =\mu (A_i)>0$ for each $i\in \{ 0,1 \}$, we may find symmetric Borel subsets, $C_0\subset \mathcal{G}_{A_0}\setminus D$ and $C_1\subset \mathcal{G}_{A_1}\setminus D$, with $\mu ^1 (C_0)=\mu ^1 (C_1)>0$ and $|C_i \cap {\mathcal{G}}_x| \leq 1$ for any $x\in {\mathcal{G}^0}$. Then the function $\eta _{\alpha}{\coloneqq}\eta _{\alpha -1}+{\varepsilon}(1_{C_0} - 1_{C_1})$ has the required properties, and the induction is complete.
By property (iv), there is some countable ordinal $\alpha _0$ such that $s_{\eta _{\alpha _0}} = 1_{\mathcal{G}^0}$, so letting $\xi {\coloneqq}\eta _{\alpha _0}$ works.
Return to the proof of Lemma \[lem:weakstar\]. By the Hahn-Banach theorem, the set $L^1(\mathcal{G},\mu ^1 )_{+,1}$ is weak${}^*$-dense in the set of all means on $(\mathcal{G},\mu )$. Since $\bm{m}$ is symmetric and $\bm{m}(D)=0$, $\bm{m}$ belongs to the weak${}^*$-closure of the set of all symmetric vectors in $L^1(\mathcal{G},\mu ^1 )_{+,1}$ that vanish on $D$. Let $(\eta _i )$ be a net weak${}^*$-converging to $\bm{m}$ and consisting of symmetric vectors in $L^1(\mathcal{G},\mu ^1 )_{+,1}$ that vanish on $D$. Then $s_{\eta _i}$ converges weakly to $1_{\mathcal{G}^0}$ in $L^1({\mathcal{G}^0},\mu )$, and $\eta _i ^{\phi} - \eta _i$ converges weakly to $0$ in $L^1({\mathcal{G}^0},\mu )$ for any $\phi \in [\mathcal{G}]$. Thus, by the Hahn-Banach theorem, after taking convex sums, we may assume without loss of generality that $\| s_{\eta _i}- 1_{\mathcal{G}^0} \|_1 \rightarrow 0$ and $\| \eta _i ^{\phi } - \eta _i \| _1 \rightarrow 0$ for any $\phi \in [\mathcal{G}]$. Applying Claim \[claim:transfin\] to each $\eta _i$, we obtain the required net $(\xi _i )$.
Permanence of inner amenability {#subsec-perma}
-------------------------------
We discuss permanence of inner amenability under inflations and restrictions, finite-index inclusions, measure-preserving extensions, ergodic decompositions, and inverse limits.
### Inflations and restrictions
\[prop:inflate\] Let $(\mathcal{G},\mu )$ be an ergodic discrete p.m.p. groupoid and let $A\subset {\mathcal{G}}^0$ be a Borel subset with positive measure. Then $(\mathcal{G},\mu )$ is inner amenable if and only if $(\mathcal{G}_A,\mu _A )$ is inner amenable, where $\mu_A$ is the normalized restriction of $\mu$ to $A$.
As seen in Remark \[rem:locsec\], if $(\mathcal{G},\mu )$ is inner amenable then so is $(\mathcal{G}_A,\mu _A )$. Conversely, assume $(\mathcal{G}_A, \mu _A)$ is inner amenable and let $\bm{m}_A$ be a mean on $(\mathcal{G}_A,\mu _A)$ as in condition (5) of Theorem \[thm:equiv\]. After shrinking $A$, we may assume that $\mu (A) = 1/n$ for some $n\in {\mathbb{N}}$. Since $(\mathcal{G},\mu )$ is ergodic, we can find some $\phi \in [\mathcal{G}]$ with $\{ (\phi^i)^0A \}_{i=0}^{n-1}$ partitioning $\mathcal{G}^0$ and $(\phi ^n)^0(x) =x$ for any $x\in \mathcal{G}^0$, where $\phi^i\in [{\mathcal{G}}]$ is the $i$-th iterate of $\phi$. For each $i\in \{ 0, 1,\ldots, n-1\}$, let $A_i {\coloneqq}(\phi ^i)^0A$. Define a mean $\bm{m}$ on $(\mathcal{G},\mu )$ by $$\bm{m}(D) = \frac{1}{n}\sum _{i=0}^{n-1} \bm{m}_A((\mathcal{G}_{A_i}\cap D)^{\phi ^i}).$$ Then it is clear that $\bm{m}$ is diffuse and balanced. Fix $\psi \in [\mathcal{G}]$ and a Borel subset $D\subset \mathcal{G}$ toward verifying conjugation-invariance of $\bm{m}$. Since $\bm{m}$ is balanced, we have $$\begin{aligned}
\bm{m}(D) &= \sum _{i, j=0}^{n-1} \bm{m} (\mathcal{G}_{A_i\cap \psi ^0A_j} \cap D)\quad \text{and} \\
\bm{m} ( D^{\psi}) &= \sum _{i, j=0}^{n-1}\bm{m} (\mathcal{G}_{(\psi ^{0})^{-1}A_i\cap A_j}\cap D^{\psi}) = \sum _{i,j=0}^{n-1} \bm{m} ((\mathcal{G}_{A_i\cap \psi ^0A_j} \cap D)^{\psi}) ,\end{aligned}$$ so it suffices to show that if $D\subset \mathcal{G}_{A_i\cap \psi ^0A_j}$, then $\bm{m}(D)= \bm{m}(D^{\psi })$. Let $\chi$ be the restriction of $(\phi^i)^{-1}\circ \psi \circ \phi^j$ to $(\phi ^0 )^{-j}((\psi ^0)^{-1}A_i\cap A_j )\subset A$. Then $\mathrm{ran}(\chi ^0) = (\phi ^0)^{-i}(A_i \cap \psi ^0A_j)\subset A$, so $\chi$ is a local section of $(\mathcal{G}_A,\mu _A )$. Since $D\subset \mathcal{G}_{A_i\cap \psi ^0A_j}$, we have $D^{\phi ^i}\subset \mathcal{G}_{\mathrm{ran}(\chi ^0)}$ and $$\begin{aligned}
\bm{m} (D) &= \bm{m}_A (D^{\phi ^i })/n = \bm{m}_A (D^{\phi ^i\circ \chi })/n = \bm{m}_A(D^{\psi \circ \phi ^j})/n = \bm{m}(D^{\psi} ).\end{aligned}$$ Thus $\bm{m}$ is conjugation-invariant, and by Theorem \[thm:equiv\], $(\mathcal{G},\mu )$ is inner amenable.
### Finite-index inclusions
Let ${\mathcal{G}}$ be a discrete p.m.p. groupoid and let ${\mathcal{H}}$ be a Borel subgroupoid of ${\mathcal{G}}$. For each $x\in {\mathcal{G}}^0$, we have the equivalence relation on ${\mathcal{G}}_x$ such that two elements $\gamma, \delta \in {\mathcal{G}}_x$ are equivalent if and only if $\gamma \delta^{-1}\in {\mathcal{H}}$. The function assigning to each $x\in {\mathcal{G}}^0$ the number of equivalence classes in ${\mathcal{G}}^0$ is Borel and ${\mathcal{G}}$-invariant, and hence constant on a conull set if ${\mathcal{G}}$ is ergodic. If ${\mathcal{G}}$ is ergodic, this constant value is called the *index* of ${\mathcal{H}}$ in ${\mathcal{G}}$. This definition extends the index of a subrelation of a discrete p.m.p. equivalence relation given in [@fsz Section 1].
\[prop-finite-index\] Let ${\mathcal{G}}$ be an ergodic discrete p.m.p. groupoid and let ${\mathcal{H}}$ be a finite-index Borel subgroupoid of ${\mathcal{G}}$. If ${\mathcal{H}}$ is inner amenable, then ${\mathcal{G}}$ is also inner amenable.
Assume first that ${\mathcal{H}}$ is ergodic. By assumption, we have a conjugation-invariant, balanced, diffuse mean $\bm{m}_0$ on ${\mathcal{H}}$. By setting $\bm{m}_0({\mathcal{G}}\setminus {\mathcal{H}})=0$, we regard $\bm{m}_0$ as a mean on ${\mathcal{G}}$. Then $\bm{m}_0$ is a balanced mean on ${\mathcal{G}}$, and is conjugation-invariant under ${\mathcal{H}}$, i.e., we have $\bm{m}_0(D^\phi)=\bm{m}_0(D)$ for any Borel subset $D\subset {\mathcal{G}}$ and any $\phi \in [{\mathcal{H}}]$. Let $N$ be the index of ${\mathcal{H}}$ in ${\mathcal{G}}$. Since ${\mathcal{H}}$ is ergodic, we may choose $N$ elements $\psi_1,\ldots, \psi_N\in [{\mathcal{G}}]$ as in [@fsz Lemma 1.3], such that for any $x\in {\mathcal{G}}^0$ the sets $\{ \, h\psi_i(x)\mid h\in {\mathcal{H}}_{r(\psi_i(x))}\, \}$ with $i=1, \ldots, N$, partition ${\mathcal{G}}_x$. We define a mean $\bm{m}$ on ${\mathcal{G}}$ by $\bm{m}(D)=N^{-1}\sum_{i=1}^N\bm{m}_0(D^{\psi_i})$. Pick a Borel subset $D\subset {\mathcal{G}}$ and $\phi \in [{\mathcal{G}}]$. Let $A_{ij}$ be the Borel subset of all points $x\in {\mathcal{G}}^0$ such that $(\phi \circ \psi_i)(x)\psi_j(x)^{-1}$ belongs to ${\mathcal{H}}$. Then the sets $A_{ij}$ with $i, j=1,\ldots, N$ partition ${\mathcal{G}}^0$, and we have $$\begin{aligned}
\bm{m}(D^\phi)&=\frac{1}{N}\sum_{i=1}^N\bm{m}_0(D^{\phi \circ \psi_i})=\frac{1}{N}\sum_{i, j=1}^N\bm{m}_0(D^{\phi \circ \psi_i}\cap {\mathcal{G}}_{A_{ij}})\\
&=\frac{1}{N}\sum_{i, j=1}^N\bm{m}_0(D^{\psi_j}\cap {\mathcal{G}}_{A_{ij}})=\frac{1}{N}\sum_{i=1}^N \bm{m}_0(D^{\psi_j})=\bm{m}(D),\end{aligned}$$ where the second and fourth equations hold because $\bm{m}_0$ is balanced, and the third equation holds because $\bm{m}_0$ is conjugation-invariant under ${\mathcal{H}}$. The mean $\bm{m}$ is therefore conjugation-invariant under ${\mathcal{G}}$. Since $\bm{m}$ is diffuse, ${\mathcal{G}}$ is inner amenable by Theorem \[thm:equiv\].
In general, since ${\mathcal{G}}$ is ergodic and ${\mathcal{H}}$ has finite index in ${\mathcal{G}}$, there is some positive measure ${\mathcal{H}}$-invariant Borel subset $A$ of ${\mathcal{G}}^0$ such that ${\mathcal{H}}_A$ is ergodic. By ergodicity of ${\mathcal{G}}$, we may find a Borel map $T\colon {\mathcal{G}}^0\rightarrow {\mathcal{G}}$, with $T(x)\in {\mathcal{G}}_x$, such that $T(x)=x$ for all $x\in A$, and $r(T(x)) \in A$ for all $x\in {\mathcal{G}}^0$. Then the Borel map $c_T\colon {\mathcal{G}}\rightarrow {\mathcal{G}}_A$, $c_T(\gamma ) {\coloneqq}T(r(\gamma ))\gamma T(s(\gamma ))^{-1}$, is a groupoid homomorphism, and the groupoid ${\mathcal{K}}{\coloneqq}c_T^{-1}({\mathcal{H}}_A )$ is an ergodic finite index subgroupoid of ${\mathcal{G}}$ with ${\mathcal{K}}_A = {\mathcal{H}}_A$. Since ${\mathcal{H}}$ is inner amenable, ${\mathcal{H}}_A$ is inner amenable by Remark \[rem:locsec\], and hence ${\mathcal{K}}$ is inner amenable by Proposition \[prop:inflate\]. The ergodic case proved above therefore implies that ${\mathcal{G}}$ is inner amenable.
The converse of Proposition \[prop-finite-index\] also holds (see Corollary \[cor-finite-index-cpt-ext\]).
### Measure-preserving extensions
Let $(\mathcal{G} ,\mu )$ be a discrete p.m.p. groupoid. Let $(Z,\zeta )$ be a standard probability space and let $\alpha \colon \mathcal{G} \rightarrow \mathrm{Aut}(Z ,\zeta )$ be a cocycle. The associated *extension groupoid* $(\mathcal{G} ,\mu )\ltimes _{\alpha}(Z ,\zeta )= (\tilde{\mathcal{G}},\tilde{\mu} )$ is the discrete p.m.p. groupoid defined as follows: The set of groupoid elements is $\tilde{\mathcal{G}} {\coloneqq}\mathcal{G}\times Z$, with unit space $\tilde{\mathcal{G}}^0 {\coloneqq}\mathcal{G}^0\times Z$ and measure $\tilde{\mu} {\coloneqq}\mu \times \zeta$ on $\tilde{{\mathcal{G}}}^0$. The source and range maps are defined by $\tilde{s}(\gamma , z ) = (s(\gamma ), z)$ and $\tilde{r}(\gamma ,z )= (r(\gamma ), \alpha (\gamma )z )$, respectively, with groupoid operations defined by $(\gamma _1, \alpha (\gamma _0)z)(\gamma _0, z) = (\gamma _1\gamma _0 , z )$ and $(\gamma ,z )^{-1} = (\gamma ^{-1}, \alpha (\gamma )z )$.
\[prop:Ginn\] Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid, let $(Z, \zeta)$ be a standard probability space, and let $\alpha \colon \mathcal{G}\rightarrow \mathrm{Aut}(Z,\zeta )$ be a cocycle. Suppose that the extension groupoid $(\mathcal{G} ,\mu )\ltimes _{\alpha}(Z ,\zeta )$ is inner amenable. Then $(\mathcal{G},\mu )$ is inner amenable.
In particular, if a countable group $G$ admits a p.m.p. action $G\curvearrowright (Z,\zeta )$ such that the associated translation groupoid $G\ltimes (Z,\zeta )$ is inner amenable, then $G$ is inner amenable.
Suppose that the groupoid $(\tilde{\mathcal{G}},\tilde{\mu} ){\coloneqq}(\mathcal{G} ,\mu )\ltimes _{\alpha}(Z ,\zeta )$ is inner amenable, and let $\tilde{\bm{m}}$ be a mean on $(\tilde{\mathcal{G}},\tilde{\mu})$ as in condition (5) of Theorem \[thm:equiv\]. Then the mean $\bm{m}$ on $(\mathcal{G},\mu )$ defined by $\bm{m}(D)= \tilde{\bm{m}}(D\times Z )$ witnesses that $(\mathcal{G},\mu )$ is inner amenable.
While the converse of Proposition \[prop:Ginn\] does not hold in general (e.g., Corollary \[cor-ber\]), it does hold for compact extensions (Corollary \[cor:cpctext\]), and more generally it holds for distal extensions (Corollary \[cor:distal\]).
### Ergodic decompositions
We refer to [@hahn] for the ergodic decomposition of discrete p.m.p. groupoids.
\[prop:ergdec\] Let $(\mathcal{G},\mu )$ be a discrete p.m.p. groupoid with ergodic decomposition map $\pi \colon (\mathcal{G}^0,\mu )\rightarrow (Z,\zeta )$ and disintegration $(\mathcal{G},\mu ) = \int _Z (\mathcal{G}_z,\mu _z )\, d\zeta (z)$. Then $(\mathcal{G},\mu )$ is inner amenable if and only if $(\mathcal{G}_z,\mu _z)$ is inner amenable for $\zeta$-almost every $z\in Z$.
Assume first that $(\mathcal{G}_z,\mu _z)$ admits an inner amenability sequence for $\zeta$-almost every $z\in Z$. The groupoid $(\mathcal{G}_z, \mu _z)$, being inner amenable, is aperiodic for $\zeta$-almost every $z\in Z$, and hence $(\mathcal{G}, \mu )$ is aperiodic as well. Let $\mu _{{\mathbb{N}}}$ denote counting measure on the natural numbers ${\mathbb{N}}$. Since $s\colon \mathcal{G}\rightarrow \mathcal{G}^0$ is a countably infinite-to-one Borel map, by the Lusin-Novikov Uniformization Theorem ([@kec-set Theorem 18.10]), we may find an isomorphism of measure spaces $$\varphi \colon ({\mathbb{N}}\times \mathcal{G}^0, \, \mu _{{\mathbb{N}}}\times \mu ) \rightarrow (\mathcal{G},\mu ^1)$$ satisfying $\varphi (0,x)=x\in \mathcal{G}^0$ and $s(\varphi (i,x)) = x$ for any $i\in {\mathbb{N}}$ and $\mu$-almost every $x\in \mathcal{G}^0$. We may therefore assume without loss of generality that there is some standard probability space $(X,\mu _X )$ such that, as measure spaces we have $(\mathcal{G},\mu ^1 )=({\mathbb{N}}\times X, \mu _{{\mathbb{N}}}\times \mu _X )$ and $(\mathcal{G}^0 , \mu ) = ( \{ 0 \} \times X, \delta _0 \times \mu _X )$, with the source map $s\colon \mathcal{G}\rightarrow \mathcal{G}^0$ being given by $s(i,x)=(0,x)$ for $\mu^1$-almost every $(i,x) \in \mathcal{G}$.
Let $Z_0$ consist of all $z\in Z$ for which the measure $\mu _z$ on $\mathcal{G}^0_z$ is atomless, and for each integer $n\geq 1$, let $Z_n$ consist of all $z\in Z$ for which the measure $\mu _z$ is uniformly distributed on $n$ points. Then $Z_0,Z_1,\dots$ partition $Z$, and $\pi ^{-1}(Z_0), \pi ^{-1}(Z_1),\dots $ partition $\mathcal{G}^0$ into $\mathcal{G}$-invariant sets, so it is enough to show that, for each $n$ with $\mu (\pi ^{-1}(Z_n))>0$, the groupoid $(\mathcal{G}_{\pi ^{-1}(Z_n) }, \mu _{\pi ^{-1}(Z_n)} )$ admits an inner amenability sequence. We may therefore assume without loss of generality that $Z=Z_n$ for some $n$, and hence (arguing as in the proof of [@gla Theorem 3.18]) we may also assume that $(X,\mu _X )=(Y\times Z ,\nu \times \zeta )$ for some standard probability space $(Y,\nu )$, with $\pi \colon \mathcal{G}^0 \rightarrow Z$ being the projection map to the $Z$-coordinate, $\pi (0,(y,z)) = z$. Then for each $z\in Z$, the measures $\mu _z$ on $\mathcal{G}_z^0= \{ 0 \} \times Y \times \{ z\} $ and $\mu ^1_z$ on $\mathcal{G}_z = {\mathbb{N}}\times Y \times \{ z \}$ are respectively given by $\mu _z = \delta _0 \times \nu \times \delta _z$ and $\mu ^1_z = \mu _{{\mathbb{N}}} \times \nu \times \delta _z$.
Fix ${\varepsilon}>0$, along with a finite subset $\Phi \subset [\mathcal{G}]$, finite Borel partitions $\mathcal{B}$ and $\mathcal{C}$ of $Y$ and $Z$, respectively, and a Borel subset $D\subset \mathcal{G}$ with $\mu ^1 (D)<\infty$. It is enough to find a symmetric, non-negative unit vector $\xi \in L^1(\mathcal{G},\mu ^1 )=L^1({\mathbb{N}}\times Y\times Z , \mu _{{\mathbb{N}}} \times \nu \times \zeta )$ satisfying
1. $| \| 1_{{\mathcal{G}}_{\{ 0 \} \times B\times C}}\xi \| _1 - (\nu \times \zeta ) (B\times C) | < {\varepsilon}$ for any $B\in \mathcal{B}$ and any $C\in \mathcal{C}$,
2. $\| \xi ^{\phi } - \xi \| _1 < {\varepsilon}$ for any $\phi \in \Phi$,
3. $\| 1_D \xi \| _1 < {\varepsilon}$, and
4. $\sum _{i \in {\mathbb{N}}} \xi (i,(y,z)) =1$ for $(\nu\times \zeta )$-almost every $(y,z)\in Y\times Z$.
For $z\in Z$ and $\xi \in L^1( {\mathbb{N}}\times Y , \mu _{{\mathbb{N}}} \times \nu )$, let $\xi ^{(z)} \in L^1(\mathcal{G}_z, \mu ^1 _z ) = L^1({\mathbb{N}}\times Y \times \{ z \} , \mu _{{\mathbb{N}}}\times \nu \times \delta _z )$ be given by $\xi ^{(z)} (i,y,z)=\xi (i,y)$. For almost every $z\in Z$, the groupoid $(\mathcal{G}_z, \mu _z )$ is inner amenable and $\mu ^1_z (D)<\infty$, so we may find some $\xi \in L^1 ({\mathbb{N}}\times Y, \mu _{{\mathbb{N}}}\times \nu )$ such that $\xi ^{(z)}\in L^1(\mathcal{G}_z, \mu ^1 _z)$ is a symmetric, non-negative unit vector satisfying
1. $| \| 1_{{\mathcal{G}}_{\{ 0 \} \times B\times C}}\xi ^{(z)} \| _{L^1(\mu ^1_z )} - \nu (B)1_C(z) | < {\varepsilon}$ for any $B\in \mathcal{B}$ and any $C\in \mathcal{C}$,
2. $\| (\xi ^{(z)} ) ^{\phi } - \xi ^{(z)} \| _{L^1(\mu ^1_z)} < {\varepsilon}$ for any $\phi \in \Phi$,
3. $\| 1_D \xi ^{(z)} \| _{L^1(\mu ^1 _z)} < {\varepsilon}$, and
4. $\sum _{i \in {\mathbb{N}}} \xi (i, y) =1$ for $\nu$-almost every $y\in Y$.
Let $\Omega$ be the set of all such pairs $(z,\xi )$, i.e., all pairs $(z,\xi )\in Z\times L^1({\mathbb{N}}\times Y ,\mu _{{\mathbb{N}}}\times \nu )$ such that $\xi ^{(z)}\in L^1(\mathcal{G}_z, \mu ^1 _z)$ is a symmetric, non-negative unit vector satisfying conditions (1.$z$)–(4.$z$). Then $\Omega$ is a Borel subset of $Z\times L^1 ({\mathbb{N}}\times Y, \mu _{{\mathbb{N}}}\times \nu )$, where the reason (2.$z$) defines a Borel property is because all the groupoid operations are by assumption Borel and each $\phi \in \Phi$ is Borel. By applying the Jankov-von Neumann Uniformization Theorem ([@kec-set Theorem 18.1]) and Lusin’s theorem that analytic sets are universally measurable ([@kec-set Theorem 21.10]), after discarding a $\zeta$-null set from $Z$, we may find a Borel map $Z\rightarrow L^1({\mathbb{N}}\times Y, \mu _{{\mathbb{N}}}\times \nu )$, $z\mapsto \xi _z$, such that $(z,\xi _z ) \in \Omega$ for almost every $z\in Z$. Define $\xi \in L^1({\mathcal{G}},\mu ^1 )$ by $\xi (i,y,z)= \xi _z (i,y)$. Then $\xi$ is a symmetric, non-negative unit vector in $L^1({\mathcal{G}},\mu ^1 )$ satisfying conditions (1)–(4).
Conversely, assume that $(\xi _n)_{n\in {\mathbb{N}}}$ is an inner amenability sequence for $(\mathcal{G},\mu )$. By properties (i) and (iv) of Definition \[def:groupoid\], for any Borel subset $B\subset \mathcal{G}^0$, we have $$\begin{aligned}
&\int _Z \biggl| \mu _z (B) - \int _{(\mathcal{G}_z)_B} \xi _n \, d\mu ^1_z\biggr| \, d\zeta(z) = \int _Z \biggl| \int _B 1 - \sum _{\gamma \in ({\mathcal{G}}_B)_x} \xi _n (\gamma )\, d\mu _z(x) \biggr| \, d\zeta(z) \\
&=\int _Z \int _B\biggl( 1 - \sum _{\gamma \in ({\mathcal{G}}_B)_x} \xi _n (\gamma )\biggr)\, d\mu _z(x) \, d\zeta(z) = \mu (B) - \int _{\mathcal{G}_B}\xi _n \, d\mu ^1 \rightarrow 0.\end{aligned}$$ Likewise, for any $\phi \in [\mathcal{G}]$, we have $\int _{Z} \| \xi _n ^{\phi} - \xi _n \| _{L^1(\mu ^1 _z)} \, d\zeta(z) \rightarrow 0$, and for any bounded Borel subset $D\subset \mathcal{G}$, we have $\int _Z \| 1_D \xi _n \| _{L^1(\mu ^1 _z)}\, d\zeta(z) \rightarrow 0$. Therefore, by separability of $L^1(\mathcal{G}^0,\mu )$ and of $[\mathcal{G}]$, we can find a single subsequence $(\xi _{n_i})$ such that for $\zeta$-almost every $z\in Z$, $(\xi _{n_i})$ is an inner amenability sequence for $(\mathcal{G}_z,\mu _z)$.
### Inverse limits
Let $(\mathcal{G}_1, \mu _1 )$ be a discrete p.m.p. groupoid. A *locally bijective extension* of $(\mathcal{G}_1 ,\mu _1 )$ is a measure-preserving groupoid homomorphism $\varphi \colon (\mathcal{G}, \mu ) \rightarrow (\mathcal{G}_1,\mu _1 )$, from a discrete p.m.p. groupoid $(\mathcal{G}, \mu )$ to $(\mathcal{G}_1,\mu _1 )$, such that for almost every $x\in \mathcal{G}^0$, its restriction $\varphi \colon {\mathcal{G}}_x \rightarrow ({\mathcal{G}}_1)_{\varphi (x)}$ is bijective. Clearly, compositions of locally bijective extensions are locally bijective.
Suppose that $I$ is a countable directed set and we have a directed family $(\varphi _{i,j} \colon (\mathcal{G}_j,\mu _j)\rightarrow (\mathcal{G}_i,\mu _i ) )_{i,j\in I , i<j}$ of locally bijective extensions of groupoids, that is, $\varphi_{i, j}$ is a measure-preserving groupoid homomorphism such that $\varphi _{i,j}\circ \varphi _{j,k}=\varphi _{i,k}$ whenever $i<j<k$. Then the *inverse limit* of this family is the discrete p.m.p. groupoid $(\mathcal{G},\mu )$ defined by $$\begin{aligned}
\mathcal{G} &= \left\{ \left. (\gamma _i )_{i\in I} \in \prod _{i\in I}\mathcal{G}_i \, \right| \, \varphi _{i,j} (\gamma _j ) = \gamma _i \text{ for any }i<j \, \right\} $$ and $$\begin{aligned}
\mathcal{G}^0 &= \left\{ \left. (x_i )_{i\in I} \in \prod _{i\in I}\mathcal{G}_i^0 \, \right| \, \varphi _{i,j}(x_j ) = x_i \text{ for any }i <j \, \right\} ,\end{aligned}$$ with $(\mathcal{G}^0, \mu )$ the inverse limit of the measure spaces $(\mathcal{G}_i^0,\mu _i^0 )$, and with source and range maps defined by $s((\gamma _i)_{i\in I}) = (s(\gamma _i))_{i\in I}$ and $r((\gamma _i)_{i\in I}) = (r(\gamma _i))_{i\in I}$, respectively. For each $i\in I$, the projection map $\varphi _i \colon (\mathcal{G},\mu )\rightarrow (\mathcal{G}_i,\mu _i )$ is a locally bijective extension of groupoids, and if $i<j$, then $\varphi _{i,j}\circ \varphi _j = \varphi _i$.
\[prop:invlim\] Let $(\varphi _{i,j} \colon (\mathcal{G}_j,\mu _j)\rightarrow (\mathcal{G}_i,\mu _i ) )_{i,j\in I , i<j}$ be a countable directed family of locally bijective extensions of groupoids, and let $(\mathcal{G},\mu )$ be its inverse limit. If each of the groupoids $(\mathcal{G}_i, \mu _i )$ is inner amenable, then $(\mathcal{G},\mu )$ is inner amenable.
Since $(\mathcal{G}_i, \mu _i )$ is inner amenable, we may find a positive linear map $P_i\colon L^{\infty}(\mathcal{G}_i,\mu ^1 _i ) \rightarrow L^{\infty}(\mathcal{G}_i^0,\mu _i )$ as in condition (6) of Theorem \[thm:equiv\], so that the mean $\bm{n}_i$ on $({\mathcal{G}}_i, \mu_i)$ defined by $\bm{n}_i(D) = \int _{\mathcal{G}_i^0} P_i(1_D) \, d\mu _i$ is a balanced, diffuse, conjugation-invariant mean on $(\mathcal{G}_i,\mu _i)$. Let $\mu = \int _{\mathcal{G}_i^0} \mu _i^z \, d\mu _i(z)$ be the disintegration of $\mu$ via $\varphi _i$. For $\mu_i^1$-almost every $\delta \in \mathcal{G}_i$, there is the bijection $\gamma _{\varphi _i, \delta} \colon \varphi _i^{-1}(s(\delta ))\rightarrow \varphi _i^{-1}(\delta )$ that sends each $x\in \varphi _i^{-1}(s(\delta ))$ to the unique element $\gamma = \gamma _{\varphi _i ,\delta}(x)$ in ${\mathcal{G}}_x$ with $\varphi _i (\gamma ) = \delta$. We then obtain the conditional expectation $E_i\colon L^{\infty}(\mathcal{G},\mu ^1 ) \rightarrow L^{\infty}(\mathcal{G}_i,\mu ^1 _i )$ given by $$E_i(F)(\delta )= \int _{\varphi _i^{-1}(s(\delta ))}F(\gamma _{\varphi _i,\delta}(x))\, d\mu _i ^{s(\delta )}(x)$$ for $F\in L^\infty({\mathcal{G}}, \mu^1)$ and $\delta \in {\mathcal{G}}_i$. Then we have the mean $\bm{m}_i$ on $(\mathcal{G} ,\mu )$ defined by $\bm{m}_i(D) = \int _{\mathcal{G}_i^0} P_i(E_i(1_D))\, d\mu _i$, which projects via $\varphi _i$ to $\bm{n}_i$. Let $\bm{m}$ be any weak${}^*$-cluster point of the net $(\bm{m}_i )_{i\in I}$. Then $\bm{m}$ is a balanced, diffuse, conjugation-invariant mean on $(\mathcal{G},\mu )$. By Theorem \[thm:equiv\], $(\mathcal{G},\mu )$ is inner amenable.
Central sequences in the full group {#subsec-schmidt}
-----------------------------------
Let $\mathcal{R}$ be an ergodic discrete p.m.p. equivalence relation on a standard probability space $(X,\mu )$. A sequence $(T_n)_{n\in {\mathbb{N}}}$ of elements of $[{\mathcal{R}}]$ is called *central* if $\mu (\{ \, x\in X \mid ST_nx= T_nSx \, \} )\rightarrow 1$ for any $S\in [\mathcal{R}]$. A central sequence $(T_n)_{n\in {\mathbb{N}}}$ in $[{\mathcal{R}}]$ is called *non-trivial* if $\liminf_n \mu (\{ \, x\in X \mid T_nx \neq x \, \} ) > 0$. We recall that a sequence $(A_n)_{n\in {\mathbb{N}}}$ of Borel subsets of $X$ is called *asymptotically invariant* for ${\mathcal{R}}$ if $\mu(TA_n\bigtriangleup A_n)\to 0$ for any $T\in [{\mathcal{R}}]$.
\[lem-ai\] For any asymptotically invariant sequence $(A_n)_n$ for ${\mathcal{R}}$ with $\mu(A_n)\to r$ for some number $r$, for any Borel subset $B\subset X$, we have $\mu(A_n\cap B)\to r\mu(B)$.
This is observed in the proof of [@js Lemma 2.3]. Since it will frequently be applied throughout the paper, we give a proof here. By compactness, it is enough to show that the convergence holds for a subsequence of $(A_n)_n$. Passing to a subsequence, we may assume that the sequence $(1_{A_n})_n$ in $L^\infty(X)$ converges to some $f\in L^\infty(X)$ in the weak${}^*$-topology. Since $(A_n)_n$ is asymptotically invariant for ${\mathcal{R}}$, the limit $f$ is invariant under ${\mathcal{R}}$. By ergodicity of ${\mathcal{R}}$, the function $f$ is constant, and since $\mu (A_n)\to r$, this constant must be $r$. For any Borel subset $B\subset X$, we have $\mu(A_n\cap B)=\int 1_{A_n}1_B\, d\mu \to \int r1_B \, d\mu =r\mu(B)$.
\[prop:iagroupoid\]$\ $ Let $\mathcal{R}$ be an ergodic discrete p.m.p. equivalence relation on a standard probability space $(X,\mu )$, and suppose that $[\mathcal{R}]$ admits a non-trivial central sequence. Then $\mathcal{R}$ is inner amenable.
Let $(T_n)_{n\in {\mathbb{N}}}$ be a non-trivial central sequence in $[{\mathcal{R}}]$. For each $n$, let $A_n {\coloneqq}\{ \, x\in X \mid T_nx \neq x \, \}$. After passing to a subsequence, we may assume that $\mu (A_n )$ converges to some $r>0$. Since $(T_n)_{n\in {\mathbb{N}}}$ is central, the sequence $(A_n)_{n\in {\mathbb{N}}}$ is asymptotically invariant. By Lemma \[lem-ai\], $\mu (A_n \cap A)\to r\mu (A)$ for any Borel subset $A\subset X$. Let $\omega$ be a non-principal ultrafilter on ${\mathbb{N}}$ and define a mean $\bm{m}$ on $(\mathcal{R}, \mu)$ by $$\bm{m}(D) = \lim _{n\to \omega} \frac{\mu ( \{ \, x\in A_n \mid (T_nx, x)\in D \, \} )}{\mu (A_n)}$$ for a Borel subset $D\subset \mathcal{R}$. Given a Borel subset $A\subset X$, since $\mu (T_nA\bigtriangleup A)\to 0$, we have $$\bm{m}(\mathcal{R}_A) = \lim _{n\to\omega}\mu (A_n)^{-1} \mu (T_n^{-1}A\cap A \cap A_n )= r^{-1}\mu (A)r = \mu (A).$$ This shows that $\bm{m}$ is balanced. For $S\in [\mathcal{R}]$ and a Borel subset $D\subset \mathcal{R}$, we have $$\begin{aligned}
|\bm{m}(D^S) - \bm{m}(D)|
&\leq \lim _{n\to \omega} \mu (A_n)^{-1} (\mu ( SA_n\bigtriangleup A_n ) + \mu (\{ \, x\in X \mid ST_nx\neq T_nSx \, \} ) ) = 0\end{aligned}$$ since $(T_n)_{n\in {\mathbb{N}}}$ is central in $[\mathcal{R}]$ and $(A_n)_{n\in {\mathbb{N}}}$ is asymptotically invariant with $\lim _n \mu (A_n)=r>0$. Thus $\bm{m}$ is conjugation-invariant. By definition, we have $\bm{m} ( \{ \, (x,x) \mid x\in X \, \} ) =0$ and hence $\bm{m}$ is diffuse by Lemma \[lem:diffuse\]. By Theorem \[thm:equiv\], ${\mathcal{R}}$ is inner amenable.
Schmidt raises the following problem, which remains open.
\[question:Schmidt\] Does every countable inner amenable group $G$ admit a free ergodic p.m.p. action $G{\curvearrowright}(X,\mu )$ such that the full group $[\mathcal{R}(G{\curvearrowright}(X,\mu ))]$ admits a non-trivial central sequence?
Let us say that an ergodic discrete p.m.p. equivalence relation ${\mathcal{R}}$ is *Schmidt* if $[{\mathcal{R}}]$ admits a non-trivial central sequence. We also say that a free ergodic p.m.p. action $G{\curvearrowright}(X,\mu )$ of a countable group is *Schmidt* if the orbit equivalence relation $\mathcal{R}(G{\curvearrowright}(X,\mu ))$ is Schmidt, and that a countable group has the *Schmidt property* if it admits a free ergodic p.m.p. action which is Schmidt. Question \[question:Schmidt\] turns out to have an affirmative answer when $G$ is linear ([@td Theorem 15]). In general though, there is much more evidence for an affirmative answer to the following question:
\[qu:oia\] Does every countable inner amenable group admit a free ergodic p.m.p. action whose orbit equivalence relation is inner amenable?
We call a countable group *orbitally inner amenable* if it admits an action as in Question \[qu:oia\]. Observe that by Proposition \[prop:iagroupoid\], any group with the Schmidt property is orbitally inner amenable, and by Proposition \[prop:Ginn\], any orbitally inner amenable group is inner amenable. In Corollary \[cor:cpctext\], we will show that any residually finite, inner amenable group is orbitally inner amenable. See Propositions \[prop:resfinex\] and \[prop:wreath\] for other classes of orbitally inner amenable groups.
Property Gamma
--------------
Let $M$ be a $\mathrm{II}_1$-factor with the tracial state $\tau$. Let $L^2(M)$ be the Hilbert space obtained as the completion of $M$ with respect to the norm $\Vert x\Vert_2=\tau(x^*x)^{1/2}$. We say that $M$ has *property Gamma* if there exists a sequence $(u_n)_n$ of unitaries of $M$ such that $\tau(u_n)=0$ and $\Vert [x, u_n]\Vert_2\to 0$ for any $x\in M$. For an ICC countable group $G$, if the group factor $LG$ has property Gamma, then $G$ is inner amenable ([@effros]), but the converse is not true ([@vaes]).
Choda [@choda] shows that for a free ergodic p.m.p. action $G{\curvearrowright}(X, \mu)$ of a countable group $G$, if the associated factor $G\ltimes L^\infty(X)$ has property Gamma and the action $G{\curvearrowright}(X, \mu)$ is strongly ergodic, then $G$ is inner amenable. Under the same assumption, we obtain the stronger conclusion that the groupoid $G\ltimes (X, \mu)$ is inner amenable. Recall that a free ergodic p.m.p. action $G{\curvearrowright}(X, \mu)$ is called *strongly ergodic* if every asymptotically invariant sequence $(A_n)_n$ for the action (i.e., sequence with $\mu(gA_n\bigtriangleup A_n)\to 0$ for all $g\in G$) satisfies $\mu(A_n)(1-\mu(A_n))\to 0$. We note that strong ergodicity is an invariant under orbit equivalence.
Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on a standard probability space $(X, \mu)$. Suppose that the factor $M$ associated to ${\mathcal{R}}$ has property Gamma, and let $(u_n)$ be a sequence of unitaries of $M$ such that $\tau(u_n)=0$ and $\Vert [x, u_n]\Vert_2\to 0$ for any $x\in M$. Choose a family $(\phi_k)_{k\in {\mathbb{N}}}$ of local sections of ${\mathcal{R}}$ such that $${\mathcal{R}}=\bigsqcup_{k\in {\mathbb{N}}} \{ \, (\phi_k(x), x)\mid x\in \mathrm{dom}(\phi_k)\, \}$$ and $\phi_1=\mathrm{id}$ with $\mathrm{dom}(\phi_1)=X$. Expand $u_n=\sum_ku_{\phi_k}f_k^n$ with $f_k^n\in L^\infty(X)$ supported on $\mathrm{dom}(\phi_k)$, where for a local section $\phi$ of ${\mathcal{R}}$, we let $u_{\phi}$ denote the associated partial isometry of $M$. Define $\xi_n\in L^1({\mathcal{R}}, \mu^1)$ by $$\xi_n(\phi_k(x), x)=|f_k^n(x)|^2$$ for $k\in {\mathbb{N}}$ and $x\in \mathrm{dom}(\phi_k)$. Then
1. each $\xi_n$ is a non-negative unit vector of $L^1({\mathcal{R}}, \mu^1)$, we have $\Vert \xi_n^\phi -\xi_n\Vert_1\to 0$ for every $\phi \in [{\mathcal{R}}]$, and we have $\sum_{y\in [x]_{\mathcal{R}}}\xi_n(y, x)=1=\sum_{y\in [x]_{\mathcal{R}}}\xi_n(x, y)$ for $\mu$-almost every $x\in X$, where $[x]_{\mathcal{R}}$ is the equivalence class of $x$ in ${\mathcal{R}}$.
2. If ${\mathcal{R}}$ is strongly ergodic, then $(\xi_n)_n$ is an inner amenability sequence for ${\mathcal{R}}$.
Since $u_n$ is a unitary, we have $\sum_k\Vert f_k^n\Vert_2^2=1$ and hence $\xi_n$ is a non-negative unit vector of $L^1({\mathcal{R}}, \mu^1)$. Pick $\phi \in [{\mathcal{R}}]$, and we show $\Vert \xi_n^\phi -\xi_n\Vert_1\to 0$. For $k, l\in {\mathbb{N}}$, set $$D_k^l=\{ \, x\in X\mid \phi(x)\in \mathrm{dom}(\phi_k),\, x\in \mathrm{dom}(\phi_l)\ \text{and}\ \phi^{-1}\phi_k\phi(x)=\phi_l(x)\, \}.$$ Then $D_1^1=X$ and we have the Borel partitions, $\phi^{-1}(\mathrm{dom}(\phi_k))=\bigsqcup_l D_k^l$ and $\mathrm{dom}(\phi_l)=\bigsqcup_kD_k^l$. We also have $$u_\phi^*u_nu_\phi =\sum_ku_{\phi^{-1}\phi_k\phi}(\phi^{-1}\cdot f_k^n)=\sum_k\sum_l u_{\phi_l}1_{D_k^l}(\phi^{-1}\cdot f_k^n),$$ where we set $\phi^{-1}\cdot f_k^n=f_k^n\circ \phi$, and thus $$\begin{aligned}
\Vert u_\phi^*u_nu_\phi-u_n\Vert_2^2&=\sum_l \biggl\Vert \sum_k 1_{D_k^l}(\phi^{-1}\cdot f_k^n)-f_l^n\biggr\Vert_2^2=\sum_{k, l}\int_{D_k^l}|f_k^n\circ \phi -f_l^n|^2\, d\mu.\end{aligned}$$ For any $x\in D_k^l$, we have $$\xi_n^\phi(\phi_l(x), x)=\xi_n(\phi \phi_l(x), \phi(x))=\xi_n(\phi_k\phi(x), \phi(x))=|f_k^n(\phi(x))|^2.$$ We therefore have $$\begin{aligned}
\Vert \xi_n^\phi-\xi_n\Vert_1&=\sum_{k, l}\int_{D_k^l}|\xi_n^\phi(\phi_l(x), x)-\xi_n(\phi_l(x), x)|\, d\mu(x)=\sum_{k, l}\int_{D_k^l}\bigl| |f_k^n\circ \phi|^2-|f_l^n|^2 \bigr|\, d\mu\\
&\leq \Biggl( \sum_{k, l}\int_{D_k^l}\bigl| |f_k^n\circ \phi|-|f_l^n|\bigr|^2\, d\mu \Biggr)^{1/2}\Biggl( \sum_{k, l}\int_{D_k^l}(|f_k^n\circ \phi|+|f_l^n|)^2\, d\mu \Biggr)^{1/2}\\
&\leq \Vert u_\phi^*u_nu_\phi -u_n\Vert_2 \Biggl( \Biggl( \sum_{k, l}\int_{D_k^l}|f_k^n\circ \phi|^2\, d\mu \Biggr)^{1/2} +\Biggl( \sum_{k, l}\int_{D_k^l}|f_l^n|^2\, d\mu \Biggr)^{1/2}\Biggr)\\
&= 2\Vert u_\phi^*u_nu_\phi -u_n\Vert_2 \to 0\end{aligned}$$ as $n\to \infty$.
We show the last equation in assertion (i). For a Borel subset $A\subset X$, let $\Delta A{\coloneqq}\{ \, (x, x)\mid x\in A\, \}$ be the diagonal set. Then its indicator function $1_{\Delta A}$ is a vector in $L^2({\mathcal{R}}, \mu^1)$. Since $u_n$ is a unitary, we have $\Vert u_n1_{\Delta A}\Vert_2^2=\Vert 1_{\Delta A}\Vert_2^2=\mu(A)$. By definition of the operator $u_\phi$ on $L^2({\mathcal{R}}, \mu^1)$ for a local section $\phi$ of ${\mathcal{R}}$, we have $\Vert u_n1_{\Delta A}\Vert_2^2=\int_A \sum_{y\in [x]_{\mathcal{R}}}\xi_n(y, x)\, d\mu(x)$. This is equal to $\mu(A)$ for any Borel subset $A\subset X$, and therefore we obtain $\sum_{y\in [x]_{\mathcal{R}}}\xi_n(y, x)=1$ for $\mu$-almost every $x\in X$. Similarly the other equation follows if $u_n$ is replaced by $u_n^*$. Assertion (i) was proved.
Suppose now that ${\mathcal{R}}$ is strongly ergodic. The space $(X,\mu )$ is atomless (since $M$ has property Gamma), so strong ergodicity implies that ${\mathcal{R}}$ is not amenable, and hence $(\xi_n)_n$ is balanced by Lemma \[lem:bal\]. Suppose toward a contradiction that $(\xi_n)$ is not asymptotically diffuse, i.e., for some Borel subset $D\subset {\mathcal{R}}$ with $\mu^1(D)<\infty$, the norm $\Vert 1_D\xi_n\Vert_1$ does not converge to $0$. Then by Lemma \[lem:diffuse\], after passing to a subsequence of $(\xi_n)$, the $\xi_n$-measure of the diagonal in ${\mathcal{R}}$ is uniformly positive, and hence $\Vert f_1^n\Vert_2$ is uniformly positive. It follows from $\tau(u_n)=0$ that $f_1^n$ belongs to $L^2_0(X)$, the orthogonal complement of the constants in $L^2(X)$, and further belongs to $L^\infty(X)$. By assertion (i), $(f_1^n)_n$ is asymptotically invariant for ${\mathcal{R}}$ and therefore ${\mathcal{R}}$ is not strongly ergodic, thanks to [@connes-oc I.1], a contradiction.
Thus $(\xi_n)_n$ is an inner amenability sequence for ${\mathcal{R}}$. Assertion (ii) was proved.
\[cor-gamma\] Let ${\mathcal{R}}$ be a strongly ergodic, discrete p.m.p. equivalence relation. If the von Neumann algebra associated to ${\mathcal{R}}$ has property Gamma, then ${\mathcal{R}}$ is inner amenable.
The converse of Corollary \[cor-gamma\] is not true. A counterexample is obtained via the Vaes group ([@vaes]), which is defined as follows. Let $(p_n)_{n=0}^\infty$ be a sequence of mutually distinct prime numbers. We set $$H_n=({\mathbb{Z}}/p_n{\mathbb{Z}})^3,\quad K=\bigoplus_{n=0}^\infty H_n \quad \text{and}\quad \Lambda =\mathit{SL}_3({\mathbb{Z}}).$$ Let $\Lambda$ act on $H_n$ by automorphisms and act on $K$ diagonally. For a non-negative integer $N$, we define the subgroup $K_N=\bigoplus_{n=N}^\infty H_n$ of $K$. We set $G_0=\Lambda \ltimes K$ and inductively define the amalgamated free product $$G_{N+1}=G_N\ast_{K_N}(K_N\times {\mathbb{Z}}).$$ Let $G_N$ include in $G_{N+1}$ as the first factor subgroup, and let $G$ be the union $\bigcup_NG_N$. The group $G$ is called the Vaes group, and remarkably it is ICC and inner amenable, and the group factor $LG$ does not have property Gamma ([@vaes]).
Let $Z=\prod_{n=0}^\infty H_n$ be the compact group, equip $Z$ with the normalized Haar measure, and regard $K$ as a subgroup of $Z$ naturally. Let $K$ act on $Z$ by translation and then co-induce the action $G{\curvearrowright}X=\prod_{G/K}Z$. After choosing a section $s\colon G/K\to G$, this action of $G$ is defined by $(gf)(b)=k^{-1}f(g^{-1}b)$ for $f\in X$, $g\in G$ and $b\in G/K$, where the element $k\in K$ is determined by $s(g^{-1}b)k=g^{-1}s(b)$. We have the probability measure $\mu$ on $X$ that is the product of the Haar measure on $Z$, and let ${\mathcal{R}}$ be the orbit equivalence relation associated with the action $G{\curvearrowright}(X, \mu)$.
We show that ${\mathcal{R}}$ is inner amenable. Define a non-negative unit vector $\xi_n\in L^1({\mathcal{R}}, \mu^1)$ by $\xi_n(gx, x)=1_{H_n}(g)/|H_n|$ for $g\in G$ and $x\in X$. We claim that $(\xi_n)_n$ is an inner amenability sequence for ${\mathcal{R}}$. For any $g\in G$, since $g$ normalizes $H_n$, the equation $\xi_n^g=\xi_n$ holds. By Remark \[rem:suffice\], it suffices to show $\sup_{h\in H_n}\mu(hA\bigtriangleup A)\to 0$ for any Borel subset $A\subset X$. For $h\in H_n$, $f\in X$ and $b\in G/K$, if $n$ is chosen to be large enough, depending only on $b$ and being independent of $f$, then $hb=b$ and the action of $h$ is given by $(hf)(b)=(s(b)^{-1}hs(b))f(b)$. The element $s(b)^{-1}hs(b)$ belongs to $H_n$ and does not change the coordinates $H_k$ in $Z=\prod_{k=0}^\infty H_k$ if $k<n$. It turns out that $\sup_{h\in H_n}\mu(hA\bigtriangleup A)\to 0$ if $A$ is a cyrindrical subset of $X=\prod_{G/K}Z=\prod_{G/K}\prod_{k=0}^\infty H_k$, and the claim follows.
We next show that the von Neumann algebra $M$ associated to ${\mathcal{R}}$ does not have property Gamma. Suppose toward a contradiction that $M$ has property Gamma, and let $(u_n)_n$ be a sequence of unitaries of $M$ such that $\tau(u_n)=0$ and $\Vert [x, u_n]\Vert_2\to 0$ for any $x\in M$. The action $G{\curvearrowright}(X, \mu)$ has stable spectral gap because its restriction to $\Lambda =\mathit{SL}_3({\mathbb{Z}})$ is mixing. Therefore if $P\colon L^2(M)\to \ell^2(G)\otimes {\mathbb{C}}1$ denotes the orthogonal projection, where $L^2(M)$ is naturally identified with $\ell^2(G)\otimes L^2(X)$, then $\Vert P(u_n)-u_n\Vert_2\to 0$ and $\Vert P(u_n)\Vert_2\to 1$. Since $P$ is $G$-equivariant, where $G$ acts on $M$ by conjugation, the sequence $(P(u_n))_n$ asymptotically commutes with any element of $G$. The restriction of $P$ to $M$ is the conditional expectation onto the factor $LG$, and hence the operator norm of $P(u_n)$ is at most $1$ and approaches to $1$ because $\Vert P(u_n)\Vert_2\to 1$. We also have $\tau(P(u_n))=\tau(u_n)=0$, and it follows from [@connes-ap Corollary 3.8] that $LG$ has property Gamma. This contradicts the result of Vaes [@vaes].
Compact extensions and inner amenability {#sec-cpt-ia}
========================================
As observed by Giordano-de la Harpe [@gdlh], if a countable group $G$ is inner amenable, then any finite index subgroup $H$ of $G$ is inner amenable as well. We can rephrase their argument as follows: let $\bm{m}$ be a conjugation-invariant, diffuse mean on $G$. We define a mean $\check{\bm{m}}$ on $G$ by $\check{\bm{m}}(D)=\bm{m}(D^{-1})$ for a subset $D\subset G$. Let $\check{\bm{m}} \ast \bm{m}$ be the convolution defined by $(\check{\bm{m}}\ast \bm{m})(D)=\int_G \bm{m}(g^{-1}D)\, d\check{\bm{m}}(g)$ for a subset $D\subset G$, which is a conjugation-invariant, diffuse mean on $G$. Since $H$ is of finite index in $G$, we have $\bm{m}(g_0H)>0$ for some $g_0\in G$. Then $$(\check{\bm{m}} \ast \bm{m})(H)=\int_G\bm{m}(gH)\, d\bm{m}(g)\geq \int_{g_0H}\bm{m}(gH)\, d\bm{m}(g)=\bm{m}(g_0H)^2>0.$$ Thus the normalization of the restriction $(\check{\bm{m}} \ast \bm{m})|_H$ is a conjugation-invariant, diffuse mean on $H$, and $H$ is inner amenable.
In this section, we generalize a version of this convolution argument to show that inner amenability is preserved under compact extension of ergodic p.m.p. groupoids.
Let $({\mathcal{G}}, \mu)$ be a discrete p.m.p. groupoid. Let $\eta ,\xi \in L^1(\mathcal{G},\mu ^1 )$. The *convolution* of $\eta$ and $\xi$, denoted $\eta \ast \xi$, is defined by $$(\eta \ast \xi )(\gamma ) =\sum _{\delta \in {\mathcal{G}}^{s(\gamma )}}\eta (\gamma \delta ) \xi (\delta ^{-1}) = \sum _{\delta \in {\mathcal{G}}^{r(\gamma )}} \eta (\delta )\xi (\delta ^{-1} \gamma ) = \sum _{\substack{\delta _1, \delta _0 \in {\mathcal{G}}\\ \delta _1\delta _0 = \gamma}} \eta (\delta _1)\xi (\delta _0) .$$ For $\xi \in L^1(\mathcal{G},\mu ^1 )$, we define $\check{\xi}\in L^1(\mathcal{G},\mu ^1 )$ by $\check{\xi}(\gamma ) = \xi (\gamma ^{-1})$. We then have $\Vert \check{\xi}\Vert_1=\Vert \xi \Vert_1$ and $(\eta \ast \xi)^\vee =\check{\xi} \ast \check{\eta}$.
\[lem-conv\] Let $\eta, \xi \in L^1({\mathcal{G}}, \mu^1)$. Then
1. if $\eta$ is non-negative and there is some $c>0$ such that $\sum_{\gamma \in {\mathcal{G}}_x}\eta(\gamma)\leq c$ and $\sum_{\gamma \in {\mathcal{G}}^x}\eta(\gamma)\leq c$ for almost every $x\in {\mathcal{G}}^0$, then $\Vert \eta \ast \xi \Vert_1=\Vert \xi \ast \eta \Vert_1\leq c\Vert \eta \Vert_1$.
2. For any $\phi \in [{\mathcal{G}}]$, we have $(\eta \ast \xi)^\phi =\eta^\phi \ast \xi^\phi$.
Assertion (i) follows from $$\begin{aligned}
\Vert \eta \ast \xi \Vert_1&\leq \int_X\sum_{\gamma \in {\mathcal{G}}_x}\sum_{\delta \in {\mathcal{G}}^x}\eta(\gamma \delta )|\xi(\delta^{-1})|\, d\mu(x)=\int_X\sum_{\delta \in {\mathcal{G}}^x}\sum_{\gamma \in {\mathcal{G}}_{s(\delta)}}\eta(\gamma)|\xi(\delta^{-1})|\, d\mu(x)\\
&\leq c\int_X\sum_{\delta \in {\mathcal{G}}}|\xi(\delta^{-1})|\, d\mu(x)=c\Vert \xi \Vert_1\end{aligned}$$ and $\Vert \xi \ast \eta \Vert_1=\Vert (\xi \ast \eta)^\vee \Vert_1=\Vert \check{\eta} \ast \check{\xi} \Vert_1\leq c\Vert \check{\xi}\Vert_1$. Putting $\psi =\phi^{-1}$, we obtain assertion (ii) from: $$\begin{aligned}
(\eta \ast \xi)^\phi(\gamma)&=(\eta \ast \xi)(\gamma^\psi)=\sum_{\delta \in {\mathcal{G}}^{\phi^0(s(\gamma))}}\eta(\gamma^\psi \delta)\xi(\delta^{-1})=\sum_{\delta \in ({\mathcal{G}}^{\phi^0(s(\gamma))})^\phi}\eta((\gamma \delta)^\psi)\xi((\delta^{-1})^\psi)\\
&=\sum_{\delta \in {\mathcal{G}}^{s(\gamma)}}\eta^\phi(\gamma \delta)\xi^\phi(\delta^{-1})=(\eta^\phi \ast \xi^\phi)(\gamma),\end{aligned}$$ where we use $(\gamma \delta)^\psi =\gamma^\psi \delta^\psi$ and $(\delta^{-1})^\psi =(\delta^\psi)^{-1}$ in the third equation.
\[lem:conv\] Let $(\eta _n)_{n\in {\mathbb{N}}}$ and $(\xi _n)_{n\in {\mathbb{N}}}$ be inner amenability sequences for $(\mathcal{G},\mu )$. Then there exists a increasing sequence $m_1<m_2<\cdots$ of positive integers such that $(\eta _n \ast \check{\xi} _{m_n})_n$ is also an inner amenability sequence for $(\mathcal{G},\mu )$.
We first show that any sequence of the form $(\eta _n \ast \check{\xi} _{m_n})_n$ satisfies conditions (i), (ii) and (iv) of Definition \[def:groupoid\]. Condition (iv) follows from direct computation. Condition (ii) follows from $$\begin{aligned}
\Vert (\eta_n\ast \check{\xi}_{m_n})^\phi -\eta_n\ast \check{\xi}_{m_n}\Vert_1\leq \Vert \eta_n^\phi -\eta_n\Vert_1+\Vert \check{\xi}_{m_n}^\phi -\check{\xi}_{m_n}\Vert_1,\end{aligned}$$ where Lemma \[lem-conv\] is applied. To check condition (i), we set $\eta_A{\coloneqq}1_{{\mathcal{G}}_A}\eta$ for $\eta \in L^1({\mathcal{G}}, \mu^1)$ and a Borel subset $A\subset {\mathcal{G}}^0$. For ${\varepsilon}>0$ and a Borel subset $A\subset {\mathcal{G}}^0$, let $E_{{\varepsilon}, A}$ be the set of non-negative unit vectors $\eta$ in $L^1({\mathcal{G}}, \mu^1)$ such that $\sum_{\gamma \in {\mathcal{G}}_x}\eta(\gamma)=1=\sum_{\gamma \in {\mathcal{G}}^x}\eta(\gamma)$ for almost every $x\in {\mathcal{G}}^0$, $|\Vert \eta_A \Vert_1-\mu(A)|<{\varepsilon}$, and $|\Vert \eta_{X\setminus A} \Vert_1-\mu(X\setminus A)|<{\varepsilon}$.
Let $\eta, \xi \in E_{{\varepsilon}, A}$. We show that $|\Vert 1_{{\mathcal{G}}_A}(\eta \ast \xi)\Vert_1-\mu(A)|<5{\varepsilon}+3{\varepsilon}^{1/2}$. This is enough to imply condition (i) for any sequence of the form $(\eta _n \ast \check{\xi} _{m_n})_n$. We have $\Vert \eta -\eta_A-\eta_{X\setminus A}\Vert_1<2{\varepsilon}$ and the similar inequality for $\xi$. Therefore $$\eta \ast \xi \approx_{4{\varepsilon}} (\eta_A+\eta_{X\setminus A})\ast (\xi_A+\xi_{X\setminus A})=\eta_A\ast \xi_A+\eta_{X\setminus A}\ast \xi_{X\setminus A},$$ where $f\approx_c g$ means $\Vert f-g\Vert_1<c$ for $f, g\in L^1({\mathcal{G}}, \mu^1)$. Then $1_{{\mathcal{G}}_A}(\eta \ast \xi)\approx_{4{\varepsilon}}\eta_A\ast \xi_A$. We also have $\Vert \eta_A\Vert_1=\int_A\sum_{\gamma \in {\mathcal{G}}_x\cap {\mathcal{G}}_A}\eta(\gamma)\, d\mu(x)$, and since this is more than $\mu(A)-{\varepsilon}$ and the integrand in the right hand side is non-negative and at most $1$ almost everywhere, there exists a Borel subset $B\subset A$ such that $\mu(A\setminus B)<{\varepsilon}^{1/2}$ and $\sum_{\gamma \in {\mathcal{G}}_x\cap {\mathcal{G}}_A}\eta(\gamma)>1-{\varepsilon}^{1/2}$ for any $x\in B$. Therefore $$\begin{aligned}
&\Vert \eta_A\ast \xi_A\Vert_1=\int_A\sum_{\gamma \in {\mathcal{G}}_x}\sum_{\delta \in {\mathcal{G}}^x\cap {\mathcal{G}}_A}\eta_A(\gamma \delta)\xi(\delta^{-1})\, d\mu(x)\\
&\approx_{{\varepsilon}^{1/2}}\int_B\sum_{\gamma \in {\mathcal{G}}_x}\sum_{\delta \in {\mathcal{G}}^x\cap {\mathcal{G}}_A}\eta_A(\gamma \delta)\xi(\delta^{-1})\, d\mu(x)\approx_{{\varepsilon}^{1/2}}\int_B\sum_{\delta \in {\mathcal{G}}^x\cap {\mathcal{G}}_A}\xi(\delta^{-1})\, d\mu(x)\\
&\approx_{{\varepsilon}^{1/2}}\int_A \sum_{\delta \in {\mathcal{G}}^x\cap {\mathcal{G}}_A}\xi(\delta^{-1})\, d\mu(x)=\Vert \xi_A\Vert_1\approx_{\varepsilon}\mu(A),\end{aligned}$$ where $a\approx_cb$ means $|a-b|<c$ for real numbers $a$, $b$. Thus $\Vert 1_{{\mathcal{G}}_A}(\eta \ast \xi)\Vert_1\approx_{5{\varepsilon}+3{\varepsilon}^{1/2}}\mu(A)$.
For condition (iii), we will need to choose $m_n$ more carefully. Let $D_0\subset D_1\subset \cdots$ be a sequence of bounded Borel subsets of $\mathcal{G}$ with $\mathcal{G}=\bigcup _n D_n$. For each $n$, let $F_n \colon \mathcal{G}\rightarrow {\mathbb{R}}$ be defined by $F_n (\delta ) = \sum _{\gamma \in D_n \cap \, {\mathcal{G}}_{r(\delta )}}\eta _n (\gamma \delta )$. Then $\| 1_{D_n} (\eta _n \ast \check{\xi} _m) \| _1 = \| F_n \xi _m \| _1$ and condition (iv) of $(\eta _n)$ implies that $\| F_n \| _1 = \mu ^1 (D_n) < \infty$. Therefore, condition (iii) of $(\xi _n)$ implies that for any large enough $m$, we have $\| 1_{D_n} (\eta _n \ast \check{\xi} _m) \| _1 = \| F_n \xi _m \| _1 < 1/n$. Thus, by choosing a sufficiently fast growing sequence of positive integers, $m_1<m_2<\cdots$, we can ensure that for any Borel subset $D\subset \mathcal{G}$ with $\mu ^1 (D)<\infty$, we have $\| 1_{D}(\eta _n \ast \check{\xi} _{m_n})\| _1 \rightarrow 0$.
\[lem-pigeon\] Let $(X, \mu)$ be a standard probability space, and let $(C_n)_n$ be a sequence of Borel subsets of $X$ having uniformly positive measure. Then after passing to a subsequence, there exists an $r>0$ such that for any $n,m\in {\mathbb{N}}$, we have $\mu(C_n \cap C_m)>r$.
By assumption, there exists a $c>0$ such that $\mu(C_n)>c$ for any $n$. After moving to a subsequence, we may assume that $1_{C_n}$ converges in the weak${}^*$-topology to some $f\in L^{\infty}(X,\mu )$. We have $f\geq 0$, and $\int f\, d\mu\geq c >0$, so we may find some $r>0$ with $\int f^2\, d\mu > r >0$. By the convergence $\int 1_{C_n}f\, d\mu \to \int f^2\, d\mu >r$, we may assume after moving to a subsequence that $\int 1_{C_n}f \, d\mu > r$ for all $n$.
It follows that for any $n$, as $m\to \infty$ the convergence $\int 1_{C_n}1_{C_m}\, d\mu \to \int 1_{C_n}f\, d\mu >r$ holds, and hence for all large enough $m$ we have $\mu (C_n\cap C_m ) >r$. We may therefore inductively find $n_0<n_1<n_2<\cdots$ with $\mu (C_{n_i}\cap C_{n_j} ) >r$ for all $i<j$.
\[thm:cpctseq\] Let $(\mathcal{G}, \mu )$ be an ergodic discrete p.m.p. groupoid which is inner amenable. Let $(Z,\zeta )$ be a standard probability space, let $\alpha \colon \mathcal{G}\rightarrow \mathrm{Aut}(Z,\zeta )$ be a cocycle, and assume that the image $\alpha (\mathcal{G})$ is contained in a compact subgroup $K$ of $\mathrm{Aut}(Z,\zeta )$. Then for any decreasing sequence $V_1\supset V_2\supset \cdots$ of neighborhoods of the identity in $K$, there exists an inner amenability sequence $(\xi _n)$ for $(\mathcal{G},\mu )$ such that for any $n$, the function $\xi _n$ vanishes outside $\alpha ^{-1}(V_n)$.
Fix a bi-invariant metric on $K$. For each ${\varepsilon}>0$, let $V_{{\varepsilon}}$ denote the open ${\varepsilon}$-ball about the identity element in $K$. By Lemma \[lem:weakstar\], it is enough to show that for any ${\varepsilon}>0$, we can find a mean $\bm{m}$ on $(\mathcal{G},\mu )$ as in condition (5) of Theorem \[thm:equiv\] such that $\bm{m}(\alpha ^{-1}(V_{{\varepsilon}}))=1$.
Toward this goal, fix ${\varepsilon}>0$ and find $0<{\varepsilon}_2< {\varepsilon}_1<{\varepsilon}$ such that $V_{{\varepsilon}_2}^2\subset V_{{\varepsilon}_1}$. Since $K$ is compact, we may find $c_1,\dots , c_N\in K$ such that $K=\bigcup _{i=1}^NV_{{\varepsilon}_2}c_i$. Let $(\eta _n )_{n\in {\mathbb{N}}}$ be an inner amenability sequence for $(\mathcal{G},\mu )$. For each non-negative unit vector $\eta \in L^1(\mathcal{G},\mu ^1 )$, let $\nu _{\eta}$ be the probability measure on $K$ given by $\nu _{\eta}(B) = \int _{\alpha ^{-1}(B)}\eta \, d\mu ^1$ for a Borel subset $B\subset K$. After passing to a subsequence of $(\eta _n )$, we may assume that for some $i\in \{ 1,\ldots, N\}$ and $r>0$, we have $\inf _{n\in {\mathbb{N}}} \nu _{\eta _n} (V_{{\varepsilon}_2}c_i )>r$. We put $c= c_i$ and define a function $f_n$ on ${\mathcal{G}}^0$ by $$f_n(x) = \sum _{\delta \in \alpha ^{-1}(V_{{\varepsilon}_2}c)_x}\eta _n (\delta ).$$ Then $\int _{\mathcal{G}^0} f_n \, d\mu = \nu _{\eta _n} (V_{{\varepsilon}_2}c)>r$ and $0\leq f_n \leq 1$, and hence the sets $C_n$ for $n\in {\mathbb{N}}$ defined by $C_n = \{ \, x\in \mathcal{G}^0 \mid f_n(x) > r^2 \, \}$ have uniformly positive measure. By Lemma \[lem-pigeon\], after passing to a subsequence of $(\eta _n )$, we may assume without loss of generality that the sets $C_n\cap C_m$ for $n,m\in {\mathbb{N}}$ have uniformly positive measure, and $\inf _{n,m\in {\mathbb{N}}} \int _{\mathcal{G}^0} f_nf_m\, d\mu > r_0$ for some $r_0 >0$. Therefore $$\begin{aligned}
&\nu _{\eta _n\ast \check{\eta} _m}(V_{{\varepsilon}_1}) = \int _{\mathcal{G}^0} \sum _{\substack{\delta _1, \delta _0 \in {\mathcal{G}}_x \\ \alpha (\delta _1\delta _0^{-1})\in V_{{\varepsilon}_1} }}\eta _n (\delta _1)\eta _m (\delta _0) \, d\mu(x) \\
&\geq \int _{\mathcal{G}^0} \sum _{\delta _1\in \alpha ^{-1}(V_{{\varepsilon}_2}c)_x}\eta _n (\delta _1) \sum _{\delta _0 \in \alpha ^{-1}(V_{{\varepsilon}_2}c)_x} \eta _m (\delta _0 ) \, d\mu(x) = \int _{\mathcal{G}^0} f_n(x)f_m(x)\, d\mu(x) > r_0 .\end{aligned}$$ Thus, by Lemma \[lem:conv\], by choosing an appropriate subsequence $m_1<m_2<\cdots$, we obtain an inner amenability sequence $\xi_n {\coloneqq}\eta _n \ast \check{\eta}_{m_n}$ satisfying $\inf _{n\in {\mathbb{N}}}\nu _{\xi_n}(V_{{\varepsilon}_1}) > r_0>0$.
We may therefore assume without loss of generality that our original sequence $(\eta _n)$ already satisfies $\inf _{n\in {\mathbb{N}}}\nu _{\eta _n}(V_{{\varepsilon}_1})>r_0>0$. Since $V_{{\varepsilon}_1}$ is symmetric, after replacing $\eta _n$ by $(\eta _n + \check{\eta}_n )/2$, we may assume that each $\eta _n$ is symmetric as well. We may also assume, after passing to a subsequence, that the sequence $(\nu _{\eta _n} )$ converges to some probability measure $\nu _{\infty}$ in the compact space of Borel probability measures on $K$. Then for any number ${\varepsilon}_0$ with ${\varepsilon}_1 <{\varepsilon}_0 < {\varepsilon}$, we have $\nu _{\infty}(V_{{\varepsilon}_0}) \geq \limsup _n \nu _{\eta _n} (\overline{V_{{\varepsilon}_1}})\geq r_0 > 0$. Since, as ${\varepsilon}_0$ varies, the boundaries $\partial V_{{\varepsilon}_0}$ are pairwise disjoint, we may find some ${\varepsilon}_0$ with ${\varepsilon}_1<{\varepsilon}_0 <{\varepsilon}$ such that $\nu _{\infty}(\partial V_{{\varepsilon}_0}) = 0$. Let $U{\coloneqq}V_{{\varepsilon}_0}$ so that $U\subset V_{{\varepsilon}}$, $\nu _{\infty}(U)>0$ and $\nu _{\infty}(\partial U ) =0$.
Since $\nu _{\infty}(\partial U )=0$, it follows that $\nu _{\eta _n}(U)\rightarrow \nu _{\infty}(U)$. Let $\omega$ be a non-principal ultrafilter on ${\mathbb{N}}$ and let $\bm{m}_1$ be the weak${}^*$-limit $\bm{m}_1 =\lim _{n\rightarrow \omega} \eta _n$ in $L^{\infty}(\mathcal{G} ,\mu ^1 )^*$, so that $\bm{m}_1$ is a mean on $(\mathcal{G},\mu )$ as in condition (5) of Theorem \[thm:equiv\] with $\bm{m}_1(\alpha ^{-1}(K))=1$ and $\bm{m}_1(\alpha ^{-1}(U))=\lim _{n\rightarrow \omega} \nu _{\eta _n}(U) =\nu _{\infty}(U)> 0$.
\[claim:alphaU\] Let $\phi \in [\mathcal{G}]$. Then $\bm{m}_1(\alpha ^{-1}(U)\setminus \alpha ^{-1}(U)^{\phi} ) = 0$.
Let $W$ be an open neighborhood of the identity in $K$. Let $W_1$ be a symmetric open neighborhood of the identity in $K$ with $W_1^2\subset W$, and let $b_1,\dots , b_M\in K$ be such that $K=\bigcup _{i=1}^M W_1b_i$. For each $i\in \{ 1,\ldots, M\}$, we set $Y_i = \{ \, x\in \mathcal{G}^0 \mid \alpha (\phi x) \in W_1b_i \, \}$ so that $\mathcal{G}^0=\bigcup _{i=1}^M Y_i$. Then $\bm{m}_1(\bigcup_{i=1}^M \mathcal{G}_{Y_i}) =1$. If $\gamma \in (\bigcup_{i=1}^M \mathcal{G}_{Y_i}) \cap (\alpha ^{-1}(U)\setminus \alpha ^{-1}(U)^{\phi})$, then there is some $i\in \{ 1,\ldots, M\}$ such that $\alpha (\phi r(\gamma ) ), \alpha (\phi s(\gamma )) \in W_1b_i$, and hence $$\begin{aligned}
\alpha (\gamma ^{(\phi ^{-1})})&= \alpha (\phi r(\gamma ) )\alpha (\gamma ) \alpha (\phi s(\gamma ))^{-1} \\
&= \alpha (\phi r(\gamma )) \alpha (\phi s(\gamma ))^{-1} \alpha (\phi s(\gamma ))\alpha (\gamma ) \alpha (\phi s(\gamma ))^{-1} \in W_1b_ib_i^{-1}W_1^{-1}U \subset WU ,\end{aligned}$$ and so $\gamma ^{(\phi ^{-1})} \in \alpha ^{-1}(WU \setminus U)$. This shows that $$\Biggl(\bigcup _{i=1}^M \mathcal{G}_{Y_i}\Biggr) \cap (\alpha ^{-1}(U)\setminus \alpha ^{-1}(U)^{\phi})\subset \alpha ^{-1}(WU\setminus U) ^{\phi },$$ and therefore $$\bm{m}_1 (\alpha ^{-1}(U)\setminus \alpha ^{-1}(U)^{\phi} ) \leq \bm{m}_1 (\alpha ^{-1}(\overline{WU}\setminus U)) = \lim _{n\rightarrow \omega}\nu _{\eta _n}(\overline{WU}\setminus U)\leq \nu _{\infty}(\overline{WU}\setminus U) .$$ Since $\nu _{\infty}(\partial U ) = 0$, we can make $\nu _{\infty}(\overline{WU}\setminus U )$ as small as we like by choosing an appropriate neighborhood $W$ of the identity in $K$. This proves the claim.
Consider now the (countably additive, finite Borel) measure $\mu _U$ on $\mathcal{G}^0$ given by $\mu _U(A)= \bm{m}_1(\mathcal{G}_A \cap \alpha ^{-1}(U))$. The measure $\mu _U$ is absolutely continuous with respect to $\mu$, and for any $\phi \in [\mathcal{G}]$ and any Borel subset $A\subset \mathcal{G}^0$, by Claim \[claim:alphaU\], we have $$\begin{aligned}
\mu _U ((\phi ^0)^{-1}A)&= \bm{m}_1(\mathcal{G}_{(\phi ^0)^{-1}A}\cap \alpha ^{-1}(U))= \bm{m}_1((\mathcal{G}_A \cap \alpha ^{-1}(U))^{\phi}) = \bm{m}_1(\mathcal{G}_A \cap \alpha ^{-1}(U))\\
& = \mu _U (A).\end{aligned}$$ Therefore, the Radon-Nikodym derivative $d\mu _U/d\mu$ must be constant by ergodicity of $(\mathcal{G},\mu )$, and hence $\bm{m}_1(\mathcal{G}_A\cap \alpha ^{-1}(U))= \mu (A)\bm{m}_1(\alpha ^{-1}(U))$ for any Borel subset $A\subset \mathcal{G}^0$. Define the mean $\bm{m}$ on $(\mathcal{G},\mu )$ by $\bm{m}(D) = \bm{m}_1(D\cap \alpha ^{-1}(U))/\bm{m}_1(\alpha ^{-1}(U))$. It is now clear that $\bm{m}$ is a mean on $(\mathcal{G},\mu )$ satisfying condition (5) of Theorem \[thm:equiv\], and moreover $\bm{m}(\alpha ^{-1}(V_{{\varepsilon}}))\geq \bm{m}(\alpha ^{-1}(U))=1$.
\[cor-finite-index-cpt-ext\] Let $(\mathcal{G}, \mu )$ be an ergodic discrete p.m.p. groupoid, and let ${\mathcal{H}}$ be a finite-index Borel subgroupoid of ${\mathcal{G}}$. If ${\mathcal{G}}$ is inner amenable, then ${\mathcal{H}}$ is also inner amenable.
Let $N$ be the index of ${\mathcal{H}}$ in ${\mathcal{G}}$ and let $\sigma \colon {\mathcal{G}}\to \Sigma$ be the index cocycle, where $\Sigma$ is the symmetric group of $N$ letters. This cocycle is constructed in [@fsz Section 1] when ${\mathcal{G}}$ is principal, and it is defined for general ${\mathcal{G}}$ as well. Then ${\mathcal{H}}=\ker \sigma$ and the corollary follows from Theorem \[thm:cpctseq\].
\[cor:cpctext\] Let $(\mathcal{G}, \mu )$ be an ergodic discrete p.m.p. groupoid which is inner amenable. Let $(Z,\zeta )$ be a standard probability space, let $\alpha \colon \mathcal{G}\rightarrow \mathrm{Aut}(Z,\zeta )$ be a cocycle, and assume that the image $\alpha (\mathcal{G})$ is contained in a compact subgroup of $\mathrm{Aut}(Z,\zeta )$. Then the extension groupoid $(\mathcal{G} ,\mu )\ltimes _{\alpha}(Z,\zeta )$ is inner amenable.
In particular, if $G$ is a countable inner amenable group which is a subgroup of a compact group $K$, $L$ is a closed subgroup of $K$, and we let $G$ act on $(K/L, \mu)$ by left multiplication, where $\mu$ is the $K$-invariant probability measure on $K/L$, then the associated translation groupoid $G\ltimes (K/L, \mu)$ is inner amenable.
An inner amenability sequence $(\xi _n)$ for $(\mathcal{G},\mu )$ as in Theorem \[thm:cpctseq\] lifts to an inner amenability sequence $(\eta _n )$ for $(\mathcal{G} ,\mu )\ltimes _{\alpha}(Z,\zeta )$ just by taking $\eta _n (\gamma , z ){\coloneqq}\xi _n (\gamma )$.
By Corollary \[cor:cpctext\], if $G$ is countable, residually finite, inner amenable group, then the translation groupoid associated with any profinite free action of $G$ is inner amenable and therefore $G$ is orbitally inner amenable.
\[cor:distal\] Let $G$ be a countable inner amenable group. Let $G\curvearrowright (X,\mu )$ be an ergodic p.m.p. action of $G$ which is measure distal. Then the translation groupoid $G \ltimes (X,\mu )$ is inner amenable.
We proceed by transfinite induction on the length of the distal tower associated to the action $\Gamma \curvearrowright (X,\mu )$. At successor stages we apply Corollary \[cor:cpctext\], and at limit stages the translation groupoid $G\ltimes (X,\mu )$ is the inverse limit of the translation groupoids of the tower, so Proposition \[prop:invlim\] applies.
Compact extensions and central sequences {#sec-cpt-cs}
========================================
Following the previous section, we investigate central sequences in the full group and stability sequences under compact extensions. Main results of this section are found in Subsections \[subsec-main\] and \[subsec-main2\]. Throughout this section, $(X, \mu)$ is a standard probability space and ${\mathcal{B}}$ denotes the algebra of Borel subsets of $X$, where two Borel subsets are identified when their symmetric difference is $\mu$-null.
Stability sequences
-------------------
Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$. Let $(T_n, A_n)_n$ be a sequence of a pair of $T_n\in [{\mathcal{R}}]$ and $A_n\in {\mathcal{B}}$. We call $(T_n, A_n)_n$ a *stability sequence* for ${\mathcal{R}}$ if the following three conditions hold:
1. For every $B\in {\mathcal{B}}$, we have $\mu(T_nB\bigtriangleup B)\to 0$.
2. For every $g\in [{\mathcal{R}}]$, we have $\mu (\{ gT_n\neq T_ng \})\to 0$.
3. The sequence $(A_n)_n$ is asymptotically invariant for ${\mathcal{R}}$, $T_n^2=\text{id}$ and $T_nA_n\bigtriangleup A_n=X$ for any $n$, and $T_nT_m=T_mT_n$ and $T_nA_m=A_m$ for any distinct $n$ and $m$.
We call ${\mathcal{R}}$ *stable* if ${\mathcal{R}}$ is isomorphic to the direct product ${\mathcal{R}}\times {\mathcal{R}}_0$, where ${\mathcal{R}}_0$ is the ergodic p.m.p. aperiodic hyperfinite equivalence relation. By [@js Theorem 3.4], ${\mathcal{R}}$ is stable if and only if it admits a stability sequence. The theorem also says that for ${\mathcal{R}}$ to be stable, it is enough to find a sequence $(T_n, A_n)_n$ satisfying conditions (1) and (2) and the following condition weaker than condition (3):
1. The sequence $(A_n)_n$ is asymptotically invariant for ${\mathcal{R}}$, and $\mu(T_nA_n\setminus A_n)$ is uniformly positive.
We call a sequence $(T_n, A_n)_n$ satisfying conditions (1), (2) and (4) a *pre-stability sequence* for ${\mathcal{R}}$.
Preliminary lemmas
------------------
Throughout this subsection, let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$.
\[lem-ai-cover\] If $(D_n)_n$ is an asymptotically invariant sequence for ${\mathcal{R}}$ such that $\mu(D_n)$ is uniformly positive, then $\bigcup_nD_n=X$.
Passing to a subsequence, we may assume that $\mu(D_n)\to d$ for some $d>0$ and also $\mu(D_n)> d/2$ for all $n$. By Lemma \[lem-ai\], passing to a subsequence further, we may assume that for any $n$, the two values $\mu(\bigcap_{k=1}^nD_k^c)$ and $\mu(D_1^c)\cdots \mu(D_n^c)$ are close. The latter value is less than $(1-d/2)^n$ and hence $\mu(\bigcap_nD_n^c)=0$.
\[lem-comp\] For any $A, A', B, B'\in {\mathcal{B}}$, the following inequality holds: $$\mu(A\setminus B)\leq 2\mu(A\bigtriangleup A')+\mu(B\bigtriangleup B')+\mu(A'\setminus B').$$
Let $\Vert \cdot \Vert_1$ denote the norm on $L^1(X, \mu)$. The inequality follows from: $$\begin{aligned}
\mu(A\setminus B)&=\Vert 1_A-1_A1_B\Vert_1 \\
&\leq \Vert 1_A -1_{A'}\Vert_1 +\Vert 1_{A'}-1_{A'}1_{B'}\Vert_1 +\Vert (1_{A'}-1_A)1_{B'}\Vert_1 +\Vert 1_A (1_{B'}-1_B)\Vert_1\\
&\leq \mu(A\bigtriangleup A')+\mu(A'\setminus B')+\mu(A\bigtriangleup A')+\mu(B\bigtriangleup B').\qedhere\end{aligned}$$
\[lem-patch\] Let $(T_n, D_n)_n$ be a sequence of a pair of $T_n\in [{\mathcal{R}}]$ and $D_n\in {\mathcal{B}}$ such that
- $(T_n)_n$ is a central sequence in $[{\mathcal{R}}]$ and $T_n^2=\mathrm{id}$ for every $n$,
- $(D_n)_n$ is an asymptotically invariant sequence for ${\mathcal{R}}$ such that $\mu(D_n)$ is uniformly positive, and
- $T_nD_n=D_n$ for every $n$.
Then for any ${\varepsilon}>0$ and any finite subset $Q\subset [{\mathcal{R}}]$, we can find an $S\in [{\mathcal{R}}]$ such that
1. the map $S$ is obtained by patching together pieces of the restrictions $T_n|_{D_n}$, $n\in {\mathbb{N}}$, along with a piece of the identity map such that the latter piece is small. More precisely: for all $x\in X$ outside a subset of measure less than ${\varepsilon}$, there exists an index $n$ with $x\in D_n$ and $Sx=T_nx$, and for every point $y$ in the excluded subset we have $Sy=y$;
2. $\mu(\{ gS\neq Sg \})<{\varepsilon}$ for every $g\in Q$.
If $(A_n)_n$ is further an asymptotically invariant sequence for ${\mathcal{R}}$ with $\mu(T_nA_n\bigtriangleup A_n)=1$ for every $n$, then after replacing $A_n$ by $X\setminus A_n$ if necessary, we may assume that $\mu(A_n\cap D_n)\geq \mu(D_n)/2$ for every $n$, and we can find a $Z\in {\mathcal{B}}$ such that
1. $\mu(SZ\setminus Z)>1/10$, and
2. $\mu(gZ\bigtriangleup Z)<{\varepsilon}$ for every $g\in Q$.
As a consequence of the former assertion of the lemma, varying ${\varepsilon}$ and $Q$, we obtain a central sequence $(S_m)_m$ in $[{\mathcal{R}}]$ such that for each $m$, the map $S_m$ is obtained by patching together pieces of the maps $T_n|_{D_n}$, $n\in {\mathbb{N}}$, along with a piece of the identity map such that the latter piece is small. The central sequence $(S_m)_m$ is hence non-trivial as long as $T_nx\neq x$ for every $n$ and every $x\in X$. Under the assumption in the latter assertion of the lemma, we further obtain a pre-stability sequence $(S_m, Z_m)_m$ for ${\mathcal{R}}$ with $\mu(S_mZ_m\setminus Z_m)>1/10$ for all $m$. In the proof of Theorem \[thm-main\] below, it will be significant that this lower bound “1/10" can be taken independently of the uniform lower bound of $\mu(D_n)$.
Passing to a subsequence of $(T_n, D_n)_n$, we may assume that the following three conditions hold:
1. $\sum_n\mu(gD_n\bigtriangleup D_n)<{\varepsilon}$ for every $g\in Q$.
2. $\sum_n\mu(\{ gT_n\neq T_ng \} )<{\varepsilon}$ for every $g\in Q$.
3. $\sum_n \sum_{k<n}\mu(T_nD_k\bigtriangleup D_k)<{\varepsilon}$.
These conditions follow from the sequence $(D_n)_n$ being asymptotically invariant for ${\mathcal{R}}$, and the sequence $(T_n)_n$ being central in $[{\mathcal{R}}]$. Under the assumption in the latter assertion of the lemma, we may further assume that $\mu(D_n)\to d_1$ and $\mu(A_n\cap D_n)\to d_2$ for some $d_1>0$ and $d_2>0$ and that the following three conditions hold:
1. For each $n$, setting $C_n=\bigcup_{k<n}D_k$, we have $\mu(D_n)\mu(C_n^c)\geq (2/3)\mu(D_n\setminus C_n)$.
2. For each $n$, setting $E_n=(A_n\cap D_n)\setminus C_n$, we have $\mu(E_n)\geq (2/3)\mu(A_n\cap D_n)\mu(C_n^c)$.
3. $\sum_n \mu(gE_n\bigtriangleup E_n)<{\varepsilon}$ for every $g\in Q$.
Indeed, condition (4) is obtained as follows: If $D_1,\ldots, D_{n-1}$ are chosen, then by Lemma \[lem-ai\], we have $\mu(D_m\cap C_n^c)\to d_1\mu(C_n^c)$ as $m\to \infty$. For all large $m$, $\mu(D_m)$ and $d_1$ are close and hence $\mu(D_m\cap C_n^c)=\mu(D_m\setminus C_n)$ and $\mu(D_m)\mu(C_n^c)$ are close. Condition (4) therefore holds after relabeling $D_m$ for a sufficiently large $m$ as $D_n$. Condition (5) is similarly obtained from Lemma \[lem-ai\] and the convergence $\mu(A_n\cap D_n)\to d_2$. Condition (6) is obtained from asymptotic invariance of the sequences $(D_n)_n$ and $(A_n)_n$.
We set $$Y_1=D_1,\ Y_n=D_n\setminus (C_n\cup T_n^{-1}C_n)\ \ \text{for}\ \ n\geq 2,\ \ \text{and} \ \ Y=\bigcup_{n=1}^\infty Y_n.$$ Note that the last union is a disjoint union. For each $n$, we have $T_nY_n=Y_n$ because $T_n$ is an involution and $T_nD_n=D_n$. The inclusion $Y_n\subset D_n\setminus C_n$ holds. By condition (3), we have $\sum_n \mu(T_nC_n\bigtriangleup C_n)<{\varepsilon}$. Therefore $\sum_n \mu((D_n\setminus C_n)\setminus Y_n)<{\varepsilon}$, and $\mu(\bigcup_n(D_n\setminus C_n)\setminus Y)<{\varepsilon}$. By the definition of $C_n$, the equation $\bigcup_n (D_n\setminus C_n)=\bigcup_nD_n$ holds, and this is equal to $X$ by Lemma \[lem-ai-cover\]. It follows that
1. $\mu(X\setminus Y)<{\varepsilon}$.
We pick $g\in Q$ and estimate $\sum_n\mu(gY_n\bigtriangleup Y_n)$. Pick $y\in Y_n\setminus gY_n$. Since $g^{-1}y\not \in Y_n$, either $g^{-1}y\not \in D_n$ or $g^{-1}y\in C_n\cup T_n^{-1}C_n$. In the former case, we have $y\in D_n\setminus gD_n$. In the latter case, we have $$\begin{aligned}
y & \in (g(C_n\cup T_n^{-1}C_n)\setminus (C_n\cup T_n^{-1}C_n))\cap Y_n\\
& \subset \Biggl( \, \bigcup_{k<n}(gD_k\setminus D_k)\cap Y_n\Biggr)\cup \Biggl(\, \bigcup_{k<n}(gT_n^{-1}D_k\setminus T_n^{-1}D_k)\cap Y_n\Biggr). \end{aligned}$$ Let $N$ be a positive integer. We have $$\begin{aligned}
&\sum_{n=1}^N\mu(Y_n\setminus gY_n)\leq \sum_{n=1}^N\mu(D_n\setminus gD_n)\\
&\hspace{5em}+\sum_{n=1}^N\sum_{k=1}^{n-1}\mu((gD_k\setminus D_k)\cap Y_n)+\sum_{n=1}^N\sum_{k=1}^{n-1}\mu((gT_n^{-1}D_k\setminus T_n^{-1}D_k)\cap Y_n).\end{aligned}$$ By condition (1), in the right hand side, the first term is less than ${\varepsilon}$, and the second term is at most $$\sum_{n=1}^N\sum_{k=1}^{N-1}\mu((gD_k\setminus D_k)\cap Y_n)\leq \sum_{k=1}^{N-1}\mu(gD_k\setminus D_k)<{\varepsilon}.$$ The third term is $$\begin{aligned}
&\sum_{n=1}^N\sum_{k=1}^{n-1}\mu((gT_n^{-1}D_k\cap Y_n)\setminus (T_n^{-1}D_k\cap Y_n))\\
& \leq \sum_{n=1}^N\sum_{k=1}^{n-1}(\mu((gD_k\cap Y_n)\setminus (D_k\cap Y_n))+3\mu(T_n^{-1}D_k\bigtriangleup D_k))\\
& <\sum_{n=1}^N\sum_{k=1}^{n-1} \mu((gD_k\setminus D_k)\cap Y_n)+3{\varepsilon}<4{\varepsilon},\end{aligned}$$ where Lemma \[lem-comp\] and condition (3) are applied in the first inequality and the second one, respectively. It follows that $\sum_{n=1}^N\mu(Y_n\setminus gY_n)<6{\varepsilon}$ and therefore
1. $\sum_n \mu(gY_n\bigtriangleup Y_n)< 12{\varepsilon}$ for any $g\in Q$.
We define $S\in [{\mathcal{R}}]$ as follows: For each $n$, we set $S=T_n$ on $Y_n$ and define $S$ on $X\setminus Y$ to be the identity map. This map $S$ is an automorphism of $X$ because $T_n$ preserves $Y_n$, and by condition (7), it satisfies condition (i). To check condition (ii), pick $g\in Q$. We have the inclusions $$\begin{aligned}
\{ gS\neq Sg\} &\subset \bigcup_n \, (\{ gS\neq Sg\} \cap Y_n)\cup (X\setminus Y),\\
\{ gS\neq Sg\} \cap Y_n &\subset (\{ gS\neq Sg\} \cap (Y_n\cap g^{-1}Y_n))\cup (Y_n\setminus g^{-1}Y_n)\ \text{and}\\
\{ gS\neq Sg\} \cap (Y_n\cap g^{-1}Y_n)&\subset \{ gT_n\neq T_ng\} \cap (Y_n\cap g^{-1}Y_n).\end{aligned}$$ It follows from conditions (2), (8) and (7) that $$\mu(\{ gS\neq Sg\})\leq \sum_n(\mu(\{ gT_n\neq T_ng\})+\mu(Y_n\setminus g^{-1}Y_n))+\mu(X\setminus Y)<{\varepsilon}+12{\varepsilon}+{\varepsilon}.$$ Condition (ii) follows after scaling ${\varepsilon}$ appropriately. The proof of the former assertion of the lemma is completed.
We prove the latter assertion of the lemma. By conditions (4) and (5) and the inequality $\mu(A_n\cap D_n)\geq \mu(D_n)/2$, for any $n$, we have
1. $\mu(E_n)\geq (2/9)\mu(D_n\setminus C_n)$.
We set $$Z=\bigcup_n \, (A_n\cap Y_n).$$ We check condition (iii). Since the set $E_n$ is defined as $E_n=(A_n\cap D_n)\setminus C_n$, we have the inclusions $A_n\cap Y_n\subset E_n$ and $E_n\setminus (A_n\cap Y_n)\subset T_n^{-1}C_n\setminus C_n$. It follows that $$0\leq \sum_n\mu(E_n)-\mu(Z)= \sum_n\mu(E_n\setminus (A_n\cap Y_n))\leq \sum_n\sum_{k<n}\mu(T_n^{-1}D_k\setminus D_k)<{\varepsilon},$$ where the last inequality follows from condition (3). On the other hand, by condition (9), we have $\sum_n\mu(E_n)\geq (2/9)\sum_n\mu(D_n\setminus C_n)$, and by condition (7), $$\sum_n\mu(D_n\setminus C_n)\geq \sum_n\mu(Y_n)=\mu(Y)>1-{\varepsilon}.$$ It follows that $\sum_n\mu(E_n)>2/9-2{\varepsilon}/9$ and $\mu(Z)>\sum_n\mu(E_n)-{\varepsilon}>2/9-11{\varepsilon}/9$. The sets $SZ$ and $Z$ are disjoint because $T_nA_n$ and $A_n$ are disjoint and $S$ is equal to $T_n$ on $Y_n$. We therefore have $\mu(SZ\setminus Z)=\mu(SZ)=\mu(Z)>1/10$, where the last inequality holds if ${\varepsilon}$ is taken to be small enough. Condition (iii) follows.
Finally we check condition (iv). Pick $g\in Q$. By Lemma \[lem-comp\], $$\mu(g(A_n\cap Y_n)\setminus (A_n\cap Y_n))\leq 3\mu(E_n\bigtriangleup (A_n\cap Y_n))+\mu(gE_n\setminus E_n).$$ Summing over $n$, we obtain $$\begin{aligned}
\mu(gZ\setminus Z)&\leq \sum_n\mu(g(A_n\cap Y_n)\setminus (A_n\cap Y_n))\leq \sum_n(3\mu(E_n\bigtriangleup (A_n\cap Y_n))+\mu(gE_n\setminus E_n))\\
&<3{\varepsilon}+{\varepsilon},\end{aligned}$$ where the last inequality follows from the inclusion $A_n\cap Y_n\subset E_n$, the inequality shown in the previous paragraph, and condition (6). Condition (iv) follows.
Without assuming that $T_n$ is an involution and that $D_n$ is invariant under $T_n$, we prove the following lemma in which conclusion (i) is slightly milder than that in Lemma \[lem-patch\]. This will be used in the proof of Lemma \[lem-c\] and Theorem \[thm-main-c\].
\[lem-patch-c\] Let $(T_n, D_n)_n$ be a sequence of a pair of $T_n\in [{\mathcal{R}}]$ and $D_n\in {\mathcal{B}}$ such that
- $(T_n)_n$ is a central sequence in $[{\mathcal{R}}]$,
- $(D_n)_n$ is an asymptotically invariant sequence for ${\mathcal{R}}$ such that $\mu(D_n)$ is uniformly positive, and
- $\mu(T_nD_n\bigtriangleup D_n)\to 0$ as $n\to \infty$.
Then for any ${\varepsilon}>0$ and any finite subset $Q\subset [{\mathcal{R}}]$, we can find an $S\in [{\mathcal{R}}]$ such that
1. a large piece of the map $S$ is obtained by patching together pieces of the restrictions $T_n|_{D_n}$, $n\in {\mathbb{N}}$. More precisely, for every $x\in X$ outside a subset of measure less than ${\varepsilon}$, there exists an index $n$ with $x\in D_n$ and $Sx=T_nx$, and
2. $\mu(\{ gS\neq Sg \})<{\varepsilon}$ for every $g\in Q$.
As in the proof of Lemma \[lem-patch\], after passing to a subsequence of $(T_n, D_n)_n$, we may assume that conditions (1)–(3) in that proof hold. By the third assumption in the present lemma, we may further assume that
1. $\sum_n\mu(T_nD_n\bigtriangleup D_n)<{\varepsilon}$.
We set $C_n=\bigcup_{k<n}D_k$, $Y_1=D_1$, $Y_n=D_n\setminus (C_n\cup T_n^{-1}C_n)$ for $n\geq 2$, and $Y=\bigcup_{n=1}^\infty Y_n$ in the same way. As in the proof of Lemma \[lem-patch\], condition (3) implies $\mu(X\setminus Y)<{\varepsilon}$, and conditions (1) and (3) imply $\sum_n(gY_n\bigtriangleup Y_n)< 12{\varepsilon}$ for any $g\in Q$. We set $$Y_1'=D_1\cap T_1^{-1}D_1,\ Y_n'=(D_n\cap T_n^{-1}D_n)\setminus (C_n\cup T_n^{-1}C_n)\ \text{for}\ n\geq 2,\ \text{and} \ Y'=\bigcup_{n=1}^\infty Y_n'.$$ For any $n$, the inclusions $Y_n'\subset Y_n$ and $Y_n\setminus Y_n'\subset D_n\setminus T_n^{-1}D_n$ hold, and hence $$\sum_n\mu(Y_n\setminus Y_n')\leq \sum_n\mu(D_n\setminus T_n^{-1}D_n)<{\varepsilon}/2$$ by condition (10). Hence $\mu(Y\setminus Y')<{\varepsilon}/2$ and $\mu(X\setminus Y')<2{\varepsilon}$. For any $g\in Q$, we have $$\sum_n\mu(gY_n'\bigtriangleup Y_n')<\sum_n(\mu(gY_n'\bigtriangleup gY_n)+\mu(gY_n\bigtriangleup Y_n)+\mu(Y_n\bigtriangleup Y_n'))<{\varepsilon}+12{\varepsilon}+{\varepsilon}.$$
We define $S\in [{\mathcal{R}}]$ as follows: We first define it on $Y'$ such that $S=T_n$ on $Y_n'$ for each $n$. This map $S\colon Y'\to X$ is injective. Indeed, if $k<n$ and $T_kx=T_ny$ with $x\in Y_k'$ and $y\in Y_n'$, then it follows from $T_kY_k'\subset D_k$ that $T_kx\in D_k$ and thus $y\in T_n^{-1}D_k\subset T_n^{-1}C_n$. This contradicts $y\in Y_n'$. It turns out that the measures of the sets $Y'$ and $SY'$ are equal. Pick a Borel isomorphism between $X\setminus Y'$ and $X\setminus SY'$ which is a local section of ${\mathcal{R}}$. We define the map $S$ on $X\setminus Y'$ to be that isomorphism. The obtained map $S\colon X\to X$ is then an automorphism of $X$ and belongs to $[{\mathcal{R}}]$. Condition (i) then follows. Along the proof in Lemma \[lem-patch\], condition (ii) also follows from the estimates for $Y_n'$ and $Y'$ obtained in the previous paragraph, after scaling ${\varepsilon}$ appropriately.
As a simple application of the last lemma, we obtain the following:
\[lem-c\] If ${\mathcal{R}}$ is Schmidt, then there exists a central sequence $(T_n)_n$ in $[{\mathcal{R}}]$ such that $T_nx\neq x$ for every $n$ and every $x\in X$.
Let $(T_n)_n$ be a non-trivial central sequence in $[{\mathcal{R}}]$. Set $D_n=\{ \, x\in X\mid T_nx\neq x\, \}$. The measure $\mu(D_n)$ is uniformly positive, and $T_nD_n=D_n$ for every $n$. We claim that $(D_n)_n$ is an asymptotically invariant sequence for ${\mathcal{R}}$. Indeed, for any $g\in [{\mathcal{R}}]$, if $n$ is large, then for every $x\in X$ outside a subset of small measure, we have $gT_nx=T_ngx$, and if furthermore $x \in X\setminus D_n$, then $gx=T_ngx$, that is, $gx\in X\setminus D_n$. The sequence $(X\setminus D_n)_n$ is therefore asymptotically invariant for ${\mathcal{R}}$ and the claim follows.
Pick ${\varepsilon}>0$ and a finite subset $Q\subset [{\mathcal{R}}]$. By Lemma \[lem-patch-c\], we can find an $S\in [{\mathcal{R}}]$ with $\mu(\{ gS\neq Sg\})<{\varepsilon}$ for any $g\in Q$ and $Sx\neq x$ for any $x\in Y$, where $Y$ is a Borel subset of $X$ with $\mu(X\setminus Y)<{\varepsilon}$. There exists an isomorphism from $X\setminus Y$ onto $S(X\setminus Y)$ which fixes no point and is a local section of ${\mathcal{R}}$. We define $R\in [{\mathcal{R}}]$ as the map equal to $S$ on $Y$ and equal to that isomorphism on $X\setminus Y$. It turns out that $\mu(\{ gR\neq Rg\})<3{\varepsilon}$ for any $g\in Q$ and $Rx\neq x$ for any $x\in X$.
The next lemma will be used in the proof of Theorem \[thm-main-c\].
\[lem-s\] Let $(T_n)_n$ be a sequence in $[{\mathcal{R}}]$ such that $\mu(T_nA\bigtriangleup A)\to 0$ for every Borel subset $A\subset X$. Then $\mu(\{ \, x\in X\mid T_nx=Sx\neq x\, \})\to 0$ for every $S\in [{\mathcal{R}}]$.
Suppose that the conclusion does not hold true. We set $Y=\{ \, x\in X\mid Sx\neq x\, \}$. Passing to a subsequence we may assume that $\mu(\{ \, x\in Y\mid T_nx=Sx \, \}) \rightarrow s$ for some $s>0$. Fix a non-principal ultrafilter $\omega$ on ${\mathbb{N}}$ and let $\bm{m}$ be the mean on ${\mathcal{R}}$ defined by $$\bm{m}(D)= \lim _{n\rightarrow \omega} \mu ( \{ x\in X \mid (T_n x, x) \in D \} ) .$$ The mean $\bm{m}$ is balanced since $\bm{m}({\mathcal{R}}_A) = \lim _{n\to\omega} \mu (T_n ^{-1}A \cap A ) = \mu (A)$. Therefore, by Lemma \[lem:diffuse\] we have $\bm{m}( \{ (Sx,x) \mid x\in Y \} ) =0$. Hence $$0 = \bm{m}( \{ (Sx,x) \mid x\in Y \} ) = \lim _{n\to\omega} \mu ( \{ x\in Y \mid T_nx = Sx \} ) = s,$$ a contradiction.
Stability under compact extensions {#subsec-main}
----------------------------------
The following is the main technical result of this subsection:
\[thm-main\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$ and let $\alpha \colon {\mathcal{R}}\to K$ be a cocycle into a compact group $K$. If ${\mathcal{R}}$ is stable, then for any decreasing sequence $V_1\supset V_2\supset \cdots$ of open neighborhoods of the identity in $K$, there exists a pre-stability sequence $(T_n, A_n)_n$ for ${\mathcal{R}}$ such that $\alpha(T_n x, x)\in V_n$ for every $n$ and almost every $x\in X$.
Throughout this proof, for $T\in [{\mathcal{R}}]$ and $x\in X$, we denote $\alpha(Tx, x)$ by $\alpha(T, x)$ for the ease of symbols. Fix a bi-invariant metric on $K$. For ${\varepsilon}>0$, let $V_{\varepsilon}$ denote the open ball of radius ${\varepsilon}$ about the identity in $K$. Pick ${\varepsilon}>0$. We will find a pre-stability sequence $(T_n, A_n)_n$ of ${\mathcal{R}}$ such that $\alpha(T_n, x)\in V$ for any $n$ and any $x\in X$.
Let $(T_n, A_n)_n$ be a stability sequence for ${\mathcal{R}}$. Find $0<{\varepsilon}_2<{\varepsilon}_1<{\varepsilon}$ such that $V_{{\varepsilon}_2}^2\subset V_{{\varepsilon}_1}$. Since $K$ is compact, there exist finitely many elements $c_1,\ldots, c_N\in K$ with $K=\bigcup_{i=1}^NV_{{\varepsilon}_2}c_i$. Passing to a subsequence of $(T_n, A_n)_n$, we can find an $i\in \{ 1,\ldots, N\}$ such that the set $$C_n=\{ \, x\in X\mid \alpha(T_n, x)\in V_{{\varepsilon}_2}c_i\, \}$$ has uniformly positive measure. Let $c$ denote this $c_i$. By Lemma \[lem-pigeon\], passing to a subsequence, we may assume that for any $n,m\in {\mathbb{N}}$, the intersection $C_n\cap C_m$ has uniformly positive measure. If $x\in C_n\cap C_m$, then $$\alpha(T_mT_n^{-1}, T_nx)=\alpha(T_m, x)\alpha(T_n, x)^{-1}\in V_{{\varepsilon}_2}c(V_{{\varepsilon}_2}c)^{-1}=V_{{\varepsilon}_2}^2\subset V_{{\varepsilon}_1}$$ and therefore $$T_n(C_n\cap C_m)\subset \{ \, x\in X\mid \alpha(T_mT_n^{-1}, x)\in V_{{\varepsilon}_1}\, \}.$$ It follows that the set in the right hand side has uniformly positive measure. Replacing the pair $(T_n, A_n)$ into the pair $(T_mT_n^{-1}, A_m)$ with sufficiently large $m>n$, we may assume that the stability sequence $(T_n, A_n)_n$ satisfies that the set $$E_n=\{ \, x\in X\mid \alpha(T_n, x)\in V_{{\varepsilon}_1}\, \}.$$ has uniformly positive measure. We should note that $T_mT_n^{-1}$ is involutive thanks to the condition, $T_nT_m=T_mT_n$ and $T_nA_m=A_m$ for any distinct $n$ and $m$, that is required for a stability sequence, while not being required for a pre-stability sequence.
Let $\nu_n$ denote the probability measure on $K$ that is the image of $\mu$ under the map sending each $x\in X$ to $\alpha(T_n, x)$. Let $\nu_\infty$ be a weak${}^*$-cluster point of $\nu_n$. We may assume that $\nu_n$ converges to $\nu_\infty$ in the weak${}^*$-topology. Since $\mu(E_n)$ is uniformly positive, we have $\nu_\infty(\overline{V_{{\varepsilon}_1}})>0$. For any ${\varepsilon}_0$ with ${\varepsilon}_1<{\varepsilon}_0<{\varepsilon}$, we have $\nu_\infty(V_{{\varepsilon}_0})>0$. Since, as ${\varepsilon}_0$ varies, the boundaries $\partial V_{{\varepsilon}_0}$ are mutually disjoint, we may find some ${\varepsilon}_0$ with ${\varepsilon}_1<{\varepsilon}_0<{\varepsilon}$ such that $\nu_\infty(\partial V_{{\varepsilon}_0})=0$. We set $U=V_{{\varepsilon}_0}$ and set $$D_n=\{ \, x\in X\mid \alpha(T_n, x)\in U\, \}.$$ Since $E_n\subset D_n$, the set $D_n$ has uniformly positive measure. The equation $T_nD_n=D_n$ follows from that $T_n$ is an involution and $U^{-1}=U$. Indeed, for any $x\in D_n$, $$\alpha(T_n, T_nx)=\alpha(T_n^{-1}, T_nx)=\alpha(T_n, x)^{-1}\in U^{-1}=U$$ and hence $T_nx\in D_n$. Using $\nu_\infty(\partial U)=0$, we show the following:
\[claim-d-ai\] The sequence $(D_n)_n$ is asymptotically invariant for ${\mathcal{R}}$.
Pick $g\in [{\mathcal{R}}]$ and ${\varepsilon}>0$. Since $\nu_\infty(\partial U)=0$, there exists an open ball $W$ in $K$ about the identity with $\nu_\infty(\overline{WU}\setminus U)<{\varepsilon}$. We claim that for any sufficiently large $n$, $$\mu(\{ \, x\in X\mid \alpha(g, T_nx)\alpha(g, x)^{-1}\not\in W\, \})<{\varepsilon}.$$ In fact, choosing an open ball $W_1$ in $K$ centered at the identity with $W_1^2\subset W$ and finitely many elements $b_1,\ldots, b_M\in K$ with $K=\bigcup_{i=1}^MW_1b_i$, we set $$Y_i=\{ \, x\in X\mid \alpha(g, x)\in W_1b_i\, \}$$ for $i\in \{ 1,\ldots, M\}$. The set $X$ is covered by the sets $Y_1,\ldots, Y_M$. If $n$ is sufficiently large, then for any $i$, we have $\mu(T_nY_i\bigtriangleup Y_i)<{\varepsilon}/M$ and for any $x\in T_n^{-1}Y_i\cap Y_i$, $$\alpha(g, T_nx)\alpha(g, x)^{-1}\in W_1b_i(W_1b_i)^{-1}=W_1^2\subset W.$$ The claim follows.
If $n$ is sufficiently large, then for any $x\in D_n$ outside a subset of small measure, we have $gT_nx=T_ngx$ and $\alpha(g, T_nx)\alpha(g, x)^{-1}\in W$, and therefore $$\begin{aligned}
\alpha(T_n, gx)&=\alpha(T_ng, x)\alpha(g, x)^{-1}=\alpha(gT_n, x)\alpha(g, x)^{-1}\\
&=\alpha(g, T_nx)\alpha(T_n, x)\alpha(g, x)^{-1}\\
&=\alpha(g, T_nx)\alpha(g, x)^{-1}\alpha(g, x)\alpha(T_n, x)\alpha(g, x)^{-1}\\
&\in W\alpha(g, x)U\alpha(g, x)^{-1}=WU.\end{aligned}$$ Since the inequality $\limsup_n \nu_n(\overline{WU}\setminus U)\leq \nu_\infty(\overline{WU}\setminus U)$ holds and the measure $\nu_\infty(\overline{WU}\setminus U)$ is small, for any $x\in D_n$ outside a subset of small measure, we have $\alpha(T_n, gx)\in U$, that is, $gx\in D_n$. It turns out that the measure of $D_n\setminus g^{-1}D_n$ is small if $n$ is sufficiently large.
By Lemma \[lem-patch\], there exists a pre-stability sequence $(S_m, Z_m)_m$ of ${\mathcal{R}}$ such that for any $m$, the map $S_m$ is obtained by patching together pieces of the restrictions $T_n|_{E_n}$, $n\in {\mathbb{N}}$, along with a piece of the identity map, and $\mu(S_mZ_m\setminus Z_m)>1/10$. The former condition implies that for any $x\in X$, the element $\alpha(S_m, x)$ belongs to $U$ and hence $V_{\varepsilon}$. This is the claim in the beginning of this subsection.
To obtain the conclusion of Theorem \[thm-main\], we vary ${\varepsilon}$. Let $V_1\supset V_2\supset \cdots$ be a decreasing sequence of open balls in $K$ about the identity. Applying the claim shown above to each $V_n$, we obtain a pre-stability sequence $(R_n, Y_n)_n$ of ${\mathcal{R}}$ such that $\alpha(R_n, x)\in V_n$ for any $n$ and any $x\in X$. We note that $\mu(R_nY_n\setminus Y_n)$ is uniformly positive because $\mu(S_mZ_m\setminus Z_m)$ is uniformly positive independently of $V_n$. This completes the proof of Theorem \[thm-main\].
We recall fundamental facts on compact extensions of equivalence relations. Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$. Let $\alpha \colon {\mathcal{R}}\to K$ be a cocycle into a compact group $K$ and $L$ a closed subgroup of $K$. We equip the space $X\times K/L$ with the product measure of $\mu$ and the $K$-invariant probability measure on $K/L$. We define an equivalence relation ${\mathcal{R}}_{\alpha, L}$ on $X\times K/L$ such that for any $(y, x)\in {\mathcal{R}}$ and any $k\in K$, the two points $(x, kL)$ and $(y, \alpha(y, x)kL)$ are equivalent. This ${\mathcal{R}}_{\alpha, L}$ is a p.m.p. discrete measured equivalence relation. We call the equivalence relation ${\mathcal{R}}_{\alpha, L}$ obtained through this procedure a *compact extension* of ${\mathcal{R}}$. When $L$ is trivial, the extension ${\mathcal{R}}_{\alpha, L}$ is simply denoted by ${\mathcal{R}}_\alpha$.
For any cocycle $\alpha \colon {\mathcal{R}}\to K$ into a compact group $K$, there exist a closed subgroup $K_0$ of $K$ and a cocycle $\alpha_0$ equivalent to $\alpha$ such that values of $\alpha_0$ are in $K_0$ and there is no cocycle equivalent to $\alpha_0$ with values in a proper closed subgroup of $K_0$. The subgroup $K_0$ is uniquely determined up to conjugacy in $K$, and it is called the Mackey range of the cocycle $\alpha$. The extension ${\mathcal{R}}_{\alpha_0}$ is then ergodic. We refer to [@z Corollary 3.8] for details.
\[cor-ext-s\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$. Let $\alpha \colon {\mathcal{R}}\to K$ be a cocycle into a compact group $K$ such that there is no cocycle equivalent to $\alpha$ with values in a proper closed subgroup of $K$. Let $L$ be a closed subgroup of $K$. If ${\mathcal{R}}$ is stable, then the extension ${\mathcal{R}}_{\alpha, L}$ is stable.
Pick a decreasing sequence $V_1\supset V_2\supset \cdots$ of open neighborhoods of the identity in $K$ such that their intersection consists of only the identity. By Theorem \[thm-main\], there exists a pre-stability sequence $(T_n, A_n)_n$ of ${\mathcal{R}}$ such that $\alpha(T_n, x)\in V_n$ for any $n$ and any $x\in X$. We note that to each element $T$ of $[{\mathcal{R}}]$, the element $\tilde{T}$ of $[{\mathcal{R}}_{\alpha, L}]$ is associated through the formula $\tilde{T}(x, kL)=(Tx, \alpha(T, x)kL)$ for $x\in X$ and $k\in K$. The sequence $(\tilde{T}_n, A_n\times K/L)_n$ is then a pre-stability sequence for ${\mathcal{R}}_{\alpha, L}$.
\[rem-erg-comp\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$ and let $\alpha \colon {\mathcal{R}}\to K$ be a cocycle into a compact group $K$. Even if $\alpha$ does not satisfy the assumption in Corollary \[cor-ext-s\], we can show that if ${\mathcal{R}}$ is stable, then almost every ergodic component of ${\mathcal{R}}_{\alpha, L}$ is stable. Indeed, take a closed subgroup $K_0$ and a cocycle $\beta$ equivalent to $\alpha$ such that there is no cocycle equivalent to $\beta$ with values in a proper closed subgroup of $K_0$. Since $\alpha$ and $\beta$ are equivalent, we have an isomorphism between ${\mathcal{R}}_{\alpha, L}$ and ${\mathcal{R}}_{\beta, L}$. Let $\beta_0\colon {\mathcal{R}}\to K_0$ be the cocycle obtained by simply replacing the range of the cocycle $\beta \colon {\mathcal{R}}\to K$ into $K_0$. Let $\theta \colon X\times K/L\to K_0\backslash K/L$ be the projection of the second coordinate. The map $\theta$ gives rise to the decomposition into ergodic components of ${\mathcal{R}}_{\beta, L}$. For almost every $c\in K$, we have an isomorphism between the ergodic component $({\mathcal{R}}_{\beta, L})|_{\theta^{-1}(c)}$ and the extension ${\mathcal{R}}_{\beta_0, K_0 \, \cap \, cLc^{-1}}$. If ${\mathcal{R}}$ is stable, then ${\mathcal{R}}_{\beta_0, K_0 \, \cap \, cLc^{-1}}$ is stable by Corollary \[cor-ext-s\], and therefore almost every ergodic component of ${\mathcal{R}}_{\alpha, L}$ is stable.
\[cor-td\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$. Let ${\mathcal{S}}$ be a finite index subrelation of ${\mathcal{R}}$. If ${\mathcal{R}}$ is stable, then each ergodic component of ${\mathcal{S}}$ is stable.
Let $N$ be the index of ${\mathcal{S}}$ in ${\mathcal{R}}$ and let $\sigma \colon {\mathcal{R}}\to \Sigma$ be the index cocycle, where $\Sigma$ is the symmetric group of $N$ letters ([@fsz Section 1]). Then ${\mathcal{S}}=\ker \sigma$, and the restriction of ${\mathcal{R}}_\sigma$ to $X\times \{ e\}$ is identified with ${\mathcal{S}}$. The corollary follows from Remark \[rem-erg-comp\]. We note that in general, for an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$ and a Borel subset $A\subset X$ of positive measure, the equivalence relation is stable if and only if its restriction to $A$ is stable.
We say that a free ergodic p.m.p. action of a countable group is *stable* if the associated orbit equivalence relation is stable, and that a countable group is *stable* if it admits a free ergodic p.m.p. action which is stable. Combining a result from [@kida-sce], we obtain the following:
\[cor-fi\] Let $G$ be a countable group. Then
1. for any finite index subgroup $H$ of $G$, $H$ is stable if and only if $G$ is stable.
2. For any finite normal subgroup $N$ of $G$, $G/N$ is stable if and only if $G$ is stable.
Let $H$ be a finite index subgroup of $G$. If $H$ has a free ergodic p.m.p. action $H{\curvearrowright}(X, \mu)$ which is stable, then the action of $G$ induced from it is stable. Conversely, if $G$ is stable, then $H$ is stable by Corollary \[cor-td\]. Assertion (i) follows.
To prove assertion (ii), let $N$ be a finite normal subgroup of $G$. If $G$ has a free ergodic p.m.p. action $G{\curvearrowright}(X, \mu)$ which is stable, then the action of the quotient, $G/N{\curvearrowright}X/N$, is also stable and hence $G/N$ is stable. Conversely, suppose that $G/N$ is stable. Let $H$ denote the centralizer of $N$ in $G$, which is of finite index in $G$ because $N$ is finite. Let $Z$ denote the center of $N$. The quotient $H/Z$ is then a finite index subgroup of $G/N$ and hence stable by assertion (i). Since $Z$ is central in $H$, by [@kida-sce Corollary 1.2], $H$ is stable. By assertion (i) again, $G$ is stable.
The Schmidt property under compact extensions {#subsec-main2}
---------------------------------------------
Following Theorem \[thm-main\], we obtain a similar result for non-trivial central sequences in the full group, in place of pre-stability sequences, while the conclusion of the following theorem is slightly weaker than that of Theorem \[thm-main\].
\[thm-main-c\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$ and let $\alpha \colon {\mathcal{R}}\to K$ be a cocycle into a compact group $K$. If ${\mathcal{R}}$ is Schmidt, then for any decreasing sequence $V_1\supset V_2\supset \cdots$ of open neighborhoods of the identity in $K$, there exists a non-trivial central sequence $(T_n)_n$ in $[{\mathcal{R}}]$ such that $\mu(\{ \, x\in X\mid \alpha(T_n, x)\not\in V_n\, \})\to 0$.
We basically follow the proof of Theorem \[thm-main\], while not a few modifications will be performed. Fix a bi-invariant metric on $K$. For ${\varepsilon}>0$, let $V_{\varepsilon}$ denote the open ball of radius ${\varepsilon}$ about the identity in $K$. Pick ${\varepsilon}>0$ and a finite subset $Q\subset [{\mathcal{R}}]$. To prove the theorem, it suffices to find an $S\in [{\mathcal{R}}]$ such that $\mu(\{ \, x\in X\mid Sx=x\, \})<{\varepsilon}$, $\mu(\{ gS\neq Sg\})<{\varepsilon}$ for any $g\in Q$, and $\mu(\{ \, x\in X\mid \alpha(S, x)\not\in V_{\varepsilon}\, \})<{\varepsilon}$.
Let $(T_n)_n$ be a non-trivial central sequence in $[{\mathcal{R}}]$. Find $0<{\varepsilon}_2<{\varepsilon}_1<{\varepsilon}$ such that $V_{{\varepsilon}_2}^2\subset V_{{\varepsilon}_1}$. Since $K$ is compact, there exist finitely many elements $c_1,\ldots, c_N\in K$ with $K=\bigcup_{i=1}^NV_{{\varepsilon}_2}c_i$. Passing to a subsequence of $(T_n)_n$, we can find an $i\in \{ 1,\ldots, N\}$ such that the set $$C_n=\{ \, x\in X\mid T_nx\neq x,\, \alpha(T_n, x)\in V_{{\varepsilon}_2}c_i\, \}$$ has uniformly positive measure. Let $c$ denote this $c_i$. By Lemma \[lem-pigeon\], passing to a subsequence, we may assume that there exists an $r>0$ such that for any $n, m\in {\mathbb{N}}$, we have $\mu(C_n\cap C_m)>r$. We have the inclusion $$T_n(C_n\cap C_m)\subset \{ \, x\in X\mid T_n^{-1}x\neq x,\, \alpha(T_mT_n^{-1}, x)\in V_{{\varepsilon}_1}\, \},$$ as shown in the proof of Theorem \[thm-main\], and hence the set in the right hand side has measure more than $r$. By Lemma \[lem-s\], if we fix $n$ and let $m$ be large, then the measure of the set $\{ \, x\in X\mid T_m^{-1}x=T_n^{-1}x\neq x\, \}$ converges to $0$. Therefore for any $n$, for any sufficiently large $m>n$, the set $$\{ \, x\in X\mid T_mT_n^{-1}x\neq x,\, \alpha(T_mT_n^{-1}, x)\in V_{{\varepsilon}_1}\, \}$$ has measure more than $r/2$. As a result, relabeling $T_mT_n^{-1}$ as $S_n$, we obtain a central sequence $(S_n)_n$ in $[{\mathcal{R}}]$ such that the set $$E_n=\{ \, x\in X\mid S_nx\neq x,\, \alpha(S_n, x)\in V_{{\varepsilon}_1}\, \}$$ has uniformly positive measure.
For each $n$, let $\mu_n$ be the restriction of $\mu$ to the set $\{ \, x\in X\mid S_nx\neq x\, \}$, whose total measure is uniformly positive. Let $\nu_n$ denote the measure on $K$ that is the image of $\mu_n$ under the map sending a point $x$ to $\alpha(S_n, x)$. Let $\nu_\infty$ be a weak${}^*$-cluster point of $\nu_n$. We may assume that $\nu_n$ converges to $\nu_\infty$ in the weak${}^*$-topology. Since $\mu(E_n)$ is uniformly positive, we have $\nu_\infty(\overline{V_{{\varepsilon}_1}})>0$. For any ${\varepsilon}_0$ with ${\varepsilon}_1<{\varepsilon}_0<{\varepsilon}$, we have $\nu_\infty(V_{{\varepsilon}_0})>0$. Since, as ${\varepsilon}_0$ varies, the boundaries $\partial V_{{\varepsilon}_0}$ are mutually disjoint, we may find some ${\varepsilon}_0$ with ${\varepsilon}_1<{\varepsilon}_0<{\varepsilon}$ such that $\nu_\infty(\partial V_{{\varepsilon}_0})=0$. We set $U=V_{{\varepsilon}_0}$ and set $$D_n=\{ \, x\in X\mid S_nx\neq x,\ \alpha(S_n, x)\in U\, \}.$$ Since $E_n\subset D_n$, the set $D_n$ has uniformly positive measure. Using that the sequence of the set $\{ \, x\in X\mid S_nx\neq x\, \}$ is asymptotically invariant for ${\mathcal{R}}$ and following the proof of Claim \[claim-d-ai\], we can show that the sequence $(D_n)_n$ is asymptotically invariant for ${\mathcal{R}}$. If $\mu(S_nD_n\bigtriangleup D_n)$ does not converge to $0$, then after passing to a subsequence of $(S_n, D_n)_n$, we obtain a pre-stability sequence for ${\mathcal{R}}$. The proof then reduces to Theorem \[thm-main\]. Otherwise, by Lemma \[lem-patch-c\], there exists an $S\in [{\mathcal{R}}]$ such that a large part of the map $S$ is obtained by patching together pieces of the restrictions $S_n|_{D_n}$, $n\in {\mathbb{N}}$, and $\mu(\{ gS\neq Sg\})<{\varepsilon}$ for any $g\in Q$. The former condition means that for any $x\in X$ outside a subset of measure less than ${\varepsilon}$, there is an $n$ such that $x\in D_n$ and $Sx=S_nx$. For any such $x\in X$, we have $Sx\neq x$ and $\alpha(S, x)\in U\subset V_{\varepsilon}$.
\[cor-ext-c\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$. Let $\alpha \colon {\mathcal{R}}\to K$ be a cocycle into a compact group $K$ such that there is no cocycle equivalent to $\alpha$ with values in a proper closed subgroup of $K$. Let $L$ be a closed subgroup of $K$. If ${\mathcal{R}}$ is Schmidt, then the extension ${\mathcal{R}}_{\alpha, L}$ is Schmidt.
Pick a decreasing sequence $V_1\supset V_2\supset \cdots$ of open neighborhoods of the identity in $K$ such that their intersection consists of only the identity. By Theorem \[thm-main-c\], there exists a non-trivial central sequence $(T_n)_n$ in $[{\mathcal{R}}]$ such that $\mu(\{ \, x\in X\mid \alpha(T_n, x)\not\in V_n\, \})\to 0$. As shown in the proof of Corollary \[cor-ext-s\], to each $T_n$, the element $\tilde{T}_n$ of $[{\mathcal{R}}_{\alpha, L}]$ is associated, and the sequence $(\tilde{T}_n)_n$ is then a non-trivial central sequence in $[{\mathcal{R}}_{\alpha, L}]$.
The same proof as that of Corollary \[cor-td\] works to obtain the following:
\[cor-td-c\] Let ${\mathcal{R}}$ be an ergodic discrete p.m.p. equivalence relation on $(X, \mu)$ and let ${\mathcal{S}}$ be a finite index subrelation of ${\mathcal{R}}$. If ${\mathcal{R}}$ is Schmidt, then each ergodic component of ${\mathcal{S}}$ is Schmidt.
We may ask whether the same result as Corollary \[cor-fi\] holds for the Schmidt property of groups in place of stability. Corollary \[cor-td-c\] immediately implies that the Schmidt property passes from a group to each of its finite index subgroups. However the following remains unsolved:
Let $G$ be a countable group with a finite central subgroup $C$. If $G/C$ has the Schmidt property, then does $G$ have the Schmidt property as well?
We note that if $C$ is an infinite central subgroup of $G$, then $G$ automatically has the Schmidt property (see Example \[ex-infinite-center\]).
Consequences under spectral-gap assumptions {#sec-sg}
===========================================
Stable spectral gap
-------------------
Recall that a unitary representation of a countable group $G$ has *spectral gap* if it does not contain the trivial representation of $G$ weakly. We say that a p.m.p. action $G{\curvearrowright}(X, \mu)$ has [*spectral gap*]{} if the Koopman representation $G{\curvearrowright}L^2_0(X)$ has spectral gap, and that a p.m.p. action $G{\curvearrowright}(X, \mu)$ has [*stable spectral gap*]{} if for any unitary representation of $G$ on a Hilbert space ${\mathcal}{H}$, the diagonal representation $G{\curvearrowright}L^2_0(X)\otimes {\mathcal}{H}$ has spectral gap. If a free p.m.p. action $G{\curvearrowright}(X, \mu)$ has stable spectral gap and the groupoid $G\ltimes (X, \mu)$ is inner amenable, then we can find the following remarkable Følner sequence for the conjugating action of $G$:
\[prop-iasgap\] Let $G{\curvearrowright}(X, \mu)$ be a free p.m.p. action with stable spectral gap, and let $\mathcal{R}$ be the associated orbit equivalence relation. If $\mathcal{R}$ is inner amenable, then there exists a sequence $(F_n)_n$ of finite subsets of $G$ such that
1. $|F_n^g \bigtriangleup F_n |/|F_n| \to 0$ for each $g\in G$.
2. $1_{F_n}(g) \to 0$ for each $g\in G$.
3. $\sup _{g\in F_n}\mu (gA\bigtriangleup A) \to 0$ for each Borel subset $A\subset X$.
Thus, if $(g_n)_n$ is a sequence in $G$ with $g_n\in F_n$ for all $n$, then $g_n$ converges to the identity transformation in $\mathrm{Aut}(X,\mu)$. In particular, the image of $G$ in $\mathrm{Aut}(X, \mu)$ is not discrete.
Before the proof, we prepare the following:
\[lem-specgap\] Let $G{\curvearrowright}(X, \mu)$ be a p.m.p. action with stable spectral gap. For each $g\in G$ let $\phi _g \in [G\ltimes (X, \mu)]$ denote the section $\phi _g (x)=g$. Then for any ${\varepsilon}>0$, there exist a finite subset $S\subset G$ and $\delta >0$ such that if $\xi \in L^1(G\times X)$ is any non-negative unit vector satisfying $\sup _{g\in S} \Vert \xi^{\phi _g} - \xi \Vert_1 < \delta$, then $\Vert \xi - P\xi \Vert _1 < {\varepsilon}$, where $P\colon L^1(G\times X)\to L^1(G\times X)$ is the projection defined by integrating functions along $X$: $$(P\xi)(g, x) = \int_X\xi(g, t) \, d\mu(t)$$ for $\xi \in L^1(G\times X)$, $g\in G$ and $x\in X$.
The action $G{\curvearrowright}G\times X$ defined by $g (h,x)=(h,x)^{\phi_g^{-1}} = (ghg^{-1},gx )$ gives rise to a unitary representation $\pi \colon G{\curvearrowright}L^2(G\times X)$ which is naturally identified with the tensor product of the conjugation representation of $G$ on $\ell ^2(G)$ with the Koopman representation of $G$ on $L^2(X)$. The projection $P$ is also defined on $L^2(G\times X)$ by the same formula, and it is exactly the orthogonal projection onto the subspace $\ell^2(G)\otimes {\mathbb{C}}1$.
Let ${\varepsilon}>0$ be given and choose ${\varepsilon}_0>0$ so that $2{\varepsilon}_0 ^{1/2}+{\varepsilon}_0 <{\varepsilon}$. Since the representation $\pi \colon G{\curvearrowright}\ell^2(G)\otimes L^2_0 (X)$ has spectral gap, there exist a finite subset $S\subset G$ and $\delta >0$ such that if $\eta \in L^2(G\times X)$ is any unit vector satisfying $\sup _{g\in S} \Vert \pi (g)\eta - \eta \Vert_2^2 <\delta$, then $\Vert \eta - P\eta \Vert _2 ^2 < {\varepsilon}_0$. Let $\xi \in L^1(G\times X)$ be any non-negative unit vector satisfying $\sup _{g\in S} \Vert \xi ^{\phi _g} - \xi \Vert _1 < \delta$, and let $\eta {\coloneqq}\xi ^{1/2}$. Then $\eta$ is a unit vector in $L^2(G\times X)$, and $$\sup _{g\in S} \Vert \pi (g)\eta - \eta \Vert _2^2 \leq \sup _{g\in S} \Vert \xi ^{\phi _g} - \xi \Vert _1 < \delta,$$ where the first inequality follows from the inequality $|a-b|^2\leq |a^2-b^2|$ for all $a, b\geq 0$. By our choice of $\delta$, we then have $\Vert \eta - P\eta \Vert _2 ^2 < {\varepsilon}_0$. It follows that $\Vert P\eta \Vert _2 ^2 > 1-{\varepsilon}_0$ and hence $$\begin{aligned}
\Vert P\xi - (P\eta )^2 \Vert _1 &= \sum _{g\in G} \Biggl( \int_X \xi(g, x) \, d\mu(x) - \biggl(\int _X \xi(g, x)^{1/2} \, d\mu(x) \biggr)^2\Biggl)\notag \\
&= 1- \Vert P\eta \Vert _2 ^2 <{\varepsilon}_0,\notag\end{aligned}$$ where we use Jensen’s inequality in the first equation. By the Cauchy-Schwarz inequality, we have $$\begin{aligned}
\Vert \xi - (P\eta ) ^2 \Vert _1 &= \Vert (\eta + P\eta ) (\eta - P\eta ) \Vert _1 \leq \Vert \eta + P\eta \Vert _2 \Vert \eta - P\eta \Vert _2 \leq 2{\varepsilon}_0^{1/2},\end{aligned}$$ and therefore $\Vert \xi - P\xi \Vert _1 < 2 {\varepsilon}_0^{1/2} + {\varepsilon}_0 < {\varepsilon}$.
Let $\alpha \colon \mathcal{R}\to G$ be the cocycle defined by the equation $\alpha (y,x)x = y$. Let $(\xi _n )_{n\in {\mathbb{N}}} $ be an inner amenability sequence for $\mathcal{R}$. For each $n\in {\mathbb{N}}$, we define $p_n \in \ell ^1(G)$ by $p_n (g) = \int _X\xi _n (gx,x) \, d\mu(x)$. Then each $p_n$ is a probability measure on $G$, and by Lemma \[lem-specgap\], we have $\Vert \xi_n-p_n\circ \alpha \Vert_1\to 0$. Therefore, for each $g\in G$, we have $\Vert p_n^g - p_n \Vert _1 \to 0$, and $p_n (g) \to 0$. For any Borel subset $A\subset X$, we have $$\begin{aligned}
\sum _{g\in G}p_n(g)\mu (g^{-1}A\cap A ) &= \int _{\mathcal{R}_A}p_n (\alpha (y,x)) \, d\mu ^1 (y,x) \\
&= \int _{\mathcal{R}_A}\xi _n \, d\mu ^1 + \int _{\mathcal{R}_A}(p_n \circ \alpha - \xi _n )\, d\mu ^1 \to \mu (A),\end{aligned}$$ where the last convergence follows because $(\xi _n)_n$ is balanced. Therefore, for every ${\varepsilon}>0$, we have $p_n (D_{A,{\varepsilon}})\to 1$, where we set $$D_{A,{\varepsilon}} = \{ \, g \in G \mid \mu (gA\bigtriangleup A ) < {\varepsilon}\, \}.$$
Let $\{ A_i \} _{i\in {\mathbb{N}}}$ be a countable collection of sets which are dense in the measure algebra of $\mu$. After passing to a subsequence of $(p_n)_{n\in {\mathbb{N}}}$, we may assume without loss of generality that $p_n (\bigcap _{i<n}D_{A_i,1/n}) > 1-1/n$ for every $n\in {\mathbb{N}}$. Let $Q_1\subset Q_2\subset \cdots$ be an increasing exhaustion of $G$ by finite subsets. Since $p_n(g)\to 0$ for each $g\in G$, after passing to a further subsequence, we can assume without loss of generality that $p_n (Q_n) < 1/n$ for every $n\in {\mathbb{N}}$. Let $q_n$ be the normalized restriction of $p_n$ to $(\bigcap _{i<n}D_{A_i, 1/n}) \setminus Q_n$. Then $\Vert q_n - p_n \Vert _1 \to 0$, so $\Vert q_n^g - q_n \Vert _1 \to 0$ for each $g\in G$. Therefore, by the Namioka trick, we may find a sequence of finite subsets $F_n \subset (\bigcap _{i<n}D_{A_i,1/n})\setminus Q_n$ such that condition (i) holds. Condition (ii) holds since $F_n\cap Q_n=\emptyset$. Given any Borel subset $A\subset X$ and ${\varepsilon}>0$, we can find some $i\in {\mathbb{N}}$ with $\mu (A\bigtriangleup A_i)<{\varepsilon}/3$. Then for any $n>i$ with $1/n <{\varepsilon}/3$, for any $g\in F_n\subset D_{A_i,1/n}$, we have $\mu (gA\bigtriangleup A ) \leq \mu (gA_i\bigtriangleup A_i ) +2\mu (A\bigtriangleup A_i ) <{\varepsilon}$. This shows that condition (iii) holds.
\[cor-ber\] The orbit equivalence relation associated to any Bernoulli shift action of a non-amenable group is not inner amenable.
Let $G{\curvearrowright}(X,\mu )$ be a Bernoulli shift action of a non-amenable group $G$. This action is mixing, so the image of $G$ in $\mathrm{Aut}(X,\mu )$ is discrete. Since $G$ is non-amenable, the action $G{\curvearrowright}(X, \mu)$ has stable spectral gap. Thus the corollary follows from Proposition \[prop-iasgap\].
We will use the following lemma and corollary, which impose constraints on central sequences in a full group, in order to construct interesting examples in Sections \[sec-finite-index\] and \[sec-ex\].
\[lem-specgap-product\] Let $G{\curvearrowright}(X, \mu)$ and $G{\curvearrowright}(Y, \nu)$ be p.m.p. actions and suppose that the action $G{\curvearrowright}(X, \mu)$ has stable spectral gap. We set $(Z, \zeta)=(X\times Y, \mu \times \nu)$ and let $G$ act on $(Z, \zeta)$ diagonally. For any ${\varepsilon}>0$, there exist a finite subset $S\subset G$ and $\delta >0$ such that if $\phi$ is any element of $[G\ltimes (Z, \zeta)]$ satisfying $$\label{eqn-phicomm}
\inf_{g\in S}\zeta(\{ \, z\in Z\mid (\phi \circ \phi_g)z=(\phi_g\circ \phi)z\, \})>1-\delta,$$ then there exists a Borel subset $Y'\subset Y$, its partition $Y'=\bigsqcup_{i=1}^m Y_i$ into finitely many Borel subsets, and elements $g_1,\ldots, g_m\in G$ such that $\nu(Y')>1-{\varepsilon}$ and for any $y\in Y_i$, we have $$\mu(\{ \, x\in X\mid \phi(x, y)=g_i\, \})>1-{\varepsilon}.$$
We may assume that ${\varepsilon}< 1/2$. Let $\pi \colon G{\curvearrowright}\ell^2(G)\otimes L^2_0 (X)\otimes L^2(Y)$ be the subrepresentation of the tensor product of the conjugation representation $G{\curvearrowright}\ell ^2(G)$ with the Koopman representation $G{\curvearrowright}L^2(Z)$. Let $P\colon L^2(G\times Z)\to \ell^2(G)\otimes {\mathbb{C}}1\otimes L^2(Y)$ be the orthogonal projection. Since $\pi$ has spectral gap, there exist a finite subset $S\subset G$ and $\delta >0$ such that if $\eta \in L^2(G\times Z)$ is any unit vector satisfying $\inf _{g\in S} \mathrm{Re}\langle \pi (g) \eta , \eta \rangle >1-\delta$, then $\Vert \eta - P\eta \Vert _2 ^2 < {\varepsilon}^2$.
Assume that $\phi \in [G\ltimes (Z, \zeta)]$ satisfies condition . We set $D = \{ \, (\phi (z), z)\in G\times Z \mid z\in Z \, \}$. Then $1_{D}$ is a unit vector in $L^2(G\times Z)$, and for any $g\in S$, we have $$\langle \pi (g)1_{D} , 1_{D} \rangle = \zeta (\{ \, z\in Z \mid (\phi \circ \phi _g )z = (\phi _g \circ \phi )z \, \} ) >1-\delta .$$ It follows from our choice of $S$ and $\delta$ that $\Vert 1_{D} - P(1_{D}) \Vert _2 ^2 < {\varepsilon}^2$ and hence $\Vert P(1_{D} ) \Vert _2^2 > 1-{\varepsilon}^2$. For $g\in G$ and $y\in Y$, we set $A_{g, y} = \{ \, x\in X \mid \phi (x, y) = g \, \}$, so that $P(1_{D})(g, (x, y)) = \mu (A_{g, y})$ for any $g\in G$ and almost every $(x, y)\in X\times Y$. Then we have $$\begin{aligned}
1-{\varepsilon}^2 & < \Vert P(1_{D})\Vert _2^2 = \int_Y \sum _{g\in G}\mu (A_{g, y})^2 \, d\nu(y)\leq \int_Y \biggl( \sup _{g\in G}\mu (A_{g, y}) \biggr) \sum_{g\in G}\mu(A_{g, y})\, d\nu(y)\\
& = \int_Y \sup_{g\in G} \mu (A_{g, y}) \, d\nu(y),\end{aligned}$$ and therefore there exists a Borel subset $Y'\subset Y$ such that $\nu(Y')>1-{\varepsilon}$ and for almost every $y\in Y'$, we have $\sup_{g\in G}\mu(A_{g, y})>1-{\varepsilon}$. Since ${\varepsilon}<1/2$, for almost every $y\in Y'$, there exists a unique $g\in G$ such that $\mu(A_{g, y})>1-{\varepsilon}$. If $Y'$ is replaced with its slightly smaller subset, there exist finitely many elements $g_1,\ldots, g_m\in G$ and a Borel partition $Y'=\bigsqcup_{i=1}^mY_i$ such that for almost every $y\in Y_i$, we have $\mu(A_{g_i, y})>1-{\varepsilon}$.
\[cor:specgap\] Let $G{\curvearrowright}(X, \mu)$ be a p.m.p. action with stable spectral gap. For any ${\varepsilon}>0$ and any finite subset $F\subset G$, there exist a finite subset $S\subset G$ and $\delta >0$ such that if $\phi$ is any element of $[G\ltimes (X, \mu)]$ satisfying $$\inf _{g\in S} \mu ( \{ \, x\in X \mid (\phi \circ \phi _g) x = (\phi _g\circ \phi ) x \, \} ) > 1-\delta,$$ then there exists an element $g_0\in G$ which commutes with every element of $F$ and satisfies $\mu (\{ \, x\in X \mid \phi (x)=\phi _{g_0}(x) \, \} ) > 1-{\varepsilon}$.
Pick $0<{\varepsilon}<1/2$ and a finite subset $F\subset G$. In the proof of Lemma \[eqn-phicomm\], suppose that $Y$ is a singleton. We may assume that $\delta <1/4$ for the number $\delta$ obtained from the assumption that the action $G{\curvearrowright}(X, \mu)$ has stable spectral gap. We may also assume that the obtained finite subset $S\subset G$ contains $F$. We set $D = \{ \, (\phi (x),x)\in G\times X \mid x\in X \, \}$. Following the proof of Lemma \[eqn-phicomm\], we obtain $1-{\varepsilon}^2 < \Vert P(1_{D})\Vert _2^2\leq \sup _{g\in G}\mu (A_g)$, where we set $A_g = \{ \, x\in X \mid \phi (x) = g \, \}$ for $g\in G$. Hence there exists some $g_0\in G$ such that $\mu (A_{g_0})>1-{\varepsilon}^2$. It remains to show that $g_0$ commutes with every element of $S$ (and hence of $F$). Fix $g\in S$. Since $\mu (A_{g_0})>1-{\varepsilon}^2 > 3/4$ and $\mu ( \{ \, x\in X \mid (\phi \circ \phi _g) x = (\phi _g \circ \phi )x \, \} ) > 1-\delta > 3/4$, the set $$g^{-1}A_{g_0}\cap A_{g_0}\cap \{ \, x\in X \mid (\phi \circ \phi _g) x = (\phi _g \circ \phi )x \, \}$$ is non-null, so fix some element $x$ in this set. Then $g_0g = (\phi \circ \phi _g)x = (\phi _g\circ \phi )x = gg_0$ and hence $g_0$ commutes with $g$.
Product actions
---------------
In this subsection, we show that if a free p.m.p. action $G{\curvearrowright}(X, \mu)$ satisfies both a spectral gap property and a mixing property, then its product with an arbitrary p.m.p. action of $G$ gives rise to the orbit equivalence relation which is not inner amenable, and moreover the associated von Neumann algebra often does not have property Gamma. This is already known for the Bernoulli shift action of a non-amenable group ([@i Lemma 2.3]).
\[prop-diagonal-action\] Let $G{\curvearrowright}(X, \mu)$ be a free, p.m.p., mildly mixing action of an infinite countable group $G$. Suppose that either
1. the action $G{\curvearrowright}(X, \mu)$ has stable spectral gap, or
2. there is an infinite subgroup $H$ of $G$ such that the pair $(G, H)$ has property (T).
Let $G{\curvearrowright}(Y, \nu)$ be an ergodic p.m.p. action and let $G$ act on $(X\times Y, \mu \times \nu)$ diagonally. Then the translation groupoid $G\ltimes (X\times Y, \mu \times \nu)$ is not inner amenable.
If the action $G{\curvearrowright}(Y, \nu)$ is further strongly ergodic, then the von Neumann algebra associated to the action $G{\curvearrowright}(X\times Y, \mu \times \nu)$ does not have property Gamma.
Recall that a p.m.p. action $G{\curvearrowright}(X, \mu)$ is called *mildly mixing* if for any Borel subset $A\subset X$ with $0<\mu(A)<1$, we have $\liminf_{g\to \infty}\mu(gA\bigtriangleup A)>0$ ([@sch-mild]). Any mildly mixing action is weakly mixing, and hence the diagonal action $G{\curvearrowright}(X\times Y, \mu \times \nu)$ in Proposition \[prop-diagonal-action\] is ergodic.
We will actually prove Proposition \[prop-diagonal-action\] in a slightly more general setting.
\[assum\] Let $G{\curvearrowright}(X, \mu)$ be a free p.m.p. action of an infinite countable group $G$ satisfying the following condition:
1. There exist a finite subset $K\subset G$ and a Borel subset $D\subset X$ such that $$\inf_{g\in G\setminus K}\mu(gD\bigtriangleup D)>0.$$
Let $G{\curvearrowright}(Y, \nu)$ be a p.m.p. action and set $(Z, \zeta)=(X\times Y, \mu \times \nu)$. Let $G$ act on $(Z, \zeta)$ diagonally and suppose that the action $G {\curvearrowright}(Z, \zeta)$ is ergodic. Let $G$ act on $G\times Z$ by $g(h, z)=(ghg^{-1}, gz)$, and then we have the Koopman representation $\pi \colon G{\curvearrowright}L^2(G\times Z)$. Suppose also the following condition:
1. The representation of $G$ on $\ell^2(G)\otimes L^2_0(X)\otimes L^2(Y)$ that is a subrepresentation of $\pi$ has spectral gap.
We fix the notation. Let $M$ be the von Neumann algebra associated with the action $G{\curvearrowright}(Z, \zeta)$. Let $\tau$ be the tracial state on $M$, and let $L^2(M)$ be the completion of $M$ with respect to the norm $\Vert x\Vert_2=\tau(x^*x)^{1/2}$. The Hilbert space $L^2(M)$ is naturally identified with $\ell^2(G)\otimes L^2(X)\otimes L^2(Y)$. Let $Q\colon L^2(M)\to \ell^2(\{ e\})\otimes \mathbb{C}1\otimes L^2(Y)$ be the orthogonal projection.
\[rem-ass\] The diagonal action $G{\curvearrowright}(X\times Y, \mu \times \nu)$ in Proposition \[prop-diagonal-action\] satisfies conditions (A) and (B) in Assumption \[assum\]. Indeed, condition (A) follows since the action $G{\curvearrowright}(X, \mu)$ is mildly mixing. If condition (1) holds, then condition (B) obviously follows. If condition (2) holds, then the restriction $H{\curvearrowright}(X, \mu)$ is mildly mixing and hence weakly mixing ([@sch-mild Section 2]). Since the representation $H{\curvearrowright}L^2_0(X)$ is weakly mixing, there is no $H$-invariant unit vector in $\ell^2(G)\otimes L^2_0(X)\otimes L^2(Y)$, and condition (B) follows from property (T) of the pair $(G, H)$.
The following is a remark due to Adrian Ioana on the first author’s earlier note.
\[lem-p\] Under Assumption \[assum\], if $(\eta_n)_n$ is a sequence of unit vectors in $L^2(M)$ such that $\Vert \pi(g)\eta_n-\eta_n\Vert_2\to 0$ for any $g\in G$ and $\Vert \eta_n1_A-1_A\eta_n\Vert_2\to 0$ for any Borel subset $A\subset Z$, then $\Vert \eta_n-Q\eta_n\Vert_2\to 0$.
Note that $\eta_n1_A$ is the vector in $L^2(M)$ obtained by multiplying $\eta_n$ by $1_A\in M$ from the right, which is identified with the function $1_{{\mathcal{G}}_{Z, A}}\eta_n\in L^2(G\times Z)$, where ${\mathcal{G}}{\coloneqq}G\ltimes (Z, \zeta)$. Similarly $1_A\eta_n$ is the vector in $L^2(M)$ which is identified with the function $1_{{\mathcal{G}}_{A, Z}}\eta_n$.
We first find a Borel subset $E\subset X$ such that $$\inf_{g\in G\setminus \{ e\}}\mu(gE\bigtriangleup E)>0.$$ Let $K_0$ be the subgroup of elements $g\in G$ with $gD=D$, which is contained in $K$ and hence finite. By condition (A), the number $c{\coloneqq}\inf_{g\in G\setminus K_0}\mu(gD\bigtriangleup D)$ is positive. Since the action $G{\curvearrowright}X$ is free, there exists a Borel subset $D_1\subset X\setminus D$ such that $\mu(D_1)<c/3$ and $\mu(gD_1\bigtriangleup D_1)>0$ for any $g\in K_0\setminus \{ e\}$. The set $E=D\cup D_1$ is a desired one. Indeed, for any $g\in G\setminus K_0$, we have $\mu(gE\bigtriangleup E)\geq \mu(gD\bigtriangleup D)-2\mu(D_1)\geq c/3$, and for any $g\in K_0\setminus \{ e\}$, we have $\mu(gE\bigtriangleup E)=\mu(gD_1\bigtriangleup D_1)>0$. We set $d=\inf_{g\in G\setminus \{ e\}}\mu(gE\bigtriangleup E)>0$.
For $g\in G$, let $u_g$ be the unitary of $M$ associated to $g$. The representation $\pi$ is given by $\pi(g)x=u_gxu_g^*$ for $x\in M$. Let $P\colon L^2(M)\to \ell^2(G)\otimes \mathbb{C}1\otimes L^2(Y)$ be the orthogonal projection. By condition (B), we have $$\begin{aligned}
\label{px}
\Vert \eta_n-P\eta_n\Vert_2\to 0.\end{aligned}$$ For each $n$, write $P\eta_n=\sum_{g\in G}u_g(1\otimes b_{n, g})$ where $b_{n, g}\in L^2(Y)$. Let $F{\coloneqq}E\times Y$ and let $1_F$ be the indicator function of $F$. We have $\Vert \eta_n1_F-1_F\eta_n\Vert_2\to 0$, and hence condition (\[px\]) implies that $\Vert P(\eta_n)1_F-1_FP(\eta_n)\Vert_2\to 0$. It follows that $$\Vert P(\eta_n)1_F-1_FP(\eta_n)\Vert_2^2=\sum_{g\in G}\mu(E\bigtriangleup g^{-1}E)\Vert b_{n, g}\Vert_2^2\geq d\sum_{g\in G\setminus \{ e\}}\Vert b_{n, g}\Vert_2^2.$$ By the definition of $P$ and $Q$, we have $\sum_{g\in G\setminus \{ e\}}\Vert b_{n, g}\Vert_2^2=\Vert P\eta_n-Q\eta_n\Vert_2^2$ and hence $\Vert P\eta_n-Q\eta_n\Vert_2\to 0$. By condition (\[px\]) again, $\Vert \eta_n-Q\eta_n\Vert_2\to 0$.
\[cor-c\] Under Assumption \[assum\], the following two assertions hold:
1. The translation groupoid $G\ltimes (Z, \zeta)$ is not inner amenable.
2. (A. Ioana) If the action $G{\curvearrowright}(Y, \nu)$ is strongly ergodic, then $M$ does not have property Gamma.
To prove assertion (i), suppose toward a contradiction that there exists an inner amenability sequence $(\xi_n)$ for the groupoid $G\ltimes (Z, \zeta)$. Let $\eta_n{\coloneqq}\xi_n^{1/2}$. Then $\eta_n$ is a unit vector in $L^2(G\times Z)$ and satisfies the assumption in Lemma \[lem-p\]. Indeed, for any $g\in G$, we have $\Vert \pi(g)\eta_n-\eta_n\Vert_2^2\leq \Vert (\xi_n)^{\phi_{g^{-1}}}-\xi_n\Vert_1\to 0$, and setting ${\mathcal{G}}=G\ltimes (Z, \zeta)$, for any Borel subset $A\subset Z$, we have $\Vert \eta_n1_A-1_A\eta_n\Vert_2^2\leq \Vert 1_{{\mathcal{G}}_{Z, A}}\xi_n-1_{{\mathcal{G}}_{A, Z}}\xi_n\Vert_1\to 0$ since $(\xi_n)$ is balanced. By Lemma \[lem-p\], we have $\Vert \eta_n-Q\eta_n\Vert_2\to 0$. The projection $Q$ is also defined on $L^1(G\times Z)$: For $\xi \in L^1(G\times Z)$, $g\in G$, $x\in X$ and $y\in Y$, we set $$(Q\xi)(g, x, y)=
\begin{cases}
\int_X\xi(e, t, y)\, d\mu(t) & \text{if} \ g=e,\\
0 & \text{if}\ g\neq e.
\end{cases}$$ As in the proof of Lemma \[lem-specgap\], by Jensen’s inequality, we have $$\begin{aligned}
\Vert Q\xi_n-(Q\eta_n)^2\Vert_1&=\int_{X\times Y}\xi_n(e, x, y)\, d\mu(x) d\nu(y)-\Vert Q\eta_n\Vert_2^2\leq \Vert \eta_n\Vert_2^2-\Vert Q\eta_n\Vert_2^2 \to 0.\end{aligned}$$ By the Cauchy-Schwarz inequality, we have $$\Vert \xi_n-(Q\eta_n)^2\Vert_1\leq \Vert \eta_n+Q\eta_n\Vert_2\Vert \eta_n-Q\eta_n\Vert_2\leq 2\Vert \eta_n-Q\eta_n\Vert_2\to 0.$$ Therefore $\Vert \xi_n-Q\xi_n\Vert_1\to 0$ and $\xi_n$ is almost concentrated on the set $\{ e\} \times Z$. This contradicts that $(\xi_n)_n$ is diffuse. Assertion (i) follows.
Suppose the action $G{\curvearrowright}(Y, \nu)$ is strongly ergodic. If $M$ had property Gamma, then we would have a central sequence $(u_n)_n$ of unitaries of $M$ such that $\tau(u_n)=0$ for any $n$. Since $Q$ is $G$-equivariant, $(Q(u_n))_n$ is asymptotically $G$-invariant. By strong ergodicity of the action $G{\curvearrowright}(Y, \nu)$, we have $\Vert Q(u_n)\Vert_2=\Vert Q(u_n)-\tau(Q(u_n))\Vert_2\to 0$. This contradicts Lemma \[lem-p\] and assertion (ii) follows.
Proposition \[prop-diagonal-action\] follows from Corollary \[cor-c\] and Remark \[rem-ass\].
Finite-index inclusions and central sequences {#sec-finite-index}
=============================================
For a finite-index inclusion ${\mathcal{S}}<{\mathcal{R}}$ of ergodic discrete p.m.p. equivalence relations, in Corollaries \[cor-td\] and \[cor-td-c\], we showed that if ${\mathcal{R}}$ is stable or Schmidt, then so is ${\mathcal{S}}$. In this section, we discuss whether the converse holds. In Subsection \[subsec-alg\], we give a sufficient condition for the converse to hold. In Subsections \[subsec-counter-schmidt\] and \[subsec-counter-stable\], we give examples for which the converse does not hold. Throughout this section, let $(X, \mu)$ be a standard probability space and ${\mathcal{B}}$ denote the algebra of Borel subsets of $X$, where two elements of ${\mathcal{B}}$ are identified when their symmetric difference is $\mu$-null.
The algebra of asymptotically invariant sequences {#subsec-alg}
-------------------------------------------------
Let ${\mathcal{S}}<{\mathcal{R}}$ be a finite-index inclusion of ergodic discrete p.m.p. equivalence relations on $(X, \mu)$. It follows from [@ha Theorem 2.11] or [@su Theorem 2] that we have an ergodic finite-index subrelation ${\mathcal{S}}_0<{\mathcal{S}}$ and a finite group $F$ acting on ${\mathcal{S}}_0$ by automorphisms such that ${\mathcal{R}}={\mathcal{S}}_0 \rtimes F$. Under the assumption that ${\mathcal{S}}$ is stable or Schmidt, since these properties pass to ${\mathcal{S}}_0$, we may assume that ${\mathcal{R}}$ is written as ${\mathcal{R}}={\mathcal{S}}\rtimes F$ for some finite group $F$ acting on ${\mathcal{S}}$.
Fix a non-principal ultrafilter $\omega$ on ${\mathbb{N}}$ and form the ultraproduct $({\mathcal{B}}^\omega, \mu^\omega)$ of the measure algebra $({\mathcal{B}}, \mu)$. The full group $[{\mathcal{S}}]$ naturally acts on ${\mathcal{B}}^\omega$, preserving $\mu^\omega$. Let ${\mathcal{A}}$ denote the fixed point algebra for this action. The group $F$ also acts on ${\mathcal{B}}^\omega$ and on ${\mathcal{A}}$.
\[prop-f-faith\] Suppose that $F$ acts on ${\mathcal{A}}$ faithfully. Then
1. if ${\mathcal{S}}$ is stable, then ${\mathcal{R}}$ is also stable.
2. If ${\mathcal{S}}$ is Schmidt, then ${\mathcal{R}}$ is also Schmidt.
We first find a non-zero $\bar{B}\in {\mathcal{A}}$ such that $\alpha(\bar{B})\cap \bar{B}=0$ for any non-trivial $\alpha \in F$. Such a $\bar{B}$ is obtained by applying the following repeatedly: For any non-trivial $\alpha \in F$, if $\bar{C}\in {\mathcal{A}}$ is non-zero, then there exists a non-zero $\bar{B}\in {\mathcal{A}}$ such that $\bar{B}\subset \bar{C}$ and $\alpha(\bar{B})\cap \bar{B}=0$. Although this is proved in [@ck Lemma 2.3], we give a proof for completeness: Otherwise we would have $\alpha (\bar{B})=\bar{B}$ for any $\bar{B}\in {\mathcal{A}}$ with $\bar{B}\subset \bar{C}$, and since $\alpha$ acts on ${\mathcal{A}}$ non-trivially, there exists a non-zero $\bar{D}\subset 1-\bar{C}$ such that $\alpha (\bar{D})\cap \bar{D}=0$. If $(C_n)_n$ and $(D_n)_n$ represent $\bar{C}$ and $\bar{D}$, respectively, then by Lemma \[lem-ai\], there exists a subsequence $(D_{k_n})_n$ of $(D_n)_n$ such that $\mu(D_n')$ is uniformly positive, where $D_n'{\coloneqq}C_n\cap D_{k_n}$. Let $\bar{D}'\in {\mathcal{A}}$ be represented by the sequence $(D_n')_n$. Then $\bar{D}'$ is non-zero, but we have $\alpha(\bar{D}')=\bar{D}'$ since $\bar{D}'\subset \bar{C}$, and we have $\alpha(\bar{D}')\cap \bar{D}'=0$ since $\alpha (\bar{D})\cap \bar{D}=0$. This is a contradiction.
Let $(B_n)_{n\in {\mathbb{N}}}$ be a sequence representing $\bar{B}$ such that $\alpha(B_n)\cap B_n=\emptyset$ for any $n$ and any non-trivial $\alpha \in F$. Take a decreasing sequence ${\varepsilon}_n\searrow 0$ of positive numbers, a sequence $(E_k)_{k\in {\mathbb{N}}}$ of elements of ${\mathcal{B}}$ which is dense in ${\mathcal{B}}$, and a sequence $(g_k)_{k\in {\mathbb{N}}}$ of elements of $[{\mathcal{S}}]$ which is dense in $[{\mathcal{S}}]$. Passing to a subsequence of $(B_n)$, we may assume that
1. $\mu(g_k\alpha(B_n)\bigtriangleup \alpha(B_n))<{\varepsilon}_n/|F|$ for any $k\leq n$ and any $\alpha \in F$.
To prove assertion (i), assume that ${\mathcal{S}}$ is stable, and let $(T_n, A_n)_{n\in {\mathbb{N}}}$ be a stability sequence for ${\mathcal{S}}$. Passing to a subsequence of $(T_n, A_n)$, we may assume that for each $n$, for any $k\leq n$ and any $\alpha \in F$, we have
1. $\mu(T_nB_n\bigtriangleup B_n)<{\varepsilon}_n/|F|$,
2. $|\mu(A_n\cap B_n)-\mu(B_n)/2|<{\varepsilon}_n$,
3. $\mu(T_n(\alpha^{-1}(E_k\cap \alpha(B_n)))\bigtriangleup \alpha^{-1}(E_k\cap \alpha(B_n))<{\varepsilon}_n/|F|$, and
4. $\mu(\{ \alpha^{-1}g_k\alpha T_n\neq T_n\alpha^{-1}g_k\alpha \})<{\varepsilon}_n/|F|$.
We define $S_n\in [{\mathcal{S}}]$ such that for each $\alpha \in F$, we have $S_n=\alpha T_n\alpha^{-1}$ on $\alpha(B_n)$ outside the set $\alpha(B_n\setminus T_n^{-1}B_n)$, which has measure less than ${\varepsilon}_n/(2|F|)$ by condition (2), and also have $S_n(\alpha(B_n))=\alpha(B_n)$, and $S_n$ is the identity outside $\bigcup_{\alpha \in F}\alpha(B_n)$. By condition (4), we have $$\mu(S_n(E_k\cap \alpha(B_n))\bigtriangleup (E_k\cap \alpha(B_n)))<2{\varepsilon}_n/|F|\quad \text{and}\quad \mu(S_nE_k\bigtriangleup E_k)<2{\varepsilon}_n$$ for any $k\leq n$. Therefore $\mu(S_nE\bigtriangleup E)\to 0$ for any $E\in {\mathcal{B}}$. By construction, we also have $\mu(\{ \alpha S_n\neq S_n\alpha \})<{\varepsilon}_n$ for any $\alpha \in F$.
Fix $k\leq n$. We claim that $\mu(\{ g_kS_n\neq S_ng_k\})<5{\varepsilon}_n$. This claim together with the facts proved in the last paragaraph implies that the sequence $(S_n)$ is central in $[{\mathcal{R}}]$. By the definition of $S_n$, for each $\alpha \in F$, we have $g_kS_n=g_k\alpha T_n\alpha^{-1}$ on $\alpha(B_n)$ outside a subset of measure less than ${\varepsilon}_n/|F|$. By condition (5), outside a subset of measure less than ${\varepsilon}_n/|F|$, we have $g_k\alpha T_n\alpha^{-1}=\alpha(\alpha^{-1}g_k\alpha)T_n\alpha^{-1}=\alpha T_n\alpha^{-1}g_k$, and the right hand side is equal to $S_ng_k$ on $\alpha(B_n)$ outside a subset of measure less than $2{\varepsilon}_n/|F|$ by condition (1) and the definition of $S_n$. As a result, $\mu(\{ g_kS_n\neq S_ng_k\} \cap \alpha(B_n))<4{\varepsilon}_n/|F|$ for each $\alpha \in F$. Our claim then follows from condition (1).
We set $C_n=F(A_n\cap B_n)$. The sequence $(A_n\cap B_n)$ is asymptotically invariant for ${\mathcal{S}}$, and the sequence $(C_n)$ is thus asymptotically invariant for ${\mathcal{R}}$. Since $S_n$ preserves $\alpha(B_n)$ for each $\alpha \in F$, we have $S_n(A_n\cap B_n)\setminus (A_n\cap B_n)\subset S_nC_n\setminus C_n$. Up to a subset of measure less than ${\varepsilon}_n/|F|$, the left hand side is equal to $T_n(A_n\cap B_n)\setminus (A_n\cap B_n)$, which is equal to $T_n(A_n\cap B_n)$ because $T_nA_n$ is disjoint from $A_n$. By condition (3), the measure of the set $T_n(A_n\cap B_n)$ is equal to $\mu(B_n)/2$ up to ${\varepsilon}_n$. Therefore $\mu(S_nC_n\setminus C_n)$ is uniformly positive, and $(S_n, C_n)$ is a pre-stability sequence for ${\mathcal{R}}$. Assertion (i) follows.
We prove assertion (ii) and assume that ${\mathcal{S}}$ is Schmidt. In the beginning of the proof of this proposition, we found a sequence $(B_n)_{n\in {\mathbb{N}}}$ of elements of ${\mathcal{B}}$ representing a non-zero element of ${\mathcal{A}}$ and satisfying $\alpha (B_n)\cap B_n=\emptyset$ for any $n$ and any non-trivial $\alpha \in F$. Let $(T_n)$ be a central sequence in $[{\mathcal{S}}]$ such that $T_nx\neq x$ for any $n$ and any $x\in X$. Such a sequence exists by Lemma \[lem-c\]. Take a sequence $(E_k)_{k\in {\mathbb{N}}}$ of elements of ${\mathcal{B}}$ which is dense in ${\mathcal{B}}$. Passing to a subsequence of $(T_n)$, we may assume that for each $n$, $T_n$ almost fixes the set $B_n$, and that for any $k\leq n$ and any $\alpha \in F$, $T_n$ almost fixes the set $\alpha^{-1}(E_k\cap \alpha(B_n))$. Define $S_n\in [{\mathcal{S}}]$ such that $S_n=\alpha T_n\alpha^{-1}$ on the set $\alpha(B_n)$ outside its some subset of small measure, $S_n$ preserves $\alpha(B_n)$ for each $\alpha \in F$, and $S_n$ is the identity outside $\bigcup_{\alpha \in F}\alpha(B_n)$. In the same manner as in the proof of assertion (i), we can show that $(S_n)$ is central in $[{\mathcal{R}}]$. Since $S_n$ is equal to $T_n$ on most part of $B_n$, the measure of the set $\{ \, x\in X\mid S_nx\neq x\, \}$ is uniformly positive. Assertion (ii) follows.
An example for the Schmidt property {#subsec-counter-schmidt}
-----------------------------------
The following is a slight refinement of [@kec Theorem 29.10]:
\[prop-t\] Let $G$ be a countably infinite, discrete group with property (T) and let $G{\curvearrowright}(X, \mu)$ a free ergodic p.m.p. action. Suppose the following two conditions:
- For any non-trivial element of $G$, its conjugacy class in $G$ consists of at least two elements.
- For any $g\in G$ whose conjugacy class in $G$ is finite, the centralizer of $g$ in $G$ acts on $(X, \mu)$ ergodically.
Then the action $G{\curvearrowright}(X, \mu)$ is not Schmidt.
Let ${\mathcal{R}}{\coloneqq}{\mathcal{R}}(G{\curvearrowright}(X, \mu))$. Suppose toward a contradiction that there exists a non-trivial central sequence $(T_n)_n$ in $[{\mathcal{R}}]$. By Lemma \[lem-c\], we may assume that $T_nx\neq x$ for any $n$ and any $x\in X$. We set $G^*=G\setminus \{ e\}$. Let $G$ act on the space $G^*\times X$ by $g(h, x)=(ghg^{-1}, gx)$ for $g\in G$, $h\in G^*$ and $x\in X$. We then have the unitary representation of $G$ on the Hilbert space $\mathcal{H}=\ell^2(G^*)\otimes L^2(X)$. Let $P\colon \mathcal{H}\to \mathcal{H}$ be the orthogonal projection onto the subspace of $G$-invariant vectors. Let $\Omega$ denote the set of finite conjugacy classes in $G^*$. Each vector $\xi$ of $\mathcal{H}$ is uniquely written as $\xi =\sum_{g\in G^*}\delta_g\otimes \xi_g$, where $\delta_g$ is the Dirac function on $g$ and $\xi_g\in L^2(X)$. For $\omega \in \Omega$, we set $c_\omega(\xi)=|\omega|^{-1}\sum_{h\in \omega}\mu(\xi_h)$ and identify it with the constant function on $X$ of that value. By our second assumption, the equation $P\xi =\sum_{\omega \in \Omega}\sum_{g\in \omega}\delta_g\otimes c_\omega(\xi)$ holds.
\[claim-ineq\] Any element $T\in [{\mathcal{R}}]$ with $Tx\neq x$ for any $x\in X$ gives rise to the unit vector of $\mathcal{H}$, $\xi_T=\sum_{g\in G^*}\delta_g\otimes \xi_{T, g}$, where $\xi_{T, g}$ is the indicator function of the set $\{ T=g\}$, and this vector satisfies the inequality $\Vert \xi_T-P\xi_T\Vert^2\geq (1/2)\Vert P\xi_T\Vert^2$.
For simplicity, we set $\xi =\xi_T$ and $\xi_g=\xi_{T, g}$. For any $\omega \in \Omega$, we have $0\leq c_\omega(\xi)\leq |\omega|^{-1}\leq 1/2$ by the first assumption in Proposition \[prop-t\]. The desired inequality is obtained as follows: $$\begin{aligned}
\Vert \xi-P\xi\Vert^2&\geq \sum_{\omega \in \Omega}\sum_{g\in \omega}\Vert \xi_g-c_\omega(\xi)\Vert^2=\sum_{\omega \in \Omega}\sum_{g\in \omega}(\mu(\xi_g)-2c_\omega(\xi)\mu(\xi_g)+c_\omega(\xi)^2)\\
&=\sum_{\omega \in \Omega}|\omega|c_\omega(\xi)(1-c_\omega(\xi))\geq \frac{1}{2}\sum_{\omega \in \Omega}|\omega|c_\omega(\xi)\geq \frac{1}{2}\Vert P\xi \Vert^2.\qedhere\end{aligned}$$
Let $\xi_n$ denote the unit vector to which the automorphism $T_n$ gives rise as in Claim \[claim-ineq\]. Since $(T_n)_n$ is a central sequence in $[{\mathcal{R}}]$, the sequence $(\xi_n)_n$ is asymptotically $G$-invariant, that is, for any $g\in G$, we have $\Vert g\xi_n-\xi_n\Vert \to 0$ as $n\to \infty$. It follows from property (T) of $G$ that $\Vert \xi_n-P\xi_n\Vert \to 0$. This contradicts the inequality in Claim \[claim-ineq\].
\[prop-t-c\] Let $G$ be a countable discrete group with property (T) and $H$ a finite index, normal subgroup of $G$. Let $C$ be an infinite central subgroup of $H$ which is normal in $G$. Suppose that for any non-trivial element of $G$, its conjugacy class in $G$ either contains an element of $C$ and at least two elements or is infinite. Then there exists a free ergodic p.m.p. action $G{\curvearrowright}(X, \mu)$ which is not Schmidt and such that the restriction $H{\curvearrowright}(X, \mu)$ is ergodic and Schmidt.
Embed the group $C$ into an abelian compact group $K$, equip $K$ with the normalized Haar measure, and let $C$ act on $K$ by translation. Choose a section $s_0\colon H/C\to H$ of the quotient map. Pick representatives $g_1,\ldots, g_N$ of left cosets of $H$ in $G$, where $N$ is the index of $H$ in $G$. We then have the section $s\colon G/C\to G$ defined by $s(g_ib)=g_is_0(b)$ for $i\in \{ 1,\ldots, N\}$ and $b\in H/C$. Let $G$ act on the product space $X{\coloneqq}\prod_{G/C}K$ by the action co-induced from the action $C{\curvearrowright}K$. This is defined by $(gf)(b)=c^{-1}f(g^{-1}b)$ for $g\in G$, $f\in X$ and $b\in G/C$, where the element $c\in C$ is determined by the equation $s(g^{-1}b)c=g^{-1}s(b)$. Let $\mu$ be the probability measure on $X$ that is the product of the Haar measure on $K$. Since $H/C$ is infinite, the restriction $H{\curvearrowright}(X, \mu)$ is ergodic. Choose a sequence $(c_n)_n$ of non-trivial elements of $C$ such that for any $i\in \{ 1,\ldots, N\}$, the sequence $(g_i^{-1}c_ng_i)_n$ converges to the identity in $K$. It follows from the definition of the co-induced action that $(c_n)_n$ is central as a sequence in the full group $[{\mathcal{R}}(H{\curvearrowright}(X, \mu))]$. Indeed, for any $b\in H/C$, $c\in C$ and $i\in \{ 1,\ldots, N\}$, the equation $$c^{-1}s(g_ib)=c^{-1}g_is_0(b)=g_ig_i^{-1}c^{-1}g_is_0(b)=g_is_0(b)g_i^{-1}c^{-1}g_i=s(g_ib)g_i^{-1}c^{-1}g_i$$ holds, where the third equation holds since $C$ is normal in $G$ and thus $g_i^{-1}c^{-1}g_i\in C$. This implies that $c$ acts on the coordinate of $X$ whose index is $g_ib$ by adding $g_i^{-1}cg_i$. On the other hand, by Proposition \[prop-t\], the action $G{\curvearrowright}(X, \mu)$ is not Schmidt.
In the notation of Proposition \[prop-t-c\], we set ${\mathcal{R}}={\mathcal{R}}(G{\curvearrowright}(X, \mu))$ and define a cocycle $\alpha \colon {\mathcal{R}}\to G/H$ by $\alpha(gx, x)=gH$ for $g\in G$ and $x\in X$. The extension ${\mathcal{R}}_\alpha$ is an equivalence relation on $X\times G/H$ and its restriction to the subset $X\times \{ e\}$ is exactly the orbit equivalence relation ${\mathcal{R}}(H{\curvearrowright}(X, \mu))$. The conclusion of Proposition \[prop-t-c\] says that ${\mathcal{R}}_\alpha$ is Schmidt, while ${\mathcal{R}}$ is not. Thus the converse of Corollary \[cor-ext-c\] does not hold.
\[ex-matrix\] We give an example of groups $G$, $H$ satisfying the assumption in Proposition \[prop-t-c\]. We define $H$ as the subgroup of $\mathit{SL}_5({\mathbb{Z}})$ consisting of matrices of the form $$\begin{pmatrix}
1 & \ast & \ast \\
0 & h & \ast \\
0 & 0 & 1
\end{pmatrix},$$ where $h$ runs through all elements of $\mathit{SL}_3({\mathbb{Z}})$. This kind of groups appears in [@cor], and as mentioned in [@cor Proposition 2.7], the group $H$ has property (T). The center of $H$, denoted by $C$, consists of matrices such that any diagonal entry is one and the $(1, 5)$-entry is the only off-diagonal entry that is possibly non-zero. We set $$u=\begin{pmatrix}
-1 & 0 & 0 \\
0 & I_3 & 0 \\
0 & 0 & 1
\end{pmatrix},$$ where $I_3$ is the $3$-by-$3$ identity matrix, and define $G$ as the subgroup of $\mathit{GL}_5({\mathbb{Z}})$ generated by $H$ and $u$. The element $u$ normalizes $H$, and the index of $H$ in $G$ is $2$.
We claim that these groups $G$, $H$ satisfy the assumption of Proposition \[prop-t-c\], that is, for any non-trivial element of $G$, its conjugacy class in $G$ either contains an element of $C$ and at least two elements or is infinite. For any non-trivial element $c\in C$, we have $ucu^{-1}=c^{-1}$, and the conjugacy class of $c$ in $G$ consists of the two distinct elements, $c$ and $c^{-1}$.
We show that the other conjugacy class except for $\{ e\}$ is infinite. Pick $g, h\in H$ and write them as $$g=\begin{pmatrix}
1 & g_{12} & g_{13} \\
0 & g_{22} & g_{23} \\
0 & 0 & 1
\end{pmatrix}, \qquad
h=\begin{pmatrix}
1 & h_{12} & h_{13} \\
0 & h_{22} & h_{23} \\
0 & 0 & 1
\end{pmatrix}.$$ We first show that the centralizer of $gu$ in $H$ is of infinite index. Otherwise, for any matrix $h$ such that $h_{22}$ belongs to some finite index subgroup of $\mathit{SL}_3({\mathbb{Z}})$ and the other $h_{ij}$ is zero, $gu$ and $h$ would commute. It follows from $$\begin{aligned}
guh&=\begin{pmatrix}
-1 & g_{12} & g_{13} \\
0 & g_{22} & g_{23} \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
0 & h_{22} & 0 \\
0 & 0 & 1
\end{pmatrix}
=
\begin{pmatrix}
-1 & g_{12}h_{22} & g_{13} \\
0 & g_{22}h_{22} & g_{23} \\
0 & 0 & 1
\end{pmatrix} \ \text{and}\\
hgu&=\begin{pmatrix}
1 & 0 & 0 \\
0 & h_{22} & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
-1 & g_{12} & g_{13} \\
0 & g_{22} & g_{23} \\
0 & 0 & 1
\end{pmatrix}
=
\begin{pmatrix}
-1 & g_{12} & g_{13} \\
0 & h_{22}g_{22} & h_{22}g_{23} \\
0 & 0 & 1
\end{pmatrix}\end{aligned}$$ that $g_{22}=I_3$ and that $g_{12}$ and $g_{23}$ are zero. For any $h$ in some finite index subgroup of the center of $H$, $gu$ and $h$ also commute. The $(1, 5)$-entries of $guh$ and $hgu$ are $g_{13}-h_{13}$ and $g_{13}+h_{13}$. This is impossible. It turns out that the centralizer of $gu$ in $H$ is of infinite index in $H$, and the conjugacy class of $gu$ in $G$ is therefore infinite.
Similarly, comparing $gh$ and $hg$ for the matrix $h$ such that $h_{22}$ belongs to some finite index subgroup of $\mathit{SL}_3({\mathbb{Z}})$ and the other $h_{ij}$ are zero, we can show that if the centralizer of $g$ in $H$ is of finite index in $H$, then $g$ belongs to the center of $H$. The conjugacy class of any non-central element of $H$ in $G$ is therefore infinite. The proof that $G$ and $H$ satisfy the assumption of Proposition \[prop-t-c\] is completed.
An example for stability {#subsec-counter-stable}
------------------------
We construct an ergodic p.m.p. equivalence relation ${\mathcal{R}}$ and its finite-index subrelation ${\mathcal{S}}$ such that ${\mathcal{S}}$ is stable, but ${\mathcal{R}}$ is not stable. Let $X_0=\prod_{\mathbb{N}}{\mathbb{Z}}/ 2{\mathbb{Z}}$ be the compact group equipped with the normalized Haar measure $\mu_0$. Let $H_0=\bigoplus_{\mathbb{N}}{\mathbb{Z}}/2{\mathbb{Z}}$ be the subgroup of $X_0$, let $H_0$ act on $X_0$ by translation, and set ${\mathcal{R}}_0={\mathcal{R}}(H_0{\curvearrowright}(X_0, \mu_0))$. Choose a non-amenable group $\Gamma$ arbitrarily. We set $Y=\prod_\Gamma X_0$ and equip $Y$ with the product measure of $\mu_0$, denoted by $\nu$. Let $H_1{\coloneqq}\Gamma \times H_0$ act on $Y$ such that $\Gamma \times \{ e\}$ acts on $Y$ by the Bernoulli shift and $\{ e\} \times H_0$ acts on $Y$ diagonally. We set ${\mathcal{R}}_1={\mathcal{R}}(H_1{\curvearrowright}(Y, \nu))$ and set ${\mathcal{S}}={\mathcal{R}}_0\times {\mathcal{R}}_1$, which is an equivalence relation on the product space $(Z, \zeta){\coloneqq}(X_0\times Y, \mu_0\times \nu)$ and is also the orbit equivalence relation associated with the coordinatewise action $G{\coloneqq}H_0\times H_1{\curvearrowright}(Z, \zeta)$.
We define an automorphism $\sigma$ of $(Z, \zeta)$ by $$\sigma(x, (y_\gamma)_{\gamma \in \Gamma})=(x, (xy_\gamma)_{\gamma \in \Gamma})$$ for $x\in X_0$ and $(y_\gamma)_\gamma \in Y$. We claim that $\sigma$ is an automorphism of ${\mathcal{S}}$. We first show that $\sigma$ is equivariant under the action of $H_1$, where $H_1$ is identified with the subgroup $\{ e\} \times H_1$ of $G$. Indeed, for any $(x, (y_\gamma)_\gamma)\in Z$ and $(\delta, h)\in H_1=\Gamma \times H_0$, we have $(\delta, h)(y_\gamma)_\gamma =(hy_{\delta^{-1}\gamma})_\gamma$. We also have $\sigma((\delta, h)(x, (y_\gamma)_\gamma))=(x, (xhy_{\delta^{-1}\gamma})_\gamma)=(\delta, h)\sigma(x, (y_\gamma)_\gamma)$. We therefore obtain the desired equivariance of $\sigma$ and then $\sigma({\mathcal{I}}_0\times {\mathcal{R}}_1)={\mathcal{I}}_0\times {\mathcal{R}}_1$, where ${\mathcal{I}}_0$ denotes the trivial equivalence relation on $(X_0, \mu_0)$. We next show that $\sigma({\mathcal{R}}_0\times {\mathcal{I}}_1)\subset {\mathcal{S}}$, where ${\mathcal{I}}_1$ denotes the trivial equivalence relation on $(Y, \nu)$. For any $(x, (y_\gamma)_\gamma)\in Z$ and $h\in H_0$, we have $\sigma(x, (y_\gamma)_\gamma)=(x, (xy_\gamma)_\gamma)$, which is ${\mathcal{S}}$-equivalent to $(hx, (hxy_\gamma)_\gamma)=\sigma(hx, (y_\gamma)_\gamma)$. The claim follows.
Since any non-trivial element of $X_0$ is of order $2$, the automorphism $\sigma$ is also of order $2$. Let ${\mathcal{R}}{\coloneqq}{\mathcal{S}}\rtimes \langle \sigma \rangle$ be the equivalence relation generated by ${\mathcal{S}}$ and $\sigma$, which contains ${\mathcal{S}}$ as a subrelation of index $2$.
The equivalence relation ${\mathcal{S}}={\mathcal{R}}_0\times {\mathcal{R}}_1$ is clearly stable. We show that ${\mathcal{R}}$ is not stable. Otherwise, by Theorem \[thm-main\], there would exist a pre-stability sequence $(T_n, A_n)$ for ${\mathcal{R}}$ with $T_n\in [{\mathcal{S}}]$. Since the action $G{\curvearrowright}(Y, \nu)$ has stable spectral gap, the unitary representation $G{\curvearrowright}L^2(X_0)\otimes L^2_0(Y)$ has spectral gap. We may therefore assume that $A_n$ is of the form $A_n=\bar{A}_n\times Y$ for some Borel subset $\bar{A}_n\subset X_0$. Furthermore by Lemma \[lem-specgap-product\], passing to a subsequence of $(T_n, A_n)$, we may assume that for any $n$, there exist finitely many $g_1, \ldots, g_m\in G$, a Borel subset $X'\subset X_0$ and its Borel partition $X'=\bigsqcup_{i=1}^mX_i$ such that $\mu_0(X')>1-1/n$ and $\nu(A_{g_i, x})>1-1/n$ for any $x\in X_i$ and any $i\in \{ 1,\ldots, m\}$, where we set $A_{g, x}=\{ \, y\in Y\mid T_n(x, y)=g(x, y)\, \}$ for $g\in G$ and $x\in X_0$. Passing to a subsequence of $(T_n, A_n)$, we may assume that $\zeta (W)>1-1/n$, where we set $$W=\{ \, z\in Z\mid (\sigma \circ T_n)(z)=(T_n\circ \sigma)(z)\, \}.$$ Let $I$ be the set of all $i\in \{ 1,\ldots, m\}$ such that the set $W\cap A_{g_i}\cap \sigma A_{g_i} \cap (X_i\times Y)$ has positive measure, where we set $A_g=\{ \, z\in Z\mid T_n(z)=gz\, \}$ for $g\in G$. Then $\zeta(\bigsqcup_{i\in I}(A_{g_i} \cap \sigma A_{g_i}\cap (X_i\times Y)))>1-4/n$. If $i\in I$, then $g_i$ belongs to $\{ e\} \times H_1$. Indeed, setting $g_i=(h_i, (\gamma_i, h_i'))\in H_0\times H_1$ and taking a point $(x, (y_\gamma)_\gamma)$ from the set $W\cap A_{g_i} \cap \sigma A_{g_i}\cap (X_i\times Y)$, we have $$(\sigma\circ T_n)(x, y)=\sigma(h_ix, (h_i'y_{\gamma_i^{-1}\gamma})_\gamma )=(h_ix, (h_ixh_i'y_{\gamma_i^{-1}\gamma})_\gamma)$$ and $$(T_n\circ \sigma)(x, y)=T_n(x, (xy_\gamma)_\gamma)=(h_ix, (h_i'xy_{\gamma_i^{-1}\gamma})_\gamma).$$ Since these two elements are equal, we have $h_i=e$. Since $H_1$ acts on $X_0$ trivially and the set $A_n$ is of the form $A_n=\bar{A}_n\times Y$, the transformation $T_n$ sends $A_n\cap A_{g_i}\cap \sigma A_{g_i}\cap (X_i\times Y)$ into $A_n$. Hence $A_n\setminus T_n^{-1}A_n\subset Z\setminus (\bigsqcup_{i\in I}(A_{g_i} \cap \sigma A_{g_i}\cap (X_i\times Y)))$, and the measure of the set in the right hand side is less than $4/n$. This contradicts that $(T_n, A_n)$ is a stability sequence.
We note that $G$ and $\sigma$ generate the semi-direct product $G\rtimes \langle \sigma \rangle$ such that $\sigma$ acts on $G$ by $\sigma(h, (\gamma, h'))=(h, (\gamma, hh'))$ for $h\in H_0$ and $(\gamma, h')\in H_1=\Gamma \times H_0$. Then ${\mathcal{R}}$ is the orbit equivalence relation generated by the action of $G\rtimes \langle \sigma \rangle$. We also note that ${\mathcal{R}}$ is not stable, but is Schmidt because the subgroup $\{ e\} \times (\{ e \} \times H_0)$ of $G$ is central in $G\rtimes \langle \sigma \rangle$ and the elements $h_n\in H_0=\bigoplus_{\mathbb{N}}{\mathbb{Z}}/2{\mathbb{Z}}$ such that the coordinate indexed by $n$ is $1$ and the other coordinates are $0$ form a central sequence in the full group of ${\mathcal{R}}$.
Miscellaneous examples {#sec-ex}
======================
Orbitally inner amenable groups
-------------------------------
Recall that a countable group is said to be orbitally inner amenable if it admits a free ergodic p.m.p. action whose orbit equivalence relation is inner amenable (Subsection \[subsec-schmidt\]). We provide several examples of such groups and free ergodic p.m.p. actions whose orbit equivalence relations are inner amenable, but not Schmidt. By Corollary \[cor:cpctext\], any countable, residually finite, inner amenable group is orbitally inner amenable. More generally, we obtain the following:
\[prop:resfinex\] Let $G$ be a countable group with normal subgroup $N$. Assume that there exists a chain $N=N_0> N_1>\cdots$ of finite index subgroups of $N$ with $\bigcap _i N_i =\{ e\}$, and with each $N_i$ normal in $G$. Assume furthermore that there exists a diffuse conjugation invariant mean $\bm{m}$ on $G$ with $\bm{m}(N)=1$. Then $G$ is orbitally inner amenable.
Let $\check{\bm{m}}$ be the mean on $G$ defined by $\check{\bm{m}}(D)=\bm{m}(D^{-1})$ for $D\subset G$. Then the convolution $\bm{n}_0{\coloneqq}\check{\bm{m}}\ast \bm{m}$ is also diffuse and conjugation-invariant with $\bm{n}_0(N)=1$. For any finite index subgroup $L$ of $N$, by finite additivity, we have $\sum _{hL \in N/L}\bm{m}(hL) =1$ and hence $$\bm{n}_0(L) = \int _N \bm{m}(k^{-1}L) \, d\check{\bm{m}}(k) = \sum _{hL\in N/L}\int _{hL}\bm{m}(kL)\, d\bm{m}(k) = \sum _{hL \in N/L}\bm{m}(hL)^2 > 0.$$ For each $i\in {\mathbb{N}}$, since $N_i$ is normal in $G$, the normalized restriction of $\bm{n}_0$ to $N_i$, denoted by $\bm{n}_i$, is a diffuse, conjugation-invariant mean on $G$ with $\bm{n}_i(N_i)=1$. Let $\bm{n}$ be any weak${}^*$-cluster point of $(\bm{n}_i ) _{i\in {\mathbb{N}}}$ in the space of means on $G$. Then $\bm{n}$ is a diffuse, conjugation-invariant mean on $G$ with $\bm{n}(N_i)=1$ for any $i\in {\mathbb{N}}$. Let $N\curvearrowright (K,\mu _K )$ be the profinite action associated to the chain $(N_i )_{i\in {\mathbb{N}}}$, i.e., the inverse limit of the finite actions $N\curvearrowright (N/N_i , \mu _{N/N_i})$ with each $\mu _{N/N_i}$ normalized counting measure on $N/N_i$. We naturally view $N$ as a subgroup of the profinite group $K$. Observe that for any open neighborhood $U$ of the identity of $K$, we have $\bm{n}(N\cap U ) =1$. Therefore, for any Borel subset $B\subset K$, we have $$\begin{aligned}
\label{eqn:Kint0}
\int _{N } \mu _K (hB\bigtriangleup B ) \, d\bm{n}(h) = 0.\end{aligned}$$ Let $G\curvearrowright (X,\mu )$ be the action co-induced from the action $N\curvearrowright (K,\mu _K)$. This is a free ergodic p.m.p. action of $G$. We set $\mathcal{R}={\mathcal{R}}(G{\curvearrowright}(X, \mu))$. Since each $N_i$ is normal in $G$, the action $N{\curvearrowright}(X,\mu )$ is isomorphic to the product of countably many copies of the action $N{\curvearrowright}(K,\mu _K)$. Therefore, implies that for any Borel subset $A\subset X$, we have $$\begin{aligned}
\label{eqn:Xint0}
\int _{N } \mu (hA\bigtriangleup A ) \, d\bm{n}(h) = 0.\end{aligned}$$ Define a mean $\bm{n}_{\mathcal{R}}$ on $(\mathcal{R}, \mu)$ by $$\bm{n}_{\mathcal{R}}(D) = \int _{N} \mu ( \{ \, x\in X \mid (hx,x) \in D \, \} ) \, d\bm{n}(h)$$ for a Borel subset $D\subset \mathcal{R}$. The mean $\bm{n}_{\mathcal{R}}$ is diffuse since $\bm{n}$ is diffuse, and $\bm{n}_{\mathcal{R}}$ is balanced by . Since $\bm{n}$ is invariant under conjugation by $G$, and $\mu$ is $G$-invariant, we have $\bm{n}_{\mathcal{R}}(D^{\phi _g})=\bm{n}_{\mathcal{R}}(D)$ for any Borel subset $D\subset \mathcal{R}$ and any $g\in G$. By Remark \[rem:suffice\], $\bm{n}_{\mathcal{R}}$ is a mean witnessing that $\mathcal{R}$ is inner amenable.
\[prop:wreath\] Let $H$ be a non-trivial countable group, let $K$ be a countable group acting on a countable set $V$, and assume that there exists a diffuse $K$-invariant mean on $V$. Then the generalized wreath product $G{\coloneqq}H\wr_V K$ is orbitally inner amenable.
By assumption, there exists a sequence $(Q_n)_{n\in {\mathbb{N}}}$ of finite subsets of $V$ satisfying
1. $|k\cdot Q_n \bigtriangleup Q_n |/|Q_n|\rightarrow 0$ for any $k\in K$, and
2. $1_{Q_n}(v)\rightarrow 0$ for any $v \in V$.
We set $N=\bigoplus _V H$ and view each element of $N$ as a function $z\colon V\rightarrow H$ whose support $\mathrm{supp}\, (z)= \{ \, v\in V \mid z(v)\neq e_H \, \}$ is finite. Let $K$ act on $N$ by $(k\cdot z )(v) =z(k^{-1}\cdot v )$, and identify $N$ and $K$ with subgroups of $G$, so that $G=NK$ is the internal semi-direct product of $N$ with $K$ and $kzk^{-1}=k\cdot z$ for $k\in K$ and $z\in Z$.
Let $H\curvearrowright ^{\alpha _0}(Y_0,\nu _0 )$ be a free ergodic p.m.p. action. We set $(Y,\nu ) = (Y_0^V,\nu _0 ^V )$ and let $N\curvearrowright ^{\alpha}(Y,\nu )$ be the componentwise action given by $(z^{\alpha}y)(v)= z(v)^{\alpha _0} y(v)$ for $z\in N$, $y\in Y$ and $v\in V$. The action $N\curvearrowright ^{\alpha}(Y,\nu )$ is free and ergodic, and naturally extends to an action of $G$, but this extension may not be free in general. Instead, we set $(X,\mu ) = (Y^K,\nu ^K )$ and let $G\curvearrowright ^{\beta} (X,\mu )$ be the action co-induced from the action $N{\curvearrowright}^\alpha (Y, \nu)$. Explicitly, $$((zk)^\beta x )(k_0)= (k_0^{-1}\cdot z)^{\alpha} x(k^{-1}k_0)$$ for $k,k_0\in K$, $z\in N$ and $x\in X$. This is a free ergodic p.m.p. action of $G$. Since $N$ is normal in $G$, the action $N{\curvearrowright}^\beta (X, \mu)$ is isomorphic to the product of countably many copies of the action $N{\curvearrowright}^\alpha (Y, \nu)$.
For $v\in V$ and $h\in H$, let $z_{v,h}\in N$ be the element determined by $\mathrm{supp}\, (z_{v,h}) = \{ v\}$ and $z_{v,h}(v)=h$. Fix any non-trivial $h _0 \in H$, and for each $n\in {\mathbb{N}}$, set $F_n = \{ \, z_{v,h _0} \mid v\in Q_n \, \}$. Then conditions (i) and (ii) imply that $|F_n^g \bigtriangleup F_n |/|F_n|\rightarrow 0$ and $1_{F_n}(g)\rightarrow 0$ for any $g\in G$. Moreover, condition (ii) implies that if $C$ is any subset of $Y=Y_0^V$ that depends on only finitely many $V$-coordinates, then for any large enough $n$, $C$ is independent of the coordinates in $Q_n$, and hence for every $z\in F_n$, we have $z^{\alpha}C = C$. It follows that for any Borel subset $B\subset Y$, we have $\lim _{n\rightarrow \infty}\sup _{z\in F_n}\nu (z^{\alpha}B\bigtriangleup B ) = 0$. Since the action $N{\curvearrowright}^\beta (X,\mu )$ is isomorphic to the product of countably many copies of the action $N{\curvearrowright}^\alpha (Y, \nu)$, it follows that for any Borel subset $A\subset X$, we have $$\label{eqn:betalim}
\lim _{n\rightarrow \infty}\sup _{z\in F_n}\mu (z^{\beta}A\bigtriangleup A ) = 0.$$
Let ${\mathcal{R}}$ denote the orbit equivalence relation of the action $G{\curvearrowright}^\beta (X, \mu)$. For each $n\in {\mathbb{N}}$, define a function $\xi _n$ on ${\mathcal{R}}$ by $\xi_n(g^\beta x, x)= 1_{F_n}(g)/|F_n|$ for $g\in G$ and $x\in X$. Then the sequence $(\xi _n )_{n\in {\mathbb{N}}}$ is balanced by . Moreover, the sequence $(\xi _n)$ inherits from $(F_n)$ the properties of being asymptotically diffuse and asymptotically invariant under conjugation by elements of $G$. Therefore, by Remark \[rem:suffice\], any weak${}^*$-cluster point of $(\xi _n )_{n\in {\mathbb{N}}}$ will be a mean witnessing that $\mathcal{R}$ is inner amenable.
In the proof of Proposition \[prop:wreath\], observe that, since $K\curvearrowright ^{\beta} (X,\mu )$ is a Bernoulli shift, if $K$ is non-amenable, then the action $G\curvearrowright ^{\beta} (X,\mu )$ has stable spectral gap. Thus, if in addition $K$ is finitely generated and center-less, and $K$ acts on $V$ with infinite orbits, then the centralizer of $K$ in $G$ is trivial, and hence Corollary \[cor:specgap\] shows that the equivalence relation $\mathcal{R}(G\curvearrowright ^{\beta} (X,\mu ))$ is not Schmidt. Examples of $K$ and $V$ satisfying those conditions are found in [@gm], and for such $K$ and $V$, ${\mathcal{R}}(G{\curvearrowright}^\beta (X, \mu))$ is inner amenable, but not Schmidt.
Product groups
--------------
Let $L$ be a non-amenable group and let $H$ be a countable group. We set $G=L\times H$ and identify $L$ and $H$ with their respective images, $L\times \{ e\}$ and $\{ e\} \times H$, in $G$. Let $H\curvearrowright (X_0,\mu _0)$ be a free ergodic p.m.p. action, and let $G\curvearrowright (X,\mu ){\coloneqq}(X_0^L, \mu_0^L)$ be the action such that $H$ acts diagonally and $L$ acts by Bernoulli shift. Explicitly the action $G{\curvearrowright}X$ is given by $(lh\cdot x)(k) = h\cdot x(l^{-1}k)$ for $l,k\in L$, $h\in H$ and $x\in X$. This is a free ergodic action and set ${\mathcal{R}}={\mathcal{R}}(G{\curvearrowright}(X, \mu))$. We discuss inner amenability and the Schmidt property of ${\mathcal{R}}$. This kind of an example is considered in [@cj] in another context.
\[prop-product\] In the above notation,
1. if $H$ is inner amenable and the action $H{\curvearrowright}(X_0, \mu_0)$ is profinite, then ${\mathcal{R}}$ is inner amenable.
2. If $H$ is finitely generated and the center of $H$ is trivial, then the action $G{\curvearrowright}(X, \mu)$ is not Schmidt.
3. If the action $H{\curvearrowright}(X_0, \mu_0)$ is mildly mixing, then for any ergodic p.m.p. action $G{\curvearrowright}(Y, \nu)$, the diagonal action $G{\curvearrowright}(X\times Y, \mu \times \nu)$ is not inner amenable.
In particular, if we let the group $H$ and the action $H{\curvearrowright}(X_0, \mu_0)$ satisfy the assumptions in assertions (i) and (ii) simultaneously, then the action $G{\curvearrowright}(X, \mu)$ is inner amenable, but not Schmidt. For example, let $H$ be the lamplighter group $H=({\mathbb{Z}}/2{\mathbb{Z}}) \wr {\mathbb{Z}}$. Then $H$ is finitely generated, residually finite, inner amenable group whose center is trivial.
The action $G{\curvearrowright}(X, \mu)$ is isomorphic to the action co-induced from the action $H{\curvearrowright}(X_0, \mu_0)$. The proof of Proposition \[prop:resfinex\] (with $H$ playing the role of $N$) therefore shows that ${\mathcal{R}}$ is inner amenable under the assumption in assertion (i).
We prove assertion (ii). The action $L{\curvearrowright}(X,\mu )$ is isomorphic to a Bernoulli shift. Since $L$ is non-amenable, the equivalence relation $\mathcal{R}_L{\coloneqq}\mathcal{R}(L\curvearrowright (X,\mu ))$ is not inner amenable, and the action $L\curvearrowright (X,\mu )$ has stable spectral gap. Therefore, the action $G \curvearrowright (X,\mu )$ has stable spectral gap as well. Since $H$ is finitely generated and the center of $H$ is trivial, Corollary \[cor:specgap\] shows that if $(\phi _n )$ is any central sequence in $[\mathcal{R}]$, then there exists a sequence $(l_n)$ in $L$ such that $\mu ( \{ \, x \in X \mid \phi_n (x)= l_n x \, \} )\rightarrow 1$. Since $[\mathcal{R}_L]$ has no non-trivial central sequence, we must have $l_n=e$ for any large enough $n$, and hence the sequence $(\phi _n)$ is asymptotically trivial.
Assertion (iii) is a direct consequence of Proposition \[prop-diagonal-action\].
We keep the notation in the beginning of this subsection. The assumption in Proposition \[prop-product\] (ii) on the center of $H$ is necessary. Indeed, let $H={\mathbb{Z}}$ and let $X_0={\mathbb{Z}}_2$, the group of 2-adic integers, with the normalized Haar measure $\mu_0$. Let the action $H{\curvearrowright}(X_0, \mu_0)$ be given by translation. Then the action $G{\curvearrowright}(X, \mu)$ is not stable (since it has stable spectral gap), but is Schmidt. Indeed, the element $2^n$ of ${\mathbb{Z}}$ for $n\in {\mathbb{N}}$ forms a central sequence in $[{\mathcal{R}}]$ (see also [@kec Section 29 (C)]).
The assumption in Proposition \[prop-product\] (iii) is also necessary. Indeed, keeping the notation in the last paragraph, we set $(Y, \nu)=(X_0, \mu_0)$ and let $G$ act on $(Y, \nu)$ through the projection from $G$ onto ${\mathbb{Z}}$ and the translation by ${\mathbb{Z}}$ on ${\mathbb{Z}}_2$. The diagonal action $G{\curvearrowright}^\alpha (X\times Y, \mu \times \nu)$ is then stable: Let ${\mathcal{R}}_\alpha$ denote the orbit equivalence relation of this action. For $n\in {\mathbb{N}}$, let $T_n\in [{\mathcal{R}}_\alpha]$ be the element $2^n$ of ${\mathbb{Z}}<G$, and let $A_n$ be the subset of $X\times Y$ consisting of all $(x, y)\in X\times Y$ with $y_n=0$ when $y$ is written as $y=\sum_{k=0}^{\infty}2^ky_k\in {\mathbb{Z}}_2$. Then the sequence $(T_n, A_n)_n$ is a pre-stability sequence for ${\mathcal{R}}_\alpha$.
We construct the following interesting examples through actions of product groups:
\[prop:noinvol\] There exists an ergodic discrete p.m.p. equivalence relation whose full group admits a non-trivial central sequence, but admits no non-trivial central sequence consisting of periodic transformations with uniformly bounded periods.
Furthermore there exists an ergodic discrete p.m.p. equivalence relation whose full group admits a non-trivial central sequence, but admits no non-trivial central sequence consisting of aperiodic transformations.
In what follows, for a group $G$, its center is denoted by $Z(G)$. Let $L_0$ and $L_1$ be a finitely generated, non-amenable group with trivial center. Let $H_0$ and $H_1$ be finitely generated groups, and suppose that $Z(H_0)$ is torsion-free and infinite (e.g., take $H_0={\mathbb{Z}}$) and that $Z(H_1)$ is isomorphic to the direct sum of infinitely many copies of ${\mathbb{Z}}/2{\mathbb{Z}}$ (e.g., by [@hall Theorem 6], any countable abelian group is isomorphic to the center of some 2-generated central-by-metabelian group). For each $i=0,1$, we set $G_i=L_i\times H_i$, so that $Z(G_i) = \{ e\} \times Z(H_i)$. Since $Z(H_i)$ is infinite, we have a free ergodic p.m.p. action $H_i\curvearrowright (Y_i, \nu _i )$ such that the image of $Z(H_i)$ in $\mathrm{Aut}(Y_i,\nu _i )$ is precompact (see the proof of [@td Theorem 15] or Example \[ex-infinite-center\] below). Set $(X_i,\mu _i ) = (Y_i^{L_i}, \nu _i ^{L_i})$ and let $G_i$ act on $(X_i,\mu _i )$ via $((l,h)\cdot x)(k)=h\cdot x(l^{-1}k)$ for $l,k\in L_i$, $h\in H_i$ and $x\in X_i$. Let ${\mathcal{R}}_{G_i}$ denote the orbit equivalence relation of this action $G_i{\curvearrowright}(X_i, \mu_i)$.
The image of $Z(G_i)=\{ e\} \times Z(H_i)$ in $\mathrm{Aut}(X_i,\mu _i)$ is precompact, so the full group $[\mathcal{R}_{G_i}]$ admits a non-trivial central sequence consisting of non-trivial elements of $Z(G_i)$ which converge to the identity transformation in $\mathrm{Aut}(X_i,\mu _i )$. The action $G_i{\curvearrowright}(X_i,\mu _i )$ is free and has stable spectral gap since $L_i$ is non-amenable and $L_i\times \{ e\}$ acts by Bernoulli shift on $(X_i,\mu _i)$. If $(T_{n,i})_{n\in {\mathbb{N}}}$ is any central sequence of elements in $[\mathcal{R}_{G_i}]$, then, since $G_i$ is finitely generated, Corollary \[cor:specgap\] shows that there exists a sequence $e\neq c_{i,n} \in Z(G_0)$, $n\in {\mathbb{N}}$, such that $\mu _i ( \{ \, x \in X_i \mid T_{n,i}(x) = c_{n,i}\cdot x \, \} ) \rightarrow 1$. Therefore, since $Z(G_0)$ is torsion-free, the sequence $(T_{n,0})$ cannot consist of involutions, or even of periodic transformations with uniformly bounded periods. Likewise, since every non-trivial element of $Z(G_1)$ has order 2, the sequence $(T_{n,1})$ cannot consist of aperiodic transformations.
Groups with Schmidt or stable actions
-------------------------------------
Some inner amenable groups considered in various contexts are shown to admit a Schmidt or stable action.
\[ex-infinite-center\] Let $G$ be a countable group with an infinite central subgroup $C$. A Schmidt action of $G$ is constructed in the proof of [@td Theorem 15] and Proposition \[prop-t-c\], obtained as follows: Embed $C$ into a compact abelian group $K$, equip $K$ with the normalized Haar measure, and let $C$ act on $K$ by translation. Let $G{\curvearrowright}X=\prod_{G/C}K$ be the action co-induced from the action $C{\curvearrowright}K$. Then $C$ acts on the product $X=\prod_{G/C}K$ diagonally since $C$ is central. Therefore if $(c_n)_n$ is any sequence of non-trivial elements of $C$ converging to the identity in $K$, we have $\mu(c_nA\bigtriangleup A)\to 0$ for any cylindrical set $A$ of $X$, where $\mu$ is the probability measure on $X$ that is the product of the Haar measure on $K$, and this also holds for any Borel subset $A$ of $X$.
Let $H$ be a countable group with property (T) and suppose that $H$ has a central element $a$ of infinite order. For non-zero integers $p$, $q$ with $|p|\neq |q|$, define $G$ to be the HNN extension $G=\langle \, H,\, t\mid ta^pt^{-1}=a^q\, \rangle$. The group $G$ is ICC, inner amenable and not stable ([@kida-inn]).
We construct a Schmidt action of $G$. Let $C$ be the group generated by $a$. Embed $C$ into a compact abelian group $K$ and construct the co-induced action $G{\curvearrowright}X=\prod_{G/C}K$, with the probability measure $\mu$ on $X$, as in Example \[ex-infinite-center\]. Let $N$ be the kernel of the homomorphism from $G$ onto ${\mathbb{Z}}$ sending each element of $H$ to $0$ and $t$ to $1$. We set $a_n=a^{p^nq^n}$. The sequence $(a_n)_n$ is asymptotically central in $N$, i.e., for any $g\in N$, for any sufficiently large $n$, we have $a_ng=ga_n$. Therefore the sequence of elements $a_ma_n^{-1}$ running through some $m>n$ converges to the identity in $K$ and is central in the full group $[N{\curvearrowright}(X, \mu)]$. Take a free ergodic p.m.p. action $G/N{\curvearrowright}(Y, \nu)$ and let $G$ act on $X\times Y$ diagonally. As noted in [@td Remark 7.4], using the above non-trivial central sequence in $[N{\curvearrowright}(X, \mu)]$ and the hyperfiniteness of the equivalence relation ${\mathcal{R}}(G/N{\curvearrowright}(Y, \nu))$, we can construct a non-trivial central sequence in $[G{\curvearrowright}(X\times Y, \mu \times \nu)]$.
For a group $G$, its *FC-radical* $R$ is defined as the set of elements whose centralizer in $G$ is of finite index in $G$, or equivalently, whose conjugacy class in $G$ is finite. Then $R$ is a normal subgroup of $G$. Popa-Vaes [@pv Theorem 6.4] show that any countable, residually finite group $G$ whose FC-radical is not virtually abelian is *McDuff*, i.e., $G$ admits a free ergodic p.m.p. action such that the associated von Neumann algebra tensorially absorbs the hyperfinite $\textrm{II}_1$ factor. A property (T) group satisfying that assumption does exist ([@ershov]). It provides an example of a non-stable McDuff group. Based on this group, Deprez-Vaes [@dv Section 3] constructed an ICC, non-stable and McDuff group.
Popa-Vaes’ construction in [@pv Theorem 6.4] shows that any residually finite group whose FC-radical is infinite has the Schmidt property. In fact, in the proof in those papers, it is shown that the sequence $(v_h)_h$ in the symbol of that proof, with $h$ running through some elements of $G$, is a non-trivial central sequence in the full group. We note that in [@pv Theorem 6.4], while the FC-radical of the group $G$ is assumed to be not virtually abelian, this condition is required for the action to be McDuff. For the action to have a non-trivial central sequence in its full group, it suffices to assume that the FC-radical of $G$ is infinite.
Similarly the group $G$ of Deprez-Vaes [@dv Section 3] also has the Schmidt property. In fact, the action of $G$ they constructed admits the central sequence $(v_h)_h$ in the symbol of their proof.
Let $A$ be a countably infinite, amenable group. Let $H$ be a countable group acting on $A$ by automorphisms and suppose that any $H$-orbit in $A$ is finite. This condition is satisfied if $A$ is an increasing union of its finite subgroups invariant under the action of $H$ (e.g., the group $\mathit{SL}_n({\mathbb{Z}})$ naturally acts on $\bigoplus_{\mathbb{N}}({\mathbb{Z}}/2{\mathbb{Z}})^n$ and on $({\mathbb{Z}}[1/2]/{\mathbb{Z}})^n$ with this condition satisfied). Let $G=H\ltimes A$ be the semi-direct product. Then the FC-radical of $G$ contains $A$ and is hence infinite. We present two constructions of a free ergodic p.m.p. action of $G$ which is stable. It is remarkable that for the first stable action of $G$, its restriction to $A$ is ergodic (or rather mixing), while it is not ergodic for the second one.
**The first construction.** We set $X=[0, 1]^A$ and let $\mu$ be the probability measure on $X$ that is the product of the Lebesgue measure on $[0, 1]$. The group $G$ acts on $A$ by affine transformations: $(h, a)\cdot b=h(ab)$ for $h\in H$ and $a, b\in A$. This action induces the action $G{\curvearrowright}(X, \mu)$, which is p.m.p. and ergodic, but is not necessarily free (e.g., if the action of $H$ on $A$ is not faithful). We show that the groupoid $G\ltimes (X, \mu)$ is stable, i.e., it absorbs the ergodic p.m.p. hyperfinite equivalence relation on a non-atomic probability measure space under the direct product of groupoids. This is enough for $G$ to admit a free ergodic p.m.p. action which is stable, thanks to [@kida-srt Theorem 1.4].
Let $K$ be the closure of the image of $H$ in the automorphism group of $A$, ${\mathrm{Aut}}(A)$, where ${\mathrm{Aut}}(A)$ is equipped with the compact-open topology. By the assumption that any $H$-orbit in $A$ is finite, the group $K$ is compact. We also have the p.m.p. action $K\ltimes A{\curvearrowright}(X, \mu)$ induced by the affine action of $K\ltimes A$ on $A$.
\[claim-kafree\] The action $K\ltimes A{\curvearrowright}(X, \mu)$ is essentially free.
Suppose that we have $x\in X$, $k\in K$ and $a\in A$ such that $(k, a)x=x$, which means that $x((k, a)^{-1}b)=x(b)$ for any $b\in A$. Letting $b=e$, we have $x(a^{-1})=x(e)$, and the set of all $x$ satisfying this equation is null unless $a=e$. We may therefore assume that $a=e$, and then we have $x(k^{-1}b)=x(b)$ for any $b\in A$. Since the set $\{ \, x\in X\mid x(b_1)=x(b_2)\, \}$ is null for any distinct $b_1, b_2\in A$, the equation $x(k^{-1}b)=x(b)$ for any $b\in A$ does not hold for almost every $x\in X$ unless $k=e$.
Let $Y=X/K$ be the quotient space by the action of $K$ on $X$, which is a standard Borel space since $K$ is compact. Let $p\colon X\to Y$ be the projection, which is injective on almost every $A$-orbit in $X$. Indeed, if $a\in A$ and $x\in X$ satisfy $p(ax)=p(x)$, then $(k, a)x=x$ for some $k\in K$, but by Claim \[claim-kafree\], this does not occur for any point $x$ in a conull set unless $a=e$.
Let ${\mathcal{R}}$ be the orbit equivalence relation generated by the action $K\ltimes A{\curvearrowright}(X, \mu)$. Then the measured equivalence relation ${\mathcal{S}}$ on $(Y, \nu)$ is induced, where we set $\nu =p_*\mu$. In fact, this is given by the quotient of the coordinatewise action of $K\times K$ on ${\mathcal{R}}$. We have the induced map $p\colon {\mathcal{R}}\to {\mathcal{S}}$ and set ${\mathcal{R}}_1={\mathcal{R}}(A{\curvearrowright}(X, \mu))$. The restriction $p\colon {\mathcal{R}}_1\to {\mathcal{S}}$ induces a bijection from $({\mathcal{R}}_1)_x$ onto ${\mathcal{S}}_{p(x)}$ for almost every $x\in X$. Since $A$ is amenable, the equivalence relation ${\mathcal{R}}_1$ is amenable, and so is ${\mathcal{S}}$, which follows by the characterization of amenability in terms of the fixed point property for affine actions ([@ar Theorem 4.2.7]) or existence of invariant states ([@ar Definition 3.2.8], [@kerr-li Definition 4.57 (i)]).
We have an action of ${\mathcal{S}}$ on the fibered space $X$ with respect to $p$: For $(y_2, y_1)\in {\mathcal{S}}$ and $x\in X$ with $p(x)=y_1$, we define a point $(y_2, y_1)\cdot x$ as the unique point of $p^{-1}(y_2)$ that is ${\mathcal{R}}_1$-equivalent to $x$. Then this action of ${\mathcal{S}}$ and the action of $K$ commute in the sense that for any $(y_2, y_1)\in {\mathcal{S}}$, $k\in K$ and $x\in X$, we have $k((y_2, y_1)\cdot x)=(y_2, y_1)\cdot (kx)$. This holds because $K$ normalizes $A$.
We construct a cocycle $\alpha \colon {\mathcal{S}}\to K$. Choose a Borel section $q\colon Y\to X$ of the quotient map $p\colon X\to Y$ ([@kec-set Theorem 12.16]). For $(y_2, y_1)\in {\mathcal{S}}$, setting $x_2=(y_2, y_1)\cdot q(y_1)$, we define an element $\alpha(y_2, y_1)$ of $K$ by the equation $\alpha(y_2, y_1)x_2=q(y_2)$. Then the map $\alpha$ is a cocycle. Indeed, for $(y_3, y_2), (y_2, y_1)\in {\mathcal{S}}$, setting $x_3=(y_3, y_1)\cdot q(y_1)$, we have $$\alpha(y_3, y_2)\alpha(y_2, y_1)x_3=\alpha(y_3, y_2)\alpha(y_2, y_1)(y_3, y_1)\cdot q(y_1)=\alpha(y_3, y_2)(y_3, y_2)\cdot q(y_2)=q(y_3).$$ By the definition of $\alpha$, this implies that $\alpha(y_3, y_2)\alpha(y_2, y_1)=\alpha(y_3, y_1)$.
Since ${\mathcal{S}}$ is amenable and hence stable, by Theorem \[thm-main\], after choosing a decreasing sequence $V_1\supset V_2\supset \cdots$ of open neighborhoods of the identity in $K$, there exists a pre-stability sequence $(T_n, B_n)$ for ${\mathcal{S}}$ such that $\alpha(T_ny, y)\in V_n$ for any $n$ and almost every $y\in Y$. We define the lift $\tilde{T}_n\in [{\mathcal{R}}_1]$ of $T_n$ by $\tilde{T}_nx=(T_np(x), p(x))\cdot x$ for $x\in X$. Then $\tilde{T}_n$ commutes with any element of $K$, asymptotically commutes with the lift of any element of $[{\mathcal{S}}]$, and satisfies $\mu(\tilde{T}_nB\bigtriangleup B)\to 0$ for any Borel subset $B\subset X$. The equivalence relation ${\mathcal{R}}_1$ is naturally identified with the subgroupoid $A\ltimes (X, \mu)$, and we regard $\tilde{T}_n$ as an element of $[A\ltimes (X, \mu)]$. Then $\tilde{T}_n$ commutes with any element of $[H\ltimes (X, \mu)]$, and $H\ltimes (X, \mu)$ and the lifts of elements in $[{\mathcal{S}}]$ generate $G\ltimes (X, \mu)$. Therefore the sequence $(\tilde{T}_n, p^{-1}(B_n))$ is a pre-stability sequence for $G\ltimes (X, \mu)$, and $G\ltimes (X, \mu)$ is stable by [@kida-srt Theorem 3.1].
**The second construction.** Let $K$ be the closure of the image of $H$ in ${\mathrm{Aut}}(A)$ again. We set $D=H\times A$ and let $K$ act on $D$ by automorphisms such that $K$ acts on $H\times \{ e\}$ trivially and acts on $\{ e\} \times A$ as elements of ${\mathrm{Aut}}(A)$ under the identification of $\{ e\} \times A$ with $A$. Then $L{\coloneqq}K\ltimes D$ is a locally compact second countable group, and clearly $D$ is a lattice in $L$. Furthermore we have the embedding $\imath \colon G\to L$ defined by $\imath(h, a)=(\jmath(h), (h, a))$ for $h\in H$ and $a\in A$, where $\jmath \colon H\to {\mathrm{Aut}}(A)$ is the homomorphism arising from the action of $H$ on $A$. The image of $\imath$ is then a lattice in $L$. Therefore $L$ is a measure-equivalence coupling of $D$ and $G$ with respect to the left and right multiplications.
We have the p.m.p. action $D{\curvearrowright}L/\imath(G)$, and its restriction to $\{ e\} \times A$ is trivial. Indeed, for any $a, a'\in A$, $k\in K$ and $h'\in H$, we have $$\begin{aligned}
&(e_K, (e_H, a))(k, (h', a'))=(k, (h', (k^{-1}\cdot a)a'))\\
&=(k, (h', a'))(e_K, (e_H, (a')^{-1}(k^{-1}\cdot a)a'))\in (k, (h', a'))\imath(G).\end{aligned}$$ Choose a free ergodic p.m.p. action $A{\curvearrowright}(X, \mu)$ arbitrarily. Let $D\times G$ act on $L\times X$ by $$((h, a), g)(l, x)=((h, a)l\imath(g)^{-1}, ax)$$ for $h\in H$, $a\in A$, $g\in G$, $l\in L$ and $x\in X$. Then $L\times X$ is a measure-equivalence coupling of $D$ and $G$ such that the groupoid $D\ltimes ((L\times X)/\imath(G))$ is stable. Therefore the groupoid $G\ltimes ((L\times X)/D)$ is also stable, and $G$ is stable by [@kida-srt Theorem 1.4].
[99999]{}
C. Anantharaman-Delaroche and J. Renault, *Amenable groupoids*, Monogr. Enseign. Math., 36. L’Enseignement Mathématique, Geneva, 2000.
M. Choda, Inner amenability and fullness, *Proc. Amer. Math. Soc.* **86** (1982), 663–666.
A. Connes, Almost periodic states and factors of type $\mathrm{III}_1$, *J. Funct. Anal.* **16** (1974), 415–445.
A. Connes, Outer conjugacy classes of automorphisms of factors, *Ann. Sci. École Norm. Sup. (4)* **8** (1975), 383–419.
A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, *Ergodic Theory Dynam. Systems* **1** (1981), 431–450.
A. Connes and V. F. R. Jones, A ${\rm II}_1$ factor with two nonconjugate Cartan subalgebras, [*Bull. Amer. Math. Soc. (N. S.)*]{} **6** (1982), 211–212.
A. Connes and W. Krieger, Measure space automorphisms, the normalizers of their full groups, and approximate finiteness, *J. Funct. Anal.* **24** (1977), 336–352.
Y. de Cornulier, Finitely presentable, non-Hopfian groups with Kazhdan’s property (T) and infinite outer automorphism group, *Proc. Amer. Math. Soc.* **135** (2007), 951–959.
T. Deprez and S. Vaes, Inner amenability, property Gamma, McDuff $\textrm{II}_1$ factors and stable equivalence relations, *Ergodic Theory Dynam. Systems* **38** (2018), 2618–2624.
E. G. Effros, Property $\Gamma$ and inner amenability, [*Proc. Amer. Math. Soc.*]{} **47** (1975), 483–486.
M. Ershov, Kazhdan groups whose FC-radical is not virtually abelian, *J. Comb. Algebra* **1** (2017), 59–62.
J. Feldman, C. E. Sutherland, and R. J. Zimmer, Subrelations of ergodic equivalence relations, *Ergodic Theory Dynam. Systems* **9** (1989), 239–269.
T. Giordano and P. de la Harpe, Groupes de tresses et moyennabilité intérieure, *Ark. Mat.* **29** (1991), 63–72.
E. Glasner, *Ergodic theory via joinings*, Math. Surveys Monogr., 101, Amer. Math. Soc., Providence, RI, 2003.
Y. Glasner and N. Monod, Amenable actions, free products and a fixed point property, *Bull. Lond. Math. Soc.* **39** (2007), 138–150.
P. Hall, Finiteness conditions for soluble groups, *Proc. Lond. Math. Soc. (3)* **4** (1954), 419–436.
P. Hahn, The regular representations of measure groupoids, *Trans. Amer. Math. Soc.* **242** (1978), 35–72.
T. Hamachi, Canonical subrelations of ergodic equivalence relations-subrelations, *J. Operator Theory* **43** (2000), 3–34.
A. Ioana, A relative version of Connes’ $\chi(M)$ invariant and existence of orbit inequivalent actions, *Ergodic Theory Dynam. Systems* **27** (2007), 1199–1213.
A. Ioana and P. Spaas, A class of $\textrm{II}_1$ factors with a unique McDuff decomposition, preprint, arXiv:1808.02965.
V. F. R. Jones and K. Schmidt, Asymptotically invariant sequences and approximate finiteness, *Amer. J. Math.* **109** (1987), 91–114.
A. S. Kechris, *Classical descriptive set theory*, Grad. Texts in Math., 156, Springer-Verlag, New York, 1995.
A. S. Kechris, *Global aspects of ergodic group actions*, Math. Surveys Monogr., 160, Amer. Math. Soc., Providence, RI, 2010.
D. Kerr and H. Li, Ergodic theory. Independence and dichotomies, Springer Monogr. Math., Springer, Cham, 2016.
Y. Kida, Inner amenable groups having no stable action, *Geom. Dedicata* **173** (2014), 185–192.
Y. Kida, Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups, *Groups Geom. Dyn.* **9** (2015), 203–235.
Y. Kida, Stable actions of central extensions and relative property (T), *Israel J. Math.* **207** (2015), 925–959.
Y. Kida, Stable actions and central extensions, *Math. Ann.* **369** (2017), 705–722.
A. Marrakchi, Stability of products of equivalence relations, *Compos. Math.* **154** (2018), 2005–2019.
M. Pimsner and S. Popa, Entropy and index for subfactors, *Ann. Sci. Éc. Norm. Supér. (4)* **19** (1986), 57–106.
S. Popa and S. Vaes, On the fundamental group of $\textrm{II}_1$ factors and equivalence relations arising from group actions, in *Quanta of maths*, 519–541, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010.
K. Schmidt, Asymptotic properties of unitary representations and mixing, *Proc. Lond. Math. Soc. (3)* **48** (1984), 445–460.
K. Schmidt, Some solved and unsolved problems concerning orbit equivalence of countable group actions, in [*Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986)*]{}, 171–184, Teubner-Texte Math., 94, Teubner, Leipzig, 1987.
C. E. Sutherland, Subfactors and ergodic theory, in *Current topics in operator algebras (Nara, 1990)*, 38–42, World Sci. Publ., River Edge, NJ, 1991.
R. Tucker-Drob, Invariant means and the structure of inner amenable groups, preprint,\
arXiv:1407.7474.
S. Vaes, An inner amenable group whose von Neumann algebra does not have property Gamma, *Acta Math.* **208** (2012), 389–394.
R. J. Zimmer, Extensions of ergodic group actions, *Illinois J. Math.* **20** (1976), 373–409.
[^1]: The first author was supported by JSPS Grant-in-Aid for Scientific Research, 17K05268. The second author was supported by NSF grant DMS 1600904.
|
---
abstract: 'We present a treatment of the high energy scattering of dark Dirac fermions from nuclei, mediated by the exchange of a light vector boson. The dark fermions are produced by proton-nucleus interactions in a fixed target and, after traversing shielding that screens out strongly interacting products, appear similarly to neutrino neutral current scattering in a detector. Using the Fermilab experiment E613 as an example, we place limits on a secluded dark matter scenario. Visible scattering in the detector includes both the familiar regime of large momentum transfer to the nucleus ($Q^2$) described by deeply inelastic scattering, as well as small $Q^2$ kinematics described by the exchanged vector mediator fluctuating into a quark-antiquark pair whose interaction with the nucleus is described by a saturation model. We find that the improved description of the low $Q^2$ scattering leads to important corrections, resulting in more robust constraints in a regime where a description entirely in terms of deeply inelastic scattering cannot be trusted.'
author:
- 'Davison E. Soper'
- 'Michael Spannowsky and Chris J. Wallace'
- 'Tim M. P. Tait'
title: Scattering of Dark Particles with Light Mediators
---
Introduction and motivation
===========================
There is compelling evidence that most of the mass in the Universe is in the form of nonbaryonic dark particles. And yet, the identity of this dark matter (DM) remains elusive. Among the many proposed candidates, weakly-interacting massive particles (WIMPs) are the most popular, due to the fact that their abundance in the Universe can be explained by virtue of their being thermal relics provided they have weak scale masses and couplings [@Bertone:2004pz].
One possibility is that the dark matter particles do not interact with ordinary matter strictly by the weak force. Rather, they may be able to exchange particles that interact with quarks or gluons. In this case, the relevant couplings would have to be small. Such particles could potentially be discovered by any of three methods. First, dark matter particles in locations in our galaxy where they are especially abundant could annihilate to form baryonic matter and, eventually, photons that might be detected (indirect detection). Second, dark matter particles in the halo of our galaxy might interact with nuclei in a detector on earth and this interaction might be observable (direct detection). Third, dark matter particles might be created in hadron collisons at an accelerator (accelerator production). If this happens often enough at a colliding beam accelerator such as the Large Hadron Collider, one might discover these events by looking, for example, for a missing energy signal. Alternatively, one might create dark matter particles in hadron collisions with nuclei in a fixed target and detect them through their interactions with nuclei in a suitable detector.
Currently, the best constraints on dark particles interacting with quarks come from a mixture of searches for direct detection and accelerator production. In a direct detection experiment, a particle $\chi$ with mass $m_\chi$ and velocity $v_\chi$ interacts with a nucleus in the detector and one looks for the nuclear recoil, where the typical magnitude of $v_\chi \simeq 10^{-3}$ is determined by the gravitational potential of the Galaxy. If $m_\chi$ is not large enough, the momentum $m_\chi v_\chi$ will not be large enough to create an observable nuclear recoil [@Akerib:2013tjd; @Aprile:2012nq; @Ahmed:2009rh; @Angloher:2011uu]. For this reason, the current generation of direct detection experiments have not been sensitive to dark matter particles with $m_\chi \lesssim 5$ GeV. However, these limits may improve in experiments using specialized detection techniques (e.g. based on measurements of ionization yield) [@Essig:2012yx; @Cushman:2013zza]. As a result, the best bounds on hadronic interactions for such light dark matter particles currently come from accelerator production at colliders [@Goodman:2010yf; @Bai:2010hh; @Goodman:2010ku; @Fox:2011pm; @Rajaraman:2011wf; @Bai:2012xg; @Aaltonen:2012jb; @Cheung:2012gi; @Chatrchyan:2012tea; @Chatrchyan:2012pa; @ATLAS:2012ky], particularly for the case in which the particles mediating these interactions are heavy compared to the momentum transfer of the production process.
Of special interest are models in which the dark sector particles that mediate the interactions between the $\chi$ and standard model particles are not heavy but rather light, in some cases even lighter than the $\chi$ particles. This is the secluded scenario of Refs. [@Batell:2009yf; @Batell:2009di]. If the dark matter particles $\chi$ are themselves light enough so that they escape from direct detection experiments, a promising way to look for them is at fixed target experiments [@Batell:2009di; @Essig:2010gu] where a beam of protons strike a target to produce a beam of $\chi$ particles which are sufficiently weakly interacting so as to pass through shielding (as do neutrinos) where they can eventually be detected via their rare scattering with the nuclei comprising a detector. The advantage of a fixed target experiment over a colliding beam experiment is the higher luminosity that a fixed target experiment can offer, a key factor when searching for extremely rare production processes. In particular, we focus on the Fermilab beam dump experiment E613, which utilized a 400 GeV incoming proton beam on a tungsten target. Future high energy beam dump experiments could potentially extend the reach of E613 [@Hewett:2014qja].
We employ a very simple model for the dark sector of the theory consisting of a single Dirac fermion dark matter particle $\chi$ and a light vector particle $V$, which couples to both $\chi$ particles as well as quarks. We refer to $V$ as the dark vector boson. The relevant interactions are $$\mathcal{L}_{\mathrm{I}}= V_\mu \left( g_{q\bar q v} ~ \sum_q ~ \bar{q} \gamma^\mu q
+ g_{\chi \bar \chi v} ~ \bar{\chi} \gamma^\mu \chi \right)~.
\label{eq:lag}
$$ This framework is similar to a “dark photon" model, in which $V$ picks up interactions to the Standard Model through kinetic mixing with hyper-charge [@Dienes:1996zr], but differs in that it has universal charges for the quarks and is agnostic concerning the coupling to leptons. We discuss the dark photon case in more detail below, but it is worth noting here that for the regions of parameter space of interest to us, $1 {\ \mathrm{MeV}}< m_\chi < 10 {\ \mathrm{GeV}}$ and $m_v \sim 1 {\ \mathrm{MeV}}$, there are much stronger constraints on a dark photon mediator from experiments with electrons on fixed targets [@Bjorken:2009mm; @Batell:2014mga] than on models interacting only with quarks [@Carone:1994aa]. Thus one might consider the interaction (\[eq:lag\]) in a leptophobic model in which the light vector particles do not couple to leptons. The leptophobic model is not really intended to be taken as a realistic model for the dark sector, but is a convenient framework to explore the degree to which non-perturbative QCD plays a role in describing how $\chi$ particles scatter off of the nuclei in a detector. The high energy of the $\chi$ particles produced by E613’s $400 {\ \mathrm{GeV}}$ beam demands this more detailed treatment of scattering than is necessary for the low energy neutrino factories discussed in the context of a similar model in [@deNiverville:2011it; @deNiverville:2012ij; @Batell:2014yra].
We will frame the discussion in terms of a dark matter search at E613 using the simple model of Eq. (\[eq:lag\]). In Section \[sec:model\] we describe the production of dark particles at proton fixed target experiments. In Section \[sec:scattering\], we calculate the rescattering rate of produced $\chi$s in the detector, using both a deeply inelastic scattering (DIS) approach, detailed in Section \[sec:DIS\], and a parton saturation approach, detailed in Section \[sec:SAT\]. We examine the connection between the two approaches in Section \[sec:connection\]. In Section \[sec:dm\], we use the results of experiment E613 to place limits on the couplings in Eq. (\[eq:lag\]) and in a closely related “minicharge” model. Finally, we present conclusions in Section \[sec:conclusions\]. Details of the kinematics are provided in an Appendix.
Production of dark matter particles {#sec:model}
===================================
![Feynman diagram for direct production of $\chi$ particles from $p\,A$ collisions.[]{data-label="fig:chiproduction"}](chiproduction.pdf "fig:"){width="5"}\
When beam protons strike the tungsten target in experiment E613, they can produce $\chi \bar \chi$ pairs through the diagram shown in Fig. \[fig:chiproduction\]. We demand that one or both of the $\chi$ particles have a high energy in the lab frame. Then this is a hard process that can be reliably calculated in lowest order perturbation theory, taking the tungsten nucleus to consist of $Z = 74$ protons and $A - Z \approx 110$ neutrons, treated as non-interacting. The interactions of Eq. (\[eq:lag\]) are implemented in Madgraph 5 [@Alwall:2011uj] with the help of FeynRules [@Christensen:2008py]. The inclusive cross sections for the process $$p p \rightarrow \bar{\chi} \chi + X
$$ for a proton of energy $E_B$ incident upon a proton at rest is simulated at the parton level in the Monte Carlo generator. In order to convert this into the number of $\chi$s or $\bar\chi$s produced with energy $E$ and angle $\theta$, we write (approximating the cross section from neutrons in the nucleus as being identical to the cross sections from protons, as is approximately true in our model (\[eq:lag\])) $$\frac{dN}{dE\,d\theta} =
A\, \frac{d\sigma (p p \rightarrow \chi \bar{\chi})}{d E\, d\theta}\,
L_T\, n_T \, {\rm POT}
~,
$$ multiplying by the length of the target $L_T$, the density of tungsten nuclei inside it, $n_T$, and the number of protons incident on the target corresponding to the data set, POT. Here the cross section is the cross section to produce either a $\chi$ or a $\bar \chi$.
The number of $\chi$s that actually make it to a detector further depends on the angular acceptance of the detector. The E613 detector geometry is somewhat complicated in this regard. The detector face was $3~{\rm m} \times 1.5~{\rm m}$, with the beam offset along the horizontal axis by $0.75$ m. To be conservative, we assume $\chi$s must be incident within the $0.75$ m radius circle centered on the beam axis, though in practice there was a larger instrumented region which could be capable of detecting additional $\chi$s with larger production angles. The produced $\chi$s are thus incident on the detector provided their production angle is less than, $$\theta_{\rm max} = \frac{0.75~{\rm m}}{55.8~{\rm m}} = 0.0134~.
$$ The number of $\chi$s per unit energy incident on the detector is then[^1] $$\label{eq:dNdE}
\frac{dN}{dE} = \int_0^{\theta_{\rm max}}\!d\theta\
\frac{dN}{dE\,d\theta}
~.
$$ In Fig. \[fig:dNdE\], we show a plot of the calculated ${dN}/{dE}$ divided by $g_{q \bar q v}^2\, g_{\chi \bar \chi v}^2$.
![Typical distribution of $\chi$ particles as a function of energy, $\frac{dN}{dE}$, divided by $g_{q \bar q v}^2\, g_{\chi \bar \chi v}^2$. The vertical scale is logarithmic. We show the distribution of all produced particles $\chi$ and $\bar \chi$ and the distribution of particles produced at angles that will result in their impacting the target. Many of the lowest energy dark particles are produced at wide angles and miss the detector.[]{data-label="fig:dNdE"}](dNdE.pdf "fig:"){width="12"}\
Structure functions for dark matter scattering in the detector {#sec:scattering}
==============================================================
The detector is made of lead plus liquid scintillator. When a $\chi$ particle enters the detector with energy $E$, it can scatter from a lead nucleus. In order for the scattering to be detected, we demand that the scattering transfer at least an amount of energy $E_{\rm cut}$ to the nucleus. We take $E_{\rm cut} = 20 {\ \mathrm{GeV}}$, corresponding to the minimum energy demanded by the detector to register a jet [@Romanowski:1985xn; @Duffy:1988rw]. Thus the expected number of events is proportional to the convolution of $dN/dE$ from Eq. (\[eq:dNdE\]) with the cross section $\sigma(E,E_{\rm cut})$ for a $\chi$ particle to deposit energy greater than $E_{\rm cut}$ in the nucleus.
How should we calculate $\sigma(E,E_{\rm cut})$? Our process is quite analogous to deeply inelastic lepton scattering. We can take advantage of that. There is a standard analysis that allows us to write the cross section for $\chi$ scattering from the nucleus via vector boson exchange in terms of two structure functions, $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$. In this section, we apply this standard analysis to $\chi$ scattering, using variables that are convenient for our present purposes. Although this analysis substantially simplifies the problem, it does not tell us what the structure functions $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$ are. We will examine two rather different models for the structure functions in the following two sections.
The $\chi$ particle exchanges a virtual dark vector boson with the nucleus, as depicted in Fig. \[fig:DIS\]. The $\chi$ particle has momentum $p_\chi$ before the scattering and momentum $p'_\chi$ after the scattering. The dark vector boson carries spacelike momentum $q = p_\chi - p'_\chi$. One defines $Q^2 = - q^2$ so that $Q^2 > 0$. We define $\nu$ to be the energy of the vector boson in the nucleus rest frame. Thus the cut on the energy delivered to the nucleus is a cut $\nu > E_{\rm cut}$. We let $P$ be the momentum of the nucleus before the scattering and $M$ be its mass. Normally, $(P+q)^2 > M^2$, so that the scattering breaks up the nucleus. We define the Bjorken scaling variable $x_{\rm bj}$ by $$x_{\rm bj} = \frac{Q^2}{2 M \nu}~.
\label{eq:bjx}
$$ We use the mass $M$ of the nucleus here. If we were to consider the nucleus as consisting of $A$ independent nucleons, then we might instead use $Ax_{\rm bj} = {Q^2}/(2 m_p \nu)$.
Using lowest order perturbation theory in the interactions of the vector boson and using Lorentz invariance, parity invariance, and current conservation for the strong interactions, the differential cross section has the form familiar from deeply inelastic lepton scattering: $$d\sigma = \frac{1}{4M[{E^2 - m_\chi^2}]^{1/2}}\,
(2\pi)^{-3}
d^4 p'_\chi\ \delta({p'_\chi}^2- m_\chi^2)\,
\frac{g^2_{\chi\bar \chi v}L^{\mu\nu}\ 4\pi g^2_{q\bar qv}W_{\mu\nu}}{(q^2 - m_v^2)^2}
\;,
$$ where $L^{\mu\nu}$ is $$L^{\mu\nu} = 4 p_\chi^\mu p_\chi^\nu
- 2 (p_\chi^\mu q^\nu + q^\mu p_\chi^\nu)
+ q^2 g^{\mu\nu}
$$ and $W_{\mu\nu}$ is the hadronic matrix element of the quark currents to which the vector particle couples, not including the coupling $g^2_{q\bar qv}$ but including a conventional factor $1/(4\pi)$, $$W_{\mu\nu} = \frac{1}{4\pi} \sum_X
{\langle P|} J_\mu(0) {|X\rangle}
{\langle X|} J_\nu(0) {|P\rangle}
(2\pi)^4\delta(P + q - p_X)
\;.
$$ With the use of Eq. (\[eq:jacobian\]) in Appendix A, this is $$d\sigma = \frac{g^2_{\chi\bar \chi v}\, g^2_{q\bar qv}}{16\pi M}\,
\frac{d\nu\, dQ^2}{E^2 - m_\chi^2}\,
\frac{L^{\mu\nu} W_{\mu\nu}}{(Q^2 + m_v^2)^2}
\;.
$$ We use $\nu$ and $Q^2$ as integration variables instead of the components of $p'_\chi$. The kinematics impose limits on $\nu$ and $Q^2$, which we derive in Appendix \[sec:kinematics\]. Defining $$\label{eq:muofnudef}
\mu^2(\nu)
=
\frac{m_\chi^2 \nu^2}
{[E(E-\nu) - m_\chi^2] + \sqrt{[E(E-\nu) - m_\chi^2]^2 - m_\chi^2\nu^2}}
$$ from Eq. (\[eq:muofnudef0\]), the limits are (Eqs. (\[eq:nurange\]), (\[eq:Qsqbounds\]), and (\[eq:QSqlessthan2mnu\])) $$\begin{split}
\label{eq:limits}
E_{\rm cut} <{}& \nu < E - m_\chi
\;,
\\
2 \mu^2(\nu) <{}& Q^2
<
4 [E (E-\nu)-m_\chi^2]
- 2 \mu^2(\nu)
\;,
\\
Q^2 <{}& 2 M \nu
\;.
\end{split}$$ Now we can write $W_{\mu\nu}$ in terms of standard structure functions, $$W^{\mu\nu} = C_{\mathrm{T}}^{\mu\nu} F_{\mathrm{T}}(x_{\rm bj},Q^2)
+ C_{\mathrm{L}}^{\mu\nu} F_{\mathrm{L}}(x_{\rm bj},Q^2)
\;,
$$ where $$\begin{split}
C_{\mathrm{T}}^{\mu\nu} ={}&
-g^{\mu\nu}
+ \frac{q^\mu q^\nu}{q^2}
+ \frac{2x_{\rm bj}}{P\cdot q + 2 x_{\rm bj} M^2}
\left(P^\mu - \frac{P\cdot q}{q^2}\,q^\mu\right)
\left(P^\nu - \frac{P\cdot q}{q^2}\,q^\nu\right)
\;,
\\
C_{\mathrm{L}}^{\mu\nu} ={}&
\frac{1}{P\cdot q + 2 x_{\rm bj} M^2}
\left(P^\mu - \frac{P\cdot q}{q^2}\,q^\mu\right)
\left(P^\nu - \frac{P\cdot q}{q^2}\,q^\nu\right)
\;.
\end{split}$$ Notice that $C_{\mathrm{T}}^{\mu\nu}q_\nu = C_{\mathrm{L}}^{\mu\nu}q_\nu = 0$ and that $C_{\mathrm{T}}^{\mu\nu}a_\nu = 0$ for any vector $a$ in the $P$-$q$ plane while $C_{\mathrm{L}}^{\mu\nu}a_\nu = 0$ for any vector orthogonal to $P$ and $q$. Thus $C_{\mathrm{T}}$ corresponds to the exchange of transversely polarized virtual vector bosons while $C_{\mathrm{L}}$ corresponds to the exchange of longitudinally polarized virtual vector bosons. The structure functions $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$ are related to the standard structure functions $F_1$ and $F_2$ by $F_{\mathrm{T}}= F_1$ and $F_{\mathrm{L}}= (1+2x_{\rm bj} M^2/P\cdot q) F_2 - 2 x_{\rm bj} F_1$.
We can thus write the cross section in terms of structure functions as $$d\sigma = \frac{g^2_{\chi\chi v}\, g^2_{qqv}}{16\pi M}\,
\frac{d\nu\, dQ^2}{E^2 - m_\chi^2}\,
\frac{1}{(Q^2 + m_v^2)^2}
\left[
C_{\mathrm{T}}^{\mu\nu} L_{\mu\nu}\,F_{\mathrm{T}}(x_{\rm bj},Q^2)
+
C_{\mathrm{L}}^{\mu\nu} L_{\mu\nu}\,F_{\mathrm{L}}(x_{\rm bj},Q^2)
\right]
\;.
$$ One finds $$\begin{split}
\label{eq:CTandCL}
C_{\mathrm{T}}^{\mu\nu} L_{\mu\nu} ={}&
\frac{Q^2 (2E-\nu)^2 }{\nu^2 + Q^2}
+ Q^2
- 4 m_\chi^2
\;,
\\
C_{\mathrm{L}}^{\mu\nu} L_{\mu\nu} ={}& M\nu\,\frac{4 E(E-\nu) - Q^2}{\nu^2 + Q^2}
\;.
\end{split}$$ Thus $$\begin{split}
\label{eq:crosssection}
d\sigma ={}& \frac{g^2_{\chi\chi v}\, g^2_{qqv}}{16\pi}\,
\frac{d\nu\, dQ^2}{E^2 - m_\chi^2}\,
\frac{\nu}{(Q^2 + m_v^2)^2}
\Bigg\{
\left[
\frac{(2E-\nu)^2}{\nu^2 + Q^2}
+ \frac{Q^2 - 4 m_\chi^2}{Q^2}
\right]
2 x_{\rm bj} F_{\mathrm{T}}(x_{\rm bj},Q^2)
\\&\quad
+
\frac{4E(E-\nu)-Q^2}{\nu^2 + Q^2}\,
F_{\mathrm{L}}(x_{\rm bj},Q^2)
\Bigg\}
\;.
\end{split}$$ The cross section that we want, $\sigma(E,E_{\rm cut})$, is then this $d\sigma$ integrated over $\nu > E_{\rm cut}$, taking into account the kinematic constraints (\[eq:limits\]). This result is exact within the approximation of considering single vector boson exchange, but, of course, we need to be able to calculate $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$. We explore this in the following two sections.
DIS model {#sec:DIS}
=========
One way is to approach this as deeply inelastic scattering, as depicted in Fig. \[fig:DIS\]. The $\chi$ exchanges a virtual $V$ that is absorbed by a quark in the nucleus. If $Q^2$ is large, there is a short distance interaction in which the vector boson interacts with a quark or gluon in the nucleus. There are also long range interactions, both in the initial state and in the final state. For an inclusive cross section like that considered here, the final state interactions do not affect the cross section. The initial state interactions do affect the cross section, but they can be factored into parton distribution functions. The short distance interaction can be calculated perturbatively. Thus $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$ are written as a convolution of parton distribution functions with the partonic structure functions $\hat F_{\mathrm{T}}$ and $\hat F_{\mathrm{L}}$.
We will work at lowest order in perturbation theory for $\hat F_{\mathrm{T}}$ and $\hat F_{\mathrm{L}}$. At lowest order, the contributions from the gluon parton distribution function vanish for both $\hat F_{\mathrm{L}}= 0$ and $\hat F_{\mathrm{T}}$. For quarks at lowest order, $\hat F_{\mathrm{L}}= 0$ and $\hat F_{\mathrm{T}}$ is simply a delta function that sets the quark momentum fraction equal to $x_{\rm bj}$. (There would be a squared charge, $g^2_{q\bar q v}$, but we have already factored that out of the hadronic matrix element.) That is, $F_{\mathrm{L}}= 0$ and $$\label{eq:FTfromfq}
F_{\mathrm{T}}= \frac{1}{2 x_{\rm bj}}\sum_q x_{\rm bj} f_{q/A}(x_{\rm bj},Q^2)
\;.
$$ Here we sum over flavors of quarks and antiquarks, $q = {\mathrm{u}},\bar{\mathrm{u}}, {\mathrm{d}}, \bar{\mathrm{d}}, {\mathrm{s}}, \bar{\mathrm{s}}$ under our assumption that the mediator particle $v$ couples equally to all the flavors. (However, we have omitted charm and bottom quarks here since the corresponding parton distribution functions are small.) We have multiplied and divided by $x_{\rm bj}$ so that one factor is $x_{\rm bj} f_{q/A}(x_{\rm bj},Q^2)$, which is relatively insensitive to $x_{\rm bj}$ at small $x_{\rm bj}$. We note that the parton distributions here are the distributions in the nucleus A. The distribution of partons in a nucleus may be related approximately to the distribution of partons in a proton. For instance, if A is a nucleus with baryon number $A$ and charge $Z$ then $$f_{{\mathrm{u}}/A}(x_{\rm bj},Q^2) dx_{\rm bj} \approx [Z f_{{\mathrm{u}}/p}(A x_{\rm bj},Q^2)
+ (A-Z) f_{{\mathrm{d}}/p}(A x_{\rm bj},Q^2) ] d(Ax_{\rm bj})
\;.
$$ That is $$\label{eq:nuclearpdfs}
f_{{\mathrm{u}}/A}(x_{\rm bj},Q^2) \approx AZ f_{{\mathrm{u}}/p}(A x_{\rm bj},Q^2)
+ A(A-Z) f_{{\mathrm{d}}/p}(A x_{\rm bj},Q^2)
\;.
$$ Note that there are two factors of $A$ or $Z$ here. However, we use parton distribution functions for the nucleus provided at leading order by Hirai-Kumano-Nagai (HKNlo) [@HKNpartons], rather than this approximate formula.
Thus in the DIS model we have $$\begin{split}
\label{eq:crosssectionpartonmodel}
d\sigma ={}& \frac{g^2_{\chi\chi v}\, g^2_{qqv}}{16\pi}\,
\frac{d\nu\, dQ^2}{E^2 - m_\chi^2}\,
\frac{\nu}{(Q^2 + m_v^2)^2}
\left[
\frac{(2E-\nu)^2}{\nu^2 + Q^2}
+ \frac{Q^2 - 4 m_\chi^2}{Q^2}
\right]
\sum_q x_{\rm bj}
f_{q/A}(x_{\rm bj},Q^2)
\;.
\end{split}$$ This approximation for the cross section should work well as long as $Q^2$ is large, say larger than a few ${\rm GeV}^2$. However, our numerical studies indicate that a good part of the cross section can come from the integration region in which $Q^2 < 1 {\ \mathrm{GeV}}^2$. For that region, we need another model.
Saturation model {#sec:SAT}
================
There is another model available that should be useful for smaller values of $Q^2$ and large values of $\nu$. In this model, we view the interaction in the rest frame of the nucleus, as illustrated in Fig. \[fig:Dipole\]. The dark vector boson, carrying a large momentum, splits into a quark-antiquark pair. Each of the quark and antiquark also carry a large momentum as they move towards the nucleus. Thus they form a color dipole that can interact with the nucleus. The dipole interacts with the nucleus via gluon exchange, as illustrated in Fig. \[fig:Dipoledetails\]. We will model this interaction.
To motivate the model, it is helpful to examine the kinematics of the interaction in a little detail. We work in the rest frame of the nucleus and align the negative $z$-axis with the momentum $\vec q$ of the dark vector boson. Then, defining $q^\pm = (q^0 \pm q^3)/\sqrt 2$, we have $q^- \approx \sqrt 2 \nu$ and $q^+ \approx - 2^{-3/2}Q^2/\nu$. Thus in this frame $q^-$ is large and $q^+$ is small. In the Feynman diagram in Fig. \[fig:Dipoledetails\], the dark vector boson couples to a quark propagator with momentum $p_q$, as in Fig. \[fig:DIS\]. We can estimate that $p_q^-$ is large while $p_q^+$ is small. Imagine writing the quark propagator in coordinate space, with the quark traveling through a space-time separation $\Delta x$ between the point where it interacts with a gluon from the nucleus and the point where it couples to the dark vector boson. Since $p_q \cdot \Delta x = p_q^+ \Delta x^- + p_q^- \Delta x^+ + p_q^\perp \cdot \Delta x^\perp$, we conclude that typically $\Delta x^-$ is large while $\Delta x^+$ is small. That is, the quark moves a long way in the minus direction. In fact, an estimate for $p_q^+$ is $p_q^+ \approx 2^{-3/2} Q^2/\nu$, so that an estimate for a typical range in the minus direction is $\Delta x^- = 2^{5/2} \pi\,\nu/Q^2$. Assuming that the first interaction of the quark with a gluon is inside the nucleus, this accounting puts the interaction of the quark with the dark vector boson well outside the nucleus when $\nu$ is large and $Q^2$ is not large.
This physical picture, depicted in Fig. \[fig:Dipole\], seems at first to be completely different from the DIS picture of the previous section. Yet, if $\nu$ is very large and also $Q^2$ is large, both pictures can be correct and we can arrive at two ways of approximating the same cross section. The difference in the pictures arises from the difference of reference frames. The DIS picture is most easily derived in a reference frame in which the nucleus has a large momentum along the positive $z$-axis. The dipole picture of this cross section is most easily derived in the rest frame of the nucleus, with the dark vector boson having a large momentum along the negative $z$-axis.
We now need a model for $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$ in the picture in which the dark vector boson turns into a quark-antiquark pair. The model, known as the saturation model, comes from the work of Nikolaev and Zakharov [@NikolaevZakharov], Golec-Biernat and Wüsthoff [@GolecBiernatWusthoff1; @GolecBiernatWusthoff2], and Mueller [@Mueller]. There is an extensive literature on the subject [@FrankfurtDipolesGluons; @ForshawDipoles; @McDermottDipoles; @GotsmanReview; @MuellerLecture; @Bartels; @GolecBiernat1; @GolecBiernat2; @HSdipole; @Venugopalan; @KowalskiDataAnalysis]. We will follow mostly Ref. [@HSdipole] and will incorporate some refinements introduced by Bartels, Golec-Biernat, and Kowalski [@Bartels].
When $Q^2$ is small, the longitudinal structure function $F_{\mathrm{L}}$ is small compared to $2 x_{\rm bj} F_{\mathrm{T}}$ since an on-shell massless vector boson does not have longitudinal polarizations. (For an analysis of $F_{\mathrm{L}}$ in the saturation picture, see ref. [@Machado:2006kd].) Thus we simply approximate $F_{\mathrm{L}}$ by zero in the saturation model, as we did in the DIS model. This leaves $F_{\mathrm{T}}$. The result [@HSdipole] in the saturation model for $F_{\mathrm{T}}$ is $$\label{eq:FTsaturation}
2 x_{\rm bj} F_{\mathrm{T}}= \frac{1}{4\pi}\sum_f
\frac{24 Q^2}{(2\pi)^3}
\int\!d\bm b \int\!d\bm\Delta\
\frac{G(\sqrt{Q^2 + \Lambda_\rho^2}\,\Delta)}{\Delta^2}\,
\Xi(\bm b, \bm \Delta)
\;.
$$ Here one sums over quark flavors $f = \{u,d,s\}$ and the parameter $ \Lambda_\rho$ is discussed below. We integrate over a two dimensional vector $\bm b$ and a two dimensional vector $\bm \Delta$. The picture as outlined above is that the dark vector boson splits into a $q$-$\bar q$ pair, both with a large momentum in the direction of the dark vector boson momentum $q$. When this $q$-$\bar q$ pair reaches the nucleus, the quark is at transverse position $\bm b + \bm \Delta/2$ and the antiquark is a position $\bm b - \bm \Delta/2$.
The function $G(\sqrt{Q^2 + \Lambda_\rho^2}\,\Delta)/\Delta^2$ represents the squared wave function for the $q$-$\bar q$ pair, integrated over the fraction $\alpha$ of the longitudinal momentum of the pair that is carried by the quark. The function $G(z)$ is $$\label{eq:Gz}
G(z) = \int_0^1\!d\alpha\ [1-2\alpha(1-\alpha)]\,
\Big[\sqrt{\alpha(1-\alpha)}\, z\, K_1(\sqrt{\alpha(1-\alpha)} z)\Big]^2
\;.
$$ Here $K_1(x)$ is the modified Bessel function of order 1, equal to $-dK_0(x)/dx$. The function $G(z)$ equals 2/3 for $z = 0$. It behaves like $8/[3 z^2]$ for $z \to \infty$. Thus a rough approximation to it is $$\label{eq:approxG}
G(z) \approx \frac{2}{3[1 + z^2/4]}
\;.
$$ This approximation is good to about 15% for all values of $z$.
We take the argument of $G$ to be $z = \sqrt{Q^2 + \Lambda_\rho^2}\,\Delta$. The perturbative calculation gives just $Q\Delta$. That means that the spatial extent of the wave function is of order $\Delta \sim 1/Q$. That should be right for large $Q$. But for small $Q$, we expect that the $q$ and $\bar q$ exchange gluons so as to bind themselves into one or more mesons – predominantly a single $\rho$ meson. The $\rho$ meson has a size, which we can denote by $1/\Lambda_\rho$. To represent this non-perturbative effect, it seems sensible to replace $Q\Delta$ by $\sqrt{Q^2 + \Lambda_\rho^2}\,\Delta$. For the inverse radius of a $\rho$ meson, an approximate first guess might be $\Lambda_\rho \approx 1/(1\ \text{fm}) \approx 200 {\ \mathrm{MeV}}$.
The function $\Xi(\bm b, \bm \Delta)$ represents the probability that the $q$-$\bar q$ pair scatters from hadron $A$. If $\Delta$ is not small, then this probability is approximately 1 if either the quark or the antiquark hits hadron $A$. But if $\Delta$ is very small, the color dipole moment of the $q$-$\bar q$ pair is small and the pair can pass right through hadron $A$ without scattering. (This effect is known as [*color transparency*]{}). This suggests the following model (from Mueller [@Mueller] and Golec-Biernat and Wüsthoff [@GolecBiernatWusthoff1; @GolecBiernatWusthoff2]). We write[^2] $$\label{eq:Xi}
\Xi(\bm b, \bm \Delta) = 1 - e^{-\Delta^2 Q_{\mathrm{s}}^2(b)/4}
\;,
$$ where $Q_{\mathrm{s}}^2$ is the [*saturation scale*]{}. Evidently if $\Delta^2\ll 1/Q_{\mathrm{s}}^2$ then $\Xi(\bm b, \bm \Delta) \propto \Delta^2$ and the scattering probability tends to zero as $\Delta^2$ decreases. There is no scattering because the gluon field in hadron $A$ does not see the $q$-$\bar q$ pair.
Before we go on to talk about the saturation scale $Q_{\mathrm{s}}^2(b)$, we should discuss Eq. (\[eq:Xi\]) and its connection to unitarity and to classical optics. Define $T(\bm b,\bm \Delta)$ by $\Xi(\bm b,\bm \Delta) = 1 - T(\bm b,\bm \Delta)$. We think of $\Xi$ as the probability for the dipole to be absorbed by the nucleus and we think of $T$ as the analogue of the transmission coefficient in optics [@HSdipole]. Let $R_A$ be the radius of the nucleus. We can then determine the necessary limiting properties of the function $T(\bm b,\bm \Delta)$. Here we follow Ref. [@HSdipole], which contains more details.
- If the dipole misses the nucleus, i.e. $|{\bm b}| > R_{A} + \Delta/2$, then $\Xi(\bm b,\bm \Delta)$ must be zero, therefore $T(\bm b,\bm \Delta) = 1$.
- If the quark and the antiquark that make up the dipole are separated from each other by zero distance then, since it is a color singlet object, it simply passes through the nucleus. Therefore, $T(\bm b,\bm \Delta)=1$ for $\Delta=0$.
- For small $\Delta$, the probability for the dipole to interact with the nucleus should be proportional to the square of the color dipole moment of the dipole: $T(\bm b,\bm \Delta) \propto \Delta^2$. We need $\Delta^2$ here because in the cut Feynman diagram for the process the dipole must exchange at least two gluons with the nucleus.
- For small $\Delta$, we can calculate the coefficient of $\Delta^2$ in $T$ using QCD perturbation theory.
- $T(\bm b,\bm \Delta)\approx 0$ for large dipoles (large $\Delta$), when $|\bm b| < R_{A}$. That is, a large, strongly interacting dipole cannot pass through the nucleus leaving it intact.
To calculate the coefficient of $\Delta^2$ in $T$, we recognize that the probability that the gluon field does see the $q$-$\bar q$ pair depends not only on how small the color dipole moment is but also on how strong the gluon field is. Thus it is not surprizing that the saturation scale $Q_{\mathrm{s}}^2(b)$ in Eq. (\[eq:Xi\]) is proportional to the density of gluons in the nucleus: $$\label{eq:Qs}
Q_{\mathrm{s}}^2(b) = \frac{2\pi^2 {\alpha_{\mathrm{s}}}(\mu^2)}{3}\,
x G(x,\mu^2)\,\phi(b)
\;.
$$ Here $\phi(b)$ is modeled as a geometrical quantity that tells how the gluons are spread in the transverse separation from the center of the nucleus: $$\label{eq:phi}
\phi(b) = \frac{3}{2\pi R_A^3}\,\sqrt{R_A^2 - b^2}\,
\Theta(b^2 < R_A^2)
\;.
$$ The function $\phi(b)$ is normalized to $\int\!d\bm b\, \phi(b) = 1$. The function $G(x,\mu^2)$ is the gluon distribution function in the nucleus. We again employ the HKNlo distribution for lead, which is defined such that the total gluon distribution for the nucleus is given by $G(x,\mu^2) = A\, G_{\rm HKN}(A\,x_{\rm bj},\mu^2)$, which we insert in place of $G(x,\mu^2)$ in equation .
We need to set $\mu^2$ in ${\alpha_{\mathrm{s}}}(\mu^2)$ and $x G(x,\mu^2)$ and we need to set $x$ in $x G(x,\mu^2)$. We follow the form of the choices of Bartels, Golec-Biernat, and Kowalski [@Bartels]. For the scale $\mu^2$, we take $$\label{eq:BartelsMusq}
\mu^2 = \frac{C}{\Delta^2} + \mu_0^2
\;.
$$ The choice of a constant divided by $\Delta^2$ is sensible in the perturbative regime of small $\Delta^2$. However, for large $\Delta^2$ we do not want $\mu^2$ to be arbitrarily small. Thus we add a constant, $\mu_0^2$ to $C/\Delta^2$. We find a reasonable fit for $C = 6.00$ and $\mu_0^2 = 2.0 {\ \mathrm{GeV}}^2$. For the momentum fraction variable in the gluon distribution, we take $$\label{eq:BartelsX}
x = \frac{Q^2 + 4 m_{\mathrm{q}}^2}{2 M \nu}
\;.
$$ This is $x_{\rm bj}$ when $Q^2$ is not too small. But for very small $Q^2$, we do not want $x$ to be arbitrarily small. Thus we add a small mass term, $4 m_{\mathrm{q}}^2$, to $Q^2$. This is in the same spirit as our adjustment of the argument of $G(z)$ in Eq. (\[eq:FTsaturation\]). Following Ref. [@Bartels], we take $m_{\mathrm{q}}= 140 {\ \mathrm{MeV}}$.
![The function $\Xi(\bm b, \bm \Delta)$ as a function of $|\bm b|/R$ and $|\bm \Delta|/R$ where $R$ is the radius of the lead nucleus. We calculate $\Xi(\bm b, \bm \Delta)$ using Eqs. (\[eq:Xi\]), (\[eq:Qs\]), and (\[eq:phi\]) with parameters given in Eqs. (\[eq:BartelsMusq\]) and (\[eq:BartelsX\]) and using HKNlo parton distributions for the distribution of gluons in a lead nucleus, with $x = 10^{-4}$ and $Q^2 = 1 {\ \mathrm{GeV}}^2$.[]{data-label="fig:Xi"}](xi_plot.pdf "fig:"){width="10"}\
We see that there is some QCD theory and some modeling in the net formula for $\Xi$. The resulting function $\Xi(\bm b, \bm \Delta)$ is illustrated in Fig. \[fig:Xi\]. We can perhaps appreciate from the figure that the model dependence is less than one might have thought. For $|\bm \Delta| > R/10$, $\Xi(\bm b, \bm \Delta)$ is very close to 1 for $|\bm b| < R$. When we get to $|\bm b| \approx R$, $\Xi$ drops very quickly to zero. The value $\Xi \approx 1$ is nonperturbative, but it is not really model dependent because 1 is the largest that $\Xi$ could be. For $|\bm \Delta| < R/10$, the behavior of $\Xi(\bm b, \bm \Delta)$ is not so trivial. However, this region is perturbative, so we have some control over the theory. In part, the shape is determined by the function $\phi(b)$ from Eq. (\[eq:phi\]). This part of the formula for $\Xi$ is simply a model for the distribution of gluons. The model is that the density of gluons is uniform throughout the nucleus. Thus there is some model dependence, but the model dependence is not too large.
There is more model dependence in the function $G(\sqrt{Q^2 + \Lambda_\rho^2}\,\Delta)/\Delta^2$ in Eq. (\[eq:Gz\]). This function is calculated using lowest order perturbation theory, so it should be accurate for large $Q^2$ and, correspondingly, small $\Delta$. For small $Q^2$ it simply represents a plausible model.
Connection between the DIS and saturation models {#sec:connection}
================================================
In Eq. (\[eq:FTsaturation\]), we can try to take the large $Q^2$ limit of $x_{\rm bj} F_{\mathrm{T}}$ by taking the large $Q^2$ limit under the integration over $\bm\Delta$. In this limit, the argument, $\sqrt{Q^2 + \Lambda_\rho^2}\,\Delta$, of the function $G$ becomes just $Q\Delta$. Then for large $Q\Delta$ we have $G(Q\Delta) \sim 8/[3Q^2\Delta^2]$, as we noted earlier. We need to enforce that $Q\Delta$ is large inside the integration over $\bm \Delta$ and we do that in a crude way by inserting a factor $\Theta(\Delta > a/Q)$ for some constant $a$. This gives the approximation
$$\label{eq:FTlargeQ}
2 x_{\rm bj} F_{\mathrm{T}}\approx \sum_f
\frac{2}{\pi^4}
\int\!d\bm b \int\!d\bm\Delta\
\frac{\Theta(\Delta > a/Q)}{\Delta^4}\,
\Xi(\bm b, \bm \Delta)
\;.
$$
This matches with our DIS formula Eq. (\[eq:FTfromfq\]) if we identify $$\label{eq:pdfdipole}
x\,f_{q/A}(x,Q^2) =
\frac{1}{\pi^4}
\int\!d\bm b \int\!d\bm\Delta\
\frac{\Theta(\Delta > a/Q)}{\Delta^4}\,
\Xi(\bm b, \bm \Delta)
\;.
$$ There is a factor of 2 in this formula that results from suming over flavors $f$ in Eq. (\[eq:FTlargeQ\]) and over flavors and antiflavors in Eq. (\[eq:FTfromfq\]). The right hand side of this equation has some $x$ dependence because the gluon distribution that appears in the exponent in $\Xi$ depends on $x$. It is independent of the choice of quark flavor or antiflavor $q \in \{{\mathrm{u}},\bar {\mathrm{u}}, {\mathrm{d}},\bar {\mathrm{d}}, {\mathrm{s}}, \bar {\mathrm{s}}\}$.
There is a more direct approach to this, which was obtained in Ref. [@HSdipole]. One starts directly with the operator definition of the parton distribution functions, $f_{q/A}(x,\mu^2)$, and analyzes the operator matrix element using the dipole picture. The operator matrix element requires ultraviolet renormalization, to eliminate a divergence from small $\Delta$ in the integration over $\bm \Delta$. To match the standard $\overline {\rm MS}$ definition of parton distribution functions, one should use dimensional regularization and an appropriate pole subtraction. However, one can obtain the same result at one loop order with a simple cut. The result of this analysis is Eq. (\[eq:pdfdipole\]) with $$a = 2 e^{1/6 - \gamma_{\mathrm{E}}} \approx 1.32657
\;.
$$ Eq. (\[eq:pdfdipole\]) is based on lowest order perturbation theory for the wave function of the quark dipole, so one expects that it should begin to be accurate for $Q^2$ large enough so that perturbation theory applies. However the formula does not properly account for DGLAP evolution, so the result should begin to fail for very large $Q^2$. In Fig. \[fig:pdfCompare\], we test how well this relationship works by ploting $A x f_{q/A}(A x,Q^2)$ versus $\log_{10}(Ax)$ for a few values of $Q^2$. We see that the approximation in Eq. (\[eq:pdfdipole\]) is only moderately successful at $Q^2 = 2 {\ \mathrm{GeV}}^2$, but that it works quite well for $Q^2 = 10 {\ \mathrm{GeV}}^2$. By $Q^2 = 50 {\ \mathrm{GeV}}^2$, it is still working quite well but is beginning to fail.
![The parton distribution function $f_{q/A}(x,Q^2)$ for $\bar {\rm u}$ quarks in a uranium nucleus according to the HKNlo parton distributions [@HKNpartons] used in this paper compared to the same distribution in the saturation model, Eq. (\[eq:pdfdipole\]). We plot $A x f_{q/A}(A x,Q^2)$ versus $\log_{10}(Ax)$ for $Q^2 = 2 {\ \mathrm{GeV}}^2$, $10 {\ \mathrm{GeV}}^2$, and $50 {\ \mathrm{GeV}}^2$.[]{data-label="fig:pdfCompare"}](pdf_comparison_x.pdf "fig:"){width="14"}\
Application to Scattering of Dark Matter {#sec:dm}
========================================
We have studied the scattering of dark Dirac fermions through a vector mediated interaction with quarks. This amounts to a neutrino–like neutral current event, with the added theoretical interest of having no heavy electroweak boson to regulate the momentum transfer of the interaction. In this section we shall apply this formalism to a model of dark matter.
Continuing from Section \[sec:model\], with the scattering cross sections now in hand, it is straightforward to calculate the number of events expected in the detector. We first calculate the mean free path of the propagating dark particle, $$\lambda = \frac{1}{\rho_{A}\,\sigma(\chi N \to \chi N)},$$ where $\rho_{A}$ is the number density of nuclei and $\sigma(\chi N \to \chi N)$ is the nuclear scattering cross section. The mean free path enters into the rescattering probability, $$P = \int^{L}_{0} dx\,\frac{1}{\lambda} e^{-\frac{x}{\lambda}} = 1 - e^{-\frac{L}{\lambda}}.$$ The final number of events expected in the detector is, $$N_{\rm det.} = \int\!dE\ (1 - P_{\rm shielding}(E))\times P_{\rm detector}(E) \times \frac{dN}{dE},
\label{eq:Nevents}$$ where $dN/dE$ is defined in equation . For scattering in the shielded region, composed of $\sim 15$ m of iron, we impose an arbitrary 1 GeV cut on the required energy transfer to prevent divergence of the deep inelastic cross section. We note that in practice, for the small values of the couplings that we can constrain, the probability of rescattering in shielding is extremely small, such that practically no scattering occurs. Further, since the probability of any given dark particle scattering is so low, one does not need to account for the degradation of the beam along the length of the detector, and can approximate the scattering probability as simply $P \sim L/\lambda$. A fully realistic treatment would include multiple rescatterings, including low energy scatters that degrade the energy of incoming particles. This is not necessary for our purposes, which are adequately modeled by a single scattering event per dark particle.
Using equation and data provided by the experimental collaboration [@Romanowski:1985xn; @Duffy:1988rw] (and interpreted as below in [@Golowich]), we may constrain our model. The E613 experiment delivered $1.8 \times 10^{17}$ protons on target (POT), and estimate that at most 100 detected events per $10^{17}$ POT represent muonless neutral current events at 90% C.L. Thus, we exclude couplings where the number of expected detector events, $N_{\chi} > 180$.
The result of this analysis as applied to the model (\[eq:lag\]) is shown in Fig. \[fig:leptophobic\], with the mediator mass set to $1 {\ \mathrm{MeV}}$ and the mediator–dark particle coupling fixed to unity. The two colored regions in the plot correspond to the scattering models described in Sections \[sec:DIS\] and \[sec:SAT\]. The “DIS only” region cuts off integration of the cross section for $Q^2 < 1 {\ \mathrm{GeV}}^2$, applying the scattering picture of Section \[sec:DIS\]. The region labeled “With saturation model” applies the same formalism, but additionally includes the dipole scattering mechanism described in Section \[sec:SAT\] for the $Q^2 < 1 {\ \mathrm{GeV}}^2$ region, resulting in a substantial improvement of the constraint. Also plotted is a mapping of the constraint on a leptophobic U(1) gauge boson, which couples to baryon number. Several constraints on such a model are described in [@Carone:1994aa], the strongest of which (plotted) arises from the contribution of the new boson to the decay width of $\Upsilon$ mesons into hadronic final states. While we emphasize that this is a toy model, very similar models are of considerable phenomenological interest, and apt to be studied at existing fixed target facilities [@Batell:2014yra].
![Exclusion limits for the leptophobic model described in the text, with $g_{\chi\bar{\chi}v} = 1$ and a mediator mass of 1 MeV. Also plotted is the region excluded by the study of $\Upsilon$ decays from [@Carone:1994aa].[]{data-label="fig:leptophobic"}](leptophobic.pdf){width="12cm"}
As another concrete example to demonstrate the impact of our formalism, we consider a “minicharged” particle scenario [@Holdom:1985ag], which is realized as the limit in which the mediator is a massless $U(1)$ vector boson which mixes kinetically with hypercharge [@Okun:1982xi; @Galison:1983pa; @Holdom:1985ag; @Dienes:1996zr; @Jaeckel:2013ija]. The dark sector matter (the Dirac fermion) that is charged under the additional U(1) interacts with the standard model only through this mixing, which is parametrized via the mixing angle, $\kappa$, in the gauge invariant Lagrangian term $\mathcal{L} \supset -\frac{\kappa}{2}F^{\mu\nu}X_{\mu\nu}$, where $F^{\mu\nu}$ and $X^{\mu\nu}$ are respectively the field strength tensors of the SM and dark U(1) gauge groups.
One can diagonalize the kinetic term in the Lagrangian with a field redefinition, the result of which is to induce electromagnetic interactions with the dark particles, which have an effective “minicharge”, $\epsilon = \kappa g_{\rm h}/e$, where $g_{\rm h}$ is the hidden photon-dark fermion coupling and $e$ is the electromagnetic coupling constant. Then the cross section for both production and scattering scale with $\epsilon^2$. For our scenario involving quarks, the appropriate quark charges must be included in the cross section, such that the coupling of the mediator to the nucleus is correctly modeled as proceeding via mixing with the photon. The exclusion limits on the minicharge $\epsilon$ are shown in Fig. \[fig:excl\_full\]. It is worth noting that the constraint from the effective number of light particle species, $N_{\rm eff}$, is strong but subject to astrophysical uncertainties that make terrestrial collider based studies worthwhile.
In Fig. \[fig:excl\_e613\], we compare (only) the constraints on the mini-charge model previously derived from E613 [@Golowich] with those derived in this work, in the plane of the mini-charged particle mass and $\epsilon$. The results from our analysis using only the deeply inelastic scattering regime are shown as the red dashed line, whereas the inclusion of the low $Q^2 < 1{\ \mathrm{GeV}}^2$ regime via dipole scattering leads to the solid red line. A large improvement in the strength of the bound from the improved treatment of the low transfer scattering is evident. The previous constraint [@Golowich] is shown as the shaded region, and shows a marked transition in the strength of the bound on $\epsilon$ by about an order of magnitude as the particle mass crosses a few hundred MeV. This sharp transition is the result of dark particle production through meson decay, which switches off around 500 MeV, leaving Drell-Yan production of the dark particles to dominate. We have not included this production mechanism in our bound, as it is model-dependent and tangential to our goal of an improved description of the $\chi$-nucleus scattering cross section. A more appropriate comparison of the impact of our improved computations is to the blue dashed curve, which extrapolates the previous bound by extending the Drell-Yan-only limit to lower masses. Of course, the actual bound on the mini-charged model at low mass would be better represented by including the $\chi$ production from meson decay together with our improved treatment of the scattering, though this is beyond the scope of this work. Clearly, fixed target experiments are a fertile ground for testing this class of models.
![Exclusion limits for minicharged particles in the MeV to GeV mass regime, including the results of this analysis. Other constraints are shown, arising from colliders [@Davidson:1991si], a SLAC beam dump [@Prinz:1998ua], the LHC [@Jaeckel:2012yz], CMB [@Dolgov:2013una; @Dubovsky:2003yn] and recent work on the number of light species, $N_{\rm eff}$ [@Vogel:2013raa].[]{data-label="fig:excl_full"}](minicharge_bounds.pdf){width="12"}
![The minicharge constraints arising only from the E613 experiment. See text for details.[]{data-label="fig:excl_e613"}](minicharge_e613.pdf){width="12"}
Conclusions {#sec:conclusions}
===========
We have investigated the detection of dark Dirac fermions in the context of the E613 beam dump experiment. The model employed gives rise to neutral current scattering, but in the absence of a heavy electroweak gauge boson to mediate the interaction. We studied the deep inelastic scattering in the detector in detail, and introduced a model valid at the low $Q^2$ values that become important in the absence of a heavy mediator to regulate the $1/Q^4$ behavior of the cross section. By including the effects of scattering at $Q^2 < 1 {\ \mathrm{GeV}}^2$ with a well theoretically motivated dipole model, we substantially improve upon constraints calculated using parton-level deep inelastic scattering alone. This could be especially relevant for new particle searches at future high energy beam dump facilities, which would allow access to regions of low $Q^2$ and small Bjorken-$x$.
Acknowledgements {#acknowledgements .unnumbered}
================
DES thanks Francesco Hautmann for helpful conversations. CJW thanks ITP Heidelberg for hospitality while performing some of this work, Joerg Jaeckel for providing questions and Céline Bœhm for now ancient but very useful conversations. MS acknowledges the hospitality of the Aspen Center for Physics while part of this work was finished. MS and CJW thank Lucian Harland-Lang for enlightenment regarding nuclear parton distribution functions. All authors are grateful to Paolo Gondolo for the suggestion to consider extending an earlier analysis of MINOS to old beam dump experiments. The research of TMPT is supported in part by NSF grant PHY-1316792 and by the University of California, Irvine through a Chancellor’s Fellowship. The research of DES was supported in part by the United States Department of Energy.
------------------------------------------------------------------------
Kinematics {#sec:kinematics}
==========
In this appendix, we present some of the details for the cross section for scattering a dark spin 1/2 particle, $\chi$, with momentum $p_\chi$, from a hadron A with momentum $P$. The hadron can be a nucleus. The dark particle has mass $m_\chi$ while the hadron has mass $M$. The dark particle mass may be of order 1 GeV or it may be smaller. The dark particle energy in the hadron rest frame, which we call $E$, is large compared to 1 GeV. We will introduce two different models for the scattering cross section.
In the hadron rest frame, we write the components of $P$ and $p_\chi$ as $$\begin{aligned}
P &=& (M,0,0,0)
\;, \\
p_\chi &=& (E,0,0,k)
\;,
$$ where $k = \sqrt{E^2 - m_\chi^2}$. The final state $\chi$ has 4-momentum $$p_\chi' = (E-\nu, k' \sin\theta \cos\phi, k' \sin\theta \sin\phi, k'\cos\theta)
\;,
$$ where $$\label{eq:kprimedef}
k' = \sqrt{(E-\nu)^2 - m_\chi^2}
\;.
$$ We suppose that the $\chi$ in the final state is not observed. Thus we will integrate over $p_\chi'$.
The momentum transfer is $$q = p_\chi - p_\chi'
$$ and is characterized by the energy transfer $\nu$, $$\label{eq:qdotP}
q \cdot P = M \nu
\;,
$$ and by the invariant $$Q^2 = - q^2
$$ with $Q^2 > 0$. In terms of the final state $\chi$ momentum, $$Q^2 =
2 E (E-\nu)
- 2 k k' \cos\theta
-2m_\chi^2
\;.
$$ We define $$x_{\rm bj} = \frac{Q^2}{2 M\nu}
\;.
$$ Note that $x_{\rm bj} \le 1$. Also note that when A is a nucleus of baryon number $A$, one often defines a scaled $x_{\rm bj}$ equal to $A Q^2/(2 M \nu)$. We do not do that here. However, we note that the ultimate limit on $x_{\rm bj}$ is $x_{\rm bj} \le 1$, but the practical limit beyond which the cross section is very small is $x_{\rm bj} \le 1/A$.
We will integrate over $Q^2$ and $\nu$ and will need the integration limits. Begin with $\nu$. We will impose a cut $$\nu > E_{\rm cut}
\;\;.
$$ That is, we wish to calculate the cross section for the process when at least a certain amount of energy $E_{\rm cut}$ is delivered to the hadron. Also Eq. (\[eq:kprimedef\]) and $k^{\prime\,2} > 0$ gives $\nu < E - m_\chi$. Thus the integration range for $\nu$ is $$\label{eq:nurange}
E_{\rm cut} < \nu < E - m_\chi
\;.
$$ Next, we need the limits on $Q^2$ at fixed $\nu$. Define a function $\mu^2(\nu)$ by $$k k' = E (E-\nu) - \mu^2(\nu) - m_\chi^2
\;\;.
$$ Then $$\label{eq:muofnudef0}
\mu^2(\nu)
=
\frac{m_\chi^2 \nu^2}
{[E(E-\nu) - m_\chi^2] + \sqrt{[E(E-\nu) - m_\chi^2]^2 - m_\chi^2\nu^2}}
\;.
$$ Then $Q^2$ has a simple form in terms of $\mu^2 (\nu )$, $$Q^2 =
2 [E (E-\nu)-m_\chi^2] [1 - \cos\theta]
+ 2 \mu^2(\nu) \cos\theta
\;\;.
$$ One boundary of the integration region is forward scattering, $\cos\theta = 1$. On this boundary we have $$Q^2 =
2 \mu^2(\nu)
\;\;,
\hskip 2 cm \cos\theta = 1
\;\;.
$$ The other boundary of the integration region is at $\cos\theta = -1$. There, we have $$Q^2 =
4 [E (E-\nu)-m_\chi^2]
- 2 \mu^2(\nu)
\;\;,
\hskip 2 cm \cos\theta = -1
\;\;.
$$ For small $m_\chi$, this is $Q^2 \approx 4 E (E-\nu)$ with small corrections. Put together, the inequalities $-1 < \cos \theta < 1$ lead to $$\label{eq:Qsqbounds}
2 \mu^2(\nu) < Q^2
<
4 [E (E-\nu)-m_\chi^2]
- 2 \mu^2(\nu)
\;\;.
$$ There is a separate upper bound for $Q^2$. The momentum $q$ is absorbed by the hadron, giving a final state with momentum $P+q$. We need $(P+q)^2 > M^2$. This condition gives $x_{\rm bj} < 1$ or $$\label{eq:QSqlessthan2mnu}
Q^2 < 2 M \nu
\;.
$$ Having found the integration limits, we translate the integration over $p_\chi'$ into integration over $\nu$ and $Q^2$ (integrating over the azimuthal angle $\phi$ to give a factor $2\pi$). We obtain, $$\begin{split}
\label{eq:jacobian}
d^4 p'_\chi\ \delta({p'_\chi}^2- m_\chi^2) ={}& \frac{k'^2 dk'}{2E'}\, d\cos\theta\, d\phi
\\
={}& \frac{\pi}{2\sqrt{E^2 - m_\chi^2}}\ d\nu\, dQ^2
\;\;.
\end{split}$$ where $E' = E - \nu$ and $k'$ is given by Eq. (\[eq:kprimedef\]).
We will introduce two different models for the structure functions, and in particular for $F_{\mathrm{T}}$. We will simply state the results of these models. However, if we want to examine the physics behind the models, it is convenient to use choose our reference frame wisely. We note that $F_{\mathrm{T}}$ and $F_{\mathrm{L}}$ depend only on $q$ and $P$. Thus we should choose a frame in which $q$ and $P$ are simple. We choose a frame in which both $q$ and $P$ have no transverse components and in which $q$ has a positive 3-component. Additionally, we now write momentum components in $(p^+,p^-,\bm p^T)$ format with $p^\pm = (p^0 \pm p^3)/\sqrt 2$. In our new frame, $$\begin{split}
P ={}& (M/\sqrt 2,M/\sqrt 2, \bm 0)
\;\;,
\\
q ={}& \left(
\frac{1}{\sqrt 2}
\left[\nu + \sqrt{\nu^2 + Q^2}\right],
-\frac{1}{\sqrt 2}\,\frac{Q^2}{\nu + \sqrt{\nu^2 + Q^2}},
\bm 0
\right)
\end{split}$$ In the kinematic region important for this paper, $Q^2 \ll \nu^2$, so that $$q \approx \left(
\sqrt 2\, \nu,
-\frac{Q^2}{2\sqrt 2\, \nu},
\bm 0
\right)
\;\;.
$$ Thus $q^+ \gg q^-$.
[99]{}
G. Bertone, D. Hooper and J. Silk, Phys. Rept. [**405**]{}, 279 (2005) \[hep-ph/0404175\]. D. S. Akerib [*et al.*]{} \[LUX Collaboration\], arXiv:1310.8214 \[astro-ph.CO\]. E. Aprile [*et al.*]{} \[XENON100 Collaboration\], arXiv:1207.5988 \[astro-ph.CO\]. Z. Ahmed [*et al.*]{} \[CDMS Collaboration\], Phys. Rev. D [**81**]{}, 042002 (2010) \[arXiv:0907.1438 \[astro-ph.GA\]\]. G. Angloher, M. Bauer, I. Bavykina, A. Bento, C. Bucci, C. Ciemniak, G. Deuter and F. von Feilitzsch [*et al.*]{}, Eur. Phys. J. C [**72**]{}, 1971 (2012) \[arXiv:1109.0702 \[astro-ph.CO\]\]. R. Essig, A. Manalaysay, J. Mardon, P. Sorensen and T. Volansky, Phys. Rev. Lett. [**109**]{}, 021301 (2012) \[arXiv:1206.2644 \[astro-ph.CO\]\]. P. Cushman, C. Galbiati, D. N. McKinsey, H. Robertson, T. M. P. Tait, D. Bauer, A. Borgland and B. Cabrera [*et al.*]{}, arXiv:1310.8327 \[hep-ex\]. J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. -B. Yu, Phys. Lett. B [**695**]{}, 185 (2011) \[arXiv:1005.1286 \[hep-ph\]\]. Y. Bai, P. J. Fox and R. Harnik, JHEP [**1012**]{}, 048 (2010) \[arXiv:1005.3797 \[hep-ph\]\]. J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. -B. Yu, Phys. Rev. D [**82**]{}, 116010 (2010) \[arXiv:1008.1783 \[hep-ph\]\]. P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D [**85**]{}, 056011 (2012) \[arXiv:1109.4398 \[hep-ph\]\]. A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangco, Phys. Rev. D [**84**]{}, 095013 (2011) \[arXiv:1108.1196 \[hep-ph\]\]. Y. Bai and T. M. P. Tait, arXiv:1208.4361 \[hep-ph\]. T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**108**]{}, 211804 (2012) \[arXiv:1203.0742 \[hep-ex\]\]. K. Cheung, P. -Y. Tseng, Y. -L. S. Tsai and T. -C. Yuan, JCAP [**1205**]{}, 001 (2012) \[arXiv:1201.3402 \[hep-ph\]\]. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], arXiv:1204.0821 \[hep-ex\]. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], arXiv:1206.5663 \[hep-ex\]. . Aad [*et al.*]{} \[ATLAS Collaboration\], JHEP [**1304**]{}, 075 (2013) \[arXiv:1210.4491 \[hep-ex\]\]. B. Batell, M. Pospelov and A. Ritz, Phys. Rev. D [**79**]{}, 115008 (2009) \[arXiv:0903.0363 \[hep-ph\]\]. B. Batell, M. Pospelov and A. Ritz, Phys. Rev. D [**80**]{}, 095024 (2009) \[arXiv:0906.5614 \[hep-ph\]\]. R. Essig, R. Harnik, J. Kaplan and N. Toro, Phys. Rev. D [**82**]{}, 113008 (2010) \[arXiv:1008.0636 \[hep-ph\]\]. J. L. Hewett, H. Weerts, K. S. Babu, J. Butler, B. Casey, A. de Gouvea, R. Essig and Y. Grossman [*et al.*]{}, arXiv:1401.6077 \[hep-ex\]. K. R. Dienes, C. F. Kolda and J. March-Russell, Nucl. Phys. B [**492**]{}, 104 (1997) \[hep-ph/9610479\]; T. G. Rizzo, Phys. Rev. D [**59**]{}, 015020 (1998) \[hep-ph/9806397\]; F. del Aguila, M. Masip and M. Perez-Victoria, Nucl. Phys. B [**456**]{}, 531 (1995) \[hep-ph/9507455\]; J. Kumar and J. D. Wells, Phys. Rev. D [**74**]{}, 115017 (2006) \[hep-ph/0606183\]. J. D. Bjorken, R. Essig, P. Schuster and N. Toro, Phys. Rev. D [**80**]{}, 075018 (2009) \[arXiv:0906.0580 \[hep-ph\]\]. B. Batell, R. Essig and Z. ’e. Surujon, arXiv:1406.2698 \[hep-ph\]. C. D. Carone and H. Murayama, Phys. Rev. Lett. [**74**]{}, 3122 (1995) \[hep-ph/9411256\]; C. D. Carone and H. Murayama, Phys. Rev. D [**52**]{}, 484 (1995) \[hep-ph/9501220\]. P. deNiverville, M. Pospelov and A. Ritz, Phys. Rev. D [**84**]{}, 075020 (2011) \[arXiv:1107.4580 \[hep-ph\]\]. P. deNiverville, D. McKeen and A. Ritz, Phys. Rev. D [**86**]{}, 035022 (2012) \[arXiv:1205.3499 \[hep-ph\]\]. B. Batell, P. deNiverville, D. McKeen, M. Pospelov and A. Ritz, arXiv:1405.7049 \[hep-ph\]. J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP [**1106**]{}, 128 (2011) \[arXiv:1106.0522 \[hep-ph\]\]. N. D. Christensen and C. Duhr, Comput. Phys. Commun. [**180**]{}, 1614 (2009) \[arXiv:0806.4194 \[hep-ph\]\]. T. A. Romanowski, Acta Phys. Polon. B [**16**]{}, 179 (1985). M. E. Duffy, G. K. Fanourakis, R. J. Loveless, D. D. Reeder, E. S. Smith, S. Childress, C. Castoldi and G. Conforto [*et al.*]{}, Phys. Rev. D [**38**]{}, 2032 (1988). M. Hirai, S. Kumano and T. -H. Nagai, [*Determination of nuclear parton distribution functions and their uncertainties in next-to-leading order*]{}, Phys. Rev. C [**76**]{}, 065207 (2007) [\[INSPIRE\]](http://inspirehep.net/record/761288?ln=en). N. N. Nikolaev and B. G. Zakharov, [*Color transparency and scaling properties of nuclear shadowing in deep inelastic scattering*]{}, Z. Phys. C [**49**]{}, 607 (1991) [\[INSPIRE\]](http://inspirehep.net/record/298382?ln=en). K. J. Golec-Biernat and M. Wusthoff, [*Saturation effects in deep inelastic scattering at low Q\*\*2 and its implications on diffraction*]{}, Phys. Rev. D [**59**]{}, 014017 (1998) [\[INSPIRE\]](http://inspirehep.net/record/473813?ln=en). K. J. Golec-Biernat and M. Wusthoff, [*Saturation in diffractive deep inelastic scattering*]{}, Phys. Rev. D [**60**]{}, 114023 (1999) [\[INSPIRE\]](http://inspirehep.net/record/496774?ln=en). A. H. Mueller, [*Parton saturation at small x and in large nuclei*]{}, Nucl. Phys. B [**558**]{}, 285 (1999) [\[INSPIRE\]](http://inspirehep.net/record/498572?ln=en). L. Frankfurt, A. Radyushkin and M. Strikman, [*Interaction of small size wave packet with hadron target*]{}, Phys. Rev. D [**55**]{}, 98 (1997) [\[INSPIRE\]](http://inspirehep.net/record/418564?ln=en). J. R. Forshaw, G. Kerley and G. Shaw, [*Extracting the dipole cross-section from photoproduction and electroproduction total cross-section data*]{}, Phys. Rev. D [**60**]{}, 074012 (1999) [\[INSPIRE\]](http://inspirehep.net/record/496674?ln=en). M. McDermott, L. Frankfurt, V. Guzey and M. Strikman, [*Unitarity and the QCD improved dipole picture*]{}, Eur. Phys. J. C [**16**]{}, 641 (2000) [\[INSPIRE\]](http://inspirehep.net/record/512804?ln=en). E. Gotsman, E. Levin, M. Lublinsky, U. Maor, E. Naftali and K. Tuchin, [*Has HERA reached a new QCD regime?: (Summary of our view)*]{}, J. Phys. G [**27**]{}, 2297 (2001) [\[INSPIRE\]](http://inspirehep.net/record/535295?ln=en). A. H. Mueller, [*Parton saturation: An Overview*]{}, in J.P.Blaizot and E. Iancu, eds. “QCD perspectives on hot and dense matter. Proceedings, NATO Advanced Study Institute”, Cargese, France, August, 2001, hep-ph/0111244 [\[INSPIRE\]](http://inspirehep.net/record/609280). K. J. Golec-Biernat, [*Physics of parton saturation*]{}, Acta Phys. Polon. B [**35**]{}, 3103 (2004) [\[INSPIRE\]](http://inspirehep.net/record/669868?ln=en). K. J. Golec-Biernat, [*Theoretical review of diffractive phenomena*]{}, Nucl. Phys. A [**755**]{}, 133 (2005) [\[INSPIRE\]](http://inspirehep.net/record/687954?ln=en). P. Tribedy and R. Venugopalan, [*Saturation models of HERA DIS data and inclusive hadron distributions in p+p collisions at the LHC*]{}, Nucl. Phys. A [**850**]{}, 136 (2011) \[Erratum-ibid. A [**859**]{}, 185 (2011)\] [\[INSPIRE\]](http://inspirehep.net/record/875812?ln=en). A. Luszczak and H. Kowalski, [*Dipole model analysis of high precision HERA data*]{}, arXiv:1312.4060 \[hep-ph\] [\[INSPIRE\]](http://inspirehep.net/record/1269467?ln=en). F. Hautmann, D. E. Soper, [*Parton distribution function for quarks in an s-channel approach*]{}, Phys. Rev. [**D75**]{}, 074020 (2007) [\[INSPIRE\]](http://inspirehep.net/record/744097?ln=en). J. Bartels, K. J. Golec-Biernat and H. Kowalski, [*A modification of the saturation model: DGLAP evolution*]{}, Phys. Rev. D [**66**]{}, 014001 (2002) [\[INSPIRE\]](http://inspirehep.net/record/584724?ln=en). M. V. T. Machado, Eur. Phys. J. C [**47**]{} (2006) 365. Golowich, E. and Robinett, R. W. Phys. Rev. D [**35**]{}, 391 (1987).
B. Holdom, Phys. Lett. B [**166**]{}, 196 (1986). L. B. Okun, Sov. Phys. JETP [**56**]{}, 502 (1982) \[Zh. Eksp. Teor. Fiz. [**83**]{}, 892 (1982)\]. P. Galison and A. Manohar, Phys. Lett. B [**136**]{}, 279 (1984). J. Jaeckel, Frascati Phys. Ser. [**56**]{}, 172 (2012) \[arXiv:1303.1821 \[hep-ph\]\]. S. Davidson, B. Campbell and D. C. Bailey, Phys. Rev. D [**43**]{}, 2314 (1991). A. A. Prinz, R. Baggs, J. Ballam, S. Ecklund, C. Fertig, J. A. Jaros, K. Kase and A. Kulikov [*et al.*]{}, Phys. Rev. Lett. [**81**]{}, 1175 (1998) \[hep-ex/9804008\]. J. Jaeckel, M. Jankowiak and M. Spannowsky, Phys. Dark Univ. [**2**]{} (2013) 111 \[arXiv:1212.3620 \[hep-ph\]\]. A. D. Dolgov, S. L. Dubovsky, G. I. Rubtsov and I. I. Tkachev, Phys. Rev. D [**88**]{}, no. 11, 117701 (2013) \[arXiv:1310.2376 \[hep-ph\]\]. S. L. Dubovsky, D. S. Gorbunov and G. I. Rubtsov, JETP Lett. [**79**]{}, 1 (2004) \[Pisma Zh. Eksp. Teor. Fiz. [**79**]{}, 3 (2004)\] \[hep-ph/0311189\]. H. Vogel and J. Redondo, JCAP [**1402**]{}, 029 (2014) \[arXiv:1311.2600 \[hep-ph\]\].
[^1]: Some dark matter particles can be lost on their way to the detector because they scatter in the rock that lies between the production point and the detector or in the iron shielding of the detector. We discuss this effect in the calculations of Sec. \[sec:dm\].
[^2]: Golec-Biernat and Wüsthoff write this in the form $2\int\! d\bm b\ \Xi(\bm b, \bm \Delta) = \sigma_0 [1 - \exp(- \Delta^2/(2 R_0^2)]$, which is approximately equivalent when $\sigma_0$ and $R_0$ are suitably adjusted.
|
---
address: |
Department of Physics, University of California,\
Riverside, CA 92521, USA
author:
- 'M. GIUNTA'
title: 'RAPIDITY GAPS IN GLUON JETS / COLOR RECONNECTION AT LEP'
---
Introduction
============
Rapidity is a standard variable used to describe the phase space distribution of particles in a multihadronic event. It is defined by $y= \frac{1}{2} \ln \left( \frac{E+p_{\parallel}}{E-p_{\parallel}} \right)$ with $E$ the energy of a particle and $p_{\parallel}$ the 3-momentum component along an axis [^1].
A rapidity gap event is an event in which two populated regions in rapidity are separated by an empty region. Color reconnection (CR), i.e a rearrangement of the color structure of an event, can be a source of this type of events. In the standard Monte Carlo models only the simplest configuration is allowed (Fig.\[fig\_diag\]a), where the color flux tube is stretched from a quark to the corresponding antiquark without crossing. If CR is included in the Monte Carlo, we can have more complex diagrams, in which string segments can either cross or appear as disconnected entities whose endpoints are gluons (Fig.\[fig\_diag\]b); these diagrams are of higher order in Quantum Chromodynamics (QCD) and are suppressed by a factor $1/N_C^2$, where $N_C=3$ is the number of colors.
In events with an isolated gluonic system (Fig.\[fig\_diag\]b) a rapidity gap can form between the particles coming from the hadronization of the isolated segment - often the leading (highest rapidity) part of a gluon jet - and the rest of the event. Thus rapidity gaps in gluon jets provide a sensitive means to search for color reconnection effects.
------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------
{width="3.cm"} {width="3.cm"}
------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------
\[fig\_diag\]
Color reconnection study
========================
To establish the sensitivity of the analysis to processes with color reconnection, multihadronic events are generated using Monte Carlo simulations, both with and without the effects of color reconnection.
The models without color reconnection used in the following studies are the Jetset [@jet], Herwig [@hw1; @hw2] and Ariadne [@ar] Monte Carlo programs, version 7.4, 6.2 and 4.11 respectively. The models which incorporate color reconnection are the model of Lönnblad [@ar-cr] implemented in the Ariadne Monte Carlo [^2], the color reconnection model [@hw2] in the Herwig Monte Carlo, and a model introduced by Rathsman [@rath] implemented in the Pythia Monte Carlo, version 5.7. In the following I refer to these as the Ariadne-CR, Herwig-CR, and Rathsman-CR models, respectively.
The general strategy used in this type of analysis is the following. After verifying that all the models, with and without CR, give a good description of the global features of hadronic events, gluon jets with a rapidity gap are selected. This is done requiring a large value for the smallest particle rapidity in a jet, $y_{min}$, or a large value for the maximum difference between the rapidities of adjacent rapidity-ordered particles, $\Delta y_{max}$. The *leading part* of the jet is defined by the particles beyond the gap. Then the predictions given by the different Monte Carlo models for the distributions of the charged particle multiplicity $n^{ch}_{leading}$ and the total electric charge $Q_{leading}$ of the leading part of the jet are compared to the experimental data.
Rapidity gap analysis
---------------------
In a recent note [@op-pn518] OPAL chooses a relatively complete and uncorrelated set of distributions sensitive to global event properties and evaluates the total $\chi^2$ value between the hadron level predictions of the different models and the corrected data. For a total of 81 bins, the $\chi^2$ is 36.9 for Ariadne and 32.4 for Ariadne-CR; 200.7 for Jetset and 243.5 for Rathsman-CR; 127.9 for Herwig and 151.6 for Herwig-CR. This result illustrates the fact that color reconnection has only a small effect on the global features of inclusive $\mathrm{e^+e^-}$ events.
All the events are then forced to three jets using the Durham [@durham] jet finder with a variable value for the resolution scale $y_{cut}$ and the two quark ($\mathrm{q~ or ~\bar{q}}$) jets are identified using a b-tagging technique [@b-tag], the remaining jet is the gluon jet.
The rapidity gap is defined using charged and neutral particles and requiring $y_{min}>1.4$, or $\Delta y_{max}>1.3$ for jets with $y_{min}<1.4$, where 1.4(1.3) is chosen because it is the value where the prediction for the $y_{min}$ ($\Delta y_{max}$) distribution given by the models with CR starts to deviate from that of the corresponding model without CR.
The $n^{ch}_{leading}$ and $Q_{leading}$ distributions are normalized to the total number of selected jets before the rapidity gap requirement. The results are shown in Fig.2. Both Rathsman-CR and Ariadne-CR predict a large excess of entries at $n^{ch}_{leading}=2$ and 4 and at $Q_{leading}=0$. These are consequences of events with an isolated gluonic system in the leading part of the gluon jet, which is neutral and decays into an even number of charged particles. Herwig-CR predicts a less striking excess for $3\leq n^{ch}_{leading}\leq 5$. Jetset and Ariadne give predictions 15-20 % too low for the $Q_{leading}=0$ bin but as there is no spiking behaviour in the data for the $n^{ch}_{leading}$ distribution – the most unambiguous signal for reconnection – it cannot be concluded this is due to CR.
{width="10.cm"} \[fig\_nch\]
ALEPH, performing a similar analysis, observes the same excess in the $Q_{leading}=0$ bin for Rathsman-CR and Ariadne-CR.
The parameter values used for the different Monte Carlo models (Jetset, Rathsman-CR, Ariadne and Ariadne-CR) are obtained from fits to global quantities. Three jet events are selected using Durham with fixed $y_{cut}=0.01$, then gluon jets are identified using energy ordering: the jet energies are recalculated using massless kinematics and ordered in energy, the lowest energy jet is identified as the gluon jet.
The rapidity gap is defined using charged particles only and requiring $y_{min}=1.5$. For comparison, quark jets from two jet events and the two higher energy jets from three jet events are also analyzed. The distributions are normalized to the number of jets with a rapidity gap.
It is observed that all the models give an excellent description of the $Q_{leading}$ distribution of quark jets. For the gluon jets, the models with CR predict an excess in the $Q_{leading}=0$ bin and the models without CR predict too few neutral jets, consistent with the result by OPAL. ALEPH performs the following checks: defines the gap using $y_{min}$=1.7 or 2.0 or using charged and neutral particles, changes some cuts used to define jets; they reach qualitatively the same conclusions.
Models re-tuning
----------------
OPAL then checks if it is possible to vary the Monte Carlo parameters in order to describe the gluon jet data while continuing to give a good description of the global features of the events.
The Rathsman-CR model describes these two distributions if the value $R_0=0.0085\pm0.0075(stat.)\pm0.0087(syst.)$ is used for the CR suppression factor, this result is consistent with $R_0=0$, that means no CR allowed.
Then OPAL tries to re-tune the models with CR so that they can describe the $n^{ch}_{leading}$ and $Q_{leading}$ data distributions. For Rathsman-CR this is only possible using $Q_0=5.5~ \mathrm{GeV/c^2}$ and $b=0.27 ~\mathrm{GeV^{-2}}$, but the description of inclusive $\mathrm{Z^0}$ measurements is severely degraded – the total $\chi^2$ increases from 243.5 to 1117.7 –. For Ariadne-CR that result is only obtained with $p_{T,min}=4.7~ \mathrm{GeV/c}$ and $b=0.17 ~\mathrm{GeV^{-2}}$, but also in this case the consequence is a bad description of the global features of hadronic events – the total $\chi^2$ goes from 32.4 to 3019.3 –. OPAL concludes that the Rathsman-CR and the Ariadne-CR models are both disfavored.
Summary and Conclusion
======================
Results by the OPAL and ALEPH collaborations have been presented.
It is observed that models with color reconnection predict a large excess of gluon jets with a rapidity gap and large spikes in some distributions sensitive to color reconnection effects, namely the charged particle multiplicity $n^{ch}_{leading}$ and the total electric charge $Q_{leading}$ of the leading part of the gluon jet, in disagreement with data. A tuning of the models with color reconnection in order to describe the $n^{ch}_{leading}$ and $Q_{leading}$ distributions results in a severe degradation of the description of the global features of the events.
It is concluded that color reconnection as currently implemented in the Rathsman-CR and Ariadne-CR models is strongly disfavored. No definite conclusion is obtained concerning the Herwig-CR model.
References {#references .unnumbered}
==========
[99]{} T. Sjöstrand, .
G. Marchesini [*et al*]{}, .
G. Corcella [*et al*]{}, .
L. Lönnblad, .
L. Lönnblad, .
J. Rathsman, .
OPAL Collab., *Rapidity gaps in gluon jets and a study of color reconnection*,\
OPAL Note PN518.
S. Catani [*et al*]{}, .
OPAL Collab., R. Akers [*et al*]{}, .
[^1]: usually the thrust, jet or beam axis
[^2]: the three different existing implementations, corresponding to settings of the parameter MSTA(35)=1, 2, or 3 are equivalent for hard processes involving a single color singlet system, such as $\mathrm{Z^0}$ decays.
|
---
abstract: 'An earlier calculation in a generalized linear sigma model showed that the well-known current algebra formula for low energy pion pion scattering held even though the massless Nambu Goldstone pion contained a small admixture of a two-quark two-antiquark field. Here we turn on the pion mass and note that the current algebra formula no longer holds exactly. We discuss this small deviation and also study the effects of an SU(3) symmetric quark mass type term on the masses and mixings of the eight SU(3) multiplets in the model. We calculate the s wave scattering lengths, including the beyond current algebra theorem corrections due to the scalar mesons, and observe that the model can fit the data well. In the process, we uncover the way in which linear sigma models give controlled corrections (due to the presence of scalar mesons) to the current algebra scattering formula. Such a feature is commonly thought to exist only in the non-linear sigma model approach.'
author:
- 'Amir H. Fariborz $^{\it \bf a}$ '
- 'Renata Jora $^{\it \bf b}$ '
- 'Joseph Schechter $^{\it \bf c}$ '
title: |
\
[ Low energy scattering with a nontrivial pion ]{}
---
Introduction
============
A linear sigma model with both quark-antiquark type fields and fields containing (in an unspecified configuration) two quarks and two antiquarks, seems useful for understanding the light scalar spectrum of QCD. In a previous treatment [@bigpaper] we considered a usual simplification in which the three light quark masses were taken to be zero. The model was seen to give a neat intuitive explanation of how “four quark" scalar states could be naturally much lighter than the conventional p-wave quark-antiquark scalars. We also verified in detail that, as long as the potential of the model satisfied SU(3)$_{\rm L}$ $\times$ SU(3)$_{\rm R}$ invariance, the massless version of the famous current algebra theorem [@W] on low energy pion pion scattering was correct.
In the present paper we introduce a common mass for the three light quarks in such a way that the pion gets its correct mass. SU(3) flavor invariance continues to hold, which is a desirable simplification. First we reexamine the masses and mixings of the particles in the model. It is seen that the natural explanation for the lightness of a “four quark" scalar remains unchanged. Then we reexamine the pion pion scattering amplitude to try to see if the low energy theorem continues to hold. Curiously we find that it does not exactly hold. What goes wrong? The algebra of the Noether currents should be good in this model so that is not the cause. It turns out that the partially conserved axial vector current, which is also required for the theorem, does not hold, unlike for the massless case. The axial vector current has a single particle contribution from the “heavy pion" in this chiral model as well as from the ordinary pion. The ordinary pion does not therefore completely saturate the axial current. Actually this is a small effect but is of conceptual interest and may be of more importance for the kaon scattering case.
A more important quantitative effect arises from the contributions of the scalar isosinglet mesons in the model. It is shown that these contributions can explain the experimental s wave isosinglet scattering length.
The notation is reviewed in Section II. The corrections to the masses and mixings, due to the quark mass term, are studied in Section III. An approximate analytic treatment of the scattering, for a general potential, is contained in Section IV. The exact numerical treatment for a “leading order" potential is presented in Section V. Some discussion and conclusions are given in Section VI. The Appendix explains the method of parameter determination from experiment and a listing of typical values for all the parameters.
Notation
========
We introduce the 3$\times$3 matrix chiral nonet fields; $$M = S +i\phi, \hskip 2cm
M^\prime = S^\prime +i\phi^\prime.
\label{sandphi}$$ Here $M$ represents scalar, $S$ and pseudoscalar, $\phi$ quark-antiquark type states, while $M^\prime$ represents states which are made of two quarks and two antiquarks. The transformation properties under SU(3)$_{\rm L}\times$ SU(3)$_{\rm R}
\times$ U(1)$_{\rm A}$ are $$M \rightarrow e^{2i\nu}
\, U_{\rm L} M U_{\rm R}^\dagger, \hskip 2cm
M^\prime \rightarrow e^{-4i\nu}
\, U_{\rm L} M^\prime U_{\rm R}^\dagger,
\label{Mchiral}$$ where $U_{\rm L}$ and $U_{\rm R}$ are unitary unimodular matrices, and the phase $\nu$ is associated with the U(1)$_{\rm A}$ transformation. The general Lagrangian density which defines our model is $${\cal L} = - \frac{1}{2} {\rm Tr}
\left( \partial_\mu M \partial_\mu M^\dagger
\right) - \frac{1}{2} {\rm Tr}
\left( \partial_\mu M^\prime \partial_\mu M^{\prime \dagger} \right)
- V_0 \left( M, M^\prime \right) - V_{SB},
\label{mixingLsMLag}$$ where $V_0(M,M^\prime) $ stands for a function made from SU(3)$_{\rm L} \times$ SU(3)$_{\rm R}$ (but not necessarily U(1)$_{\rm A}$) invariants formed out of $M$ and $M^\prime$. The quantity $V_{SB}$ stands for chiral symmetry breaking terms which transform in the same way as the quark mass terms in the fundamental QCD Lagrangian. In our previous paper [@bigpaper], we focused on general properties which continued to hold when $V_{SB}$ was set to zero. Here, we include the SU(3) symmetric mass term: $$V_{SB} = - 2\, A\, {\rm Tr} (S)
\label{vsb}$$ where $A$ is a real parameter. A characteristic feature of the model is the presence of “two-quark” and “four-quark” condensates: $$\left\langle S_a^b \right\rangle = \alpha_a \delta_a^b,
\quad \quad \left\langle S_a^{\prime b} \right\rangle =
\beta_a \delta_a^b.
\label{vevs}$$ We shall assume the vacuum to be SU(3)$_{\rm V}$ invariant, which implies $$\alpha_1 = \alpha_2 = \alpha_3 \equiv \alpha, \hskip 2cm
\beta_1 = \beta_2 = \beta_3 \equiv \beta.$$ The SU(3) particle content of the model consists of two pseudoscalar octets, two pseudoscalar singlets, two scalar octets and two scalar singlets. This gives us eight different masses and four mixing angles. We next give the notations for resolving the nonets into SU(3) octets and singlets. Note the matrix convention $\phi_a^b \to
\phi_{ab}$. The properly normalized singlet states are: $$\begin{aligned}
\phi_0=\frac{1}{\sqrt{3}}{\rm Tr}(\phi),\hspace{1cm}
\phi_0^\prime=\frac{1}{\sqrt{3}}{\rm Tr} (\phi^\prime),
\nonumber \\
S_0=\frac{1}{\sqrt{3}}{\rm Tr}(S),\hspace{1cm}
S_0^\prime=\frac{1}{\sqrt{3}}{\rm Tr}(S^\prime).
\label{singlets}\end{aligned}$$ Then we have the matrix decompositions: $$\begin{aligned}
\phi={\hat \phi}+\frac{1}{\sqrt{3}}\phi_{0}1,\hspace{1cm}
\phi^\prime={\hat \phi^\prime}+\frac{1}{\sqrt{3}}\phi_{0}^\prime1,
\nonumber \\
S={\hat S}+\frac{1}{\sqrt{3}}S_{0}1,\hspace{1cm}
S^\prime={\hat S^\prime}+\frac{1}{\sqrt{3}}S_{0}^{\prime}1,
\label{octets}\end{aligned}$$ wherein ${\hat \phi}$, ${\hat \phi^\prime}$, ${\hat S}$ and ${\hat S^\prime}$ are all 3 $\times $ 3 traceless matrices. The singlet scalar fields may be further decomposed as: $$S_0=\sqrt{3}\alpha+{\tilde S_0},\hspace{1cm}
S_0^\prime=\sqrt{3}\beta+{\tilde S_0}^\prime.
\label{scalarsinglets}$$ Here ${\tilde S_0}$ and ${\tilde S_0}^\prime$ are the fluctuation fields around the true ground state of the model. The breaking of SU(3) to the isospin group SU(2) will be examined in the future. In that case there are 16 different masses, four 2$\times$2 mixing matrices and two 4$\times$4 mixing matrices. To fully characterize the system we will also require some knowledge of the axial vector and vector currents obtained by Noether’s method: $$\begin{aligned}
(J_\mu^{axial})_a^b &=&(\alpha_a+\alpha_b)\partial_\mu\phi_a^b +
(\beta_a+\beta_b)\partial_\mu{\phi'}_a^b+ \cdots,
\nonumber \\
(J_\mu^{vector})_a^b &=&i(\alpha_a-\alpha_b){\partial_\mu} S_a^b +
i(\beta_a-\beta_b)\partial_\mu {S'}_a^b+ \cdots,
\label{currents}\end{aligned}$$ where the dots stand for terms bilinear in the fields.
In our model we use a previously discussed scheme to select the most important terms in the potential, $V_0(M,M')$. The favored terms which are SU(3)$_{\rm L}\times$SU(3)$_{\rm R}$ invariant but violate U(1)$_{\rm
A}$ are: $$V_\eta=c_3\, [F_\eta(M,M')]^2,
\label{veta}$$ in which $c_3$ is a coupling constant and $$F_\eta(M,M')=
\gamma_1\,
{\rm ln} \left(
{
{ {\rm det}(M)}
\over
{ {\rm det}(M^\dagger)}
}
\right)
+(1-\gamma_1) \,
{\rm ln}\left(
{
{ {\rm Tr}(MM'^\dagger) }
\over
{ {\rm Tr}(M'M^\dagger) }
}
\right),
\label{gamma1}$$ where $\gamma_1$ is a dimensionless parameter. This form exactly mocks up the U(1)$_{\rm
A}$ anomaly of QCD. Information about the pseudoscalar particles which is independent of the choice of the U(1)$_{\rm A}$ invariant terms in $V_0$ may be obtained by differentiating the following matrix equation representing the response of the potential to an infinitesimal axial transformation: $$\left[\phi,\frac{\partial V_0}{\partial S}\right]_+ -
\left[S,\frac{\partial V_0}{\partial \phi}\right]_+ +
(\phi,S)\rightarrow(\phi',S') =
1\left[ 2\, {\rm Tr} \left(\phi'\frac{\partial V_0}{\partial S'}-
S'\frac{\partial V_0}{\partial \phi'}\right) - 8\,c_3\, i\,
F_\eta(M,M')\right].
\label{geneq}$$ To get general constraints on the pseudoscalar particle masses we differentiate this equation once with respect to each of the two matrix fields: $\phi,\phi'$ and evaluate the equation in the ground state. Thus we also need the “minimum" condition, $$\left< \frac{\partial V_0}{\partial S}\right> + \left< \frac{\partial
V_{SB}}{\partial
S}\right>=0,
\quad \quad \left< \frac{\partial V_0}{\partial S'}\right> +
\left<\frac{\partial
V_{SB}}{\partial S'}\right> =0.
\label{mincond}$$
Masses and mixings
==================
As we previously discussed, the leading choice of terms corresponding to eight or fewer quark plus antiquark lines at each effective vertex reads: $$\begin{aligned}
V_0 =&-&c_2 \, {\rm Tr} (MM^{\dagger}) +
c_4^a \, {\rm Tr} (MM^{\dagger}MM^{\dagger})
\nonumber \\
&+& d_2 \,
{\rm Tr} (M^{\prime}M^{\prime\dagger})
+ e_3^a(\epsilon_{abc}\epsilon^{def}M^a_dM^b_eM'^c_f + h.c.)
\nonumber \\
&+& c_3\left[ \gamma_1 {\rm ln} (\frac{{\rm det} M}{{\rm det}
M^{\dagger}})
+(1-\gamma_1)\frac{{\rm Tr}(MM'^\dagger)}{{\rm Tr}(M'M^\dagger)}\right]^2.
\label{SpecLag}\end{aligned}$$ All the terms except the last two have been chosen to also possess the U(1)$_{\rm A}$ invariance.
The minimum equations for this potential are: $$\left\langle { {\partial V_0} \over {\partial S_a^a} } \right\rangle =
2 \,\alpha\, \left( - c_2 + 2\, c_4^a\, \alpha^2 + 4\, e_3^a \, \beta
\right) = 2A,
\label{mealpha}$$
$$\left\langle { {\partial V_0} \over
{\partial {S'}_a^a} } \right\rangle
=
2 \left( d_2\, \beta + 2\, e_3^a\, \alpha^2 \right) = 0.
\label{mebeta}$$
Differentiating the potential in Eq.(\[SpecLag\]) twice will yield four 2$\times$2 mass matrices denoted as $(M_\pi^2)$, $(M_0^2)$, $(X_a^2)$ and $(X_0^2)$ respectively for the pseudoscalar octets, the pseudoscalar singlets, the scalar octets and the scalar singlets. These may be brought to diagonal (hatted) form by the following 2$\times$2 orthogonal transformations: $$\begin{aligned}
\sum_{B,C}(R_{\pi}^{-1})_{AB}(M_{\pi}^2)_{BC}(R_{\pi})_{CD}&=&({\hat
M}_{\pi}^2)_{AD},\hskip 0.5cm
\sum_{B,C}(R_0^{-1})_{AB}(M_0^2)_{BC}(R_0)_{CD}=({\hat
M}_0^2)_{AD},
\nonumber \\
\sum_{B,C}(L_a^{-1})_{AB}(X_a^2)_{BC}(L_a)_{CD}&=&({\hat
X}_a^2)_{AD}, \hskip 0.5cm
\sum_{B,C}(L_0^{-1})_{AB}(X_0^2)_{BC}(L_0)_{CD}=({\hat
X}_0^2)_{AD},
\label{simtransf}\end{aligned}$$ Notice that the four mass matrices are identical to those given in section IV of [@bigpaper] for the zero quark mass case. The numerical values of the entries will however differ because the relations among the coefficients are different due to the presence of 2A rather than zero on the right hand side of Eq.(\[mealpha\]).
In the massive case there are 9 parameters (A, $\alpha$, $\beta$, $c_2$, $d_2$, $c_4^a$, $e_3^a$, $c_3$ and $\gamma_1$). These can be reduced to seven by use of the two minimum equations just given. We note that the parameters $c_3$ and $\gamma_1$, associated with modeling the U(1)$_{\rm A}$ anomaly, do not contribute to either the minimum equations or to the mass matrices of the particles which are not $0^-$ singlets. Thus it is convenient to first determine the other five independent parameters. As the corresponding experimental inputs [@ropp] we take the non-strange quantities: $$\begin{aligned}
m(0^+ {\rm octet}) &=& m[a_0(980)] = 984.7 \pm 1.2\, {\rm MeV}
\nonumber \\
m(0^+ {\rm octet}') &=& m[a_0(1450)] = 1474 \pm 19\, {\rm MeV}
\nonumber \\
m(0^- {\rm octet}') &=& m[\pi(1300)] = 1300 \pm 100\, {\rm MeV}
\nonumber \\
m(0^- {\rm octet}) &=& m_\pi = 137 \, {\rm MeV}
\nonumber \\
F_\pi &=& 131 \, {\rm MeV}
\label{inputs1}\end{aligned}$$
Evidently, a large experimental uncertainty appears in the mass of $\pi(1300)$; we shall initially take the other masses as fixed at their central values and vary this mass in the indicated range. Essentially $m[\pi(1300)]$ is being treated as an arbitrary parameter of our model. As shown in Eq.(\[lagpara\]) in Appendix A, it is straightforward to determine the five independent parameters in terms of these five inputs. This determination is a generalization of the one in the previous zero mass pion case in which four parameters were determined from four inputs.
The effects of adding a non zero quark mass term on the masses of the two predicted scalar singlets are displayed in Fig.\[ms0vsmpip\]. It is clear that the small mass term has a negligible effect on the mass of the heavier scalar singlet. On the other hand, there is a larger effect on the mass of the lighter scalar singlet. Still this singlet is exceptionally light so there is no qualitative difference in the result.
= 7.5cm
In Fig. \[4qp\] we display a comparison of the four quark percentages of the $\pi$ meson, the lighter $a_0$ meson and the lighter scalar singlet with the corresponding values in the model with zero quark masses. (These are, of course, equal to the two quark percentages of the heavier particles with the same quantum numbers). It is clear that there is not much change compared to the zero quark mass case.
= 7.5cm
It remains to discuss the four quark percentages of the two SU(3) singlet pseudoscalars. The lightest is the $\eta$(958) while candidates for the heavier one include $\eta$(1295), $\eta$(1405), $\eta$(1475) and $\eta$(1760). As in the zero mass case, the first two candidates are ruled out because they do not lead to positive eigenvalues of the prediagonal squared mass matrix $(M_0^2)$. For the other two scenarios we may find the numerical values of the remaining parameters $c_3$ and $\gamma_1$ by using Eqs. (\[phizeromixing\]), (\[K\]), (\[findgamma1\]) and (\[findc3\]) given in the Appendix. The four quark contents for the $\eta$(958) are being compared between the massive and massless quark cases in Fig. \[4qpeta\]. Note that there are two solutions for each scenario corresponding to Eq. (\[findgamma1\]) being of quadratic type. The lower four quark percentage curves seem the most plausible. Again there seems to be little difference between the zero and non-zero quark mass cases. This is understandable by comparing the values of the Lagrangian parameters found in Appendix A with those found in Appendix B of [@bigpaper].
= 7.5cm
Pion scattering: approximate analytic treatment for general V$_0$
=================================================================
We have seen that there are not very big changes when using an SU(3) symmetric quark mass term of proper strength to give the experimental mass value to the pion. However, for the discussion of the pion pion scattering amplitude near threshold, which is one of the most important applications of the chiral approach, the correct value of the pion mass is important. In the zero quark mass case we showed that for any choice of terms in the potential V$_0$, the current algebra expression for the threshold pion pion amplitude held exactly. This was understandable since the algebra of chiral currents held by construction and furthermore the pion completely saturated the axial current. In the present case, the pion does not, as we shall see, completely saturate the axial current. Thus it is interesting to study this case in more detail.
Note that the transformation between the diagonal fields ($\pi^+$ and $\pi'^+$) and the original pion fields is given as: $$\left[
\begin{array}{c} \pi^+ \\
\pi'^+
\end{array}
\right]
=
R_\pi^{-1}
\left[
\begin{array}{c}
\phi_1^2 \\
{\phi'}_1^2
\end{array}
\right]=
\left[
\begin{array}{c c}
\cos\theta_\pi & -\sin \theta_\pi
\nonumber \\
\sin \theta_\pi & \cos \theta_\pi
\end{array}
\right]
\left[
\begin{array}{c}
\phi_1^2 \\
{\phi'}_1^2
\end{array}
\right].
\label{mixingangle}$$ The value of the mixing angle, $\theta_\pi$ was previously written for an arbitrary potential V$_0$ and with the symmetry breaker, Eq.(\[vsb\]) as: $$\tan(2\theta_\pi)=\frac{-2y_\pi z_\pi}{y_\pi(1-z_\pi^2)-x_\pi},
\label{thetasubpi}$$ wherein, $$x_\pi = \frac{2A}{\alpha},\hskip .5cm
y_\pi =\left< \frac{\partial^2V}{\partial
{\phi'}_2^1\partial{\phi'}_1^2}
\right>, \hskip .5cm
z_\pi= \frac{\beta}{\alpha}.
\label{xyzpi}$$ The specific values of $x_\pi$, $y_\pi$ and $z_\pi$ will depend on the particular potential. In the case of no symmetry breaking, there was a big simplification and tan$\theta_\pi$ was given by just -$\beta/\alpha$. Furthermore, $x_\pi$ and $y_\pi$ would be respectively the squared pion mass and the squared $\pi(1300)$ mass in the absence of mixing. Clearly, the ratio $x_\pi/y_\pi$ is a very small number in the general case. We now make use of this fact to solve for the first correction to the mixing angle obtained from Eq.(\[thetasubpi\]): $$\begin{aligned}
\sin\theta_\pi&=&\frac{-\beta}{\sqrt{\alpha^2+\beta^2}}
\left[
1+\frac{\alpha^4}{\left(\alpha^2+\beta^2\right)^2}\frac{x_\pi}{y_\pi}
\right]
+\cdots,
\nonumber \\
\cos\theta_\pi&=&\frac{\alpha}{\sqrt{\alpha^2+\beta^2}}
\left[
1-\frac{\alpha^2\beta^2}{\left(\alpha^2+\beta^2\right)^2}
\frac{x_\pi}{y_\pi}
\right] +\cdots.
\label{sincos}\end{aligned}$$ In these equations the first terms correspond to the massless pion case and the second terms to the leading $x_\pi/y_\pi$ corrections when the pion mass is turned on.
An important application of Eq.(\[sincos\]) is to the axial vector current with the pion’s quantum numbers. After taking account of the mixing in Eq.(\[mixingangle\]), the right hand side of the first Eq.(\[currents\]) may be rewritten as: $$(J_\mu^{axial})_1^2=F_\pi\partial_\mu\pi^+
+F_{\pi'}\partial_\mu\pi'^+ +\cdots,
\label{axcur}$$ where, $$\begin{aligned}
F_\pi&=&2\, \alpha \cos\theta_\pi -2\, \beta \sin\theta_\pi,
\nonumber \\
F_{\pi'}&=&2\, \alpha \sin\theta_\pi + 2\, \beta \cos \theta_\pi.
\label{dconstants}\end{aligned}$$ In the zero pion mass case, $F_{\pi'}$ is seen to vanish. For the non zero pion mass case, we find, using Eq.(\[sincos\]), $$\begin{aligned}
F_\pi&=&2\sqrt{\alpha^2+\beta^2}+
{\cal O}
\left(
\frac{x_\pi}{y_\pi}
\right)^2,
\nonumber \\
F_{\pi'}&=&\frac{-2\alpha^3\beta}
{\left(\alpha^2+\beta^2\right)^{3/2}}
\left(
\frac{x_\pi}{y_\pi}
\right) +
{\cal O}
\left(\frac{x_\pi}{y_\pi}\right)^2.
\label{2Fs}\end{aligned}$$ It is seen that $F_\pi$ does not change much while $F_{\pi'}$ picks up a non zero value. Thus the $\pi'$ does not decouple from the axial vector current in the massive pion case. This means that PCAC does not strictly hold in the massive case and hence there is no reason to expect that the current algebra threshold theorem should be exactly correct in the present model.
As a check of the accuracy of the approximate formula (\[2Fs\]) for $F_{\pi'}$ we made an exact numerical calculation using the specific potential in Eq.(\[SpecLag\]) and found $F_{\pi'}=-6.937\times 10^{-4}$ GeV while, for the same parameters, the approximate formula gave $F_{\pi'}=-6.864\times 10^{-4}$ GeV.
Now let us discuss the pion pion scattering in the threshold region. Our initial goal will be to see what results may be obtained for a general choice of chiral invariant potential, $V_0$ in Eq.(\[mixingLsMLag\]). The general pattern of this discussion and the notation is given in sections V, VI and VII of [@bigpaper] for the massless pion case. We start with the conventional $\pi-\pi$ scattering amplitude at tree level: $$A(s,t,u)=-\frac{g}{2} +\sum_D
\left(
\frac{g_{8D}^2}{({\hat X}_a^2)_{DD}-s}
+\frac{g_{0D}^2}{({\hat X}_0^2)_{DD}-s}
\right).
\label{wholeamp1}$$ In this equation, $g$ denotes the coefficient of the four point contact interaction among the physical (mass diagonal) pions. Furthermore $g_{0D}$ denotes the three point coupling constants connecting the physical pions to the two physical SU(3) singlet scalar mesons. Similarly $g_{8D}$ stands for the coupling constants connecting the physical pions to the two physical scalar mesons which transform as the eighth component of an SU(3) octet. The usual Mandelstam variables, $s,t,u$ are being employed. It is straightforward to numerically calculate the coupling constants just mentioned and the amplitude if the form of $V_0$ is specified \[e.g. Eq. (\[SpecLag\])\]; this will be discussed in the next section. To proceed with the general case we note that, for example, the coupling constant $g_{0D}$ may be written as: $$g_{0D}=
\left<
{
{{\partial^3V}}
\over
{\partial{\pi}^+\partial{\pi}^- \partial(S_{0p})_D }
}
\right> =\sum_{A,B,C}(R_\pi)_{A1}(R_\pi)_{B1} (L_0)_{CD}
\left< {{\partial^3V}\over{\partial({\phi}_1^2)_A\partial({\phi}_2^1)_B
\partial(S_0)_C}}
\right>.
\label{chainrule}$$ As discussed in [@bigpaper], there is a relation between the three point coupling constants on the right hand side and two point elements of the squared mass matrices. Such a relation follows from differentiation of Eq. (\[geneq\]) together with the use of Eq.(\[mincond\]). In fact it is similar in form to the relation obtained for the zero mass pion case: $$g_{0D}=\frac{2}{\sqrt{3}F_{\pi}}(R_{\pi})_{A1} (L_0)_{HD}
\left[ (X_0^2)_{AH}-(M_{\pi}^2)_{AH} \right],
\label{physicalcoupling}$$ wherein we have now adopted the convention of summing over repeated indices. The elements of the pion transformation matrix, $(R_\pi)_{A1}$ are the angles $\sin \theta_\pi$ and $\cos \theta_\pi$ given in Eq. (\[sincos\]) for the present non zero pion mass case. In the zero pion mass case, $\sin \theta_\pi$ and $\cos \theta_\pi$ may be rewritten in terms of just $\alpha$ and $\beta$ as we see by setting the second and higher order terms on the right hand sides of Eq. (\[sincos\])to zero. That is the form in which Eq. (\[physicalcoupling\]) exactly holds also for the massive pion case. Since Eq. (\[physicalcoupling\])and its analog for the four point vertices play an important role in the proof of the current algebra theorem, we can only prove the theorem in the massive case when we make the (not too bad) approximation that $x_\pi/y_\pi$ is zero.
With this approximation we can show, by generalizing the treatment given in [@bigpaper], that the usual “current algebra" formula holds for the massive pion case. In the previous treatment, the second term on the right hand side of Eq.(\[physicalcoupling\]) made no contribution. Now we must take this term’s contribution into account. To get the same delicate cancellation due to chiral symmetry we must expand the amplitude in powers of $s-m_\pi^2$ instead of simply powers of $s$. Explicitly, $$\begin{aligned}
A(s,t,u)&=&-\frac{g}{2} +
\left(
\frac{g_{8D}^2}{({\hat
X}_a^2)_{DD}-({\hat M}_{\pi}^2)_{11}} +\frac{g_{0D}^2}{({\hat
X}_0^2)_{DD}-({\hat M}_{\pi}^2)_{11}}
\right)
\nonumber \\
&+& \left( s-({\hat M}_{\pi}^2)_{11} \right)
\left(
\frac{g_{8D}^2}{ \left[ ({\hat X}_a^2)_{DD}-({\hat
M}_{\pi}^2)_{11}\right]^2} +\frac{g_{0D}^2}{\left[({\hat
X}_0^2)_{DD}-({\hat
M}_{\pi}^2)_{11}\right]^2}
\right) +\cdots.
\label{expandedamp1}\end{aligned}$$ Note that $({\hat X}_a^2)_{DD}$ is a single number indexed by $D$. There is a huge simplification of the coefficients of the $(s-m_\pi^2)$ term: $$\begin{aligned}
&&\frac{g_{0D}^2}{[({\hat X}_0^2)_{DD}-({\hat M}_{\pi}^2)_{11}]^2}
\nonumber\\
&&=\frac{4}{3F_\pi^2}(R_\pi)_{A1}
\left[(X_0^2)_{AH}-(M_{\pi}^2)_{AH}\right]
(L_0)_{HD}
\frac{1}{\left[ ({\hat X}_0^2)_{DD}-({\hat
M}_{\pi}^2)_{11}\right]^2}(R_{\pi})_{C1}[(X_0^2)_{CK}-(M_{\pi}^2)_{CK}](L_0)_{KD}
\nonumber \\
&&=\frac{4}{3F_{\pi}^2}(R_{\pi}^{-1})_{1G}(L_0)_{GE}(L_0^{-1})_{EA}(X_0^2)_
{AH}
(L _0)_{HD}\frac{1}{ \left[({\hat X}_0^2)_{DD}-({\hat
M}_{\pi}^2)_{11}\right]^2}
(L_0^{-1})_{DK}(X_0^2)_{KC}(L_0)_{CF}(L_0^{-1})_{FJ}(R_{\pi})_{J1}-
\nonumber\\
&&
\frac{8}{3F_\pi^2}
(R_\pi)_{A1}(M_{\pi}^2)_{AM}(R_{\pi})_{MP}(R_{\pi}^{-1})_{
PH}(L_0)_{HD}
\frac{1}{\left[ ({\hat X}_0^2)_{DD}-({\hat M}_{\pi}^2)_{11}\right]^2}
(L_0^{-1})_{DK}(X_0^2)_{KC}(L_0)_{CF}(L_0^{-1})_{FJ}(R_{\pi})_{J1}+
\nonumber\\
&&\frac{4}{3F_{\pi}^2}(R_{\pi})_{A1}(M_{\pi}^2)_{AM}(R_{\pi})_{MP}(R_{\pi}^{-1})
_{PH}(L_0)_{HD}
\frac{1}{\left[({\hat X}_0^2)_{DD}-({\hat M}_{\pi}^2)_{11}\right]^2}
(R_{\pi}^{-1})_{1C}(M_{\pi}^2)_{CN}(R_{pi})_{NF}(R_{\pi}^{-1})_{FK}(L_0)_{KD}
\nonumber\\
&&=\frac{4}{3F_{\pi}^2}(R_{\pi}^{-1})_{1G}(L_0)_{GE}
\frac{\left[({\hat X}_0^2)_{DD}-({\hat M}_{\pi}^2)_{11}\right]^2}
{\left[({\hat X}_0^2)_{DD}-({\hat
M}_{\pi}^2)_{11}\right]^2}
(L_0^{-1})_{EJ}(R_{\pi})_{J1}=
\frac{4}{3F_{\pi}^2}.
\label{g2m402}\end{aligned}$$ Similarly, $$\frac{g_{8D}^2}{\left[({\hat X}_a^2)_{DD}-({\hat
M}_{\pi}^2)_{11}\right]^2}
=\frac{2}{3F_{\pi}^2}.
\label{g2m48}$$
Next we must consider the contribution of the terms independent of $(s-m_\pi^2)$. The four point coupling constant $g$ is approximately related to the matrix elements of the squared mass matrices (again with the proviso that the pion transformation matrices be similarly approximated) as: $$g=\frac{8}{F_{\pi}^2}
\left[
\frac{1}{3}(R_{\pi}^{-1})_{1D} \left[(
X_0^2)_{DJ}-(M_{\pi}^2)_{DJ}\right]
(R_{\pi})_{J1}+\frac{1}{6}(R_{\pi}^{-1})_{1D}
\left[(
X_0^2)_{DJ}-(M_{\pi}^2)_{DJ}\right](R_{\pi})_{J1}
\right].
\label{gfinal1}$$ (The analogous calculation for the zero mass pion case is given in section VII of [@bigpaper].) Using calculations similar to Eq.(\[g2m402\]) we also get: $$\begin{aligned}
\frac{g_{0D}^2}{ ({\hat X}_0^2)_{DD}
-({\hat M}^2_\pi)_{11}
}=
\frac{4}{3F_{\pi}^2}(R_{\pi}^{-1})_{1D}\left[(
X_0^2)_{DJ}-(M_{\pi}^2)_{DJ}\right](R_{\pi})_{J1} \label{g2m20}\end{aligned}$$
$$\begin{aligned}
\frac{g_{8D}^2}{({\hat X}_a^2)_{DD}-({\hat M}_{\pi}^2)_{11}}
&=&
\frac{4}{6F_{\pi}^2}(R_{\pi}^{-1})_{1D} \left[(
X_a^2)_{DJ}-(M_{\pi}^2)_{DJ}\right](R_{\pi})_{J1} \label{g2m28}\end{aligned}$$
Putting the last three equations into Eq.(\[expandedamp1\]) we see that the sum of the terms independent of $(s-m_\pi^2)$ vanishes in the given approximation. The usual formula, $$A(s,t,u)=\frac{2}{F_\pi^2}(s-m_\pi^2)+\cdots,
\label{caform}$$ is thus obtained as an approximation.
It should be remarked that Eq. (\[caform\]) corresponds to keeping only terms up to linear order in the expansion for $A(s,t,u)$. That means there are corrections, even at threshold, due to the masses of the scalars not being infinite. To see this and to summarize in a simple way, the preceding steps let us expand to one higher order, adopting a condensed notation in which $m_i$ stands for the mass of any of the four scalars while $g_i$ stands for the corresponding trilinear coupling constant of that scalar with two pions. The first four terms in the expansion of $A(s,t,u)$, as exactly obtained from Eq.(\[wholeamp1\]), are $$\begin{aligned}
A(s,t,u)&=&-\frac{g}{2}+
\sum_i\frac{g_i^2}{m_i^2-m_\pi^2}\left[
1+\frac{s-m_\pi^2}{m_i^2-m_\pi^2}+
(\frac{s-m_\pi^2}{m_i^2-m_\pi^2})^2+\cdots\right]
\nonumber \\
&\approx&(s-m_\pi^2)\left[\frac{2}{F_\pi^2}
+(s-m_\pi^2)\sum_i\frac{g_i^2}{(m_i^2-m_\pi^2)^3}+\cdots\right].
\label{correction}\end{aligned}$$ The exact first equation contains, for each $m_i$, a geometrical expansion in the quantity $(s-m_\pi^2)/
(m_i^2-m_\pi^2)$. Thus the radius of convergence in s for this expression is the squared mass of the lightest scalar singlet. To apply this expression in the resonance region we must, of course, unitarize the formula in some way. Here we will be content to look at the threshold region. In going from the first to the second equation we used the facts established above that (in the approximation where $F_{\pi'}=0$): 1) the sum of the first two terms of the first equation vanishes and 2) the third term of the first equation simplifies to becomes the first, current algebra, term of the second equation. The third term of the second equation represents the leading correction to the usual current algebra formula. It depends on the masses of the scalar mesons and would vanish in a hypothetical limit (often used) in which the scalar meson masses are taken to infinity. Note that every term in the approximate amplitude vanishes for $s=m_\pi^2$, an unphysical point called the Adler zero [@adler]. Our derivation shows that the Adler zero follows from the generating equation (\[geneq\]), which in turn expresses the chiral invariance of the potential, $V_0$ and from the saturation of the axial vector current by the pion field (so-called partial conservation of the axial current). The second equation in (\[correction\]) is an approximation, though a numerically good one, because the saturation of the axial current has been seen to be not strictly accurate in the present model.
The situation in the case of zero pion mass [@bigpaper] is slightly different. There the amplitude is proportional to $s$ so the Adler zero occurs at $s=0$, which is also the threshold. Thus the current algebra amplitude as well as the corrections due to non-infinite mass scalar mesons vanish at threshold in the zero pion mass case.
In the above we found that the current algebra theorem for a general potential does not seem to be exactly correct. This small deviation and in addition the more important effect of the scalar mesons will next be calculated exactly, by numerical means, for the scattering amplitude using the leading choice of $V_0$ discussed in section III.
Pion scattering: exact numerical treatment
==========================================
In the exact numerical treatment we do not need to make use of the relations between two point and three point functions and between three point and four point functions since we adopt the specific potential $V_0$ given in Eq. (\[SpecLag\]). The needed quantities for calculating the scattering amplitude are displayed in Eq. (\[wholeamp1\]): the four physical scalar singlet masses, the four three-point coupling constants conecting these scalars to two pions and the four-point pion physical coupling constant, $g$. These are obtained by, in turn, differentiating the potential twice with respect to two scalar fields ($\frac{{\partial}^2}
{\partial S\partial S}$) , two pseudoscalar fields as well as one scalar field ($\frac{{\partial}^3}
{\partial S\partial \phi\partial\phi}$) and four pseudoscalar fields ($\frac{{\partial}^4}
{\partial \phi\partial\phi\partial\phi\partial\phi}$). Furthermore we must use equations like Eq. (\[chainrule\]) to relate the “bare" amplitudes obtained by such differentiations to the physical ones (ie, in mass diagonal bases for the fields). The matrices transforming the fields to mass diagonal bases are defined in Eq. (\[simtransf\]) and are obtained by diagonalizing the relevant squared mass matrices.
For our purpose we [*define*]{} the current algebra result in terms of the expansion of the tree level amplitude $A(s,t,u)$ in powers of $(s-m_\pi^2)$, as displayed in Eq. (\[correction\]). Specifically, $$A(s,t,u)=C_0 +C_1(s-m_\pi^2)+ \cdots,
\label{Aexpansion}$$ where, $$\begin{aligned}
C_0&=&-\frac{g}{2}+\sum_i\frac{g_i^2}{m_i^2-m_\pi^2},
\nonumber \\
C_1&=&\sum_i\frac{g_i^2}{(m_i^2-m_\pi^2)^2}.
\label{C0C1} \end{aligned}$$ The current algebra result requires $C_0$ to vanish and $C_1=2/F_\pi^2$.
Plots of $C_0$ and $C_1$ as functions of the model parameter $m[\pi(1300)]$ are shown in Figs. \[C0plot\] and \[C1plot\] respectively. Even though $C_0$ is small it is clearly non vanishing. Also $C_1$ deviates by a few percent from the current algebra prediction. To estimate the numerical accuracy of this calculation it was repeated for the case of zero pion mass. There it was found that $C_0={\cal O}(10^{-8})$ whereas it should be exactly zero. Thus the accuracy of the calculation method is several orders of magnitude more sensitive than the indicated effect. In this model, the Adler zero is shifted (by about -$C_0/C_1$) very slightly to the left of $m_\pi^2$.
= 7.5cm
= 7.5cm
The small deviations from the current algebra result just discussed seem to be beyond present experimental accuracy. On the other hand, the “beyond" current algebra contributions to $A(s,t,u)$ due to the higher than linear terms in the expansion shown in Eq. (\[correction\]) seem to be highly relevant for comparison with present day experiments [@NA48; @E865; @dirac; @an]. These corrections would vanish in a limit where the scalar masses all go to infinity, which essentially corresponds to the use of a non-linear rather than the present linear type of sigma model.
It is usual to discuss the amplitudes near threshold in terms of their partial wave scattering lengths. The $J=0$ scattering lengths are of course especially affected by the presence of light scalar mesons. Using the compact notation in Eq. (\[correction\]), the (dimensionless) partial wave scattering lengths may be calculated to be: $$\begin{aligned}
m_\pi a_0^0&=&\frac{1}{32\pi}\left[-\frac{5g}{2}
+\sum_{i}g_i^2\left(\frac{3}{m_i^2-4m_\pi^2}
+\frac{2}{m_i^2
}\right)\right],
\nonumber \\
m_\pi a_0^2&=&\frac{1}{32\pi}\left[-g
+2\sum_{i}\frac{g_i^2}{m_i^2}\right].
\label{scatlen}\end{aligned}$$ These formulas are expressed in terms of the physical masses and coupling constants which are being computed exactly by numerical means. Note that the isospin label, $I$ and the angular momentum label, $J$ appear as $a_J^I$. For comparison, we may give the usual current algebra results [@W]: $$\begin{aligned}
m_\pi a_0^0 &=& \frac{7m_\pi^2}{16{\pi}F_\pi^2},
\nonumber \\
m_\pi a_0^2 &=& \frac{-2m_\pi^2}{16{\pi}F_\pi^2}.
\label{cascatlen}\end{aligned}$$
The results of our numerical calculation are shown in Fig. \[scatle\].
= 7.5cm
It is seen that the numerical calculation for the scattering length in the non resonant $I=2$ channel gives about the same value, -0.04 as the current algebra result. In the resonant $I=0$ s-wave channel the current algebra result of 0.15 is smaller than the result of the exact calculation for the range shown of the model parameter $m[\pi(1300)]$. The exact calculation result in this resonant channel varies strongly with $m[\pi(1300)]$ in contrast to the case in the non resonant channel. To understand what this means we should ask what is the significance of varying $m[\pi(1300)]$. Such a variation might be associated with variations in the masses of the four iso-singlet scalars. But the set up of the model as shown in Eq. (\[inputs1\]) fixes the physical masses of the two octet isosinglets (at 985 MeV and 1474 MeV) . Furthermore we see in Fig. \[ms0vsmpip\] that varying $m[\pi(1300)]$ leaves the mass of the heavier SU(3) singlet scalar essentially unchanged (at about 1500 MeV) while it changes the mass of the lightest SU(3) singlet scalar. Thus it seems unavoidable to interpret the variation of $m[\pi(1300)]$ as being associated with the variation of the mass of the lightest scalar. This is confirmed by noticing that the largest change in $a_0^0$ occurs in the region of $m[\pi(1300)]$ where the mass of the lightest scalar is changing most rapidly.
The correction to the current algebra result for $a_0^0$ due to the finite masses of light scalars was already discussed and noted to be positive in [@SU1], some years ago. The contribution of a light scalar meson to the scattering length was recently calculated in [@skr].
It is very interesting to examine the recent experimental data on the s-wave scattering lengths $a_0^0$ and $a_0^2$; these include the following.
NA48/2 collaboration [@NA48]: $$m_{\pi^+}(a_0^0-a_0^2)=0.264 \pm 0.015$$ $$m_{\pi^+}a_0^0 = 0.256 \pm 0.011$$
E865 Collaboration [@E865]: $$m_{\pi^+}a_0^0 = 0.216 \pm 0.015$$
DIRAC Collaboration [@dirac] $$m_{\pi^+}a_0^0 =0.264_{-0.020}^{+0.038}$$
A general discussion of these experiments is given in [@an].
Comparison of experiment with theory shows that the larger values of $a_0^0$ predicted by the numerical calculation when $m[\pi(1300)]$ is greater than about 1215 MeV, give good agreement. (This corresponds to the lightest scalar singlet lighter than about 460 MeV). In contrast, the current algebra prediction for $a_0^0$ is clearly too low. The nonresonant channel with ($I=2$, $J=0$) is not so well determined from experiment but seems to be consistent with the common prediction of current algebra or the numerical calculation.
The indicated value of the lightest scalar mass in our model is consistent with recent results [@roy] obtained by using Roy dispersion relation sum rules. The typical values obtained for the mass and width of the lightest scalar are M = 441 MeV and $\Gamma$ = 544 MeV. These are also similar to what is obtained [@BFMNS01], M = 457 MeV and $\Gamma$ = 632 MeV, by using a K-matrix unitarized three flavor linear sigma model (with just one chiral nonet). A unitarized two flavor linear sigma model was earlier given in [@AS94]. The $s$ wave pion pion interaction has recently [@ay] been discussed using the Adler sum rule. Actually, a long time ago the Adler sum rule was used [@cs] to suggest a light scalar with a similar mass to the above.
= 7.5cm
= 7.5cm
The encouraging result for $a_0^0$ with inclusion of scalar meson corrections corresponds to a tree level treatment of this linear sigma model. One may justifiably wonder whether the agreement would be spoiled by inclusion of loops, i.e. by a unitarization of the model. While this is a more complicated matter we may note that the simplest unitarization, the K matrix approach, does not change the result at all. In this approach the corrected scattering length would be,
$$m_\pi a_0^0 ({\rm corrected}) = \frac{m_\pi a_0^0}{1-im_\pi a_0^0
\sqrt{1-4m_\pi^2/s}},
\label{correctedsl}$$
where the kinematical square root is to be evaluated at threshold, $s=4m_\pi^2$. Of course this is just an indication rather than a proof that the effect of unitarization is expected to be small for the scattering lengths.
Discussion and conclusions
==========================
We have seen that the intuitive explanation for the existence of a very light scalar meson as well as light scalar mesons with large “four quark" content given in [@bigpaper] (which used the simplifying assumption of zero pion mass) still is good when the physical pion mass is used. The main physical input is (almost) spontaneously broken chiral symmetry. Our model treats the chiral symmetry in its linear realization. This is normally considered inconvenient for the treatment of the scattering problem since it is known that there are very large cancellations between the four pion contact and scalar meson pole terms due to the symmetry. However it has the nice feature that it retains the light scalar mesons which give crucial corrections to the current algebra results. The inconvenience due to the large cancellations can be removed by using the Taylor expansion in Eq. (\[correction\]). The third term in the second line of that equation shows that the beyond current algebra corrections can be neatly displayed in a physical way; they are seen to be of order $(m_\pi/m_i)^2$ compared to the current algebra term. Here $m_i$ is a scalar meson mass. Still higher order corrections are suppressed by additional orders of $(m_\pi/m_i)^2$.
It is very interesting to illustrate how the use of the expanded form of the amplitude given in Eq. (\[correction\]) leads to a nice picture. In Fig. \[ICwholeamp1\] the individual contributions to the starting form of the amplitude, Eq. (\[wholeamp1\]), associated with each scalar exchange as well as with the four point contact term are displayed. Evidently the contact term is the largest in magnitude and cancels off most of the other contributions. In addition the contribution from the lightest scalar $S_0^{(1)}$ exchange, using this starting equation, has a very small magnitude, although its effect is expected to be the largest. Evidently, the intricate cancellations completely distort the underlying physics when expressed in this form. On the other hand, the use of the second of Eq.(\[correction\]) clears things up, as may be seen from Fig. \[ICcorrection\]. There the correction to the current algebra result is seen to be completely dominated by the lightest scalar.
Clearly the $s$-wave pion pion scattering is rather nicely described by a linear sigma model. At the present stage it does not seem to matter which one is chosen for this aspect. The virtue of the present toy model is that it accomodates both 2 quark and 4 quark scalar nonets in a consistent way and may help in understanding the relation to QCD as earlier discussed [@bigpaper]. Another interesting consequence of this linear formulation is the presence of heavy, largely four quark pseudoscalars, of which the $\pi(1300)$ is a possible candidate.
An amusing feature of the present model with a massive pion is that, due to a small deviation from exact PCAC, the current algebra result for near threshold pion pion scattering no longer holds exactly. Associated with this is the feature that the amplitude no longer vanishes at the unphysical point (Adler zero), $s=m_\pi^2$. The zero is shifted somewhat. The measure of this effect is the deviation of the coefficient, $F_{\pi'}$ from zero. We have seen that this is quite small since $F_{\pi'}$ is of the order $x_\pi/y_\pi\approx
m_\pi^2/m_{\pi'}^2 \approx$ 0.01, as may be seen from Eq. (\[2Fs\]). On the other hand the similar effect should be more noticeable for the $K-K'$ system and for $K-{\bar K}$ scattering. In [@toyfor2] it is shown that this system is formally analogous to the $\pi-\pi'$ system with the substitutions $x_\pi\rightarrow x_K$ and $y_\pi\rightarrow y_K$. But in this case $F_{K'}$ is expected to be of the order $m_K^2/m_{K'}^2\approx$ 0.1, ten times greater than for the pion case. A full calculation would require setting up the model with the inclusion of SU(3) symmetry breaking quark mass terms. This will be studied elsewhere.
A possible general question about the present model is that it introduces both states made of a quark and an antiquark as well as states with two quarks and two antiquarks. According to the usual ’t Hooft large $N_c$ extrapolation [@th] of QCD the “four quark" states are expected to be suppressed. However, it was recently pointed out [@ss07] that the alternative, mathematically allowed, Corrigan Ramond [@cr] extrapolation does not suppress the multiquark states. This kind of extrapolation may be relevant for understanding the physics of the light scalar mesons.
Acknowledgments {#acknowledgments .unnumbered}
===============
-.5cm We are happy to thank A. Abdel-Rehim, D. Black, M. Harada, S. Moussa, S. Nasri and F. Sannino for many helpful related discussions. The work of A.H.F. has been partially supported by the NSF Award 0652853. The work of R.J. and J.S. is supported in part by the U. S. DOE under Contract no. DE-FG-02-85ER 40231.
Parameter determination
=======================
Given the inputs: the pion decay constant, $F_\pi$; the mass of the pion, $m_\pi$; the mass of the $a_0(980)$, $m_a$; the mass of the $a_0(1450)$, $m_{a'}$; the mass of the $\pi(1300)$, $m_{\pi'}$, the independent model parameters which don’t involve the $U(1)_A$ violating terms can be successively determined (in the order given) by the equations: $$\begin{aligned}
2 d_2 &=& { {m_a^2 m_{a'}^2 - m_\pi^2\, m_{\pi'}^2} \over
{m_a^2 +m_{a'}^2 - m_\pi^2 - m_{\pi'}^2} }
\nonumber \\
{A\over \alpha} &=& {{m_\pi^2\, m_{\pi'}^2}\over {4\, d_2}}
\nonumber \\
(\alpha e_3^a)^2 &=&\frac{1}{64}\left( (m_a^2 - m_{a'}^2)^2 - [4 d_2 -
(m_a^2
+m_{a'}^2)]^2\right)
\nonumber \\
4 c_2 &=& m_a^2 + m_{a'}^2 - 2d_2 - {{ 56(\alpha e_3^a)^2} \over d_2}
- {3\over 2}\, m_\pi^2\, m_{\pi'}^2
\nonumber \\
{\beta\over \alpha} &=& {{-2 (\alpha e_3^a)} \over d_2}
\nonumber \\
{\rm cos}\, 2\theta_\pi &=&
{
{2\, d_2 - 2\, d_2\, \left({\beta\over \alpha}\right)^2
- 2\,\left({A\over \alpha}\right)
}
\over
{
\sqrt{
16\, d_2^2\left(\beta\over \alpha\right)^2
+\left[ 2\, d_2 - 2\, d_2 \left({\beta\over \alpha}\right)^2
- 2\,\left({A\over \alpha}\right)
\right]^2
}
}
}
\nonumber \\
\alpha &=&
{1\over 2}\,
{
{F_\pi}
\over
{ {\rm cos}\, \theta_\pi - \left({\beta\over \alpha}\right) \,
{\rm sin}\, \theta_\pi
}
}
\nonumber \\
c_4^a &=& {1 \over {2 \alpha^2}}
\left[ c_2 + {{8 (\alpha e_3^a)^2} \over d_2} + \left({A\over
\alpha}\right)\right]
\label{lagpara}\end{aligned}$$
Once the above parameters are determined, the parameters $\gamma_1$ and $c_3$ of the $U(1)_A$ violating sector are obtained in terms of the mass of the $\eta(958)$, $m_{\eta 1}$ and the mass of a suitable heavier $0^-$ isosinglet, $m_{\eta 2}$ using the following procedure. The 2$\times$2 prediagonal mass-squared matrix of the two SU(3) singlet pseudoscalars is written in the form: $$(M^2_0)=
\left[ \begin{array}{c c}
-\frac{8c_3(2\gamma_1+1)^2}{3\alpha^2} +K_{11} &
\frac{8c_3(1-\gamma_1)(2\gamma_1+1)}{3\alpha\beta}+K_{12}
\nonumber \\
\frac{8c_3(1-\gamma_1)(2\gamma_1+1)}{3\alpha\beta}+K_{12} &
-\frac{8c_3(1-\gamma_1)^2}{3\beta^2}+K_{22}
\end{array} \right] ,
\label{phizeromixing}$$ where $K_{ij}$ is a real symmetric matrix involving the coefficients of the terms in V$_0$ which are U(1)$_{\rm A}$ invariant . With the choice of invariant terms in Eq.(\[SpecLag\]) we have: $$\begin{aligned}
K_{11} &=& - 2\, (c_2 - 2\, c_4^a\, \alpha^2 + 4\, e_3^a\, \beta)
\nonumber \\
K_{12} &=& - 8 \, e_3^a \, \alpha
\nonumber \\
K_{22} &=& 2 \, d_2
\label{K}\end{aligned}$$
Then, $\gamma_1$ is found as a solution of the quadratic equation: $$\begin{aligned}
0&=&S\gamma_1^2+T\gamma_1+U,
\nonumber \\
S&=&\frac{R}{\alpha^2}
\left(
4+\frac{\alpha^2}{\beta^2}
\right)
-\frac{K_{11}}{\beta^2}
+\frac{4K_{12}}{\alpha\beta}-\frac{4K_{22}}{\alpha^2},
\nonumber \\
T&=&\frac{R}{\alpha^2}
\left(
4-2\frac{\alpha^2}{\beta^2}
\right)
+\frac{2K_{11}}{\beta^2}
-\frac{2K_{12}}{\alpha\beta}-\frac{4K_{22}}{\alpha^2},
\nonumber \\
U&=&\frac{R}{\alpha^2}
\left(
1+\frac{\alpha^2}{\beta^2}
\right)
-\frac{K_{11}}{\beta^2}
-\frac{2K_{12}}{\alpha\beta}-\frac{K_{22}}{\alpha^2},
\nonumber \\
R&=&\frac{4m_{\eta
1}^2m_{\eta 2}^2- {\rm det} (K)}
{m_{\eta 1}^2+ m_{\eta 2}^2-{\rm Tr} (K)}.
\label{findgamma1}\end{aligned}$$ In addition, $$c_3=\frac{\frac{3}{8}
\left(
-m_{\eta 1}^2m_{\eta 2}^2+ {\rm det} (K)
\right)
}
{K_{11}
\left(\frac{1-\gamma_1}{\beta}
\right)^2
+2K_{12}
\left(
\frac{1-\gamma_1}{\beta}
\right)\left(\frac{1+2\gamma_1}{\alpha}\right)
+K_{22}\left(\frac{1+2\gamma_1}{\alpha}\right)^2}
\label{findc3}$$
Next we give the numerical values of the parameters for the central values of all the listed input masses except for $m[\pi(1300)]$ which instead will take the typical value allowed by both the data and by the model, 1215 MeV. Table \[T\_6param\] shows the results for the parameters which are not associated with the U(1)$_{\rm A}$ violating part of the Lagrangian.
----------------------- -----------------------
$c_2 ({\rm GeV}^2)$ 8.79 $\times 10^{-2}$
$d_2 ({\rm GeV}^2)$ 6.30 $\times 10^{-1}$
$e_3^a ({\rm GeV})$ $-2.13$
$c_4^a $ 42.4
$\alpha ({\rm GeV})$ 6.06 $\times 10^{-2}$
$\beta ({\rm GeV})$ 2.49 $\times 10^{-2}$
$ A ({\rm GeV}^3)$ 6.66 $\times 10^{-4}$
----------------------- -----------------------
: Calculated Lagrangian parameters:$c_2$, $d_2$, $e_3^a$, $c_4^a$ and vacuum values: $\alpha$, $\beta$. []{data-label="T_6param"}
Table \[T\_2param\] shows the calculated Lagrangian parameters associated with the U(1)$_{\rm A}$ violating terms. Two “scenarios" associated with different identifications of the heavy $\eta$ which is the partner of the $\eta(958)$ are shown (I assumes $\eta(1475)$ to be chosen while II assumes $\eta(1760)$ to be chosen.) For each scenario, the two solutions (labeled 1 and 2) are shown.
--------------------- ------------------------ ------------------------ ------------------------ ------------------------
I1 I2 II1 II2
$c_3 ({\rm GeV}^4)$ $-2.39 \times 10^{-4}$ $-2.38 \times 10^{-4}$ $-3.42 \times 10^{-4}$ $-3.37 \times 10^{-4}$
$\gamma_1$ 5.33 $\times 10^{-1}$ $2.52 \times 10^{-1}$ 8.68 $\times 10^{-1}$ $-8.65 \times 10^{-2}$
--------------------- ------------------------ ------------------------ ------------------------ ------------------------
: Calculated parameters: $c_3$ and $\gamma_1$. []{data-label="T_2param"}
Using these parameters we next list the mixing matrices for, respectively, the two $0^-$ octet states, the two $0^+$ octet states and the two $0^+$ singlet states:
$$(R_\pi^{-1}) =
\left[
\begin{array}{cc}
0.923 & $0.385$ \\
-$0.385$ & 0.923\\
\end{array}
\right], \hspace{.3cm}
(L_a^{-1}) =
\left[
\begin{array}{cc}
-$0.486$ & $0.874$ \\
0.874 & 0.486\\
\end{array}
\right], \hspace{.3cm}
(L_0^{-1}) =
\left[
\begin{array}{cc}
0.706 & 0.708 \\
$0.708$ & -0.706\\
\end{array}
\right].
\label{mms}$$
Similarly, the mixing matrices for the two solutions for scenario I of the $0^-$ singlet states are:
$$I\,\,1:(R_0^{-1}) =
\left[
\begin{array}{cc}
-$0.671$ & 0.742 \\
0.742 & 0.671\\
\end{array}
\right], \hspace{.3cm}
I\,\,2: (R_0^{-1})=
\left[
\begin{array}{cc}
0.853 & -$0.522$ \\
0.522 & 0.853\\
\end{array}
\right].
\label{R0_num_I}$$
Finally, the mixing matrices for the two solutions for scenario II of the $0^-$ singlet states are:
$$II\,\,1:(R_0^{-1}) =
\left[
\begin{array}{cc}
-$0.411$ & 0.912 \\
0.912 & 0.411\\
\end{array}
\right], \hspace{.3cm}
II\,\,2: (R_0^{-1}) =
\left[
\begin{array}{cc}
0.972 & -$0.235$ \\
0.235 & 0.972\\
\end{array}
\right].
\label{R0_num_II}$$
[10]{}
A.H. Fariborz, R. Jora and J. Schechter, arXiv:0707.0843 \[hep-ph\]. This “companion" paper contains the appropriate background references.
S. Weinberg, Phys. Rev. Lett. [**17**]{}, 616 (1966).
W-M Yao et al, J. Phys. G: Nucl. Part. Phys. [**33**]{}, 1 (2006).
S.L. Adler, Phys. Rev. [**137**]{}, B1022 (1965); [**139**]{}, B1638 (1965).
J.R. Betley et al \[NA48/2 Collaboration\], Phys. Lett. B [**633**]{}, 173 (2006) \[arXiv:hep-ex/0511056\].
S. Pislak et al \[E865 Collaboration\], Phys. Rev. D [**67**]{}, 072004 (2003) \[arXiv: hep-ex/0301040\].
B. Adeva et al \[DIRAC Collaboration\], Phys. Lett. B [**619**]{}, 50 (2005) \[arXiv:hep-ex/0504044\].
B. Ananthanarayan, arXiv:physics/0703141.
J. Schechter and Y. Ueda, Phys. Rev. [**D3**]{}, 2874, 1971; Erratum D [**8**]{} 987 (1973). See sections V and VIB.
M.D. Scadron, F. Kleefeld and G. Rupp, arXiv: hep-ph/0601196.
The Roy equation for the pion amplitude, S.M. Roy, Phys. Lett. B [**36**]{}, 353 (1971), has been used by several authors to obtain information about the $f_0(600)$ resonance. See T. Sawada, page 67 of S. Ishida et al “Possible existence of the sigma meson and its implication to hadron physics", KEK Proceedings 2000-4, Soryyushiron Kenkyu 102, No. 5, 2001 , I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. [**96**]{}, 132001 (2006). A similar approach has been employed to study the putative light kappa by S.Descotes-Genon and B. Moussallam, Eur. Phys. J. C [**48**]{}, 553 (2006) Further discussion of the approach in ref. [@roy] above is given in D.V. Bugg, J. Phys. G [**34**]{}, 151 (2007) \[hep-ph/0608081\].
See Table V of D. Black, A.H. Fariborz, S. Moussa, S. Nasri and J. Schechter, Phys.Rev. [**D64**]{}, 014031 (2001).
N.N. Achasov and G.N. Shestakov, Phys. Rev. [**D49**]{}, 5779 (1994).
S.L. Adler and F.J. Yndurain, arXiv:0704.1201 \[hep-ph\].
Y.T. Chiu and J. Schechter, Nuovo Cimento, [**46**]{}, 548 (1966)). The Regge form for the high energy behavior used was also suggested by H.J. Schnitzer (private communication).
A.H. Fariborz, R. Jora and J. Schechter, Phys. Rev. D [**72**]{}, 034001 (2005).
G.’t Hooft, Nucl. Phys. [**B 72**]{}, 461 (1974).
F. Sannino and J. Schechter, arXiv:0704.0602 \[hep-ph\].
E. Corrigan and P. Ramond, Phys. Lett. [**87 B**]{}, 73 (1979).
|
---
address: |
School of Physics, Monash University, VIC 3800, Australia\
$^*$ Present address: Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland.
author:
- 'V. Negnevitsky$^*$ and L. D. Turner'
title: Wideband laser locking to an atomic reference with modulation transfer spectroscopy
---
We demonstrate that conventional modulated spectroscopy apparatus, used for laser frequency stabilization in many atomic physics laboratories, can be enhanced to provide a wideband lock delivering deep suppression of frequency noise across the acoustic range. Using an acousto-optic modulator driven with an agile oscillator, we show that wideband frequency modulation of the pump laser in modulation transfer spectroscopy produces the unique single lock-point spectrum previously demonstrated with electro-optic phase modulation. We achieve a laser lock with 100kHz feedback bandwidth, limited by our laser control electronics. This bandwidth is sufficient to reduce frequency noise by 30dB across the acoustic range and narrows the imputed linewidth by a factor of five.
[10]{}
J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. **39**, 680–682 (1981).
G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation [(FM)]{} spectroscopy,” Appl. Phys. B **32**, 145–152 (1983).
S. C. Bell, D. M. Heywood, J. D. White, J. D. Close, and R. E. Scholten, “Laser frequency offset locking using electromagnetically induced transparency,” Appl. Phys. Lett. **90**, 171120 (2007).
C. P. Pearman, C. S. Adams, S. G. Cox, P. F. Griffin, D. A. Smith, and I. G. Hughes, “Polarization spectroscopy of a closed atomic transition: applications to laser frequency locking,” J. Phys. B **35**, 5141–5151 (2002).
N. P. Robins, B. J. J. Slagmolen, D. A. Shaddock, J. D. Close, and M. B. Gray, “Interferometric, modulation-free laser stabilization,” Opt. Lett. **27**, 1905–1907 (2002).
G. Camy, C. Bordé, and M. Ducloy, “Heterodyne saturation spectroscopy through frequency modulation of the saturating beam,” Opt. Commun. **41**, 325–330 (1982).
J. H. Shirley, “Modulation transfer processes in optical heterodyne saturation spectroscopy,” Opt. Lett. **7**, 537–539 (1982).
D. J. [McCarron]{}, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. **19**, 105601 (2008).
J. Eble and F. [Schmidt-Kaler]{}, “Optimization of frequency modulation transfer spectroscopy on the calcium [$4^{1}$]{}S[$_0$]{} to [$4^{1}$]{}P[$_1$]{} transition,” Appl. Phys. B **88**, 563–568 (2007).
F. du Burck, O. Lopez, and A. El Basri, “Narrow-band correction of the residual amplitude modulation in frequency-modulation spectroscopy,” [IEEE]{} T. Instrum. Meas. **52**, 288 – 291 (2003).
Q. [Xiang-Hui]{}, C. [Wen-Lan]{}, Y. Lin, Z. [Da-Wei]{}, Z. Tong, X. Qin, D. Jun, Z. [Xiao-Ji]{}, and C. [Xu-Zong]{}, “[Ultra-stable]{} [rubidium-stabilized]{} [external-cavity]{} diode laser based on the modulation transfer spectroscopy technique,” Chinese Phys. Lett. **26**, 044205 (2009).
F. du Burck, G. Tetchewo, A. N. Goncharov, and O. Lopez, “Narrow band noise rejection technique for laser frequency and length standards: application to frequency stabilization to I[$_2$]{} lines near dissociation limit at 501.7 nm,” Metrologia **46**, 599–606 (2009).
AOM: Crystal Tech. 3080-122; Laser controller: MOGlabs DLC-202.
C. J. Hawthorn, K. P. Weber, and R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. **72**, 4477–4479 (2001).
Four times the geometric mean of the horizontal and vertical standard deviations of the beam intensity distribution. D4$\sigma$ diameter is equivalent to $1/e^2$ diameter for Gaussian beams, and is less affected by noise for non-Gaussian beams such as those used in the spectrometer.
E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, “Double-pass acousto-optic modulator system,” Rev. Sci. Instrum. **76**, 063112 (2005).
E. Jaatinen, “An iodine stabilized laser source at two wavelengths for accurate dimensional measurements,” Rev. Sci. Instrum. **74**, 1359–1361 (2003).
J. Zhang, D. Wei, C. Xie, and K. Peng, “Characteristics of absorption and dispersion for rubidium D[$_2$]{} lines with the modulation transfer spectrum,” Opt. Express **11**, 1338–1344 (2003). VCO: Mini-Circuits (MCL) ZX95-78-S+; PLL board: Analog Devices EVAL-ADF411X-EB1; DDS board: Novatech DDS9m; Bias tee: MCL ZFBT-4R2GW+; Phase detector: MCL ZRPD-1.
Photodetector: Thorlabs PDA36A, +10dB gain setting, 12MHz bandwidth.
L. Mudarikwa, K. Pahwa, and J. Goldwin, “[Sub-Doppler]{} modulation spectroscopy of potassium for laser stabilization,” J. Phys. B **45**, 065002 (2012).
Z. Zhou, R. Wei, C. Shi, and Y. Wang, “Observation of modulation transfer spectroscopy in the deep modulation regime,” Chinese Phys. Lett. **27**, 124211 (2010).
E. Jaatinen and D. J. Hopper, “Compensating for frequency shifts in modulation transfer spectroscopy caused by residual amplitude modulation,” Opt. Laser. Eng. **46**, 69–74 (2008).
D. A. Smith and I. G. Hughes, “The role of hyperfine pumping in multilevel systems exhibiting saturated absorption,” Am. J. Phys. **72**, 631–637 (2004).
The peak-to-peak height and width of each spectral feature were obtained by numerically locating the outermost pair of stationary points within the expected frequency range of the spectral feature and above a threshold amplitude, then calculating the frequency and amplitude difference between them. This method is immune to RAM, which causes a distortion with even symmetry around the transition frequency. The MTS features are odd-symmetric around this frequency, thus RAM distortion shifts both stationary points by a common distance in frequency and amplitude. Other stationary points, such as the central trough in the 7MHz closed transition feature (due to high levels of RAM), are ignored by the algorithm.
H. Noh, S. E. Park, L. Z. Li, J. Park, and C. Cho, “Modulation transfer spectroscopy for [$^{87}$]{}Rb atoms: theory and experiment,” Opt. Express **19**, 23444–23452 (2011).
E. Jaatinen, “Theoretical determination of maximum signal levels obtainable with modulation transfer spectroscopy,” Opt. Commun. **120**, 91–97 (1995).
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, “Numerical Recipes in C: The Art of Scientific Computing” (Cambridge University Press, 1992).
L. D. Turner, K. Weber, C. Hawthorn, and R. E. Scholten, “Frequency noise characterisation of narrow linewidth diode lasers,” Opt. Commun. **201**, 391–397 (2002).
D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A **26**, 12–18 (1982).
G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Optics **49**, 4801–4807 (2010).
S. Hädrich, P. Jauernik, L. [McCrumb]{}, and P. Feru, “Narrow linewidth ring laser with frequency doubling for titanium:sapphire and dye operation,” Proc. [SPIE]{} **6871**, 68711S (2008).
D. J. Thompson and R. E. Scholten, “Narrow linewidth tunable external cavity diode laser using wide bandwidth filter,” Rev. Sci. Instrum. **83**, 023107 (2012).
Introduction
============
Closed-loop laser frequency stabilization — ‘laser locking’ — is required for laser cooling, precision spectroscopy and atomic clocks. These experiments demand laser light frequency-locked to an atomic transition to an absolute accuracy of better than 1MHz. Such accuracy may be achieved by stabilizing the laser to a sub-Doppler spectroscopic feature of an atomic reference, with several locking schemes in common use. Schemes where the laser is frequency-modulated [@hall_optical_1981; @bjorklund_frequency_1983; @bell_laser_2007] are often preferred to baseband schemes [@pearman_polarization_2002; @robins_interferometric_2002] as they operate at frequencies above most laser technical noise and are less sensitive to alignment drift, temperature variations and stray light. The most common modulation schemes are frequency modulation spectroscopy (FMS) [@hall_optical_1981] and modulation transfer spectroscopy (MTS) [@camy_heterodyne_1982]. We show that modulation transfer spectroscopy is particularly suited to wideband locking, providing strong acoustic noise rejection and appreciable linewidth narrowing, advantages typically associated with systems using an optical cavity as a reference. The system retains the environmental immunity, robustness and absolute accuracy of a direct atomic lock. Using only the optics of a commonly used acousto-optic modulator (AOM) lock setup, we show that enhanced electronics delivers feedback bandwidths over 100kHz, suppressing frequency noise across the acoustic range by 30dB. Furthermore, we demonstrate that as the modulation frequency is increased into the megahertz range, the error signal evolves from the familiar cluster of overlapping dispersion curves into a characteristic modulation transfer spectrum with an unambiguous lock point located only at the closed atomic transition.
There are several reasons to prefer MTS over FMS for locking purposes. In both schemes the probe is coherently demodulated to obtain the error signal. The FMS lineshape arises due to the direct probing of the vapor absorption and dispersion by the probe beam [@bjorklund_frequency_1983], while MTS relies on the frequency-dependent *transfer* of modulation from the pump to the probe beam [@shirley_modulation_1982]. The modulation frequency $f_m$ of an MTS setup is limited roughly to the natural linewidth, above which the error signal lock point develops a ‘kink’ [@mccarron_modulation_2008]. FMS avoids this limitation, offering higher modulation frequencies and potentially higher signal-noise ratio (SNR). However, as it is still a linear process, FMS is encumbered with an undesirable Doppler background, and the amplitudes of its sub-Doppler features match those of the saturated absorption spectrum without emphasizing the closed transitions; which can lead to the lock point ‘hopping’ from one transition to another, among other pathologies. FMS is also sensitive to any dispersive element, including parasitic etalons, which may change the lock point by adding offsets to the demodulated error signal. In contrast, MTS produces symmetric sub-Doppler features free of Doppler background that are strong at closed transitions, such as $^{87}$Rb $F=2 \rightarrow F'=3$. Undesirable crossovers and other transitions are suppressed (given the correct selection of MTS parameters) and parasitic etalons do not contribute offsets.
Many laboratories employ a low-frequency variant of MTS, frequency modulating the pump beam with an acousto-optical modulator at relatively low modulation frequencies of order 100kHz. This spectroscopy mode yields spectra without the Doppler background and free of etalon offsets, but the spectra otherwise resemble FM spectra showing all transitions and crossovers, and the usable servo bandwidth is less than 10kHz. Recent work has revived interest in MTS-based laser locking, in particular due to the clean spectra and wide control bandwidths made possible by MTS at high modulation frequencies [@mccarron_modulation_2008; @eble_optimization_2007]. These workers used resonant electro-optical modulators (EOMs) to phase modulate the pump beam at high modulation frequencies above 1MHz. While AOMs have been used in both FMS and MTS systems [@camy_heterodyne_1982; @du_burck_narrow-band_2003] to provide improved optical isolation and intensity noise suppression [@xiang-hui_ultra-stable_2009; @du_burck_narrow_2009], to our knowledge there has been no detailed presentation of wideband AOM-based MTS locking. The key result of this paper is that a conventional modulated spectrometer using an acousto-optic modulator can deliver wideband modulation and coherent modulation transfer spectra using little, if any, additional optics. This AOM-based MTS spectrometer inherits the unambiguous spectrum, Doppler-free background, immunity to etalon interference effects and wide control bandwidth of EOM-based spectrometers, and requires only changes to the electronics operating a conventional AOM-based pump-probe spectrometer. We discuss the design and performance of such a system, and the various considerations in using it for robust laser locking.
Experimental setup
==================
![Electro-optical schematic of our acousto-optic MTS system. Frequency modulation is imparted to the pump beam via a double pass of the acousto-optic modulator (AOM). Modulation is transferred from the pump to the probe within the rubidium vapor cell. The beat of the probe frequency components at photodetector PD1 (12MHz bandwidth) is demodulated to produce an error signal. F1, F2, F3 = +100, -35, +100 mm.[]{data-label="fig:mts_optical_layout"}](mts_optical_layout)
In our spectrometer (Fig. \[fig:mts\_optical\_layout\]) an AOM [@equipment_list_optical] shifts the frequency of the pump beam by 160MHz and imposes a fast frequency modulation. This results in a lock point 80MHz below the rubidium D$_2$ $F=2 \rightarrow F'=3$ cycling transition, which is optimal for subsequent acousto-optical generation of several near-resonant beams. The AOM shift has the addition benefit of providing optical isolation between the diode laser and the spectrometer. We use an external-cavity diode laser [@hawthorn_littrow_2001], with a commercial laser controller [@equipment_list_optical] for temperature stabilization and control of the diode injection current and laser cavity length. The laser produces a 40mW elliptical beam with a large horizontal eccentricity of 0.8 and a D4$\sigma$ (second moment) diameter of 1.4mm [@d4sigma_definition]. We direct 2.7mW to the spectrometer, of which 250W is used for the probe beam. The pump beam passes through a telescope (F1, F2) to reduce its diameter to 500m with a vertical eccentricity of 0.8, and is double-passed through the AOM using a ‘cat-eye’ retroreflector made up of F3 and M4 [@donley_double-pass_2005]. The double pass reduces unwanted wavevector modulation and amplitude modulation compared to single-pass arrangements such as those described in Refs. [@jaatinen_iodine_2003] and [@zhang_characteristics_2003]. Narrowing the beam improves AOM modulation bandwidth, and collimation minimizes the range of Bragg angles, reducing spatial inhomogeneity in the modulation of the pump beam. The frequency-modulated pump beam incident on the room-temperature rubidium vapor cell is 900W with the same vertical eccentricity, and a diameter of 2.4mm.
The 80MHz rf drive for the AOM is generated by an agile voltage-controlled oscillator (VCO), locked to a 10MHz reference using a phase-locked loop (PLL) evaluation board [@equipment_list_electronic]. A direct digital synthesizer (DDS) board generates the modulating signal. An $f_m=3$MHz modulation was used for the laser lock results presented in Section 5, added to the PLL control signal on a bias tee [@equipment_list_electronic]. The PLL loop bandwidth is 5kHz, being deliberately low to avoid the PLL feedback interfering with the modulation. The system is easily adaptable to other AOM center frequencies, provided a VCO is available with sufficient modulation bandwidth.
The pump and probe counter-propagate through the vapor cell. Two processes — modulated optical pumping and four-wave mixing — compete to transfer modulation from the pump to the probe [@shirley_modulation_1982]; these are discussed in more detail in the next section. The result of the modulation transfer is the generation of sidebands on the probe beam, offset from the optical carrier by the modulation frequency. Probe sidebands beat with the carrier at a photodetector [@photodetector_details], producing a 3MHz signal at approximately -50dBm. After amplification by 30dB the probe signal is demodulated by mixing with the modulating signal, phase-shifted by demodulation phase $\phi$, on a phase detector [@equipment_list_electronic]. Accurate phase shifts were set digitally using a second channel of the DDS. After the mixer, a fifth-order modified Chebyshev low-pass filter removes up-converted noise and image components, yielding a clean wideband error signal. The Chebyshev filter provides a rapid transition from passband to stopband, at the expense of non-monotonic response in the stopband. We exploit this non-monotonicity by placing two notches at 3 and 6MHz ($f_m$ and $2f_m$). The filter gain remains nearly flat until 2MHz, providing a much wider passband than simpler filter designs while ensuring adequate attenuation in the key regions of the stopband. Finally the error signal is fed back to the laser controller, where the external-cavity diode current (proportional-integral) gain and piezo (double integrator) gain are individually adjustable.
Modulation transfer lineshapes
==============================
Our use of acousto-optic modulation makes it much easier to vary the modulation frequency $f_m$ than with a resonant EOM, helping us find the modulating frequency giving the optimal lock and a clean MTS spectrum. We studied the MTS lineshape as $f_m$ was varied from 100kHz to 8MHz, limited by the mixer below 100kHz and the AOM modulation bandwidth above 8MHz. This provided novel information on MTS in rubidium below 1MHz, a region where it is difficult to construct high-Q resonant electro-optic modulators [@mudarikwa_sub-doppler_2012], bridging the gap between AOM-based schemes with $f_m < 200$kHz [@zhang_characteristics_2003; @zhou_observation_2010] and EOM-based schemes with $f_m$ above several megahertz [@mccarron_modulation_2008]. We maintained a *constant frequency deviation* of 2.4MHz (i.e. maximum deviation of the rf from 80MHz, limited by our VCO); larger deviations produce monotonically larger locking features, but the spectra are not significantly altered otherwise. Recent studies of MTS have used a *constant modulation index* [@mccarron_modulation_2008; @mudarikwa_sub-doppler_2012], which precludes direct comparison of the results due to the nonlinear dependence of feature size upon frequency deviation. Demodulated spectra presented were filtered through a 20kHz Bessel low-pass filter, rather than the wideband loop filter used for locking; Fig. \[fig:mts\_spectra\](a) displays several example spectra.
![Variation of MTS spectra with modulation frequency: **a)** solid (dashed) spectra represent 0 (90) components, with **b)** amplitudes and **c)** trough-to-peak widths of each spectral feature at 0. Solid lines in **b)** and **c)** represent the predictions of Eq. 1 in using a linewidth value of 9.5MHz; the line in **b)** is scaled to the experimental amplitudes using a least-squares fit. Widths are omitted where signal amplitude is too small for their reliable determination. \[blue circle\] ($F=2 \rightarrow F'= \{ 1,2 \}$) and \[green triangle\] ($F=2 \rightarrow F'= \{ 1,3 \}$) are crossover resonances. \[red square\] ($F=2 \rightarrow F'=3)$ is the closed ‘cooling’ transition.[]{data-label="fig:mts_spectra"}](mts_spectra)
In MTS, frequency modulation on the pump beam is converted to amplitude modulation (AM) on the probe beam through the mutual interaction of the beams with the atomic reference. The frequency-dependent AM phase is recovered via demodulation to produce a symmetric error signal. Residual amplitude modulation (RAM) of the pump beam distorts the error signal, and occurs due to the asymmetric frequency responses of both the AOM and VCO. The distortion is an evenly symmetric sum of Lorentzians centered around the transition frequency [@jaatinen_compensating_2008], shifting the zero crossing of the error signal away from resonance. RAM may be minimized at a chosen modulation frequency by careful optical alignment, as discussed in the next section. For the data presented in Fig. \[fig:mts\_spectra\], RAM was minimized once at $f_m = $3MHz, and so spectra are increasingly distorted by RAM for higher $f_m$. The spectra for lower $f_m$ are much less affected by RAM, and the distortions present especially in the cycling transition are discussed later in this section.
Contributions to MTS spectra may be broadly classified as incoherent (modulated hole burning) or coherent (four-wave mixing). The most familiar process is hole burning, whereby the pump beam creates a Bennett hole in the ground state population, and the probe beam measures reduced absorption when both probe and pump beam are resonant with atoms of the same velocity class. This process occurs with unmodulated probe beams, and is commonly known as ‘saturated absorption’; it is well-known that in the alkalis the reduction of absorption on open transitions and crossovers is in fact largely due to the incoherent process of ‘velocity selective optical pumping’ into the other hyperfine ground state [@smith_role_2004].
At low modulation frequencies, the pump sweeps across the ground state velocity distribution, modulating the location of the Bennett hole on the distribution. The absorption of the velocity class resonant with the probe is thus modulated, causing in-phase amplitude modulation on the probe beam. This process is clearest in the $f_m=100$kHz trace uppermost in Fig. \[fig:mts\_spectra\]. As a simple sub-Doppler modulation spectroscopy, the spectrum resembles the derivative of the unmodulated absorption spectrum and shows strong signals at crossover resonances (leftmost two features) and closed transitions (rightmost feature). It thus resembles an FMS spectrum, though without the Doppler background. At much higher modulation frequencies, the period of the frequency modulation becomes short compared to the mean transit time of an atom across the probe beam. In this limit there will be atoms still transiting the beam which have been optically pumped by the pump beam at any given optical frequency within its deviation range, and so there will be vanishing oscillation of the probe absorption at the modulation frequency. For our beam diameters and a room-temperature thermal distribution, the mean transit time is equal to the modulation period at approximately $f_m=150$kHz. Figure \[fig:mts\_spectra\](b) shows the heights of the crossover resonances falling sharply [@amp_width_calculation] and becoming unmeasurable by $f_m=400$kHz, consistent with this simple model of modulated optical pumping.
The coherent component of a modulation transfer spectrum has been attributed to four-wave mixing, whereby the pump carrier, probe carrier and a pump sideband interact via the $\chi^3$ susceptibility of the near-resonant vapor to create a new probe sideband [@shirley_modulation_1982]. This process is inoperative at DC, is largely (if not entirely [@noh_modulation_2011]) suppressed at crossover resonances, and is only efficient at closed transitions. The incoherent process weakens as the modulation frequency is increased, revealing the coherent single-feature spectra at frequencies above 1MHz. Below 1MHz modulated optical pumping is much more significant than four-wave mixing for the crossover lineshapes, however both processes contribute significantly to the closed transition. For example, at 100kHz the crossovers are relatively undistorted, while the closed transition shows strong asymmetry. This distortion appears to be independent of RAM effects, which should affect all spectral features equally. We cannot conclusively pinpoint its cause, although it appears to be due to the interplay between the two MTS processes.
The spectra (Fig. \[fig:mts\_spectra\](a)) and peak-to-peak heights (Fig. \[fig:mts\_spectra\](b)) show that while the absolute height of the closed transition feature falls relatively slowly (approximately as the negative logarithm) with modulation frequency, the closed transition completely dominates the spectrum for $f_m>500$kHz, providing an unambiguous lock point. The roll-off of the closed transition height in Fig. \[fig:mts\_spectra\](b) is steeper than predicted by Eq. (1) in Ref. [@jaatinen_theoretical_1995] using a simple model of the atomic four-wave mixing process. We attribute this discrepancy to the limited modulation bandwidth of the AOM, discussed in the following section.
The peak-trough widths of the spectral features are shown in Fig. \[fig:mts\_spectra\]. We fit these to the same simple model, using the effective linewidth as the free parameter. Agreement is best with an effective linewidth of 9.5MHz (1.6 times the natural linewidth). As noted elsewhere, the four-wave mixing process cannot be modeled as a simple two-level system [@mccarron_modulation_2008]. This is clearly illustrated by the fact that a two-level model predicts a power-broadened linewidth of around 20MHz. We found that $f_m=3$MHz was optimal for locking purposes, providing spectra with weak crossovers, a single clear lock point at the closed transition and a high SNR. Frequencies well below 3MHz were insufficient to escape $1/f$ environmental and laser noise, and the limited modulation bandwidth of the AOM reduced the signal amplitude above 3MHz. Modulating at 3MHz should give a usable servo bandwidth of order 1MHz for a laser controller able to respond sufficiently fast.
Optimizing the spectrometer
===========================
Our MTS-based lock employs a high modulation frequency of 3MHz, delivering a usable feedback bandwidth of several hundred kHz. Working at such modulation frequencies places more critical demands on AOM alignment than those of commonly-used narrowband modulation at 100kHz or below. We briefly summarize techniques we found useful in achieving a symmetrical error signal at these high frequencies.
AOMs diffract efficiently when the Bragg criterion is satisfied, connecting angles of incident, diffracted and radiofrequency waves. Conventionally this match is best made with the widest possible laser beam, limited by the ‘acoustic aperture’ of the AOM. Such wide beams present no problems at conventional low modulation frequencies: at $f_m=100$kHz the acoustic wave travels several cm over a modulation cycle, and the laser beam illuminates a moving grating of spatial homogeneous period. At our high modulation frequency of $f_m=3$MHz in our AOM crystal (TeO$_2$, $v_\text{sound}=4.2$mm/s), the wave propagates only 1.4mm during a cycle, so that the a 1mm wide diffracted beam contains almost the full cycle of frequency deviations across its transverse profile. This ‘modulation bandwidth’ limit reduces contrast of the MTS error signal, and we reduce the incident beam diameter to 500m to ameliorate this effect. Focusing the pump beam through the AOM tends to produce too narrow a waist for reasonable focal lengths, and instead we found that forming a collimated beam of diameter 500m with a telescope (as shown in Fig. \[fig:mts\_optical\_layout\]) yields the best compromise of sufficient diffraction efficiency with only slight compression of the MTS error signal due to the acoustic transit-time effect.
Operation of the cat-eye AOM double pass has been thoroughly described elsewhere [@donley_double-pass_2005]. A convenient focal length for cat-eye lens F3 was 100mm, with F3 placed 100mm after the AOM and the retroreflection mirror M4 a further 100mm after the lens. In a variation on the layout of Ref. [@donley_double-pass_2005], the lens axis and mirror normal are co-linear with the diffracted beam axis. This alignment is easily made using an optical cage bench attached to a kinematic mount. During this alignment, attention was paid to (i) maximizing double-pass efficiency and (ii) minimizing the deflection and distortion of the double-passed beam induced by varying the rf carrier frequency (by several MHz).
After minimizing the angular deviation of the double-passed beam, we next measured the residual amplitude (intensity) modulation using a photodiode (PD2 in Fig. \[fig:mts\_optical\_layout\]). On a DC-coupled oscilloscope, residual amplitude modulation (RAM) appears as a small $f_m = 3$MHz signal, superposed on a large DC offset corresponding to the diffracted beam intensity. Slight rotations of the cage bench and M3, and small adjustments of the carrier radiofrequency, minimize the RAM, while aiming to preserve the diffracted beam power. When the optics and electronics were optimally adjusted the RAM minimum coincided with the diffracted intensity (DC) maximum. The constant time delay due to the AOM, caused by the travel time of the acoustic wave from the piezoelectric transducer to the point of interaction with the light, can be accurately measured by amplitude modulating the AOM and demodulating the signal from PD2.
The final optical step is the careful superposition of the modulated pump beam with the probe beam in the vapor cell. This alignment was best made using the MTS signal as a diagnostic, adjusting beam overlap until a symmetric error signal was obtained. The signal was observed at several settings of the demodulation phase. We found that imperfect suppression of amplitude modulation in the AOM could be largely compensated with beam overlap [@jaatinen_compensating_2008].
Electronic adjustments complete the optimization. The demodulation phase was chosen to maximize the error signal amplitude, and the lineshape was balanced using the above techniques for this particular phase. Finally, we note that small frequency shifts in the lock point of up to several megahertz may be realized with great precision and stability by altering the AOM carrier frequency. We anticipate that such ‘frequency shimming’ will be useful to calibrate the MTS lock point to a more precise atomic reference, such as absorption measurements of cold atoms in regions of minimal stray magnetic field.
Lock performance
================
The major aims of this work were to produce a stable, reliable lock, with strong frequency noise suppression and significant linewidth reduction. Reliability was verified empirically over a period of months: typically the laser remains locked for many hours, and often days, at a time, with loss of lock caused most often by laser mode hops or temperature fluctuations exceeding the controller range. The lock is difficult to dislodge without mechanically disturbing the laser itself, thus no special care needs to be taken when working near the system.
To quantify noise suppression and linewidth, we measured laser frequency noise at Fourier frequencies up to 30kHz using an optical spectrum analyzer (Sirah EagleEye) based on a high-finesse cavity. The EagleEye signal processor servos the cavity length to maintain the laser frequency half-way up an Airy fringe of the cavity transmission spectrum, and records time series of the transmission. Welch’s average periodogram method [@press_numerical_1992] was used to obtain the power spectral density (PSD) $S_{\delta\nu}(f)$ of the laser frequency noise, shown in Fig. \[fig:freq\_noise\](a). From 30kHz to 1.5MHz the pre-filter error signal spectrum was measured electronically, and calibrated to the EagleEye data by measuring the response of both to modulation of the controller input.
The frequency noise spectrum is more fundamental than the Allan variance and generally a more helpful diagnostic [@turner_frequency_2002], and was used to optimize the feedback gains.
![Laser stabilization properties. **a)** Composite $S_{\delta\nu}(f)$, showing both EagleEye and MTS error signal frequency noise, for (top to bottom) piezo feedback with minimal gain, piezo and current feedback with minimal gains, piezo and current with optimal gains. Dark (light) traces represent EagleEye (MTS error signal) data. Below 30kHz both are displayed, while above 30kHz only MTS error signal data are shown. The $\beta$-separation line (red) and off-resonant MTS error signal noise floor (black) are shown (EagleEye photodetector noise floor is below this by 30dB). **b)** (outer to inner) corresponding inferred lineshapes, calculated from spectra in (a) using $S_{\delta\nu}(f)$ from EagleEye data below 30kHz, and MTS error signal data from 30kHz to 1.5MHz.[]{data-label="fig:freq_noise"}](freq_noise)
Figure \[fig:freq\_noise\] shows the noise suppression and imputed linewidth narrowing of the MTS system. A low-gain piezoelectric-only laser lock was used to estimate the free-running laser frequency noise, holding the laser frequency steady during measurements (top trace in Fig. \[fig:freq\_noise\](a)). To leverage the wideband design of the spectrometer, we added injection current feedback (second trace), which strongly attenuated noise below 2kHz but failed to suppress mechanical resonances at 2 and 4kHz. Piezo and current gains were then optimized for maximum noise suppression, raising the bandwidth up to 100kHz with over 30dB suppression across acoustic frequencies (third trace). The peaks in the noise spectrum below 5kHz are due to mechanical resonances in the laser grating mount.
A common figure of merit for laser frequency stability is full-width at half-maximum (FWHM) linewidth, typically measured for sub-megahertz lines using a beatnote between two similar laser systems. Lacking two lasers, we instead inferred the optical spectrum from the PSD by determining the autocorrelation function, then applying the Wiener-Khinchin theorem to reconstruct the lineshape [@elliott_extracavity_1982]. The effect on the FWHM linewidth of noise at Fourier frequency $f$ hinges primarily on the strength of the noise $S_{\delta\nu}(f)$ relative to the line $8 \ln(2) f / \pi^2$, which separates the PSD into regions of high and low modulation index [@di_domenico_simple_2010]. Noise above this ‘$\beta$-separation’ line (i.e. at frequencies where $S_{\delta\nu}(f) > 8 \ln(2) f / \pi^2$) causes a relatively high frequency deviation compared to its Fourier frequency, and contributes significantly to broadening the line at its shoulders. The further $S_{\delta\nu}(f)$ is above the line, the more the FWHM linewidth is increased. The converse applies to noise ‘below the line’, which affects only the wings of the lineshape, and its suppression is unimportant for reducing the FWHM linewidth.
The control bandwidth of our system was sufficient to suppress noise across the entire frequency range where it exceeded the $\beta$-separation line. Figure \[fig:freq\_noise\](a) shows the frequency noise suppressed to almost the system noise floor from near DC up to 60kHz. As with all stable controllers to which the Bode sensitivity integral applies, a distinct ‘servo bump’ appeared above the 0dB point (100kHz) as a necessary consequence of noise suppression at lower frequencies. This bump is visible as a broadening of the base in Fig. \[fig:freq\_noise\](b), however because it falls largely below the $\beta$-separation line the FWHM linewidth is not significantly increased by its presence.
For controller gains higher than those used for the ‘optimal’ trace in Fig. \[fig:freq\_noise\], the noise in the error signal fell below the EagleEye frequency noise floor, as photodetector noise and laser intensity noise began to dominate frequency noise in the control loop. In this limit the error signal was an unreliable measure of linewidth. To further reduce the linewidth our system would require significantly higher SNR of the error signal; this could be achieved by increasing the power diverted to the spectrometer, increasing beam diameters, reducing noise by using intensity noise cancellation [@du_burck_narrow_2009] and/or heating the vapor cell.
Laser systems are often compared using the root mean square (RMS) of their frequency noise to obtain a ‘linewidth’, especially where a frequency discriminator is used [@hadrich_narrow_2008; @thompson_narrow_2012]. As shown in Tab. \[tab:locking\_linewidths\], RMS linewidths poorly reflect the valuable FWHM linewidth reduction visible in Fig. \[fig:freq\_noise\](b). RMS figures underestimate the linewidth when $1/f$ noise dominates and overestimate it when it is suppressed, due to the dominance of the servo bump.
Lock actuators (gain) FWHM RMS
--------------------------- ------ ----- -- --
Piezo (minimal) 867 389
Piezo + current (minimal) 547 275
Piezo + current (optimal) 165 216
: FWHM and RMS linewidths under the locking circumstances of . All quantities in kilohertz.[]{data-label="tab:locking_linewidths"}
Our system has thus reduced the imputed laser linewidth by a factor of 5, limited by the controller bandwidth of 100kHz and the noise floor of the system. We hope to increase this by extending the control bandwidth to at least several hundred kilohertz; the wideband MTS electronics should support this without modification. The PSD and laser lineshape are both useful for designing and characterizing such an extension.
Conclusions
===========
In summary, this paper has outlined our approach for designing, characterizing and optimizing an AOM-based laser lock using modulation transfer spectroscopy. We have combined conventional, inexpensive AOM-based optics with a wideband electronic modulation and demodulation scheme, and shown that the error signal is of sufficient quality to reduce the inferred linewidth of a typical external-cavity diode laser by a factor of five. We have also demonstrated that to obtain the full range of benefits an MTS spectrometer provides, modulation frequencies above 500kHz must be used. Heuristics for designing and optimizing such a system have been presented, which should help improve performance for adopters of this system. The laser frequency PSD has been used to quantify lock performance, and numerical calculation and analysis of the laser lineshape will inform future improvements to the controller.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by a Monash University Faculty of Science ECR Grant, the J.L. William Fund and the Australian Research Council (DP1094399). R.P. Anderson, M. Jasperse, A.A. Wood and L.M. Bennie assisted with the construction and optimization of the laser system. We are obliged to R.E. Scholten for loan of the EagleEye optical spectrum analyzer. The manuscript benefited greatly from a thorough review by R.P. Anderson and R.E. Scholten, for which we are very grateful. We thank A. Slavec for implementing the Chebyshev filter and A. Benci for electronics support.
|
---
abstract: 'How to generate instances with relevant properties and without bias remains an open problem of critical importance for a fair comparison of heuristics. In the context of scheduling with precedence constraints, the instance consists of a task graph that determines a partial order on task executions. To avoid selecting instances among a set populated mainly with trivial ones, we rely on properties that quantify the characteristics specific to difficult instances. Among numerous identified such properties, the *mass* measures how much a task graph can be decomposed into smaller ones. This property, together with an in-depth analysis of existing random task graph generation methods, establishes the sub-exponential generic time complexity of the studied problem. Empirical observations on the impact of existing generation methods on scheduling heuristics concludes our study.'
author:
- 'Louis-Claude Canon'
- Mohamad El Sayah
- 'Pierre-Cyrille Héam'
bibliography:
- 'biblio-taskgraph.bib'
title: A Comparison of Random Task Graph Generation Methods for Scheduling Problems
---
Introduction
============
How to correctly evaluate the performance of computing systems has been a central question since several decades[@jain1990art]. Among the arsenal of available evaluation methods, relying on random instances allows comparing strategies in a large variety of situations. However, random generation methods are prone to bias, which prevents a fair empirical assessment. It is thus crucial to provide guarantees on the random distribution of generated instances by ensuring, for instance, a uniform selection of any instance among all possible ones. Yet, for some problems, such uniformly generation instances are easy to solve and thus uninteresting. For instance, in uniformly distributed random graphs, the probability that the diameter is 2 tends exponentially to 1 as the size of the graph tends to infinity[@DBLP:journals/jsyml/Fagin76]. Studying the problem characteristics to constrain the uniform generation on a category of instances is thus critical.
In the context of parallel systems, instances for numerous multiprocessor scheduling problems contain the description of an application to be executed on a platform[@leung2004handbook]. This study focuses on scheduling problems requiring a Directed Acyclic Graph (DAG) as part of the input. Such a DAG represents a set of tasks to be executed in a specific order given by precedence constraints: the execution of any task cannot start before all its predecessors have completed their executions. Scheduling a DAG on a platform composed of multiple processors consists in assigning each task to a processor and in determining a start time for each task. While this work studies the DAG structure for several scheduling problems, it illustrates and analyzes existing generation methods in light of a specific problem with unitary costs and no communication. This simple yet difficult problem emphasizes the effect of the DAG structure on the performance of scheduling heuristics.
Some pathological instances are straightforward to solve. For instance, if the width (i.e. maximum number of tasks that may be run in parallel) is lower than the number of processors, then the problem can be solved in polynomial time. To avoid such instances, multiple DAG properties are proposed and analyzed. In particular, the mass measures the degree to which an instance can be decomposed into smaller independent sub-instances. In the absence of communication, this property has an impact on scheduling algorithms. The purpose of this work is to identify such properties to determine how the uniform generation of DAGs should be constrained and how existing generation methods perform relatively to these properties. As a major contribution of this work, we determine the generic time complexity to be sub-exponential for uniform instances for a large class of scheduling problems (i.e. those that can be decomposed into smaller problems).
After exposing related works in Section \[section:related\], Section \[section:background\] lists DAG properties and covers scheduling and random generation concepts. Section \[sec:analys-spec-dag\] motivates the focus on a selection of properties by analyzing all the proposed DAG properties on a set of special DAGs. Section \[sec:existinggeneration\] provides an in-depth analysis of existing random DAG generation methods supported by consistent empirical observations. Finally, Section \[sec:eval-sched-algor\] studies the impact of these methods and the DAG properties on scheduling heuristics. The algorithms are implemented in R and Python and the related code, data and analysis are available in[@figshare].
Related Work {#section:related}
============
Analysis of Generation Methods
------------------------------
Our approach is similar to the one followed in[@cordeiro2010a] and[@martinez2018b], which consists in studying the properties of randomly generated DAGs before comparing the performance of scheduling heuristics. In[@cordeiro2010a], three properties are measured and analyzed for each studied generation method: the length of the longest path, the distribution of the output degrees and the number of edges. We describe 15 such properties in Table \[tab:properties\]. They consider five random generation methods (described in this section and Section \[sec:existinggeneration\]): two variants of the [Erdős-Rényi]{}algorithm, one layer-by-layer variant, the random orders method and the Fan-in/Fan-out method. Finally, for each generation method, the paper compares the performance of four scheduling heuristics. The results are consistent with the observations done in Section \[sec:existinggeneration\] (Figures \[fig:erdos\_proba\], \[fig:poset\_perm\] and \[fig:layer\_layer\]) for the length and the number of edges. A similar approach is undertaken in[@martinez2018b]. First, three characteristics are considered: the number of vertices in the critical path, the width (or maximum parallelism) and the density of the DAG in terms of edges. These characteristics are studied on DAGs generated by two main approaches (the [Erdős-Rényi]{}algorithm and a MCMC approach) with sizes between 5 and 30 vertices. Finally, although no DAG property is studied, scheduling heuristics are compared using a variety of random and non-random DAGs in[@kwok1999a].
We describe below generation tools, data sets and random generation methods.
Generation Tools
----------------
Many tools have been proposed in the literature to generate DAGs in the context of scheduling in parallel systems. TGFF (Task Graphs For Free)[^1] is the first tool proposed for this purpose[@dick1998a]. This tool relies on a number of parameters related to the task graph structure: maximum input and output degrees of vertices, average for the minimum number of vertices, etc. The task graph is constructed by creating a single-vertex graph and then incrementally augmenting it. This approach randomly alternates between two phases until the number of vertices in the graph is greater than or equal to the minimum number of vertices: the expansion of the graph and its contraction. The main goal of TGFF is to gain more control over the input and output degrees of the tasks.
DAGGEN[^2] was later proposed to compare heuristics for a specifc problem[@dutot2009a]. This tool relies on a layer-by-layer approach with five parameters: the number of vertices, a width and regularity parameters for the layer sizes, and a density and jump parameters for the connectivity of the DAG. The number of elements per each layer is uniformly drawn in an interval centered around an average value determined by the width parameter and with a range determined by the regularity parameter. Lastly, edges are added between layers separated by a maximum number of layers determined by the jump parameter (edges only connect consecutive layers when this parameter is one). For each vertex, a uniform number of predecessors is added between one and a maximum value determined by the density parameter.
GGen[^3] has been proposed to unify the generation of DAGs by integrating existing methods[@cordeiro2010a]. The tool implements two variants of the [Erdős-Rényi]{}algorithm, one layer-by-layer variant, the random orders method and the Fan-in/Fan-out method. It also generates DAGs derived from classical parallel algorithms such as the recursive Fibonacci function, the Strassen multiplication algorithm, the Cholesky factorization, etc.
The Pegasus workflow generator[^4] can be used to generate DAGs from several scientific applications[@juve2013a] such as Montage, CyberShake, Broadband, etc. XL-STaGe[^5] produces layer-by-layer DAGs using a truncated normal distribution to distribute the vertices to the layers[@campos2016a]. This tool inserts edges with a probability that decreases as the number of layers between two vertices increases. A tool named RandomWorkflowGenerator[^6] implements a layer-by-layer variant[@gupta2017a]. Other tools have also been proposed but are no longer available as of this writing: DAGEN[@amalarethinam2011a], RTRG[^7][@shafik2012a], MRTG[@ashish2016a].
Finally, other fields such as electronic circuit design or dataflow also use DAGs. In this last field, however, requirements differ: the acyclicity is no longer relevant, while ensuring a strong connectedness is important. Two noteworthy generators have been proposed SDF$^3$ inspired from TGFF[^8][@stuijk2006a] and Turbine[^9][@bodin2014a].
Instance Sets
-------------
The STG (Standard Task Graph) set[^10] has been specifically proposed for parallel systems[@tobita2002a] and is frequently used to compare scheduling heuristics[@aggarwal2005genetic; @davidovic2012bee]. The DAG structures of STG relies on four different methods. Two methods, [`sameprob`]{}and [`samepred`]{}, rely on the [Erdős-Rényi]{}algorithm, while the other two, [`layrprob`]{}and [`layrpred`]{}, constitute layer-by-layer variants. A connection probability is given to [`sameprob`]{}and [`layrprob`]{}, while an average number of predecessors is given to [`samepred`]{}and [`layrpred`]{}. With these last two methods, the parameter is apparently converted to a connection probability inferred from the size of the DAG. Any layer-by-layer variant proceeds by first distributing vertices into layers such that the average layer size is 10. Then, edges between any pair of vertices from distinct layers are added from top to bottom according to the connectivity parameter. The size of the DAGs varies from 50 to . For each size, the data set contains 15 instances for each combination of a method among the four ones and a value for the connectivity parameter among three possible ones (leading to 180 instances). Both layer-by-layer variants do not guarantee that the layer of any vertex equals its depth. As a consequence, the length is not necessarily $\frac{n}{10}+2$ (2 dummy vertices are always added) where $n$ is the number of vertices[^11] and this problem becomes more apparent with large DAGs generated by [`layrpred`]{}because there are not enough inserted edges to ensure the layered structure. The STG set also contains costs and real DAGs such as robot control, sparse matrix solver and SPEC fpppp program.
PSPLIB[^12] contains difficult instances for RCPSP (Resource-Constrained Project Scheduling Problems)[@kolisch1995a], a scheduling problem in the field of project management. Finally, in the graph drawing context, a set of 112 real-life graphs were proposed[^13][@battista1997a] but are no longer available.
In addition to those implemented in GGen and the ones in STG, other DAGs from real-cases can be used such as the LU decomposition[@lord1983a], the parallel Gaussian elimination algorithm[@cosnard1988a], the parallel Laplace equation algorithm[@wu1990a], the mean value analysis (MVA)[@almeida1992a], which has a diamond-like structure, the FFT algorithm[@cormen2009a], which has a butterfly structure, the QR factorization, etc.
Layer-by-Layer Methods {#sec:layer-layer-method}
----------------------
The layer-by-layer method was first proposed by[@adam1974a] but popularized later by the introduction of the STG data set[@tobita2002a]. This method produces DAGs in which vertices are distributed in layers and vertices belonging to the same layer are independent. The method consists in three steps: determining the number of layers; distributing the vertices to the layers; connecting the vertices from different layers. In most proposed methods, there is at least one parameter for each step. For instance, the shape parameter controls the number of layers and is related to the ratio of $\sqrt{n}$ to the number of layers[@topcuoglu2002a; @ilavarasan2007a; @gupta2017a].
The number of layers can be drawn from a parameterized uniform distribution[@adam1974a; @topcuoglu2002a; @ilavarasan2007a; @saovapakhiran2011a], given as a parameter[@cordeiro2010a; @gupta2017a] or generated in a non-parameterized way[@ahmad1998a; @tobita2002a; @campos2016a].
Similarly, vertices can be distributed by generating a number of vertices at each layer with a parameterized uniform distribution[@adam1974a; @topcuoglu2002a; @ilavarasan2007a; @dutot2009a; @saovapakhiran2011a], by selecting a layer for each vertex with a parameterized normal distribution[@campos2016a], by using a balls into bins approach[@cordeiro2010a; @gupta2017a] or in a non-parameterized way[@ahmad1998a]. Note that generating a uniform number of vertices per layer may lead to a different number of vertices $n$ than expected. Also, using a balls into bins strategy may lead to empty layers.
Finally, the connection between vertices can depend on a connection probability[@tobita2002a; @dutot2009a; @cordeiro2010a; @saovapakhiran2011a; @campos2016a] or an average number of predecessors or successors for each vertex[@adam1974a; @tobita2002a; @topcuoglu2002a]. Although vertices in the same layer may have different depth (e.g.this occurs in the STG data set), adding specific edges prevents this situation[@dutot2009a; @gupta2017a]. The layer-by-layer approach can also lead to DAGs with multiple connected components except for[@adam1974a]. Finally, some methods allow edges between non-consecutive layers[@adam1974a; @tobita2002a; @cordeiro2010a], while others limit them[@dutot2009a; @campos2016a; @gupta2017a].
Uniform Random Generation
-------------------------
Many works address the problem of randomly generating DAGs with a known distribution. Uniform random generation of DAGs can be done using counting approaches[@roblab] based on generating functions. Many exisiting methods have been developped in the literature and the most important ones are described in Section \[sec:existinggeneration\].
While previous uniform approaches consider only the size of the DAG $n$ as a parameter, other studies have proposed to generate directed graphs from a prescribed degree sequence[@milo2003a; @karrer2009a; @ajwani2013a]. A uniform method is proposed in[@milo2003a] but may produce cyclic graphs. In contrast, the method proposed in[@karrer2009a] forbids cyclicity but has no uniformity guarantee. Last, in the context of sensor streams, several methods has been proposed[@ajwani2013a] to generate DAGs with a prescribed degree distribution.
Finally, a multitude of related approaches has been proposed but are discarded in this study because of their specificity. For instance, specific structures may be used to assess the performance of scheduling methods[@li2013a; @canon2018a] or special DAGs with known optimal solutions relatively to a given platform may also be built[@kwok1999a; @oppermann2018a].
Background {#section:background}
==========
Directed Acyclic Graphs {#sec:direct-acycl-graphs}
-----------------------
All graphs considered throughout this paper are finite. A *directed graph* is a pair $(V,E)$ where $V$ is a finite set of *vertices* and $E\subseteq V\times V$ is the set of *edges*. A *path* is a finite sequence of consecutive edges, that is a sequence of the form $(v_1,v_2),(v_2,v_3),\ldots,(v_{k-1},v_k)$; $k$ is the *length* of the path, i.e. the number of vertices on this path.
The *output degree* of a vertex $v$ is the cardinal of the set $\{(v,w)\mid w\in V,\ (v,w)\in E\}$. Similarly the *input degree* of a vertex $v$ is the cardinal of the set $\{(w,v)\mid
w\in V,\ (w,v)\in E\}$. The *output* (resp. *input*) *degree* of a directed graph is the maximum value of the output (resp. input) degrees of its vertices. The degree of a vertex is the sum of its input and output degrees.
A directed graph is *acyclic* (DAG for short) if there is no path of strictly positive length $k$ such that $v_1=v_k$ (with the above notation). Let ${\mathcal{D}^n}$ be the set of all DAGs whose set of vertices is $\{1,2,\ldots,n\}$. In a DAG, if $(v,w)$ is an edge, $v$ is a *predecessor* of $w$ and $w$ a *successor* of $v$.
In a DAG $D$ with $n$ vertices, all paths have a length less than or equal to $n$. The *length* of a DAG is defined as the maximum length of a path in this DAG. The *depth* of a vertex $v$ in a DAG is inductively defined by: if $v$ has no predecessor, then its depth is $1$; otherwise, the depth of $v$ is one plus the maximum depth of its predecessors. The *shape decomposition* of a DAG is the tuple $(X_1,X_2,\ldots,X_k)$ where $X_i$ is the set of vertices of depth $i$. Note that $k$ is the length of the DAG. The *shape* of the DAG is the tuple $(|X_1|,\ldots,|X_k|)$. The maximum (resp. minimum) value of the $|X_i|$ is called the *maximum shape* (resp. *minimum shape*) of the DAG. Computing the shape decomposition and the shape of a DAG is easy. If $|X_i|=1$, the unique vertex of $X_i$ is called a *bottleneck vertex*. A *block* is a subset of vertices of the form $\cup_{i<j<i+\ell} X_j$ with $\ell>1$ where $X_i$ is either a singleton or $i=0$, $X_{i+\ell}$ is either a singleton or $i+\ell=k+1$, and for each $i< j<i+\ell$, $|X_j|\neq 1$. We denote by ${\textnormal{mass}^{\textnormal{abs}}}(B)$ the cardinal of $B=\cup_{i<j<i+\ell} X_j$ and by ${\textnormal{mass}^{\textnormal{abs}}}(D)=\max\{{\textnormal{mass}^{\textnormal{abs}}}(B)\mid B \text{ is a block}\}$ the *absolute mass* of $D$. The *relative mass*, or simply the *mass*, is given by ${\textnormal{mass}}(D)=\frac{{\textnormal{mass}^{\textnormal{abs}}}(D)}{n}$.
For example, the DAG on Fig. \[fig:DAG:examplea\] has for shape decomposition the tuple $(\{1,5\},\{2,4\},\allowbreak\{6,8\},\{7\},\{3\})$ and for shape the tuple $(2,2,2,1,1)$. A longest path is $(5,4),(4,6),(6,7),(7,3)$. It has two bottleneck vertices $7$ and $3$. Its absolute mass is $2+2+2+1=7$.
In a DAG, two distinct vertices $v$ and $w$ are *incomparable* if there is neither a path from $v$ to $w$, nor from $w$ to $v$. The *width* of a graph is the maximum size of the subset of vertices whose elements are pair-wise incomparable. Since vertices of same depth are incomparable, the maximum shape of a DAG is less than or equal to its width. The width is also the size of the largest antichain, which can be computed in polynomial time using Dilworth’s theorem and a technique developed by Ford and Fulkerson[@ford2015flows]. The methodology is conjectured to have a time complexity of $O(n^{5/2})$[@plotnikov2007experimental]. In some cases (for instance the *comb* DAG, see Section \[sec:analys-spec-dag\]), the width can be much larger than the maximum shape. Table \[tab:comparison\] compares the width and the maximum shape on the DAGs obtained with two random generators explored in this paper.
$n$ [Erdős-Rényi]{} Uniform
----- ----------------- -----------------
10 2.95 – 0.34 – 2 2.35 – 0.09 – 1
20 3.52 – 0.45 – 2 2.77 – 0.14 – 1
30 3.62 – 0.46 – 2 3.13 – 0.23 – 1
: \[tab:comparison\]Comparison of width and maximum shape of randomly generated DAGs with different methods: “[Erdős-Rényi]{}” for the so-called algorithm with parameter $p=0.5$ (see Section \[sec:rand-gener-triang\]) and “Uniform” for the recursive random generator (see Section \[sec:recursive\]). Reported numbers $x - y - z$ correspond respectively to the average width, the average difference between width and shape width, and the maximum difference pointed out. Each experiment is performed by sampling 100 DAGs.
Two DAGs $(V_1,E_1)$ and $(V_2,E_2)$ are isomorphic, denoted $(V_1,E_1)\sim(V_2,E_2)$, if there exists a bijective map $\varphi$ from $V_1$ to $V_2$ such that $(x,y)\in E_1$ iff $(\varphi(x),\varphi(y))\in E_2$. The relation $\sim$ is an equivalence relation. Intuitively, two DAGs are isomorphic if they are equal up to vertices names. For example, the DAGs on Fig. \[fig:DAG:example\] are isomorphic.
The *transitive reduction* of a DAG $D$[@aho1972transitive] is the DAG $D^T$ for which: $D^T$ has a directed path between $u$ and $v$ iff $D$ has a directed path between $u$ and $v$; there is no graph with fewer edges than $D^T$ that satisfies the previous property. Intuitively, this operation consists in removing redundant edges. The *reversal* of a DAG $D$ is the DAG $D^R$ for which there is an edge between $u$ and $v$ iff there is an edge between $v$ and $u$ in $D$. Intuitively, this operation consists in reversing the DAG.
Finally, Table \[tab:properties\] presents some of the DAG properties that may impact the performance of scheduling algorithms. We discard the minimum input and output degrees because they are always ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{min}}}={\textnormal{deg}_{\textnormal{out}}^{\textnormal{min}}}=0$. We also discard the mean input and output degrees because they are always equal to half the mean degree (${\textnormal{deg}_{\textnormal{in}}^{\textnormal{mean}}}={\textnormal{deg}_{\textnormal{out}}^{\textnormal{mean}}}=\frac{{\textnormal{deg}_{\textnormal{}}^{\textnormal{mean}}}}{2}$). For all nine edge-related properties ($m$ and the degree-based properties) applied to a DAG $D$, we can also compute them on the transitive reduction $D^T$. The vertex-related properties ($n$, the width and the shape-based ones) remain the same on the transitive reduction. For all seven shape-based properties on a DAG $D$, we can also compute them on the reversal $D^R$. The edge-related properties remain the same through the reversal with the inversion of ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{max}}}$ and ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{sd}}}$ with ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{max}}}$ and ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{sd}}}$, respectively. Finally, some of these properties are related: $n\times{\textnormal{deg}_{\textnormal{}}^{\textnormal{mean}}}=\frac{m}{2}$ and ${\textnormal{len}}\times{\textnormal{sh}^{\textnormal{mean}}}=n$.
[rm[0.6]{}]{} Symbol & Definition\
$n$ & number of vertices\
$m$ & number of edges\
${\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}} ({\textnormal{deg}_{\textnormal{in}}^{\textnormal{max}}},{\textnormal{deg}_{\textnormal{out}}^{\textnormal{max}}}$) & maximum (input, output) degree\
${\textnormal{deg}_{\textnormal{}}^{\textnormal{min}}}$ & minimum degree\
${\textnormal{deg}_{\textnormal{}}^{\textnormal{mean}}}$ & mean (input, output) degree\
${\textnormal{deg}_{\textnormal{}}^{\textnormal{sd}}} ({\textnormal{deg}_{\textnormal{in}}^{\textnormal{sd}}},{\textnormal{deg}_{\textnormal{out}}^{\textnormal{sd}}})$ & standard deviation of the (input, output) degrees\
${\textnormal{len}}$ or $k$ & length (also called height, number of levels, longest path or critical path length)\
${\textnormal{width}}$ & width\
${\textnormal{sh}^{\textnormal{max}}}$ & maximum shape\
${\textnormal{sh}^{\textnormal{min}}}$ & minimum shape\
${\textnormal{sh}^{\textnormal{mean}}}$ & mean shape (parallelism in[@tobita2002a])\
${\textnormal{sh}^{\textnormal{sd}}}$ & standard deviation of the shape\
${\textnormal{sh}^{\textnormal{1}}}$ & number of source vertices (vertices with null input degree)\
${\textnormal{sh}^{\textnormal{k}}}$ & last element of the shape\
${\textnormal{mass}}$ & (relative) mass\
$p$ & connectivity probability\
$K$ & number of permutations (for the random orders method in Section \[sec:random-orders\])\
$P$ & set of processors\
Scheduling {#sec:scheduling}
----------
We consider a classic problem in parallel systems noted [$P|p_j=1,prec|C_{\max}$]{}in Graham’s notation[@graham79a]. The objective consists in scheduling a set of tasks on homogeneous processors such as to minimize the overall completion time. The dependencies between tasks are represented by a precedence DAG $(V,E)$ where $|V|=n$ is the number of tasks and $|E|=m$ the number of edges. Before starting the execution of a task, all its predecessors must complete their executions. The execution cost $p_j$ of task $j$ on any processor is unitary and there is no cost on the edges (i.e. no communication). A schedule defines on which processor and at which date each task starts its execution such that no processor executes more than one task at any time and all precedence constraints are met. The problem consists in finding the schedule with the minimum makespan, i.e. overall completion time before the first task starting its execution and the last one completing its execution.
A possible schedule for the DAG of Figure \[fig:DAG:examplea\] on two processors $P_1$ and $P_2$, assuming costs are unitary, consists in starting executing tasks 1 and 2 on processor $P_1$ as soon as possible (i.e. at times 0 and 1), while processor $P_2$ processes tasks 5, 4, 8, 7 and 3 similarly. The execution of task 6 follows the termination of task 2 on processor $P_1$ to satisfy the precedence constraint of task 7. The makespan of this schedule is 5.
This problem is strongly NP-hard[@Ullman:75:NP-complete-scheduling], while it is polynomial when there are no precedence constraints ($P|p_j=1|C_{\max}$), which means the difficulty comes from the dependencies. Many polynomial heuristics have been proposed for this problem (see Section \[sec:eval-sched-algor\]). With specific instances, such heuristics may be optimal. This is the case when the width does not exceed the number of processors, which leads to a potentially large length. Any task can thus start its execution as soon as it becomes available. The problem is also polynomial when edges only belong to the critical path (i.e. $m={\textnormal{len}}-1$ and the width equal $n-{\textnormal{len}}+1$, which is large when the length is small). In this case, any heuristic prioritizing critical tasks and scheduling all other tasks as soon as possible will be optimal. This paper explores how DAG properties are impacted by the generation method with the objective to control them to avoid easy instances.
Although this paper studies random DAGs with heuristics for the specific problem [$P|p_j=1,prec|C_{\max}$]{}, generated DAGs can be used for any scheduling problem with precedence constraints. While avoiding specific instances depending on their width and length is relevant for many scheduling problems, it is not necessary the case for all of them. For instance, with non-unitary processing costs, instances with large width and small length are difficult because the problem is strongly NP-Hard even in the absence of precedence constraints ($P||C_{\max}$)[@GareyJohnson:78:Strong-NP-completeness].
Mass and Scheduling {#sec:mass-scheduling}
-------------------
The proposed mass measure has a direct implication in this scheduling context. Consider a DAG $D=(V,E)$ whose minimum shape is $1$; there exists a bottleneck vertex $v$ such that the shape of the DAG is of the form $(X_1,\ldots,X_{\ell},\{v\},X_{\ell+1},\ldots,X_k)$. The scheduling problem for $D$ can be decomposed into two subproblems, one for the sub-DAG of $D$ whose set of vertices is $\{v\}\cup\bigcup_{i\leq
\ell}X_i$ and one for the sub-DAG of $D$ whose set of vertices is $\{v\}\cup\bigcup_{i>
\ell} X_i$. Using recursively this decomposition, the initial problem can be decomposed into $n_c+1$ independent scheduling problems, where $n_c$ is the number of bottleneck vertices.
Applying a brute force algorithm for the scheduling problems computes the optimal results in a time $T\leq n_c T_m$, where $T_m$ is the maximum time required to solve the problem on a DAG with ${\textnormal{mass}^{\textnormal{abs}}}(D)$ vertices. Since exponential brute force exact approaches exist, it follows that if ${\textnormal{mass}^{\textnormal{abs}}}(D)=O(\log^k n)$ for a constant $k$, then an optimal solution of the scheduling problem can be computed in sub-exponential time. Consequently, scheduling heuristics are irrelevant for task graph with logarithmic absolute mass. Similarly, the same arguments work to claim that interesting instances for the scheduling problem must have quite a large absolute mass (not in $o(n)$). It is therefore preferable to have instances with no or few bottleneck vertices, that is a unitary mass.
The relevance of the mass property is limited to a specific class of scheduling problems that contains all problems for which the instance can be cut into independent instances. While the mass is still relevant with non-unitary processing costs, it is no longer the case when there are communication costs.
Uniformity of the Random Generation {#sec:unif-rand-gener}
-----------------------------------
[cccc]{} Isom. classes & Matrices & ER & Labeling\
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{};
& $\left(\begin{matrix}0 & 0 \\ & 0\end{matrix}\right)$ & $\frac{1}{8}$ & 1\
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; (A) – (B);
& $\left(\begin{matrix}1 & 0 \\ & 0\end{matrix}\right)$ $\left(\begin{matrix}0 & 1 \\ & 0\end{matrix}\right)$ $\left(\begin{matrix}0 & 0 \\ & 1\end{matrix}\right)$ & $\frac{3}{8}$ & 6\
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; (B) – (C); (A) to \[bend left\] (C);
& $\left(\begin{matrix}0 & 1 \\ & 1\end{matrix}\right)$ & $\frac{1}{8}$ & 3\
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; (A) – (B); (A) to \[bend left\] (C);
& $\left(\begin{matrix}1 & 1 \\ & 0\end{matrix}\right)$ & $\frac{1}{8}$ & 3\
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; (A) – (B); (B) – (C);
& $\left(\begin{matrix}1 & 0 \\ & 1\end{matrix}\right)$ & $\frac{1}{8}$ & 6\
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; (B) – (C); (A) – (B); (A) to \[bend left\] (C);
& $\left(\begin{matrix}1 & 1 \\ & 1\end{matrix}\right)$ & $\frac{1}{8}$ & 6\
This work focuses on the importance generating DAGs uniformly. We discuss the notion of uniformity through the example with 3 vertices given in Table \[tab:dag3\]. In this instance, there are six isomorphism classes (i.e. six different unlabeled DAGs) for a total of 25 different (labeled) DAGs. A generator is thus uniform up to isomorphism if it generates each isomorphism class (or unlabelled DAGs) with a probability $\frac1{6}$ or uniform on all (labelled) DAGs if it generates each DAG with a probability $\frac1{25}$. We also say that we generate non-isomorphic DAGs in the former case. Finally, when considering only transitive reductions, we discard the complete DAG. The probability to generate each of the remaining isomorphism classes (resp. labeled DAGs) with a uniform generator becomes $\frac1{5}$ (resp. $\frac1{19}$). This leads to four different uniformity definitions.
Analysis of special DAGs {#sec:analys-spec-dag}
========================
[m[4cm]{}m[6cm]{}>m[4cm]{}]{} Name & description & representation\
Empty ($D_{\textnormal{empty}}$) & no edge &
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; [(D) at (3,0);]{};
\
Complete ($D_{\textnormal{complete}}$) & maximum number of edges &
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; [(D) at (3,0);]{}; (A) – (B); (B) – (C); (C) – (D); (B) to\[bend left\] (D); (A) to\[bend right\] (C); (A) to\[bend right\] (D);
\
Chain ($D_{\textnormal{chain}}$) & transitive reduction of the complete DAG &
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; [(D) at (3,0);]{}; (A) – (B); (B) – (C); (C) – (D);
\
Complete binary tree ($D_{\textnormal{out-tree}}$) & each non-leaf/non-root vertex has a unique predecessor and two successors &
[(A) at (0,0);]{}; [(B) at (1,0.5);]{}; [(C) at (1,-0.5);]{}; [(D) at (2,0.25);]{}; [(E) at (2,0.75);]{}; [(F) at (2,-0.25);]{}; [(G) at (2,-.75);]{}; (A) – (B); (A) – (C); (B) – (D); (B) – (E); (C) – (F); (C) – (G);
\
Comb ($D_{\textnormal{comb}}$) & a chain where each non-leaf vertex has an additional leaf successor &
[(A) at (0,-1);]{}; [(B) at (1,-1);]{}; [(C) at (2,-1);]{}; [(D) at (3,-1);]{}; [(E) at (1,-0.5);]{}; [(F) at (2,-0.5);]{}; [(G) at (3,-0.5);]{}; (A) – (B); (B) – (C); (C) – (D); (A) – (E); (B) – (F); (C) – (G);
\
Complete bipartite ($D_{\textnormal{bipartite}}$) & $\frac{n}{2}$ vertices connected to $\frac{n}{2}$ vertices &
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (2,0);]{}; [(D) at (0,-1);]{}; [(E) at (1,-1);]{}; [(F) at (2,-1);]{}; (A) – (D); (A) – (E); (A) – (F); (B) – (D); (B) – (E); (B) – (F); (C) – (D); (C) – (E); (C) – (F);
\
Complete layer-by-layer square ($D_{\textnormal{square}}$) & similar to the complete bipartite with $\sqrt{n}$ layers of size $\sqrt{n}$ &
[(A) at (0,0.6);]{}; [(B) at (0,0);]{}; [(C) at (0,-0.6);]{}; [(D) at (1,0.6);]{}; [(E) at (1,0);]{}; [(F) at (1,-0.6);]{}; [(G) at (2,0.6);]{}; [(H) at (2,0);]{}; [(I) at (2,-0.6);]{}; (A) – (D); (A) – (E); (A) – (F); (B) – (D); (B) – (E); (B) – (F); (C) – (D); (C) – (E); (C) – (F); (D) – (G); (D) – (H); (D) – (I); (E) – (G); (E) – (H); (E) – (I); (F) – (G); (F) – (H); (F) – (I);
\
Complete layer-by-layer triangular ($D_{\textnormal{triangular}}$) & similar to the complete layer-by-layer square but the size of each new layer increases by 1 &
[(A) at (0,0);]{}; [(B) at (1,0);]{}; [(C) at (1,0.6);]{}; [(D) at (2,0.6);]{}; [(E) at (2,1.2);]{}; [(F) at (2,0);]{}; (A) – (B); (A) – (C); (B) – (F); (B) – (D); (B) – (E); (C) – (F); (C) – (D); (C) – (E);
\
To analyse the properties described in the previous section, we introduce in Table \[tab.DAG\] a collection of special DAGs. The first three DAGs ($D_{\textnormal{empty}}$, $D_{\textnormal{complete}}$ and $D_{\textnormal{chain}}$) constitutes extreme cases in terms of precedence. The next two DAGs ($D_{\textnormal{out-tree}}$ and $D_{\textnormal{comb}}$), to which we can add the reversal of the complete binary tree ($D_{\textnormal{in-tree}}=D_{\textnormal{out-tree}}^R$), are examples of binary tree DAGs. The last three DAGs ($D_{\textnormal{bipartite}}$, $D_{\textnormal{square}}$ and $D_{\textnormal{triangular}}$) are denser with more edges and with a compromise between the length and the width for the last two DAGs.
[m[0.08]{}ccccccccc]{} DAG & $m$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}}$ & ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{max}}}$ & ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{max}}}$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{min}}}$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{mean}}}$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{sd}}}$ & ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{sd}}}$ & ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{sd}}}$\
$D_{\textnormal{empty}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\
$D_{\textnormal{complete}}$ & $\frac{n^2}{2}$ & $n$ & $n$ & $n$ & $n$ & $n$ & 0 & $\frac{n}{\sqrt{12}}$ & $\frac{n}{\sqrt{12}}$\
$D_{\textnormal{chain}}$ & $n$ & $2$ & $1$ & $1$ & $1$ & $2$ & $\sqrt{\frac{2}{n}}$ & $\frac{1}{\sqrt{n}}$ & $\frac{1}{\sqrt{n}}$\
$D_{\textnormal{out-tree}}$ $D_{\textnormal{comb}}$ & $n$ & $3$ & $1$ & $2$ & $1$ & $2$ & $1$ & $\frac{1}{\sqrt{n}}$ & $1$\
$D_{\textnormal{in-tree}}$ $D_{\textnormal{comb}}^R$ & $n$ & $3$ & $2$ & $1$ & $1$ & $2$ & $1$ & $1$ & $\frac{1}{\sqrt{n}}$\
$D_{\textnormal{bipartite}}$ & $\frac{n^2}{4}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 0 & $\frac{n}{4}$ & $\frac{n}{4}$\
$D_{\textnormal{square}}$ & $n\sqrt{n}$ & $2\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $2\sqrt{n}$ & $\sqrt{2\sqrt{n}}$ & $\sqrt{\sqrt{n}}$ & $\sqrt{\sqrt{n}}$\
$D_{\textnormal{triangular}}$ & $\frac{2n\sqrt{2n}}{3}$ & $2\sqrt{2n}$ & $\sqrt{2n}$ & $\sqrt{2n}$ & 2 & $\frac{4}{3}\sqrt{2n}$ & $\frac{2}{3}\sqrt{n}$ & $\frac{\sqrt{n}}{3}$ & $\frac{\sqrt{n}}{3}$\
[m[0.08]{}ccccccccc]{} DAG & ${\textnormal{len}}$ & ${\textnormal{width}}$ & ${\textnormal{sh}^{\textnormal{max}}}$ & ${\textnormal{sh}^{\textnormal{min}}}$ & ${\textnormal{sh}^{\textnormal{mean}}}$ & ${\textnormal{sh}^{\textnormal{sd}}}$ & ${\textnormal{sh}^{\textnormal{1}}}$ & ${\textnormal{sh}^{\textnormal{k}}}$ & ${\textnormal{mass}}$\
$D_{\textnormal{empty}}$ & 1 & $n$ & $n$ & $n$ & $n$ & 0 & $n$ & $n$ & 1\
$D_{\textnormal{complete}}$ $D_{\textnormal{chain}}$ & $n$ & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0\
$D_{\textnormal{out-tree}}$ & $\log_2(n)$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 1 & $\frac{n}{\log_2(n)}$ & $\frac{n}{\sqrt{3\log_2(n)}}$ & 1 & $\frac{n}{2}$ & 1\
$D_{\textnormal{in-tree}}$ & $\log_2(n)$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 1 & $\frac{n}{\log_2(n)}$ & $\frac{n}{\sqrt{3\log_2(n)}}$ & $\frac{n}{2}$ & 1 & 1\
$D_{\textnormal{comb}}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 2 & 1 & $2$ & $\sqrt{\frac{2}{n}}$ & 1 & 2 & 1\
$D_{\textnormal{comb}}^R$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 1 & $2$& $\sqrt{\frac{n}{2}}$ & $\frac{n}{2}$ & 1 & $\frac1{2}$\
$D_{\textnormal{bipartite}}$ & 2 & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 0 & $\frac{n}{2}$ & $\frac{n}{2}$ & 1\
$D_{\textnormal{square}}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & 0 & $\sqrt{n}$ & $\sqrt{n}$ & 1\
$D_{\textnormal{triangular}}$ & $\sqrt{2n}$ & $\sqrt{2n}$ & $\sqrt{2n}$ & 1 & $\sqrt{\frac{n}{2}}$ & $\sqrt{\frac{n}{6}}$ & 1 & $\sqrt{2n}$ & 1\
Table \[tab:prop\] illustrates the properties for these special DAGs. To discuss them, we analyze the most extreme values for each property. They are reached with the empty and complete DAGs except for the maximum standard deviations. The maximum value for the shape standard deviation is $\frac{n-1}{2}$ (reached with an empty DAG to which a single edge is added). When considering only transitive reductions (i.e. when discarding the complete DAG), the maximum value for the maximum degrees remains $n$ with either a fork (a single source vertex is the predecessor of all other vertices) or a join (the reversed fork). Proposition \[prop:bipartite\] states that the maximum number of edges among all transitive reductions is $\left\lfloor {\frac{n^2}{4}} \right\rfloor$ (reached with the bipartite DAG). As a corollary, the maximum value for the minimum and mean degrees is $\frac{n}{2}$. Studying the maximum achievable values for the degree standard deviations is left to future work.
\[prop:bipartite\] The maximum number of edges among all transitive reductions of size $n$ is $\left\lfloor \frac{n^2}{4} \right\rfloor$.
Transitive reductions do not contain triangle (i.e. clique of size three), otherwise there is either a cycle or a redundant edge. By Mantel’s Theorem[@mantel1907problem], the maximum number of edges in a $n$-vertex triangle-free graph is $\left\lfloor \frac{n^2}{4} \right\rfloor$. This is the case for the complete bipartite DAG because the number of edges is $\frac{n^2}{4}=\left\lfloor \frac{n^2}{4} \right\rfloor$ when $n$ is even and $\frac{n^2-1}{4}=\left\lfloor \frac{n^2}{4} \right\rfloor$ when $n$ is odd.
The edge-related properties are considerably affected when considering the transitive reduction of the complete DAG, i.e. the chain. Except for the standard deviations, all such properties are divided by $O(n)$. Considering transitive reductions can thus lead to different conclusions. The edge-related properties also highlight the asymmetry of both trees through the difference between input and output degrees. Moreover, the density of a DAG appears to be quantified by the edge-related properties (e.g. the complete DAG and last three DAGs). Small values for the degree standard deviations characterize DAGs in which every vertex shares a similar structure (e.g. the empty DAG, chain, trees and combs). The length and shape-based properties show whether the DAG is short (empty and bipartite DAGs), balanced (the trees, square triangular DAGs) or long (the complete DAG, chain and combs). The maximum shape equals the width except for the reversed comb, which confirms the results shown in Table \[tab:comparison\] on the similarity between the maximum shape and the width. Finally, large values for the shape standard deviation characterize DAGs for which the parallelism varies significantly. This is the case for the trees and triangular DAG.
The analysis of these special DAGs provides some insight to select the relevant properties in the rest of this paper. Each given DAG possesses 18 properties, to which we add 9 properties by considering the transitive reduction and 7 properties by considering the reversal (for a total of 34 properties, $n$ excluded). We limit the scope of our study to discard some properties for simplicity. First, we assume that the generated DAGs are symmetrical and have similar properties through the reversal operation (which is the case for all special DAGs except for the trees and combs). This eliminates the 7 properties on the reversal. Moreover, the following properties become redundant with ${\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}}$ and ${\textnormal{deg}_{\textnormal{}}^{\textnormal{sd}}}$: ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{max}}}$, ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{max}}}$, ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{sd}}}$ and ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{sd}}}$ (which eliminates 8 additional properties). Second, we assume that only transitive reductions are meaningful in the context of scheduling without communication. This eliminates only 4 other properties because we keep the number of edges in the initial DAG because it provides meaningful information on the generation method. Moreover, we discard the mean degree because it is redundant with $n$ and $m$, and provides little insight. Similarly, the minimum degree is not kept because it may be uninformative as it is low for source and sink vertices. We also discard the width and maximum shape because the mean shape provides a more global information. The mass already takes into account the minimum shape, which we discard. The last two shape properties (${\textnormal{sh}^{\textnormal{1}}}$ and ${\textnormal{sh}^{\textnormal{k}}}$) provides only local information and are thus not kept.
This leaves 8 properties. In particular, we measure the following edge-related properties on the transitive reduction of any DAG: the number of edges, maximum degree and degree standard deviation. Additionally, we keep the length, the mean shape (even though it is redundant with $n$ and ${\textnormal{len}}$, it provides essential information on the global parallelism of the DAG), the shape standard deviation and the mass. The final property is the number of edges in the initial DAG.
![\[fig:spe\_schedule\]Makespan obtained with three heuristics (described in Section \[sec:eval-sched-algor\]) on all special DAGs of Section \[sec:analys-spec-dag\] for [$P|p_j=1,prec|C_{\max}$]{}. The number of vertices is $n=128$ for the empty DAG, complete and bipartite DAGs, $n=127$ for the trees and combs, $n=121$ for the square DAG, and $n=120$ for the triangular DAG ($1+\cdots+15=120$). ](1_special_schedule)
Figure \[fig:spe\_schedule\] shows the makespan obtained with three scheduling heuristics with all special DAGs as the number of processors varies. HEFT is always optimal because of the regularity of the DAG structures and because costs are unitary. This is also the case for the other heuristics most of the time. A zero mass, for long DAGs such as the complete DAG and chain, leads to an even easier scheduling problem where the number of processors has no impact. This confirms the discussion in Section \[sec:mass-scheduling\] stating that low mass is characteristic of easy instances. For the other DAGs, increasing the number of processors decreases the makespan until it reaches 1, 2, 7, 11, 15 and 50 for the empty DAG, bipartite DAG, trees, square DAG, triangular DAG and combs, respectively. Note that the stairs for the square are due to its layered structure. For the reversed comb, MinMin behaves poorly because this simple heuristic does not take into account the critical path and fill the processors with any of the initial source vertices. Finally, the sub-optimal schedule produced by HCPT for the comb DAG is because, contrarily to HEFT with its insertion mechanism, this heuristic does not rely on backfilling and cannot schedule a task before any other already scheduled tasks.
Analysis of Existing Generation Methods {#sec:existinggeneration}
=======================================
This section covers and analyzes existing generation methods: the classic [Erdős-Rényi]{}algorithm; a uniform random generation method via a recursive approach; a poset-based method; and, an ad-hoc method frequently used in the scheduling literature.
Random Generation of Triangular Matrices {#sec:rand-gener-triang}
----------------------------------------
This approach is based on the [Erdős-Rényi]{}algorithm[@erdos1959a] with parameter $p$ (noted $G(n,p)$ in[@bollobas2001random]): an upper-triangular adjacency matrix is randomly generated. For each pair of vertices $(i,j)$, with $i<j$, there is an edge from $i$ to $j$ with an independent probability $p$. The expected number of edges is therefore $p\frac{n(n-1)}{2}$.
The approach is not uniform (nor uniform up to isomorphism). For instance, a generator that is uniform up to isomorphism picks up the empty DAG with probability $1/6$ (see Table \[tab:dag3\]). Moreover, a random generator that is uniform over all the DAGs (see Section \[sec:unif-rand-gener\] for the distinction) generates the empty DAG with probability $1/25$. With $p=0.5$, the [Erdős-Rényi]{}algorithm generates the DAG with no edges with probability $1/8$.
![\[fig:erdos\_proba\] Properties of 300 DAGs of size $n=100$ generated by the [Erdős-Rényi]{}algorithm with probability $p$ uniformly drawn between 0 and 1. The smoothed line is obtained with a linear regression using a polynomial spline with 10 degrees of freedom. The degree CV (Coefficient of Variation) is the ratio of the mean degree to the degree standard deviation. Red lines correspond to formal results for the length and mean shape (Proposition \[prop:ERmean\]), the number of edges, and the number of edges in the transitive reduction (Proposition \[prop:ERET\]). ](2_erdos_analysis_proba)
Figures \[fig:erdos\_proba\] and \[fig:erdos\_node\] show the effect of both parameters, probability $p$ and size $n$, on the properties of the generated DAGs. For readability of both figures, each standard deviation is replaced by a CV (Coefficient of Variation), which is the ratio of the standard deviation to the mean. The most evident effect on both figures is that the number of edges $m$ increases linearly as $p$ increases and quadratically as $n$ increases, which is a direct consequence of the algorithm and the expected number of edges. Similarly, but with more variation, the length also increases as either parameter increases. This effect also concerns the mean shape because ${\textnormal{sh}^{\textnormal{mean}}}=\frac{n}{{\textnormal{len}}}$ (for instance, the length is close to 20 when $p=0.125$, whereas the mean shape is close to 5). Therefore, on Figure \[fig:erdos\_proba\], the mean shape decreases as the inverse function of the probability $p$ because the length increases quasi-linearly with $p$. This effect is consistent with Proposition \[prop:ERmean\] in Appendix \[sec:annexe:probaER\], which suggests that the expected mean shape is no greater than $\frac1{p}$.
A more remarkable effect can be seen for the number of edges in the transitive reduction $m(D^T)$. This property shows that after a maximum around $p=0.10$, adding more edges with higher probabilities leads to redundant dependencies and simplifies the structure of the DAG by making it longer. The same observation can be done with ${\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}}(D^T)$. This is consistent with the fact that the algorithm generates the empty DAG when $p=0$ and the complete DAG when $p=1$. Proposition \[prop:ERET\] in Appendix \[sec:annexe:probaER\] also confirms this effect.
We rely on this apparent threshold around $p=10\%$ to characterize three probability intervals: below 5%, between 5% and 15%, and above 15%. DAGs generated with a probability in the first interval are almost empty (hence a length lower than 10 and a mean shape higher than 10) with few vertices having some edges and many with no edges (hence the high degree standard deviation). For these DAGs, most edges are not redundant. Given the high shape standard deviation, many tasks must be available at first. As mentioned in Section \[sec:scheduling\], these DAGs lead to a simplistic scheduling process that consists in starting each task on a critical path as soon as possible and then distributing a large number of independent tasks. Analogously, DAGs generated with probabilities $p$ greater than 15% contain many edges that simplify the DAG structure by increasing the length and thus reducing the mean shape (recall that with a small width, the problem is easy, see Section \[sec:scheduling\]). At the same time, the mass decreases continuously, allowing the problem to be divided into smaller problems. In particular, for probability $p$ greater than 90%, DAGs are close to the chain, which is trivial to schedule. Therefore, most interesting DAGs are generated with probabilities between 5% and 15%.
![\[fig:erdos\_node\] Properties of 191 DAGs generated by the [Erdős-Rényi]{}algorithm with probability $p=0.15$ and for each size $n$ between 10 and 200. The smoothed line is obtained with a linear regression using a polynomial spline with 4 degrees of freedom. Red lines correspond to formal results for the length and mean shape (Proposition \[prop:ERmean\]), the number of edges, and the number of edges in the transitive reduction (Proposition \[prop:ERET\]). ](2_erdos_analysis_node)
As shown on Figure \[fig:erdos\_node\], the size of the DAG $n$ has a simpler effect on the number of edges in the transitive reduction $m(D^T)$ than the probability $p$: $m(D^T)$ increases linearly with $n$ (see Proposition \[prop:ERET\]). Moreover, the length increases with $n$ as the shape mean remains constant (see Proposition \[prop:ERmean\]). As a consequence, the mass decreases with $n$ because the probability to obtain the value 1 increases in a vector with constant mean but increasing size. It is thus advisable to lower the probability with large sizes to maintain a constant mass.
The analysis of the [Erdős-Rényi]{}algorithm provides some insight on the desirable characteristics for the purpose of comparing scheduling heuristics. The effect of probability $p$ illustrates the compromise between the length and mean shape to avoid simplistic instances that are easily tackled (see Section \[sec:scheduling\]). Moreover, the maximum number of edges in the transitive reduction $m(D^T)$ is around $\frac5{2}n$ in both figures. However, we know that reaching $\frac{n^2}{4}$ is possible (Proposition \[prop:bipartite\]) and layer-by-layer DAGs (square and triangular) are in $O(n^{\frac{3}{2}})$. Therefore, the [Erdős-Rényi]{}algorithm fails to generate DAGs with such large $m(D^T)$.
Uniform Random Generation {#sec:recursive}
-------------------------
There are two main ways to provide a uniform random generator to uniformly generate elements of ${\mathcal{D}^n}$ (uniform over all labelled DAGs, see Section \[sec:unif-rand-gener\]). The first one consists in using a classical recursive/counting approach[@roblab]. This counting approach relies on recursively counting the number of DAGs with a given number of source vertices, that is vertices with no in-going edges. See[@DBLP:journals/sac/KuipersM15 Section 4] for a complete algorithm that uniformly generates random DAGs with this approach. The second one relies on MCMC approaches[@DBLP:journals/endm/MelanconDB01; @DBLP:conf/sbia/IdeC02; @DBLP:journals/ipl/MelanconP04]. We describe below the recursive approach.
Let $a_n=|{\mathcal{D}^n}|$, $a_{n,s}$ be the number of DAGs of ${\mathcal{D}^n}$ having exactly $s$ source vertices (${\textnormal{sh}^{\textnormal{1}}}=s$). It is proved in[@roblab] that: $$a_n=\sum_{k=1}^na_{n,k}\quad \text{and}\quad
a_{n,k}={\binom{n}{k}}b_{n,k} \quad \text{with}\quad
b_{n,k}=\sum_{s=1}^{n-k}(2^k -1)^s2^{k(n-k-s)}a_{n-k,s}.$$
First, we compute all values $a_i$ and $a_{i,k}$ for $1\leq i\leq n$ and $1\leq k\leq i$ with the initial conditions $a_{i,i}=1$ for $1\leq i\leq n$. Next, a shape is generated using Algorithm \[algo:shaperec\], where $\oplus$ is the concatenation of vectors.
Randomly generate $s\in \{1,n\}$ with distribution ${\mathbb{P}}(s=j)=\frac{a_{n,j}}{a_n}$
Finally, Algorithm \[algo:shapeDAG\] builds the final DAG by adding random edges.
![\[fig:recursive\] Properties of 191 DAGs generated by the recursive algorithm for each size $n$ between 10 and 200. The smoothed line is obtained with a linear regression using a polynomial spline with 4 degrees of freedom. Red lines correspond to formal results for the length and mean shape, and the number of edges (the bound from Theorem \[theorem:mass\] is discarded because it is too far). ](3_recursive_analysis)
Figure \[fig:recursive\] depicts the effect of the number of vertices on the selected DAG properties. Three effects are noteworthy: the length closely follows the function $\frac{3n}{2}$, the number of edges $m$ is almost indistinguishable from the function $\frac{n^2}{4}$ and the number of edges in the transitive reduction $m(D^T)$ closely follows $1.4n$. The first effect is consistent with a theoretical result stating that the expected number of source vertices ${\textnormal{sh}^{\textnormal{1}}}$ in a uniform DAG is asymptotically 1.488 as $n\to\infty$[@liskovetsmaixmal]. This implies that the expected value for each shape element is close to this value by construction of the shape. Proposition \[prop:expectshape\] in Appendix \[sec:annexe:proba\] confirms this expectation is no larger than 2.25, which makes the DAG an easy instance for scheduling problems (see Section \[sec:scheduling\]). For the second effect, we know that the average number of edges in a uniform DAG is indeed $\frac{n^2}{4}$[@DBLP:journals/endm/MelanconDB01 Theorem 2]. Despite the large amount of studies dedicated to formally analyzing uniform random DAGs, to the best of our knowledge, the last effect has not been formally considered. We finally observe that the mass decreases as the size $n$ increases. This is confirmed by the following result, proved in Appendix \[sec:annexe:proba\]:
\[theorem:mass\] Let $D$ be a DAG uniformly and randomly generated among the labeled DAGs with $n$ vertices. One has ${\mathbb{P}}({\textnormal{mass}^{\textnormal{abs}}}(D)\geq \log^4(n))\to 0$ when $n\to+\infty$.
Therefore, the mass converges to zero as the size $n$ tends to infinity. As shown in Section \[sec:mass-scheduling\], such instances can be decomposed into independent problems and efficiently solved with a brute force strategy. This leads to a sub-exponential generic time complexity with uniform instances.
To obtain a similar average number of edges $m$ with the [Erdős-Rényi]{}algorithm, we must choose a probability $p=0.5$. We can compare both methods by considering $p=0.5$ and $n=100$ on Figures \[fig:erdos\_proba\] and \[fig:recursive\], respectively. We observe that DAGs generated by both methods share similar properties. This leads to similar conclusions as in Section \[sec:rand-gener-triang\].
Random Orders {#sec:random-orders}
-------------
The random orders method derives a DAG from randomly generated orders[@winkler1985random]. The first step consists in building $K$ random permutations of $n$ vertices. Each of these permutations represents a total order on the vertices, which is also a complete DAG with a random labeling. Intersecting these complete DAGs by keeping an edge iff it appears in all DAGs with the same direction leads to the final DAG. This is a variant of the algorithm presented in[@cordeiro2010a] where the transitive reduction in the last step is not performed because we already measure the properties on the transitive reduction.
![\[fig:poset\_perm\] Properties of 420 DAGs of size $n=100$ generated by the random orders algorithm for each number of permutations $K$ between 2 and 15 (30 DAGs per boxplot). Red lines correspond to formal results for the length and mean shape. ](4_poset_analysis_perm)
Figure \[fig:poset\_perm\] shows the effect of the number of permutations $K$ on the DAG properties with boxplots[^14]. The extreme cases $K=1$ and $K\to\infty$ are discarded from the figure for clarity. They correspond to the chain and the empty DAG, respectively. Recall that for the chain, $m(D^T)\approx{\textnormal{len}}=100$, $m\approx\numprint{5000}$, ${\textnormal{sh}^{\textnormal{mean}}}={\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}}(D^T)=1$ and the CVs and mass are zero. Similarly, for the empty DAG, ${\textnormal{len}}={\textnormal{mass}}=1$, the mean shape is 100 and all the other properties are zero.
The number of permutations quickly constrains the length. For instance, the length is already between 15 and 20 when $K=2$ and at most 5 when $K\ge5$. A formal analysis suggests that the length is almost surely in $O(n^{1/K})$[@winkler1985random Theorem 3], which is consistent with our observation. The number of edges and the maximum degree in the transitive reduction reach larger values than with previous approaches for any size $n$ (twice larger than with the [Erdős-Rényi]{}algorithm). Moreover, the mass is always close to one for $K>1$. Some specific values can finally be explained. First, the maximum value for ${\textnormal{deg}_{\textnormal{}}^{\textnormal{CV}}}(D^T)$ is exactly 7 and corresponds to DAGs of size $n=100$ with a single edge (2 vertices have degree 1 and 98 others have 0). Also, the shape CV is at most 0.98 when the length is 2 (which frequently when $K\ge10$). This CV corresponds to a shape with values 99 and 1.
![\[fig:poset\_node\] Properties of 191 DAGs generated by the random orders algorithm with $K=3$ permutations and for each size $n$ between 10 and 200. The smoothed line is obtained with a linear regression using a polynomial spline with 4 degrees of freedom. Red lines correspond to formal results for the length and mean shape. ](4_poset_analysis_node)
Figure \[fig:poset\_node\] shows the effect of the number of vertices $n$ for a fixed number of permutations $K$. We selected $K=3$ to have the maximum number of edges in the transitive reduction. The sublinear relation between the length and size $n$ is again consistent with the previously cited result (i.e. $O(n^{1/K})$). Even though $K=3$ is small, the length is already low, leading to line patterns for both the length and the mean shape. Note that the mass is frequently either 1 or almost 1 (i.e., $1-\frac1{k}$), which corresponds to cases where only the last value of the shape ${\textnormal{sh}^{\textnormal{k}}}$ is one.
The random orders method can generate denser DAGs than [Erdős-Rényi]{}or uniform DAGs without the mass issue, but with difficult control over the compromise between the length and the mean shape.
Layer-by-Layer
--------------
Many variants of the layer-by-layer principle have been used throughout the literature to assess scheduling algorithms and are covered in Section \[sec:layer-layer-method\]. This section analyzes the effect of three parameters (size $n$, number of layers $k$ and connectivity probability $p$) using the following variant inspired from[@cordeiro2010a; @gupta2017a]. First, $k$ vertices are affected to distinct layers to prevent any empty layer. Then, the remaining $n-k$ vertices are distributed to the layers using a balls into bins approach (i.e. a uniformly random layer is selected for each vertex). For each vertex not in the first layer, a random parent is selected among the vertices from the previous layer to ensure that the layer of any vertex equals its depth (similar to[@dutot2009a; @gupta2017a] and the recursive method in Section \[sec:recursive\]). Finally, random edges are added by connecting any pair of vertices from distinct layers from top to bottom with probability $p$.
This variant departs from[@cordeiro2010a; @gupta2017a] to ensure generated DAGs have a length equal to $k$ and mean shape equal to $n/k$. Moreover, with some parameter values, this method produces some of the special DAGs covered in Section \[sec:analys-spec-dag\]. It generates the empty DAG when $k=1$, whereas it generates the complete DAG with $k=n$ and $p=1$. To interpret the number of edges depicted in Figures \[fig:layer\_proba\] to \[fig:layer\_node\], we study the case (called *regular*) when all layers have the same size $n/k$, which constitutes an approximation of the DAGs generated by the layer-by-layer variant studied in this section. When $p=1$, the DAG is the bipartite one for $k=2$ and the square one for $k=\sqrt{n}$. In such DAGs and when $n$ is a multiple of $k$, the expected number of edges is $$\label{eq:edges_regular}
{\mathbb{E}}(m)=n\left(1-\frac{1}{k}\right)\left(p\left(\frac{n}{2}-1\right)+1\right)$$ and the expected number of edges in the transitive reduction is $$\label{eq:edges_red_regular}
{\mathbb{E}}(m(D^T))\ge p(k-1)\left(\frac{n}{k}\right)^2+(1-p)n\left(1-\frac{1}{k}\right).$$
![\[fig:layer\_proba\] Properties of 300 DAGs of size $n=100$ generated by the layer-by-layer algorithm with $k=10$ layers and probability $p$ uniformly drawn between 0 and 1. The smoothed line is obtained with a linear regression using a polynomial spline with 4 degrees of freedom. Red lines correspond to formal results for the length and mean shape, the number of edges (Equation \[eq:edges\_regular\]), the number of edges in the transitive reduction (Equation \[eq:edges\_red\_regular\]), and the mass. ](5_layer_analysis_proba)
Figure \[fig:layer\_proba\] shows the effect of the probability $p$. The analysis for *regular* layer-by-layer DAGs closely approximates the results. The number of edges $m$ is predicted to increase linearly from 90 to (Equation \[eq:edges\_regular\]), while this quantity in the transitive reduction $m(D^T)$ is expected to increase from 90 to 900 (Equation \[eq:edges\_red\_regular\]). Remark that this last property undergoes a steeper increase for probability $p<0.1$ than for larger $p$. With many edges ($p>0.1$), adding a new one is likely to result into the introduction of redundant edges, which is not the case for $p<0.1$. More generally, the layered structure ensures a steady increase of $m(D^T)$ as the probability $p$ increases because any edge between two consecutive layers cannot become redundant through the insertion of any edge. The mass is always close to one because the probability to have a layer with one vertex is close to zero with $k=10$ layers.
![\[fig:layer\_layer\] Properties of 300 DAGs of size $n=100$ generated by the layer-by-layer algorithm with probability $p=0.5$ and a number of layers $k$ randomly drawn between 1 and 100 ($k=\lfloor {e^{\mathcal{U}(\log(1),\log(101))}} \rfloor$ where $\mathcal{U}(a,b)$ is a uniform distribution between $a$ and $b$). The smoothed line is obtained with a linear regression using a polynomial spline with 5 degrees of freedom. Red lines correspond to formal results for the length and mean shape, the number of edges (Equation \[eq:edges\_regular\]), the number of edges in the transitive reduction (Equation \[eq:edges\_red\_regular\]), and the mass. ](5_layer_analysis_layer)
Figure \[fig:layer\_layer\] represents the effect of the number of layers $k$. With *regular* layer-by-layer DAGs, the expected number of edges ${\mathbb{E}}(m)$ goes from 0 to for $k=1$ to 100 (Equation \[eq:edges\_regular\]), which is close to the results with our layer-by-layer variant. The increase is steep because it is already for $k=10$, which is consistent with Figure \[fig:layer\_layer\]. The number of edges in the transitive reduction ${\mathbb{E}}(m(D^T))$ decreases from an expected value of to 99 as the number of layers goes from $k=2$ to 100 (Equation \[eq:edges\_red\_regular\]). The expected value for $k=10$ is 495 and is consistent with both Figures \[fig:layer\_proba\] and \[fig:layer\_layer\]. Finally, the mass is unitary when there are at least two balls in each bin. Since there is initially one ball per bin, this occurs when there is at least one of the $n-k$ additional balls in each of the $k$ bin. To compute if there are enough additional balls to have a unitary mass with probability greater than $0.5$, we can use a bound for the coupon collector problem[@levin2017markov Proposition 2.4]. This occurs when $\left\lceil k\log(2k) \right\rceil+k<n$, which is the case for $k\le20$ with $n=100$. This is consistent with Figure \[fig:layer\_layer\] where the mass becomes non-unitary around this value.
![\[fig:layer\_node\] Properties of 191 DAGs generated by the layer-by-layer algorithm with probability $p=0.5$, $k=\sqrt{n}$ (rounded to closest integer) layers and for each size $n$ between 10 and 200. The smoothed line is obtained with a linear regression using a polynomial spline with 4 degrees of freedom. Red lines correspond to formal results for the length and mean shape, the number of edges (Equation \[eq:edges\_regular\]), the number of edges in the transitive reduction (Equation \[eq:edges\_red\_regular\]), and the mass. ](5_layer_analysis_node)
When varying the number of vertices $n$, we expect the number of edges $m$ to increase quadratically from 20 to around (Equation \[eq:edges\_regular\]), which is consistent with the results on Figure \[fig:layer\_node\]. Similarly, the number of edges in the transitive reduction $m(D^T)$ is expected to increase quadratically from around 14.4 to around (Equation \[eq:edges\_red\_regular\]).
In Figures \[fig:layer\_proba\] to \[fig:layer\_node\], the length and mean shape show stable behavior consistent with our expectation. In all figures, the shape CV can formally be analyzed using the balls into bins model and we refer the interested reader to the specialized literature[@kolchin1978random]. Finally, in the transitive reduction, the maximum degree ${\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}}(D^T)$ has a similar trend as the number of edges $m(D^T)$.
To avoid non-unitary mass, the layer-by-layer method can be adapted to ensure that each layer has two vertices initially. For instance, we can rely on a uniform distribution between two and a maximum value, or on a balls into bins approach with two balls per bin initially. It is also possible to use the method described in[@canon2018markov Section III] to have a uniform distribution of the vertices in the layers over all possible distributions and with a constraint on the minimum value.
Evaluation on Scheduling Algorithms {#sec:eval-sched-algor}
===================================
Generating random task graphs allows the assessment of existing scheduling algorithms in different contexts. Numerous heuristics have been proposed for the problem denoted [$P|p_j=1,prec|C_{\max}$]{}(homogeneous tasks and processors, see Section \[sec:scheduling\]) or generalization of this problem. Such heuristics rely on different principles. Some simple strategies, like MinMin, execute available tasks on the processors that minimize completion time without considering precedence constraints. In contrast, many heuristics sort tasks by criticality and schedule them with the Earliest Finish Time (EFT) policy (e.g. HEFT and HCPT). Finally, other principles may be also used: migration for BSA[@kwok2000a], clustering for DSC[@yang1994dsc], etc. We focus on the impact of generation methods on the performance of a selection of three heuristics for this problem: MinMin, HEFT and HCPT.
HEFT[@topcuoglu2002a] (Heterogeneous Earliest Finish Time) first computes the upward rank of each task, which can be seen as a reversal depth (depth in the reversal DAG). It then consider tasks by decreasing order of their upward ranks and schedules them with the EFT policy. Backfilling is performed following an insertion policy that tries to insert a task at the earliest idle time between two already scheduled tasks on a processor if the slot is large enough to accommodate it. The time complexity of this approach is dominated by the insertion policy in $O(n^2)$. Numerous heuristics are equivalent to HEFT when tasks and processors are homogeneous: PEFT[@arabnejad2014list], HLEFT[@adam1974a], HBMCT[@sakellariou2004hybrid].
HCPT[@hagras2003simple] (Heterogeneous Critical Parent Trees) starts by considering any task on a critical path by decreasing order of their depth. The objective is to prioritize the ancestors of such tasks and in particular when their depth is large. This process generates a priority list of tasks that are then scheduling with the EFT policy. The time complexity is $O(m+n\log(n)+n|P|)$ where $|P|$ is the number of processors.
Finally, MinMin[@ibarra1977 Algorithm D][@freund1998a minmin] considers all available tasks any time a processor becomes idle and schedules any task on any available processor. With homogeneous tasks and processors, this algorithm is equivalent to MaxMin[@ibarra1977 Algorithm E][@freund1998a maxmin]. The time complexity is $O(m)$.
![\[fig:comparison\] Difference between the makespan obtained with any heuristic and the best value among the three heuristics for each instance. Each boxplot represents the results for 300 DAGs of size $n=100$ built with one of the following methods: the [Erdős-Rényi]{}algorithm with probability $p=0.15$, the recursive algorithm, the random orders algorithm with $K=3$ permutations and the layer-by-layer algorithm with probability $p=0.5$ and a number of layers $k=10$. Costs are unitary and $|P|$ represents the number of processors. ](6_heuristics_comparison)
Figure \[fig:comparison\] shows the absolute difference between HEFT, HCPT and MinMin for each generation method covered in Section \[sec:existinggeneration\]. Despite guaranteeing an unbiased generation, instances built with the recursive algorithm fail to discriminate heuristics except when there are two processors. Recall that the mean shape is close to $1.5$ for such DAGs and few processors are sufficient to obtain a makespan equal to the DAG length (i.e. an optimal schedule). In contrast, instances built with the random orders algorithm lead to difference performance for each scheduling heuristics. However, this generation method has no uniformity guarantee and its discrete parameter $K$ limits the diversity of generated DAGs. Finally, the last two algorithms, [Erdős-Rényi]{}and layer-by-layer, fail to highlight a significant difference between MinMin and HEFT even though the former scheduling heuristic can be expected to be inferior to the latter because it discards the DAG structure.
To support these observations, we analyse below the maximum difference between the makespan obtained with HEFT and the ones obtained with the other two heuristics. Because it lacks any backfilling mechanism, HCPT performs worse than HEFT with an instance composed of the following two elements. First, a chain of length $k$ with $|P|-1$ additional tasks with predecessor the $(k-2)$th task of the chain and successor the $k$th task of the chain. Alternatively, this first element can be seen as a chain of length $k-3$ connected to a fork-join with width $|P|$. The second element is a chain of length $k-1$. HCPT schedules the first element and then the second one afterward, leading to a makespan of $2k-1$ whereas the optimal one is $k$. With $n=100$ tasks and $|P|\le10$, the difference from HEFT with this instance is greater than or equal to 45. Moreover, MinMin also performs worse with specific instances. Consider the ad hoc instances considered in[@canon2018a] each consisting of one chain of length $k$ and a set of $k(|P|-1)$ independent tasks. Discarding the information about critical tasks prevents MinMin from prioritizing tasks from the chain. With $n=100$ tasks and with $|P|\le10$, the worst-case absolute difference can be greater than or equal to 9 (when MinMin completes first the independent before starting the chain). While the difficult instances for HCPT rely on a specific weakness, it is interesting to analyse the properties of the difficult instances for MinMin. Each DAG is characterized by a length equal to ${\textnormal{len}}=\frac{n}{|P|}$ and a number of edges in the transitive reduction $m(D^T)={\textnormal{len}}-1$ (leading to a large width and a large shape standard deviation). With $n=100$ tasks, with both HCPT and MinMin, the absolute difference from HEFT can be greater than or equal to 9.
Theses experiments illustrate the need for better generation methods that control multiple properties while avoiding any generation bias. An ideal generation method would uniformly select a DAG over all existing DAGs having a given number of tasks $n$, number of edges $m$ and/or $m(D^T)$, length and/or width, and with a unitary mass.
Conclusion
==========
This work contributes in three ways to the final objective of uniformly generating random DAGs belonging to a category of instances with desirable characteristics. First, we identify a list of 34 DAG properties and focus on a selection of 8 such properties. Among these, the mass quantifies how much an instance can be decomposed into smaller ones. Second, existing random generation methods are formally analyzed and empirically assessed with respect to the selected properties. Establishing the sub-exponential generic time complexity for decomposable scheduling problems with uniform instances constitutes the most noteworthy result of this paper. Last, we study how the generation methods impact scheduling heuristics with unitary costs.
The relevance and impact of many other properties need to be investigated. For instance, the number of tasks present on a critical path can exceed the length and even reach $n$. Also, we could measure the distance of a DAG from a serie-parallel one by counting with the minimum number of edges to remove in the former DAG to obtain the latter one. Both these measures may impact the performance of scheduling heuristics.
Adapting current results to instances with communication costs requires some adaptations that need to be explored. For instance, each edge with a cost could be discarded when there is another path of higher processing cost (i.e. assuming all communication costs are null on this path). The definition of the mass could state that a vertex is a bottleneck vertex when no edge connect a preceding vertex to a following one.
Finally, extending properties to instances with non-unitary costs is left to future work. For instance, the shape could be replaced by the continuous occupation of the DAG when scheduled on an infinite number of processors (i.e.the number of occupied processors at each time step). As a result, the length would be the critical path length and the mean shape would be the sum of all costs (called the work) divided by the critical path length (called parallelism in[@tobita2002a]).
Exact Properties of Special DAGs {#sec:exact-prop-spec}
================================
Tables \[tab:DAG-prop-prec-edge\] and \[tab:DAG-prop-prec-node\] synthesize the exact properties of the special DAGs presented in Section \[sec:analys-spec-dag\].
[m[0.08]{}ccccccccc]{} DAG & $m$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{max}}}$ & ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{max}}}$ & ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{max}}}$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{min}}}$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{mean}}}$ & ${\textnormal{deg}_{\textnormal{}}^{\textnormal{sd}}}$ & ${\textnormal{deg}_{\textnormal{in}}^{\textnormal{sd}}}$ & ${\textnormal{deg}_{\textnormal{out}}^{\textnormal{sd}}}$\
$D_{\textnormal{empty}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\
$D_{\textnormal{complete}}$ & $\frac{n(n-1)}{2}$ & $n-1$ & $n-1$ & $n-1$ & $n-1$ & $n-1$ & 0 & $\sqrt{\frac{n^2-1}{12}}$ & $\sqrt{\frac{n^2-1}{12}}$\
$D_{\textnormal{chain}}$ & $n-1$ & $2$ & $1$ & $1$ & $1$ & $2(1-\frac{1}{n})$ & $\sqrt{\frac{2}{n}(1-\frac{2}{n})}$ & $\sqrt{\frac{1}{n}(1-\frac{1}{n})}$ & $\sqrt{\frac{1}{n}(1-\frac{1}{n})}$\
$D_{\textnormal{out-tree}}$ $D_{\textnormal{comb}}$ & $n-1$ & $3$ & $1$ & $2$ & $1$ & $2(1-\frac{1}{n})$ & $\sqrt{1-\frac1{n}-\frac4{n^2}}$ & $\sqrt{\frac{1}{n}(1-\frac{1}{n})}$ & $\frac{\sqrt{n-1}\sqrt{n+1}}{n}$\
$D_{\textnormal{in-tree}}$ $D_{\textnormal{comb}}^R$ & $n-1$ & $3$ & $2$ & $1$ & $1$ & $2(1-\frac{1}{n})$ & $\sqrt{1-\frac1{n}-\frac4{n^2}}$ & $\frac{\sqrt{n-1}\sqrt{n+1}}{n}$ & $\sqrt{\frac{1}{n}(1-\frac{1}{n})}$\
$D_{\textnormal{bipartite}}$ & $\frac{n^2}{4}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 0 & $\frac{n}{4}$ & $\frac{n}{4}$\
$D_{\textnormal{square}}$ & $n(\sqrt{n}-1)$ & $2\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $2(\sqrt{n}-1)$ & $\sqrt{2\sqrt{n}-4}$ & $\sqrt{\sqrt{n}-1}$ & $\sqrt{\sqrt{n}-1}$\
$D_{\textnormal{triangular}}$ & $\frac{k(k+1)(k-1)}{3}$ & $2(k-1)$ & $k-1$ & $k$ & 2 & $\frac{4}{3}(k-1)$ & $\frac{\sqrt{2}}{3}(k-1)$ & $\frac{\sqrt{(k-1)(k+2)}}{3\sqrt{2}}$ & $\frac{\sqrt{(k-1)(k+14)}}{3\sqrt{2}}$\
[m[0.08]{}ccccccccc]{} DAG & ${\textnormal{len}}$ & ${\textnormal{width}}$ & ${\textnormal{sh}^{\textnormal{max}}}$ & ${\textnormal{sh}^{\textnormal{min}}}$ & ${\textnormal{sh}^{\textnormal{mean}}}$ & ${\textnormal{sh}^{\textnormal{sd}}}$ & ${\textnormal{sh}^{\textnormal{1}}}$ & ${\textnormal{sh}^{\textnormal{k}}}$ & ${\textnormal{mass}}$\
$D_{\textnormal{empty}}$ & 1 & $n$ & $n$ & $n$ & $n$ & 0 & $n$ & $n$ & 1\
$D_{\textnormal{complete}}$ $D_{\textnormal{chain}}$ & $n$ & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0\
$D_{\textnormal{out-tree}}$ & $\log_2(n+1)$ & $\frac{n+1}{2}$ & $\frac{n+1}{2}$ & 1 & $\frac{n}{\log_2(n+1)}$ & $\sqrt{\frac{n}{\log_2(n+1)}\left(\frac{n+2}{3}-\frac{n}{\log_2(n+1)}\right)}$ & 1 & $\frac{n+1}{2}$ & $1-\frac1{n}$\
$D_{\textnormal{in-tree}}$ & $\log_2(n+1)$ & $\frac{n+1}{2}$ & $\frac{n+1}{2}$ & 1 & $\frac{n}{\log_2(n+1)}$ & $\sqrt{\frac{n}{\log_2(n+1)}\left(\frac{n+2}{3}-\frac{n}{\log_2(n+1)}\right)}$ & $\frac{n+1}{2}$ & 1 & $1-\frac1{n}$\
$D_{\textnormal{comb}}$ & $\frac{n+1}{2}$ & $\frac{n+1}{2}$ & 2 & 1 & $2(1-\frac{1}{n+1})$ & $\sqrt{\frac{2}{n+1}(1-\frac{2}{n+1})}$ & 1 & 2 & $1-\frac1{n}$\
$D_{\textnormal{comb}}^R$ & $\frac{n+1}{2}$ & $\frac{n+1}{2}$ & $\frac{n+1}{2}$ & 1 & $2(1-\frac{1}{n+1})$& $\frac{n-1}{n+1}\sqrt{\frac{n-1}{2}}$ & $\frac{n+1}{2}$ & 1 & $\frac1{2}+{1}{2n}$\
$D_{\textnormal{bipartite}}$ & 2 & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & $\frac{n}{2}$ & 0 & $\frac{n}{2}$ & $\frac{n}{2}$ & 1\
$D_{\textnormal{square}}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & $\sqrt{n}$ & 0 & $\sqrt{n}$ & $\sqrt{n}$ & 1\
$D_{\textnormal{triangular}}$ & $k$ & $k$ & $k$ & 1 & $\frac{k+1}{2}$ & $\sqrt{\frac{k^2-1}{12}}$ & 1 & $k$ & $1-\frac1{n}$\
Probabilistic Properties of Random Triangular Matrices {#sec:annexe:probaER}
======================================================
We investigate in this section some probabilistic results on DAG generated by the [Erdős-Rényi]{}approach.
\[prop:ERmean\] Let $D$ be a DAG with $n$ vertices randomly generated by the [Erdős-Rényi]{} algorithm with parameter $p$. Denoting by $(X_1,\ldots,X_k)$ its shape decomposition, one has, for each $1\leq i\leq k$, ${\mathbb{E}}(|X_i|)\leq\frac{1}{p}$.
Let $M_{i,j}$ be the upper triangular matrix corresponding to $D$. Let $Y_i=\cup_{j<i} X_j$. If $j\in X_i$, then, for all $r <j$ such that $r\notin Y_i$, $M_{r,j}=0$. Therefore, since the $M_{r,j}$ are independent Bernoulli random variables of parameter $p$, ${\mathbb{P}}(j\in X_i)\leq (1-p)^{j-|Y_i|}$. Consequently, ${\mathbb{E}}(x_i)\leq
1+(1-p)+\ldots+(1-p)^{n-|Y_i|}\leq \frac{1}{1-(1-p)}=\frac{1}{p}$.
\[prop:ERET\] Let $D$ be a DAG with $n$ vertices randomly generated by the [Erdős-Rényi]{} Algorithm with parameter $p$. One has ${\mathbb{E}}(m(D^T))\leq \frac{n-1}{p}-\frac{1-p^2}{p^3}(1-(1-p^2)^{n-1})$.
Let $M_{i,j}$ be the upper triangular matrix corresponding to $D$. $$A=\{(i,j)\mid 1\leq i <j \leq n,\ (i,j)\in E\text{ s.t. } \forall
i<r<j,\ (i,r)\notin E \text{ or }(r,j)\notin E\},$$ where $E$ is the set of edges of $D$. By definition of $D^T$, if $(i,j)\in D^T$, then $(i,j)\in A$. Consequently, $$\label{eq:prop:ERET1}
|D^T|\leq |A|.$$
Moreover, for every $i<j$, ${\mathbb{P}}((i,j)\in A)={\mathbb{P}}(M_{i,j}=1\text{
and } \forall i<r<j,\ M_{i,r}=0\text{ or } M_{r,j}=0)$. Since the $M_{i,j}$ are independent Bernoulli random variables, $$\begin{aligned}
{\mathbb{P}}((i,j)\in A)&={\mathbb{P}}(M_{i,j}=1)\Pi_{r=i+1}^{j-1}{\mathbb{P}}( M_{i,r}=0\text{ or } M_{r,j}=0)\\
&=p \Pi_{r=i+1}^{j-1}(1- {\mathbb{P}}( M_{i,r}=1\text{ and } M_{r,j}=1))\\
&=p\Pi_{r=i+1}^{j-1}(1-p^2)=p(1-p^2)^{j-i-1}.\end{aligned}$$
Let $A_{i,j}$ be the Bernoulli random variable encoding that $(i,j)\in A$. One has $|A|=\sum_{i<j} A_{i,j}$. Consequently, $$\begin{aligned}
{\mathbb{E}}(|A|)&=\sum_{j=2}^n\sum_{i=1}^{j-1} {\mathbb{E}}(A_{i,j})=\sum_{j=2}^n\sum_{i=1}^{j-1}p(1-p^2)^{j-i-1}\\
&=\sum_{j=2}^n\sum_{r=0}^{j-2}p(1-p^2)^{r}\quad \text{with }r=j-i-1\\
&=p\sum_{j=2}^n\frac{1-(1-p^2)^{j-1}}{1-(1-p^2)}\\
&=\frac{1}{p}\sum_{j=2}^n(1-(1-p^2)^{j-1})\\
&=\frac{n-1}{p}-\frac{1}{p}\sum_{j=2}^n(1-p^2)^{j-1}\\
&=\frac{n-1}{p}-\frac{1-p^2}{p}\sum_{j=0}^{n-2}(1-p^2)^{j}\\
&=\frac{n-1}{p}-\frac{1-p^2}{p}\frac{1-(1-p^2)^{n-1}}{1-(1-p^2)}\\
&=\frac{n-1}{p}-\frac{1-p^2}{p^3}(1-(1-p^2)^{n-1}).\end{aligned}$$
One can conclude using Equation (\[eq:prop:ERET1\]).
Probabilistic Properties of Random Uniform DAGs {#sec:annexe:proba}
===============================================
We are interested in this section in the probabilistic properties of the shape of a DAG $D$ randomly generated with the uniform distribution as exposed in Section \[sec:recursive\]. In this context, we consider a random shape $(x_1,\ldots,x_k)$ generated by Algorithm \[algo:shaperec\] (for a DAG with $n$ vertices). Note that the length $k$ of the shape is a random variable and that the $x_i$’s are dependent random variables. However, the distribution of $x_i$ only depends on the sum of the $x_j$’s, with $j<i$ (formally, ${\mathbb{P}}(x_i=r\mid x_1,\ldots,x_i-1)={\mathbb{P}}(x_i=r\mid
s_i)=\frac{a_{n-s_i,k}}{a_{n-s_i}}$). Let $s_1=0$ and for $i\geq 1$, $s_i=\sum_{j<i}x_j$. The following result is proved in[@liskovetsmaixmal Proposition 3]: if $n-s_i\geq 2$ and $r\geq
2$, $$\label{eq:liskovets}
{\mathbb{P}}(x_i\leq r\mid s_i)\geq
1-\frac{n-s_i-r}{n-s_i+1}\frac{2}{(r+1)!\ 2^{\frac{r(r-1)}{2}}}.$$
It follows from Equation (\[eq:liskovets\]), for $r\geq 2$ and if $n-s_i\geq 2$, $${\mathbb{P}}(x_i> r\mid s_i)\leq
\frac{n-s_i-r}{n-s_i+1}\frac{2}{(r+1)!\ 2^{\frac{r(r-1)}{2}}}\leq
\frac{2}{(r+1)!\ 2^{\frac{r(r-1)}{2}}}.$$
The above equation still holds if $n-s_i< 2$ since the probability is null.
These upper bounds show that the probability of having large values in the shape is very small. For instance, the probability that $x_i\geq 9$ is less than $10^{-11}$.
Moreover, since for $r >2$, $(r+1)!\geq
2^{r+1}$, one has for every $r>2$,
$$\label{eq:preuve1}
{\mathbb{P}}(x_i> r\mid s_i)\leq
2^{-\frac{r(r+1)}{2}}.$$
The following lemma will be useful.
\[lemma:max\] One has, for every $r>2$, $n\geq 2$, ${\mathbb{P}}(\max(x_i)>r)\leq n2^{-\frac{r(r+1)}{2}}$.
Let $A_{i,r}$ denotes the event *$x_i\leq r$*. One has $$\begin{aligned}
&{\mathbb{P}}(\max(x_i)\leq r)={\mathbb{P}}(\bigcap_{1\leq i \leq k} A_{i,r})\\
&={\mathbb{P}}(A_{1,r}){\mathbb{P}}(A_{2,r}\mid A_{1,r})\ldots
{\mathbb{P}}(A_{i,r}\mid A_{1,r},A_{2,r},\ldots,A_{i-1,r})\ldots
{\mathbb{P}}(A_{k,r}\mid A_{1,r},A_{2,r},\ldots,A_{k-1,r})
\end{aligned}$$
Using Equation (\[eq:preuve1\]) and Bernoulli’s inequality, it follows that ${\mathbb{P}}(\max(x_i)\leq r)\geq (1-2^{-\frac{r(r+1)}{2}})^k\geq
(1-2^{-\frac{r(r+1)}{2}})^n\geq 1-n2^{-\frac{r(r+1)}{2}}.$ Now, ${\mathbb{P}}(\max(x_i)>r)=1-{\mathbb{P}}(\max(x_i)\leq r)\leq
n2^{-\frac{r(r+1)}{2}}$.
We can now claim an upper bound for the expected value of ${\textnormal{sh}^{\textnormal{max}}}(D)=\max(x_i)$.
\[prop:max\] One has ${\mathbb{E}}(\max(x_i))=O(\log n)$.
Let $h=\left\lfloor \sqrt{6\log_2 n}\right\rfloor+1$. $$\begin{aligned}
{\mathbb{E}}(\max(x_i))&=\sum_{r=1}^n {\mathbb{P}}(\max(x_i)=r).r\\
&=\sum _{r=1}^hr{\mathbb{P}}(\max(x_i)=r) \quad+\quad
\sum _{r=h+1}^nr{\mathbb{P}}(\max(x_i)=r)\\
&\leq \sum _{r=1}^hh{\mathbb{P}}(\max(x_i)=r)+
\sum _{r=h+1}^n r {\mathbb{P}}(\max(x_i)=r)\\
&\leq h^2+\sum _{r=h+1}^n r {\mathbb{P}}(\max(x_i)\geq r)\\
&\leq h^2+\sum _{r=h+1}^n r {\mathbb{P}}(\max(x_i> r-1)\\
&\leq h^2+\sum _{r=h+1}^n r n 2^{-\frac{r(r-1)}{2}}\\
&\leq h^2+n 2^{-\frac{h(h+1)}{2}}\sum _{r=h+1}^n r\\
&\leq 6\log_2 n +n^3 2^{-\frac{h(h+1)}{2}}
\end{aligned}$$
Since $r>3$ for $n\ge2$, we can apply Lemma \[lemma:max\] to eliminate the probability in the second term. Note that $2^{-\frac{h(h+1)}{2}}\leq 2^{-\frac{h^2}{2}}\leq
2^{-3\log_2 n}$. Since $n^3 2^{-3\log_2 n}=1$, we have ${\mathbb{E}}(\max(x_i))\leq 6\log_2 n +1$, proving the result.
It is proved in[@liskovetsmaixmal] that ${\mathbb{E}}(x_1)$ converges to a constant (approximately 1.488) when $n$ grows to infinity. One can easily obtain a bound for each level and each $n$.
\[prop:expectshape\] One has, for every $n\geq 2$, every $1\leq i\leq k$, ${\mathbb{E}}(x_i)\leq 2+\frac1{4}$.
$$\begin{aligned}
{\mathbb{E}}(x_i)&=\sum_{r=1}^n j {\mathbb{P}}(x_i=r)=
{\mathbb{P}}(x_i=1)+2{\mathbb{P}}(x_i=2)
+\sum_{r=3}^n r{\mathbb{P}}(x_i=r)\\
\end{aligned}$$
Note that ${\mathbb{P}}(x_i=1)+{\mathbb{P}}(x_i=2)\le1$. $$\begin{aligned}
{\mathbb{E}}(x_i)&\leq 2+\sum_{r=3}^n \frac{r}{2^{r(r+1)/2}} \leq
2+\sum_{r=3}^n \frac{1}{2^r}\frac{r}{2^{(r+1)/2}}\\
&\leq 2+\sum_{r=3}^n \frac{1}{2^r}.
\end{aligned}$$ Since $r\ge3$ for $n\ge2$, we can apply Lemma \[lemma:max\] to eliminate the probability in the last term. Since $\sum_{r=1}^n \frac{1}{2^r}\leq 1$, ${\mathbb{E}}(x_i)\leq 2+\frac1{4}$.
Previous results confirm experimental results in Section \[sec:recursive\] and show that the values of the shape are all quite small. In order to evaluate the mass of a random DAG, we will now investigate the lengths of the bloc. More precisely, let $$\ell_{\max}=\max \{\ell\mid \exists i\text{ s. t. }
(1-x_i)(1-x_{i+1})\ldots(1-x_{i+\ell-1})\neq 0\},$$ the maximum length of a sequence of consecutive $x_i$ non equal to $1$.
It is proved in[@liskovetsmaixmal page 407] that there exists a constant $0< \alpha_0<2/3$ such that for all $n$, ${\mathbb{P}}(x_i=1\mid
s_i)\geq \alpha_0$ (the constant proposed in[@liskovetsmaixmal] is $\frac{1}{96}$ but practical evaluation leads to claim that the probability to have only a single vertex in a level is greater than or equal to $1/3$).
\[lemma:run\] For every $r>0$, ${\mathbb{P}}(\ell_{\max}\geq \ell)\leq n(1-\alpha_0)^\ell$.
One has $$\begin{aligned}
{\mathbb{P}}((1-x_i)&(1-x_{i+1})\ldots(1-x_{i+\ell-1})\neq 0\mid s_i)\\
&={\mathbb{P}}((1-x_i)\neq 0\mid s_i)
{\mathbb{P}}((1-x_{i+1})\ldots(1-x_{i+\ell-1})\neq
0\mid s_i\text{ and } x_1\neq 1)\\
&\leq(1-\alpha_0){\mathbb{P}}((1-x_{i+1})
\ldots(1-x_{i+\ell-1})\neq 0\mid s_{i+1})
\end{aligned}$$ By a direct induction, one get ${\mathbb{P}}((1-x_i)(1-x_{i+1})\ldots(1-x_{i+\ell-1})\neq 0\mid
s_i)\leq (1-\alpha_0)^\ell$. Now let $\overline{x}_i$ be defined, for $1\leq i\leq n$ by $\overline{x}_i=x_i$ if $i\leq k$ and $\overline{x}_i=1$ otherwise. We also denote by $\overline{s}_i$ the sum $\sum_{j=1}^{-1}
\overline{x}_i$. $$\begin{aligned}
{\mathbb{P}}(\ell_{\max}\geq \ell)
&={\mathbb{P}}(\cup_{i=1}^{k-\ell+1}\{(1-x_i)(1-x_{i+1})
\ldots(1-x_{i+\ell-1})\neq 0\mid s_i\})\\
&\leq {\mathbb{P}}(\cup_{i=1}^{n-\ell+1}\{(1-\overline{x}_i)(1-\overline{x}_{i+1})
\ldots(1-\overline{x}_{i+\ell-1})\neq 0\mid \overline{s}_i\})\\
&\leq \sum_{i=1}^{k-n+1}{\mathbb{P}}((1-x_i)(1-x_{i+1})
\ldots(1-x_{i+\ell-1})\neq 0\mid s_i)\\
&\leq n(1-\alpha_0)^\ell,
\end{aligned}$$ proving the result.
One can now prove Theorem \[theorem:mass\].
Consider the event $A_n={\textnormal{sh}^{\textnormal{max}}}(D)\geq \log^2(n)$ and $B_n=\ell_{\max}(D)\geq\log^2(n)$. Using Markov Inequality and Proposition \[prop:max\], ${\mathbb{P}}(A_n)\leq \frac{1}{\log n}$. Therefore, ${\mathbb{P}}(A_n)\to 0$ when $n\to+\infty$.
Moreover ${\mathbb{P}}(B_n)\leq n(1-\alpha_0)^{\log^2n}$ by Lemma \[lemma:run\]. But $\log(n(1-\alpha_0)^{\log^2n})=\log n + \log(1-\alpha_0) \log^2n$. Since $0<1-\alpha_0<1$, $\log n + \log(1-\alpha_0) \log^2n\to -\infty$ when $n\to+\infty$. Consequently $n(1-\alpha_0)^{\log^2n}\to 0$ when $n\to+\infty$.
Therefore ${\mathbb{P}}(A_n\cup B_n)\to 0$ when $n\to +\infty$.
[^1]: <http://ziyang.eecs.umich.edu/projects/tgff/index.html>
[^2]: <https://github.com/frs69wq/daggen>
[^3]: <https://github.com/perarnau/ggen>
[^4]: <https://confluence.pegasus.isi.edu/display/pegasus/WorkflowGenerator>
[^5]: <https://github.com/nizarsd/xl-stage>
[^6]: <https://github.com/anubhavcho/RandomWorkflowGenerator>
[^7]: <http://users.ecs.soton.ac.uk/ras1n09/rtrg/index.html> (unavailable as of this writing)
[^8]: <http://www.es.ele.tue.nl/sdf3/>
[^9]: <https://github.com/bbodin/turbine>
[^10]: <http://www.kasahara.elec.waseda.ac.jp/schedule/>
[^11]: This is the case for the instance `rand0038.stg` for size 50.
[^12]: <http://www.om-db.wi.tum.de/psplib/>
[^13]: <ftp://infokit.dis.uniromal.it/public/> (unavailable as of this writing)
[^14]: Each boxplot consists of a bold line for the median, a box for the quartiles, whiskers that extend at most to 1.5 times the interquartile range from the box and additional points for outliers.
|
November 1994\
Submitted to Phys. Rev. E
[**Exact Exponent $\lambda$ of the Autocorrelation Function for a Soluble Model of Coarsening**]{}\
A. J. Bray$^1$ and B. Derrida$^{2,3}$\
$^1$Department of Physics and Astronomy, The University, Manchester M13 9PL, UK.\
$^2$Laboratoire de Physique Statistique, Ecole Normale Supérieure, 24 Rue Lhomond, 75231 Paris cedex 05, France.\
$^3$Service de Physique Théorique, CE Saclay, 91191 Gif sur Yvette, France.\
\
The exponent $\lambda$ that describes the decay of the autocorrelation function $A(t)$ in a phase ordering system, $A(t) \sim L^{-(d-\lambda)}$, where $d$ is the dimension and $L$ the characteristic length scale at time $t$, is calculated exactly for the time-dependent Ginzburg-Landau equation in $d=1$. We find $\lambda = 0.399\,383\,5\ldots$. We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as $M(t) \sim L^\mu$ and prove that for the $1d$ Ginzburg-Landau equation, $\mu=\lambda$, exemplifying a general result.
The field of phase ordering kinetics has seen a number of new developments in recent years [@BrayRev]. In particular the values of the growth exponents $z$, which describe the time-dependence of the characteristic scale $L(t)$ via $L \sim t^{1/z}$, are known exactly for most models with purely dissipative dynamics [@BrayRev; @Exponents]. For systems with short-range interactions and dynamics which are either nonconserved or obey a local conservation law, the exponent $z$ is usually a dimension-independent integer [@Exponents]. Recently, however, it has been realized that for nonconserved dynamics the description of two-time correlations requires a new exponent, whose dependence on the spatial dimension $d$ and on the symmetry of the order parameter is nontrivial [@FH88; @Newman].
The exponent $\lambda$ can be defined in terms of the general two-point correlation function $C(r;t_1,t_2) = \langle \phi({\bf x},t_1)\,\phi({\bf x}+{\bf r},t_2)\rangle$, where $\phi$ is the order parameter field. In the scaling regime, this is expected to have the scaling form $C(r;t_1,t_2) = f(r/L_1,r/L_2)$, where $L_1$, $L_2$ are the characteristic length scales at times $t_1$ and $t_2$ [@Furukawa; @Rutenberg]. In the limit of well-separated times, $L_2 \gg L_1$, one anticipates [@Furukawa] the power-law form $C(r;t_1,t_2) \sim (L_1/L_2)^{d-\lambda} f(r/L_2)$, defining the exponent $\lambda$. An especially simple case is where we take $r=0$, and the initial time $t_1=0$. Then the general form reduces to $$A(t) \equiv C(0;0,t) \sim [\xi_0/L(t)]^{d-\lambda}\ ,
\label{AUTO}$$ where $\xi_0$ is some fixed length related to the initial conditions. The ‘autocorrelation function’ $A(t)$ has been measured in simulations of $O(n)$ models for various spatial dimensions $d$ [@FH88; @Sims], and in experiments on twisted nematic liquid crystals films [@Yurke], and the exponent $\lambda$ deduced. It generally has a nontrivial value.
There are a few analytical results for $\lambda$ – the nonconserved $O(n)$ model for $n=\infty$ ($\lambda=d/2$ [@Newman]), and the $d=1$ Glauber model ($\lambda=0$ [@Bray]), while for nonconserved scalar fields in $d=2$ Fisher and Huse [@FH88] have conjectured that $\lambda=3/4$ exactly. In general, however, $\lambda$ appears to be a nontrivial exponent associated with ordering dynamics, although it is known to satisfy the bound (in our notation) $\lambda \le d/2$ for nonconserved dynamics [@FH88; @Yeung].
In this paper we calculate $\lambda$ exactly for a soluble model corresponding to the late-time, zero-temperature coarsening dynamics of the time-dependent Ginzburg-Landau (TDGL) equation for a scalar field in $d=1$. The equation of motion is $\partial_t \phi = \partial_x^2 \phi
- dV/d\phi$, where $V(\phi)$ is a symmetric, double-well potential with minima at $\phi=\pm 1$ (e.g. $V(\phi) = (1-\phi^2)^2$). At late times, when the mean separation $L$ of domain walls is large compared to their intrinsic width $\xi$ ($=[V''(1)]^{-1/2}$), the walls only interact weakly, through the exponential tails of the wall profile function. Then the dynamics is very simple [@Kawasaki; @RB; @BDG]. The closest pair of walls move together and annihilate, while the other walls hardly move at all, and the system coarsens by successively eliminating the smallest domains. It is found that the distribution of domain sizes $l$ approaches a scaling form, $P(l) = L^{-1}f(l/L)$. The scaling function $f(x)$ can be exactly calculated [@Kawasaki; @RB; @BDG].
In an earlier work [@BDG], we have shown that there is a nontrivial exponent associated with the fraction of the line that has never been traversed by a domain wall (i.e. the fraction of the line where the order parameter $\phi$ has never changed its sign [@DBG]). This fraction decays as $L^{-(1-\beta)}$, with $\beta = 0.824\,924\,12\dots$. Here we show that the approach developed in [@BDG] can be generalised to calculate $\lambda$ for this model. The result is $\lambda = 0.399\,383\,5\ldots$. A recent simulation of the same model [@MH] gave the estimate $\lambda = 0.43 \pm 0.01$, which, we think, is in reasonable agreement with our exact result, given that the extrapolation to large $L$ was not straightforward.
The exponent $\lambda$ can also be obtained from the rate at which a small initial bias in the order parameter grows with time [@Kiss], $\langle \phi \rangle \sim L^\lambda$. We demonstrate this explicitly within the present model in the second part of this work.
The calculation of the autocorrelation exponent $\lambda$ follows closely that presented in reference [@BDG]. One starts with random intervals on the line. Each interval $I$ is characterised by its length $l(I)$ and by its overlap $q(I)$ with its initial condition (initially $q(I)=l(I)$ for all $I$). At each iteration step, the smallest interval $I_{\rm min}$ is removed (i.e. the field $\phi$ is replaced by $- \phi$ in this interval). So three intervals (the smallest interval $I_{\rm min}$ and its two neighbors $I_1$ and $I_2$) are replaced by a single interval $I$. The length and the overlap of the new interval $I$ are given by $$\begin{aligned}
l(I) & = & l(I_1) +l(I_{\rm min}) + l(I_2)\ , \\
q(I) & = & q(I_1) + q(I_2) -q(I_{\rm min})\ .\end{aligned}$$ Then the average length $L$ of domains and the autocorrelation function $A$ are given by $$\begin{aligned}
L & = & \sum_I l(I) / \sum_I 1 \ , \nonumber \\
A & = & \sum_I q(I) / \sum_I l(I) \ .
\label{sum}\end{aligned}$$ where the sums are over all the intervals $I$ present in the system.
The argument showing that no correlations develop if none are present initially was given earlier [@BDG] and the calculation is then very similar to that for the evaluation of the exponent $\beta$. We take, for simplicity, the lengths of the intervals to be integers and $i_0$ to be the minimal length in the system. We also assume that the total number $N$ of intervals is very large. We call $n_i$ the number of intervals of length $i$ and $q_i$ the average overlap of the intervals of length $i$. At the beginning, $q_i= i$.
We denote with a prime the values of these quantities after all the $n_{i_0}$ intervals of length $i_0$ have been eliminated, so that the minimal length has become $i_0 + 1$. Then the time evolution is given by (compare equation (2) of [@BDG]) $$\begin{aligned}
N' & = & N- 2 n_{i_0} \nonumber \\
n_i' & = & n_i(1 - {2 n_{i_0} \over N}) + n_{i_0} \sum_{j=i_0}^{i-2i_0}
{ n_{j} \over N} \ { n_{i-j-i_0} \over N} \nonumber \\
n_i' q_i' & = & n_i q_i(1 - {2 n_{i_0} \over N}) +
n_{i_0} \sum_{j=i_0}^{i-2i_0}{ n_{j} \over N} \
{ n_{i-j-i_0} \over N} \ ( q_j + q_{i-j-i_0} -q_{i_0})\ .
\label{EQ:ITERATE}\end{aligned}$$ This is only valid under the condition that $n_{i_0} \ll N$ which is indeed valid when $i_0$ becomes large and as long as the system consists of a large number of intervals.
We assume that after many iterations, i.e. when $i_0$ becomes large, a scaling limit is reached where $$\begin{aligned}
n_i & = & \frac{N}{i_0} f \left( {i \over i_0} \right) \nonumber \\
n_i q_i & = & N (i_0)^{\lambda-1} g \left( {i \over i_0} \right)\ ,
\label{SCALING}\end{aligned}$$ where $\lambda$ is the exponent we want to calculate (\[sum\]). Because $i_0$ is so large, we can consider $x=i/i_0$ as a continuous variable. This gives $$\begin{aligned}
n_i' & = & \frac{N'}{i_0+1} f \left( {i \over i_0+1} \right) =
{N \over i_0} \left[f(x) - {2 \over i_0} f(1)f(x) - {1 \over i_0} f(x)
-{1 \over i_0} x f'(x) \right] \nonumber \\
n_i'q_i' & = & N' (i_0+1)^{\lambda-1} g \left( {i \over i_0+1} \right) \\
& = & {N i_0^{\lambda-1}} \left[g(x) - {2 \over i_0} f(1)g(x)
+ {\lambda -1 \over i_0} g(x) -{1 \over i_0} x g'(x) \right]\ .\end{aligned}$$ Inserting these expressions in the time evolution equations (\[EQ:ITERATE\]) gives $$\begin{aligned}
i_0 {\partial f \over \partial i_0} & = &
f(x) + x f'(x) + \theta(x-3) f(1) \int_1^{x-2} dy \ f(y) f(x-y-1)
\nonumber \\
i_0 {\partial g \over \partial i_0} & = &
(1 -\lambda)g(x) + x g'(x) + 2 \theta(x-3) f(1) \int_1^{x-2} dy
\ g(y) f(x-y-1) \nonumber \\
&& -g(1) \theta(x-3) \int_1^{x-2} dy \ f(y) f(x-y-1) \ .
\label{9}\end{aligned}$$ In (\[SCALING\]), both $n_i$ and $n_i q_i$ are functions of $x=i/i_0$ and of $i_0$, and the partial derivatives in (\[9\]) mean the derivative with respect to $i_0$, keeping $x$ fixed. Demanding that the system is self-similar, i.e. that the functions $f(x)$ and $g(x)$ do not change with time (i.e. replacing the left-hand sides of (\[9\]) by zero), one finds that the Laplace transforms $$\begin{aligned}
\phi(p) & = & \int_1^\infty e^{-px} \ f(x) \ dx\ , \nonumber \\
\psi(p) & = & \int_1^\infty e^{-px} \ g(x) \ dx\ ,
\label{EQ:LAPLACE}\end{aligned}$$ satisfy the following equations (where primes now indicate derivatives) $$\begin{aligned}
\label{PHI}
-f(1) e^{-p} - p \phi'(p) + f(1) e^{-p} \phi^2(p) & = & 0 \\
-\lambda \psi(p) - g(1) e^{-p} - p \psi'(p)
+ 2 f(1) e^{-p} \phi(p) \psi(p) -g(1) e^{-p} \phi^2(p) & = & 0\ .
\label{PSI}\end{aligned}$$ Defining the function $h(p)$ by $$h(p) = 2 f(1) \int_p^\infty {e^{-t} \over t} dt\ ,$$ the solutions of the above equations are $$\begin{aligned}
\label{EQ:PHI}
\phi(p) & = & \tanh [h(p)/2] \\
\psi(p) & = & g(1) \int_p^\infty (1 + \phi^2(q)) { 1-\phi^2(p) \over
1-\phi^2(q) } \ {q^{\lambda -1} \over p^\lambda} \ e^{-q} dq\ .
\label{EQ:PSI}\end{aligned}$$ The constants of integration implied by these forms were fixed by the requirement that both $\phi$ and $\psi$ decay fast enough for large $p$, as is clear from the definitions (\[EQ:LAPLACE\]). So far the parameters $f(1)$, $g(1)$ and $\lambda$ are arbitrary. We shall see that they are fixed by physical considerations.
Eq. (\[EQ:PHI\]) for $\phi$, which determines the domain size distribution, is of course identical to that obtained in previous work [@Kawasaki; @RB; @BDG]. Eq. (\[EQ:PSI\]) for $\psi$ can be rewritten in the more convenient form $$\psi(p) = 2 g(1) \int_p^\infty { e^{h(q)} + e^{-h(q)} \over
e^{h(p)} +2 + e^{-h(p)} } \
{q^{\lambda -1} \over p^\lambda} \ e^{-q} dq \ .
\label{CONV}$$ It is helpful to introduce the expansion $$\int_p^\infty {e^{-q} \over q} dq = -\log p
-\gamma -\sum_{n=1}^{\infty} {(-p)^n \over n \ n!}\ ,
\label{EXPANSION}$$ where $\gamma = -\int_0^\infty dt \ e^{-t} \ \log t
= .577\,215\,6... $ is Euler’s constant.
From the small-$p$ expansion of (\[EQ:PHI\]), it is easy to show that, provided the first moment of the domain size distribution exists, one must have $f(1)=1/2$ [@Kawasaki; @RB; @BDG]. From now on, we will consider only this case(see [@DGY] for the discussion of cases where the stationary distribution has long tails). Defining the function $r(p)$ by $$r(p) = h(p)+\log p = \int_p^\infty {e^{-q} \over q} dq + \log p\ ,
\label{r}$$ one obtains, using (\[CONV\]), $$\psi(p) = 2 g(1) \int_p^\infty { e^{r(q)} +q^2 e^{-r(q)} \over
e^{r(p)} +2 p + p^2 e^{-r(p)} } \
{q^{\lambda -2} \over p^{\lambda-1}} \ e^{-q} dq\ .$$ Now $r(p)$ can be expanded in powers of $p$, using (\[EXPANSION\]), and so this last form makes it easier to analyse the singular behavior of $\psi(p)$ at $p=0$. One finds that, for small $p$, $$\psi(p) = A + B p^{1-\lambda}[1+O(p)]\ ,
\label{EQ:SING}$$ where $A=2g(1)/(1 - \lambda)$ and $$\begin{aligned}
B & = & 2 g(1) e^{-r(0)} \left[ \int_0^\infty {q^{\lambda-1} e^{-q}
\over 1 - \lambda} (r'(q) -1) e^{r(q)} dq
+ \int_0^\infty q^\lambda e^{-q} e^{-r(q)} dq \right]
\label{Cond1}
\\
& = & 2 g(1) e^{\gamma}(1-\lambda)^{-1} \int_0^\infty q^{\lambda-2} e^{-q}
\left[ (1-q-e^{-q}) e^{r(q)} +q^2 (1 - \lambda) e^{-r(q)} \right] dq \ .
\nonumber\end{aligned}$$
Now compare (\[EQ:SING\]) with a direct expansion of (\[EQ:LAPLACE\]), namely $\psi(p) = \int_1^\infty dx g(x)(1-px +O(p^2))$. If the function $g(x)$ is to have a finite first moment then we must have $B=0$ in (\[EQ:SING\]). This condition determines $\lambda$ as $$\lambda = .399\,383\,5\ldots\ .
\label{lambda}$$ From numerical simulations of the same model, Majumdar and Huse [@MH] found the power-law decay $A(t) \sim L^{-\bar{\lambda}}$, with $\bar{\lambda} = 0.57 \pm 0.01$, corresponding to $\lambda \equiv d-\bar{\lambda} = 0.43 \pm .01$. There were, however, large corrections to scaling in their numerical data, which we think are the origin of the disagreement between their numerical estimate and our exact result.
As in [@BDG], one can show that $B \neq 0$ would correspond to a power law decay in $g(x)$ and that such a power law cannot be produced if it is not present in the initial condition. Note that $g(1)$ cannot be determined as one can always multiply all the $q_i$ by a constant without changing our results.
For the remainder of this paper we will look at a related quantity, the growth of an initially small bias in the order parameter, and show that the bias grows as $L^\mu$ as the system coarsens (while the bias remains small). Furthermore, we will show explicitly that $\mu=\lambda$ for this model, exemplifying a general result [@Kiss].
Consider a sequence of positive and negative domains on a line. We call $n_i$ ($m_i$) the number of positive (negative) domains of length $i$. The total number $N$ of positive domains is of course equal to the total number of negative domains, $ N = \sum_i n_i = \sum_i m_i $. When the domains of size $i_0$ are removed, the new values of $n_i$, $m_i$ and $N$ are given by $$\begin{aligned}
n_i' & = & \left( 1 - {2 m_{i_0} \over N} \right)
+ m_{i_0} \sum_{j=i_0}^{i-2i_0} {n_j \ n_{i-j-i_0} \over N^2} \nonumber \\
m_i' & = & \left( 1 - {2 n_{i_0} \over N} \right)
+ n_{i_0} \sum_{j=i_0}^{i-2i_0} {m_j \ m_{i-j-i_0} \over N^2} \nonumber \\
N' & = & N- n_{i_0} - m_{i_0}\ .\end{aligned}$$ Let us write forms for $n_i$ and $m_i$ analogous to the first of equations (\[SCALING\]): $$n_i = {N \over i_0} f_1\left( {i \over i_0} \right)\ ,\ \ \ \ \
m_i = {N \over i_0} f_2\left( {i \over i_0} \right)\ .
\label{PSEUDOSCALING}$$ Then one has $$n_i' = {N- n_{i_0} - m_{i_0} \over i_0 +1} f_1 \left( {i \over i_0 +1}
\right)\ ,$$ which gives, for $i_0$ large (when $x=i/i_0$ can be treated as a continuous variable), $$n_i' = {N \over i_0} \left[ f_1(x) +{1 \over i_0}
\left\{ - f_1(1) f_1(x) - f_2(1)f_1(x) - f_1(x) - x f_1'(x) \right\}
\right]\ ,
\label{ASYMMETRIC}$$ and a similar expression for $m_i'$.
Inserting the forms (\[PSEUDOSCALING\]) into (\[ASYMMETRIC\]) gives coupled evolution equations for $f_1$ and $f_2$: $$\begin{aligned}
i_0 { \partial f_1(x) \over \partial i_0} & = &
[f_1(1)-f_2(1)]f_1(x) +f_1(x) + x f_1'(x) \nonumber \\
& & + \theta(x-3) f_2(1) \int_1^{x-2} dy \ f_1(y) f_1(x-y-1)\ ,\end{aligned}$$ and a second equation obtained by interchanging the subscripts ‘1’ and ‘2’. Note that the derivatives on the left-hand sides are with respect to the (implicit) second argument $i_0$. Introducing the Laplace transforms with respect to the first argument, $$\psi_n(p) = \int_1^\infty f_n(x) e^{-px} dx\ ,\ \ \ \ \ (n=1,2)
\label{PSILAPLACE}$$ one finds that their evolution is given by $$i_0 { \partial \psi_1(p) \over \partial i_0}
= [f_1(1)-f_2(1)]\psi_1(p) -p \psi_1'(p) -
f_1(1) e^{-p} + f_2(1) e^{-p} \psi_1^2(p)\ ,
\label{BIAS}$$ and a second equation with ‘1’ and ‘2’ interchanged.
So far this is completely general. The basic idea is to perform a linear stability analysis around the ‘symmetric’ solution $\psi_1(p)=\psi_2(p)=\phi(p)$, where $\phi(p)$ satisfies (\[PHI\]) with $f(1)=1/2$, in order to determine the rate at which a small perturbation will grow. We therefore take $\psi_1(p)$ and $\psi_2(p)$ to have the forms $$\psi_n(p) = \phi(p) \pm \epsilon \ \sigma(p)\ ,
\label{SIGMA}$$ with $$f_n(1) = {1 \over 2} \pm \epsilon \ a\ ,
\label{a}$$ with $\epsilon$ small and the $+$ ($-$) sign corresponding to $n=1$ ($n=2$). If the bias represented by the terms in $\epsilon$ is a relevant perturbation, $\sigma(p)$ will grow under iteration: $\sigma \sim (i_0)^\mu$ with $\mu>0$ (and similarly, $a \sim (i_0)^\mu$ in (\[a\])). Subtracting from (\[BIAS\]) its counterpart with ‘1’ and ‘2’ interchanged, and putting $i_0\,\partial\sigma(p)/\partial i_0 =
\mu\sigma(p)$, yields the eigenvalue equation $$\mu\sigma = 2 a \phi - p \sigma' -a e^{-p} - a e^{-p} \phi^2
+ e^{-p} \phi \sigma\ ,$$ with solution $$\sigma(p) = a \int_p^\infty {\phi^2(q) e^{-q} + e^{-q}
- 2 \phi(q) \over q} \left( q \over p \right)^\mu {1 - \phi^2(p)
\over 1 - \phi^2(q)} dq\ .$$ The integration constant was fixed as before by the requirement that $\sigma(p)$ decrease as $\exp(-p)/p$ for large $p$, which follows from (\[PSILAPLACE\]), (\[SIGMA\]) and (\[a\]). Demanding once more that $\sigma(p)$ be regular at $p=0$ (so that the first moments of $f_1(x)$ and $f_2(x)$ exist) yields the following equation for $\mu$: $$\int_0^\infty dq \ [ e^{-q} \phi^2(q) + e^{-q} - 2 \phi(q) ]
\ { q^{\mu-1} \over 1 - \phi^2(q) } =0\ .$$ Using $\phi(q) = (e^{r(q)} - q)/(e^{r(q)} +q)$, which follows from (\[EQ:PHI\]) and (\[r\]), gives the condition $$\int_0^\infty dq \ [\ (e^{-q} -1 )\ e^{r(q)}
+ q^2 (e^{-q} +1)\ e^{-r(q)}\ ] q^{\mu-2} =0
\label{Cond2}$$ for $\mu$, with solution $\mu \simeq .399\,38...$. Comparison with (\[lambda\]) suggests that $\mu=\lambda$. In fact, using integration by parts one can show that condition (\[Cond2\]) for $\mu$ is identical to (\[Cond1\]) (with $B=0$) for $\lambda$, and so $\mu=\lambda$ exactly.
The result $\mu=\lambda$ is, in fact, quite general. For TDGL dynamics, it has been discussed elsewhere [@Kiss]. Let us derive it for any kind of dynamics of an Ising model. Consider a system of $N$ Ising spins in dimension $d$. We call $P(\,\{S_i(t)\}\,|\,\{S_i(0)\}\,)$ the probability of findingthe system in the spin configuration $\{S_i(t)\}$ at time $t$ given that it was in configuration $\{S_i(0)\}$ at time $0$. We assume that the system evolves in a zero magnetic field and that the dynamics preserves the $\pm$ symmetry, namely $P(\,\{S_i(t)\,\}|\,\{S_i(0)\}\,) = P(\,\{-S_i(t)\}\,|\,\{- S_i(0)\}\,)$.
Suppose that one starts with an initial condition $\{S_i(0)\}$ chosen completely at random, then the correlation $\langle S_i(t) S_j(0) \rangle $ is given by $$\langle S_i(t) S_j(0) \rangle = {1 \over 2^N} \sum_{\{S(t)\}}
\sum_{\{S(0)\}}S_i(t)S_j(0)\,P(\,\{S_i(t)\}\,|\,\{S_i(0)\}\,)\ .$$ where $\sum_{\{S(t)\}}$ indicates a sum over the $2^N$ configurations at time $t$.
Suppose on the other hand that one starts with a weakly magnetized initial condition, i.e. the initial configuration $\{S_i(0)\}$ is chosen with probability $$Q(\{S_i(0)\}) = \prod_{i=1}^N {1 + m(0) S_i(0) \over 2}
\simeq {1 + m(0) \sum_j S_j(0) \over 2^N }$$ when $m(0)$ is infinitesimal. Then the magnetization $m(t)$ per spin at time $t$ is a function of $m(0)$, and to first order in powers of $m(0)$ one has $$\begin{aligned}
m(t) & = & \sum_{\{S(t)\}} \sum_{\{S(0)\}}
P(\,\{S_i(t)\}\,|\,\{S_i(0)\}\,)\,Q(\{S_i(0)\})\,
{\sum_j S_j(t) \over N} \nonumber \\
&\simeq &
m(0)\,{\sum_i \sum_j \langle S_i(t) S_j(0) \rangle \over N}\ .\end{aligned}$$ Therefore if one assumes that due to some coarsening phenomenon the two-point function scales as $$\langle S_i(0) S_j(t) \rangle \simeq L^{-(d-\lambda)} f({R_{ij} \over L})$$ where $R_{ij}$ is the distance between sites $i$ and $j$, one finds that $$m(t) \simeq L^\lambda m(0) \int d^d R f(R)$$ which means that the magnetization and the autocorrelation exponents are the same.
To summarise, we have derived a non-trivial value for the exponent $\lambda$ within an exactly soluble model, and shown explicitly that the growth of an initial bias in the order parameter is controlled by the same exponent.
We thank the Isaac Newton Institute, Cambridge, where this work was carried out, for its hospitality.
[99]{} For a recent review see A. J. Bray, Advances in Physics, to appear. I. M. Lifshitz, Zh. Theor. Fiz. [**42**]{}, 1354 (1962) \[Sov. Fiz. JETP [**15**]{}, 939 (1962)\]; S. M. Allen and J. W. Cahn, Acta. Metall. [**27**]{}, 1085 (1979); I. M. Lifshitz and V. V. Slyozov, J. Chem. Phys. Solids [**19**]{}, 35 (1961); A. J. Bray, Phys. Rev. Lett. [**62**]{}, 2841 (1989); A. J. Bray and A. D. Rutenberg, Phys. Rev. E [**49**]{}, R27 (1994). D. S. Fisher and D. A. Huse, Phys. Rev. B [**38**]{}, 373 (1988). Note that our exponent $\bar{\lambda} \equiv d-\lambda$ is called $\lambda$ in this paper. T. J. Newman and A. J. Bray, J. Phys. A [**23**]{}, 4491 (1990); J. G. Kissner and A. J. Bray, J. Phys. A [**26**]{}, 1571 (1993). H. Furukawa, J. Phys. Soc. Jpn. [**58**]{}, 216 (1989); Phys. Rev. B [**40**]{}, 2341 (1989). This generalized scaling assumption does not always hold however. An explicit counterexample is the $O(2)$ model for $d=1$: see A. J. Bray, reference [@BrayRev]; T. J. Newman et al., reference [@Sims]; A. D. Rutenberg and A. J. Bray, preprint. T. J. Newman, A. J. Bray and M. A. Moore, Phys. Rev. B [**42**]{}, 4514 (1990); A. J. Bray and K. Humayun, J. Phys. A [**23**]{}, 5897 (1990); K. Humayun and A. J. Bray, J. Phys. A [**24**]{}, 1915 (1991); F. Liu and G. F. Mazenko, Phys. Rev. B [**44**]{}, 9185 (1991). N. Mason, A. N. Pargellis and B. Yurke, Phys. Rev.Lett. [**70**]{}, 190 (1993); A. N. Pargellis, S. Green and B. Yurke, Phys. Rev. E [**49**]{}, 4250 (1994). A. J. Bray, J. Phys. A [**22**]{}, L67 (1990). C. Yeung, M. Rao and R. C. Desai, preprint. T. Nagai and K. Kawasaki, Physica A [**134**]{}, 483 (1986). A. D. Rutenberg and A. J. Bray, Phys. Rev. E [**50**]{}, 1900 (1994). A. J. Bray, B. Derrida and C. Godrèche, Europhys.Lett. [**27**]{}, 175 (1994). B. Derrida, A.J. Bray and C. Godrèche, J. Phys. A [**27**]{}, L357 (1994). S. N. Majumdar and D. A. Huse, preprint. Note that our exponent $\bar{\lambda} \equiv d-\lambda$ is called $\lambda$ in this paper. A. J. Bray and J. G. Kissner, J. Phys. A [**25**]{}, 31 (1992). B. Derrida, C. Godrèche and I. Yekutieli, Phys.Rev. A [**44**]{}, 6241 (1991).
|
---
abstract: 'In a recent result, Frauchiger and Renner argue that if quantum theory accurately describes complex systems like observers who perform measurements, then “we are forced to give up the view that there is one single reality.” Following a review of the Frauchiger-Renner argument, I argue that quantum mechanics should be understood *probabilistically*, as a new sort of non-Boolean probability theory, rather than *representationally*, as a theory about the elementary constituents of the physical world and how these elements evolve dynamically over time. I show that this way of understanding quantum mechanics is not in conflict with a consistent “single-world” interpretation of the theory.'
author:
- |
Jeffrey Bub\
Philosophy Department\
Institute for Physical Science and Technology\
Joint Center for Quantum Information and Computer Science\
University of Maryland, College Park, MD 20742, USA
title: 'In Defense of a “Single-World” Interpretation of Quantum Mechanics'
---
Introduction
============
In a recent “no go” result, Frauchiger and Renner,[@FrauchigerRenner] argue that no “single-world” interpretation of quantum mechanics can be self-consistent, where a single-world interpretation is any interpretation that asserts, for a measurement with multiple possible outcomes, that just one outcome actually occurs. The argument is a novel re-formulation of the “Wigner’s friend” argument,[@Wigner] with a twist that exploits Hardy’s paradox.[@Hardy] Frauchiger and Renner conclude that if quantum theory accurately describes complex systems like observers who perform measurements, then “we are forced to give up the view that there is one single reality.”[@FrauchigerRenner2]
Following a review of the Frauchiger-Renner argument in §[2]{}, I argue in §[3]{} that quantum mechanics should be understood *probabilistically*, as a new sort of non-Boolean probability theory, rather than *representationally*, as a theory about the elementary constituents of the physical world, standardly particles and fields of a certain sort, and how these elements evolve dynamically as they interact over time.[@Wallace] In §[4]{}, I show that this way of understanding quantum mechanics is not in conflict with a consistent “single-world” interpretation of the theory.
The Frauchiger-Renner Argument
==============================
Here’s how the Frauchiger-Renner argument goes: Alice measures an observable $A$ with eigenstates ${| h \rangle}_{A}, {| t \rangle}_{A}$ on a system in the state $\frac{1}{\sqrt{3}}{| h \rangle}_{A} + \frac{\sqrt{2}}{\sqrt{3}}{| t \rangle}_{A}$. One could say that Alice “tosses a biased quantum coin” with probabilities 1/3 for heads and 2/3 for tails. She prepares a qubit in the state ${| 0 \rangle}_{B}$ if the outcome is $h$, or in the state $\frac{1}{\sqrt{2}}({| 0 \rangle}_{B} + {| 1 \rangle}_{B})$ if the outcome is $t$, and sends it to Bob. When Bob receives the qubit, he measures a qubit observable $B$ with eigenstates ${| 0 \rangle}_{B}, {| 1 \rangle}_{B}$. After Alice and Bob obtain definite outcomes for their measurements, the quantum state of the combined quantum coin and qubit system is ${| h \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 1 \rangle}_{B}$, with equal probability. At least, that’s the state of quantum coin and qubit system from the perspective of Alice and Bob.
Now, the quantum coin and the qubit, as well as Alice and Bob, their measuring instruments and all the systems in their laboratories that become entangled with the measuring instruments in registering and recording the outcomes of the quantum coin toss and the qubit measurement, including the entangled environments, are just two big many-body quantum systems $S_{A}$ and $S_{B}$, which are assumed to be completely isolated from each other after Bob receives Alice’s qubit. Consider two super-observers, Wigner and Friend, with vast technological abilities, who contemplate measuring a super-observable $X$ of $S_{A}$ with eigenstates ${| \mbox{fail} \rangle}_{A} = \frac{1}{\sqrt{2}}({| h \rangle}_{A} + {| t \rangle}_{A}), {| \mbox{ok} \rangle}_{A} = \frac{1}{\sqrt{2}}({| h \rangle}_{A} - {| t \rangle}_{A})$, and a super-observable $Y$ of $S_{B}$ with eigenstates ${| \mbox{fail} \rangle}_{B} = \frac{1}{\sqrt{2}}({| 0 \rangle}_{B} + {| 1 \rangle}_{B}),{| \mbox{ok} \rangle}_{B} = \frac{1}{\sqrt{2}}({| 0 \rangle}_{B} - {| 1 \rangle}_{B})$.
To avoid unnecessarily complicating the notation by introducing new symbols for super-observables corresponding to observables, I’ll use the same symbols $A$ and $B$ to represent super-observables of the composite systems $S_{A}$ and $S_{B}$ that end up with definite values corresponding to the outcomes of Alice’s and Bob’s measurements on the quantum coin and the qubit, and I’ll denote eigenstates of the super-observables $A$ and $B$ by the same symbols ${| h \rangle}_{A},{| t \rangle}_{A}$ and ${| 0 \rangle}_{B}, {| 1 \rangle}_{B}$ I used to represent eigenstates of the observables measured by Alice and Bob, with $\{h,t\}$ and $\{0,1\}$ representing the corresponding eigenvalues. (The alternative would be to use primes or some other notational device to denote super-observables and their eigenstates and eigenvalues, but this seems unnecessary, especially since the following argument concerns only super-observables. If the reader finds this confusing, simply add primes to all symbols from this point on.)
Of course, such a measurement by the super-observers Wigner and Friend would be extraordinarily difficult to carry out in practice on the whole composite system, including Alice and Bob and their brain states, and all the systems in their environments, but nothing in quantum mechanics precludes this possibility. From the perspective of Wigner and Friend, $S_{A}$ and $S_{B}$ are just composite many-body entangled quantum systems that have evolved unitarily to a combined entangled state: $${| \psi \rangle} = \frac{1}{\sqrt{3}}({| h \rangle}_{A}{| 0 \rangle}_{B} + {| t \rangle}_{A}{| 0 \rangle}_{B} +{| t \rangle}_{A}{| 1 \rangle}_{B}) \label{eqn:entangled}$$ The outcomes of Alice’s and Bob’s measurements simply don’t appear anywhere in the super-observers’ description of events, so Wigner and Friend see no reason to conditionalize the state to one of the product states ${| h \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{1}{| 1 \rangle}_{B}$. For Wigner and Friend, this would seem to require a suspension of unitary evolution in favor of an unexplained “collapse” of the quantum state.
But now we have a contradiction. The state ${| \psi \rangle}$ can also be expressed as: $$\begin{aligned}
{| \psi \rangle} & = & \frac{1}{\sqrt{12}}{| \mbox{ok} \rangle}_{A}{| \mbox{ok} \rangle}_{B} - \frac{1}{\sqrt{12}}{| \mbox{ok} \rangle}_{A}{| \mbox{fail} \rangle}_{B} \nonumber\\
&& + \frac{1}{\sqrt{12}}{| \mbox{fail} \rangle}_{A}{| \mbox{ok} \rangle}_{B} + \sqrt{\frac{3}{4}}{| \mbox{fail} \rangle}_{A}{| \mbox{fail} \rangle}_{B}\\
& = & \sqrt{\frac{2}{3}}{| \mbox{fail} \rangle}_{A}{| 0 \rangle}_{B} + \frac{1}{\sqrt{3}}{| t \rangle}_{A}{| 1 \rangle}_{B}\\
& = & \frac{1}{\sqrt{3}}{| h \rangle}_{A}{| 0 \rangle}_{B} + \sqrt{\frac{2}{3}}{| t \rangle}_{A}{| \mbox{fail} \rangle}_{B}\end{aligned}$$ From the first expression for ${| \psi \rangle}$, the probability is $1/12$ that Wigner and Friend find the pair of outcomes $\{\mbox{ok, ok}\}$ in a joint measurement of $X$ and $Y$ on the two systems. But this outcome is *inconsistent with any pair of outcomes for Alice’s and Bob’s measurements*. From the second expression, the pair $\{\mbox{ok}, 0\}$ has zero probability, so $\{\mbox{ok}, 1\}$ is the only possible pair of values for the super-observables $X, B$ if $X$ has the value $\mbox{ok}$. From the third expression, the pair $\{t, \mbox{ok}\}$ has zero probability, so $\{h, \mbox{ok}\}$ is the only possible pair of values for the super-observables $A, Y$ if $Y$ has the value $\mbox{ok}$. But the pair of values $\{h, 1\}$ for the super-observables $A$ and $B$ has zero probability in the state ${| \psi \rangle}$, so it does not correspond to a possible pair of measurement outcomes for Alice and Bob.
Both the observers, Alice and Bob, and the super-observers, Wigner and Friend, apply quantum mechanics correctly. The argument depends only on (i) the one-world assumption, that a measurement has a single outcome, (ii) the assumption that quantum mechanics applies to systems of any complexity, including observers, and (iii) self-consistency, in particular agreement between an observer and a super-observer. The surprising conclusion is that there is no consistent story that includes observers and super-observers: a pair of outcomes with finite probability, according to quantum mechanics, of the super-observers’ measurements on the composite observer system is inconsistent with the observers obtaining definite (single) outcomes for their measurements.
The Alice-Bob measurements and the Wigner-Friend measurements could be separated by any time interval. As far as we know, there are no super-observers, but the actuality of a measurement outcome can’t depend on whether or not a super-observer turns up at some point. It’s the theoretical possibility of a super-observer that shows the inconsistency of the theory. One could put the problem this way: according to quantum mechanics, there can be no quantum measurements with definite (single) outcomes, because it is always possible that super-observers could turn up at some point, perhaps centuries after the Alice-Bob measurements when technology is sufficiently advanced to generate a Frauchiger-Renner contradiction.
The Quantum Revolution
======================
Before considering the options in the light of the Frauchiger-Renner result, I want to review the genesis of quantum mechanics and argue that the theory should be understood *probabilistically*, as a new sort of non-Boolean probability theory, rather than *representationally*, as a theory about the elementary constituents of the physical world and their dynamical evolution.
Quantum mechanics began with Heisenberg’s “Umdeutung” paper, [@Heisenberg] his proposed “reinterpretation” of physical quantities at the fundamental level as noncommutative. To say that the algebra of physical quantities is commutative is equivalent to saying that the idempotent elements form a Boolean algebra. For the physical quantities or observables of a quantum system represented by self-adjoint Hilbert space operators, the idempotent elements are the projection operators, with eigenvalues 0 and 1. They represent yes-no observables, or properties (for example, the property that the energy of the system lies in a certain range of values), or propositions (the proposition asserting that the value of the energy lies in this range), with the two eigenvalues corresponding to the truth values, true and false.
Heisenberg’s insight amounts to the proposal that certain phenomena in our Boolean macro-world that defy a classical physical explanation can be explained probabilistically as a manifestation of collective behavior at a non-Boolean microlevel. The Boolean algebra of physical properties of classical mechanics is replaced by a family of “intertwined” Boolean algebras, one for each set of commuting observables, to use Gleason’s term.[@Gleason] The intertwinement precludes the possibility of embedding the whole collection into one inclusive Boolean algebra, so you can’t assign truth values consistently to the propositions about observable values in all these Boolean algebras. Putting it differently: there are Boolean algebras in the family of Boolean algebras of a quantum system, notably the Boolean algebras for position and momentum, or for spin components in different directions, that don’t fit together into a single Boolean algebra, unlike the corresponding family for a classical system.
The intertwinement of commuting and noncommuting observables in Hilbert space imposes objective pre-dynamic probabilistic constraints on correlations between events, analogous to the way in which Minkowski space-time imposes kinematic constraints on events. The probabilistic constraints encoded in the geometry of Hilbert space provide the framework for the physics of a *genuinely indeterministic universe*. They characterize the way probabilities fit together in a world in which there are nonlocal probabilistic correlations that violate Bell’s inequality up to the Tsirelson bound, and these correlations can only occur between intrinsically random events.[@Bub] As von Neumann put it,[@vonNeumann2] quantum probabilities are “sui generis.” They don’t quantify incomplete knowledge about an ontic state (the basic idea of “hidden variables”), but reflect the irreducibly probabilistic relation between the non-Boolean microlevel and the Boolean macrolevel.
This means that quantum mechanics is quite unlike any theory we have dealt with before in the history of physics, and there is no reason, apart from tradition, to assume that the theory can provide the sort of representational explanation we are familiar with in a theory that is commutative or Boolean at the fundamental level. Quantum probabilities can’t be understood in the Boolean sense as quantifying ignorance about the pre-measurement value of an observable, but cash out in terms of what you’ll find if you “measure,” which involves considering the outcome, at the Boolean macrolevel, of manipulating a quantum system in a certain way.
A quantum “measurement” is a bit of a misnomer and not really the same sort of thing as a measurement of a physical quantity of a classical system. It involves putting a microsystem, like a photon, in a situation, say a beamsplitter or an analyzing filter, where the photon is forced to make an intrinsically random transition recorded as one of two macroscopically distinct alternatives in a device like a photon detector. The registration of the measurement outcome at the Boolean macrolevel is crucial, because it is only with respect to a suitable structure of alternative possibilities that it makes sense to talk about an event as definitely occurring or not occurring, and this structure is a Boolean algebra.
From this perspective, Heisenberg’s theory does not, without embellishment (e.g., as in Bohm’s theory or the Everett interpretation) provide a representational story, but rather a way of deriving probabilities and probabilistic correlations with no causal explanation. They are “uniquely given from the start” as a feature of the non-Boolean structure, to quote von Neumann,[@vonNeumann1] related to the angles in Hilbert space, not measures over states as they are in a classical or Boolean theory.
There is a rival way of thinking about quantum mechanics in terms of Schrödinger’s wave-mechanical version of the theory[@Schrodinger1] that lends itself to a representational interpretation. Here the notion of “superposition” appears as a new ontological category: propositions can true, false, and in the case of superpositions, indeterminate, or neither true nor false. The measurement problem then arises as the problem of explaining the transition from indeterminate to determinate as a dynamical evolution. But as Schrödinger himself pointed out in a lecture to the Royal Institution in London in March, 1928, the wave associated with a quantum system evolves in an abstract, multi-dimensional representation space, not real physical space, so “it is merely an adequate mathematical description of what happens”:[@Schrodinger2]
> The statement that what *really* happens is correctly described by describing a wave-motion does not necessarily mean exactly the same thing as: what *really* exists is a wave-motion. We shall see later on that in generalizing to an *arbitrary* mechanical system we are led to describe what really happens in such a system by a wave-motion in the generalized space of its co-ordinates ($q$-space). Though the latter has quite a definite physical meaning, it cannot very well be said to ‘exist’; hence a wave-motion in this space cannot be said to ‘exist’ in the ordinary sense of the word either. It is merely an adequate mathematical description of what happens. It may be that also in the case of a single mass-point, with which we are now dealing, the wave-motion must not be taken to ‘exist’ in *too* literal a sense, although the configuration space happens to coincide with ordinary space in this particular simple case.
The idea of a wave as a representation of quantum reality, and the associated measurement problem as the problem of accounting for the transition from indeterminate to determinate, “from the limbo of potentialities to the clarity of actualities,”[@Ghirardi] as a dynamical process, continues to shape contemporary discussions of conceptual issues in the foundations of quantum mechanics. From the perspective adopted here, the later formalization of quantum mechanics by Dirac[@Dirac] in 1930 and von Neumann[@vonNeumann3] in 1932 as a theory of observables represented by operators on a space of quantum states is fundamentally an elaboration of Heisenberg’s “Umdeutung” rather than a wave theory.
The really significant thing about a noncommutative mechanics is the novel possibility of correlated events that are *intrinsically random*, not merely apparently random like coin tosses, where the probabilities of “heads” and “tails” represent an averaging over differences among individual coin tosses that we don’t keep track of for practical reasons. This intrinsic randomness allows *new sorts of nonlocal probabilistic correlations* for “entangled” quantum states of separated systems. Schrödinger, who coined the term, referred to entanglement (“Verschränkung” in German) as “*the* characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”[@Schrodinger3]
The view that Hilbert space is fundamentally a theory of probabilistic correlations that are structurally different from correlations that arise in Boolean theories is, in effect, an information-theoretic interpretation of quantum mechanics. The classical theory of information was initially developed by Shannon to deal with certain problems in the communication of messages as electromagnetic signals along a channel such as a telephone wire. An information source produces messages composed of sequences of symbols from an alphabet, with certain probabilities for the different symbols. The fundamental question for Shannon was how to quantify the minimal physical resources required to represent messages produced by a source, so that they could be communicated via a channel and reconstructed by a receiver:[@Shannon]
> The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design.
A theory of information in Shannon’s sense is about the “engineering problem” of communicating messages over a channel efficiently. In this sense, the concept of information has nothing to with anyone’s knowledge and everything to do with the stochastic or probabilistic process that generates the messages. What we have discovered in quantum phenomena is that the possibilities for representing, manipulating, and communicating information, encoded in the geometry of Hilbert space, are different than we thought, irrespective of what the information is about.
On this non-representational way of understanding quantum mechanics, as a non-classical theory of information or a new way of generating probabilities and probabilistic correlations between intrinsically random events, probabilities are defined with respect to *a single Boolean frame*, the Boolean algebra generated by the “pointer-readings” of what Bohr referred to as the “ultimate measuring instruments,” which are “kept outside the system subject to quantum mechanical treatment”:[@Bohr1]
> In the system to which the quantum mechanical formalism is applied, it is of course possible to include any intermediate auxiliary agency employed in the measuring processes. …The only significant point is that in each case some ultimate measuring instruments, like the scales and clocks which determine the frame of space-time coordination—on which, in the last resort, even the definition of momentum and energy quantities rest—must always be described entirely on classical lines, and consequently be kept outside the system subject to quantum mechanical treatment.
Bohr did not, of course, refer to Boolean algebras, but the concept is simply a precise way of codifying a significant aspect of what Bohr meant by a description “on classical lines” or “in classical terms” in his constant insistence that (his emphasis)[@Bohr2]
> *however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms.*
by which he meant “unambiguous language with suitable application of the terminology of classical physics”—for the simple reason, as he put it, that we need to be able to “tell others what we have done and what we have learned.” Formally speaking, the significance of “classical” here as being able to “tell others what we have done and what we have learned” is that the events in question should fit together as a Boolean algebra. George Boole, who came up with the idea in the mid-1800’s, introduced Boolean constraints on probability as “conditions of possible experience.”[@Boole]
It’s not that unitarity is suppressed at a certain level of complexity, where non-Booleanity becomes Booleanity and quantum becomes classical. Rather, there is a macrolevel, which is Boolean, and there are actual events at the macrolevel. Any system, of any complexity, is fundamentally a quantum system and can be treated as such, in principle, which is to say that a unitary dynamical analysis can be applied to whatever level of precision you like. But at the end of the day, so to speak, some particular system, $M$, counts as the “ultimate measuring instrument” with respect to which an event corresponding to a definite measurement outcome occurs in an associated Boolean frame whose selection is not the outcome of a dynamical evolution described by the theory. The system $M$, or any part of $M$, can be treated quantum mechanically, but then some other system, $M^{\prime}$, treated as classical or commutative or Boolean, plays the role of the ultimate measuring instrument in any application of the theory.
The crucial assumption in this probabilistic interpretation of the theory is that the outcome of a measurement is an intrinsically random event at the macrolevel, *something that actually happens*, not described by the deterministic unitary dynamics, so outside the theory, or “irrational” as Pauli characterizes it (his emphasis):[@Pauli]
> Observation thereby takes on the character of *irrational, unique actuality* with unpredictable outcome. …Contrasted with this *irrational aspect* of concrete phenomena which are determined in their *actuality*, there stands the *rational aspect* of an abstract ordering of the *possibilities* of statements by means of the mathematical concept of probability and the $\psi$-function \[I would say ‘by means of the geometry of Hilbert space’\].
Putting it differently, the “collapse,” as a conditionalization of the quantum state, is something you put in by hand after recording the actual outcome. The physics doesn’t give it to you.
The Options
===========
What are the options in the light of the Frauchiger-Renner result?
If the quantum state is interpreted representationally, as the analogue of the classical state in stipulating what’s true and what’s false, and we accept assumption (ii) of §[2]{} and hence the universality of unitarity (so no “collapse” of the quantum state), the correct description of the composite system $S_{A} + S_{B}$ just before the super-observers’ measurements is the entangled state (1), a superposition with several components, each associated with a different measurement outcome for Alice’s and Bob’s measurements. The entangled state is the source of the measurement problem, here presented as an inconsistency in the theory, given the other assumptions in the Frauchiger-Renner argument. Dropping the “one-world” assumption (i) then leads to Everett’s many-worlds interpretation.
If we interpret the quantum state probabilistically, we seem to be forced to QBism, the quantum Bayesianism of Christopher Fuchs and Ruediger Schack.[@Fuchs1] The QBist rejects assumption (iii), the self-consistency assumption. On this view, all probabilities, including quantum probabilities, are understood in the subjective sense as the personal judgements of an agent, based on how the external world responds to actions by the agent. For QBists, the Born rule “is a normative statement …about the decision-making behavior any individual agent should strive for …not a “law of nature” in the usual sense,” and “measurement outcomes *just are* personal experiences for the agent gambling upon them.”[@Fuchs2] So there is no requirement that the perspective of an observer and a super-observer should be consistent.
There is another option, which is to reject assumption (ii)—not by restricting the universality of the unitary dynamics or any part of quantum mechanics, but by interpreting the quantum state probabilistically rather than representationally in the sense of §3. Quantum probabilities don’t quantify incomplete knowledge about an ontic state, but reflect the irreducibly probabilistic relation between the non-Boolean microlevel and the Boolean macrolevel, expressed through the intrinsic randomness of events associated with the outcomes of quantum measurements. On this option, what the Frauchiger-Renner argument shows is that *quantum mechanics, as it stands without embellishment, is self-contradictory if the quantum state is interpreted representationally.* The conclusion is avoided if we interpret the state probabilistically, with respect to a Boolean frame defined with respect to an “ultimate measuring instrument” or “ultimate observer.”
In a situation, as in the Frauchiger-Renner argument, where there are multiple candidate observers, there is a question as to whether Alice and Bob are “ultimate observers,” or whether only Wigner and Friend are “ultimate observers.” The difference has to do with whether Alice and Bob perform measurements of the observables $A$ and $B$ with definite outcomes at the Boolean macrolevel, or whether they are manipulated by Wigner and Friend in unitary transformations that entangle Alice and Bob with systems in their laboratories, with no definite outcomes for the observables $A$ and $B$. What actually happens to Alice and Bob is different in the two situations.
If there are events at the macrolevel corresponding to definite measurement outcomes for Alice and Bob, then Alice and Bob represent “ultimate observers” and the final state of the combined quantum coin and qubit system is ${| h \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 1 \rangle}_{B}$, depending on the outcomes. If Wigner and Friend subsequently measure the super-observables $X, Y$ on the whole composite Alice-Bob system (so they are “ultimate observers” in the *subsequent* scenario), the probability of obtaining the pair of outcomes $\{\mbox{ok}, \mbox{ok}\}$ is 1/4 for any of the product states ${| h \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 0 \rangle}_{B}$ or ${| t \rangle}_{A}{| 1 \rangle}_{B}$. After the measurement, the super-observables $A, B$ are indefinite, and so are the corresponding quantum coin and qubit observables. There is no contradiction because the argument from the entangled state no longer applies. If Wigner and Friend are “ultimate observers” but not Alice and Bob, there are no events at the macrolevel corresponding to definite measurement outcomes for Alice and Bob and the state is the entangled state (1). The probability of Wigner and Friend finding the pair of outcomes $\{\mbox{ok}, \mbox{ok}\}$ is 1/12, but there is no contradiction because there are no measurement outcomes for Alice and Bob.
There is a factual difference at the Boolean macrolevel between the two cases. If Alice and Bob are “ultimate observers” in the application of quantum mechanics to the scenario, the probability of the super-observers finding the pair of outcomes $\{\mbox{ok}, \mbox{ok}\}$ is 1/4. If Wigner and Friend are “ultimate observers” but not Alice and Bob, then the probability of Wigner and Friend finding the pair of outcomes $\{\mbox{ok}, \mbox{ok}\}$ is 1/12. The difference between the two cases— the difference between probability 1/4 and probability 1/12—is an objective fact at the macrolevel.
Special relativity, as a theory about the structure of space-time, provides an explanation for length contraction and time dilation through the geometry of Minkowski space-time, but that’s as far as it goes. This explanation didn’t satisfy Lorentz, who wanted a dynamical explanation in terms of forces acting on physical systems used as rods and clocks.[@Janssen] Quantum mechanics, as a theory about randomness and nonlocality, provides an explanation for probabilistic constraints on events through the geometry of Hilbert space, but that’s as far as it goes. This explanation doesn’t satisfy Bohmians or Everettians, who insist on a representational story about how nature pulls off the trick of producing intrinsically random events at the macrolevel, with nonlocal probabilistic correlations constrained by the Tsirelson bound.
Such a representational story comes at a price that shapes the future direction of physics. Do we really want to give up the concept of measurement as a procedure that provides information about the actual value of an observable of a system to preserve the ideal of representational explanation in physics, as the Everett interpretation proposes? It seems far more rational to accept that if current physical theory has it right, the nature of reality, the way things are, limits the sort of explanation that a physical theory provides.
Acknowledgements {#acknowledgements .unnumbered}
================
Although we probably don’t agree, thanks to Matt Leifer for clarification of the Frauchiger-Renner argument, and to Michel Janssen and Michael Cuffaro for some really helpful input on early drafts of this paper.
[99]{}
Daniela Frauchiger and Renato Renner, “Single-World interpretations of quantum mechanics cannot be self-consistent,” arXiv eprint quant-ph/1604.07422.
Eugene Wigner, “Remarks on the mind-body question,” in I.J. Good (ed.), *The Scientist Speculates,* (Heinemann, London, 1961).
Lucien Hardy, “Quantum mechanics, local realistic theories, and Lorentz-invariant physical theories,” *Physical Review Letters* 68, 2981–2984 (1992). See also Paul G. Kwiat and Lucien Hardy, “The mystery of the quantum cakes,” *American Journal of Physics* 68, 33–36 (2000).
Frauchiger-Renner, *op.cit.*, p. 22.
David Wallace distinguishes between *representational* and *probabilistic* interpretations of the quantum state in “What is orthodox quantum mechanics?” arXiv eprint quant- ph/1604.05973.
Werner Heisenberg, “Über Quantentheoretischer Umdeutung kinematischer und mechanischer Beziehungen,” *Zeitschrift für Physik* 33, 879–893 (1925). In the same year, Max Born and Pascual Jordan published the first part of a two-part paper “Zur Quantenmechanik” in *Zeitschrift für Physik* 34, 858–888 (1925). Part II of this paper, referred to by historians as the ‘three-man paper,’ was co-authored with Heisenberg and published in *Zeitschrift für Physik* 35, 557-615 (1926).
Andrew N. Gleason, “Measures on the Closed Subspaces of Hilbert Space,” *Journal of Mathematics and Mechanics* 6, 885–893 (1957). See p. 886.
Jeffrey Bub, *Bananaworld: Quantum Mechanics for Primates* (Oxford University Press, Oxford, 2016), Chapter 4.
John von Neumann, “Quantum Logics: Strict- and Probability-Logics,” a 1937 unfinished manuscript published in A.H. Taub (ed.), *Collected Works of John von Neumann*, Volume 4 (Pergamon Press, Oxford and New York, 1961), pp. 195–197.
John von Neumann, “Unsolved problems in mathematics,” an address to the International Mathematical Congress, Amsterdam, September 2, 1954. In Miklós Rédei and Michael Stöltzner (eds.), *John von Neumann and the Foundations of Quantum Physics*, pp. 231–245 (Kluwer Academic Publishers, Dordrecht, 2001). The quotation is from p. 245. Gleason’s theorem, *op. cit*, is usually referenced as proving the uniqueness of the Born rule for quantum probabilities from structural features of Hilbert space. Von Neumann proved a similar result in 1927 in “Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik,” *Nachrichten von der Gesellschaft der Wissenschaften zu G¬öttingen. Mathematisch-Physikalische Klasse 1927*, 245–272 (1927). See section 6 in Anthony Duncan and Michel Janssen, “(Never) Mind your pÕs and q’s: von Neumann versus Jordan on the foundations of quantum theory,” *The European Physical Journal* H 38, 175–259 (2013).
Erwin Schrödinger’s papers on wave mechanics were published as ‘Quantisierung als Eigenwertproblem,’ *Annalen der Physik* 79, 361–376 (1926) and ‘An undulatory theory of the mechanics of atoms and molecules,’ *Physical Review* 28, 1049–1070 (1926). His proof of the equivalence of his wave mechanics with Heisenberg’s matrix mechanics was published as ‘Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen,’ *Annalen der Physik* 79, 734–756 (1926).
Schrödinger gave his Royal Institution (London) lecture in 1928. It was published as ‘Four lectures on wave mechanics,’ in *Collected Papers on Wave Mechanics* (Chelsea Publishing Company, New York, 1982). The quotation about a wave as a representation of quantum reality being ‘merely an adequate mathematical description of what happens’ is on p. 160.
Giancarlo Ghirardi, *Sneaking a Look at God’s Cards: Unraveling the Mysteries of Quantum Mechanics* (Princeton University Press, Princeton, 2007). On p. 402, Ghirardi refers to Wigner’s view, following his account of the “Wigner’s friend” thought experiment, as follows: “It is the act of becoming conscious that makes reality pass from the limbo of potentialities to the clarity of actualities.”
Paul Dirac, *The Principles of Quantum Mechanics* (Clarendon Press, Oxford, 1958).
John von Neuman, *Mathematical Foundations of Quantum Mechanics* (Princeton University Press, Princeton, 1955).
Erwin Schrödinger, ‘Discussion of Probability Relations Between Separated Systems,’ *Proceedings of the Cambridge Philosophical Society*, 31, 555–563 (1935). The quotation is from p. 555.
Claude Shannon, “A mathematical theory of communication,” *The Bell System Technical Journal* 27, 379–423, 623-656 (1948). The statement about the semantic aspects of communication being irrelevant to the engineering problem is on p. 379.
Niels Bohr, “The causality problem in atomic physics,” in *New Theories in Physics* (International Institute of Intellectual Cooperation, Warsaw,1939), pp. 11–30. The quotation is from pp. 23–24.
Niels Bohr, “Discussions with Einstein on epistemological problems in modern physics,” in P. A. Schilpp (ed.), *Albert Einstein: Philosopher-Scientist*, The Library of Living Philosophers, Volume 7 (Open Court, Evanston, 1949), pp. 201–241. The quotation is from p. 209.
Itamar Pitowsky, “George Boole’s “conditions of possible experience” and the quantum puzzle,” *British Journal for the Philosophy of Science* 45, 95–125 (1994).
Wolfgang Pauli, “Probability and physics,” in Charles P. Enz and Karl von Meyenn (eds.), *Wolfgang Pauli: Writings on Physics and Philosophy* (Springer, Berlin, 1994), p. 46. The article was first published in *Dialectica* 8, 112–124 (1954).
Christopher A. Fuchs, N. David Mermin, and Ruediger Schack, “An introduction to QBism with an application to the locality of quantum mechanics,” *American Journal of Physics* 82, 749–754 (2014). Also Christopher Fuchs and Ruediger Schack, “Quantum measurement and the Paulian idea,” in H. Atmanspacher and C.A. Fuchs (eds.), *The Pauli–Jung Conjecture and its Impact Today* (Imprint Academic, Exeter, 2014).
Christopher A. Fuchs, “Notwithstanding Bohr, the reasons for QBism,” arXiv eprint quant- ph/1705.03483.
See Michel Janssen, “Drawing the line between kinematics and dynamics in special relativity,” *Studies in History and Philosophy of Modern Physics* 40, 26–52 (2009).
|
---
abstract: 'In this survey we catalogue the many results of the past several decades concerning bounds on the cardinality of a topological space with homogeneous or homogeneous-like properties. These results include van Douwen’s Theorem, which states $|X|\leq 2^{\pi w(X)}$ if $X$ is a power homogeneous Hausdorff space [@VD78], and its improvements $|X|\leq d(X)^{\pi\chi(X)}$ [@Rid06] and $|X|\leq 2^{c(X)\pi\chi(X)}$ [@CR08] for spaces $X$ with the same properties. We also discuss de la Vega’s Theorem, which states that $|X|\leq 2^{t(X)}$ if $X$ is a homogeneous compactum [@DLV2006], as well as its recent improvements and generalizations to other settings. This reference document also includes a table of strongest known cardinality bounds on spaces with homogeneous-like properties. The author has chosen to give some proofs if they exhibit typical or fundamental proof techniques. Finally, a few new results are given, notably (1) $|X|\leq d(X)^{\pi n\chi(X)}$ if $X$ is homogeneous and Hausdorff, and (2) $|X|\leq \pi\chi(X)^{c(X)q\psi(X)}$ if $X$ is a regular homogeneous space. The invariant $\pi n\chi(X)$, defined in this paper, has the property $\pi n\chi(X)\leq\pi\chi(X)$ and thus (1) improves the bound $d(X)^{\pi\chi(X)}$ for homogeneous Hausdorff spaces. The invariant $q\psi(X)$, defined in [@ism81], has the properties $q\psi(X)\leq\pi\chi(X)$ and $q\psi(X)\leq\psi_c(X)$ if $X$ is Hausdorff, thus (2) improves the bound $2^{c(X)\pi\chi(X)}$ in the regular, homogeneous setting.'
address: 'Department of Mathematics, California Lutheran University, 60 W. Olsen Rd, MC 3750, Thousand Oaks, CA 91360 USA'
author:
- Nathan Carlson
title: A survey of cardinality bounds on homogeneous topological spaces
---
Introduction
============
A topological space $X$ is *homogeneous* if for every $x,y\in X$ there exists a homeomorphism $h:X\to X$ such that $h(x)=y$. Roughly, $X$ is homogeneous if the topology at every point is “identical” to that of every other point. $X$ is *power homogeneous* if there exists a cardinal $\kappa$ such that $X^\kappa$ is homogeneous. Many commonly studied spaces are homogeneous (for example, $\mathbb{R}^2$, the unit circle, all connected manifolds in general, and topological groups) and as such are ubiquitous across fields of mathematics. In particular, those homogeneous spaces that are compact play a prominent role. In 1931 Keller [@Keller] showed that the Hilbert Cube $[0,1]^\omega$ is homogeneous. As $[0,1]$ is not homogeneous, this was an early example of a compact power homogeneous space that is not homogenous. Another such example is the ordinal space $\omega+1$, as $(\omega+1)^\omega$ is homogeneous. The author refers the reader to the 2014 book chapter *Topological Homogeneity* by A.V. and J. van Mill [@av14] for a broad and extensive reference on the theory of general homogeneous topological spaces.
[@arh69] showed in 1969 that the cardinality of any compact, first countable space is at most $\cont$, the cardinality of the continuum ${\mathbb{R}}$, thus answering a 50 year old question of Alexandroff and Urysohn. Soon afterwards, in 1970, he showed that any compact, homogeneous, sequential space also has cardinality at most $\cont$ [@arh70a][@arh70b]. This demonstrated that in the presence of homogeneity the first countable condition can be relaxed to the weaker sequential condition. This might be regarded as the first example of cardinality bound that can be improved if a space is additionally known to be homogeneous. In the decades that followed, and in recent years, many well-known cardinality bounds on topological spaces have been improved with homogeneity, or homogeneous-like properties.
The purpose of this survey is to give a thorough account of subsequent results concerning the cardinality of a homogeneous topological space. While there are important open problems in the more general theory of cardinal functions on homogeneous spaces (such as van Douwen’s Problem, which asks if the cellularity $c(X)$ of a homogeneous compactum is at most $\mathfrak{c}$), in this survey we confine ourselves only to cardinality considerations.
Several proofs are given in this survey. The ones that are chosen were chosen for their illustrative nature, as they are fundamental to the theory of cardinality bounds on homogeneous spaces. They were also chosen for their simplicity and elegance, in the author’s opinion. Theorems that have proofs that are more involved and complicated are simply cited in this survey. In addition, there are a few new proofs given in this paper that represent mostly minor improvements of known results.
This paper is organized as follows. In §2 we consider van Douwen’s Theorem, a groundbreaking 1978 result that established the first cardinality bound for a general homogeneous Hausdorff space. In §3 and §4, we explore two bounds for homogeneous Hausdorff spaces that improved van Douwen’s Theorem: $d(X)^{\pi\chi(X)}$ and $2^{c(X)\pi\chi(X)}$, respectively. In §5 we consider compact homogeneous spaces, de la Vega’s Theorem, and improvements to that theorem, while in §6 we look at the many extensions and generalizations of de la Vega’s Theorem to the Hausdorff and other settings. In §7 we consider other results not in these categories, in §8 we compile a list of questions and give a table of cardinality bounds for homogeneous-like spaces that are strongest, as known to the author.
In this survey we do not assume any additional separation axioms on a topological space. All topological spaces are, in fact, topological spaces. We refer the reader to [@EN] and [@Juhasz] for all undefined terms.
van Douwen’s Theorem
====================
In 1978 Eric van Douwen [@VD78] showed that $|X|\leq 2^{\pi w(X)}$ for a Hausdorff homogeneous space $X$. This was first cardinality bound for a general homogeneous space $X$. Homogeneity, or homogeneous-like properties are necessary in this result; for example, the non-homogeneous space $\beta\omega$ does not satisfy this bound. In fact, van Douwen showed that this bound holds for general power homogeneous spaces using sophisticated “clustering” techniques that encode information about projection maps of the form $\pi: X^\kappa\to X$. Van Douwen made extensive use of collections of sets invariant under homeomorphisms. As one can see from the next theorem, his paper was primarily focused on results that imply a space is *not* homogeneous, or not power homogeneous. His cardinality bound was indeed simply just a consequence of this sophisticated theorem.
\[VDdeep\] If the space $X$ admits a continuous map onto a Hausdorff space $Y$ with $|Y|>2^{\pi w(Y)}$, then no power of $X$ is homogeneous in each of the following cases:
- $f$ is open or is a retraction and $d(X)\leq \pi w(Y)$,
- $f$ is perfect, $X$ is regular, and $d(X)\leq \pi w(Y)$, or
- $X$ is compact Hausdorff and $w(X)\leq 2^{\pi w(Y)}$.
As $d(X)\leq\pi w(X)$ for any space $X$, we have the following corollary, which we will refer to as van Douwen’s Theorem.
\[VD\] If $X$ is power homogeneous then $|X|\leq 2^{\pi w(X)}$.
In fact, it also follows from Theorem \[VDdeep\] that no power of $\beta\omega{\backslash}\omega$ is homogeneous, answering a question of Murray Bell. (Frolik [@Fro] had previously shown this space is not homogeneous in ZFC).
While van Douwen’s work was groundbreaking and answered important questions, it turns out that proofs that $|X|\leq 2^{\pi w(X)}$ when $X$ is *homogeneous* are, by comparison, straightforward. In this survey we’ll see several proofs that imply this result (see Theorems \[aut\], \[density\], and \[ER\]). Furthermore, the full version of van Douwen’s Theorem, in the power homogeneous case, has been improved upon in different directions in subsequent decades. See Theorems \[ridHausdorff\], \[open\], and \[ERPH\].
Corollary \[VD\] has a straightforward improvement in the homogeneous case by considering the group $H(X)$ of autohomeomorphisms on a space $X$. This was shown by Frankiewicz. The proof we give here is adapted from that given in [@Juhasz], 2.38. The author considers this proof to involve basic techniques that are typically used when considering bounds on the cardinality of a homogeneous space and the group $H(X)$.
\[aut\] If $X$ is Hausdorff then $|H(X)|\leq 2^{\pi w(X)}$.
Let $\kappa =\pi w(X)$ and let ${\ensuremath{\mathcal{B}}}$ be a $\pi$-base for $X$ such that $|{\ensuremath{\mathcal{B}}}|\leq\kappa$. We show that map $\phi:H(X)\to{\ensuremath{\mathcal{P}}}({\ensuremath{\mathcal{B}}})^{\ensuremath{\mathcal{B}}}$ defined by $\phi(h)(B)=\{C\in{\ensuremath{\mathcal{B}}}:C{\subseteq}h[B]\}$ is one-to-one. Suppose we have $f,g\in H(X)$ such that $f\neq g$. Then there exists $x\in X$ such that $f(x)\neq g(x)$. Let $U$ and $V$ be disjoint neighborhoods of $f(x)$ and $g(x)$, respectively. As $x\in f^{\leftarrow}[U]{\cap}g^{\leftarrow}[V]$, there exists $B\in{\ensuremath{\mathcal{B}}}$ such that $B{\subseteq}f^{\leftarrow}[U]{\cap}g^{\leftarrow}[V]$. Thus $f[B]{\subseteq}U$ and $g[B]{\subseteq}V$. There exists $C\in{\ensuremath{\mathcal{B}}}$ such that $C{\subseteq}f[B]$, and thus $C$ is not a subset of $g[B]$. This shows $\phi(f)(B)\neq\phi(g)(B)$ and $\phi(f)\neq\phi(g)$. We conclude that $\phi$ is one-to-one and therefore $|H(X)|\leq|{\ensuremath{\mathcal{P}}}({\ensuremath{\mathcal{B}}})^{\ensuremath{\mathcal{B}}}|\leq (2^\kappa)^\kappa=2^\kappa$.
As it is easily seen that $|X|\leq |H(X)|$ if $X$ is homogeneous, the homogeneous case of \[VD\] follows. Years later, in 2008, an improved bound for $|H(X)|$ was given by the author and Ridderbos using the Erdös-Rado theorem from partition theory.
\[CRH\] If $X$ is Hausdorff then $|H(X)|\leq 2^{c(X)\pi\chi(X)sd(X)}$.
The *separation degree* $sd(X)$ in the cardinal inequality above is defined as follows. We say that a subset $Z$ of $X$ *separates* a subset ${\ensuremath{\mathcal{G}}}$ of $H(X)$, if for all $f,g\in{\ensuremath{\mathcal{G}}}$ with $f\not= g$ there is some $z\in Z$ with $f(z)\not= g(z)$. $sd(X)$ is defined by $sd(X) = \min\{ |Z| : Z~\mbox{separates}~H(X)\}$. It is always the case that $sd(X)\leq{\ensuremath{d(X)}}$, and thus \[CRH\] is a logical improvement of \[aut\].
The cardinality bound $d(X)^{\pi\chi(X)}$
=========================================
The proof of Theorem \[aut\] gives a straightforward way to demonstrate van Douwen’s Theorem in the homogeneous case. A few years later, in 1981, Ismail [@ism81] gave another relatively simple proof with a slightly stronger conclusion that used the notion of a $q$-pseudo base. For a point $x$ in a space $X$, a family ${\ensuremath{\mathcal{B}}}$ of nonempty open subsets of $X$ is a $q$*-pseudo base* of $x$ in $X$ if for each $y\in X$ such that $y\neq x$, there is a subfamily ${\ensuremath{\mathcal{C}}}$ of ${\ensuremath{\mathcal{B}}}$ such that $x\in\overline{{\bigcup}{\ensuremath{\mathcal{C}}}}$ and $y\notin\overline{{\bigcup}{\ensuremath{\mathcal{C}}}}$. Ismail defined the *$q$-pseudo character* of $x$ in $X$ by $q\psi(x,X)=\min\{|{\ensuremath{\mathcal{B}}}|:{\ensuremath{\mathcal{B}}}\textup{ is a }q\textup{-pseudo base of }x\textup{ in }X\}$ and the $q$-pseudo character of $X$ by $q\psi(X)=\sup\{q\psi(x,X):x\in X\}$. It was shown in [@ism81] that if $X$ is Hausdorff then $q\psi(X)\leq |X|$, $q\psi(X)\leq \pi\chi(X)$, and that $q\psi(X)\leq\psi_c(X)$.
Recall a set $U$ in a space $X$ is regular open if $U=intclU$ and that $RO(X)$ denotes the collection of regular open subsets of $X$. Ismail showed the following fundamental result, for which we provide a proof. Integral to the proof is the fact that in a homogeneous space $X$ if one fixes a point $p\in X$ there exist homeomorphisms $h_x:X\to X$ such that $h_x(p)=x$ for each $x\in X$. These homeomorphisms play an important role in most proofs of cardinality bounds on homogeneous spaces. The proof exhibits how these homeomorphisms interact with the invariant family $RO(X)$. Upon examination, the proof does not require the Hausdorff property, despite the fact that Ismail listed that property as an hypothesis.
\[ismailq\] If $X$ is a homogeneous space, then $|X|\leq |RO(X)|^{q\psi(X)}$.
Fix a point $p\in X$ and let ${\ensuremath{\mathcal{B}}}$ be a q-pseudo base at $x$ in $X$ such that $|{\ensuremath{\mathcal{B}}}|=q\psi(X)$. Without loss of generality we can assume that ${\ensuremath{\mathcal{B}}}{\subseteq}RO(X)$, for otherwise we could consider the q-pseudo base $\{int\overline{B}:B\in{\ensuremath{\mathcal{B}}}\}{\subseteq}RO(X)$, which has the same cardinality as ${\ensuremath{\mathcal{B}}}$. For all $x\in X$, there exists a homeomorphism $h_x:X\to X$ such that $h_x(p)=x$.
Define a function $\phi:X\to RO(X)^{\ensuremath{\mathcal{B}}}$ by $\phi(x)(B)=h_x[B]$. (Note that if $B\in RO(X)$ then $h_x[B]\in RO(X)$). We show $\phi$ is one-to-one. Let $x,y\in X$ such that $x\neq y$. Then $h_x^{\leftarrow}(y)\neq p$. As ${\ensuremath{\mathcal{B}}}$ is a q-pseudo base at $p$, there exists ${\ensuremath{\mathcal{C}}}{\subseteq}{\ensuremath{\mathcal{B}}}$ such that $p\in\overline{{\bigcup}{\ensuremath{\mathcal{C}}}}$ and $h_x^{\leftarrow}(y)\notin \overline{{\bigcup}{\ensuremath{\mathcal{C}}}}$. Therefore, $y=h_y(p)\in\overline{{\bigcup}\{h_y[C]:C\in{\ensuremath{\mathcal{C}}}\}}$ and $y\notin\overline{{\bigcup}\{h_x[C]:C\in{\ensuremath{\mathcal{C}}}\}}$. It follows that there exists a $C\in{\ensuremath{\mathcal{C}}}{\subseteq}{\ensuremath{\mathcal{B}}}$ such that $h_x[C]\neq h_y[C]$, and thus $\phi(x)(C)\neq \phi(y)(C)$ and $\phi(x)\neq\phi(y)$. This shows $\phi$ is one-to-one and $|X|\leq\left|RO(X)^{{\ensuremath{\mathcal{B}}}}\right|\leq |RO(X)|^{q\psi(X)}$.
To see that the above result is logically stronger than Theorem \[VD\] in the homogeneous case, recall that $|RO(X)|\leq 2^{d(X)}$ for an arbitrary space $X$ (see, for example, 2.6d in [@Juhasz]) and that $\pi w(X) = d(X)\pi\chi(X)\geq d(X)q\psi(X)$ if $X$ is Hausdorff.
\[ismail\] If $X$ is a homogeneous Hausdorff space, then $|X|\leq |RO(X)|^{\pi\chi(X)}$.
It was noted independently by de la Vega [@DLVthesis Theorem 1.14] and Ridderbos [@RidMasters Proposition 2.2.7] that $|RO(X)|$ can be replaced in \[ismail\] by the invariant $d(X)$. The author considers the proof of this result to be elegant, representative, and a fundamental model for more sophisticated related cardinality bounds on spaces with homogeneous-like properties. One might consider this result the analogue of the cardinality bound $|X|\leq d(X)^{\chi(X)}$ for a general Hausdorff space $X$ and, in fact, involves a simpler one-to-one map argument using the homeomorphisms $h_x$. We give this proof below and the reader should view it as fundamental in the theory of cardinality bounds on homogeneous spaces.
\[density\] If $X$ is homogeneous and Hausdorff then $|X|\leq d(X)^{{\ensuremath{\pi\chi(X)}}}$.
Fix a point $p\in X$ and a local $\pi$-base ${\ensuremath{\mathcal{B}}}$ at $p$ consisting of non-empty open sets such that $|{\ensuremath{\mathcal{B}}}|\leq{\ensuremath{\pi\chi(X)}}$. Let $D$ be a dense subset of $X$ such that $|D|=d(X)$. For all $x\in X$ let $h_x:X\to X$ be a homeomorphism such that $h_x(p)=x$. For all $x\in X$ and $B\in{\ensuremath{\mathcal{B}}}$, $h_x[B]$ is a non-empty open set and thus there exists $d(x,B)\in h_x[B]{\cap}D$. Define $\phi:X\to D^{{\ensuremath{\mathcal{B}}}}$ by $\phi(x)(B)=d(x,B)$.
We show $\phi$ is one-to-one. Let $x\neq y\in X$. Separate $x$ and $y$ by disjoint open sets $U$ and $V$, respectively.Then $p\in h_x^{\leftarrow}[U]{\cap}h_y^{\leftarrow}[V]$, an open set. There exists $B\in{\ensuremath{\mathcal{B}}}$ such that $B{\subseteq}h_x^{\leftarrow}[U]{\cap}h_y^{\leftarrow}[V]$. It follows that $\phi(x)(B)=d(x,B)\in h_x[B]{\subseteq}U$ and $\phi(y)(B)=d(y,B)\in h_y[B]{\subseteq}V$. Thus $\phi(x)(B)\neq\phi(y)(B)$ and $\phi(x)\neq\phi(y)$. This shows $\phi$ is one-to-one and $|X|\leq |D|^{|{\ensuremath{\mathcal{B}}}|}\leq d(X)^{{\ensuremath{\pi\chi(X)}}}$.
It was shown in [@car07] that the semiregularization $X_s$ of a space $X$ is homogeneous if $X$ is homogeneous. (See, for example, [@por88] for a thorough discussion of the semiregularlization of a space). Using results in [@car07], Theorem \[density\], and the fact that $d(X_s)\leq RO(X)$, we have $|X| = |X_s|\leq d(x_s)^{\pi\chi(X_s)}\leq RO(X)^{\pi\chi(X)}$. Therefore Ismail’s result \[ismail\] above follows from \[density\]. However, Theorem \[ismailq\] and Theorem \[density\] appear to be incomparable.
Theorem \[density\] has a minor but interesting improvement by replacing $\pi\chi(X)$ with a smaller cardinal function the author will call $\pi n\chi(X)$. We define $\pi n\chi(X)$ as follows. For a point $x$ in a space $X$, we define a *local $\pi$-network* at $x$ to be a collection ${\ensuremath{\mathcal{N}}}$ of sets (not necessarily open) such that if $x\in U$ and $U$ is open, then there exists $N\in{\ensuremath{\mathcal{N}}}$ such that $N\in U$. Denote by $\pi n\chi(x,X)$ the least infinite cardinal $\kappa$ such that $x$ has a local $\pi$-network ${\ensuremath{\mathcal{N}}}$ of cardinality $\kappa$ and $\chi(N,X)\leq\kappa$ for all $N\in{\ensuremath{\mathcal{N}}}$. Define $\pi n\chi(X)=\sup\{\pi n\chi(x,X):x\in X\}$. Observe that $\pi n\chi(X)\leq\pi\chi(X)$.
The following result appears to be new in the literature. The proof also involves the construction of a one-to-one map, however there is another “layer” in this construction above and beyond what is done in the proof of Theorem \[density\].
\[densityn\] If $X$ is homogeneous and Hausdorff then $|X|\leq d(X)^{\pi n\chi(X)}$.
Let $\kappa = \pi n\chi(X)$. As in the proof of Theorem \[density\], fix $p\in X$ and for every $x\in X$ fix a homeomorphism $h_x:X\to X$ such that $h_x(p)=x$. There exists a local $\pi$-network ${\ensuremath{\mathcal{N}}}$ at $p$ such that $|{\ensuremath{\mathcal{N}}}|\leq\kappa$ and $\chi(N,X)\leq\kappa$ for all $N\in{\ensuremath{\mathcal{N}}}$. Let $D$ be dense in $X$ such that $|D|=d(X)$. Let $\{U(N,\alpha):\alpha<\kappa\}$ be a neighborhood base at $N$ for each $N\in{\ensuremath{\mathcal{N}}}$.
As $D$ is dense, for all $x\in X$, for all $N\in{\ensuremath{\mathcal{N}}}$, and for all $\alpha<\kappa$, there exists a point $d(x,N,\alpha)\in h_x[U(N,\alpha)]{\cap}D$. We define a function $\phi: X\to (D^\kappa)^{\ensuremath{\mathcal{N}}}$ by $\phi(x)(N)(\alpha)=d(x,N,\alpha)$ and show $\phi$ is one-to-one. Let $x\neq y\in X$ and separate $x$ and $y$ by disjoint open sets $U$ and $V$, respectively. Then $p\in h_x^\leftarrow[U]{\cap}h_y^\leftarrow[V]$. As ${\ensuremath{\mathcal{N}}}$ is a local $\pi$-network for $p$, there exists $N\in{\ensuremath{\mathcal{N}}}$ such that $N{\subseteq}h_x^\leftarrow[U]{\cap}h_y^\leftarrow[V]$. As $\{U(N,\alpha):\alpha<\kappa\}$ is a neighborhood base at $N$, there exists $\alpha<\kappa$ such that $U(N,\alpha){\subseteq}h_x^\leftarrow[U]{\cap}h_y^\leftarrow[V]$. Thus, $h_x[U(N,\alpha)]{\subseteq}U$ and $h_y[U(N,\alpha)]{\subseteq}V$, showing $d(x,N,\alpha)\neq d(y,N,\alpha)$. It follows that $\phi(x)(N)(\alpha)\neq\phi(y)(N)(\alpha)$, $\phi(x)(N)\neq\phi(y)(N)$, and finally that $\phi(x)\neq \phi(y)$. This shows $\phi$ is one-to-one, and $$|X|\leq\left|(D^\kappa)^{\ensuremath{\mathcal{N}}}\right|\leq (|D|^\kappa)^\kappa=|D|^\kappa\leq d(X)^{\pi n\chi(X)}.$$
We turn now to the setting in which a space $X$ is power homogeneous and not necessarily homogeneous. In the full power homogeneous setting, van Douwen’s Theorem \[VD\] has been improved in variety of ways using differing techniques. In the study of power homogeneous spaces $X$, information on the homogeneity of $X^\kappa$ for a cardinal $\kappa$, and the autohomeomorphisms on that space, must be utilized in some way at the level of the space $X$. This information must be captured in such a way as to generate inequalities involving cardinal functions on $X$. The projection maps $\pi:X^\kappa\to X$ are typically, and necessarily, used in this process. In 2006, Ridderbos [@Rid06] used new techniques involving projection maps to give the first improvement to \[VD\] in the full power homogeneous setting.
\[ridHausdorff\] If $X$ is a power homogeneous Hausdorff space, then $|X|\leq d(X)^{\pi\chi(X)}$.
If $X^\kappa$ is homogeneous then, as in any homogeneous space, after fixing a point $p\in X^\kappa$, there exist homeomorphisms $h_x:X^\kappa\to X^\kappa$ such that $h_x(p)=x$ for every $x\in X^\kappa$. However, a critical ingredient in the proof of Theorem \[ridHausdorff\] is demonstrating the existence of such homeomophisms with additional important properties relating to local $\pi$-bases in $X^\kappa$ and $X$. This is shown in Corollary 3.3 in [@Rid06].
Recently, Bella and the author extended \[ridHausdorff\] to give a bound for the cardinality of any open set in a power homogeneous Hausdorff space.
\[open\] Let X be a power homogeneous space. If $D{\subseteq}X$ and $U$ is an open set such that $U{\subseteq}\overline{D}$, then $|U|\leq |D|^{\pi\chi(X)}$.
Recall that a subset $D$ of a space $X$ is $\theta$-dense in $X$ if $\overline{U}{\cap}D\neq{\varnothing}$ for every non-empty open set $U$ of $X$. The $\theta$-*density* of $X$ is defined by $d_\theta(X)=\textup{min}\{|D|:D\textup{ is }\theta\textup{-dense in }X\}$. A variation of the proof of \[ridHausdorff\] in the Urysohn setting was given by the author in [@car07].
\[Ury\] If $X$ is power homogeneous and Urysohn then $|X|\leq d_\theta(X)^{{\ensuremath{\pi\chi(X)}}}$.
The cardinality bound $2^{c(X)\pi\chi(X)}$
==========================================
Van Douwen’s Theorem \[VD\] also has an improvement in a different direction. Theorem \[density\], coupled with the fact that $d(X)\leq\pi\chi(X)^{c(X)}$ for any regular space $X$ ( [@Sap1974]), shows that $|X|\leq 2^{c(X)\pi\chi(X)}$ for regular homogeneous spaces $X$. This was observed by in [@arh87]. In [@car07], these results we modified to show $|X|\leq 2^{c(X)\pi\chi(X)}$ for Urysohn homogeneous spaces $X$. This follows from Theorem \[Ury\] and the fact that $d_\theta(X)\leq \pi\chi(X)^{c(X)}$ for any space $X$ [@car07].
As $c(X)\pi\chi(X)\leq \pi w(X)$ for any space, we see that $2^{c(X)\pi\chi(X)}$ is an improved bound over $2^{\pi w(X)}$. It is a real improvement, even in the compact case, as the compact right topological group $X$, constructed under CH by Kunen [@Kunen], satisfies $c(X)\pi\chi(X) = \omega$ and $\pi w(X) = \omega_1$.
The question remained open whether this bound was valid in the Hausdorff case. Using entirely different techniques, Ridderbos and the author answered this in the affirmative in [@CR08]. While relatively simple, it represented the first use of the Erdös-Rado Theorem in the proof of a cardinal inequality involving homogeneous spaces. It is related to the proof of the Hajnal- theorem $|X|\leq 2^{c(X)\chi(X)}$ for general Hausdorff spaces that uses the Erdös-Rado theorem (see [@Juhasz 2.15b]). Indeed, one may view this result as the homogeneous analogue of the Hajnal- theorem. We give this proof below as our next fundamental proof.
\[ER\] If $X$ is homogeneous and Hausdorff then $|X|\leq 2^{c(X){\ensuremath{\pi\chi(X)}}}$.
Let $\kappa=c(X){\ensuremath{\pi\chi(X)}}$. Fix a point $p\in X$ and a local $\pi$-base ${\ensuremath{\mathcal{B}}}$ at $p$ consisting of non-empty sets such that $|{\ensuremath{\mathcal{B}}}|\leq\kappa$. For all $x\in X$ let $h_x:X\to X$ be a homeomorphism such that $h_x(p)=x$. Define $B:[X]^2\to{\ensuremath{\mathcal{B}}}$ as follows. For all $x\neq y$, there exist disjoint open sets $U(x,y)$ and $V(x,y)$ containing $x$ and $y$, respectively. For each $x\neq y\in X$ the open set $h_x^{\leftarrow}[U]{\cap}h_y^{\leftarrow}[V]$ contains $p$. Thus there exists $B(x,y)\in{\ensuremath{\mathcal{B}}}$ such that $B(x,y){\subseteq}h_x^{\leftarrow}[U]{\cap}h_y^{\leftarrow}[V]$. Note that $h_x[B(x,y)]{\cap}h_y[B(x,y)]={\varnothing}$ for all $\{x,y\}\in [X]^2$.
By way of contradiction suppose that $|X|>2^\kappa$. By the Erdös-Rado Theorem there exists $Y\in[X]^{\kappa^+}$ and $B\in{\ensuremath{\mathcal{B}}}$ such that $B=B(x,y)$ for all $x\neq y\in Y$. For $x\neq y\in Y$ we have $h_x[B]{\cap}h_y[B]=h_x[B(x,y)]{\cap}h_y[B(x,y)]={\varnothing}$. This shows that ${\ensuremath{\mathcal{C}}}=\{h_x[B]:x\in Y\}$ is a cellular family. But $|{\ensuremath{\mathcal{C}}}|=|Y|=\kappa^+>c(X)$, a contradiction. Therefore $|X|\leq 2^\kappa$.
Using Ismail’s invariant $q\psi(X)$, one observes that the above theorem has a improvement in the case when the space $X$ is additionally regular. The proof is a simple matter of lining up a few results, although the result appears to be new in the literature.
\[regular\] If $X$ is regular and homogeneous, then $|X|\leq\pi\chi(X)^{c(X)q\psi(X)}$.
Since $X$ is regular and homogeneous, we have $$|X|\leq |RO(X)|^{q\psi(X)}\leq (\pi\chi(X)^{c(X)})^{q\psi(X)}\leq\pi\chi(X)^{c(X)q\psi(X)}.$$ The first inequality above is Theorem \[ismailq\], and the second inequality follows from the inequality $|RO(X)|\leq\pi\chi(X)^{c(X)}$ for regular spaces (see [@Juhasz] 2.37).
This is an actual improvement over the bound $2^{c(X){\ensuremath{\pi\chi(X)}}}$ because $q\psi(X)\leq\pi\chi(X)$ for a Hausdorff space $X$. Furthermore, it improves the cardinality bound $\pi\chi(X)^{c(X)\psi(X)}$ for regular spaces $X$ given by [@Sap1974], as $q\psi(X)\leq \psi(X)$ if $X$ is regular. One may view Theorem \[regular\] as the homogeneous analogue of ’s result.
We turn now to the case where the space $X$ is power homogeneous. In this case, van Mill [@VM05] first demonstrated the bound $2^{c(X)\pi\chi(X)}$ holds under the assumption of compactness, using a variation of van Douwen’s clustering techniques.
\[vMcompact\] If $X$ is a power homogeneous compactum, then $|X|\leq 2^{c(X)\pi\chi(X)}$.
Van Mill’s result follows in fact as a corollary to this result in the same paper: $|X|\leq w(X)^{\pi\chi(X)}$ for a power homogeneous compactum $X$. (Recall that later it was shown that $|X|\leq d(X)^{\pi\chi(X)}$ for any power homogeneous Hausdorff space by Ridderbos (Theorem \[ridHausdorff\])). We will see, however, in Theorem \[ERPH\] below that the cardinality bound $2^{c(X)\pi\chi(X)}$ holds for any power homogeneous Hausdorff space.
Soon after van Mill’s result, Bella gave an improvement of Theorem \[VD\] for regular power homogenous spaces using the cardinal function $c^*(X)$. Recall that if $X=\prod_{i\in T}X_i$ is an arbitrary product of spaces and $c(\prod_{i\in F}X_i)\leq\kappa$ for each finite subset $F$ of $T$, then $c(X)\leq\kappa$. If follows that if $X$ is a space and $\lambda$ is an infinite cardinal then $c(X^\lambda)=c^*(X)$.
If $X$ is a power homogeneous $T_3$ space, then $|X|\leq 2^{c^*(X)\pi\chi(X)}$.
Note that $c^*(X)\pi\chi(X)\leq \pi w(X)$ for any space. Bella’s result is also a real improvement of Theorem \[VD\], as the same space $X$ in [@Kunen] satisfies $c^*(X)\pi\chi(X) = \omega$ and $\pi w(X) = \omega_1$.
In 2008 it was finally shown that $2^{c(X)\pi\chi(X)}$ is a bound for the cardinality of any power homogeneous Hausdorff space. This represents a second full improvement of van Douwen’s theorem alongside Theorem \[ridHausdorff\]. The proof of this is a sophisticated application of the Erdös-Rado theorem.
\[ERPH\] If $X$ is power homogeneous and Hausdorff then $|X|\leq 2^{c(X){\ensuremath{\pi\chi(X)}}}$.
A decade later a variation of this result was shown for spaces that are Urysohn or quasiregular. Recall that a space is *quasiregular* if every nonempty open set contains a nonempty regular closed set. A collection of nonempty open sets is a *Urysohn cellular family* if the closures of any two are disjoint. We define the *Urysohn cellularity* of a space $X$ as $Uc(X)=\sup\{|{\ensuremath{\mathcal{C}}}|: {\ensuremath{\mathcal{C}}}\textup{ is a Urysohn cellular family}\}$. It is clear that $Uc(X)\leq c(X)$ for any space $X$.
\[phUry\] If $X$ is a power homogeneous space that is Urysohn or quasiregular, then $|X|\leq 2^{Uc(X)\pi\chi(X)}$.
Compact homogeneous spaces and de la Vega’s Theorem
===================================================
In 1970 showed that the cardinality of a sequential, homogeneous compactum is at most $\cont$ [@arh70a][@arh70b]. (By *compactum* we mean a compact Hausdorff space). He then asked if the sequential property can be relaxed to countably tight (see [@arh70a]). In 2006, de la Vega [@DLV2006] answered this long-standing question by showing that the cardinality of a homogeneous compactum is bounded by $2^{t(X)}$. Previously Dow [@Dow88] had shown under PFA that any compact space $X$ of countable tightness contains a point of countable character; thus if the space is additionally homogeneous then $|X|\leq\mathfrak{c}$.
De la Vega’s original proof involved the elementary submodels technique and, in fact, showed that $|X|\leq 2^{L(X)t(X)pct(X)}$ for any regular homogeneous space. (See the next section for the definition of $pct(X)$ and a discussion of this and other generalizations of de la Vega’s Theorem). It was observed in [@CR2012] that much of the work of de la Vega’s elementary submodel proof can be replaced by a theorem of Pytkeev concerning covers by $G_\kappa$-sets. If $X$ is a space and $\kappa$ an infinite cardinal, the $G_\kappa$-*modification* $X_\kappa$ of $X$ is the space formed on the underlying set $X$ by taking the collection of $G_\kappa$-sets as a basis.
\[pyt\] Let $X$ be a compactum and $\kappa$ an infinite cardinal. Then $L(X_\kappa)\leq 2^{t(X)\cdot\kappa}$.
Another crucial ingredient in the proof of de la Vega’s theorem is a result of ’s from [@arh78].
\[arh2.2.4\] Let $X$ be a compactum and let $\kappa=t(X)$. There exists a non-empty $G_\kappa$-set $G$ and a set $H{\subseteq}X$ such that $|H|\leq 2^{\kappa}$ and $G{\subseteq}\overline{H}$.
Using Theorems \[density\], \[arh2.2.4\], and \[pyt\] a simplified proof of de la Vega’s Theorem was given in [@CR2012]. We give this below as our third fundamental proof.
\[de la Vega [@DLV2006], 2006\]\[DLV\] If $X$ is a homogeneous compactum then $|X|\leq 2^{t(X)}$.
([@CR2012]) Let $\kappa=t(X)$. By Theorem \[arh2.2.4\] there exists a non-empty $G_\kappa$-set contained in the closure of a set of size at most $\kappa$. Fix a point $p\in G$ and, as in previous proofs, we obtain homeomorphisms $h_x:X\to X$ such that $h_x(p)=x$ for all $x\in X$. ${\ensuremath{\mathcal{G}}}=\{h_x[G]:x\in X\}$ is a cover of $X$ consisting of $G_\kappa$-sets. There exists a family ${\ensuremath{\mathcal{H}}}=\{H_G:G\in{\ensuremath{\mathcal{G}}}\}$ such that $G{\subseteq}{\ensuremath{\overline{H_G}}}$ and $|H_G|\leq\kappa$ for all $G\in{\ensuremath{\mathcal{G}}}$.
By Pytkeev’s Theorem \[pyt\] there exists ${\ensuremath{\mathcal{G}}}^\prime{\subseteq}{\ensuremath{\mathcal{G}}}$ such that ${\ensuremath{\mathcal{G}}}^\prime$ covers $X$ and $|{\ensuremath{\mathcal{G}}}^\prime|\leq 2^\kappa$. It follows that $X={\bigcup}{\ensuremath{\mathcal{G}}}^\prime{\subseteq}{\bigcup}_{G\in{\ensuremath{\mathcal{G}}}^\prime}{\ensuremath{\overline{H_G}}}{\subseteq}{\ensuremath{\overline{{\bigcup}_{G\in{\ensuremath{\mathcal{G}}}^\prime}H_G}}}$. Thus, $H={\bigcup}_{G\in{\ensuremath{\mathcal{G}}}^\prime}H_G$ is dense in $X$ and $|H|\leq 2^\kappa\cdot\kappa=2^\kappa$. Therefore $d(X)\leq 2^\kappa$. By Theorem \[density\] above and ’s result that ${\ensuremath{\pi\chi(X)}}\leq t(X)$ for a compact space $X$, we have $|X|\leq d(X)^{{\ensuremath{\pi\chi(X)}}}\leq (2^\kappa)^\kappa=2^\kappa$.
While much of the work in this proof is done by Theorem \[pyt\], which itself is an elaborate closing-off argument, the homogeneity of the space is not utilized in \[pyt\]. Instead the homogeneity is applied in two straightforward and elegant ways: first by using Theorem \[arh2.2.4\] and homeomorphisms to cover the space by non-empty $G_\kappa$-sets, and second through the use of Theorem \[density\].
The compactness condition is necessary in de la Vega’s Theorem. Indeed, it does not hold for all countably compact homogeneous spaces, nor all H-closed homogeneous spaces, as the next example from [@CPR2017] shows.
There exists a countably compact, H-closed, Urysohn, separable, countably tight, homogeneous space $X$ such that $|X|=2^\mathfrak{c}$.
Let $Y$ be the Cantor Cube $2^\mathfrak{c}$ with it usual topology and let $X$ be the countable tightness modification of $Y$. That is, the closure of a set $A$ in $X$ is given by $$cl_X(A)={\bigcup}_{B\in[A]^{\leq\omega}}cl_Y(B).$$ $X$ has a finer topology than $Y$, demonstrating that $X$ is not compact as compact spaces are minimal Hausdorff. However, by Theorem 4.2 in [@CPR2017], $X$ is countably compact, H-closed, countably tight, and separable. Furthermore, since $Y$ is the semiregularization of $X$, $X$ is also Urysohn. (See [@por88]).
De la Vega’s Theorem was extended to power homogeneous compacta in [@avr07].
If $X$ is a power homogeneous compactum then $|X|\leq 2^{t(X)}$.
In 2018 and van Mill introduced new techniques and improved de la Vega’s Theorem in the countable case. Considering compact homogenous spaces that are $\sigma$-CT (a countable union of countably tight subspaces), they obtained the following two results.
\[jvm\] If a compactum $X$ is the union of countably many dense countably tight subspaces and $X^\omega$ is homogeneous, then $|X|\leq\cont$.
\[finite\] If $X$ is an infinite homogeneous compactum that is the union of finitely many countably tight subspaces, then $|X|\leq\cont$.
A crucial ingredient in these theorems was a strengthening of ’s Theorem \[arh2.2.4\] in the countable case. (Recall Theorem \[arh2.2.4\] played a central role in proving de la Vega’s Theorem). The strengthening has a deep and sophisticated proof. A subset of a space is *subseparable* if it is contained in the closure of a countable set.
\[subseparable\] Every $\sigma$-CT compactum $X$ has a non-empty subseparable $G_\delta$-set.
Soon afterwards the “$X^\omega$ is homogeneous” condition in Theorem \[jvm\] was generalized to “$X$ is power homogeneous” in [@Carlson2018].
\[phctblytight\] If a power homogeneous compactum $X$ is the union of countably many dense countably tight subspaces, then $|X|\leq\cont$.
Motivated by the results of and van Mill, the author introduced a cardinal invariant known as $wt(X)$, the weak tightness, in [@Carlson2018]. To define it we need the notion of the *$\kappa$-closure* $cl_\kappa A$ of a set $A$ in a space $X$ for a cardinal $\kappa$. This is defined by $cl_\kappa A={\bigcup}\{\overline{B}:B\in [A]^{\leq\kappa}\}$. The *weak tightness* $wt(X)$ of $X$ is defined as the least infinite cardinal $\kappa$ for which there is a cover ${\ensuremath{\mathcal{C}}}$ of $X$ such that $|{\ensuremath{\mathcal{C}}}|\leq 2^\kappa$ and for all $C\in{\ensuremath{\mathcal{C}}}$, $t(C)\leq\kappa$ and $X=cl_{2^\kappa}C$. We say that $X$ is *weakly countably tight* if $wt(X)=\omega$. It is clear that $wt(X)\leq t(X)$. Example 2.3 in [@BC2020a] provides a straightforward example of a compact group of tightness $\omega_1$ such that, under $2^{\omega}=2^{\omega_1}$, $X$ is weakly countably tight.
The condition “$X=cl_{2^\kappa}C$” in the above definition can be difficult to work with. The next proposition gives additional conditions under which this condition can be relaxed to “$C$ is dense in $X$”.
\[wteasy\] Let $X$ be a space, $\kappa$ a cardinal, and ${\ensuremath{\mathcal{C}}}$ a cover of $X$ such that $|{\ensuremath{\mathcal{C}}}|\leq 2^\kappa$, and for all $C\in{\ensuremath{\mathcal{C}}}$, $t(C)\leq\kappa$ and $C$ is dense in $X$. If $t(X)\leq 2^\kappa$ or $\pi_\chi(X)\leq 2^\kappa$ then $wt(X)\leq\kappa$.
Pytkeev’s Theorem \[pyt\] has an improvement using $wt(X)$.
\[pytimprove\] Let $X$ be a compactum and $\kappa$ an infinite cardinal. Then $L(X_\kappa)\leq 2^{wt(X)\cdot\kappa}$.
Additionally, Bella and the author were able to give a result that amounts to a variation of both Theorem \[arh2.2.4\] and \[subseparable\].
\[variation\] Let $X$ be a compactum and let $\kappa=wt(X)$. Then there exists a non-empty closed set $G{\subseteq}X$ and a ${\ensuremath{\mathcal{C}}}$-saturated set $H\in[X]^{\leq 2^\kappa}$ such that $G{\subseteq}\overline{H}$ and $\chi(G,X)\leq\kappa$.
Using Theorems \[pytimprove\] and \[variation\], Bella and the author were able to give a full improvement to de la Vega’s Theorem in [@BC2020a]. Recall that $\pi\chi(X)\leq t(X)$ for a compactum $X$.
\[cpthomog\] If $X$ is a homogeneous compactum then $|X|\leq 2^{wt(X)\pi\chi(X)}$.
Below we isolate the case of Theorem \[cpthomog\] where all cardinal invariants involved are countable. It follows directly from Proposition \[wteasy\] and the above. Compare Corollary \[hcountable\] with Theorem \[jvm\].
\[hcountable\] Let $X$ be a homogeneous compactum of countable $\pi$-character with a cover ${\ensuremath{\mathcal{C}}}$ such that $|{\ensuremath{\mathcal{C}}}|\leq\cont$ and for all $C\in{\ensuremath{\mathcal{C}}}$, $C$ is countably tight and dense in $X$. Then $|X|\leq\cont$.
Another corollary to Theorem \[cpthomog\] follows directly from the fact that in an compact, $T_5$ space there exists a point of countable $\pi$-character. (This is due to ). If the space $X$ is additionally homogeneous then $\pi\chi(X)=\omega$. This corollary has not been previously mentioned in the literature.
\[T5\] If $X$ is compact, $T_5$, and homogeneous, then $|X|\leq 2^{wt(X)}$.
Generalizations of de la Vega’s Theorem
=======================================
This section is devoted to extensions of de la Vega’s Theorem; that is, results that directly imply that theorem in a more generalized setting. Natural questions arise, such as, does Lindelöf suffice instead of the compactness property? The answer to this question is no. In [@CR2012], an example of a $\sigma$-compact, homogeneous space $X$ was constructed with the property $|X|>2^{L(X)\pi\chi(X)t(X)}$. This shows $2^{L(X)t(X)}$ is not a bound for the cardinality of every Hausdorff homogeneous space.
Exactly what are the necessary properties of compactness needed in this theorem? It turns out that one pair of necessary properties are Lindelöf and countable point-wise compactness type. The *point-wise compactness type* $pct(X)$ of a space $X$ is the least infinite cardinal $\kappa$ such that $X$ can be covered by compact sets $K$ such that $\chi(K,X)\leq\kappa$. Clearly compact spaces are of countable point-wise compactness type. Also, all locally compact spaces have this property. In [@DLVthesis], de la Vega showed that $|X|\leq 2^{L(X)t(X)pct(X)}$ for any regular homogeneous space, and this bound was shown to be valid for regular power homogeneous spaces in [@Rid06]. In [@CR2012], the regularity property was shown to be unnecessary.
\[ltpct\] If $X$ is a power homogeneous Hausdorff space then $|X|\leq 2^{L(X)t(X)pct(X)}$.
Thus, for example, the cardinality bound $2^{t(X)}$ holds for all locally compact, Lindelöf homogeneous Hausdorff spaces.
The next five theorems represent slight improvements of Theorem \[ltpct\]. For a cardinal $\kappa$ and a space $X$, a subset $\{x_\alpha:\alpha\leq\kappa\}{\subseteq}X$ is a *free sequence of length* $\kappa$ if for every $\beta<\kappa$, $cl_X\{x_\alpha:\alpha<\beta\}{\cap}cl_X\{x_\alpha:\alpha\geq\beta\}={\varnothing}$. The free sequence number $F(X)$ is the supremum of the lengths of all free sequences in $X$. It is well-known that $t(X)=F(X)$ if $X$ is a compactum. In addition, as $F(X)\leq L(X)t(X)$ for any space $X$, the following theorem improves Theorem \[ltpct\].
\[F(X)\] If $X$ is a power homogeneous Hausdorff space then $|X|\leq 2^{L(X)F(X)pct(X)}$.
The invariant $aL_c(X)$, the *almost Lindelöf degree with respect to closed sets*, is the smallest infinite cardinal $\kappa$ such that for every closed subset $C$ of $X$ and every collection ${\ensuremath{\mathcal{U}}}$ of open sets in $X$ that cover $C$, there is a subcollection ${\ensuremath{\mathcal{V}}}$ of ${\ensuremath{\mathcal{U}}}$ such that $|{\ensuremath{\mathcal{V}}}|\leq\kappa$ and $\{\overline{U}:U\in{\ensuremath{\mathcal{V}}}\}$ covers $C$. It is clear that $aL_c(X)\leq L(X)$.
\[alc\] If $X$ is a power homogeneous Hausdorff space then $|X|\leq 2^{aL_c(X)t(X)pct(X)}$.
Recently in [@BC2020b], the Lindelöf degree $L(X)$ in Theorem \[ltpct\] was replaced by the cardinal invariant $pwL_c(X)$, introduced by Bella and Spadaro in [@BS2020]. The *piecewise weak Lindelöf degree for closed sets* $pwL_c(X)$ of $X$ is the least infinite cardinal $\kappa$ such that for every closed set $F{\subseteq}X$, for every open cover ${\ensuremath{\mathcal{U}}}$ of $F$, and every decomposition $\{{\ensuremath{\mathcal{U}}}_i:i\in I\}$ of ${\ensuremath{\mathcal{U}}}$, there are families ${\ensuremath{\mathcal{V}}}_i\in[{\ensuremath{\mathcal{U}}}_i]^{\leq\kappa}$ for every $i\in I$ such that $F{\subseteq}{\bigcup}\{\overline{{\bigcup}{\ensuremath{\mathcal{V}}}_i}:i\in I\}$. It is clear that $pwL_c(X)\leq L(X)$ and, importantly, it can be shown that $pwL_c(X)\leq c(X)$.
\[Bella, C. [@BC2020b], 2020\]\[pwlc\] If $X$ is homogeneous and Hausdorff then $|X|\leq 2^{pwL_c(X)t(X)pct(X)}$.
While it is clear that Theorem \[pwlc\] is an improvement of Theorem \[ltpct\], it is also an improvement of Theorem \[alc\] as it can be shown that $pwL_c(X)\leq aL_c(X)$. Furthermore, it is a variation of Theorem \[ER\]; that is, $2^{c(X)\pi\chi(X)}$ is a bound for the cardinality of every homogeneous Hausdorff space. This is because $pwL_c(X)\leq c(X)$ and $\pi\chi(X)\leq t(X)pct(X)$ for Hausdorff spaces.
In [@BC2020b], a consistent improvement of Theorem \[ltpct\] was given using the linearly Lindelöf degree $lL(X)$. A space $X$ is *linearly Lindelöf* provided that every increasing open cover of $X$ has a countable subcover. More generally, we define the *linear Lindelöf degree* $lL(X)$ of $X$ as the smallest cardinal $\kappa$ such that every increasing open cover of $X$ has a subcover of size not exceeding $\kappa$. Equivalently, $lL(X)\leq\kappa$ if every open cover of $X$ has a subcover ${\ensuremath{\mathcal{U}}}$ such that $|{\ensuremath{\mathcal{U}}}|$ has cofinality at most $\kappa$.
\[lL\] Assume $2^\kappa<\kappa^{+\omega}$ or $2^{<2^\kappa}=2^\kappa$. If $X$ is Hausdorff, homogeneous, and $\kappa=lL(X)F(X)pct(X)$, then $|X|\leq 2^\kappa$.
Our last improvement of Theorem \[ltpct\] gives a bound for the cardinality of an open set in a power homogeneous space.
\[openset\] If $X$ is a power homogeneous Hausdorff space and $U{\subseteq}X$ is an open set, then $|U|\leq 2^{L(\overline{U})t(X)pct(X)}$
The next four results from [@SS2018], [@BC], and [@BCG2020] represent extensions of de la Vega’s Theorem in a different direction using the invariants $wL(X)$ or $wL_c(X)$. The *weak Lindelöf degree* of a space $X$ is the least infinite cardinal $\kappa$ such that every open cover ${\ensuremath{\mathcal{U}}}$ of $X$ has a subfamily ${\ensuremath{\mathcal{V}}}$ such that $|{\ensuremath{\mathcal{V}}}|\leq\kappa$ and $X=\overline{{\bigcup}{\ensuremath{\mathcal{V}}}}$. The invariant $wL_c(X)$, the *weak Lindelöf degree with respect to closed sets*, is the smallest infinite cardinal $\kappa$ such that for every closed subset $C$ of $X$ and every collection ${\ensuremath{\mathcal{U}}}$ of open sets in $X$ that cover $C$, there is a subcollection ${\ensuremath{\mathcal{V}}}$ of ${\ensuremath{\mathcal{U}}}$ such that $|{\ensuremath{\mathcal{V}}}|\leq\kappa$ and $C{\subseteq}\overline{{\bigcup}{\ensuremath{\mathcal{V}}}}$. It is clear that $wL(X)\leq wL_c(X)\leq aL_c(X)$.
\[initial\] If $X$ is an initially $\kappa$-compact power homogeneous $T_3$ space then $|X|\leq 2^{F(X)wL_c(X)}$.
\[piLindelof\] If $X$ is a regular power homogeneous space and with a $\pi$-base ${\ensuremath{\mathcal{B}}}$ such that $\overline{B}$ is Lindelöf for all $B\in{\ensuremath{\mathcal{B}}}$, then $|X|\leq 2^{wL(X)t(X)pct(X)}$.
As locally compact spaces satisfy the hypotheses in Theorem \[piLindelof\], we have the following corollary.
\[lcphup\] If $X$ is a locally compact power homogeneous space then $|X|\leq 2^{wL(X)t(X)}$.
The above theorem indicates that the compactness condition in de la Vega’s Theorem can be replaced with another pair of conditions: locally compact and weakly Lindelöf. It turns out that Corollary \[lcphup\] can be given an improved conclusion. This was demonstrated in [@BCG2020].
\[lcphdown\] If $X$ is a locally compact power homogeneous space then $|X|\leq wL(X)^{t(X)}$.
Other results
=============
Recently it was shown in [@BCG2020] that if $X$ is an extremally disconnected space then $c(X)\leq w(X)\pi\chi(X)$. Using Theorem \[ERPH\], the following is an immediate consequence.
\[ed\] If $X$ is power homogeneous and extremally disconnected then $|X|\leq 2^{wL(X)\pi\chi(X)}$.
One should regard the bound in Theorem \[ed\] as being “small” as $wL(X)$ and $\pi\chi(X)$ are generally thought of as small cardinal invariants. Observe that it follows from Theorem \[ed\] that an H-closed, extremally disconnected, power homogeneous space has cardinality at most $2^{\pi\chi(X)}$. However, it was shown in [@car07] that an infinite H-closed extremally disconnected space cannot be power homogeneous. This latter result is an extension of a result of Kunen [@K90] that an infinite compact F-space is not power homogeneous.
Given a space $X$, the *diagonal* of $X^2$, denoted by $\Delta_X$, is the set $\{(x,x):x\in X\}$. $X$ is said to have a *regular $G_\delta$-diagonal* if there exists a countable family ${\ensuremath{\mathcal{U}}}$ of open sets in $X^2$ such that $\Delta_X={\bigcap}{\ensuremath{\mathcal{U}}}={\bigcap}\{\overline{U}:U\in{\ensuremath{\mathcal{U}}}\}$. A cardinality bound for homogeneous spaces with a regular $G_\delta$-diagonal was given in [@BBR2014].
\[regdiag\] If $X$ is a homogeneous space with a regular $G_\delta$-diagonal, then $|X|\leq wL(X)^{\pi\chi(X)}$.
A notion related to homogeneity is known as countable dense homogeneity. A separable space $X$ is *countable dense homogeneous* (CDH) if given any two countable dense subsets $D$ and $E$ of $X$, there is a homeomorphism $h:X\to X$ such that $h[D]=E$. Separability is included in the definition as clearly this notion is of interest only if $X$ has a countable dense subset. Not every CDH space is homogeneous, however every connected CDH space is homogeneous [@FL1987].
In [@av14b] it was shown that the cardinality of a CDH space is as most $\cont$.
\[CDH\] The cardinality of a CDH space is at most $\cont$.
Questions and a Table of Bounds
===============================
Recall that in Theorem \[CRH\] it was shown that $|H(X)|\leq 2^{c(X)\pi\chi(X)sd(X)}$ for a Hausdorff space $X$. In addition, in Theorem \[ER\] it was shown that $|X|\leq 2^{c(X)\pi\chi(X)}$ if $X$ is Hausdorff and homogeneous. In light of these theorems, the following was asked by the author and Ridderbos in [@CR08].
Is $|H(X)|\leq 2^{c(X)\pi\chi(X)}$ for a Hausdorff space $X$?
As it was shown in [@BC2020a] that the cardinality of a homogeneous compactum is at most $2^{wt(X)\pi\chi(X)}$ (Theorem \[cpthomog\]), it is natural to ask if either of the two cardinal invariants $wt(X)$ and $\pi\chi(X)$ can be removed from this bound. The next two questions were asked in [@BC2020a]. The second was additionally asked by de la Vega in [@DLVthesis]. These two questions appear to be quite challenging. Answering either in the affirmative would likely require new breakthrough techniques, while counter-examples would likely be complicated and have intriguing properties.
Is the cardinality of a homogeneous compactum at most $2^{wt(X)}$?
Is the cardinality of a homogeneous compactum at most $2^{\pi\chi(X)}$?
The power homogeneous case of Theorem \[cpthomog\] is also an open question.
Is the cardinality of a power homogeneous compactum at most $2^{wt(X)\pi\chi(X)}$?
By results of , every $T_5$ compactum has a point of countable $\pi$-character. It follows from Theorem \[ER\] that the cardinality of a homogeneous $T_5$ compactum is at most $2^{c(X)}$. This was observed by van Mill in [@VM05] and was proved for power homogeneous $T_5$ compacta in [@Rid09]. Note additionally that Corollary \[T5\] states that the cardinality of a homogeneous $T_5$ compactum is at most $2^{wt(X)}$. Van Mill asked if the cardinality of such spaces is in fact at most $\mathfrak{c}$.
Is the cardinality of every $T_5$ homogeneous compactum at most $\mathfrak{c}$?
In light of the various cardinality bounds for homogeneous-like spaces using the weak Lindelöf degree $wL(X)$, the following was asked in [@BC2018].
If $X$ is power homogeneous and Tychonoff, is $|X|\leq 2^{wL(X)t(X)pct(X)}$?
**Bound on $|X|$** **Hypotheses on $X$** **Proved in** **Year** **Thm**
----------------------------- ------------------------------------------------------------------ ------------------------------- ---------- ------------------
$|RO(X)|^{q\psi(X)}$ homog., Hausdorff Ismail [@ism81] 1981 \[ismailq\]
$d(X)^{\pi n\chi(X)}$ homog., Hausdorff (current paper) 2020 \[densityn\]
$d(X)^{\pi\chi(X)}$ power homog., Hausdorff Ridderbos [@Rid06] 2006 \[ridHausdorff\]
$d_\theta(X)^{\pi\chi(X)}$ power homog., Urysohn C. [@car07] 2007 \[Ury\]
$\pi\chi(X)^{c(X)q\psi(X)}$ homog., $T_3$ (current paper) 2020 \[regular\]
$2^{c(X)\pi\chi(X)}$ power homog., Hausdorff C., Ridderbos [@CR08] 2008 \[ERPH\]
$2^{Uc(X)\pi\chi(X)}$ power homog., Bonanzinga, C., 2018 \[phUry\]
(Urysohn or quasiregular) Cuzzupé, Stavrova [@BCCS2018]
$\mathfrak{c} $ homog. compactum that , van Mill [@JVM2018] 2018 \[finite\]
is the union of finitely many
countably tight subspaces
$\mathfrak{c} $ power homog. compactum C. [@Carlson2018] 2018 \[phctblytight\]
that is the union of
countably many dense
countably tight subspaces
$2^{wt(X)}$ homog., compact, $T_5$ (current paper) 2020 \[T5\]
$2^{wt(X)\pi\chi(X)}$ homog., compactum Bella, C. [@BC2020a] 2020 \[cpthomog\]
$2^{L(X)F(X)pct(X)}$ power homog., Hausdorff C., Porter, 2012 \[F(X)\]
Ridderbos [@CPR2012]
$2^{pwL_c(X)t(X)pct(X)}$ homog., Hausdorff Bella, C., [@BC2020b] 2020 \[pwlc\]
$2^{lL(X)F(X)pct(X)}$ homog., Hausdorff Bella, C., [@BC2020b] 2020 \[lL\]
($2^\kappa<\kappa^{+\omega},$ or $2^{<2^\kappa}=2^\kappa$
$\kappa= lL(X)F(X)pct(X))$
$2^{F(X)wL_c(X)}$ power homog., $T_3$, Spadaro, 2018 \[initial\]
initially $\kappa$-compact Szeptycki [@SS2018]
$2^{wL(X)t(X)pct(X)}$ power homog., $T_3$, Bella, C., [@BC] 2018 \[piLindelof\]
$\pi$-base ${\ensuremath{\mathcal{B}}}$ such that $\overline{B}$
is Lindelöf for all $B\in{\ensuremath{\mathcal{B}}}$
$wL(X)^{t(X)}$ power homog., loc. compact Bella, C., Gotchev [@BCG2020] 2020 \[lcphdown\]
$2^{wL(X)\pi\chi(X)}$ power homog., Bella, C., Gotchev [@BCG2020] 2020 \[ed\]
extremally disconnected
$wL(X)^{\pi\chi(X)}$ homog, D. Basile, Bella, 2014 \[regdiag\]
regular $G_\delta$ diagonal Ridderbos [@BBR2014]
$\mathfrak{c} $ countable dense homog. , 2014 \[CDH\]
separable van Mill [@av14b]
: Strongest known cardinality bounds on spaces with homogeneous-like properties.[]{data-label="tab:table1"}
[21]{}
A. V. , *On the cardinality of bicompacta satisfying the first axiom of countability*, Soviet Math. Dokl. **10** (1969), no. 4, 951–955.
A. V. , *A survey of some recent advances in general topology, old and new problems*, in: Proc. International Congress of Mathematicians, Nice, 1970, Tome 2, pp. 19–26.
A. V. , *The Suslin number and cardinality. Character of points in sequential bicompacta*, Soviet Math. Dokl. 11 (1970) 597–601.
A. V. , *The structure and classification of topological spaces and cardinal invariants*, Uspekhi Mat. Nauk **33** (1978), no. 6 (204), 29–84, 272.
A. V. , *Topological homogeneity. Topological groups and their continuous images* Uspekhi Mat. Nauk **42** (2(254)) (1987) 69–105, 287.
A. V. , J. van Mill, *On the cardinality of countable dense homogeneous spaces* Proc. Amer. Math. Soc. **141** (2013), no. 11, 4031–4038.
A. V. , J. van Mill, *Topological Homogeneity*, in: Recent Progress in General Topology III (2014), Hart K., van Mill J., Simon P. (eds), Atlantis Press, Paris.
A. V. , J. van Mill, and G. J. Ridderbos, *A new bound on the cardinality of power homogeneous compacta*, Houston J. Math. **33** (2007), no. 3, 781–793.
D. Basile, A. Bella, G. J. Ridderbos, *Weak extent, submetrizability and diagonal degrees*, Houston J. Math. (2014), no. 1, 255-266.
A. Bella, *Remarks on the cardinality of a power homogeneous space*, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 463–468.
A. Bella, N. Carlson, *On cardinality bounds involving the weak Lindelöf degree*, Quest. Math., **41(1)** (2018), 99–113.
A. Bella, N. Carlson, *On weakening tightness to weak tightness*, Monatsh Math. **192**, no. 1 (2020), 39–48.
A. Bella, N. Carlson, *Cardinality bounds via covers by compact sets*, submitted.
A. Bella, N. Carlson, I. Gotchev, *More on cardinality bounds involving the weak Lindelöf degree*, preprint.
M. Bonanzinga, N. Carlson, M. V. Cuzzupè, D. Stavrova, *More on the cardinality of a topological space*, Appl. Gen. Topol. 19, no. 2 (2018) 269-280.
A. Bella, S. Spadaro, *A common extension of ’s theorem and the Hajnal- inequality*, Canad. Math. Bull. 63 (2020), no. 1, 197–203.
N. Carlson, *Non-regular power homogeneous spaces*, Topology Appl. **154** (2007), no. 2, 302–308.
N. Carlson, G. J. Ridderbos, *Partition relations and power homogeneity*, Top. Proc. **32** (2008), 115–124.
N. Carlson, G. J. Ridderbos, *On several cardinality bounds on power homogeneous spaces*, Houston Journal of Mathematics, **38** (2012), no. 1, 311-332.
N. Carlson, J. Porter, G.J. Ridderbos, *On cardinality bounds for homogeneous spaces and the $G_\kappa$-modification of a Space*, Topology Appl. **159** (2012), no. 13, 2932–2941.
N. Carlson, J.R. Porter, G.J. Ridderbos, *On Homogeneity and the H-closed Property*, Top. Proc. **49** (2017), 153–164.
N. Carlson, *On the weak tightness and power homogeneous compacta*, Topology Appl. **249** (2018), 103–111.
R. de la Vega, *Homogeneity properties on compact spaces*, Doctoral Thesis, University of Wisconsin-Madison.
R. de la Vega, *A new bound on the cardinality of homogeneous compacta*, Topology Appl. **153** (2006), 2118–2123.
E. K. van Douwen, *Nonhomogeneity of products of preimages and [$\pi
$]{}-weight*, Proc. Amer. Math. Soc. **69** (1978), no. 1, 183–192.
A. Dow, *An introduction to applications of elementary submodels to topology*, Topology Proc. 13 (1988) 17–72.
R. Engelking, *General Topology*, Heldermann Verlag, Berlin, 1989.
B. Fitzpatrick, Jr. and N. F. Lauer, *Densely homogeneous spaces. I*, Houston J. Math. **13** (1987), 19–25
R. Frankiewicz, *A short proof of a cardinal inequality involving homogeneous spaces*, Proc. Amer. Math. Soc. **75** (1979), no. 2, 372.
Z. Frolik, *Nonhomogeneity of $\beta P-P$*, Comment. Math. Univ. Carolinae **8** (1967), 705–709.
I. Gotchev, *Cardinalities of weakly Lindelöf spaces with regular $G_\delta$-diagonal*, ArXiv: 1504.01785.
M. Ismail, *Cardinal functions of homogeneous spaces and topological groups*, Math. Japon. 26 (1981), no. 6, 635?646.
I. Juh[á]{}sz, *Cardinal functions in topology—ten years later*, second ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980.
I. , J. van Mill, *On $\sigma$-countably tight spaces*, Proc. Amer. Math. Soc. **146** (2018), no. 1, 429–437.
O. H. Keller, *Die homoiomorphie der kompakten konvexen Mengen im Hilbertschen Raum*, Math. Ann. **105** (1931), 748–758.
K. Kunen, Large homogeneous compact spaces, in *Open Problems in Topology*, 261–270, North-Holland, Amsterdam, 1990.
K. Kunen, *Compact L-spaces and right topological groups*, Topology Proc. 24 (1999), 295–327.
J. van Mill, *On the cardinality of power homogeneous compacta*, Topology Appl. **146/147** (2005), 421–428.
J. R. Porter, G. Woods, *Extensions and Absolutes of Hausdorff Spaces*, Springer, Berlin, 1988.
E. G. Pytkeev, *About the $G_\lambda$-topology and the power of some families of subsets on compacta*, Colloq. Math. Soc. Janos Bolyai **41** (1985), 517–522.
G. J. Ridderbos, *Notes on cardinal functions and homogeneity*, Master Thesis, Vrije Universiteit, Amsterdam (unpublished).
, *On the cardinality of power homogeneous [H]{}ausdorff spaces*, Fund. Math. **192** (2006), 255–266.
G. J. Ridderbos, *Cardinality restrictions on power homogeneous $T_5$ compacta*, Studia Sci. Math. Hungar. **46** (2009), no. 1, 113–120.
B. , *Canonical sets and character, density and weight in compact spaces*, Soviet Math. Dokl. 15 (1974), 1282–1287.
S. Spadaro, P. Szeptycki, *$G_\delta$ covers of compact spaces*, Acta Math. Hungar., 154 (2018), no. 1, 252–263.
|
---
abstract: 'We introduce a general geometric framework for the construction of polyhedra and polyhedral complexes that are [*bifoldable*]{}, i.e. foldable into two different planes. This vastly generalizes origami folds known as the Miura pattern, the origami tube and the Eggbox pattern. Our polyhedra are generalized zonohedra based on 1-parameter families stars of vectors in ${\mathbb{R}}^3$ that deform in specific ways while the polyhedra are folded. After describing the framework, its basic features and the general design process, we give several new examples of infinite doubly periodic, triply periodic and fractal bifoldable polyhedra.'
address:
- |
Matthias Weber\
Department of Mathematics\
Indiana University\
Bloomington, IN 47405\
USA
- |
Jiangmei Wu\
School of Art, Architecture, and Design\
Indiana University\
Bloomington, IN 47405\
USA
author:
- Matthias Weber
- Jiangmei Wu
bibliography:
- 'foldings.bib'
---
[Biplanar Foldings]{}
Introduction
============
In recent years, origami has found a wide range of applications in material sciences, engineering, cell biology, art and other areas. At a micro scale, origami design has been used in medical devices such as heart stents [@Kuribayashi2006]. At a large scale, origami design has been applied in art installation [@Wu2018], aerospace [@Barbarino2011], and architectural facade [@Grosso2010]. In more recent years, origami has also been used to create programmable systems that can change shape, functionality and material properties [@Hawkes2010]. While many of these applications are concerned with origami that is folded from a flat sheet material to create three-dimensional depth, a few origami researchers have been focusing on creating origami tubes or honeycomb-like origami structures, such as foldable polyhedral origami that can be repeated periodically in space. Such origami structures can be used as inflatable and deployable structures [@Schenk2014] or as metamaterials [@filipov2015].
Several of these structures, have the remarkable feature that they can be folded together in [*two different*]{} ways into perpendicular planes. However, most of the examples given in the literature are general polyhedral complexes, i.e. have edges where more than two facets meet. For construction purposes, it is often desirable to have purely polyhedral structures. Moreover, the fundamental geometric construction methods that are used to generate these origami structures are somewhat limited and appear to be coincidental. Many of the resulting honeycomb origami structures are closely related to Miura folds [@Schenk2013] or space-filling polyhedra [@Tachi2012].
We address these two issues by first describing a simple mathematical framework (Section \[sec:star\]) that explains the bifoldability and allows for a very flexible construction method of general bifoldable complexes ($\Sigma$-comoplexes) in which no more than two facets meet at any given edge. In section \[sec:vertices\] we list all fourteen types of polyhedral vertices that can occur in our construction. After reviewing the basic examples (Section\[sec:simple\]), we illustrate our method to construct a bifoldable fractal (Section \[sec:fractal\]). We briefly discuss the design process to build more complicated examples in Section \[sec:design\]. We then demonstrate this method in Section \[sec:weave\] to construct a doubly periodic Miura Weave. In Section \[sec:links\] we discuss simple examples of triply periodic bifoldable surfaces with dodecahedral cavities, and in Section \[sec:dosequis\] we construct a triply periodic Miura pattern.
We believe our construction method of bifoldable complexes will have implications in the designing and building of smart metamaterials, air or hydraulic filtration systems, robots, large-scale inflatable structures, breathable architectural skins, and many more.
Initially, our investigation began with the question when polygonal approximations of classical triply periodic minimal surfaces can be folded into a plane. A simple and well known example is the mucube [@coxeter1938], an approximation of the P-surface of Schwarz. After discovering Theorem \[thm:key\], the focus quickly shifted away from minimal surfaces. In fact, some of our examples like the Fractal are distinctively non-minimal (minimal surfaces have quadratic area growth in balls). On the other hand, we observed three paradigms that triply periodic minimal surfaces and bifoldable polyhedra have in common: First, there are examples where triply periodic minimal surfaces can be collapsed into different planes through a minimal (albeit not isometric) deformation. Secondly, the flexibility one has in constructing examples is similar. Lastly, certain topological paradigms occur in both situations: The Link has a vague resemblance to Scherk’s singly periodic surface, one can create a bifoldable version of Scherk’s doubly periodic surface, and the surfaces discussed in section \[sec:dosequis\] can be modified to resemble the doubly periodic Karcher-Meeks-Rosenberg surfaces.
Stars and Foldings {#sec:star}
==================
In this section, we introduce a general framework to construct a class of bifoldable Euclidean polyhedral complexes that we call $\Sigma$-complexes. To make this notion rigorous, we introduce the following concepts.
A (geometric) polyhedral complex in ${\mathbb{R}}^n$ is a locally finite set $P$ of closed convex polytopes in ${\mathbb{R}}^n$ such that
- with each polytope, all of its facets belong to the set;
- the intersection between two polytopes is either empty or is a facet of both of them;
- as a topological space, $P$ is connected.
The dimension $d$ of a polyhedral complex is the maximal dimension of its polytopes. We then speak of polyhedral $d$-complexes.
We will only consider 2- and 3-dimensional polyhedral complexes in ${\mathbb{R}}^3$.
A polyhedral 2-complex in ${\mathbb{R}}^3$ is [*biplanar*]{} if all its edges lie in one of the two orthogonal planes which we will assume to be the two vertical coordinate planes of ${\mathbb{R}}^3$.
A [*folding*]{} is a continuous 1-parameter family of polyhedral 2-complexes $C_t$ so that the facets remain congruent. A folding [*collapses*]{} into a plane for $t\to t_0$ if the limits of all vertices of $C_t$ exist and lie in that plane.
Our goal is to construct biplanar polyhedral foldings that collapse at the end points of the parameter interval into the two vertical coordinate planes.
The polyhedral complexes we consider are generalizations of [*zonohedra*]{} [@coxeter1973] and are based on the concept of a [*4-star*]{}:
A 4-star $\Sigma$ is a set of four vectors $v_1,\ldots v_4\in {\mathbb{R}}^3$ so that no three of them are linearly dependent. A polyhedral complex is [*based*]{} on a star $\Sigma$ if all of its edge vectors lie in $\Sigma$, up to orientation.
![The star $\Sigma$[]{data-label="fig:star"}](star.pdf){width="3in"}
The most regular star is the [*tetrahedral star*]{}, given by $$\begin{aligned}
v_1 = {}& (1,1,1)\\
v_2 = {}& (1,-1,-1)\\
v_3 = {}& (-1,1,-1)\\
v_4 = {}& (-1,-1,-1) \ .\end{aligned}$$ These are the vectors that point from the center to the vertices of a regular tetrahedron placed inside a cube.
There is an abundance of polyhedral complexes based on any given star. To see this, recall that the rhombic dodecahedron tiles space and itself can be tiled in two different ways by four congruent rhomboids. Subdividing each rhombic dodecahedron of the space tiling one way or the other into four rhomboids gives an infinitude of polyhedral 3-complexes whose 2-facets form polyhedral 2-complexes based on $\Sigma$. This construction works in fact for any star, not only the tetrahedral star. There are, of course, many more examples, some of which we will encounter below.
We call the six parallelograms $\Pi_{ij}$ (or translational copies of them) that are spanned by $v_i$ and $v_j$, $i<j$, the [*facets*]{}. If a polyhedral complex is based on a star, its facets are necessarily one of these six facets. We also denote the plane spanned by $v_i$ and $v_j$ by $v_i\wedge v_j$. The parallelepiped spanned by three different star vectors $v_i$, $v_j$ and $v_k$ will be denoted by $R_{ijk}$.
Our goal is to deform a polyhedral complex that is based on a star by deforming the star. Here we will face two difficulties: The short time [*existence problem*]{} of deformations, and the long time [*embeddedness problem*]{} that during a deformation the polyhedral complexes might self-intersect.
We address the existence problem by introducing a homological condition:
Let $\eta$ be a path of edges in a polyhedral complex based on a star. We represent this path as a sequence of edges $(\pm v_{i_1},\ldots, \pm v_{i_n})$, where the sign indicates the orientation of the edge. Such a path $\eta$ is closed if $\sum_{k=1}^n \pm v_{i_k} = 0$. We say that $\eta$ is [*generically closed*]{} if each $v_i$ occurs in $\eta$ equally often with positive and negative orientation. This makes the path closed not only for the star $\Sigma$ but for any choice of star. We call a polyhedral complex [*generic*]{} if every closed path is generically closed.
Then we have:
Suppose a finite polyhedral complex $P$ is generically based on a star $\Sigma$. Let $\Sigma_t$ be a continuous deformation of $\Sigma$. Then, for $t$ small enough, there is a continuous family $P_t$ of polyhedral complexes generically based on $\Sigma_t$.
We begin by defining the vertex set of $P_t$ in ${\mathbb{R}}^3$. Let $p$ be a fixed vertex of $P$. We define the corresponding vertex of $P_t$ by letting $p_t=p$. Now let $q$ be an arbitrary vertex of $P$, and $e_1,\ldots e_n$ a path of edges from $p$ to $q$ in $P$. As $P$ is based on $\Sigma$, each edge $e_k$ corresponds to a star vector $\sigma_k=\pm v_k$, where $v_k\in \Sigma$ and the sign depends on the orientation of the edge. Then $q=p+\sum \sigma_k$. We define $q_t = p_t +\sum \sigma_k(t)$, where $\sigma_k(t)=\pm v_k(t)$, $v_k(t)\in \Sigma_t$ and the signs are chosen consistently. This definition is independent of the chosen edge path from $p$ to $q$, as we assumed that $P$ is generic.
The new vertices depend continuously on $t$. By definition of a polyhedral complex, a facet of $P$ is the convex hull of some of its vertices. We use the same vertex sets to define the facets of $P_t$. This will be possible for small $t$ as the intersection condition for the polytopes is an open condition. Here we use the finiteness assumption: It is conceivable that further and further away polytopes need smaller and smaller open neighborhoods in order to remain disjoint.
The second problem about the long-term existence of deformations is more subtle, as it can easily happen that vertices or edges from different polytopes become incident during the deformation (see Remark \[rem:embed\]). There is a special situation that applies to most of our examples where this can be avoided. This occurs when the 2-complex in question is in fact part of the 2-skeleton of an infinite 3-complex without boundary:
\[lem:embed\] Suppose that for $t\in[0,1)$, $P_t$ is a continuous family of finite polyhedral 3-complexes satisfying the following conditions:
- Each $P_t$ is a subcomplex of a polyhedral 3-complex without boundary.
- These 3-complexes without boundary are generically based on a family $\Sigma_t$ of stars.
- The stars are continuous even for $t\in[0,1]$.
Then $P_1$, defined as above, is a polyhedral 3-complex based on $\Sigma_1$.
Note that polyhedral 3-complexes based on stars have parallelepipeds as their 3-polytopes. These are non-degenerate, as we have assumed that no three star vectors are coplanar. This holds for the 3-polytopes of the limit star $\Sigma_1$ as well. In order to see that $P_1$ is a polyhedral 3-complex, we have to show that the intersection of any two polytopes is a facet. Near any point $p$ of $P_1$, the polyhedral 3-complexes $P_t$ are the union of non-degenerate parallelepipeds. These either intersect in facets or are disjoint. The same must hold in for the limit parallelepipeds, as these are non-degenerate. Hence the polytopes of $P_1$ are either disjoint or share a facet as well.
We now come to the construction of bifoldable polyhedral complexes. Our method will be based on the 1-parameter family of stars $\Sigma= \Sigma(r_1,r_2,r_3,r_4,\alpha,\beta)$ given by the vectors
$$\begin{aligned}
v_1 = {}& r_1 (\cos(\alpha), 0, \sin(\alpha))\\
v_2 = {}& r_2 (-\cos(\alpha), 0, \sin(\alpha))\\
v_3 = {}& r_3 (0,\cos(\beta), - \sin(\beta))\\
v_4 = {}& r_4 (0,-\cos(\beta), -\sin(\beta)) \ .\end{aligned}$$
Here the $r_i$ are arbitrary but fixed positive real numbers, and $\alpha, \beta\in(0,\pi/2)$ are angle parameters that will vary dependent on a single parameter. See Figure \[fig:star\]. The tetrahedral star can be recovered (up to similarity) by letting $r_i=1$ and $\alpha=\beta=\arccos(\sqrt{2/3})$.
Usually, we expect a polyhedral complex that is based on any given star to be rigid. Our key observation is that if we disallow the usage of two of the six facets, the polyhedral complex becomes foldable.
More precisely we have:
\[thm:key\] A polyhedral complex $P$ based on $\Sigma$ is biplanar. If it doesn’t have any facets of type $\Pi_{12}$ and $\Pi_{34}$, it can be collapsed into any of the two vertical coordinate planes.
That $P$ is biplanar is trivial, as the star vectors lie in either of the two vertical coordinate planes. As $P$ is based on $\Sigma$, every vertex of it is an integral linear combination of star vectors. We will deform $P$ by changing the angle parameters $\alpha$ and $\beta$, but keeping both the $r_i$ and the combinatorial information of $P$. We need to verify that under this deformation the facets of $P$ remain congruent.
A facet parallel to the plane $v_i\wedge v_j$ of the deformed complex will be congruent to the original facet if and only if the dot products $v_i\cdot v_i$, $v_j\cdot v_j$ and $v_i\cdot v_j$ remains unchanged. The first two dot products remain unchanged as we do not change the length of the star vectors during the deformation. As facets in the planes $v_1\wedge v_2$ and $v_3\wedge v_4$ are forbidden, the third dot product will be equal to $-r_i r_j \sin(\alpha)\sin(\beta)$. Thus in order to have a folding of $P$, we need to deform $\Sigma$ so that $\sin(\alpha)\sin(\beta)$ remains constant. As $\alpha, \beta\in(0,\pi/2)$ we have $\lambda=\sin(\alpha)\sin(\beta)\in(0,1)$. This shows that we can fold $P$ by letting $$\beta = \arcsin(\lambda/\sin(\alpha))$$ for $\alpha \in (\arcsin(\lambda),\pi/2)$. Note that when $\alpha=\pi/2$, $P$ has collapsed into the plane $x=0$. When $\alpha=\arcsin(\lambda)$ we have $\beta=\pi/2$, and $P$ has collapsed into the plane $y=0$.
We call the facets $\Pi_{12}$ and $\Pi_{34}$ [*forbidden facets*]{} and the facets $\Pi_{13}$, $\Pi_{14}$, $\Pi_{23}$ and $\Pi_{24}$ [*admissible facets*]{}.
We call a generic polyhedral complex based on $\Sigma$ that does not contain any forbidden facets a $\Sigma$-complex.
We note that the four admissible facets of $\Sigma$ will be parallelograms with acute angle $\gamma$ given by $$\cos\gamma = \sin\alpha\sin\beta \ .$$ In case that all star vectors have the same length, i.e. that $r_1=r_2=r_3=r_3$, these parallelograms will therefore be congruent rhombi. Observe that necessarily $0<\gamma<\pi/2$, so that squares can never be facets, but all other rhombi can occur.
As explained above, it is very easy to construct polyhedral complexes based on $\Sigma$, and by removing the forbidden facets we obtain a large number of bifoldable $\Sigma$-complexes. In the following sections, we will focus on polyhedral examples.
A polyhedral 2-complex is called a [*polyhedron*]{} if each edge belongs to at most two facets and each vertex is a manifold point, i.e. the intersection of the polyhedron with a small ball centered at that vertex is homeomorphic to a disk. A $\Sigma$-polyhedron is a $\Sigma$-complex that is also a polyhedron.
Note that we made the assumption that the two planes in which the star vectors lie are orthogonal. This is crucial: If they make another angle, Theorem \[thm:key\] fails, because the congruence condition for the admissible facets then forces $\alpha$ and $\beta$ to be constant.
\[rem:embed\] It is possible for $\Sigma$-polyhedra to self-intersect during a deformation, see Figure \[fig:Intersect\]. The only sufficient condition we know that prevents this from happening is given by Lemma \[lem:embed\].
We conclude this section by showing:
\[thm:finite\] There is no finite $\Sigma$-polyhedron without boundary.
![Intersecting side panels[]{data-label="fig:FourWays"}](FourWays.jpg){width="3in"}
We will exploit the fact that $\Sigma$-polyhedra without boundary are (generalized) zonohedra: A [*zone*]{} is a sequence of facets so that consecutive facets of the sequence share a fixed star vector. Each facet defines two different zones. In general, zones can contain infinitely many facets, but for closed zonohedra without boundary, this number is obviously finite, and the zone is topologically a cylinder. We now assume that we have a finite $\Sigma$-polyhedron, rotate it so that one type of facet is horizontal, and choose a lowest facet of that type. We can also assume that the facet in question is of type $\Pi_{13}$.
The zone through this facet with common edge vector $v_1$ consists only of facets of type $\Pi_{13}$ and type $\Pi_{14}$. As there are no facets of type $\Pi_{13}$ below the selected facet, its neighbors are either copies of $\Pi_{13}$, or type $\Pi_{14}$ facets pointing up. A similar statement holds for the other zone with common edge vector $v_3$. Thus near the selected facet, the polyhedron consists of a horizontal finite polygon made up of copies of $\Pi_{13}$ and bounded by facets of type $\Pi_{14}$ and $\Pi_{23}$ all pointing up.
Now consider a single $\Pi_{13}$ bounded by facets of type $\Pi_{14}$ and $\Pi_{23}$ all pointing up, as shown in Figure \[fig:FourWays\] (where “up” means “towards the viewer”). The two $\Pi_{14}$ bound a parallelepiped $R_{134}$ and the two $\Pi_{234}$ another parallelepiped $R_{123}$. The two parallelepipeds share their bottom facets and hence some of their other facets intersect. They cannot be equal, because this would imply $v_2=v_4$. Thus, for at least for one vertex of $\Pi_{13}$, the two adjancent $\Pi_{14}$ and $\Pi_{23}$ facets intersect. Such a vertex must also occur for the bottom polygon made up of copies of $\Pi_{13}$ and bounded by facets of type $\Pi_{14}$ and $\Pi_{23}$ all pointing up. Hence the polyhedron has self-intersections, a contradiction.
Vertex Types {#sec:vertices}
============
In this section, we list the 14 possible vertex types, i.e. the ways in which the four admissible facets can be grouped around a single vertex in a $\Sigma$-polyhedron. We have arrived at this list through an exhaustive and systematic enumeration, which we do not reproduce here.
For each case, we denote by $(a,b)$ the number $a$ of acute angles and the number $o$ of obtuse angles that occur. The sum $a+o$ is the valency of the vertex, and $\kappa = 2\pi -a\gamma -o(\pi-\gamma)$ is the (Gauss) curvature associated to the vertex. The latter quantity is useful to determine the genus $g$ of a closed polyhedral surface, because the Gauss-Bonnet formula states that $$2-2g = \sum_v \kappa_v \ ,$$ where the sum is taken over all vertices $v$.
The only possible valencies that can occur are 4, 6, and 8. We prove without relying on the enumeration below:
For a $\Sigma$-polyhedron, the valency of a vertex is even.
The sequence $\Pi_{ij}$ of facets around a vertex gives rise to a sequence of pairs $(ij)$ of indices of edges. The pairs that can occur belong to the set $${\mathcal{T}}= \{(13), (31), (14), (41), (23), (32), (24), (42) \} \ .$$ We distinguish here for once $(ij)$ from $(ji)$ to indicate the order in which the edges occur when following the facets around the vertex. Two pairs can be adjacent in such a sequence if and only if the last index of the first pair equals the first index of the second pair. From a given sequence, we now eliminate pairs of the form $(ij)(ji)$, thereby reducing the number of pairs by an even number. We are then left with a sequence that periodically contains repetitions of $(13)(32)(24)(41)$, its reversion, or any of its cyclic permutations, because when $(ij)(ji)$ pairs are eliminated, any pair determines is successors uniquely.
The fact that the valency has to be even follows from the genericity of the stars: The boundary of a vertex figure has as edges the vectors $v_i$, and as the sum has to be zero, each $v_i$ has to occur an even number of times.
We begin with valency 4:
Next, there are five possible vertex types with valency 6:
Finally, there are three possible vertex types with valency 8:
We have found that any of these vertex types can be used in an infinite doubly or triply periodic $\Sigma$-polyhedron.
Simple Doubly Periodic Examples {#sec:simple}
===============================
In this section, we review two well known doubly periodic examples. The first is the [*Eggbox pattern*]{} [@alfred1968].
In our context a translational fundamental piece is the saddle consisting of $\Pi_{13}$, $\Pi_{14}$, $\Pi_{23}$ and $\Pi_{24}$, and the translation vectors are $v_1-v_3$ and $v_3-v_4$.
Another example is the [*Miura pattern*]{} [@miura1969].
Here, a translational fundamental piece is also the union of the four admissible facets, but arranged differently, namely as $\Pi_{13}$, $\Pi_{14}$, $\Pi_{23}-v_2$ and $\Pi_{24}-v_2$, where we use addition of a vector to denote translation. The translational periods are given by $v_1+v_3$ and $v_3-v_4$.
A Bifoldable Fractal {#sec:fractal}
====================
Let $H_i$ be the 2-skeleton of the parallelepiped spanned by the four star vectors except $v_i$, with the forbidden facets removed. We call $H_i$ a [*hollowped*]{}. Each $H_i$ is a polyhedral cylinder bounded by four parallelograms. Note that the four corresponding [*solid*]{} parallelepipeds tile a solid dodecahedron, which we call the [*standard dodecahedron*]{}. In the case of the tetrahedral star this is the well-known decomposition of the rhombic dodecahedron into four congruent rhomboids.
We now use the hollowpeds to inductively construct a fractal bifoldable polyhedron.
The generation 0 fractal ${\mathfrak{F}}_0$ is the union $H_1\cup H_2\cup H_2\cup H_2$ of all four hollowpeds, see Figure \[fig:fractal0\].
This shape and its deformability has been discussed before in [@cheung2014], where translational copies of it are used to build a polyhedral complex.
We can think of ${\mathfrak{F}}_0$ as consisting of a roof, a congruent bottom, and a central saddle. Note that this is a polyhedron with octagonal boundary which is up to similarity identical to the boundary of the saddle. This suggests that whenever we cut a smaller saddle out of a larger saddle, we can fill the hole with a scaled copy of ${\mathfrak{F}}_0$.
To make this concrete, we subdivide each parallelogram of ${\mathfrak{F}}_0$ into four smaller parallelograms, and eliminate the four small parallelograms at the center of the saddle. This turns the saddle into a polyhedral annulus, with two saddle shaped boundary octagons. We can fit a scaled copy of ${\mathfrak{F}}_0$ into the inner boundary and arrive at the generation 1 fractal ${\mathfrak{F}}_1$, shown in Figure \[fig:Fractal-1\].
This process can be repeated indefinitely, creating an infinite polyhedron that is the union of polyhedral annuli. By letting the size of these annuli grow exponentially instead of linearly, one can obtain a $\Sigma$-polyhedron that is self-similar by a homothety and that has a singular point at the origin.
This fractalization procedure can be carried out whenever a bifoldable polyhedral complex contains a vertex of saddle type. It is one of the few examples of local surgery that we found possible with $\Sigma$-polyhedra.
The Design of Bifoldable Polyhedral Complexes {#sec:design}
=============================================
To design a $\Sigma$-complex, the following construction method can be used:
Use a small $\Sigma$-polyhedron like the generation 0 fractal as a template that shows how the facets can be attached. Distinguish the four admissible facets by color, both for the template and the polyhedron you are building. Thus each color also represents one of the four planes spanned by $v_1\wedge v_3$, $v_1\wedge v_4$, $v_2\wedge v_3$, and $v_2\wedge v_4$. Note that the facets parallel to $v1 \wedge v2$ and to $v_3 \wedge v_4$ are the forbidden facets.
Then begin with any of the fourteen vertex types from section \[sec:vertices\] and build the vertex figure by placing a colored facet at an existing edge so that it is parallel to the facet of the same color in your template.
Notice that at any given edge of a facet, there are at only four facets that can use this edge, of only two different colors. Thus to extend a $\Sigma$-polyhedron across one edge of a specific facet, there are at most three different ways to do so. Some of these might be impossible as they would cause the added facet to intersect with other parts of the polyhedron, as shown in Figure \[fig:FourWays\].
This procedure will ensure that the polyhedral complex will be built using only star vectors as edges and using only the admissible facets.
Whenever your polyhedron or polyhedral complex closes up (i.e. has vertices are edges meet), you need to test its flexibility by verifying the homological condition. This arises when adding a facet creates a new closed loop of edges, and you need to verify that this loop uses each type of edge as often forward as backward, when followed once around.
We also mention a second way to design $\Sigma$-complexes. You can begin with a tiling of a portion of space by the four parallelepipeds $R_{ijk}$, where $i,j,k$ designate three different numbers from $\{1,2,3,4\}$, and $R_{ijk}$ is spanned by $v_{i_1}$, $v_{i_2}$ and $v_{i_3}$. Then remove from all parallelepipeds the forbidden facets to obtain the four hollowpeds. The result is always a $\Sigma$-complex. By removing more facets where more than two facets meet at the same edge, you can then try to make the $\Sigma$-complex a $\Sigma$-polyhedron.
A $\Sigma$-complex becomes periodic when it becomes invariant under translations in one ore more dimensions. We will present examples of bifoldable periodic $\Sigma$-complexes in the following sections.
The Origami Tube and Weave {#sec:weave}
==========================
In this section, we create a doubly periodic version of the Origami tube. The Origami tube [@miura1969; @filipov2015] is the union of $\Pi_1$ and $\Pi_3$ and invariant under translations by $v_3-v_4$, see Figure \[fig:MiuraTube\].
![The Origami Tube[]{data-label="fig:MiuraTube"}](MiuraTube.jpg){width="4in"}
Using our framework, it is easy to create a doubly periodic pattern of interwoven Origami tubes. To see this, we begin by arranging three facets of type $\Pi_{14}$ into an L-shape, and we do likewise with three facets of type $\Pi_{23}$. These two $L$-shapes are then joined using two single facets $\Pi_{13}$ and $\Pi_{24}$ to create what we called the [*Double L*]{}, an arrangement of eight facets around a single vertex, see Figure \[fig:LL\].
Four of the Double L shapes can then be combined (using mirrored copies) to create a translational fundamental piece of the [*Miura Weave*]{}.
![A translational fundamental piece[]{data-label="fig:MiuraWeave-2"}](MiuraWeave-2.jpg){width="3in"}
Replicating the fundamental piece using translations by $2(v_3-v_4)$ and $2(v_1-v_2)$ creates a doubly periodic polyhedral carpet without boundary. Remarkable, the folding happens in the two translational directions, allowing to compress the entire carpet into a thin strip (Figure \[fig:MiuraWeave-4\]).
Doubly and Triply Periodic Links {#sec:links}
================================
In this section, we describe examples of doubly and triply periodic $\Sigma$-polyhedra. The basic building block consists of two different hollowpeds sharing an admissible facet. For the sake of concreteness and reproducibility, we use the hollowpeds $H_2$ and $H_4$ with common facet $\Pi_{13}$, which we call the [*link*]{}, see Figure \[fig:linksingle\]. Note that the link only utilizes three of the four admissible facets, all except for $\Pi_{24}$.
Let $\Lambda$ be the lattice spanned by $v_1-v_3$, $v_2-v_4$ and $v_1+v_3$. Then translating the link by lattice vectors produces a triply periodic bifoldable polyhedron supported by $\Sigma$, see Figure \[fig:linktriply\]. The lack of the fourth facet further increases the flexibility of this polyhedron, which might be an undesirable feature.
The genus of a triply periodic surface is commonly defined as the genus of the quotient by the maximal group of orientation preserving translations that leave the surface invariant. Note that in this case, the translation by $v_2-v_4$ is orientation reversing, so two links are needed for a fundamental domain. The quotient surface then has eight vertices with valency 6, four of type \[fig:AcuteX\] and four of type \[fig:ObtuseX\], with total curvature $-16\pi$ so that the genus is 5.
There is a more interesting variation of this construction that also employs the fourth facet $\Pi_{24}$: We first combine a link and its mirror image (which also consists of $\Pi_1$ and $\Pi_3$ but translated differently) and connect them with copy of the facet $\Pi_{24}$. We call this piece the [*butterfly*]{}:
Translating the butterfly by $v_1-v_3$ leaves room for a standard dodecahedron between the four links.
The butterfly can be translated further to create the following triply periodic example:
![The triply periodic butterfly[]{data-label="fig:ButterflyTriply"}](ButterflyTriply.jpg){width="4in"}
This surface has also genus 5. One can see this either similar as above, or by looking at the standard dodecahedra which have six facets missing. Two of them appear in the quotient, and the missing facets are used to form six handles, giving genus 5, as one of them is used to connect the dodecahedra.
Finally, in Figure \[fig:LinkDoubly\] is a simple doubly periodic $\Sigma$-polyhedron where links are first translated into chains that are then connected by two “snakes” of parallelograms.
![A doubly periodic $\Sigma$-polyhedron built with links[]{data-label="fig:LinkDoubly"}](LinkDoubly.jpg){width="2.5in"}
The Dos Equis Pattern {#sec:dosequis}
=====================
In this section, we construct a doubly periodic bifoldable $\Sigma$-polyhedron with boundary, called a “layer”. Two such layers can be combined in two different ways by parallel translation. Repeating this allows to construct an infinitude of doubly periodic polyhedral surfaces without boundary. If the combination of layers is carried out periodically, the result will be a triply periodic $\Sigma$-polyhedron. In its simplest form, this surface has the appearance of a triply periodic Miura pattern.
We begin with a vertex of valency eight arranged in an $X$-shape, see Figure \[fig:X\]. On top of it we put a mirror image of the $X$. The result is clearly translation invariant in the vertical direction, see Figure \[ig:dosequis\].
![Two stacked X[]{data-label="ig:dosequis"}](TripleX2){width="2.5in"}
We add a second such double $X$ with the order of the two $X$s switched and obtain a translational fundamental piece of the doubly periodic layer. In Figure \[fig:XX3\], the translational directions are up/down and forward/backward.
We see now that there are two different ways to identify one such layer with a translational copy (left/right) as indicated by the arrows in Figure \[fig:TripleX3carrow\]. One of these translations is orientation preserving, the other reverses orientation. Repeating this allows for an (uncountable) number of possible surfaces.
|
---
abstract: 'The graph isomorphism problem has a long history in mathematics and computer science, with applications in computational chemistry and biology, and it is believed to be neither solvable in polynomial time nor NP-complete. E. Luks proposed in 1982 the best algorithm so far for the solution of this problem, which moreover runs in polynomial time if an upper bound for the degrees of the nodes in the graphs is taken as a constant. Unfortunately, Luks’ algorithm is purely theoretical, very difficult to use in practice, and, in particular, we have not been able to find any implementation of it in the literature. The main goal of this paper is to present an efficient implementation of this algorithm for ternary graphs in the SAGE system, as well as an adaptation to fully resolved rooted phylogenetic networks on a given set of taxa.'
author:
- 'Adrià Alcalá Mena , Francesc Rosselló'
title: 'Ternary graph isomorphism in polynomial time, after Luks'
---
Introduction
============
The *graph isomorphism problem* —the problem of deciding whether two finite graphs are isomorphic or not— has a long history in mathematics and computer science. It is one of the most important decision problems for which the computational complexity is not known yet [@goldberg03; @KST93], and it is believed to be neither in P nor NP-complete. Its uncertain status has led to the definition and study of the class GI of the decision problems that are polynomial-time Turing reducible to it [@Hoff82].
The best current theoretical algorithm for the solution of the graph isomorphism problem is due to Eugene Luks [@Luks81; @Luks83]. His algorithm runs in time $2^{O(\sqrt{n}\log(n))}$, where $n$ is the number of nodes in the graphs, and it relies on the classification of finite simple groups [@FSG]. Without using this major result, whose proof consists of tens of thousands of pages in several hundred journal articles and for which a unified, simplified and revised version is still in progress, Babai and Luks [@BabLuks83] gave an algorithm that runs in $2^{O(\sqrt{n}\log(n)^2)}$ time.
The graph isomorphism problem has many practical applications outside mathematics. Our interest in it stems from its applications in computational biology and bioinformatics. Graphs are ubiquitous in biology as models of different complex systems: molecular structures, phylogenetic trees and networks, metabolical pathways, protein-protein interaction (PPI) networks, gene expression networks, etc. [@AS; @Hub; @Huson; @MV]. In all these fields, the comparison of graphs is an important computational problem. For instance, a protein’s function is closely related to its three-dimensional shape, which at an intermediate level of detail is modeled by a contact graph, and thus the comparison of such contact graphs is a key tool in the prediction of proteins’ function [@GI; @TA]; the comparison of phylogenies is acknowledged to be one of the main problems in phylogenetics [@fel:04; @Huson; @steelpenny:sb93]; and the comparison of metabolic pathways and PPI networks is currently one of the hot tools in the study of evolution [@FS; @KK]. The most basic graph comparison problem in all these contexts is, of course, the detection of equalities. Without being able to decide efficiently whether two such graphs are isomorphic or not, we cannot expect to define metrics, matchings or alignments that can be computed efficiently.
So, it is important to design efficient algorithms to test if two graphs are isomorphic, at least for some special classes of graphs with practical applications. The aforementioned algorithm by Luks [@Luks83] is fixed-parameter tractable, in the sense that it runs in polynomial time if we consider an upper bound $d$ for the degrees of the graphs under comparison as a constant. Such an algorithm, even for *ternary graphs* (graphs with all their nodes of degree at most 3), has relevant applications in computational biology: for instance, in phylogenetics, in the comparison of arbitrary fully resolved rooted phylogenetic networks, and in the comparison of split networks [@Huson]; in both types of networks, all nodes have total degrees at most 3. For these networks, no specific polynomial-time isomorphism test has been devised yet, and the authors working with them simply quote Luks’ result when they need to state that their isomorphism can be decided in polynomial time; see, for instance, [@Huson p. 168] or [@Nak §V].
The problem with Luks’ algorithm is that it is purely theoretical, very difficult to use in practice, and, in particular, we have not been able to find any implementation of it in the literature. The main goal of this paper is, then, to present an efficient implementation of this algorithm in the SAGE system. We also present an implementation of an adaptation of this algorithm that solves the isomorphism problem for fully resolved rooted phylogenetic networks. This paper is based on the first author’s Master Thesis [@MasterThesis], where we refer the reader for more details.
Luks’ algorithm for ternary graphs
==================================
In this section we explain Luks’ algorithm for *ternary* graphs, that is, for connected graphs with all their nodes of degree at most $3$. We shall omit all proofs, which can be found in the original paper by Luks [@Luks83] or, with more detail, in the first author’s Master Thesis [@MasterThesis].
We begin by reducing the graph isomorphism problem to a problem about isomorphisms of groups. Given two ternary graphs $X_1=(V_1,E_1)$ and $X_2=(V_2,E_2)$, and two edges $e_1\in E_1$ and $e_2\in E_2$, consider the ternary graph $X=\mathtt{BuildX}(X_1, X_2, e_1, e_2)$ obtained by splitting $e_1$ and $e_2$ by means of new nodes $v_1$ and $v_2$, respectively, and then connecting these new nodes $v_1$ and $v_2$ by means of a new edge $e$. Then, $X_1$ and $X_2$ are isomorphic if, and only if, given any $e_1\in E_1$, there exist some $e_2\in E_2$ and some automorphism $\sigma$ of $X$ such that $\sigma(v_1)=v_2$, and in particular such that $\sigma(e)=e$. And then, if any such automorphism of $X$ does exist, then any set of generators of $\mathrm{Aut}_e(X)$, the group of automorphisms of $G$ that fixes $e$, will contain some. This provides the following algorithm.
\[IsotoAute\]
So, we are reduced to compute a set of generators of ${\mathrm{Aut}}_e(X)$, for a ternary graph $X$ and any $e\in E(X)$. This group is determined through a natural sequence of successive “approximations” ${\mathrm{Aut}}_e(X_r)$, where each $X_r$ is the subgraph consisting of all nodes and all edges of $X$ which appear in paths of length $\leq r$ containing $e$. More formally, if $e= (a,b)$, let $$X_1=(\{a,b\},\{(a,b)\})$$ and, for every $r\geq 2$, $$\begin{array}{l}
V ( X_r) = \{ y \in V( X)\mid \exists\, x \in V(X_{r-1} ) \mbox{ such that } (x,y) \in E(X) \} \\
E(X_r) = \{ ( x,y) \in E( X) \mid \exists\, x \in V(X_{r-1} ) \mbox{ such that } (x,y) \in E(X) \}
\end{array}$$ There exist natural homomorphisms $$\pi_r : {\mathrm{Aut}}_e(X_{r+1}) \rightarrow {\mathrm{Aut}}_e(X_r)$$ defined by the restriction of the automorphisms. These homomorphisms allow us to construct recursively a generating set for ${\mathrm{Aut}}_e(X_{r+1})$ from one for ${\mathrm{Aut}}_e(X_r)$ by solving, for every $r\geq 1$, the following two problems:
1. Compute a set $\mathcal{K}_{r}$ of generators of $\ker \pi_r$.
2. Compute a set $\mathcal{S}_{r+1}$ of generators of $\pi_r({\mathrm{Aut}}_e(X_{r+1}))$.
Indeed, if $\mathcal{S}_{r+1}'$ is any set of pre-images of $\mathcal{S}_{r+1}$ in ${\mathrm{Aut}}_e(X_{r+1})$, then $\mathcal{K}_{r} \cup \mathcal{S}_{r+1}'$ generates ${\mathrm{Aut}}_e(X_{r+1})$.
So, if we know how to solve Problems I and II, the algorithm to compute a set of generators of ${\mathrm{Aut}}_e(X)$ is the following. In it, and in the sequel, given two elements $a,b$ of a set $Y$, we denote by $(a\; b)\in {\mathrm{Sym}}(Y)$ the transposition $a\leftrightarrow b$.
\[Aute\]
Set now $V_{r+1} = V(X_{r+1}) \setminus V(X_{r})$ and $A_r=\{a\subseteq V_r\mid 0<|a|\leq 3\}$, and consider the mapping $$f: V_{r+1} \rightarrow A_r$$ defined by $f(v)= \{ w \in V(X_r) \mid (v,w) \in E(X) \}$; we call $f(v)$ the *neighbor set* of $v$. The following result solves Problem I.
\[K\] $\mathcal{K}_{r}=\{(u\; v)\in {\mathrm{Sym}}(V_{r+1})\mid u\neq v,\ f(u)=f(v) \}.$
Moreover, the fact that each $\ker \pi_r$ is generated by transpositions implies, by induction, the following result, which will be useful later.
\[Tutte\] For each $r$, ${\mathrm{Aut}}_e(X_r)$ is a $2$-group.
So, our graph isomorphism problem is reduced to solve Problem II. Now, consider the following three subsets of $A_r$: $$\begin{array}{l}
A_r' = \big\{ \{v_1, v_2\} \in A_r\mid ( v_1, v_2 ) \in E(X_{r+1}) \big\}\\
A_{r,1} = \big\{ a \in A_r\mid a = f(v) \mbox{ for some unique } v \in V_{r+1} \big\}\\
A_{r,2} = \big\{ a \in A_r\mid a=f(v_1) = f(v_2 ) \mbox{ for some } v_1 \neq v_2 \big\}
\end{array}$$ We have the following result.
$\pi_r({\mathrm{Aut}}_e(X_{r+1}))$ is the set of those $\sigma \in {\mathrm{Aut}}_e (X_r)$ that stabilize the sets $A_{r,1}$, $A_{r,2}$, and $A_r'$.
Now, set $B_r = V(X_{r-1}) \cup A_r$ and extend the action of ${\mathrm{Aut}}_e(X_r)$ on $V(X_r)$ to $B_r$ in the natural way: if $a \in A_r$, then $\sigma(a) = \{ \sigma(w)\mid w \in a \}$. Color each element of $B_r$ with one of five colors that distinguish, on the one hand, whether or not it is in $A_r'$, and, on the other hand, whether it is in $A_{r,1}$, or $A_{r,2}$, or neither. Only five colors are needed, since $A_r'\cap A_{r,2}=\emptyset$.
By the previous proposition, $\sigma \in \pi_r ( {\mathrm{Aut}}_e(X_{r+1}))$ if and only if $\sigma$ preserves these colors in $A_r$. Therefore, the isomorphism problem for ternary graphs is polynomial-time reducible to the following color automorphism problem (taking $G={\mathrm{Aut}}_e(X_{r})$, $B=A=A_r$, $\sigma=\mathrm{Id}$).
\[Problem4\] Given
- A set of generators for a $2$-subgroup $G$ of the symmetric group ${\mathrm{Sym}}(A)$ of a colored set $A$
- A $G$-stable subset $B\subseteq A$
- A permutation $\sigma \in Sym(A)$
find $C_B( \sigma G)$, where, for every $T\subseteq {\mathrm{Sym}}(A)$ $$C_B(T)= \{ \tau \in T\mid \tau \mbox{ preserves the color of every } b \in B \}$$
Now, the following three lemmas are the basis of Algorithm \[CB(G)\] that solves Problem \[Problem4\] in polynomial-time, and thus it completes the graph isomorphism test we were looking for.
\[LeftCoset\] Let $G$ be a subgroup of ${\mathrm{Sym}}(A)$, $\sigma \in {\mathrm{Sym}}(A)$ and $B$ a $G$-stable subset of $A$ such that $C_B( \sigma G)$ is not empty. Then, $C_B(G)$ is a subgroup of $G$ and $C_B( \sigma G)$ it is a left coset of the subgroup $C_B(G)$.
\[LemaFilter\] Given a set of generators of a subgroup $G$ of a symmetric group, one can compute in polynomial time a set of generators of any subgroup of $G$ that is known to have polynomially bounded index in $G$ and for which a polynomial-time membership test is available.
\[LemaBlockSystem\] Given a set of generators of a subgroup $G$ of a symmetric group and a $G$-orbit $B$, one can determine in polynomial time a minimal $G$-block system in $B$.
\[CB(G)\]
\[th:cost\] Algorithm \[IsotoAute\] solves the ternary graph isomorphism in time $O(n^{10})$, with $n$ the size of the input graphs.
The cost of computing `BuildX` applied to a pair of graphs of size $n$ and a pair of edges is in $O(n)$ time. For every $r$, a set of generators of $\ker \pi_r$ can be computed as explained in Lemma \[K\] in $O(n^2)$ time, a set of generators of $ \pi_r ( {\mathrm{Aut}}_e(X_{r+1}))$ can be computed, by reducing this computation to Problem \[Problem4\] and then using Algorithm \[CB(G)\], in $O(n^8)$ time, and a set of preimages in ${\mathrm{Aut}}_e(X_{r+1})$ of the latter can be computed in $O(n^2)$ time. Since there are at most $O(n)$ indices $r$, we conclude that the cost of computing a set of generators of ${\mathrm{Aut}}_e(X)$ by means of Algorithm \[Aute\] is in $O(n^9)$. Since Algorithm \[IsotoAute\] calls Algorithm \[Aute\] $O(n)$ times, the final cost of Algorithm \[IsotoAute\] is in $O(n^{10})$.
Implementation
==============
Improvements
------------
In our implementation we have improved the efficiency of Luks’ algorithm by:
(a) Reducing the size of the set $A_r$.
(b) Representing the groups by means of smooth generating sequences.
(c) Precomputing the blocks.
(d) Running initial tests, to avoid trivial cases.
(e) Removing, for every $r$, the permutations that do not swap the two “parts” $X_1$ and $X_2$ of $X$.
With these improvements, the cost of the computation decreases to $O(n^4)$. We explain now in some detail the improvements (a)–(c), which were inspired by [@GHL83]. For more details, see [@MasterThesis].
\[Triplets\] 
As far as improvement (a) goes, we have been able to remove the triplets from $B_r$, by replacing each node $v$ with a 3-elements neighbor set by a triangle with nodes at “level” $r+1$ and labeled edges: cf. . In this way, we can replace the graph $X$ by a new graph $\tilde{X}$ with some edges labeled. The automorphisms of $\tilde{X}$ must preserve labels, and therefore the computation of ${\mathrm{Aut}}_e(\tilde{X})$ is the same as that of ${\mathrm{Aut}}_e(X)$, except for the following facts: $B_r$ needs only to include the subsets of $V_r$ of size 1 or 2; we split $A'_r$ into $$\begin{array}{l}
A'_{r,a}= \big\{ \{v_1, v_2\} \in A_r\mid ( v_1, v_2 ) \in E(X_{r+1})\mbox{ is unlabeled} \big\},\\
A'_{r,b}= \big\{ \{v_1, v_2\} \in A_r\mid ( v_1, v_2 ) \in E(X_{r+1})\mbox{ is labeled} \big\};
\end{array}$$ and we modify the set of colors on $B_r$ to distinguish, on the one hand, whether or not an element is in $A'_{r,a}$, or $A'_{r,b}$, or neither, and, on the other hand, whether it is in $A_{r,1}$, or $A_{r,2}$, or neither.
As to (b), we represent 2-groups in a way that makes easier several key computations.
Let $G$ be a 2-group generated by $\{g_1, \ldots, g_k \}$. The sequence $( g_1, \ldots, g_k )$ is a *smooth generating sequence* (*SGS*, for short) for $G$ if $[ \langle g_1, \ldots,
g_i \rangle : \langle g_1, \ldots,
g_{i-1} \rangle] \leq 2$, for $i=1, \ldots, k$.
SGS are preserved by homomorphisms and liftings, and if we know a SGS for a 2-group $G$, then it is easy construct an SGS for a subgroup $H$ of index 2 [@GHL83].
Let $G$ be a 2-group, $( g_1, \ldots, g_k )$ a SGS for it, and $H$ a subgroup of index 2. Let $j = \min \{ i\mid g_i \notin H \}$ and set, for $ i = 1, \ldots, k$, $$\beta_i = \left\{ \begin{array}{lcl} g_i & \mbox{ if } & g_i \in H \\ g_j ^{-1} g_i & \mbox{ if } & g_i \notin H \end{array}
\right.$$ Then $(\beta_1, \ldots, \beta_k )$ is a SGS for $H$, and this sequence is computed in $O(k)$ times the required time of a membership test for $H$.
Let us consider finally improvement (c). As we have seen in the proof of Theorem \[th:cost\], the most expensive part of Luks’ algorithm is the recursive calls performed by Algorithm \[CB(G)\]. The task carried out by this algorithm can be reorganized so as to limit the number of different blocks visited. These blocks form a tree that is precomputed and guides the recursion.
Let $G$ be a 2-group acting on a stable subset $B$ of a colored set $A$. We call a binary tree $T$ a *structure tree* for $B$ with respect to $G$, $T=T(B,G)$, if the set of leaves of $T$ is $B$, and the action of any $\sigma \in G$ on $B$ can be lifted to an automorphism of $T$.
Let $Q$ be a fixed color of $A$ (in our application, it will be the color of the elements of $A_r$ that do not belong to $A_r'\cup A_{r,1}\cup A_{r,2}$). A node $\tilde{B}$ of $T = T(B,G)$ will be called *active* if $\tilde{B} \cap Q^c \neq \emptyset$. An active node $\tilde{B}$ is *facile* if $G$ is intransitive on $\tilde{B}$ and the latter has exactly one active child. Let $\Delta ( \tilde{B} )$ denote the closest non facile descendant of $\tilde{B}$ in $T$.
We can precompute the entire structure tree $T(B,G)$ for a given pair $(B,G)$ using Algorithm \[T(B,G))\] below. With this algorithm, we can construct every structure tree $T(B_r, G_r)$ in $O(n^2)$ time, and we can compute the mapping $\Delta$ and the set of active nodes in $O(n \log n)$ time. Using structure trees to guide the recursion, the cost of computing $C_B(\sigma G)$ decreases to $O(n^4)$.
\[T(B,G))\]
Implementation details
----------------------
We have implemented Luks’ algorithm using the language Python and some specific SAGE libraries for handling graphs and groups of permutations.
Besides the obvious classes and functions necessary to implement the algorithm, we use a new class, called `Node`, for the structure tree. This class has four attributes:
- `Node`, the content of the node.
- `Left`, the left child.
- `Right`, the right child.
- `Parent`, the parent.
When we build a new `Node` without some attributes, they will be empty arrays. This class has the necessary functions to modify its attributes, as well as the following functions:
- `IsLeaf()`, to know whether the node is a leaf.
- `Istransitive()`, to know whether $G$ is transitive on the node.
- `Isactive(Q)`, to know whether the node is active w.r.t. $Q$.
- `Isfacile(Q)`, to know whether the node is facile w.r.t. $Q$.
- `Delta(Q)`, to know the nearest non facile descendant of the node.
To test whether two graphs are isomorphic or not, we can use two different functions, `Isomorphism` and `Isomorphism2`. Both functions answer the question whether the input graphs are isomorphic, but moreover the first function returns the whole group of automorphisms that fix the distinguished edge $e$ of the graph $X=\mathtt{BuildX}(X_1, X_2, e_1, e_2)$, while the second one only returns the subgroup of those automorphisms of $X$ that swap the parts $X_1$ and $X_2$.
Finally, we have adapted Luks’ algorithm to test the isomorphism of fully resolved rooted phylogenetic networks on a given set of taxa. A *rooted phylogenetic network*, on a set of taxa $S$ is a rooted, directed, acyclic graph with its leaves bijectively labeled in $S$. These graphs are used as explicit models of evolutionary histories that, besides mutations, include reticulate evolutionary events like genetic recombinations, lateral gene transfers or hybridizations. An evolutionary network is *fully resolved* , or binary, when, for every node $v$ in it, the ordered pair $(d_{in}(v),d_{out}(v))$ is either $(0,2)$ (the root), $(1,0)$ (the leaves), $(1,2)$ (the *tree nodes*) or $(2,1)$ (the *reticulate nodes*). Two evolutionary networks on $S$ are *isomorphic* when they are isomorphic as directed graphs and the isomorphism preserves the leaves’ labels. For more on phylogenetic networks, see [@Huson].
The main difference between the case of rooted phylogenetic networks and the general case is that, in the former, the isomorphisms map the root to the root, and therefore the graph $X=\mathtt{BuildX}(X_1, X_2)$ can be simply obtained by connecting the roots by an edge $e$. In particular, Algorithm \[IsotoAute\] needs not to call $O(n)$ times Algorithm \[Aute\], but only once, and the resulting cost is then in $O(n^3)$.
To test the isomorphism of fully resolved rooted phylogenetic networks we have defined the function `IsomorphismPhilo`, which has the following parameters:
- `X1`, the first phylogenetic network.
- `X2`, the second phylogenetic network.
- `n`, the number of nodes of X1.
- `dic1`, the dictionary with the labels of the nodes of the first graph.
- `dic2`, the dictionary with the labels of the nodes of the second graph.
- `r1`, the root of the first graph.
- `r2`, the root of the second graph.
`IsomorphismPhilo` accepts networks with *inner taxa* (with internal labeled nodes) as well as *multilabeled networks* (where different nodes can have the same label), although these features are not used (yet) in the phylogenetic networks literature. For the moment, the algorithm only accepts phylogenetic networks created with SAGE’s function `Graph`, but in the near future we plan to adapt it so that it accepts networks described in the format eNewick [@eNewick].
We have performed several tests on our implementation, which we report in the next subsection. In the first and the third tests, we have used SAGE functions to generate random graphs and we have then made these graphs connected and trivalent, by randomly adding or removing edges when it was necessary. In the second test, we have used the function `graphs.DegreeSequence` that, given a sequence of degrees, returns a graph whose nodes have this sequence of degrees, if some exists. This has allowed us to know the probability that the graphs under comparison were isomorphic.
To perform the tests on phylogenetic networks we defined some functions to create random fully resolved rooted phylogenetic networks:
- `createDic(nodes,n)`, which, given a set of nodes, returns the following dictionary: $$d(i) = \left\lbrace \begin{array}{ll} \mbox{``\emph{i}''} & \mbox{ if }i \in nodes \\ \mbox{``\ ''} & \mbox{ if }i \notin nodes \end{array} \right.$$
- `createDic2(nodes,n)`, as above, but the selected nodes are labeled “a” or “b” equiprobably.
- `leaves(X1)`, returns the set of nodes of the network X1.
Then, we have developed an algorithm `RandomTree(n)` that returns a fully resolved rooted phylogenetic network with $n$ nodes. By default, each internal node has a probability of $0.5$ of being hybrid, but it can be changed by simply changing the probability parameter from $0.5$ to the desired probability.
The documentation of the whole module can be found in <http://www.alumnos.unican.es/aam35/sage-epydoc/index.html> and the code in <http://www.alumnos.unican.es/aam35/IsoTriGraph.py>.
Tests
-----
The first two examples show that the code works correctly, and then the tests prove that it runs in a reasonable time, comparable to the speed of the own SAGE algorithm to test isomorphisms of graphs.
\[ImpTestEx1\] Consider the graphs $X_1$ and $X_2$ depicted in Fig. \[Example1\]. These graphs are created in SAGE with:
sage: X3=Graph([(1, 7), (1, 10), (2, 3), (2, 4), (3, 4),(4, 9),
(5,6),(6, 8), (7, 8), (7, 9),(8, 9)])
sage: X4=Graph([(2, 3), (2, 10), (1, 7), (1, 4), (7, 4),(4, 9),
(5, 6),(6, 8), (3, 8), (3, 9),(8, 9)])
![\[Example1\] The graphs $X_1$ (left) and $X_2$ (right) in Example \[ImpTestEx1\]](graphics/Example1X1.png "fig:")![\[Example1\] The graphs $X_1$ (left) and $X_2$ (right) in Example \[ImpTestEx1\]](graphics/Example1X2.png "fig:")
We test whether they are isomorphic:
sage: Isomorphism2(X3,X4,10,Iso=True)
1 --> 2
2 --> 1
3 --> 7
4 --> 4
5 --> 5
6 --> 6
7 --> 3
8 --> 8
9 --> 9
10 --> 10
True
And, indeed, it is obvious that this is an isomorphism between $X_1$ and $X_2$.
\[ImpTestEx2\] Consider now the graphs $X_1$ and $X_2$ depicted in Fig. \[Example2\].
sage: X1=Graph([(1, 7), (1, 8), (1, 10), (2, 3), (3, 6),
(4, 5), (5, 6), (6, 10), (7,9), (7, 10), (8, 9)])
sage: X2=Graph([(1, 7), (1, 9), (2, 3), (2, 5), (2, 10),
(4, 5), (4, 6), (4, 10), (6,8), (7, 8), (7, 10)])
sage: Isomorphism(X1,X2,10)
False
And, indeed, they are clearly non isomorphic.
![\[Example2\] The graphs $X_1$ (left) and $X_2$ (right) in Example \[ImpTestEx2\]](graphics/Example2X1.png "fig:")![\[Example2\] The graphs $X_1$ (left) and $X_2$ (right) in Example \[ImpTestEx2\]](graphics/Example2X2.png "fig:")
Fig. \[Graphic1\] shows the time needed by `Isomorphism2` to test whether two random graphs are isomorphic, as a function of the numbers of nodes in the graphs. The times are so small because two random graphs with the same number of nodes have probably different numbers of edges, a property that our program checks before proceeding with Luks’ algorithm. Although in most cases the algorithm detected non-isomorphism by trivial reasons, the algorithm is also relatively efficient in non trivial cases.
![Average time (in seconds) for different numbers of nodes when testing the isomorphism of random graphs[]{data-label="Graphic1"}](graphics/Random0500.png)
In this test we fix the degrees of $n-1$ nodes in the graphs, and the degree of the last node is chosen at random. In this way we guarantee that the probability of two graphs being isomorphic is $1/3$. In this case, our algorithm also runs in reasonable time: see Figs. \[Graphic2\] and \[Graphic3\].
![Average time (in seconds) for different numbers of nodes when testing the isomorphism of semirandom graphs[]{data-label="Graphic2"}](graphics/PseudoRandom0500.png)
![The band under 2 seconds of the graphic in Fig. \[Graphic2\][]{data-label="Graphic3"}](graphics/PseudoRandom0500Ymax2.png)
Fig. \[Graphic4\] shows the times (red dots) needed by the algorithm to detect the isomorphism between pairs of isomorphic graphs with $n$ nodes, and it compares this time with the functions $(n/10)^4$, $(n/10)^3$, $(n/10)^2\log (n/10)$, and $(n/10)^2$.
![Comparison between the algorithm and the functions $(n/10)^4, (n/10)^3, (n/10)^2 \log (n/10), (n/10)^2$[]{data-label="Graphic4"}](graphics/IsoGraphs.png)
Our last test deals with rooted phylogenetic networks.
Fig. \[Graphic5\] displays the relation running time-number of nodes for our algorithm when applied to random fully resolved rooted phylogenetic networks on the same sets of taxa, and Fig. \[Graphic6\] shows this relation for phylogenetic networks that are isomorphic as undirected graphs but need not be isomorphic as phylogenetic networks.
![Average time (in seconds) for different numbers of nodes when testing the isomorphism of fully resolved rooted phylogenetic networks[]{data-label="Graphic5"}](graphics/Phylo1.png)
![Average time (in seconds) for different numbers of nodes when testing the isomorphism of fully resolved rooted phylogenetic networks that are isomorphic as graphs[]{data-label="Graphic6"}](graphics/Phylo2.png)
Conclusions
===========
In this paper we have presented our implementation in SAGE of Luks’ polynomial-time algorithm for testing the isomorphism of ternary graphs. This algorithm has interesting applications in phylogenetics, as it allows, for instance, to detect whether two fully resolved rooted phylogenetic networks are isomorphic. Therefore, we have adapted and implemented Luks’ algorithm for this type of graphs. Our adaptation has been, except for one point, a direct translation of Luks’ algorithm. Fully resolved rooted phylogenetic networks have specific characteristics that could be used to improve the algorithm to make it still more efficient in this specific application. It is in our research agenda to develop such an adaptation, and we hope to present it elsewhere.
**Acknowledgements.** This research has been partially supported by the Spanish government and the UE FEDER program, through project MTM2009-07165. We thank Prof. T. Recio for encouraging us to write this report, and for his comments on a first version of it.
[99]{}
T. Aittokallio, B. Schwikowski. “Graph-based methods for analysing networks in cell biology.” *Briefings in Bioinformatics* 7 (2006), 243–255.
L. Babai, E. M. Luks. “Canonical labeling of graphs.” *15th STOC* (ACM, 1983), 171–183.
G. Cardona, M. Llabrés, F. Rosselló, G. Valiente. “On Nakhleh’s metric for reduced phylogenetic networks”. IEEE/ACM Transactions on Computational Biology and Bioinformatics 6 (2009), 629–638
G. Cardona, F. Rosselló, G. Valiente. “Extended Newick: it is time for a standard representation of phylogenetic networks”. *BMC Bioinformatics* 9:532 (2008).
D. Gorenstein, R. Lyons, R. Solomon. *The classification of the finite simple groups*. Mathematical Surveys and Monographs 40 (AMS, 1994).
J. Felsenstein. *Inferring Phylogenies*. Sinauer Associates Inc. (2004).
C. Forst, K. Schulten. “Phylogenetic analysis of metabolic pathways.” *J. Mol. Evol.* 52 (2001), 471–489.
M. L. Furst, J. E. Hopcroft, E. M. Luks. “Polynomial-time algorithms for permutation groups.” *Proceedings 21st FOCS* (IEEE Computer Society, 1980), 36–41.
Z. Galil, C. M. Hoffmann, E. M. Luks, C.-P. Schnorr, A. Weber. “An O(n$^3\log$(n)) deterministic and an O(n$^3$) Las Vegas isomorphism test for trivalent graphs.” *J. ACM* 34 (1987), 513–53.
M. Goldberg. “The graph isomorphism problem.” In: *Handbook of Graph Theory* (J. L. Gross and J. Yellen eds.) (CRC Press, 2003) 68–78.
C. Guerra, S. Istrail (eds). *Mathematical Methods for Protein Structure Analysis and Design*. Lecture Notes in Computer Science 2666 (Springer, 2003)
C. M. Hoffmann, *Group-theoretic algorithms and graph isomorphism*. Lecture Notes in Computer Science 136, Springer-Verlag (1982).
W. Huber, V. J. Carey, L. Long, S. Falcon, R. Gentleman. “Graphs in molecular biology.” *BMC Bioinformatics* 8(Suppl 6):S8 (2007).
D. H. Huson, R. Rupp, C. Scornavacca. *Phylogenetic Networks: Concepts, Algorithms and Applications*. Cambridge University Press (2011).
J. Köbler, U. Schöning, J. Torán. *The graph isomorphism problem: Its structural complexity*. Birkhäuser (1993).
M. Koyorturk, Y. Kim, U. Topkara, S. Subramaniam, W. Szpankowski, A. Grama. “Pairwise alignment of protein interaction networks.” *J. Comp. Biol.* 13 (2006), 182–199.
E. M. Luks. “Isomorphism of graphs of bounded valence can be tested in polynomial time.” *Proceedings 21th FOCS* (IEEE Computer Society, 1980), 42–49.
E. M. Luks. “Isomorphism of graphs of bounded valence can be tested in polynomial time.” *Journal of Computer and System Sciences* 25 (1982), 42–65.
O. Mason, M. Verwoerd. “Graph theory and networks in biology.” *IET In Systems Biology*, 1 (2007), 89–119
D. A. Spielman. “Faster isomorphism testing of strongly regular graphs.” *Proceedings 28th STOC* (ACM, 1996), 576–584.
M. A. Steel, D. Penny, “Distributions of tree comparison metrics—some new results.” *Systematic Biology* 42 (1993), 126–141.
W. Taylor, A. Aszódi. *Protein Geometry: Classification, Topology and Symmetry*. Inst. Physics Publ. (2005).
|
---
abstract: 'Based on the finite $U$ slave boson method, we have investigated the effect of Rashba spin-orbit(SO) coupling on the persistent charge and spin currents in mesoscopic ring with an Anderson impurity. It is shown that the Kondo effect will decrease the magnitude of the persistent charge and spin currents in this side-coupled Anderson impurity case. In the presence of SO coupling, the persistent currents change drastically and oscillate with the strength of SO coupling. The SO coupling will suppress the Kondo effect and restore the abrupt jumps of the persistent currents. It is also found that a persistent spin current circulating the ring can exist even without the charge current in this system.'
author:
- 'Guo-Hui Ding and Bing Dong'
title: 'Spin-orbit coupling effect on the persistent currents in mesoscopic ring with an Anderson impurity'
---
introduction
============
Recently the spin-orbit(SO) interaction in semiconductor mesoscopic system has attracted a lot of interest[@1]. Due to the coupling of electron orbital motion with the spin degree of freedom, it is possible to manipulate and control the electron spin in SO coupling system by applying an external electrical field or a gate voltage, and it is believed that the SO effect will play an important role in the future spintronic application. Actually, various interesting effects resulting from SO coupling have already been predicted, such as the Datta-Das spin field-effect transistor based on Rashba SO interaction[@2] and the intrinsic spin Hall effect[@3].
In this paper we shall focus our attention on the persistent charge current and spin current in mesoscopic semiconductor ring with SO interaction. The existence of a persistent charge current in a mesoscopic ring threaded by a magnetic flux has been predicted decades ago[@4], and has been extensively studied in theory[@5; @6; @7; @8; @9] and also observed in various experiments[@10; @11; @12]. The reason that a persistent charge current exists may be interpreted as that the magnetic flux enclosed by the ring introduces an asymmetry between electrons with clockwise and anticlockwise momentum, thus leads to a thermodynamic state with a charge current without dissipation. For a mesoscopic ring with a texture like inhomogeneous magnetic field, D. Loss et al.[@13] predicted that besides the charge current there are also a persistent spin current. The origin of the persistent spin current can be related to the Berry phase acquired when the electron spin precesses during its orbital motion. The persistent spin current has also been studied in semiconductor system with Rashba SO coupling term[@14; @15; @16]. Recently it is shown that a semiconductor ring with SO coupling can sustain a persistent spin current even in the absence of external magnetic flux[@17].
For the system of a mesoscopic ring with a magnetic impurity, the persistent charge current has been investigated in the context of a mesoscopic ring coupled with a quantum dot[@18; @19; @20; @21; @22; @23; @24], where the quantum dot acts as an impurity level and will introduce charge or spin fluctuations to the electrons in the ring. The Kondo effect arising from a localized electron spin interacting with a band of electrons will be essential in the charge transport in the ring. But to our knowledge in these systems the SO effect hasn’t been considered. It might be expected that the interplay between the Kondo effect and the SO coupling in the ring can give new features in the persistent currents. In this paper we shall address this problem and investigate the SO effect on persistent charge and spin currents in the ring system with an Anderson impurity. The Anderson impurity can act as a magnetic impurity when the impurity level is in single electron occupied state and as well as a barrier potential in empty occupied regime.
The outline of this paper is as follows. In section II we introduce the model Hamiltonian of the system and also the method of calculation by finite-U slave boson approach[@25; @26; @27; @28]. In section III the results of persistent charge current and spin current are presented and discussed. In Section IV we give the summary.
Mesoscopic ring with an Anderson impurity
=========================================
The electrons in a closed ring with SO coupling of Rashba term can be described by following Hamiltonian in the polar coordinates[@14; @29] $$H_{ring}=\Delta(-i{\partial\over{\partial\varphi}}+{\Phi\over\Phi_0})^2
+{\alpha_R\over
2}[(\sigma_x\cos\varphi+\sigma_y\sin\varphi)(-i{\partial\over{\partial\varphi}}+{\Phi\over\Phi_0})+h.c.]\;,$$ where $\Delta=\hbar^2/(2m_ea^2)$, $a$ is the radius of the ring. $\alpha_R$ will characterize the strength of Rashba SO interaction. $\Phi$ is the external magnetic flux enclosed by the ring, and $\Phi_0=2\pi\hbar c/e$ is the flux quantum.
We can write the above Hamiltonian in terms of creation and annihilation operators of electrons in the momentum space, $$H_{ring}=\sum_{m,\sigma}\epsilon_m
c^\dagger_{m\sigma}c_{m\sigma}+1/2\sum_m[t_m(c^\dagger_{m+1\downarrow}c_{m\uparrow}
+c^\dagger_{m-1\uparrow}c_{m\downarrow})+h.c.]\;,$$ where $\epsilon_m=\Delta(m+\phi)^2$, $t_m=\alpha_R
(m+\phi)$,($m=0,\pm 1,\cdots,\pm M$) with $\phi=\Phi/\Phi_0$. One can see that the SO interaction causes the $m$ mode electrons coupled with $m+1$ and $m-1$ mode electrons and spin-flip process. We consider the system with a side-coupled impurity which can be described by the Anderson impurity model, $$H_d=\sum_\sigma\epsilon_d d^\dagger_\sigma
d_\sigma+Un_{d\uparrow}n_{d\downarrow}\;.$$ The tunneling between the impurity level and the ring are given by $$H_{d-ring}=t_D\sum_{m\sigma}(d^\dagger_\sigma c_{m\sigma}+h.c)\;.$$ Then the total Hamiltonian for the system should be $$H=H_{ring}+H_d+H_{d-ring}\;.$$
In order to treat the strong on-site Coulomb interaction in the impurity level. we adopt the finite-U slave boson approach[@25; @26]. A set of auxiliary bosons $e, p_{\sigma}, d$ is introduced for the impurity level, which act as projection operators onto the empty, singly occupied(with spin up and spin down), and doubly occupied electron states on the impurity, respectively. Then the fermion operators $d_{\sigma}$ are replaced by $d_{\sigma}\rightarrow
f_{\sigma}z_{\sigma} $, with $z_{\sigma}=e^\dagger
p_{\sigma}+p^\dagger_{\bar\sigma}d$. In order to eliminate un-physical states, the following constraint conditions are imposed :$\sum_{\sigma} p^\dagger_{\sigma}p_{\sigma}+e^\dagger
e+d^\dagger d=1$, and $f^\dagger_{\sigma}f_{\sigma}=p^\dagger_{\sigma}p_{\sigma}+d^\dagger
d(\sigma=\uparrow, \downarrow)$. Therefore, the Hamiltonian can be rewritten as the following effective Hamiltonian in terms of the auxiliary boson $e, p_{\sigma}, d$ and the pesudo-fermion operators $f_{\sigma}$: $$\begin{aligned}
H_{eff}&=&\sum_{m,\sigma}\epsilon_m
c^\dagger_{m\sigma}c_{m\sigma}+1/2\sum_m[t_m(c^\dagger_{m+1\downarrow}c_{m\uparrow}
+c^\dagger_{m-1\uparrow}c_{m\downarrow})+h.c.]
\nonumber\\
&+&\sum_{\sigma}\epsilon_d f^\dagger_{\sigma}f_{\sigma}+
Ud^\dagger d
\nonumber\\
& +&\sum_{m,\sigma } (t_Dz^\dagger_{\sigma}
f^\dagger_{\sigma}c_{m\sigma}+h.c.) + \lambda^{(1)}(\sum_{\sigma}
p^\dagger_{\sigma}p_{\sigma}+e^\dagger e+d^\dagger d-1)
\nonumber\\
&+&\sum_{\sigma}\lambda^{(2)}_{\sigma}(f^\dagger_{\sigma}f_{\sigma}-p^\dagger_{\sigma}p_{\sigma}-d^\dagger
d )\;,\end{aligned}$$ where the constraints are incorporated by the Lagrange multipliers $\lambda^{(1)}$ and $\lambda^{(2)}_{\sigma}$. The first constraint can be interpreted as a completeness relation of the Hilbert space on the impurity level, and the second one equates the two ways of counting the fermion occupancy for a given spin. In the framework of the finite-U slave boson mean field theory[@25; @26], the slave boson operators $e, p_{\sigma}, d $ and the parameter $z_\sigma$ are replaced by real c numbers. Thus the effective Hamiltonian is given as $$\begin{aligned}
H^{MF}_{eff}&=&\sum_{m,\sigma}\epsilon_m
c^\dagger_{m\sigma}c_{m\sigma}+1/2\sum_m[t_m(c^\dagger_{m+1\downarrow}c_{m\uparrow}
+c^\dagger_{m-1\uparrow}c_{m\downarrow})+h.c.]
\nonumber\\
&+&\sum_{\sigma}{\tilde\epsilon_{d\sigma}}f^\dagger_{\sigma}f_{\sigma}
+\sum_{m\sigma } ({\tilde t_{D\sigma}}
f^\dagger_{\sigma}c_{m\sigma}+h.c.)+E_g\;,\end{aligned}$$ where ${\tilde t_{D\sigma}}=t_Dz_\sigma$ represents the renormalized tunnel coupling between the impurity and the mesoscopic ring. $z_\sigma$ can be regarded as the wave function renormalization factor. ${\tilde\epsilon_{d\sigma}}=\epsilon_d+\lambda^{(2)}_{\sigma}$ is the renormalized impurity level and $E_g= \lambda^{(1)}(
\sum_{\sigma}
p_{\sigma}^2+e^2+d^2-1)-\sum_{\sigma}\lambda^{(2)}_\sigma(p_{\sigma}^2+d^2)+Ud^2$ is an energy constant.
In this mean field approximation the Hamiltonian is essentially that of a non-interacting system, hence the single particle energy levels can be calculated by numerical diagonalization of the Hamiltonian matrix. Then the ground state of this system $|\psi_0>$ can be constructed by adding electrons to the lowest unoccupied energy levels consecutively . By minimizing the ground state energy with respect to the variational parameters a set of self-consistent equations can be obtained as in Ref.\[27,28\], and they can be applied to determine the variational parameters in the effective Hamiltonian.
the persistent charge current and spin current
===============================================
In this section we will present the results of our calculation of the persistent charge current and spin current circulating the mesoscopic ring. Since there is still some controversial in the literature for the definition of the spin current operator in the ring system with SO coupling term[@30]. We give both the formula of charge and spin currents used in this paper explicitly. It is easy to obtain that the $\varphi$ component of electron velocity operator in this SO coupled ring is $$v^\varphi={a\over\hbar}[2\Delta(-i{\partial\over{\partial\varphi}}+\phi)
+\alpha_R(\sigma_x\cos\varphi+\sigma_y\sin\varphi)]\;.$$ Thereby the charge current operator is define as $\hat I=-e
v^\varphi$, and in terms of creation and annihilation operator it can be written as $$\hat I=-{e a\over \hbar}[2\Delta\sum_{m,\sigma}\
c^\dagger_{m\sigma}c_{m\sigma}(m+\phi)+\alpha_R\sum_m(c^\dagger_{m+1\downarrow}c_{m\uparrow}
+c^\dagger_{m-1\uparrow}c_{m\downarrow})]\;.$$ At zero temperature, the persistent charge current is given by the expectation value of the above charge current operator in the ground state, $I={1\over{2\pi a}}<\psi_0|\hat I|\psi_0>$, and it can also be calculated from the expression $$I=-c{\partial E_{gs}\over{\partial\Phi}}=-{e\over
h}<\psi_0|{\partial H\over{\partial\phi}}|\psi_0>\;,$$ where $E_{gs}$ is the ground state energy.
In Fig.1 the persistent charge current vs. the enclosed magnetic flux is plotted for a set of values for the SO coupling strength. Here we have taken the model parameters $\Delta=0.01, t_D=0.3,
U=2.0$ and the total number of electrons $N$ is around $100$. In this case one can obtain the Fermi energy of the system $E_F=6.25$ and the level spacing $\delta=0.5$ around the Fermi surface. We consider the energy level of the Anderson impurity is well below the Fermi energy( with $\epsilon_d-E_F=-1.0$), therefore the Anderson impurity is in the Kondo regime. One can see in Fig.1 that the characteristic features of persistent charge current depends on the parity of the total number of electrons($N$), and can be distinguished by two cases with $N$ odd and $N$ even. This is attributed to the different occupation patterns of the highest occupied single particle energy level in the mean field effective Hamiltonian. The persistent charge current for the system with $N+2$ electrons is different from that with $N$ electrons by a $\pi$ phase shift $I^{N+2}(\phi)=I^{N}(\phi+\pi)$. In case (I) where the electron number is odd($N=4n-1$ and $N=4n+1$), one electron is almost localized on the impurity level and forming a singlet with electron cloud in the conducting ring. This phenomena leads to the well known Kondo effect. Fig.1 shows that the Kondo effect decreases the magnitude of the persistent charge current, and also makes its curve shape resemble sinusoidal. In the presence of finite SO coupling($\alpha_R<\Delta$), the spin-up and spin-down electrons are coupled and it causes the splitting of the twofold degenerated energy levels in the effective Hamiltonian. It turns out that the Kondo effect is suppressed and the abrupt jumps of the persistent charge current with similarity to that of ideal ring case appears. It is explained in Ref.\[14\] that the jumps of the persistent charge current in the case of odd number of electrons are due to a crossing of levels with opposite spin. In case (II) where $N$ is even ($N=4n$ and $N=4n+2$), The Kondo effect is manifested that the magnitude of persistent charge current is significantly suppressed compared with ideal ring case and the rounding of the jumps of persistent charge current due to the level crossing. In the presence of finite SO coupling, the persistent charge current decreases with increasing the SO coupling strength when $\alpha_R<\Delta$.
Fig.2 displays the persistent charge current as a function of the SO coupling strength $\alpha_R$ at different enclosed magnetic flux. The persistent charge current exhibits oscillations with increasing the value of $\alpha_R$ for both the systems with even or odd number of electrons. Therefore by tuning the SO coupling strength, the magnetic response of this system can change from paramagnetic to diamagnetic and vice versa. It indicates that SO coupling can play a important role in electron transport in this mesoscopic ring. The curve of the persistent charge current for odd number of electrons shows discontinuity in its derivation, this can be attributed the level crossing in the energy spectrum by changing $\alpha_R$. It is also noted that the position of this discontinuity for odd $N$ also corresponds to the peak or valley in even N case.
Since the electron has the spin degree of freedom as well as the charge, the electron motion in the ring may give rise to a spin current besides the charge current. Now we turn to study the persistent spin current in the ground state. The spin current operator is defined by $\hat J_v=(v^\varphi\sigma_v+\sigma_v
v^\varphi)/2$, which can be written explicitly as $$\hat
J_v={a\over\hbar}\{2\Delta(-i{\partial\over{\partial\varphi}}+\phi)\sigma_v+{\alpha_R\over
2} [ (\sigma_x\cos\varphi+\sigma_y\sin\varphi)\sigma_v+h.c.]\}\;,$$
Therefore the three component of spin current operator in terms of creation and annihilation operators are given by $$\hat J_z={a\over\hbar}[2\Delta\sum_{m}\
(c^\dagger_{m\uparrow}c_{m\uparrow}-c^\dagger_{m\downarrow}c_{m\downarrow})(m+\phi)]\;,$$
$$\hat
J_x={a\over\hbar}[2\Delta\sum_{m}(c^\dagger_{m\uparrow}c_{m\downarrow}
+c^\dagger_{m\downarrow}c_{m\uparrow})(m+\phi) +{\alpha_R\over
2}\sum_{m,\sigma}(c^\dagger_{m+1\sigma}+c^\dagger_{m-1\sigma})c_{m\sigma}]\;,$$
$$\hat J_y={a\over\hbar}[-2i\Delta\sum_{m}(
c^\dagger_{m\uparrow}c_{m\downarrow}-
c^\dagger_{m\downarrow}c_{m\uparrow})(m+\phi) -i{\alpha_R\over
2}\sum_{m,\sigma}(c^\dagger_{m+1\sigma}-c^\dagger_{m-1\sigma})c_{m\sigma}]\;,$$
The expectation value of the spin current $J_v={1\over {2\pi
a}}<\psi_0|\hat J_v|\psi_0>$.
In our calculation we find that only the $z$ component of the spin current is nonzero in the ground state. Fig.3 shows the persistent spin current $J_z$ vs. magnetic flux at different SO coupling strength. The persistent spin current is a periodic function of the magnetic flux $\phi$, which has the even parity symmetry $J_z(-\phi)=J_z(\phi)$ and also an additional symmetry $J_z(\phi)=J_z(\pi-\phi)$. It is noted that the persistent spin current has quite different dependence behaviors on magnetic flux compared with the persistent charge current in Fig.1. In the presence of finite SO coupling, the persistent spin current is nonzero both for the systems with odd $N$ and even $N$ at zero magnetic flux, it indicates that a persistent spin current can be induced solely by SO interaction without accompany a charge current. This phenomena is also shown in Ref.\[17\] where a SO coupling/normal hybrid ring was considered.
In Fig.4 the persistent spin current $J_z$ as a function of SO coupling strength is plotted. In the absence of SO coupling $\alpha_R=0$, the persistent spin current is exactly zero for both even and odd number electron system. In the presence of SO coupling, The persistent spin current becomes nonzero and shows oscillations with increasing $\alpha_R$. It can change from positive to negative values or vice versa by tuning the SO coupling strength. The sign of the persistent spin current also shows dependence on the enclosed magnetic flux. For the system with odd $N$, there is abrupt jumps in the curve of persistent spin current at certain value of $\alpha_R$, the reason for the jump is the same as that in the charge current, and is due to the level crossing in the energy spectrum. It is noted that the position of the jump coincides with that of the persistent charge current. This kind of characteristic feature of the persistent currents might provide a useful way to detect the SO coupling effects in semiconductor ring system.
conclusions
===========
In summary, we have investigated the Rashba SO coupling effect on the persistent charge current and spin current in a mesoscopic ring with an Anderson impurity. The Anderson impurity leads to the Kondo effect and decreases the amplitude of the persistent charge and spin current in the ring. In the semiconducting ring with SO interaction, the persistent charge current changes significantly by tuning the SO coupling strength, e.g. from the paramagnetic to diamagnetic current. Besides the persistent charge current, there also exists a persistent spin current, which also oscillates with the SO coupling strength. It is shown that at zero magnetic flux a persistent spin current can exist even without the charge current. Since the persistent spin current can generate an electric field[@31], one might expect that experiments on semiconductor ring with Rashba SO coupling can detect the persistent spin current.
[ This project is supported by the National Natural Science Foundation of China, the Shanghai Pujiang Program, and Program for New Century Excellent Talents in University (NCET). ]{}
[ ]{} I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. [**76**]{}, 323 (2004). S. Datta and B. Das, Appl. Phys. Lett. [**56**]{}, 665(1990). S. Murakami, N. Nagaosa, and S. C. Zhang, Science [**301**]{}, 1348 (2003); J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T.Jungwirth, and A. H. MacDonald, Phys. Rev. Lett., [**92**]{}, 126603(2004). M. B$\ddot u$ttiker, Y. Imry, and R. Landauer, Phys. Lett.[**96A**]{}, 365 (1983). H. F. Cheung, Y. Gefen, E. K. Riedel, and W. H. Shih, Phys. Rev. B [**37**]{}, 6050 (1988). D. Loss and P. Goldbart, Phys. Rev. B [**43**]{}, 13762 (1991). G. Montambaux, H. Bouchiat, D. Sigeti, and R. Friesner, Phys. Rev. B [**42**]{}, 7647 (1990). Y. Meir, Y. Gefen, and O. Entin-Wohlman, Phys. Rev. Lett. [**63**]{}, 798 (1989). B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. [**66**]{}, 88(1991). L. P. L$\acute{e}$vy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. [**64**]{}, 2074 (1990). V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gallagher, and A. Kleinsasser, Phys. Rev. Lett. [**67**]{}, 3578 (1991). D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. [**70**]{}, 2020 (1993). D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev. Lett., [**65**]{}, 1655 (1990); D. Loss and P. M. Goldbart, Phys. Rev. B [**45**]{}, 13544 (1992). J. Splettstoesser, M. Governale, and U. Z$\ddot u$licke, Phys. Rev. B [**68**]{}, 165341 (2003). J. S. Shen and K. Chang, Phys. Rev. B [**74**]{},235315(2006). R. Citro and F. Romeo, Phys. Rev. B[**75**]{},073306(2007). Q. F. Sun, X. C. Xie, and J. Wang, cond-mat/0605748. M. B$\ddot u$ttiker and C. A. Stafford, Phys. Rev. Lett. [**76**]{}, 495 (1996). V. Ferrari, G. Chiappe, E. V. Anda, and M. A. Davidovich, Phys. Rev. Lett. [**82**]{}, 5088 (1999). I. Affleck and P. Simon, Phys. Rev. Lett.[**86**]{}, 2854 (2001); Phys. Rev. B[**64**]{}, 085308 (2001). K. Kang and S. C. Shin, Phys. Rev. Lett. [**85**]{}, 5619 (2000). S. Y. Cho, K. Kang, C. K. Kim, and C. M. Ryu, Phys. Rev. B [**64**]{}, 033314 (2001) H. P. Eckle, H. Johannesson, and C. A. Stafford, Phys. Rev. Lett. [**87**]{}, 016602 (2001). H. Hu, G. M. Zhang, and L. Yu, Phys. Rev. Lett. [**86**]{}, 5558 (2001). G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. [**57**]{}, 1362 (1986). V. Dorin and P. Schlottmann, Phys. Rev. B [**47**]{}, 5095 (1993). B. Dong and X. L. Lei, Phys. Rev. B [**63**]{}, 235306 (2001); Phys. Rev. B [**65**]{}, R241304 (2002). G. H. Ding and B. Dong, Phys. Rev. B [**67**]{}, 195327 (2003). F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, Phys. Rev. B [**66**]{}, 033107 (2002). Q. F. Sun and X. C. Xie, Phys. Rev. B [**72**]{}, 245305(2005). F. Meir and D. Loss, Phys. Rev. Lett. [**90**]{}, 167204 (2003).
{width="0.8\columnwidth"}
{width="0.8\columnwidth"}
{width="0.8\columnwidth"}
{width="0.8\columnwidth"}
|
---
abstract: |
We present BeppoSAX X-ray spectra of the remnant of the supernova of AD 1006, which cover a broad spectral range of 0.1-10 keV. In our analysis we concentrate on the thermal emission from the central region of the remnant. For this purpose we fitted the spectra of the whole remnant using a new version of the spectral program SPEX, which takes into account the interdependence of spectra of different regions as the result of the point spread function of the instruments. This is in particular important for separating the synchrotron emission, coming from the bright rims of the remnant, from the thermal emission from the central regions.
The thermal emission appears best fitted with a combination of two spectral components with electron temperatures of $\sim$0.7 keV and $\simgt$3 keV. The plasma in SN 1006 appears to be very far out of ionization equilibrium ($\sim 2\ 10^{15}$ m$^{-3}$s), which means that the L-shell emission of magnesium, silicon and sulphur contribute significantly to the flux below 0.5 keV. We also note the presence of Fe K line emission at $6.3\pm0.2$ keV, in agreement with an origin from inner shell ionizations and excitations. We confirm that the abundances in SN 1006 are clearly non-solar. In particular silicon is very abundant. We did not find substantial variations in plasma properties over the face of the remnant, although the hotter component seems more dominant in the Northern half of the remnant.
author:
- Jacco Vink
- 'Jelle S. Kaastra'
- 'Johan A. M. Bleeker'
- 'Andrea Preite-Martinez'
date: 'Received / Accepted '
title: 'The BeppoSAX X-ray spectrum of the remnant of SN 1006'
---
Introduction
============
The historical supernova SN 1006 was probably a Type Ia supernova (Clark & Stephenson [@Clark], Schaefer [@Schaefer]). Its remnant has now a diameter of 30’ and in recent years three interesting findings were reported:
- The ASCA X-ray spectrum indicates that the emission above $\sim 1$ keV is dominated by synchrotron radiation from electrons with energies up to $\sim$100 TeV, which are accelerated at the shock front (Koyama et al. [@Koyama95], Reynolds [@Reynolds98]). The detection of TeV gamma ray emission confirms the presence of extremely relativistic electrons (Tanimori et al. [@Tanimori]).\
- Modeling far ultra-violet spectra of a Northwestern filament has revealed that the post shock plasma has not reached ion-electron temperature equilibration. The electron temperature was found to be only $10\%$ of the ion temperature (Laming et al. [@Laming96]).\
- Ultraviolet absorption features in the spectrum of the Schweizer-Middleditch ([@Schweizer]) star, a blue subdwarf lying behind the remnant, have revealed the presence of blue and red-shifted unshocked silicon and iron (Hamilton et al. [@Hamilton97]). Remarkably, no evidence was found for the presence of $\sim 0.5$ of iron, which is likely to have been synthesized during the carbon deflagration of the white dwarf progenitor (Nomoto et al. [@Nomoto]). Carbon deflagration is the currently favored model for Type Ia supernovae.
The presence of a synchrotron component contaminates the thermal X-ray emission and the non-equilibration of ion and electron temperatures obscures the underlying shock hydrodynamics, which might otherwise have been inferred from the temperature structure. On the other hand, in X-rays other ion species can be studied than in the UV and this may help verifying the models for nucleosynthesis of Type Ia supernovae.
In this paper we present BeppoSAX spectra of the remnant of SN 1006. We will show that the thermal emission from the central region of the remnant is best described by a two component non-equilibrium ionization model with a very low ionization parameter.
8.2truecm
The data and method
===================
The Italian-Dutch satellite BeppoSAX (Boella et al. [@Boella97a]) observed SN 1006 in April 1997. The observation consists of three different pointings, which together cover the whole remnant by the imaging instruments LECS (Parmar et al. [@Parmar]) and MECS (Boella et al. [@Boella97b]). The total observation time is 91 ks for the MECS and 37 ks for LECS, which has a lower observation efficiency owing to UV leakage.
Both the LECS ($\sim 0.1 - 10$ keV) and the MECS ($\sim 2-10$ keV) are gas scintillation counters with a spectral resolution of $\sim 8$% at 6 keV, a spatial resolution at 6 keV of $\sim 2$ FWHM and a half power width[^1] of $\sim 2.5$. The spatial resolution is strongly energy dependent: with increasing energy the core of the point spread function decreases, whereas the wings of the point spread function (determined mostly by the mirror properties) become more dominant (Parmar et al. [@Parmar], Boella et al. [@Boella97b], cf. Vink et al. [@Vink99]). The extended wings of the point spread function make that in the case of SN 1006 the spectra of the low brightness central region, which emits mostly thermal emission (Koyama et al. [@Koyama95], Willingale et al. [@Willingale], the latter based on ROSAT PSPC data), is contaminated by the radiation from the rims of the remnant, which is predominantly synchrotron radiation.
We circumvented this problem by using a new version (v2.0) of the X-ray spectral fitting program SPEX (Kaastra et al. [@Kaastra], cf. Kaastra et al. [@Kaastra99]), with which one can fit several spectra from the same object simultaneously. In this version the concept of the spectral redistribution matrix is extended to include also a spatial redistribution part. The spectral model is calculated on a spatial/spectral input grid and a model of the instrument properties describes what fraction of the incoming photons with a certain energy, coming from a certain region of the sky will end up in a given energy bin (channel) and spatial bin (spectral extraction region). This means in practice that two sets of spatial regions have to be provided (in the program these are specified using images compliant with the FITS format). One set specifies the sectors for which the model is calculated, the other set specifies the spectral extraction regions. For stable calculations, the extraction regions should roughly correspond to the spatial model sectors, but a detailed correspondence is not necessary. For example, some regions of SN 1006 were only partially covered by some observations, but this is accounted for in the spectral/spatial redistribution matrix. The X-ray emission models used for this paper are the same as in the standard SPEX program.
We divided the remnant into six regions. Apart from a central region and two X-ray bright rims, we divided these regions further into a Northern and Southern half in order to see if the thermal emission from the brighter Southern half is different from the emission from the Northern halve. With North and South we mean here, and throughout the rest of the text, North and South with respect to the principle axis of the remnant, which is tilted with respect to the North-South direction in equatorial coordinates. The spatial model grid for which the models were calculated are shown in Fig. \[sectors\]. We chose spectral extraction regions roughly corresponding to the model regions, but the extraction regions corresponding to the bright rims were larger, in order to reduce the contamination of the central regions by emission coming from the bright rims. We used archival ROSAT HRI images (Fig. \[sectors\], see also Winkler & Long [@Winkler]) to define the spectral extraction regions and the model sectors. For the spectral extraction regions we took also into account the extend of the synchrotron rims in the BeppoSAX images, as they seem broader than on the HRI images due to the point spread function.
The extraction of the spectra and the generation of the spectral-spatial response matrix was done with a program specifically made to extract BeppoSAX spectra. It incorporates the current knowledge of the detectors, such as the point spread function, vignetting and absorption by the strong-backs. We used the response matrices of September 1997 and the standard background data (November 1998). However, there are still some uncertainties in the detector calibration, especially for off axis positions. These uncertainties are larger for the LECS, which has a more complicated design than the MECS. For this reason we left the normalization of the LECS spectra and some off-axis MECS spectra free with respect to the on-axis MECS spectra. The relative LECS normalizations turned out to be between 0.6 to 0.9, consistent with other BeppoSAX results (e.g. Favata et al. [@Favata]). The spectra were binned to a bin size of roughly 1/3rd the spectral resolution and some further rebinning was done for channels with low count rates. In order to circumvent statistical problems with bins with very few counts we used a method proposed by Wheaton et al. ([@Wheaton]). This means that after obtaining a good fit, we used the best fit model to calculate the expected error per bin, instead of the observed counts. Using this method with two or three extra iterations gives in general a stable and in principle more reliable $\chi^2$ value.
[l ll]{} & North & South\
I ($10^{62}$m$^{-3}$kpc$^{-2}$) & $2.3 \pm 1.0$ & $3.9\pm 1.0$\
I (keV) & $0.71 \pm 0.15$ & $0.78 \pm 0.09$\
I ($10^{15}$ m$^{-3}$s) & 3.3 ($<4.6$) & 2.3 ($<3.3$)\
II ($10^{62}$m$^{-3}$kpc$^{-2}$) & $0.1\pm 0.04$ & $0.14 \pm 0.03$\
II (keV) & 3.4 ($> 3$) & 8.0 ($> 4$)\
II ($10^{15}$ m$^{-3}$s) & 0.7 ($< 2$) & 0.7 ($<$3 )\
C &\
O &\
Ne &\
Mg &\
Si &\
S &\
Fe &\
($10^{20}$ cm$^{-2}$) &\
$\chi^2/\nu$ &\
Spectral fitting
================
Fitting six spectral regions simultaneously has the obvious disadvantage that the spectral model can become very complex. So we took care to constrain the spectral model as far as possible without loosing too much of its heuristic qualities. For each sector we chose to have three to four spectral components. A typical configuration consisted of the following components for each sky region: a power law component, one or two non-equilibrium ionization (NEI) thermal components and an absorption component (Morrison & McCammon [@Morrison]). We only looked for spectral differences in the thermal emission between the Northern and Southern halves of the remnant. We assumed uniform abundances for SN 1006.
Based on their deprojection of the ROSAT PSPC image of SN 1006, Willingale et al. ([@Willingale]) reported that the synchrotron emission does not seem to originate from all around the remnant, but only from incomplete shells at the Northwest and Southeast of the remnant. Consequently, in our simplest model we assume that no synchrotron emission is originating from the central region of the remnant. This model produces a reasonable fit to the LECS spectra, but it does not fit the MECS spectra of the central regions above $\sim$4 keV, where an excess in the observed spectra with respect to the model exists (Fig. \[hard\_excess\]). The temperature of the thermal components was $\sim 1.5$ keV. So clearly an additional emission component is needed in order to fit the emission from the central regions. This can be either a thermal component with an higher temperature, but it could also mean that the non-thermal emission observed to come from the rims has in reality cylindrical symmetry, in which case the apparent structure of the remnant in X-rays may be due to an extreme case of limb brightening. The latter possibility would be in disagreement with the above mentioned ROSAT PSPC findings, but it is conceivable that the synchrotron emission is coming from such a thin layer that the deprojection scheme of Willingale et al. ([@Willingale]) may not have worked adequately.
We investigated both possibilities. In the case of an additional power law component we fixed the power law index of the central regions to a value of 2.8, similar to the the values found for the rims (see Table \[powerlaw\]). We found, however, that an additional thermal component offers a better explanation. Only a hot thermal component fits adequately the Fe K emission seen in Fig. \[sn1006\_spectra\] (most clearly in the right bottom panel). This emission implies that there is some plasma present with a temperature in excess of $\sim 2$ keV. An additional power law component has, moreover, the disadvantage that the result is at a closer look inconsistent: it results in very high abundances for the thermal component (e.g. 148 times solar for silicon and 356 times solar for iron), whereas the emission measure indicated a total mass of $\sim 5$ for a distance of 1.8 kpc (Laming et al. [@Laming96]). This implies that most of the shock heated plasma is swept up material, if SN 1006 was a Type Ia with a mass of $\sim$ 1.4, but then it is not clear why the abundances are so high. High abundances are also hard to reconcile with the evolved dynamical status of the remnant, which implies that the remnant consists mostly of swept up interstellar matter (Moffet et al. [@Moffet], Long et al. [@Long88]). However, X-ray synchrotron models for supernova remnants (Reynolds,[@Reynolds98]) show that the synchrotron component is likely to have a more curved spectrum, which predicts less flux at lower energies. It may well be that such more advanced models do not have the problems mentioned above, although an additional thermal component will still be needed for producing the Fe K emission. From a statistical point of view the two models fit the data equally well, with slightly better fits for the thermal model (the model with an additional power law has $\chi^2/\nu = 1689/1287$).
The best fit model with an additional thermal component (Table \[parameters\] and \[powerlaw\]) is shown in Fig. \[sn1006\_spectra\]. The total emission measure of $= (7.4\pm 1.4)\ 10^{62}$ m$^{-3}$kpc$^{-2}$ implies a mass of $M = 8.3\pm 0.8 f_{0.4}^{3/2}d_{1.8}^{3/2}$, where $f_{0.4}$ is the volume filling factor divided by 0.4 (cf. Willingale et al. [@Willingale]) and $d_{1.8}$ is the distance in units of 1.8 kpc (Laming et al. [@Laming96]). The volume estimate assumes a spherical remnant with a diameter of 30. For this model the implied silicon mass is 0.05 and the iron mass is 0.06, which is much lower than the 0.5 of iron expected in remnants of Type Ia supernovae. The iron abundance reported here is, however, higher than reported by Koyama et al. ([@Koyama95]). We do not find significant variations in temperature or ionization between the Northern and Southern half of the remnant, but, as Fig. \[sn1006\_spectra\] shows, the hottest component seems more dominant in the Northern region. An interesting feature in the spectra of the central regions is the iron K-shell emission around 6.4 keV. This is especially apparent in the lower right panel of Fig. \[sn1006\_spectra\]. The centroid of this emission ($6.3\pm 0.2$ keV) is consistent with emission from iron in low ionization stages (caused by inner shell ionizations and excitations). In the next section we will discuss the very low value of the ionization parameter ().
[lll]{} & norm & index\
& (s$^{-1}$m$^{-2}$keV$^{-1}$ @ 1 keV) &\
Northeast & $181\pm 10$ & $2.81 \pm 0.05$\
Southeast & $83 \pm 10$ & $2.76 \pm 0.08$\
Northwest & $86 \pm 13$ & $2.90 \pm 0.16$\
Southwest & $76 \pm 9$ & $2.83 \pm 0.12$\
Our best fit absorption column is a factor two lower than the value found by Koyama et al. ([@Koyama95]), but is higher than the value reported by Willingale et al. ([@Willingale]). The higher ASCA value is not very surprising, because ASCA was less reliably calibrated below $\sim 1$ keV and ASCA is not sensitive below 0.4 keV. The fact that the ROSAT PSPC and BeppoSAX LECS show that there is emission at 0.2 keV is, however, a clear indication that the absorption column is not as high as indicated by the ASCA spectrum. We think that our absorption estimate is consistent with the ROSAT PSPC spectra if one takes into account that we use non-equilibrium ionization models and in this case such a model produces more line radiation below 1 keV than an equilibrium model, which implies a higher absorption to explain the flux levels.
The power law index of the non-thermal spectra are somewhat lower than indicated by the ASCA spectra and higher than found for the ROSAT PSPC data (see Table \[powerlaw\]). We checked for gradual steepening of the spectrum, which is expected on theoretical grounds (Reynolds [@Reynolds98]). Indeed, we found some evidence that on average the power law index changes from $2.65 \pm 0.21$ below 2 keV to $2.81\pm 0.05$ above 2 keV. However, the scatter in the four points is rather large and we can, furthermore, not exclude that the difference in slope is due to calibration uncertainties, as there are still problems with the intercalibration of the LECS and MECS instruments.
Interpretation
==============
Our fits indicate that the spectra of the central region are best fitted with two NEI components with a very low value for the ionization parameter. The very low ionization is not surprising as previous studies indicate that the density of the ISM surrounding the remnant is very low: estimates of the pre-shock hydrogen number density vary from 0.04 cm$^{-3}$ (Laming et al. [@Laming96]) to 1 cm$^{-3}$ (Winkler & Long [@Winkler]). Our mass estimate implies a pre-shock density of 0.1 cm$^{-3}$. For the Sedov model the average ionization parameter () is approximately $n_0t$, where $n_0$ is the pre-shock hydrogen density and $t$ the age of the remnant. So for the shocked interstellar gas we estimate $\sim 3\, 10^{15}$ m$^{-3}$s. For the ionization parameter of the shocked ejecta we can find an upper limit by assuming that 1.4 of ejecta is completely ionized and that it has a volume filling factor of 0.4. This gives an electron density of 0.04 cm$^{-3}$ and implies $\simlt 1.2\, 10^{15}$ m$^{-3}$s. The value found by fitting the X-ray spectra agrees with our current understanding of SN 1006.
The low ionization has, however, some intriguing aspects. For example, our models indicate that most of the silicon is in the form of Si V to Si XII. So with the silicon lines around 1.8 keV we are only observing the top of the iceberg. However, the silicon L-shell emission contributes significantly to the flux between 0.1 keV and 0.4 keV. Also L-shell emission from magnesium, sulphur and argon, and K-shell emission from carbon contribute to the flux below 0.5 keV. Unfortunately, L-shell emission is notoriously complicated; only the iron L-shell emission has been investigated in substantial detail. The LECS spectral resolution below 0.5 keV is not good enough to see possible discrepancies in the atomic data, but the resulting uncertainties in the flux below 0.5 keV may have affected the spectral modeling. This demonstrates the significance of the broad spectral range covered by the two BeppoSAX instruments. The observation of emission from iron around 6.4 keV illustrates that even ions in low ionization stages emit some X-ray line emission, because of inner shell ionizations and excitations (in this respect SN1006 seems to be similar to RCW 86, see Vink et al. [@Vink97]). It would be interesting to search for similar emission features from Ar and Ca with future X-ray detectors.
The apparent two temperature structure is not surprising. SN 1006 is a large remnant and it would be more surprising if the whole remnant could be characterized by a single temperature. Furthermore, there are several possible explanations for the two temperature structure. Hydrodynamical shock models, such as the Sedov model (Sedov [@Sedov]) predict temperature gradients. Models including a reverse shock (e.g. Chevalier [@Chev82]) even predict a two temperature regime. The lowest temperature is associated with the ejecta heated by the reverse shock, whereas the highest temperatures are attained by the shocked interstellar medium. However, hydrodynamical models predict the ion temperature, whereas the shape of the thermal X-ray continuum is determined by the electron temperature, which, in absence of electron-ion equilibration, can in principle be as much as a factor 1835 (the electron/proton mass ratio) lower than the ion temperature. It can be even more if the plasma consist of highly enriched supernova ejecta. In these extreme cases, however, we would not observe thermal X-ray emission at all. The equilibration process is still poorly understood. In the case of SN 1006 there is strong evidence for poor equilibration of the electron and ion temperature (Laming et al. [@Laming96]). The evidence comes from UV spectroscopy of the Northern shock front, but the equilibration may very well vary over the remnant. Another possible cause for temperature variations over the remnant are density differences. There is some evidence that SN 1006 has a very strong density gradient from the front to the back of the remnant (Hamilton et al. [@Hamilton97]). Yet another interesting possibility is that the shock heating process does not produce a Maxwellian electron distribution, since the low density in SN 1006 may prevent a rapid thermalization of the electron distribution (Laming [@Laming98], see also Itoh [@Itoh], Hamilton & Sarazin [@Hamilton84]). The hot component could then be associated with bremsstrahlung from the tail of the non-thermal electron distribution.
For practical reasons our fitting model has been largely a phenomenological model. It is important to realize this when interpreting the best fit parameters. For instance, our use of single/double NEI components does not account for the fact that there are temperature and ionization gradients. Or to be more precise: we suppose that these gradients can be approximated by two thermal plasma components. On the other hand models incorporating temperature and ionization gradients, such as spectral models based on the Sedov model (e.g. Kaastra & Jansen [@Kaastra93]), suppose that the gradients are well defined, whereas in reality the gradients may very well be more complicated, because of the pre-supernova density structure of the interstellar medium or because of the non-equilibration of ion and electron temperature, as discussed above. In our opinion a more severe problem with our modeling is that we use uniform abundances for the whole remnant, which is probably not correct, as there is very likely freshly shocked ejecta present consisting entirely of metals, on the one hand, and shocked interstellar gas with solar abundances, on the other hand. This might for example affect our total mass estimate, which may be lower because a pure metal plasma is an efficient bremsstrahlung emitter. However, for more realistic models we need either data from more sensitive X-ray instruments, or we need a clear understanding of the thermal and abundance structure of the remnant. For instance, with future X-ray missions we may be able to identify spatially or spectrally an ejecta emission component and observe the (projected) temperature structure of the remnant directly.
The overall structure of SN 1006, as emerging from other observations, appears complicated and is not very well understood. Some observations seem even contradictory. For example, UV absorption measurements indicate that a lot of the ejecta are still in free expansion (Hamilton et al. [@Hamilton97]), but the kinematics of the remnant, on the other hand, show that SN 1006 is dynamically evolved implying that most of the ejecta has been shocked by now (Moffet et al. [@Moffet], Long et al. [@Long88]). Finally, it is peculiar that a remnant that has such a high degree of symmetry with respect to the Southeast/Northwest axis (Roger et al. [@Roger]) seems to have a strong front-back asymmetry (Hamilton et al. [@Hamilton97]). A strong deviation from cylindrical symmetry has also been reported by Willingale et al. ([@Willingale]).
Conclusion
==========
Our analysis indicates that a one temperature model does not fit the thermal spectrum of SN 1006 adequately. An additional thermal component with a temperature in excess of 3 keV is needed. The alternative, an additional power law component, cannot be excluded, but in this case the abundances of the low temperature thermal component become exceptionally high, which is inconsistent with the idea that SN 1006 must have swept up a considerable amount of interstellar material.
The values for the ionization parameters for the thermal components appear to be very low ($\simlt 3\ 10^{15}$ m$^{-3}$s), but are consistent with the electron densities that we derive from the emission measure. Note, however, that in reality a range of ionization values will be present. For instance, the Sedov model (see Kaastra & Jansen [@Kaastra93]) predicts a superposition of plasma ionization values ranging from roughly 0.2 in the center of the remnant up to 2 times the average value at a radius of 0.9 times the shock radius. Inhomogeneities in the pre-shock interstellar medium may extend this range upward and downward. For such a low ionization parameter the L-shell emission of Mg, Si and S contributes substantially to the flux below 0.5 keV. This explains that our best fit hydrogen absorption column ($= (8.8\pm 0.5)\ 10^{20}$ cm$^{-2}$) turns out to be higher than found by Willingale et al. ([@Willingale]).
Noteworthy is an iron K shell emission feature in the spectrum. The centroid of the line emission is $6.3\pm0.2$ keV, which indicates that the emission is the result of inner shell ionizations and excitations. The fitted abundances clearly indicate non-solar abundances with, in particular, a very high silicon abundance. This is in agreement with the analysis of ASCA data (Koyama et al. [@Koyama95]). We did not find significant differences in the temperatures and ionization parameters for the spectra of the Northwestern and Southeastern halves of the remnant, but the hottest component seems more dominant in the Northern region.
Observations in the near future with Chandra (AXAF), XMM and Astro E will be able to reveal new information on the remnant which is needed in order to obtain a consistent model for this remnant. In particular, a spatial or a spectral separation of emission originating from shocked ejecta and shocked interstellar gas will be very useful for understanding more about the structure and abundances of this interesting but rather enigmatic supernova remnant.
We thank Martin Laming for stimulating discussions on SN 1006. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. This work was financially supported by NWO, the Netherlands Organization for Scientific Research.
Anders E., Grevese N., 1989, Geochimica et Cosmochimica Acta 53, 197 Boella G., Butler R.C., Perola G.C., et al., 1997a, A&AS 122, 299 Boella G., Chiappetti L., Conti G., et al., 1997b, A&AS 122, 327 Chevalier R.A., 1982, ApJ 258, 790 Clark D.H., Stephenson F.R., “The Historical Supernovae”, 1977, Pergamon Press Ltd Oxford Favata F., Vink J., Dal Fiume D., et al., 1997, A&A 324, L49 Hamilton A.J.S., Sarazin C.L., 1984, ApJ 287, 282 Hamilton A.J.S., Fesen R.A., Wu C.-C., Crenshaw D.M., Sarazin C.L., 1997, ApJ 482, 838 Itoh, H., 1984, ApJ 285, 601 Kaastra J.S., Jansen F., 1993, A&AS 97, 873 Kaastra J.S, Mewe R., Nieuwenhuijzen H., 1996, In: Watanabe, T. (ed.) The 11$^{th}$ colloquium on UV and X-ray Spectroscopy of Astrophysical and Laboratory Plasmas, p. 411 Kaastra J.S., Lieu R., Mittaz J.P.D., et al., 1999, ApJ 519, L119 Koyama K., Petre R., Gotthelf E., et al., 1995, Nat 378, 255 Laming J.M., Raymond J.C., McLaughlin B.M., Blair W.P., 1996, ApJ 472, 267 Laming J.M., 1998, ApJ 499, 309 Long K.S, Blair W.P., van den Bergh S., 1988, ApJ 333, 749 Moffet D.A., Goss W.M., Reynolds S.P., 1993, AJ 106, 1566 Morrison R., McCammon D., 1983, ApJ 270, 119 Nomoto K., Thielemann F.K., Yokoi K., 1984, ApJ 286, 644 Parmar A.N., Martin D.D.E., Bavdaz M., et al., A&AS 122, 309 Reynolds S.P., 1998, ApJ 493, 375 Roger R.S., Milne D.K., Kesteven M.J., 1988, ApJ 332, 940 Schaefer B.E., 1996, ApJ 459, 438 Schweizer F., Middleditch J., 1980, ApJ 241, 1039 Sedov L.I., 1959, “Similarity and Dimensional Methods in Mechanics”, Academic Press, New York Tanimori T., Hayami Y., Kamei S., et al. , 1998, ApJ 497, L25 Thielemann F.K., Nomoto K., Yokoi K., 1986, A&A 158, 17 Vink J., Kaastra J.S., Bleeker J.A.M., 1997, A&A 328, 628 Vink J., Maccarone, M.C., Kaastra J.S., et al., 1999, A&A 344, 289 Wheaton W., Dunklee A.L., Jacobsen A.S., et al., 1995, ApJ 438 , 322 Willingale R., West R.G. Pye J.P., Stewart, G.C., 1996, MNRAS 278, 749 Winkler P.F., Long K.S., 1997, ApJ 491, 829
[^1]: The half power width gives the width of the area in which 50% of the photons of a point source are expected to end up; this measure is sensitive to the broad wings of the point spread function, whereas the FWHM is more determined by the core of the point spread function.
|
---
abstract: 'We report the detection of spatially extended and emission in the $z$=2.49 submillimeter galaxy (SMG) J123707+6214, using the Expanded Very Large Array and the Plateau de Bure Interferometer. The large molecular gas reservoir is spatially resolved into two components (north-east and south-west; previously identified in emission) with gas masses of 4.3 and 3.5$\times$10$^{10}\,(\alpha_{\rm CO}/0.8)$. We thus find that the optically invisible north-east component slightly dominates the gas mass in this system. The total molecular gas mass derived from the observations is $\gtrsim$2.5$\times$ larger than estimated from . The two components are at approximately the same redshift, but separated by $\sim$20kpc in projection. The morphology is consistent with that of an early-stage merger. The total amount of molecular gas is sufficient to maintain the intense 500starburst in this system for at least $\sim$160Myr. We derive line brightness temperature ratios of $r_{31}$=0.39$\pm$0.09 and 0.37$\pm$0.10, and $r_{51}$=0.26$\pm$0.07 and 0.25$\pm$0.08 in the two components, respectively, suggesting that the $J$$\geq$3 lines are substantially subthermally excited. This also suggests comparable conditions for star formation in both components. Given the similar gas masses of both components, this is consistent with the comparable starburst strengths observed in the radio continuum emission. Our findings are consistent with other recent studies that find evidence for lower CO excitation in SMGs than in high-$z$ quasar host galaxies with comparable gas masses. This may provide supporting evidence that both populations correspond to different evolutionary stages in the formation of massive galaxies.'
author:
- |
Dominik A. Riechers, Christopher L. Carilli, Fabian Walter, Axel Weiss,\
Jeff Wagg, Frank Bertoldi, Dennis Downes, Christian Henkel, and Jacqueline Hodge
title: |
Imaging the Molecular Gas Properties of a Major Merger\
Driving the Evolution of a $z$=2.5 Submillimeter Galaxy
---
Introduction
============
Detailed studies of submillimeter galaxies (SMGs; see review by Blain [-@bla02]) have revealed that they represent a relatively rare, but cosmologically important high redshift population of massive galaxies. They harbor intense ($>$500yr$^{-1}$), often heavily obscured, but rather short-lived ($<$100Myr) starbursts that rapidly consume their gas content through star formation at high efficiencies. SMGs may trace a common phase in the formation and evolution of massive galaxies in the early universe, making them the likely progenitors of today’s massive spheroidal galaxies.
Given their substantial dust obscuration, the most insightful way to study SMGs and their star formation properties is through the dust-reprocessed emission at rest-frame far-infrared (FIR) wavelengths. While continuum diagnostics at such wavelengths are particularly useful to determine star formation rates (SFRs) unaffected by obscuration, the most insightful way to study the fate of such galaxies is through emission line fluxes, morphology and dynamics of the material that fuels the star formation, i.e., molecular gas (typically CO). Molecular gas was detected in $>$30 SMGs to date, revealing large gas reservoirs of $>$10$^{10}$ in most cases (see Solomon & Vanden Bout [-@sv05] for a review). However, most of these studies were carried out in mid- to high-$J$ CO transitions, rather than the fundamental transition.
Recent studies of emission indicate that (in contrast to high-$z$ quasar hosts) this line appears to carry a higher brightness temperature than the mid- to high-$J$ lines in several SMGs, suggesting relatively low gas excitation (e.g., Hainline et al.[-@hai06]; Riechers et al. [-@rie06], [-@rie10]; Carilli et al. [-@car10]; Ivison et al.[-@ivi10]; Harris et al. [-@har10]). Therefore, earlier studies may systematically underestimate the total amount of molecular gas that is present in SMGs. The low excitation in at least part of the molecular gas reservoirs raises the question whether the commonly used $\alpha_{\rm CO}$ conversion factor from CO luminosity to gas mass ($M_{\rm gas}$) for ultra-luminous infrared galaxies (ULIRGs) in the nearby universe (Downes & Solomon [-@ds98]) is applicable, or if this practice leads to an underprediction of $M_{\rm
gas}$.
To further investigate this issue, we have started a systematic Expanded Very Large Array (EVLA) survey of emission in high-$z$ SMGs and quasar host galaxies that were previously detected in higher-$J$ CO lines. In this Letter, we report the first results from this study, i.e., the detection of spatially extended emission toward the $z$=2.49 SMG J123707+6214 (GN19; HDF242; Borys et al.[-@bor03]). We also report the detection of emission in this system, using the Plateau de Bure Interferometer (PdBI). We use a concordance, flat $\Lambda$CDM cosmology throughout, with $H_0$=71Mpc$^{-1}$, $\Omega_{\rm M}$=0.27, and $\Omega_{\Lambda}$=0.73 (Spergel [-@spe03], [-@spe07]).
Observations
============
EVLA
----
We observed the ($\nu_{\rm rest} = 115.2712$GHz) emission line toward J123707+6214 using the EVLA. At $z$=2.49, this line is redshifted to 33.029GHz (9.08mm). Observations were carried out under excellent weather conditions (typical atmospheric phase rms:2.3$^\circ$ on a 300m baseline) in D array on 2010 April 11 (NRAO Legacy ID: AR708), resulting in 2.9hr on-source time with 18antennas (equivalent to 1.2hr with 27antennas) after rejection of bad data. The nearby ($5.4^\circ$ distance) quasar J1302+5748 was observed every 7.5minutes for pointing, secondary amplitude and phase calibration. For primary flux calibration, the standard calibrator 3C286 was observed, leading to a calibration that is accurate within $\lesssim$10%. Observations were set up using a total bandwidth of 252MHz (dual polarization; after rejection of overlapping edge channels between sub-bands; corresponding to $\sim$2300 at 9.08mm) with the WIDAR correlator.
For data reduction and analysis, the AIPS package was used. All data were mapped using ‘natural’ weighting. Maps of the velocity-integrated CO $J$=1$\to$0 line emission yield a synthesized clean beam size of 2.8$''$$\times$2.1$''$ at an rms noise level of 39$\mu$Jybeam$^{-1}$ over 490 (54MHz).
PdBI
----
We observed the ($\nu_{\rm rest} = 576.2679$GHz) emission line toward J123707+6214 using the IRAM PdBI. At $z$=2.49, this line is redshifted to 165.1198GHz (1.82mm). Observations were carried out under good weather conditions with five antennas in the compact D configuration during 3tracks on 2008 July 10, and August 09 and 14 (IRAM program ID: SC47), for a total of 14.9hr, resulting in 8.1hr of 6 antenna equivalent on-source time after rejection of bad data. The nearby quasars B0954+658 and B1418+546 (distance to J123707+6214: $17.4^\circ$ and $15.4^\circ$) were observed every 20minutes for pointing, secondary amplitude and phase calibration. For primary flux calibration, the standard calibrators MWC349 and 3C454.3 were observed. Observations were set up using a total spectrometer bandwidth of 1GHz (dual polarization; corresponding to $\sim$1800 at 1.82mm).
For data reduction and analysis, the GILDAS package was used. All data were mapped using ‘natural’ weighting. Maps of the velocity-integrated CO $J$=5$\to$4 line emission yield a synthesized clean beam size of 3.7$''$$\times$3.0$''$ at an rms noise level of 0.39/0.54mJybeam$^{-1}$ over 563/309 (320/180MHz).
Results
=======
Gas Morphology and Emission Line Properties
-------------------------------------------
We have detected spatially resolved line emission toward the $z$=2.49 SMG J123707+6214, measuring two components at 7$\sigma$ (north-east; ‘ne’)[^1] and 6$\sigma$ (south-west; ‘sw’) significance in the velocity-integrated emission line map (Fig.\[f1\], [*left*]{}). We do not detect the underlying continuum emission at a 3$\sigma$ upper limit of 54$\mu$Jybeam$^{-1}$ at 9.08mm (rest-frame 2.6mm).
From Gaussian fitting to the line profiles of the ne and sw components (Fig. \[f1\], [*right*]{}),[^2] we obtain line peak strengths of $S_{\nu}$=375$\pm$57 and 340$\pm$64$\mu$Jy at line FWHMs of d$v$=454$\pm$87 and 414$\pm$92, centered at redshifts of $z$=2.4879$\pm$0.0005 and 2.4873$\pm$0.0005, respectively. The line widths and redshifts are consistent with those measured for the emission line (Tacconi et al. [-@tac06]), and are equal for both spatial components within the uncertainties. From the spatially-integrated emission, we determine a systemic redshift of $z$=2.4876$\pm$0.0004, which we adopt as the nominal value for the system in the following. The line parameters for J123707+6214ne and sw correspond to velocity-integrated emission line strengths of $I_{\rm
CO(1-0)}$=0.180$\pm$0.029 and 0.149$\pm$0.029Jy, i.e., line luminosities of $L'_{\rm CO(1-0)}$=(5.36$\pm$0.86) and (4.43$\pm$0.85)$\times$10$^{10}$.
We have also detected spatially resolved line emission at $\gtrsim$5$\sigma$ significance toward both components of J123707+6214 (Fig. \[f1x\]). The ne component dominates the integrated line emission (Fig. \[f1x\], [*left*]{}), extracted over a velocity range comparable to the map. The maximum signal-to-noise ratio on the sw component is obtained over a narrower velocity range (Fig.\[f1x\], [*middle*]{}). In this map, the sw component is brighter than the ne component, comparable to what is seen in the maps of Tacconi et al. ([-@tac06], [-@tac08]). Only $\sim$60% of the emission from the ne component are seen over this narrower velocity range. We do not detect the underlying continuum emission at a 3$\sigma$ upper limit of 0.8mJybeam$^{-1}$ at 1.82mm (rest-frame 520$\mu$m).
From Gaussian fitting to the integrated line profile (Fig.\[f1x\], [*right*]{}), we obtain $S_{\nu}$=4.1$\pm$0.8mJy at d$v$=485$\pm$110, centered at $z$=2.4875$\pm$0.0006. This corresponds to $I_{\rm CO(5-4)}$=2.12$\pm$0.51Jy. Fitting the ne and sw components individually yields $S_{\nu}$=2.4$\pm$0.5 and 2.1$\pm$0.5mJy, d$v$=467$\pm$124 and 432$\pm$130, and $I_{\rm CO(5-4)}$=1.17$\pm$0.33 and 0.94$\pm$0.29Jy, respectively. We thus derive $L'_{\rm
CO(5-4)}$=(1.39$\pm$0.32) and (1.12$\pm$0.29)$\times$10$^{10}$, respectively.
This implies CO $J$=3$\to$2/1$\to$0 line brightness temperature ratios of $r_{31}$=0.39$\pm$0.09 (ne)[^3] and 0.37$\pm$0.10 (sw), and CO $J$=5$\to$4/1$\to$0 line brightness temperature ratios of $r_{51}$=0.26$\pm$0.07 and 0.25$\pm$0.08. The and emission lines are clearly subthermally excited toward both components ($r_{31}$$<$1 and $r_{51}$$<$1). Interestingly, both components appear to have comparable gas excitation. The ne component is brighter in all CO transitions. This suggests that the ne component carries the dominant fraction of the molecular gas mass in this system. Based on a ULIRG conversion factor $\alpha_{\rm
CO}$=0.8()$^{-1}$ to derive $M_{\rm gas}$ from $L'_{\rm CO(1-0)}$ (Downes & Solomon [-@ds98]), we determine the total molecular gas masses of J123707+6214ne and sw to be $M_{\rm
gas}$=4.3 and 3.5$\times$10$^{10}$,[^4] i.e., by more than a factor of 2 higher than previously found based on the data (scaled to the same $\alpha_{\rm CO}$), and corresponding to $\sim$2/3 of the stellar mass in this system (Tacconi et al. [-@tac06], [-@tac08]).
Dynamical Structure of the Gas Reservoir
----------------------------------------
In Figure \[f2\], maps of the emission are shown in 182wide velocity channels. The emission toward J123707+6214ne appears dynamically resolved on $\sim$1.5$''$ ($\sim$12kpc) scales, which may suggest that the emission is more spatially extended than in the line (0.5$''$$\pm$0.2$''$, or 4.1$\pm$1.6kpc; Tacconi et al.[-@tac06]). J123707+6214sw appears marginally spatially resolved in position-velocity space at best, consistent with the size measured in emission within the relative uncertainties (0.9$''$$\pm$0.3$''$, or 7.4$\pm$2.5kpc; Tacconi et al. [-@tac06]). Assuming radii of 6 and 3.7kpc for J123707+6214ne and sw, this yields dynamical masses of $M_{\rm dyn}$sin$^2$$i$=2.9 and 1.5$\times$10$^{11}$ (which we estimate to be reliable within a factor of 2). This is about twice as high as previous estimates based on emission (Tacconi et al. [-@tac08]), and corresponds to gas mass fractions of $f_{\rm gas}$=0.15 and 0.23 for J123707+6214ne and sw, respectively. Higher resolution observations are required to better constrain how the merger dynamics impact the CO line profiles and the morphology of the gas reservoir, which is necessary to determine more precise dynamical masses.
Analysis and Discussion
=======================
Origin of the CO Emission
-------------------------
SMGs are commonly associated with heavily obscured starbursts. J123707+6214 is a particularly insightful example of this population, as the ne component remains undetected at all wavelengths shortward of 3.6$\mu$m (rest-frame 1.0$\mu$m). As shown in Figure \[f4\], the peak of the emission of the ne component is clearly associated with peaks in the mid-infrared (8.0$\mu$m; rest-frame 2.3$\mu$m) and radio continuum (20cm; rest-frame 6cm; see also Tacconi et al. [-@tac08]). The ne component also slightly dominates the radio emission ($\sim$55%), which suggests that it contributes the dominant fraction to the source’s SFR. As also shown in Fig. \[f4\], J123707+6214sw consists of multiple components in the optical (606nm; rest-frame 174nm) that are separated by a few kpc (see also Swinbank et al.[-@swi04]). This may correspond to multiple star-forming clumps, embedded in a more complex, extended molecular gas reservoir, but may also reflect the high degree of obscuration in this source.
Gas Surface Densities and Star Formation Timescales
---------------------------------------------------
Estimating that the emission in J123707+6214ne and sw is distributed over 6 and 3.7kpc radius regions, the surface-averaged gas densities are $\Sigma_{\rm gas}$=3.8 and 8.1$\times$10$^8$kpc$^{-2}$, i.e., comparable to but somewhat lower than estimates based on the (assuming the above size estimates) more compact emission (Tacconi et al. [-@tac06], [-@tac08]). This would be consistent with some of the emission being in a diffuse, low surface brightness component.
Based on the SFR of 500$\pm$250yr$^{-1}$ determined by Tacconi et al. ([-@tac08]), we derive a gas depletion timescale of $\tau_{\rm dep}$=$M_{\rm gas}$/SFR $\sim$160Myr for J123707+6214. This is consistent with but on the high end of what is found for other SMGs (for which $M_{\rm gas}$ are inferred from mid-$J$ CO lines; e.g., Greve [-@gre05]).
Conclusions
===========
We have detected spatially resolved and emission toward the $z$=2.49 SMG J123707+6214. We resolve the emission into two components previously detected in emission (Tacconi et al.[-@tac06], [-@tac08]), which are likely merging galaxies. Both components show similar CO excitation properties, with moderate $J$=3$\to$2/1$\to$0 line ratios of $r_{31}$$\sim$0.38, and relatively low $J$=5$\to$4/1$\to$0 line ratios of $r_{51}$$\sim$0.25. The implied $J$=5$\to$4/3$\to$2 line ratios of $r_{53}$$\sim$0.66 are comparable to those found in other SMGs that show evidence for mergers (e.g., Weiß et al. [-@wei05]). On the other hand, the low $r_{31}$ are comparable to those found in massive gas-rich star-forming galaxies with much lower SFRs (Dannerbauer et al. [-@dan09]; Aravena et al.[-@ara10]). This may suggest that, in addition to the highly-excited gas associated with the starburst, J123707+6214 hosts a substantial amount of low-excitation gas.
The emission suggest the presence of $\gtrsim$2.5$\times$ more molecular gas than expected if assuming a constant brightness temperature from . The optically detected merger component (sw) carries $\sim$45% of the gas mass in this system, suggesting comparable amounts of gas in both components, with a slightly higher contribution coming from the optically invisible component (ne; $\sim$55%). The radio continuum emission consistently indicates a comparable starburst strength in both components. Assuming that none is substantially contaminated by an obscured AGN, and given the high SFR of 500$\pm$250yr$^{-1}$, this provides supporting evidence for a ULIRG-like $\alpha_{\rm CO}$ in both components. The emission in J123707+6214 likely arises from the same gas phase detected in the higher-$J$ lines, but the emission appears somewhat more spatially extended. This yields a revised, $\sim$2$\times$ higher estimate for the dynamical mass of the system. Also, this finding would be consistent with the presence of some diffuse, low-excitation gas (which may have a higher $\alpha_{\rm
CO}$ than the highly-excited gas). Such a low-excitation component could be associated with gas that is redistributed by mechanical energy input from the starburst, or with tidal structure in the ongoing, gas-rich merger in this system.
Our findings highlight the importance of observing multiple CO lines including to determine the total molecular gas mass and gas properties in SMGs (as already acknowledged by Tacconi et al.[-@tac08] in the initial observations of this source). Our results are consistent with those found for other SMGs observed in emission (Hainline et al. [-@hai06]; Carilli et al.[-@car10]; Ivison et al. [-@ivi10]; Harris et al.[-@har10]), which commonly show lower CO line excitation than typically found in FIR-luminous quasar host galaxies at comparable redshifts and with comparable gas masses (e.g., Riechers et al.[-@rie06]; [-@rie09]; Weiß et al. [-@wei07]). This provides supporting evidence that both populations trace different evolutionary stages of the same massive galaxy population, as would be expected in the ULIRG-quasar transition scenario proposed by Sanders et al. ([-@san88]).
J123707+6214 is a prototypical example of an SMG during an early merger stage, found in the peak epoch of galaxy formation. Higher resolution, dynamical mapping of emission in this intriguing system (and others) is desirable to narrow down $\alpha_{\rm CO}$ through dynamical mass measurements over several resolution elements, as possible with the full EVLA in the future. A more complete census of observations of SMGs will provide the necessary context to interpret the results of such investigations. Such studies provide the most direct means to constrain the gas fraction, total mass and evolutionary state of SMGs, which is necessary to better understand the evolutionary path of massive galaxies through their most active phases, and to constrain the molecular gas mass density of the universe.
We thank the referee for a critical reading of the manuscript and for a helpful report. DR acknowledges support from NASA through Hubble Fellowship grant HST-HF-51235.01 awarded by STScI, operated by AURA for NASA, under contract NAS5-26555. The EVLA is a facility of NRAO, operated by AUI, under a cooperative agreement with the NSF.
Aravena, M., 2010, ApJ, 718, 177 Blain, A. W., Smail, I., Ivison, R. J., Kneib, J.-P., & Frayer, D. T. 2002, PhR, 369, 111 Borys, C., Chapman, S., Halpern, M., & Scott, D. 2003, MNRAS, 344, 385 Carilli, C. L., 2010, ApJ, 714, 1407 Coppin, K., 2008, MNRAS, 389, 45 Daddi, E., 2010, ApJ, 713, 686 Dannerbauer, H., Daddi, E., Riechers, D., Walter, F., Carilli, C. L., Dickinson, M., Elbaz, D., & Morrison, G. E. 2009, ApJ, 698, L178 Downes, D., & Solomon, P. M. 1998, ApJ, 507, 615 Giavalisco, M., et al. 2004, ApJ, 600, L93 Greve, T. R., 2005, MNRAS, 359, 1165 Hainline, L. J., Blain, A. W., Greve, T. R., Chapman, S. C., Smail, I., & Ivison, R. J. 2006, ApJ, 650, 614 Harris, A. I., Baker, A. J., Zonak, S. G., Sharon, C. E., Genzel, R., Rauch, K., Watts, G., & Creager, R. 2010, ApJ, 723, 1139 Ivison, R. J., Smail, I., Papadopoulos, P. P., Wold, I., Richard, J., Swinbank, A. M., Kneib, J.-P., & Owen, F. N. 2010, MNRAS, 404, 198 Ivison, R. J., Papadopoulos, P. P., Smail, I., Greve, T. R., Thomson, A. P., Xilouris, E. M., & Chapman, S. C. 2011, MNRAS, in press Morrison, G. E., Owen, F. N., Dickinson, M., Ivison, R. J., & Ibar, E. 2010, ApJS, 188, 178 Riechers, D. A., 2006, ApJ, 650, 604 Riechers, D. A., Walter, F., Carilli, C. L., & Lewis, G. F. 2009, ApJ, 690, 463 Riechers, D. A., 2010, ApJ, 720, L131 Sanders, D. B., Soifer, B. T., Elias, J. H., Madore, B. F., Matthews, K., Neugebauer, G., & Scoville, N. Z. 1988, ApJ, 325, 74 Solomon, P. M., & Vanden Bout, P. A. 2005, ARA&A, 43, 677 Spergel, D. N., 2003, ApJS, 148, 175 Spergel, D. N., 2007, ApJS, 170, 377 Swinbank, A. M., Smail, I., Chapman, S. C., Blain, A. W., Ivison, R. J., & Keel, W. C. 2004, ApJ, 617, 64 Tacconi, L. J., 2006, ApJ, 640, 228 Tacconi, L. J., 2008, ApJ, 680, 246 Weiß, A., Downes, D., Walter, F., & Henkel, C. 2005, A&A, 440, L45 Weiß, A., Downes, D., Neri, R., Walter, F., Henkel, C., Wilner, D. J., Wagg, J., & Wiklind, T. 2007, A&A, 467, 955
[^1]: Following the nomenclature by Tacconi et al. ([-@tac06]).
[^2]: spectra are Hanning-smoothed.
[^3]: We recomputed the fluxes based on the data presented in Tacconi et al. ([-@tac06]) by extracting the emission over the same (broader) velocity range as the and emission. This yields $I_{\rm CO(3-2)}$=0.63$\pm$0.10 and 0.50$\pm$0.10Jy for the ne and sw components, respectively. The different methods for extracting fluxes are likely responsible for differences between our analysis and an independent study carried out in parallel by Ivison et al. ([-@ivi11]).
[^4]: A Milky-Way-like $\alpha_{\rm CO}$=3.5()$^{-1}$ (e.g., Daddi et al.[-@dad10]) would increase $M_{\rm gas}$ by a factor of 4.4.
|
---
abstract: 'Multiwavelength data for Stephan’s Quintet (SQ) are consistent with the following model for this compact galaxy group. (1) Discordant redshift NGC 7320 is an unrelated foreground galaxy. (2) In the past SQ was an accordant redshift quartet involving NGC 7317, 18A, 19 and 20C. NGC 7320C collided (probably not for the first time) with the group a few times 10$^8$ years ago and stripped the interstellar matter from NGC 7319. (3) In the present SQ is again an accordant quartet involving NGC 7317, 18A,B, and 19. NGC 7318B is now entering the group at high velocity for the first time, giving rise to a shock zone. If most compact groups are like SQ, then they are frequently visited by infalling neighbors that perturb the group and themselves. SQ represents strong evidence for secondary infall in a small group environment. Tidal stripping reduces the mass of the infalling galaxies, thereby increasing the timescale for their orbital decay. There is little evidence that these high velocity “intruders” are rapidly captured and/or merge with the system. Instead they are the mechanism that sustains compact groups against collapse. Efficient gas stripping may account for the low star formation rate observed in compact groups and infall of residual gas into galactic nuclei may also foster the onset of active galactic nucleus activity.'
author:
- 'M. Moles$^{1,2}$, J. W. Sulentic$^3$, and I. Márquez$^{4,5}$'
title: 'The dynamical status of Stephan’s Quintet'
---
Introduction
============
Stephan’s Quintet (SQ) epitomizes the problems that compact groups pose for our ideas about galaxy formation and evolution. It is one of the most luminous and high surface brightness aggregates included in the first reasonably complete catalog (Hickson 1982; hereafter HCG) of compact groups (HCG92). Questions center around how many of the galaxies are in close proximity to one another, as well as how long and violently they have been interacting. These questions are particularly relevant because one of the components, NGC 7320, shows a discordant redshift. Finally, there is the question of whether SQ is representative of the compact group phenomenon. It is relevant for the latter question to point out that the accordant redshift part of SQ satisfies the HCG selection criteria - Stephan’s Quartet is also a compact group.
SQ is composed of a [*kernel*]{} of three galaxies (NGC 7317, 18A and 19) with very low velocity dispersion (cz= 6563, 6620 and 6650 respectively; see Moles et al. 1997; hereafter paper I). NGC 7320 (800 ) and NGC 7318B (5765 ) complete the apparent compact group. Another accordant redshift galaxy (NGC 7320C, cz$\sim$6000 ) lies 3 arcmin ENE of NGC 7319. NGC 7320C is sufficiently faint that its proximity does not violate the HCG isolation criteria. A much brighter spiral galaxy NGC 7331 lies 30 arcmin NE of SQ and shows a redshift (cz= 821 ) similar to NGC 7320. All seven of these galaxies play a role in our interpretation of the dynamical state of SQ. Figure 1a shows a schematic of SQ and environs, while Figure 2a shows a wide band image of SQ proper. Wide field images can be found in the plates of Arp and Kormendy (1972).
We discuss new optical observations (presented in paper I), as well as published X-ray and radio data. We use them to infer a dynamical history for SQ. Section 2 considers the relationship of discordant-redshift NGC 7320 and the remaining SQ members. We consider the past and present dynamical states of the accordant group in sections 3 and 4 respectively. In section 5 we consider the implications of SQ as representative of the compact group phenomenon.
NGC 7320: A Late Type Spiral Projected on an Accordant Quartet
==============================================================
Several investigators have pointed out the low probability for NGC 7320 to be a chance projection on the accordant quartet in SQ (Burbidge and Burbidge 1961; Arp 1973). Little weight would be given to such an [*a posteriori*]{} calculation were it not for the fact that discordant components are so numerous in compact groups ($\sim$43/100 groups in HCG contain at least one discordant-redshift member; see Sulentic 1994). NGC 7320 shows some peculiarities (see paper I) that could be interpreted as evidence for interaction with the higher redshift SQ members. The most significant involves a blue tidal tail that emerges from the SE end of NGC 7320 (Arp and Kormendy 1972). If we assume that the tidal tail is evidence for recent galaxy-galaxy interaction, we are left with three possible interpretations:
\(1) it involves only the high redshift members of SQ. In this case NGC 7320 is a foreground galaxy projected on one of the background manifestations of this interaction; or
\(2) it involves a past encounter between NGC 7320 and NGC 7331. In this case, it is a foreground tail projected near the background high redshift quartet; or
\(3) it involves direct interaction between NGC 7320 and the accordant SQ. This would require the assumption of non-Doppler redshifts in one (or four) components because the velocity difference between the components is too high for a pure dynamical explanation.
The tail emerging from NGC 7320 is parallel to a second narrower and brighter tail that emerges from one of the the spiral arms in NGC 7319 (see Figure 1b) and extends in the direction of similar redshift NGC 7320C. This fact clearly favors hypothesis 1 because it relates the “NGC7320 tail” to one that is unambiguously related to interaction involving only higher redshift members. The tail emerging from NGC 7320 can actually be traced even farther than the brighter one, and it extends directly to NGC 7320C (Arp and Lorre 1976). Ambiguity in the hypothesis 1 interpretation stems from the fact that NGC 7331 is located in the same direction, but beyond NGC 7320C. Further support for hypothesis 1 comes from a low-redshift H$\alpha$ image (provided by W. Keel) shown in Figure 2b. The distribution of HII regions in NGC 7320 is symmetric about its nucleus with no evidence for any low redshift H$\alpha$ emission in the tidal tail.
Standard redshift-independent distance estimates for NGC 7320 as well as NGC 7317 and 18A,B have been attempted (paper 1; see also Kent 1981; Sulentic 1994). It is unclear whether techniques calibrated with normal galaxies can be reasonably applied to galaxies that are suspected to be, or are manifestly abnormal. The debate over the interpretation of the HI data illustrates this point very well (Sulentic 1994). On the other hand, we show in paper I that some of the galaxies in SQ show only small disturbances (including NGC 7320, if the tail does not belong to it) and, therefore employ normal distance estimators. Indeed, those standard techniques yield distances consistent with the redshift-implied values, i. e., $\sim$ 10 Mpc for the low- and $\sim$ 65 Mpc for the high-redshift components.
An independent argument for two distances in SQ involves consistency of SQ redshifts with those measured in the surrounding field. Galaxies projected within about one degree of SQ show two preferred redshifts near cz= 800 and 6500 (Lynds 1972; Materne and Tammann 1974; Allen and Sullivan 1980 and Shostak et al. 1984). The low redshift galaxies are shaded in Figure 1a to distinguish them from galaxies with redshifts near 6500 . Two additional galaxies with redshifts similar to NGC 7320 were identified in Shostak et al. (1984) and lie outside the field shown in Figure 1a. The redshift data suggest that the two nearest supercluster structures in the direction of SQ lie at distances of about 10 and 65 Mpc respectively. Rejection of the Doppler interpretation for the high or low redshift parts of SQ would require nearby galaxies to also show discordant redshifts or a fortuitous match of discordant redshifts in SQ with cosmological shifts in the neighboring field.
Past History: Encounters with a Nearest Neighbor NGC 7320C
==========================================================
We interpret the kernel of three similar redshift members in SQ as a physical triplet and the core of SQ. These galaxies and NGC 7320C are hatched in Figure 1b. The morphological evidence for physical membership in the triplet is least ambiguous for NGC 7319. NGC 7317 and 7318A show early-type (E2) morphologies and are therefore less sensitive to the effects of gravitational encounters. It is somewhat surprising to find an E-dominated triplet in the field at all, but this morphology might reflect past secular evolutionary effects within the group. However, neither of these ellipticals show an unusual color or signs of geometrical distortion (Schombert et al. 1990; Zepf et al 1991; paper I). The strongest evidence for physical membership of NGC 7317 and 18A comes from the diffuse optical light that surrounds them and that is probably caused by dynamical stripping processes (Arp 1973; Arp and Lorre 1976; Schombert et al. 1990; paper I). NGC 7318B with $\Delta$V$\sim -$850 relative to the triplet mean velocity is assumed to be a recent arrival and not relevant for the past dynamical history of SQ (see next section).
The past dynamical history must account for evidence that points to past interactions, especially the following: (1) parallel tidal tails (Arp and Kormendy 1972); (2) stripping of most of the HI from NGC 7319 (Shostak et al. 1984) and (3) diffuse optical light that surrounds the triplet kernel. The first two observations are probably related because an extrapolation backwards of the brighter tail passes very close to the projected center of N7319. The parallel extension of the tails and the fact that both point towards NGC7320C suggest a common origin involving the galaxy. Simulations of galaxy interactions do not usually produce parallel tails (e.g. Howard et al. 1993) leading us to propose that we may be observing the remnants of two past encounters which suggest that NGC 7320C may be a loosely bound member of the system. The tails can reasonably be interpreted as mapping the trajectory of these past encounters if they were caused by NGC 7320C, an hypothesis that we favor here. In the recent past NGC 7320C would have been inside the group and would have formed an accordant (defined as $\Delta$V$<$10$^3$ km s$^{-1}$from group median) quartet with the triplet kernel. That accordant quartet would have satisfied the HCG selection criteria only if NGC 7320 were not superposed. We assume that NGC 7318B had not yet arrived on the scene.
Naturally the spiral member of the SQ kernel (NGC 7319) shows the most dramatic evidence for tidal disruption: 1) the spiral arms are asymmetric, with the eastern arm split into two concentric arcs and the western one ending in one of the tidal tails; 2) the spiral arms show no HII regions on H$\alpha$ images that show them in and near NGC 7318A,B and NGC 7320 (Arp 1973; paper I); 3) NGC 7319 has been stripped of essentially all HI and 4) it shows a Seyfert nucleus and associated radio/optical jet (see paper I and Aoki et al. 1996). Figure 2c shows an H$\alpha$ image (again thanks to W. Keel) centered on NGC7319 that illustrates points 2 and 4. High resolution 21cm observations (Shostak et al. 1984) reveal two extended clouds of HI with redshift similar to the stable kernel (see Figure 1b). The total mass of stripped HI with velocity near 6500 km/s is 1.4$\times$10$^{10}$ h$^{-1}$ M$_{\odot}$ ($ h$ = H$_o$/100). This exceed the HI mass of luminous Sb spirals like NGC 7319 by more than a factor of three and implies that some of the stripped gas may have come from other galaxies. The proposed intruder, NGC 7320C, shows a spiral or ringed morphology without evidence for any non-stellar material. The HI velocity listed for NGC 7320C in RC3 catalog is taken from old grid mapping with the Arecibo telescope. Higher resolution and sensitivity observations (Shostak et al. 1984) report no HI at the position of NGC 7320C, therefore some of the stripped HI within SQ may have originated from that galaxy as well. We conclude that most of the damage to NGC 7319 was caused by a direct collision with NGC 7320C (see also Shostak et al. 1984).
The twin tails and diffuse light in SQ show blue colors consistent with recent or ongoing star formation. Grid photometry obtained for SQ by Schombert et al. (1990) find B$-$V $=$ 0.57 for the brighter tail which is almost as blue as the foreground Sd spiral NGC 7320. The luminosity of this tail represents about 18% of the luminosity of NGC 7319. The brightest part of the fainter tail was also detected and showed colors in the range 0.1-0.7. Similarly the diffuse light shows colors similar to the disks of spiral galaxies B-V= 0.5-0.7. The interstellar matter (ISM) in NGC 7319 and possibly other components were stripped and heated to $\sim$10$^6$ K (by analogy to the similar ongoing event involving NGC 7318B, see next section). The clouds would then expand, recombine and cool over a time-scale $\sim$5$\times$10$^8$ years. Deep images reveal condensations consistent with the size, color and brightness of HII regions (or blue clusters) scattered throughout the halo (see e.g. Arp and Lorre 1976). The data supports the hypothesis that much of the stripping occurred in the past few 10$^8$ years and also that some tidally induced star formation is occurring or has recently occurred, in the halo and tails.
If the tidal tails map the trajectory of NGC 7320C then it has traveled $\sim$105 h$^{-1}$ kpc on the plane of the sky since the most recent passage through SQ. NGC 7320C shows an approximate line of sight velocity 700 lower than NGC 7319 (a ratio of apo- to pericenter of 5-6 assuming NGC 7318A is near the group center of mass). Assuming a transverse velocity equal to the line of sight value suggests that the collision occurred as recently as 1.5$\times$10$^8$h$^{-1}$ years ago. The fainter tail is more diffuse (2$\times$ broader than the brighter one) which is consistent with the idea that it represents an earlier passage at least t$\geq$ 5$\times$10$^8$ years ago. The roles of NGC 7317 and 7318A in the past dynamical activity or even their past morphologies are impossible to ascertain. An earlier encounter involving NGC 7320C and the southwestern-most tail however would provide a mechanism to account for much of the $\sim$6500 HI and blue halo condensations that are seen west of NGC 7319. The implied trajectory of that tail passes under NGC 7320 and is consistent with a past encounter involving NGC 7318A. We cannot rule out the possibility that NGC 7318A is the remnant bulge of an early spiral whose disk was disrupted in the past (see paper I). In such a view the high early-type galaxy fraction would be a product of dynamical evolution in SQ.
Present History: A Penetrating Encounter with NGC 7318B
=======================================================
The present dynamical history of SQ centers on NGC 7318B which shows an 850 radial velocity difference relative to the SQ kernel (these four galaxies are hatched in Figure 1c). This is so large that one might argue that this galaxy is unrelated to the group (an accordant redshift projection) or that it has been dynamically ejected by the triplet kernel. Two observational clues, however, favor the interpretation that NGC 7318B is currently colliding with SQ and for the first time. NGC 7320C is not actively involved in the dynamical evolution of the group at this time.
First, we have reanalyzed the lower velocity HI clouds detected in SQ. Shostak et al. (1984) found four, spatially-distinct HI clouds in SQ with velocities near 5700, 6000 and 6600 (two clouds) . The latter clouds were discussed in the previous section. We interpret the two lower velocity clouds as a single feature. They are adjacent to one another and are centered on the nucleus of NGC 7318B (indicated schematically in Figure 1c or see Figure 4 in Shostak et al. 1984 and paper I). These clouds were previously assumed to be stripped material like the more massive and extensive $\sim$6500 HI clouds. Our reanalysis suggests that they are still associated with the spiral disk of NGC 7318B. That galaxy can plausibly be assigned an Sb or SBb type and therefore would be expected to have a considerable ($\sim$5$\times$10$^9$M$_{\odot}$) HI mass. They show a velocity gradient consistent with rotation and a central hole coincident with the nuclear bulge region of that galaxy (see Figure 6 in the previous reference). Such holes in the 2D distribution of HI in spiral galaxies with a prominent population II bulge are often seen. The velocity profile of these two clouds taken together show a slightly double-horn structure characteristic of a rotating inclined spiral galaxy. The total HI mass would be approximately 4.1$\times$10$^9$ h$^{-1}$ M$_{\odot}$ with a FWZI velocity gradient of 400 km/s consistent with values for spiral galaxies of similar type and luminosity. The association of this much neutral gas with NGC 7318B suggests that it still retains a significant part of its ISM and, therefore has not previously passed through SQ.
The second evidence for a direct collision between NGC 7318B and SQ involves VLA radio and recent X-ray data of SQ. ROSAT-HRI mapping (Pietsch et al. 1997) reveals an elongated structure on the east side of NGC 7318B partly coincident with one of its spiral arms. This structure shows a rather close spatial correspondence with a radio continuum arc (Van der Hulst and Rots 1981). This structure appears to be sharply bounded and is most easily interpreted as a shock front (it is indicated in Figure 1c and is also well seen in the H$\alpha$ image shown in Figure 2c). The origin of the shock would most plausibly be ascribed to a high velocity ($\Delta$V$\approx$10$^3$ ) collision between the ISM in NGC 7318B and the stripped gas in SQ. The collision must be ongoing and in its early stages because NGC 7318B retains most of its HI, many HII regions and a reasonably symmetric spiral pattern. Time limits on the duration of this event come from two sources: 1) the lifetime of synchrotron electrons in the radio arc must be a few $\times$ 10$^7$ years without a local source of acceleration (van der Hulst and Rots 1981) and 2) the observed line of sight velocity difference between SQ and NGC 7318B give a similar timescale for the duration of the intruder passage through the group.
In our view, NGC 7318B has replaced NGC 7320C as a high velocity intruder in SQ. The shock is assumed to be associated with the ongoing stripping of NGC 7318B. Optical data also support this view, and the support becomes even more striking when the image of NGC 7318A is subtracted (see paper I). Most of the complex structure around NGC 7318A and B is consistent with spiral structure associated with the latter high velocity intruder. HII regions can also be seen in the spiral arms as a further indication that this galaxy has not previously sustained a major stripping encounter. The radio/Xray feature interpreted as a shock front indicates the interface between the unstripped gas disk of NGC 7318B and the ISM in SQ. The narrowness of the shock front suggests that the collision is most easily modeled with NGC 7318B entering the far side of SQ oriented nearly edge-on.
The Implications of SQ as a Typical Compact Group
=================================================
The twin problems posed by compact groups center around the large number of discordant redshift components that they contain, and their gravitational stability. In SQ, at least, the evidence favors the chance projection explanation for discordant NGC 7320. The gravitational stability issue is a problem because attempts at modeling the groups (e.g. Mamon 1986; Barnes 1989) indicate that they should be unstable to collapse on short time scales. In this case the number of compact groups observed today implies that a large merger post-cursor population should exist. Little or no evidence is found for either the mergers in progress (Zepf et al 1991; Moles et al 1994) or for the merger post-cursor population (Sulentic and Rabaça 1994). This has led some to propose that the groups are not real physical systems at all (e.g. Mamon 1986; Hernquist et al. 1995). The large volume of data for SQ can shed some light on these problems, particularly, if our interpretation of its dynamical history is correct [**and**]{} if it is representative of the compact group phenomenon.
Our analysis indicates that SQ is not simply a projection of accordant redshift galaxies. It is a dynamically active system and evidence for this activity is virtual proof that it is a physical aggregate. However the compact group aspect (defined as N$\geq$ 4 accordant members) is transient in the sense that no fourth tightly bound component can be identified at this time. Instead SQ contains a kernel of three galaxies that must have sufficient mass to attract near neighbors into high velocity encounters. Our best estimate (paper 1) for the combined stellar (galaxy+halo), HI and X-ray gas mass is M$_{SQ}$$\sim$ 0.75-1.5$\times$10$^{12}$ h$^{-1}$M$_\odot$. Infall velocities $\sim$800 km s$^{-1}$ were found in n-body simulations of groups with total mass $\sim$2$\times$10$^{13}$M$_\odot$ (Governato et al. 1996). The infalling galaxies result in dynamical activity that is distinctly episodic. Any catalog of isolated groups will be biased against many of the groups with a currently infalling intruder because the intruder will often “bridge” the group into its environment and prevent it from satisfying an isolation criterion. NGC7318B is an example of a “safe” (because it is internal) intruder and NGC 7320C is one that almost prevented SQ from satisfying the HCG sample selection criteria.
Is SQ a stable virialized system? This question was first directed towards SQ by Limber and Mathews (1960) before the redshift of NGC 7320 was known. If one repeats this calculation (with M/L$_B$= 13h for NGC 7317, 18A and 20C as well as M/L$_B$= 8h for NGC 7318B and 19) it suggests that the triplet kernel is stable with 2T/-$\Omega$$\sim$0.8 but that inclusion of NGC 7318B and 20C yields 2T/-$\Omega$$\sim$ 11. NGC7320C forms a marginally stable quartet with the triplet kernel with 2T/-$\Omega$$\sim$ 2. These estimates do not take into account the large amount of (baryonic) mass present in the halo or the possible role of non-baryonic matter that might be needed to produce the observed infall velocities. The virial calculation and optical images (the twin tidal tails) interpreted as evidence for recent passage(s) through the group suggests that NGC 7320C has been recently captured by the kernel. Finally, NGC 7318B is almost certainly entering the group for the first time. Thus, SQ is probably a bound triplet that has captured a fourth member (NGC 7320C) within the past Gyr and is now being visited by another (NGC 7318B).
Governato et al. (1996) have advanced one of the most complete scenarios for the formation and evolution of compact groups. They create groups in a critical universe by 1) seeding them with primordial merger events and 2) growing present day groups with secondary infall onto these seeds. Our study of SQ strongly supports the second part of this model including the conclusion that the infall will be high velocity and will resist rapid merging. It also supports the qualitative discussion of Moore et al. (1996) that sees high velocity intruders as an effective means to dynamically evolve galaxies and create diffuse halos. They consider random high velocity encounters in clusters while compact groups are found in non-cluster environments (Sulentic 1987). While high relative velocities in clusters are due to the high internal velocity dispersion, the lower mass of small groups such as SQ implies that only infalling galaxies can achieve velocities V$>$ 500 km s$^{-1}$.
In our case the seed is not a single merger remnant as proposed by Governato et al. (1996) but a triplet. The over-representation of luminous elliptical galaxies in the triplet suggests that it may also have experienced strong dynamical evolution as it formed perhaps by a random initial encounter between three spirals(?). It is not clear if dynamical evolutionary effects have significantly altered the properties of NGC 7317 or NGC 7318A because their morphologies, colors and kinematics appear rather normal. Thus they could be ellipticals or remnant spiral bulges. Existing images do not allow a distinction between these possibilities. An extrapolation of the tail under NGC 7320 is consistent with an encounter trajectory where NGC 7320C would have passed very near NGC 7318A as recently as $\sim$ 1Gyr ago. There is sufficient stripped gas to account for three or four spiral galaxies. The presence of radio continuum and X-ray emission in the nuclear regions of NGC 7318A may indicate an active past history.
A luminous halo surrounds the triplet which indicates that it is a dynamically evolved physical system. Our conservative estimate of the halo luminosity (V band: see paper 1) gives M$_{halo}$= -20.9 + 5log(h)$\geq$ M$_*$. This is almost ten times the luminosity of the tail created in the most recent passage of NGC7320C. Neither tail was included in this estimate but each will increase the halo luminosity by $\sim$ 10%. It is the high velocity intruders that cause SQ to grow (in galaxy population and halo mass) and that prevents the triplet kernel from coalescing by injecting kinetic energy into the group. In this view SQ must be a relatively young group unless NGC7320C has been perturbing it for a longer time; otherwise the triplet should have merged. This would require the triplet to form and dynamically evolve in the past 1-2 Gyr. Assuming that the halo was created by similar processes (a tidal tail at a time) and at about the same rate (one M$_V$= -18.4 + 5log(h) tail per 5$\times$10$^8$ years), would imply an age of several Gyr for its formation. SQ may not be primordial but the stable-kernel part of it is at least 2 Gyr old.
Neither of the elliptical components in the kernel show luminosities or other properties consistent with having been recent mergers. In this respect SQ is typical of other compact groups where no evidence for ongoing merging is seen (Zepf and Whitmore 1991; Moles et al. 1994). If our view is correct, then rather than mergers, these ellipticals may be the dynamically evolved remnants of spiral galaxies. SQ is also similar to other compact groups in the sense that its optical and FIR emission properties indicate a lower level of current star formation activity than is observed in pairs (Sulentic and de Mello Rabaça 1994; Moles et al. 1994). SQ suggests that the lack of starburst activity in compact groups is due to the lack of bound gaseous disks in many component galaxies. In SQ a major stripping event happened in the recent past and another is in progress. The gas is either stripped and neutral or shocked and hot. A large intergroup star forming region is observed in H$\alpha$ emission within the group but outside of the galaxies (see Figure 2c). However so much of the gas in SQ is stripped or shocked that it is too diffuse(cool or hot) to form large numbers of stars. The FIR emission from SQ is not strong especially when allowances are made for possible contributions from NGC7320 and the Seyfert nucleus in NGC7319. The excess FIR emission expected from these sources apparently cannot compensate for the deficit emission from SQ component galaxies. The transition from a normal star forming disk to the shocked state should be quite sudden given the high velocity of this intruder. Given the efficiency and quasi-periodic nature of the tidal perturbation in compact group, it is tempting to ascribe the Seyfert activity in NGC7319 to the past intruders as well. Whatever gas was not stripped may have been rapidly channeled into the nucleus to fuel active galactic nucleus (AGN) activity.
SQ suggests that compact groups consist of a tightly bound subsystem (kernel or seed) plus a loosely bound, or even unbound, population of infalling neighbors. The most common situation for HCG groups would involve a triplet or quartet acting as a seed plus 1 or two intruders. Indeed, it is easier to form a bound pair than a triplet, but given the n$\geq$4 number criterion used in HCG, triplets would be more often selected because it is easier to have a 3+1 rather than 2+2 configuration. The kind of dynamical encounters do not lead to rapid merging suggested by many models. The intruders are stripped which reduces their mass and cross section to frictional effects. At the same time they inject energy into the kernel which sustains it against collapse. Some stars form but not as many as commonly observed in more dynamically stable pairs. The more frequent and high impulse events in compact groups may also foster AGN activity at a higher rate than pairs do. It is perhaps appropriate that the first compact group discovered more than a century ago provides the clearest clues to their origin and evolution.
Allen, R.J., and Sullivan III, W.T. 1980, , 184, 181
Aoki, K., Ohtani, H., Yoshida, M., and Kosugi, G. 1996, , 111, 140
Arp, H. 1973, , 183, 411
Arp, H., and Kormendy, J. 1972, , 178, L111
Arp, H., and Lorre, J. 1976, , 210, 58
Barnes, J.E. 1989, Nature, 338, 123
Burbidge, E.M., and Burbidge, G.R. 1961, , 134, 244
Bushouse, H. 1987, , 320, 49
Governato, F., Tozzi, P. and Cavaliere, A. 1996, , 458, 18
Hernquist, L., Katz, N. and Weinberg, D. 1995, , 442, 57
Hickson, P. 1982 , 255, 382
Howard, S., Keel, W., Byrd, G. and Burkey, J., 1993, , 417, 502
Kent, S. 1981 , 93, 554
Limber, D. and Mathews, W. 1960, , 132, 286
Lynds, C.R. 1972, [*In External Galaxies and Quasi-Stellar Objects*]{}, ed. D.S. Evans, (Reidel:Dordrecht), p. 376
Mamon, G.A. 1986, , 307,426
Materne, J. and Tammann, G. 1974, , 35, 441
Moles, M., del Olmo, A., Perea, J., Masegosa, J., Márquez, I., and Costa, V. 1994, , 285, 404
Moles, M., Márquez, I., and Sulentic, J.W., 1997, A & A submitted, paper I.
Moore, B., Katz, N., Lake, G., Dressler, A., and Oemler Jr., A. 1996, Nature, 379, 613
Pietsch, W., Trinchieri, G., Arp, H. and Sulentic, J. 1997, , in press
Schombert, J.M., Wallin, J.F. and Struck-Marcell, C. 1990, , 99, 497
Shostak, G.S., Sullivan III, W.T., and Allen, R.J. 1984, , 139, 15
Sulentic, J.W. 1987, , 322, 605
Sulentic, J.W. 1994, In [*Progress in New Cosmologies*]{}, eds. H. C. Arp et al., (Plenum) p. 49
Sulentic, J.W., and de Mello Rabaça, D.F. 1994, , 410, 520
Sulentic, J.W., and Rabaça, C. 1994, , 429, 531
van der Hulst, J.M., and Rots, A.H. 1981, , 86, 1775
Zepf, S.E., Whitmore, B.C., and Levison, H.F. 1991, , 383, 524
|
---
abstract: 'We propose to construct a nonreciprocal single-photon frequency converter via multiple semi-infinite coupled-resonator waveguides (CRWs). We first demonstrate that the frequency of a single photon can be converted nonreciprocally through two CRWs, which are coupled indirectly by optomechanical interactions with two nondegenerate mechanical modes. Based on such nonreciprocity, two different single-photon circulators are proposed in the T-shaped waveguides consisting of three semi-infinite CRWs, which are coupled in pairwise by optomechanical interactions. One circulator is proposed by using two nondegenerate mechanical modes and the other one is proposed by using three nondegenerate mechanical modes. Nonreciprocal single-photon frequency conversion is induced by breaking the time-reversal symmetry, and the optimal conditions for nonreciprocal frequency conversion are obtained. These proposals can be used to realize nonreciprocal frequency conversion of single photons in any two distinctive waveguides with different frequencies and they can allow for dynamic control of the direction of frequency conversion by tuning the phases of external driving lasers, which may have versatile applications in hybrid quantum networks.'
author:
- 'Xun-Wei Xu'
- 'Ai-Xi Chen'
- Yong Li
- 'Yu-xi Liu'
bibliography:
- 'ref.bib'
title: 'Nonreciprocal single-photon frequency converter via multiple semi-infinite coupled-resonator waveguides'
---
Introduction
============
To build a hybrid quantum network, by harnessing advantages of different systems [@WallquistPS09; @XiangZLRMP13], we have to tackle an important problem: how to integrate different components that don’t operate at the same frequency. One solution is to build a photon frequency converter which converts the input photons of one frequency into the output photons of another frequency. Traditionally, photon frequency conversion is demonstrated by three-wave mixing in second-order nonlinear materials [KumarOL90,JHuangPRL92,RakherNPT10,IkutaNC11,ZaskePRL12,AtesPRL12,AbdoPRL13]{} or four-wave mixing in third-order nonlinear materials [McGuinnessPRL10,RadnaevNPy10,FarnesiPRL14]{}. With the development of circuit quantum electrodynamics, frequency conversion was even proposed in a single three-level superconducting quantum circuit by three-wave mixing [YXLiuSR14,YJZhaoArx15]{} or a single qubit in the ultrastrong coupling regime [@KockumArx17]. Moreover, single-photon frequency converters have been proposed in the one-dimensional (1D) linear waveguide [BradfordPRL12,BradfordPRA12,WBYanSR13]{} or 1D coupled-resonator waveguides (CRWs) [@LZhouPRL13; @ZHWangPRA14] with a three-level system coupled to different channels. Since the mechanical resonators can be coupled to various electromagnetic fields with distinctively different wavelengths through radiation pressure (for reviews, see Refs. [KippenbergSci08,MarquardtPhy09,AspelmeyerPT12,AspelmeyerARX13,MetcalfeAPR14]{}), frequency conversion has been demonstrated via two optical cavities with different frequencies, coupled by a single mechanical resonator via optomechanical interactions [SafaviNaeiniNJP11,YDWangPRL12,LTianPRL12,JTHillNC12,YLiuPRL13,CDongAP15,LecocqPRL16]{}. Recently, the conversion between microwave and optical frequencies has been implemented in the electro-optomechanical systems [BochmannNPy13,BagciNat14,AndrewsNPy14,RuedaOptica16]{}.
Besides frequency converters, isolators and circulators are also dispensable elements in constructing hybrid quantum networks for protecting some elements from unwanted noises or retracing fields [@JalasNPT13]. It is well known that, the systems with broken time-reversal symmetry can be used to construct isolators or circulators. In recent years, as a non-magnetic strategy, optical nonreciprocity in the coupled cavity modes with relative phase has drawn more and more attentions, and many different structures have been proposed theoretically [KochPRA10,HabrakenNJP12,RanzaniNJP14a,XuXWPRA15,RanzaniNJP15a,YPWangSR15a,SliwaPRX15,SchmidtOpt15,MetelmannPRX15,FangKArx15,XXuarX17a,FXSunarX17]{} and demonstrated experimentally [@RuesinkNC16a; @KFangNPy17a].
In a recent work, we have proposed a nonreciprocal frequency converter in an electro-optomechanical system with a microwave mode and an optical mode, coupled indirectly via two nondegenerate mechanical modes [@XWXuPRA16a]. Due to the broken time-reversal symmetry, the nonreciprocity is obtained when the transmission of photons from one mode to the other one is enhanced for constructive quantum interference while the transmission in the reversal direction is suppressed with destructive quantum interference. Based on a similar mechanism, nonreciprocal frequency conversion was explored theoretically [MetelmannarX16a,LTianarx16a,MiriarX16a]{} and realized experimentally [@BernierarX16a; @PetersonarX17; @BarzanjeharX17] in many different systems.
In this paper, we propose a nonreciprocal single-photon frequency converter, consisting of two or three 1D semi-infinite CRWs with different frequencies, which are coupled indirectly by nondegenerate mechanical modes via optomechanical interactions. In quantum networks, this system can also be viewed as quantum channels (1D semi-infinite CRWs) connected by a quantum node (the optomechanical systems [@XWXuPRA16a]). Different from the previous studies on nonreciprocal frequency conversion [XWXuPRA16a,MetelmannarX16a,LTianarx16a,MiriarX16a,BernierarX16a]{}, we consider the dispersion relations of the quantum channels, which play an important role in single-photon frequency conversion. Also unlike the previous studies on single-photon nonreciprocity in 1D CRWs [@XWXuPRA17; @SYangArx09; @XQLiPRA15], in this work, the frequencies of the CRWs are very different and they can not be coupled together directly. The addition of optomechanical systems (or mechanical modes) to the frequency converter offers the possibility to enable nonreciprocal frequency transduction between two CRWs with distinctively different frequencies and allows for dynamic control of the direction of frequency conversion by tuning the phases of external driving lasers.
The paper is organized as follows: In Sec. II, we propose a single-photon frequency converter using two CRWs, coupled indirectly by two nondegenerate mechanical modes via optomechanical interactions. In Secs. III and IV, two different single-photon circulators are proposed in the T-shaped waveguides consisting of three semi-infinite CRWs, which are mutually coupled by optomechanical interactions. One circulator uses two nondegenerate mechanical modes and the other one uses three nondegenerate mechanical modes. Finally, we summarize our results in Sec. V.
Nonreciprocal single-photon frequency converter
===============================================
Theoretical model and scattering matrix
---------------------------------------
![(Color online) Schematic diagram of a waveguide consisting of two semi-infinite CRWs ($a_{j}$ and $b_{j}$ for $j\geq 0$) with different frequencies (e.g., one is optical CRW and the other one is microwave CRW) coupled indirectly by the two mechanical modes ($d_{1}$ and $d_{2}$).[]{data-label="fig1"}](fig1.eps){width="8.5"}
As schematically shown in Fig. \[fig1\], a waveguide consists of two semi-infinite coupled-resonator waveguides (CRWs) with different frequencies (e.g., one is optical CRW and the other one is microwave CRW), in which both end side cavities are coupled to two mechanical modes via optomechanical interactions. The semi-infinite CRWs, as quantum channels for single-photon transmission, are made by infinite identical single-mode cavities, which are coupled to each other through coherent hopping of photons between neighboring cavities [WallraffNat04,HartmannNP06,NotomiNPT08,CastellanosAPL07]{}. The two end side cavities ($a_{0}$ and $b_{0}$), coupled indirectly by the two mechanical modes ($d_{1}$ and $d_{2}$), are served as a quantum node for single-photon frequency conversion. The total system can be described by the Hamiltonian $$\label{eq1}
H_{0}=\sum_{l=a,b}H_{l}+H_{m}+H_{\mathrm{int}}$$with the Hamiltonian $\sum_{l=a,b}H_{l}$ for the two CRWs $$\label{eq2}
H_{l}=\sum_{j=0}^{+\infty }\left[ \omega _{l}l_{j}^{\dag }l_{j}-\xi
_{l}\left( l_{j}^{\dag }l_{j+1}+\mathrm{H.c.}\right) \right] ,$$the Hamiltonian $H_{m}$ for the mechanical modes$$H_{m}=\omega _{1}d_{1}^{\dag }d_{1}+\omega _{2}d_{2}^{\dag }d_{2},$$and the interaction terms $H_{\mathrm{int}}$ for single-photon frequency conversion$$\begin{aligned}
H_{\mathrm{int}} &=&\left( g_{a,1}a_{0}^{\dag }a_{0}+g_{b,1}b_{0}^{\dag
}b_{0}\right) \left( d_{1}+d_{1}^{\dag }\right) \notag \\
&&+\left( g_{a,2}a_{0}^{\dag }a_{0}+g_{b,2}b_{0}^{\dag }b_{0}\right) \left(
d_{2}+d_{2}^{\dag }\right) \notag \\
&&+\sum_{l=a,b}\sum_{i=1}^{2}\left( l_{0}\Omega _{l,i}e^{i\omega _{l,i}t}+%
\mathrm{H.c.}\right) ,\end{aligned}$$where $l_{j}$ ($l_{j}^{\dag }$, $l=a,b$) is the bosonic annihilation (creation) operator of the $j$th cavity with the same resonant frequency $%
\omega _{l}$ and the same coupling strength $\xi _{l}$ between two nearest neighboring cavities in the CRW-$l$. $\omega _{i}$ ($i=1,2$) is the resonant frequency of the mechanical mode with the bosonic annihilation (creation) operator $d_{i}$ ($d_{i}^{\dag }$). $g_{l,i}$ is the optomechanical coupling strength between cavity $l_{0}$ ($l_{0}=a_{0},b_{0}$) and mechanical mode $d_{i}$ ($%
d_{i}=d_{1},d_{2}$). The cavity $a_{0}$ ($b_{0}$) is driven by a two-tone laser at frequencies $\omega _{a,1}=\omega _{a}-\omega _{1}+\Delta _{a,1}$ and $%
\omega _{a,2}=\omega _{a}-\omega _{2}+\Delta _{a,2}$ ($\omega _{b,1}=\omega
_{b}-\omega _{1}+\Delta _{b,1}$ and $\omega _{b,2}=\omega _{b}-\omega
_{2}+\Delta _{b,2}$) with amplitudes $\Omega _{a,1}$ and $\Omega _{a,2}$ ($%
\Omega _{b,1}$ and $\Omega _{b,2}$). For simplicity, we assume that $\Delta _{1}\equiv \Delta _{a,1}=\Delta _{b,1}$ and $\Delta _{2}\equiv \Delta _{a,2}=\Delta _{b,2}$. Thus the operators for the cavity modes can be rewritten as the sum of the quantum fluctuation operators and classical mean values, i.e., $a_{j}\rightarrow a_{j}+\sum_{i=1}^{2}\alpha^{a}
_{j,i}e^{-i\omega_{a,i}t}$ and $b_{j}\rightarrow
b_{j}+\sum_{i=1}^{2}\alpha^{b} _{j,i}e^{-i\omega _{b,i}t}$, where $a_{j}$ and $b_{j}$ on the right side of the arrow symbols describe the quantum fluctuation operators of the cavity modes, and the classical amplitude $\alpha^{l} _{j,i}$ is determined by the amplitudes $%
\Omega _{l,i}$, the frequency $\omega _{l,i}$, the damping rates $\kappa
_{a,j}$ and $\kappa _{b,j}$ of the cavities and the damping rates $\gamma
_{1}$ and $\gamma _{2}$ of the mechanical modes.
To obtain a linearized Hamiltonian, we assume that the external driving is strong, i.e. $|\alpha^{l} _{0,i}|\gg 1$, the system works in the resolved-sideband limit with respect to both mechanical modes, i.e. $\min
\left\{ \omega _{1},\omega _{2}\right\} \gg \max \left\{ \kappa
_{a,j},\kappa _{b,j}\right\} $, and the two mechanical modes are well separated in frequency, i.e. $\min \left\{ \omega _{1},\omega
_{2},\left\vert \omega _{1}-\omega _{2}\right\vert \right\} \gg \max \left\{
|g_{l,i}\alpha^{l} _{0,i}|,\gamma _{1},\gamma _{2}\right\} $. After making the standard linearization under the rotating-wave approximation, in the rotating reference frame with respect to $H_{\mathrm{rot}}=\sum_{l=a,b}\sum_{j=0}^{+%
\infty }\omega _{l}l_{j}^{\dag }l_{j}+\sum_{i=1,2}\left( \omega _{i}-\Delta
_{i}\right) d_{i}^{\dag }d_{i}$, the linearized Hamiltonian with time-independent terms becomes $$\label{eq5}
H_{\mathrm{fc}}=\sum_{l=a,b}H_{l}+H_{m}+H_{\mathrm{int}},$$where $H_{l}$, $H_{m}$, and $H_{\mathrm{int}}$ are replaced by $$\label{eq6}
H_{l}=-\xi _{l}\sum_{j=0}^{+\infty }\left( l_{j}^{\dag }l_{j+1}+\mathrm{H.c.}%
\right) ,$$$$\label{eq7}
H_{m}=\Delta _{1}d_{1}^{\dag }d_{1}+\Delta _{2}d_{2}^{\dag }d_{2},$$$$\begin{aligned}
\label{eq8}
H_{\mathrm{int}} &=&J_{a,1}(a_{0}^{\dag }d_{1}+a_{0}d_{1}^{\dag }) \notag \\
&&+J_{b,1}(e^{-i\phi }b_{0}^{\dag }d_{1}+e^{i\phi }b_{0}d_{1}^{\dag })
\notag \\
&&+J_{a,2}(a_{0}^{\dag }d_{2}+a_{0}d_{2}^{\dag }) \notag \\
&&+J_{b,2}(b_{0}^{\dag }d_{2}+b_{0}d_{2}^{\dag }).\end{aligned}$$Here $J_{l,i}e^{i\phi _{l,i}}=g_{l,i}\alpha^{l} _{0,i}$ is the effective optomechanical coupling strength between the cavity $l_{0}$ ($%
l_{0}=a_{0},b_{0}$) and mechanical mode $d_{i}$ ($d_{i}=d_{1},d_{2}$) with real strength $J_{l,i}=\left\vert g_{l,i}\alpha^{l} _{0,i}\right\vert $ and phase $\phi _{l,i}$. As only the total phase $\phi =\phi
_{a,1}+\phi _{a,2}+\phi _{b,1}+\phi _{b,2}$ has physical effects, without loss of generality, $\phi $ is only kept in the terms of $%
b_{0}d_{1}^{\dag }$ and $b_{0}^{\dag }d_{1}$ in Eq. (\[eq8\]) and the following derivation. It should be noted that $\phi $ and $J_{l,i}$ are dynamically tunable parameters, which can be controlled by tuning the strengths and phases of the external driving fields. The time-reversal symmetry of the whole system is broken when we choose the phase $\phi \neq n\pi $ ($n$ is an integer). As we will show later, the direction of frequency conversion can be controlled dynamically by tuning the value of the total phase $\phi$.
In this paper, we assume that the damping rates of the cavities in the CRWs are much smaller than the coupling strength between two nearest neighboring cavities and the effective optomechanical coupling strength, i.e. $%
\left\{\xi _{l},J_{l,i}\right\}\gg \max \left\{ \kappa _{a,j},\kappa
_{b,j}\right\} $, so that we can only consider the coherent scattering in the CRWs. Moreover, we assume that $\left\{\xi
_{l},J_{l,i},\gamma _{2}\right\}\gg \gamma _{1}$, so that $\gamma _{1}$ can be neglected in the following calculations and the Hamiltonian for two mechanical modes with $\gamma \equiv \gamma _{2}$ is described by $$H_{m}=\Delta _{1}d_{1}^{\dag }d_{1}+\left( \Delta _{2}-i\gamma \right)
d_{2}^{\dag }d_{2}.$$The mechanical damping rate $\gamma $ can be controlled by coupling the mechanical mode to an auxiliary cavity [@WilsonRaePRL07; @MarquardtPRL07; @LiYPRB08; @XWXuPRA15], and a suitable mechanical damping is another crucial condition to obtain desired nonreciprocal single-photon frequency conversion in this model [@XWXuPRA17].
To derive the sacttering matrix between different CRWs, we consider the stationary eigenstate of a single photon in the whole system as $$\begin{aligned}
\label{eq10}
\left\vert E\right\rangle &=&\sum_{j=0}^{+\infty }\left[ u_{a}\left(
j\right) a_{j}^{\dag }\left\vert 0\right\rangle +u_{b}\left( j\right)
b_{j}^{\dag }\left\vert 0\right\rangle \right] \notag \\
&&+u_{d1}d_{1}^{\dag }\left\vert 0\right\rangle +u_{d2}d_{2}^{\dag
}\left\vert 0\right\rangle ,\end{aligned}$$where $\left\vert 0\right\rangle $ indicates the vacuum state of the whole system, $u_{l}\left( j\right) $ denotes the probability amplitude in the state with a single photon in the $j$th cavity of the CRW-$l$, and $u_{d1}$ ($%
u_{d2}$) denotes the probability amplitude with a single phonon in the mechanical mode $d_{1}$ ($d_{2}$). The dispersion relation of the semi-infinite CRW-$l$ in the rotating reference frame is given by [@LZhouPRL13] $$\label{eq11}
E_{l}=-2\xi _{l}\cos k_{l},\quad 0<k_{l}<\pi ,$$where $E_{l}$ is the energy and $k_{l}$ is the wave number of the single photon in the CRW-$l$. Without loss of generality, we assume that $\xi _{l}>0$. Substituting the stationary eigenstate in Eq. (\[eq10\]) and the Hamiltonian in Eq. (\[eq5\]) into the eigenequation $H_{\mathrm{fc}%
}\left\vert E\right\rangle =E\left\vert E\right\rangle $, we can obtain the coupled equations for the probability amplitudes as$$\label{eq12}
J_{a,1}u_{d1}+J_{a,2}u_{d2}-\xi _{a}u_{a}\left( 1\right) =Eu_{a}\left(
0\right) ,$$$$J_{b,1}e^{-i\phi }u_{d1}+J_{b,2}u_{d2}-\xi _{b}u_{b}\left( 1\right)
=Eu_{b}\left( 0\right) ,$$$$J_{a,1}u_{a}\left( 0\right) +J_{b,1}e^{i\phi }u_{b}\left( 0\right) =\left(
E-\Delta _{1}\right) u_{d1},$$$$\label{eq15}
J_{a,2}u_{a}\left( 0\right) +J_{b,2}u_{b}\left( 0\right) =\left( E-\Delta
_{2}+i\gamma \right) u_{d2},$$$$\label{eq16}
Eu_{l}\left( j\right) +\xi _{l}u_{l}\left( j+1\right) +\xi _{l}u_{l}\left(
j-1\right) =0$$with $j>0$ and $l=a,b$.
If a single photon with energy $E$ is incident from the infinity side of CRW-$l$, the photon-phonon interactions in the quantum node will result in photon sacttering between different CRWs or photon absorbtion by the dissipative mechanical mode. The general expressions of the probability amplitudes in the CRWs ($j\geq 0$) are given by $$\label{eq17}
u_{l}\left( j\right) =e^{-ik_{l}j}+s_{ll}e^{ik_{l}j},$$$$\label{eq18}
u_{l^{\prime }}\left( j\right) =s_{l^{\prime }l}e^{ik_{l^{\prime }}j},$$where $s_{l^{\prime }l}$ denotes the single-photon scattering amplitude from CRW-$l$ to CRW-$l^{\prime }$ ($l,l^{\prime }=a,b$). Substituting Eqs. (\[eq17\]) and (\[eq18\]) into Eqs. (\[eq12\])-(\[eq16\]), then we obtain the scattering matrix as$$S=\left(
\begin{array}{cc}
s_{aa} & s_{ab} \\
s_{ba} & s_{bb}%
\end{array}%
\right) ,$$where $$\label{eq20}
s_{aa}=D^{-1}\left[ J_{ab}J_{ba}-\left( \xi _{a}e^{ik_{a}}+\Delta %
_{a}\right) \left( \xi _{b}e^{-ik_{b}}+\Delta _{b}\right) \right]
,$$$$\label{eq21}
s_{ba}=i2D^{-1}J_{ba}\xi _{a}\sin k_{a},$$$$\label{eq22}
s_{ab}=i2D^{-1}J_{ab}\xi _{b}\sin k_{b},$$$$s_{bb}=D^{-1}\left[ J_{ab}J_{ba}-\left( \xi _{a}e^{-ik_{a}}+\Delta
_{a}\right) \left( \xi _{b}e^{ik_{b}}+\Delta _{b}\right) \right],$$$$\label{eq24}
D=\left( \xi _{a}e^{-ik_{a}}+\Delta _{a}\right) \left( \xi
_{b}e^{-ik_{b}}+\Delta _{b}\right) -J_{ab}J_{ba}$$with the effective coupling strengths $J_{ll^{\prime }}$ and frequency shifts $\Delta _{l}$ induced by the two mechanical modes defined by $$\label{eq25}
J_{ab}\equiv\frac{J_{a,1}J_{b,1}e^{i\phi }}{\left( E-\Delta _{1}\right) }+\frac{%
J_{a,2}J_{b,2}}{\left( E-\Delta _{2}+i\gamma \right) },$$$$\label{eq26}
J_{ba}\equiv\frac{J_{a,1}J_{b,1}e^{-i\phi }}{\left( E-\Delta _{1}\right) }+\frac{%
J_{a,2}J_{b,2}}{\left( E-\Delta _{2}+i\gamma \right) },$$$$\label{eq27}
\Delta _{a}\equiv\frac{\left( J_{a,1}\right) ^{2}}{\left( E-\Delta _{1}\right) }+%
\frac{\left( J_{a,2}\right) ^{2}}{\left( E-\Delta _{2}+i\gamma \right) },$$$$\label{eq28}
\Delta _{b}\equiv\frac{\left( J_{b,1}\right) ^{2}}{\left( E-\Delta _{1}\right) }+%
\frac{\left( J_{b,2}\right) ^{2}}{\left( E-\Delta _{2}+i\gamma \right) }.$$To quantify nonreciprocity conversion, we define the scattering flows of the single photons from CRW-$l$ to CRW-$l^{\prime }$ as [@ZHWangPRA14] $$I_{l^{\prime }l}=|s_{l^{\prime }l}|^{2}\frac{\xi _{l^{\prime }}\sin
k_{l^{\prime }}}{\xi _{l}\sin k_{l}},$$where $\xi _{l}\sin k_{l}$ ($\xi _{l^{\prime }}\sin
k_{l^{\prime }}$) is the group velocity in the CRW-$l$ (CRW-$l^{\prime }$). In our model, $I_{ba}\neq I_{ab}$ implies the appearance of nonreciprocal single-photon frequency conversion, and the perfect nonreciprocal single-photon frequency conversion is obtained when $I_{ba}=1$ and $I_{ab}=0$, or $I_{ba}=0$ and $I_{ab}=1$.
Nonreciprocal single-photon frequency converter
-----------------------------------------------
![(Color online) Scattering flows $\log_{10} [I_{ab}]$ (black solid curves) and $\log_{10} [I_{ba}]$ (red dashed curves) are shown as functions of the wave number $k/\protect\pi$ for: (a) $\protect\phi=\protect\pi/2$ and $J_{2}=2\protect\xi$, (b) $\protect\phi=3\protect\pi/2$ and $J_{2}=2\protect%
\xi$, (c) $\protect\phi=\protect\pi/2$ and $J_{2}=4\protect\xi$, (d) $%
\protect\phi=3\protect\pi/2$ and $J_{2}=4\protect\xi$. The other parameters are $J_{1}=\protect\xi $, $\Delta _{1}=\Delta _{2}=0$, and $\protect\gamma$ is obtained from Eq. (\[eq32\]).[]{data-label="fig2"}](fig2.eps){width="8.5"}
![(Color online) Scattering flows $\log_{10} [I_{ab}]$ (black solid curves) and $\log_{10} [I_{ba}]$ (red dashed curves) are plotted as functions of $\Delta _{1}/\protect\xi$ in (a) and (c), and as functions of $%
\Delta _{2}/\protect\xi$ in (b) and (d) with wave numbers: (a) and (b) $k=%
\protect\pi/4$, (c) and (d) $k=3\protect\pi/4$. The other parameters are $%
\protect\phi=\protect\pi/2$, $J_{1}=\protect\xi $, $J_{2}=4\protect\xi$, and $%
\protect\gamma$ is obtained from Eq. (\[eq32\]).[]{data-label="fig3"}](fig3.eps){width="8.5"}
Before the numerical calculations of the scattering flows $I_{ab}$ and $%
I_{ba}$, it is instructive to find the optimal conditions for nonreciprocal single-photon frequency conversion analytically. For simplicity, we assume that the detunings $\Delta _{1}=\Delta _{2}=0 $, the two CRWs have the same parameters (i.e., $\xi \equiv \xi _{a}=\xi _{b} $, and $k\equiv
k_{a}=k_{b}$) and they are symmetrically coupled to the two mechanical modes ($J_{1}\equiv J_{a,1}=J_{b,1}$, $J_{2}\equiv J_{a,2}=J_{b,2} $) with $%
J_{1}=\xi $. Under the condition that $\gamma \gg \xi $, the optimal conditions for nonreciprocal single-photon frequency conversion obtained from Eqs. (\[eq20\])-(\[eq24\]) are $$\label{eq30}
\phi \approx \frac{\pi }{2}\quad \mathrm{or}\quad \frac{3\pi }{2},$$$$\label{eq31}
k\approx \frac{\pi }{4}\quad \mathrm{or}\quad \frac{3\pi }{4},$$$$\label{eq32}
\frac{\gamma }{\xi }\approx \sqrt{2\left( \frac{J_{2}}{\xi }\right) ^{4}+2}.$$In order to satisfy the condition $\gamma \gg \xi $, we should choose $J_{2}\gg \xi $ in Eq. (\[eq32\]).
Scattering flows $I_{ab}$ (black solid curve) and $I_{ba}$ (red dashed curve) as functions of the wave number $k/\pi $ are shown in Fig. \[fig2\]. The optimal nonreciprocity appears around the point $k\approx \pi/4$ and $%
3\pi/4$ for $\phi \approx \pi/2$ or $3\pi/2$, which exhibits good agreement with the analytical result shown in Eqs. (\[eq30\]) and (\[eq31\]). Specifically, when $\phi = \pi/2$, in Figs. \[fig2\](a) and \[fig2\](c), we show the reciprocal transmission from CRW-$b$ to CRW-$a$ (CRW-$a$ to CRW-$b$) at $k\approx \pi/4$ ($k\approx 3\pi/4$). In contrast, when $\phi = 3\pi/2$, we see the reciprocal transmission from CRW-$a$ to CRW-$b$ (CRW-$b$ to CRW-$a$) at $k\approx \pi/4$ ($k\approx 3\pi/4$). These imply that we can reverse the direction of frequency conversion by tuning the phase from $\phi=\pi/2$ to $\phi=3\pi/2$. Scattering flows $I_{ab}$ and $I_{ba}$ for different $J_{2}$ are shown in Figs. \[fig2\](a) and \[fig2\](c) \[or Figs. \[fig2\](b) and \[fig2\](d)\], which demonstrate that the nonreciprocity of the system improves dramatically if we take a larger value of $J_{2}/\xi$ (as well as $%
\gamma/\xi$).
The optimal conditions for nonreciprocal single-photon frequency conversion given in Eqs. (\[eq30\])-(\[eq32\]) are only applicable for zero frequency detunings $\Delta _{1}=\Delta
_{2}=0 $. The following discussions based on numerical calculations will show the effects of the frequency detunings ($\Delta _{1}$ and $\Delta_{2}$) on frequency conversion. Scattering flows $I_{ab}$ (black solid curve) and $%
I_{ba}$ (red dashed curve) as functions of the detunings $\Delta _{1}/\xi $ and $\Delta _{2}/\xi $ are shown in Fig. \[fig3\]. From these figures, we can see two interesting phenomena. (i) Besides $\Delta _{1}=0$, there is another optimal detuning $\Delta _{1}= 2\sqrt{2}\xi$ ($\Delta _{1}= -2\sqrt{2%
}\xi$) for observing nonreciprocity in the oppose direction with the wave number $k=\pi/4$ ($k=3\pi/4$). Thus we can change the direction of the scattering flows from $b\rightarrow a
$ to $a\rightarrow b$ (from $a\rightarrow b $ to $b\rightarrow a$) by tuning the detuning from $\Delta _{1}=0$ to $\Delta
_{1}= 2\sqrt{2}\xi$ ($\Delta _{1}=- 2\sqrt{2}\xi$). This phenomenon can be simply understood by plugging $\Delta _{1}= 2\sqrt{2}\xi$ and $k=\pi/4$ ($%
\Delta _{1}= -2\sqrt{2}\xi$ and $k=3\pi/4$) into Eqs. (\[eq21\]) and (\[eq22\]), then we obtain $I_{ab} \approx 0$ and $I_{ba} \approx 0.258$ ($I_{ba} \approx 0$ and $I_{ab} \approx 0.258$) for $\phi=\pi/2$. (ii) We can improve the nonreciprocity by taking $\Delta _{2}=\sqrt{2}\xi$ for $k=\pi/4$ ($\Delta
_{2}=-\sqrt{2}\xi$ for $k=3\pi/4$). This phenomenon corresponds to the condition $%
E=\Delta_{2}$ for $k=\pi/4$ (or $k=3\pi/4$) in Eqs. (\[eq25\])-(\[eq28\]), so that the optimal phase is $\phi=\pi/2$ (or $\phi=3\pi/2$).
Single-photon circulator in T-shaped waveguide
==============================================
Theoretical model and scattering matrix
---------------------------------------
![(Color online) Schematic diagram of a T-shaped waveguide consisting of three semi-infinite CRWs ($a_{j}$, $b_{j}$ and $c_{j}$ for $j\geq 0$) coupled indirectly by two mechanical modes ($d_{1}$ and $d_{2}$).[]{data-label="fig4"}](fig4.eps){width="8.5"}
Based on the nonreciprocal single-photon frequency conversion discussed in Sec. II, we propose a three-port single-photon circulator in a dissipation-free T-shaped waveguide, i.e. $\gamma =0$, as schematically shown in Fig. \[fig4\], which is made up by coupling an additional semi-infinite CRW (CRW-$c$) to the mechanical mode $d_{2}$ in Fig. \[fig1\] through optomechanical interaction. The T-shaped waveguide can be described by the Hamiltonian $$H_{\mathrm{T,I}}=H_{0}+H_{c}+H_{c,\mathrm{int}},$$where $H_{0}$ is given in Eq. (\[eq1\]), and the two additional terms $H_{c}$ and $%
H_{c,\mathrm{int}}$ are$$H_{c}=\sum_{j=0}^{+\infty }\left[ \omega _{c}c_{j}^{\dag }c_{j}-\xi
_{c}\left( c_{j}^{\dag }c_{j+1}+\mathrm{H.c.}\right) \right] ,$$$$H_{c,\mathrm{int}} =g_{c,2}c_{0}^{\dag }c_{0}\left( d_{2}+d_{2}^{\dag
}\right)+\left( c_{0}\Omega _{c,2}e^{i\omega _{c,2}t}+\mathrm{H.c.}\right).$$Here $c_{j}$ ($c_{j}^{\dag }$) is the bosonic annihilation (creation) operator of the $j$th cavity with the same resonant frequency $\omega _{c}$ and the same coupling strength $\xi_{c}$ between two nearest neighboring cavities in the CRW-$c$. $g_{c,2}$ is the optomechanical coupling strength between the cavity $c_{0}$ and the mechanical mode $d_{2}$. Cavity $c_{0}$ is driven by a laser at frequency $\omega _{c,2}=\omega _{c}-\omega _{2}+\Delta _{c,2}$ with $\Delta _{2}=\Delta _{c,2}$ and amplitude $\Omega _{c,2}$. Similarly, the operator for the cavity $c_{j}
$ can also be rewritten as the sum of its quantum fluctuation operator and classical mean value as $c_{j}\rightarrow c_{j}+\alpha ^{c}_{j,2}e^{-i\omega
_{c,2}t}$, where the classical amplitude $\alpha^{c} _{j,2}$ is determined by the amplitude $\Omega _{c,2}$, the frequency $\omega _{c,2}$, the damping rates $\kappa _{a,j}$, $\kappa _{b,j}$ and $\kappa _{c,j}$ of the cavities and the mechanical damping rates $\gamma _{1}$ and $\gamma _{2}$.
To obtain a linearized Hamiltonian, we assume that the external driving is strong, i.e. $|\alpha^{l} _{0,i}|\gg 1$, the system works in the resolved-sideband limit with respect to both mechanical modes, i.e. $\min
\left\{ \omega _{1},\omega _{2}\right\} \gg \max \left\{ \kappa
_{a,j},\kappa _{b,j},\kappa _{c,j}\right\} $, and the two mechanical modes are well separated in frequency, i.e. $\min \left\{ \omega _{1},\omega
_{2},\left\vert \omega _{1}-\omega _{2}\right\vert \right\} \gg \max \left\{
|g_{l,i}\alpha^{l} _{0,i}|,\gamma _{1},\gamma _{2}\right\} $. After making the standard linearization under the rotating-wave approximation, in the rotating reference frame with respect to $H_{\mathrm{rot}}=\sum_{l=a,b,c}\sum_{j=0}^{+%
\infty }\omega _{l}l_{j}^{\dag }l_{j}+\sum_{i=1,2}\left( \omega _{i}-\Delta
_{i}\right) d_{i}^{\dag }d_{i}$, the linearized Hamiltonian for the T-shaped waveguide is described by$$H_{\mathrm{cir,I}}=\sum_{l=a,b,c}H_{l}+H_{m}+H_{\mathrm{int,I}},$$where $H_{l}$ and $H_{m}$ have been given in Eqs. (\[eq6\]) and (\[eq7\]), and the interaction term $H_{\mathrm{int,I}}$ is given by $$\begin{aligned}
H_{\mathrm{int,I}} &=&J_{a,1}\left( a_{0}^{\dag }d_{1}+a_{0}d_{1}^{\dag
}\right) \notag \\
&&+J_{b,1}\left( e^{-i\phi }b_{0}^{\dag }d_{1}+e^{i\phi }b_{0}d_{1}^{\dag
}\right) \notag \\
&&+J_{a,2}\left( a_{0}^{\dag }d_{2}+a_{0}d_{2}^{\dag }\right) \notag \\
&&+J_{b,2}\left( b_{0}^{\dag }d_{2}+b_{0}d_{2}^{\dag }\right) \notag \\
&&+J_{c,2}\left( c_{0}^{\dag }d_{2}+c_{0}d_{2}^{\dag }\right) .\end{aligned}$$Here $J_{c,2}=\left\vert g_{c,2}\alpha^{c} _{0,2}\right\vert $ is the effective optomechanical coupling strength between the cavity $c_{0}$ and mechanical mode $d_{2}$. We assume that the damping rates $%
\kappa _{l,j}$ of the cavities in the CRWs and the damping rates $\gamma _{i}
$ of the mechanical modes are much smaller than the coupling strengths between two nearest neighboring cavities and the effective optomechanical coupling strengths, i.e. $\left\{\xi_{l},J_{l,i}\right\}\gg \max \left\{
\kappa _{l,j},\gamma _{i}\right\} $, so that we can only consider the coherent scattering in the CRWs.
The stationary eigenstate of a single-photon scattering in the T-shaped waveguide is given by $$\left\vert E\right\rangle =\sum_{l=a,b,c}\sum_{j=0}^{+\infty }u_{l}\left(
j\right) l_{j}^{\dag }\left\vert 0\right\rangle +u_{d1}d_{1}^{\dag
}\left\vert 0\right\rangle +u_{d2}d_{2}^{\dag }\left\vert 0\right\rangle .$$The dispersion relation of the CRW-$c$ can be obtained from Eq. (\[eq11\]) by setting the superscript $l=c$. Substituting the stationary eigenstate and the Hamiltonian into the eigenequation $H_{\mathrm{cir,I}}\left\vert
E\right\rangle =E\left\vert E\right\rangle $, we can obtain the coupled equations for the probability amplitudes as in Eqs. (\[eq12\])-(\[eq16\]) but with Eq. (\[eq15\]) replaced by the following two equations $$J_{a,2}u_{a}\left( 0\right) +J_{b,2}u_{b}\left( 0\right) +J_{c,2}u_{c}\left(
0\right) =\left( E-\Delta _{2}\right) u_{d2},$$$$J_{c,2}u_{d2}-\xi _{c}u_{c}\left( 1\right) =Eu_{c}\left( 0\right) ,$$and the subscript $l$ in Eq. (\[eq16\]) is replaced by $l=a,b,c$.
If a single photon with energy $E$ is incident from the infinity side of CRW-$l$, the interactions between cavity $l_{0}$ and $l_{0}^{\prime }$ ($%
l_{0},l_{0}^{\prime }=a_{0},b_{0},c_{0}$) mediated by two mechanical modes will result in photon sacttering between different quantum channels. The general expressions for the probability amplitudes in three quantum channels ($l=a,b,c$) are given by ($j\geq 0$) $$\label{eq41}
u_{l}\left( j\right) =e^{-ik_{l}j}+s_{ll}e^{ik_{l}j},$$$$u_{l^{\prime }}\left( j\right) =s_{l^{\prime }l}e^{ik_{l^{\prime }}j},$$$$\label{eq43}
u_{l^{\prime \prime }}\left( j\right) =s_{l^{\prime \prime
}l}e^{ik_{l^{\prime \prime }}j},$$where $s_{l^{\prime }l}$ ($s_{l^{\prime \prime }l}$) denotes the scattering amplitude from CRW-$l$ to CRW-$l^{\prime }$ (CRW-$l^{\prime \prime }$). Substituting Eqs. (\[eq41\])-(\[eq43\]) into the coupled equations for the probability amplitudes, we can obtain the scattering matrix as$$\label{eq44}
S=M^{-1}N,$$with $$S=\left(
\begin{array}{ccc}
s_{aa} & s_{ab} & s_{ac} \\
s_{ba} & s_{bb} & s_{bc} \\
s_{ca} & s_{cb} & s_{cc}%
\end{array}%
\right) ,$$$$M=\left(
\begin{array}{ccc}
\xi _{a}^{\prime }e^{-ik_{a}^{\prime }} & J_{ab}e^{i\phi ^{\prime }} & J_{ca}
\\
J_{ab}e^{-i\phi ^{\prime }} & \xi _{b}^{\prime }e^{-ik_{b}^{\prime }} &
J_{bc} \\
J_{ca} & J_{bc} & \xi _{c}^{\prime }e^{-ik_{c}^{\prime }}%
\end{array}%
\right) ,$$$$N=-\left(
\begin{array}{ccc}
\xi _{a}^{\prime }e^{ik_{a}^{\prime }} & J_{ab}e^{i\phi ^{\prime }} & J_{ca}
\\
J_{ab}e^{-i\phi ^{\prime }} & \xi _{b}^{\prime }e^{ik_{b}^{\prime }} & J_{bc}
\\
J_{ca} & J_{bc} & \xi _{c}^{\prime }e^{ik_{c}^{\prime }}%
\end{array}%
\right) ,$$where the renormalized coupling strength $\xi _{l}^{\prime }$ and wave number $k_{l}^{\prime }$ of the single photon in the CRW-$l$ are defined by $$\label{eq48}
\xi _{l}^{\prime }e^{ik_{l}^{\prime }}\equiv \xi _{l}e^{ik_{l}}+\Delta _{l}.$$The effective coupling strengths $J_{ll^{\prime }}$, phase $\phi ^{\prime }$, and frequency shifts $\Delta _{l}$ induced by the two mechanical modes are defined by$$J_{ab}e^{i\phi ^{\prime }}\equiv \frac{J_{a,1}J_{b,1}e^{i\phi }}{\left( E-\Delta
_{1}\right) }+\frac{J_{a,2}J_{b,2}}{\left( E-\Delta _{2}\right) },$$$$J_{ca}\equiv \frac{J_{a,2}J_{c,2}}{\left( E-\Delta _{2}\right) },$$$$\label{eq51}
J_{bc}\equiv \frac{J_{b,2}J_{c,2}}{\left( E-\Delta _{2}\right) },$$$$\Delta _{a} \equiv \frac{\left( J_{a,1}\right) ^{2}}{\left( E-\Delta _{1}\right) }+%
\frac{\left( J_{a,2}\right) ^{2}}{\left( E-\Delta _{2}\right) },$$$$\Delta _{b} \equiv \frac{\left( J_{b,1}\right) ^{2}}{\left( E-\Delta _{1}\right) }+%
\frac{\left( J_{b,2}\right) ^{2}}{\left( E-\Delta _{2}\right) },$$$$\Delta _{c} \equiv \frac{\left( J_{c,2}\right) ^{2}}{\left( E-\Delta _{2}\right) }.$$
Single-photon circulator
------------------------
![(Color online) Scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) as functions of the wave number $k_{l}/\protect\pi$ of a single photon incident from CRW-$l$ for (a)-(c) $\protect\phi=\protect\pi/2$, (d)-(f) $\protect\phi=3\protect\pi/2$. $J_{c,2}$ and $\protect\xi _{c}$ are obtained from Eqs. (\[eq57\]) and (\[eq58\]), and the other parameters are $\Delta
_{1}=\Delta _{2}=0$, $J_{a,1}=J_{b,1}=\protect\xi$, and $J_{a,2}=J_{b,2}=1.2%
\protect\xi$.[]{data-label="fig5"}](fig5.eps){width="8.5"}
![(Color online) Scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) are plotted as functions of (a)-(c) $\Delta _{1}/\protect\xi$ ($%
\Delta _{2}=0$) and (d)-(f) $\Delta _{2}/\protect\xi$ ($\Delta _{1}=0$) for $%
\protect\phi=\protect\pi/2$ and $k=\protect\pi/4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$). $J_{c,2} $ and $\protect\xi _{c}$ are obtained from Eqs. (\[eq57\]) and (\[eq58\]), and the other parameters are $%
J_{a,1}=J_{b,1}=\protect\xi$ and $J_{a,2}=J_{b,2}=1.2\protect\xi$.[]{data-label="fig6"}](fig6.eps){width="8.5"}
Let us give the optimal conditions for observing perfect circulators first. A perfect circulator is obtained when we have $I_{ba}=I_{cb}=I_{ac}=1$ or $%
I_{ab}=I_{bc}=I_{ca}=1$ and the other scattering flows are equal to zero. For the sake of simplicity, we assume that the detunings $\Delta _{1}=\Delta
_{2}=0$, the CRW-$a$ and CRW-$b$ have the same parameters (i.e. $\xi \equiv
\xi _{a}=\xi _{b}$, $k\equiv k_{a}=k_{b}$), and they are symmetrically coupled to the two mechanical modes (i.e. $J_{1}\equiv J_{a,1}=J_{b,1}$, $%
J_{2}\equiv J_{a,2}=J_{b,2}$) with the coupling strength $J_{1}=\xi $. Based on these assumptions, the perfect circulator appears with parameters satisfying the optimal conditions$$\label{eq55}
\phi =\frac{\pi }{2}\quad \mathrm{or}\quad \frac{3\pi }{2},$$$$\label{eq56}
k=\frac{\pi }{4}\quad \mathrm{or}\quad \frac{3\pi }{4},$$$$\label{eq57}
\frac{J_{c,2}}{\xi }=\sqrt{\left( \frac{J_{2}}{\xi }\right) ^{4}+1},$$$$\label{eq58}
\xi _{c}=\left|\frac{\left(J_{bc}\right)^{2}}{\xi e^{-ik}+\Delta _{a}}-\Delta _{c}\right|,$$where $J_{bc}$ has been given in Eq. (\[eq51\]).
In Fig. \[fig5\], the scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) are plotted as functions of the wave number $k_{l}/\protect\pi$ of a single photon incident from CRW-$l$ for (a)-(c) $%
\phi =\pi /2$ and (d)-(f) $\phi =3\pi /2$. As shown in Figs. \[fig5\](a)-(c), when $\phi =\pi /2$, we obtain that $I_{ba}=I_{cb}=I_{ac}=1$ and the other scattering flows are equal to zero for the wave number $k=\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$), or obtain that $I_{ab}=I_{bc}=I_{ca}=1$ and the other scattering flows are equal to zero for the wave number $k=3\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$). As shown in Figs. \[fig5\](d)-(f), when $%
\phi =3\pi /2$, we get $I_{ab}=I_{bc}=I_{ca}=1$ with the other zero scattering flows for the wave number $k=\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$) or get $%
I_{ba}=I_{cb}=I_{ac}=1$ with the other zero scattering flows for the wave number $k=3\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$). In other words, when $\phi =\pi /2$, the signal is transferred from one CRW to another clockwise ($a\rightarrow
b\rightarrow c\rightarrow a$) for the wave number $k=3\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$) or counterclockwise ($a\rightarrow c\rightarrow b\rightarrow a$) for the wave number $%
k=\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$). In contrast, when $\phi =3\pi /2$, the signal is transferred from one CRW to another counterclockwise ($a\rightarrow
c\rightarrow b\rightarrow a$) for the wave number $k=3\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$) or clockwise ($a\rightarrow b\rightarrow
c\rightarrow a$) for the wave number $k=\pi /4$ ($k_{a}=k_{b}=k$, $k_{c}=\arccos[\xi\cos(k)/\xi_{c}]$). Thus, we can reverse the direction of the circulator by tuning the phase from $\phi=\pi/2$ to $\phi=3\pi/2$.
Scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime }=a,b,c$) as functions of the detunings $\Delta _{1}/\xi $ and $\Delta _{2}/\xi $ are shown in Fig. \[fig6\]. Overall, the large detunings (both $\Delta _{1}$ and $\Delta _{2}
$) are destructive for the circulator. Similar to Fig. \[fig3\](a), when the detuning is tuned from $\Delta _{1}=0$ to $\Delta _{1}=2\sqrt{2}\xi $ as shown in Figs. \[fig6\](a)-(c), the scattering flows $I_{ba}$ and $I_{ab}$ change from ($I_{ba}=0$, $I_{ab}=1$) to ($I_{ba}=0.25$, $I_{ab}=0$), i.e., the direction of the scattering flows change from $b\rightarrow a$ to $a\rightarrow b$. What’s more, when $%
I_{ab}=1$ and $I_{bb}=I_{cb}=0$ as shown in Fig. \[fig3\](e), the scattering flows $I_{ab}$, $I_{bb}$, and $I_{ab}$ remain constant for different $\Delta _{2}$.
As shown in Eq. (\[eq58\]), i.e., $\xi _{c}\neq \xi $, the band widths of CRW-$a$ and CRW-$b$ are different from the band width of CRW-$c$, and nonreciprocity ($I_{l^{\prime }l}\neq
I_{ll^{\prime }}$) can only be obtained in the overlap band regime between the three CRWs. As shown in Fig. \[fig5\](c) and \[fig5\](f), the single photon incident from the infinity side of CRW-$c$ will be reflected totally ($I_{cc}=1$) in the regimes of $0<k_{c}<\arccos(\xi/\xi_{c})$ and $\pi-\arccos(\xi/\xi_{c})<k_{c}<\pi $. Moreover, as shown in Eqs. (\[eq55\]) and (\[eq56\]), we can have perfect circulator only with wave number $k=\pi /4$ and $k=3\pi /4$ for $\phi =\pi /2$ or $3\pi
/2$. To improve tunability of the circulator, e.g., the perfect circulator can be obtained with the wave number in a tunable regime, we can use one more mechanical mode to connect the three CRWs and this is the focus of the next section.
Single-photon circulator with three mechanical modes
====================================================
Theoretical model and scattering matrix
---------------------------------------
![(Color online) Schematic diagram of a T-shaped waveguide consisting of three semi-infinite CRWs ($a_{j}$, $b_{j}$ and $c_{j}$ for $j\geq 0$) coupled indirectly by three mechanical modes ($d_{1}$, $d_{2}$ and $d_{3}$).[]{data-label="fig7"}](fig7.eps){width="8.5"}
In this section, we propose another single-photon circulator by a different T-shaped waveguide, schematically shown in Fig. \[fig4\], which is made up of three semi-infinite CRWs coupled indirectedly by three mechanical modes via optomechanical interactions. The T-shaped waveguide can be described by the Hamiltonian$$H_{\mathrm{T,II}}=\sum_{l=a,b,c}H_{l}+H_{m}+H_{\mathrm{om,int}}+H_{\mathrm{dri}}$$with the Hamiltonian $H_{l}$ for the CRWs given in Eq. (\[eq2\]), the Hamiltonian $%
H_{m}$ for three mechanical modes$$H_{m}=\omega _{1}d_{1}^{\dag }d_{1}+\omega _{2}d_{2}^{\dag }d_{2}+\omega
_{3}d_{3}^{\dag }d_{3},$$the interactions $H_{\mathrm{om,int}}$ mediated by three mechanical modes $$\begin{aligned}
H_{\mathrm{om,int}} &=&\left( g_{a,1}a_{0}^{\dag }a_{0}+g_{b,1}b_{0}^{\dag
}b_{0}\right) \left( d_{1}+d_{1}^{\dag }\right) \notag \\
&&+\left( g_{a,2}a_{0}^{\dag }a_{0}+g_{c,2}c_{0}^{\dag }c_{0}\right) \left(
d_{2}+d_{2}^{\dag }\right) \notag \\
&&+\left( g_{b,3}b_{0}^{\dag }b_{0}+g_{c,3}c_{0}^{\dag }c_{0}\right) \left(
d_{3}+d_{3}^{\dag }\right) ,\end{aligned}$$and the externally driving terms$$\begin{aligned}
H_{\mathrm{dri}} &=&\sum_{i=1,2}a_{0}\Omega _{a,i}e^{i\omega
_{a,i}t}+\sum_{i=1,3}b_{0}\Omega _{b,i}e^{i\omega _{b,i}t} \notag \\
&&+\sum_{i=2,3}c_{0}\Omega _{c,i}e^{i\omega _{c,i}t}+\mathrm{H.c.},\end{aligned}$$where $\omega _{i}$ ($i=1,2,3$) is the resonant frequency of $i$th mechanical mode with the bosonic annihilation (creation) operator $d_{i}$ ($d_{i}^{\dag }$). $g_{l,i}$ with $l=a,b,c$ is the optomechanical coupling strength between cavity $l_{0}$ ($%
l_{0}=a_{0},b_{0},c_{0}$) and mechanical mode $d_{i}$ ($%
d_{i}=d_{1},d_{2},d_{3}$). Cavity $a_{0}$ ($b_{0}$, $c_{0}$) is driven by a two-tone laser at frequencies $\omega _{a,1}=\omega _{a}-\omega _{1}+\Delta
_{a,1}$ and $\omega _{a,2}=\omega _{a}-\omega _{2}+\Delta _{a,2}$ ($\omega
_{b,1}=\omega _{b}-\omega _{1}+\Delta _{b,1}$ and $\omega _{b,3}=\omega
_{b}-\omega _{3}+\Delta _{b,3}$, $\omega _{c,2}=\omega _{c}-\omega _{2}+\Delta
_{c,2}$ and $\omega _{c,3}=\omega _{c}-\omega _{3}+\Delta _{c,3}$), and the amplitudes are $\Omega _{a,1}$ and $\Omega _{a,2}$ ($\Omega _{b,1}$ and $\Omega _{b,3}$, $\Omega _{c,2}$ and $\Omega _{c,3}$). Here, we assume that the detunings satisfy the conditions $\Delta_{1}\equiv\Delta_{a,1}=\Delta_{b,1}$, $\Delta_{2}\equiv\Delta_{a,2}=\Delta_{c,2}$, and $\Delta_{3}\equiv\Delta_{b,3}=\Delta_{c,3}$. Thus the operators for the cavity modes can be rewritten as the sum of its quantum fluctuation operators and classical mean values as $a_{j}\rightarrow a_{j}+\sum_{i=1,2}\alpha
_{j,i}^{a}e^{-i\omega _{a,i}t}$, $b_{j}\rightarrow b_{j}+\sum_{i=1,3}\alpha
_{j,i}^{b}e^{-i\omega _{b,i}t}$ and $c_{j}\rightarrow
c_{j}+\sum_{i=2,3}\alpha _{j,i}^{c}e^{-i\omega _{c,i}t}$, where the classical amplitude $\alpha _{j,i}^{l}$ is determined by the amplitudes $%
\Omega _{l,i}$, the frequencies $\omega _{l,i}$, the damping rates $\kappa _{l,j}$ of the cavities, and the mechanical damping rates $\gamma _{i}$.
To obtain a linearized Hamiltonian, similar to the previous assumptions that the external driving is strong, i.e. $|\alpha _{0,i}^{l}|\gg 1$, the system works in the resolved-sideband limit with respect to all three mechanical modes, i.e. $\min
\left\{ \omega _{i}\right\} \gg \max \left\{ \kappa _{l,j}\right\} $, and the three mechanical modes are well separated in frequency, i.e. $\min
\left\{ \omega _{i},\left\vert \omega _{i}-\omega _{i^{\prime }\neq
i}\right\vert \right\} \gg \max \left\{ |g_{l,i}\alpha _{0,i}^{l}|,\gamma
_{i}\right\} $. After doing the standard linearization under the rotating-wave approximation, in the rotating reference frame with respect to $H_{\mathrm{rot}%
}=\sum_{l=a,b,c}\sum_{j=0}^{+\infty }\omega _{l}l_{j}^{\dag
}l_{j}+\sum_{i=1,2,3}\left( \omega _{i}-\Delta _{i}\right) d_{i}^{\dag }d_{i}
$, the linearized Hamiltonian is obtained $$\label{eq63}
H_{\mathrm{cir,II}}=\sum_{l=a,b,c}H_{l}+H_{m}+H_{\mathrm{int,II}}$$with the Hamiltonian $H_{l}$ of the CRWs given in Eq. (\[eq6\]), the Hamiltonian of the mechanical modes $$H_{m}=\Delta _{1}d_{1}^{\dag }d_{1}+\Delta _{2}d_{2}^{\dag }d_{2}+\Delta
_{3}d_{3}^{\dag }d_{3},$$and the interaction terms $$\begin{aligned}
\label{eq65}
H_{\mathrm{int,II}} &=&J_{a,1}\left( a_{0}^{\dag }d_{1}+a_{0}d_{1}^{\dag
}\right) \notag \\
&&+J_{b,1}\left( e^{-i\phi }b_{0}^{\dag }d_{1}+e^{i\phi }b_{0}d_{1}^{\dag
}\right) \notag \\
&&+J_{a,2}\left( a_{0}^{\dag }d_{2}+a_{0}d_{2}^{\dag }\right) \notag \\
&&+J_{c,2}\left( c_{0}^{\dag }d_{2}+c_{0}d_{2}^{\dag }\right) \notag \\
&&+J_{b,3}\left( b_{0}^{\dag }d_{3}+b_{0}d_{3}^{\dag }\right) \notag \\
&&+J_{c,3}\left( c_{0}^{\dag }d_{3}+c_{0}d_{3}^{\dag }\right) .\end{aligned}$$Here $J_{l,i}e^{i\phi _{l,i}}=g_{l,i}\alpha _{0,i}^{l}$ with $l=a,b,c$ is the effective optomechanical coupling strength between the cavity $l_{0}$ ($%
l_{0}=a_{0},b_{0},c_{0}$) and mechanical mode $d_{i}$ ($%
d_{i}=d_{1},d_{2},d_{3}$) with real strength $J_{l,i}=\left\vert
g_{l,i}\alpha _{0,i}^{l}\right\vert $ and phase $\phi _{l,i}$. As only the total phase $\phi =\phi _{a,1}+\phi _{a,2}+\phi
_{b,1}+\phi _{b,3}+\phi _{c,2}+\phi _{c,3}$ among them has physical effects, without loss of generality, $\phi $ is only kept in the terms of $%
b_{0}d_{1}^{\dag }$ and $b_{0}^{\dag }d_{1}$ in Eq. (\[eq65\]) and the following derivation. Similarly, $J_{l,i}$ and $\phi $ can be controlled dynamically by tuning the strengths and phases of the external driving fields.
The stationary eigenstate of a single photon scattering in the T-shaped waveguide with three mechanical modes is given by $$\label{eq66}
\left\vert E\right\rangle =\sum_{l=a,b,c}\sum_{j=0}^{+\infty }u_{l}\left(
j\right) l_{j}^{\dag }\left\vert 0\right\rangle
+\sum_{i=1}^{3}u_{di}d_{i}^{\dag }\left\vert 0\right\rangle.$$Substituting the stationary eigenstate in Eq. (\[eq66\]) and the Hamiltonian in Eq. (\[eq63\]) into the eigenequation $H_{\mathrm{cir,II}}\left\vert E\right\rangle =E\left\vert
E\right\rangle $, we can obtain the coupled equations for the probability amplitudes as $$J_{a,1}u_{d1}+J_{a,2}u_{d2}-\xi _{a}u_{a}\left( 1\right) =Eu_{a}\left(
0\right) ,$$$$J_{b,1}e^{-i\phi }u_{d1}+J_{b,3}u_{d3}-\xi _{b}u_{b}\left( 1\right)
=Eu_{b}\left( 0\right) ,$$$$J_{c,2}u_{d2}+J_{c,3}u_{d3}-\xi _{c}u_{c}\left( 1\right) =Eu_{c}\left(
0\right) ,$$$$J_{a,1}u_{a}\left( 0\right) +J_{b,1}e^{i\phi }u_{b}\left( 0\right) =\left(
E-\Delta _{1}\right) u_{d1},$$$$J_{a,2}u_{a}\left( 0\right) +J_{c,2}u_{c}\left( 0\right) =\left( E-\Delta
_{2}\right) u_{d2},$$$$J_{b,3}u_{b}\left( 0\right) +J_{c,3}u_{c}\left( 0\right) =\left( E-\Delta
_{3}\right) u_{d3},$$and the same equation as Eq. (\[eq16\]) with the subscript $l=a,b,c$.
If a single photon with energy $E$ is incident from the infinity side of CRW-$l$, following similar steps given in Section II, we can obtain the scattering matrix given in Eqs. (\[eq44\])-(\[eq48\]) with the effective coupling strengths $%
J_{ll^{\prime }}$ , phase $\phi ^{\prime }$, and frequency shifts $\Delta
_{l}$ induced by the three mechanical modes redefined by $$\label{eq73}
J_{ab}e^{i\phi ^{\prime }}\equiv\frac{J_{a,1}J_{b,1}}{\left( E-\Delta _{1}\right)
}e^{i\phi },$$$$J_{ca}\equiv\frac{J_{a,2}J_{c,2}}{\left( E-\Delta_{2}\right) },$$$$J_{bc}\equiv\frac{J_{b,3}J_{c,3}}{\left( E-\Delta_{3}\right) },$$$$\Delta _{a}\equiv\frac{\left( J_{a,1}\right) ^{2}}{\left( E-\Delta_{1}\right) }+%
\frac{\left( J_{a,2}\right) ^{2}}{\left( E-\Delta_{2}\right) },$$$$\Delta _{b}\equiv\frac{\left( J_{b,1}\right) ^{2}}{\left( E-\Delta_{1}\right) }+%
\frac{\left( J_{b,3}\right) ^{2}}{\left( E-\Delta_{3}\right) },$$$$\label{eq78}
\Delta _{c}\equiv\frac{\left( J_{c,2}\right) ^{2}}{\left( E-\Delta_{2}\right) }+%
\frac{\left( J_{c,3}\right) ^{2}}{\left( E-\Delta_{3}\right) }.$$
Single-photon circulator
------------------------
![(Color online) Scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) as functions of the wave number $k/\protect\pi$ for $\protect\phi=\protect\pi/3$ in (a)-(c) and $\protect\phi=5\protect\pi/3$ in (d)-(f). The other parameters are $\protect\xi _{c}=\protect\xi$, $k _{c}=k$, $\Delta
_{1}=\Delta _{2}=\Delta _{3}=0$, $J_{1}=J_{2}=J_{3}=J$, and $J$ is obtained from Eq. (\[eq79\]).[]{data-label="fig8"}](fig8.eps){width="8.5"}
![(Color online) Scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) are plotted as functions of (a)-(c) $\Delta _{1}/\protect\xi$ ($%
\Delta _{2}=\Delta _{3}=0$) and (d)-(f) $\Delta _{2}/\protect\xi$ ($\Delta
_{1}=0,\Delta _{3}=\Delta _{2}$) for $\protect\phi=\protect\pi/3$ and $k=0.5236$ \[obtained from Eq. (\[eq80\])\]. The other parameters are $\protect\xi _{c}=\protect%
\xi $, $k _{c}=k$, $J_{1}=J_{2}=J_{3}=J$, and $J$ is obtained from Eq. (\[eq79\]).[]{data-label="fig9"}](fig9.eps){width="8.5"}
![(Color online) Scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) are plotted as functions of the wave number $k_{l}/\protect\pi$ of a single photon incident from CRW-$l$ for the perfect circulator appearing at (a)-(c) $k=0.1\protect\pi$ and $0.9\protect\pi$, (d)-(f) $k=0.2\protect%
\pi $ and $0.8\protect\pi$. $J_{i}$ ($i=1,2,3$) and $\protect\xi_{c}$ are obtained from Eqs. (\[eq83\])-(\[eq86\]), and the other parameters are $\protect\phi=\pi/2$ and $\Delta _{1}=\Delta _{2}=\Delta _{3}=0$.[]{data-label="fig10"}](fig10.eps){width="8.5"}
The optimal conditions to obtain a perfect circulator can be derived analytically from Eqs. (\[eq44\])-(\[eq48\]) \[with $J_{ll^{\prime }}$ , $\phi ^{\prime }$, and $\Delta
_{l}$ defined in Eqs. (\[eq73\])-(\[eq78\])\] by setting $I_{ba}=I_{cb}=I_{ac}=1$ or $%
I_{ab}=I_{bc}=I_{ca}=1$ and the other zero scattering flows. For the sake of simplicity, we assume that the detunings $\Delta _{1}=\Delta
_{2}=\Delta _{3}=0$, CRW-$a$ and CRW-$b$ have the same parameters (i.e. $%
\xi \equiv \xi _{a}=\xi _{b}$, $k\equiv k_{a}=k_{b}$) with the coupling strength $J_{1}\equiv J_{a,1}=J_{b,1}$, $J_{2}\equiv J_{a,2}=J_{b,3}$, $%
J_{3}\equiv J_{c,2}=J_{c,3}$. If $\xi _{c}=\xi $ and $J\equiv J_{1}=J_{2}=J_{3}$, the perfect circulator can be obtained when the parameters satisfy the conditions $$\label{eq79}
\frac{J^{2}}{\xi ^{2}}=\frac{2\left( 2-\cos \phi \right) }{\left( 5-4\cos
\phi \right) },$$$$\begin{aligned}
\label{eq80}
k &=&\frac{1}{2}\arcsin \left\vert \frac{4\sin \phi -\sin 2\phi }{5-4\cos
\phi }\right\vert \text{ }, \notag \\
&&\mathrm{or} \quad \text{ }\pi -\frac{1}{2}\arcsin \left\vert \frac{4\sin \phi
-\sin 2\phi }{5-4\cos \phi }\right\vert .\end{aligned}$$If $\xi _{c}\neq \xi $ and $J_{1}\neq J_{2}\neq J_{3}$, the perfect circulator can also be obtained but the conditions are changed to $$\phi =\frac{\pi }{2}\quad \mathrm{{or}\quad }\frac{3\pi }{2},$$$$k\in \left( 0,\frac{\pi }{4}\right) \cup \left( \frac{3\pi }{4},\pi \right) ,$$and the optimal coupling strengths are given by $$\label{eq83}
\frac{J_{1}}{\xi }=\sqrt{\left\vert \sin 2k\right\vert },$$$$\frac{J_{2}}{\xi }=\sqrt{2\cos ^{2}k-\left\vert \sin 2k\right\vert },$$$$\frac{J_{3}}{\xi }=\left\vert \cos k\right\vert ,$$$$\label{eq86}
\frac{\xi _{c}}{\xi }=\left\vert \frac{\cos k}{\cos \left[ \arctan \left(
\frac{2\cos ^{2}k-\left\vert \sin 2k\right\vert }{4\left\vert \cos
k\right\vert \sin k}\right) \right] }\right\vert .$$From Eqs. (\[eq83\]) and (\[eq86\]), if the wave number $k$ to obtain optimal circulator satisfies the equation $$\label{eq87}
2\cos ^{2}k-\left\vert \sin 2k\right\vert =4\sin ^{2}k,$$we obtain $\xi =\xi _{c}$ and $J_{1}=J_{2}=J_{3}$, and this is consistent with the condition given in Eq. (\[eq80\]) for $\phi =\pi/2$.
In Fig. \[fig8\], the scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) are plotted as functions of the wave number $k/\pi $ for (a)-(c) $%
\phi =\pi /3$, (d)-(f) $\phi =5\pi /3$ with $\xi
_{c}=\xi $ and $J\equiv J_{1}=J_{2}=J_{3}$. As shown in Figs. \[fig8\](a)-(c), when $\phi =\pi /3$, we obtain that $I_{ab}=I_{bc}=I_{ca}=1$ and the other scattering flows are equal to zero for wave number $k=\pi /6$, or obtain $%
I_{ba}=I_{cb}=I_{ac}=1$ and the other scattering flows are equal to zero for wave number $k=5\pi /6$. As shown in Figs. \[fig8\](d)-(f), when $\phi =5\pi /3$, we get $I_{ba}=I_{cb}=I_{ac}=1$ with the other zero scattering flows for wave number $k=\pi /6$, or get $I_{ab}=I_{bc}=I_{ca}=1$ with the other zero scattering flows for wave number $k=5\pi /6$. In other words, when $\phi =\pi /3$, the signal is transferred from one CRW to another counterclockwise ($a\rightarrow c\rightarrow b\rightarrow a$) for wave number $%
k=\pi /6$ or clockwise ($a\rightarrow b\rightarrow c\rightarrow a$) for wave number $k=5\pi /6$. In contrast, when $\phi =5\pi /3$, the signal is transferred from one CRW to another clockwise ($%
a\rightarrow b\rightarrow c\rightarrow a$) for wave number $k=\pi /6$ or counterclockwise ($a\rightarrow
c\rightarrow b\rightarrow a$) for wave number $k=5\pi /6$. In other words, we can tune the phase from $\phi$ to ($\pi-\phi$) to reverse the direction of the circulator.
The effects of the detunings on the single-photon transmission can be seen from Fig. \[fig9\], where the scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime }=a,b,c$) are plotted as functions of the detunings (a)-(c) $\Delta _{1}/\xi $ ($\Delta _{2}=\Delta _{3}=0$) and (d)-(f) $\Delta _{2}/\xi $ ($\Delta _{1}=0$ and $\Delta _{3}=\Delta _{2}$). It is interesting that, we obtain $I_{l^{\prime
}l}=1/3$ ($l,l^{\prime }=a,b,c$) with the detunings $\Delta _{1}=\sqrt{3}\xi
$ ($\Delta _{2}=\Delta _{3}=0$) or $\Delta _{3}=\Delta _{2}=\sqrt{3}\xi $ ($%
\Delta _{1}=0$). This interesting phenomenon can be used to design three-way single-photon beam splitter.
If we have different coupling strengths $\xi_{c}\neq \xi $, the band widths of CRW-$a$ and CRW-$b$ are different from the band width of CRW-$c$, and nonreciprocity ($I_{ll^{\prime }}\neq
I_{l^{\prime }l}$) can only be obtained in the overlap band regime among the three CRWs. In Fig. \[fig10\], the scattering flows $I_{l^{\prime }l}$ ($l,l^{\prime
}=a,b,c$) are plotted as functions of the wave number $k_{l}/\pi $ of a single photon incident from CRW-$l$ with different coupling strengths $\xi
_{c}\neq \xi $. Different from the case of T-shaped waveguide with two mechanical modes as discussed in Sec. III, we can have perfect circulator with wave number $k\in \left( 0,\pi/4\right) \cup \left( 3\pi/4,\pi \right) $. From Eq. (\[eq87\]) \[or Eq. (\[eq80\]) for $\phi =\pi/2$\], to make the perfect circulator behavior appearing at $k=0.1476\pi$ and $0.8524\pi$, we should choose $\xi
_{c}=\xi $ and $J_{1}=J_{2}=J_{3}$. As shown in Figs. \[fig10\](a)-(c), when the perfect circulator appears at $k=0.1\pi$ ($<0.1476\pi$) and $0.9\pi$ ($>0.8524\pi$), we have $\xi _{c}>\xi $. In Figs. \[fig10\](d)-(f), when the perfect circulator behavior appears at $k=0.2\pi$ ($>0.1476\pi$) and $0.8\pi$ ($<0.8524\pi$), we have $%
\xi _{c}<\xi $.
Conclusions
===========
In summary, we have studied the nonreciprocal single-photon frequency conversion in multiple CRWs, which are coupled indirectly by optomechanical interactions with two nondegenerate mechanical modes. We have demonstrated that the frequency of a single photon can be converted nonreciprocally in two CRWs. Moreover, two different single-photon circulators are proposed in the T-shaped waveguides with two or three mechanical modes. The optomechanical systems (or mechanical modes) offers the possibility to enable nonreciprocal frequency transduction between two CRWs with distinctively different frequencies, and they allow for dynamic control of the direction of frequency conversion by tuning the phases of external driving lasers. All the proposals can be applied to integrate devices with different frequencies and simplify the construction of hybrid quantum networks.
For simplicity, in this work we do not consider the dissipative effects of the cavities. However, in reality, all the optical or microwave cavities in the CRWs interact with the environment, resulting in the reduction of the nonreciprocal single-photon frequency conversing efficiency. To lower this effect, we should enhance both the coupling strength between two nearest neighboring cavities in the CRWs and the effective optomechanical coupling strengths. The photon hopping can be enhanced by decreasing the distance between neighboring cavities, and one of the most common ways to enhance effective optomechanical coupling strengths is to increase the powers of the external driving fields. In this case, the frequencies of the mechanical modes we choose must be high enough to ensure that the rotating-wave approximation is valid in the derivations. Thus, the microwave-frequency mechanical modes coupled to superconducting quantum circuit [@OConnellNat10] and optomechanical crystal [Safavi-NaeiniPRL14]{} are suitable to realize our nonreciprocal single-photon frequency converters.
X.-W.X. is supported by the National Natural Science Foundation of China (NSFC) under Grant No.11604096 and the Startup Foundation for Doctors of East China Jiaotong University under Grant No. 26541059. A.-X.C. is supported by NSFC under Grant No. 11365009. Y.L. is supported by NSFC under Grant No. 11421063. Y.L. and Y.-X.L. are supported by the National Basic Research Program of China (973 Program) under Grant No. 2014CB921400. Y.-X.L. is also supported by NSFC under Grants No. 61328502 and 61025022, the Tsinghua University Initiative Scientific Research Program, and the Tsinghua National Laboratory for Information Science and Technology (TNList) Cross-Discipline Foundation.
[99]{} M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, Hybrid quantum devices and quantum engineering, Phys. Scr. **T137**, 014001 (2009).
Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. **85**, 623 (2013).
P. Kumar, Quantum frequency conversion, Opt. Lett. **15**, 1476 (1990).
J. Huang and P. Kumar, Observation of quantum frequency conversion, Phys. Rev. Lett. **68**, 2153 (1992).
M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion, Nat. Photon. **4**, 786 (2010).
R. Ikuta, Y. Kusaka, T. Kitano, H. Kato, T. Yamamoto, M. Koashi, and N. Imoto, Wide-band quantum interface for visible-to-telecommunication wavelength conversion. Nat. Commun. **2**, 537 (2011).
S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, Visible-to-Telecom Quantum Frequency Conversion of Light from a Single Quantum Emitter, Phys. Rev. Lett. **109**, 147404 (2012).
S. Ates, I. Agha, A. Gulinatti, I. Rech, M. T. Rakher, A. Badolato, and K. Srinivasan, Two-Photon Interference Using Background-Free Quantum Frequency Conversion of Single Photons Emitted by an InAs Quantum Dot, Phys. Rev. Lett. **109**, 147405 (2012).
B. Abdo, K. Sliwa, F. Schackert, N. Bergeal, M. Hatridge, L. Frunzio, A. D. Stone, and M. Devoret, Full Coherent Frequency Conversion between Two Propagating Microwave Modes, Phys. Rev. Lett. **110**, 173902 (2013).
H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Quantum Frequency Translation of Single-Photon States in a Photonic Crystal Fiber, Phys. Rev. Lett. **105**, 093604 (2010).
A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, A quantum memory with telecom-wavelength conversion, Nat. Phys. **6**, 894 (2010).
D. Farnesi, A. Barucci, G. C. Righini, S. Berneschi, S. Soria, and G. N. Conti, Optical Frequency Conversion in Silica-Whispering-Gallery-Mode Microspherical Resonators, Phys. Rev. Lett. **112**, 093901 (2014).
Y. X. Liu, H. C. Sun, Z. H. Peng, A. Miranowicz, J. S. Tsai, and F. Nori, Controllable microwave three-wavemixing via a single three-level superconducting quantum circuit, Sci. Rep. **4**, 7289 (2014).
Y. J. Zhao, J. H. Ding, Z. H. Peng, Y. X. Liu, Realization of microwave amplification, attenuation, and frequency conversion using a single three-level superconducting quantum circuit, Phys. Rev. A **95**, 043806 (2017).
A. F. Kockum, V. Macrì, L. Garziano, S. Savasta, and F. Nori, Frequency conversion in ultrastrong cavity QED, Sci. Rep. **7**, 5313 (2017).
M. Bradford, K. C. Obi, and J. T. Shen, Efficient Single-Photon Frequency Conversion Using a Sagnac Interferometer, Phys. Rev. Lett. **108**, 103902 (2012).
M. Bradford and J. T. Shen, Single-photon frequency conversion by exploiting quantum interference, Phys. Rev. A **85**, 043814 (2012).
W. B. Yan, J. F. Huang, and H. Fan, Tunable single-photon frequency conversion in a Sagnac interferometer, Sci. Rep. **3**, 3555 (2013).
L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, Quantum Routing of Single Photons with a Cyclic Three-Level System, Phys. Rev. Lett. **111**, 103604 (2013).
Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, Controllable single-photon frequency converter via a one-dimensional waveguide, Phys. Rev. A **89**, 053813 (2014).
T. J. Kippenberg and K. J. Vahala, Cavity Optomechanics: Back-Action at the Mesoscale, Science **321**, 1172 (2008).
F. Marquardt and S. M. Girvin, Optomechanics, Physics **2**, 40 (2009).
M. Aspelmeyer, P. Meystre, and K. Schwab, Quantum optomechanics, Phys. Today **65**(7), 29 (2012).
M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity Optomechanics, Rev. Mod. Phys. **86**, 1391 (2014).
M. Metcalfe, Applications of cavity optomechanics, Appl. Phys. Rev. **1**, 031105 (2014).
A. H. Safavi-Naeini and O. Painter, Proposal for an optomechanical traveling wave phonon/photon translator, New J. Phys. **13**, 013017 (2011).
Y. D. Wang and A. A. Clerk, Using Interference for High Fidelity Quantum State Transfer in Optomechanics, Phys. Rev. Lett. **108**, 153603 (2012). L. Tian, Adiabatic State Conversion and Pulse Transmission in Optomechanical Systems, Phys. Rev. Lett. **108**, 153604 (2012).
J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, Coherent optical wavelength conversion via cavity optomechanics, Nat. Commun. **3**, 1196 (2012).
Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, Electromagnetically Induced Transparency and Wideband Wavelength Conversion in Silicon Nitride Microdisk Optomechanical Resonators, Phys. Rev. Lett. **110**, 223603 (2013).
C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, Optical wavelength conversion via optomechanical coupling in a silica resonator, Ann. Phys. (Berlin) **527**, 100 (2015).
F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Mechanically Mediated Microwave Frequency Conversion in the Quantum Regime, Phys. Rev. Lett. **116**, 043601 (2016).
J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, Nanomechanical coupling between microwave and optical photons, Nature Phys. **9**, 712 (2013);
T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. S[ø]{}rensen, K.Usami, A. Schliesser, and E. S. Polzik, Optical detection of radio waves through a nanomechanical transducer, Nature (London) **507**, 81 (2014).
R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, Bidirectional and efficient conversion between microwave and optical light, Nature Phys. **10**, 321 (2014).
A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, Efficient microwave to optical photon conversion: an electro-optical realization, Optica **3**, 000597 (2016).
D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, What is - and what is not - an optical isolator, Nat. Photon. **7**, 579 (2013).
J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, Time-reversal-symmetry breaking in circuit-QED-based photon lattices, Phys. Rev. A **82**, 043811 (2010).
S. J. M. Habraken, K. Stannigel, M. D. Lukin, P. Zoller, and P Rabl, Continuous mode cooling and phonon routers for phononic quantum networks, New J. Phys. **14**, 115004 (2012).
L. Ranzani and J. Aumentado, A geometric description of nonreciprocity in coupled two-mode systems, New J. Phys. **16**, 103027 (2014).
L. Ranzani and J. Aumentado, Graph-Based Analysis of Nonreciprocity in Coupled-Mode Systems, New J. Phys. **17**, 023024 (2015).
Y. P. Wang, W. Wang, Z. Y. Xue, W. L. Yang, Y. Hu, and Y. Wu, Realizing and characterizing chiral photon flow in a circuit quantum electrodynamics necklace, Sci. Rep. **5**, 8352 (2015).
K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Reconfigurable Josephson Circulator/Directional Amplifier, Phys. Rev. X **5**, 041020 (2015).
M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, Optomechanical creation of magnetic fields for photons on a lattice, Optica **2**, 635 (2015).
A. Metelmann and A. A. Clerk, Nonreciprocal Photon Transmission and Amplification via Reservoir Engineering, Phys. Rev. X **5**, 021025 (2015).
X. W. Xu and Y. Li, Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems, Phys. Rev. A **91**, 053854 (2015).
K. Fang, M. H. Matheny, X. Luan, and O. Painter, Optical transduction and routing of microwave phonons in cavity-optomechanical circuits, Nat. Photon. **10**, 489 (2016).
X. Xu and J. M. Taylor, Optomechanically-induced chiral transport of phonons in one dimension, arXiv:1701.02699 \[quant-ph\].
F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, Phase control of entanglement and quantum steering in a three-mode optomechanical system, arXiv:1706.04474 \[quant-ph\].
F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, Nonreciprocity and magnetic-free isolation based on optomechanical interactions. Nat. Commun. **7**, 13662 (2016).
K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, O. Painter, Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering, Nature Physics (2017) (Published online) doi:10.1038/nphys4009.
X. W. Xu, Y. Li, A. X. Chen, and Y. X. Liu, Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems, Phys. Rev. A **93**, 023827 (2016).
A. Metelmann and A. A. Clerk, Non-reciprocal quantum interactions and devices via autonomous feed-forward, Phys. Rev. A **95**, 013837 (2017).
L. Tian and Z. Li, Nonreciprocal quantum-state conversion between microwave and optical photons, Phys. Rev. A **96**, 013808 (2017).
M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, Optical Nonreciprocity Based on Optomechanical Coupling, Phys. Rev. Applied **7**, 064014 (2017).
N. R. Bernier, L. D. Tóth, A. Koottandavida, A. Nunnenkamp, A. K. Feofanov, T. J. Kippenberg, Nonreciprocal reconfigurable microwave optomechanical circuit, arXiv:1612.08223 \[quant-ph\].
G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Demonstration of efficient nonreciprocity in a microwave optomechanical circuit, Phys. Rev. X **7**, 031001 (2017).
S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, Mechanical On-Chip Microwave Circulator, arXiv:1706.00376 \[quant-ph\].
X. W. Xu, A. X. Chen, Y. Li, and Yu-xi Liu, Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides, Phys. Rev. A **95**, 063808 (2017). S. Yang, Z. Song, and C. P. Sun, Asymmetric transmission through a flux-controlled non-Hermitian scattering center, arXiv:0912.0324 \[quant-ph\]. X. Q. Li, X. Z. Zhang, G. Zhang, and Z. Song, Asymmetric transmission through a flux-controlled non-Hermitian scattering center, Phys. Rev. A **91**, 032101 (2015).
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature (London) **431**, 162 (2004).
M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, Strongly interacting polaritons in coupled arrays of cavities, Nat. Phys. **2**, 849 (2006).
M. A. Castellanos-Beltran and K. W. Lehnert, Widely tunable parametric amplifier based on a superconducting quantum interference device array resonator, Appl. Phys. Lett. **91**, 083509 (2007)
M. Notomi, E. Kuramochi, and T. Tanabe, Large-scale arrays of ultrahigh-Q coupled nanocavities, Nat. Photon. **2**, 741 (2008).
I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, Theory of Ground State Cooling of a Mechanical Oscillator Using Dynamical Backaction, Phys. Rev. Lett. **99**, 093901 (2007). F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Quantum Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion, Phys. Rev. Lett. **99**, 093902 (2007). Y. Li, Y. D. Wang, F. Xue, and C. Bruder, Quantum theory of transmission line resonator-assisted cooling of a micromechanical resonator, Phys. Rev. B **78**, 134301 (2008). X. W. Xu, Y. X. Liu, C. P. Sun, and Y. Li, Mechanical $%
\mathcal{PT}$ symmetry in coupled optomechanical systems, Phys. Rev. A **92**, 013852 (2015).
A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator, Nature (London) **464**, 697 (2010).
A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, Two-Dimensional Phononic-Photonic Band Gap Optomechanical Crystal Cavity, Phys. Rev. Lett. **112**, 153603 (2014).
|
---
author:
- |
Marco Billó, Marialuisa Frau, Igor Pesando\
Dipartimento di Fisica Teorica, Università di Torino\
Istituto Nazionale di Fisica Nucleare - sezione di Torino\
via P. Giuria 1, I-10125 Torino
- |
Francesco Fucito\
Dipartimento di Fisica, Università di Roma Tor Vergata\
Istituto Nazionale di Fisica Nucleare - sezione di Roma 2\
Via della Ricerca Scientifica, I-00133 Roma, Italy
- |
Alberto Lerda\
Dipartimento di Scienze e Tecnologie Avanzate\
Università del Piemonte Orientale, I-15100 Alessandria, Italy\
Istituto Nazionale di Fisica Nucleare - sezione di Torino, Italy
- |
Antonella Liccardo\
Dipartimento di Scienze Fisiche, Università di Napoli\
Istituto Nazionale di Fisica Nucleare - sezione di Napoli\
Complesso Universitario “Monte Sant’Angelo”, via Cintia, I-80126 Napoli, Italy
title: Classical gauge instantons from open strings
---
Introduction {#intro}
============
Recently a lot of effort has been put in investigating various properties of (supersymmetric) field theories using string theory and in particular D-branes. At the same time, a similar effort has been devoted to extend and “lift” to string theory many of the methods that have been developed over the years to study field theories. As a result of these investigations, a strong and fruitful relation between string and field theory has been established.
Quite generally one can say that in the limit of infinite tension ($\alpha'\to 0$) a string theory reduces to an effective field theory with gauge interactions unified with gravity. Even if the precise dictionary between string and field theory is not always straightforward, the simple idea of taking $\alpha'\to 0$ has been throroughly exploited to investigate the perturbative sector of various field theories using string techniques which, indeed, turned out to be very efficient computational tools (see [*e.g.*]{} Ref. [@reviews]). In this perturbative framework, one typically starts from string scattering amplitudes computed on a Riemann surface $\Sigma$ of a given topology. In general, a $N$-point string amplitude ${\cal A}_N$ is obtained from the correlation function among $N$ vertex operators $V_{\phi_1},\ldots, V_{\phi_N}$, each of which describes the emission of a field $\phi_i$ of the string spectrum from the world-sheet. Schematically, we have $${\cal A}_N = \int_{\Sigma}
\big\langle V_{\phi_1}\cdots V_{\phi_N}\big\rangle_{\Sigma}
\label{npoint}$$ where the integral is over the positions of the vertex operators and the moduli of $\Sigma$ with an appropriate measure, and the symbol $\big\langle\cdots\big\rangle_{\Sigma}$ denotes the vacuum expectation value with respect to the (perturbative) vacuum represented by $\Sigma$.
Let us now focus on the simplest world-sheets, namely the sphere for closed strings and the disk for open strings, and let us distinguish in the vertex $V_\phi$ the polarization $\phi$ from the operator part by writing $$V_\phi = \phi~{\cal V}_\phi~~.
\label{vertex}$$ Then, for any closed string field $\phi_{\rm closed}$ we have $$\big\langle\,{\cal V}_{\phi_{\rm \,closed}}\,\big\rangle_{\rm sphere}
= 0~~,
\label{vev0}$$ and for any open string field $\phi_{\rm open}$ we have $$\big\langle\,{\cal V}_{\phi_{\rm \,open}}\,\big\rangle_{\rm disk}
= 0~~.
\label{vev}$$ The relations (\[vev0\]) and (\[vev\]) imply that the closed and open strings do not possess tadpoles on the sphere and the disk respectively; hence these are the appropriate world-sheets to describe the [classical]{} trivial vacua around which the ordinary perturbation theory is performed, but clearly they are inadequate to describe classical non-perturbative backgrounds.
However, after the discovery of D-branes [@Polchinski:1995mt] the perspective has drastically changed and nowadays also some non-perturbative properties can be studied in string theory. The key point is that the D$p$ branes are $p$-dimensional extended configurations of Type II and Type I string theory that, despite their non-perturbative nature, admit a perturbative description. In fact, they can be represented by closed strings in which the left and right movers are suitably identified [@Polchinski:1996na]. Such an identification is equivalent to insert a boundary on the closed string world-sheet and prescribe suitable boundary reflection rules for the string coordinates [@Callan:1988wz]. Thus, the simplest world-sheet topology for closed strings in the presence of a D$p$ brane is that of a disk with $(p+1)$ longitudinal and $(9-p)$ transverse boundary conditions. Moreover, due to the boundary reflection rules, on such a disk we have, in general, $$\big\langle\,{\cal V}_{\phi_{\rm \,closed}}\,\big\rangle_{{\rm disk}_p}
\not= 0~~.
\label{vev1}$$ A D$p$ brane can also be represented by a boundary state $|{\rm
D}p\rangle$, which is a non-perturbative state of the closed string that inserts a boundary on the world-sheet and enforces on it the appropriate identifications between left and right movers (for a review on the boundary state formalism, see for example Ref. [@DiVecchia:1999rh]). If we denote by $|\phi_{\rm
closed}\rangle$ the physical state associated to the vertex operator ${\cal V}_{\phi_{\rm closed}}$, we can rewrite (\[vev1\]) as follows $$\langle \phi_{\rm \,closed}|{\rm D}p\rangle \not= 0~~.
\label{vev2}$$ Thus, the boundary state, or equivalently its corresponding disk, is a classical source for the various fields of the closed string spectrum. In particular, it is a source for the massless fields (like for instance the graviton $h_{\mu\nu}$) which acquire a non-trivial profile and therefore describe a non-trivial classical background. A precise relation between such a background and the boundary state has been established in Refs. [@DiVecchia:1997pr; @DiVecchia:1999uf]. There it has been shown that if one multiplies the massless tadpoles of $|{\rm D}p\rangle$ by free propagators and then takes the Fourier transform, one gets the leading term in the large distance expansion of the classical $p$-brane solutions carrying Ramond-Ramond charges which are non-perturbative configurations of Type II or Type I supergravity. For example, applying this procedure to the graviton tadpole $$\big\langle\,{\cal V}_{h_{\mu\nu}}\,\big\rangle_{{\rm disk}_p} =
\langle h_{\mu\nu}|{\rm D}p\rangle~~,
\label{graviton}$$ one obtains the metric of the D$p$ brane in the large distance approximation from which the complete supergravity solution can eventually be reconstructed. These arguments show that in order to describe closed strings in a D-brane background it is necessary to modify the boundary conditions of the string coordinates and, at the lowest order, consider disks instead of spheres.
A natural question at this point is whether this approach can be generalized to open strings, and in particular whether one can describe in this way the instantons of four dimensional gauge theory. To show that this is possible is one of the purposes of this paper. The crucial point is that the instantons of the (supersymmetric) gauge theories in four dimensions are non-perturbative configurations which admit a perturbative description within the realm of string theory. Thus, in a certain sense, they are the analogue for open strings of what the supergravity branes with Ramond-Ramond charges are for closed strings. In this analysis a key role is again played by the D-branes; this time, hovever, they are regarded from the open string point of view, namely as hypersurfaces spanned by the string end-points on which a (supersymmetric) gauge theory is defined. For definiteness, let us consider a stack of $N$ D3 branes of Type IIB string theory which support on their world-volume a ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM) with gauge group ${\mathrm{U}}(N)$ (or ${\mathrm{SU}}(N)$ if we disregard the center of mass). Then, as shown in Refs. [@Witten:1995im; @Douglas], in order to describe instantons of this gauge theory with topological charge $k$, one has to introduce $k$ D$(-1)$ branes (D-instantons) and thus consider a D3/D$(-1)$ brane system. The role of D-instantons and their relation to the gauge theory instantons have been intensively studied from many different points of view in the last years (see for example Refs. [@Polchinski:fq; @Green:1997tv; @banksgreen; @Kogan; @Chu; @bianchi; @Dorey; @Dorey:1999pd; @Green:2000ke]; for recent reviews on this subject see Refs. [@Vandoren_TO; @Dorey:2000ww; @Dorey:2002ik] and references therein). In the D3/D$(-1)$ brane system, besides the ordinary perturbative gauge degrees of freedom represented by open strings stretching between two D3 branes, there are also other degrees of freedom that are associated to open strings with at least one end-point on the D-instantons. These extra degrees of freedom are non-dynamical parameters which, at the lowest level, can be interpreted as the moduli of the gauge (super)instantons in the ADHM construction [@Atiyah:ri]. Furthermore, in the limit $\alpha'\to 0$ the interactions of these parameters reproduce exactly the ADHM measure on the instanton moduli space [@Dorey:2002ik].
In this paper we further elaborate on this D-brane description of instantons and show that it is not only an efficient book-keeping device to account for the multiplicities and the transformation properties of the various instanton moduli, but also a powerful tool to extract from string theory a detailed information on the gauge instantons. First of all, we observe that the presence of different boundary conditions for the open strings of the D3/D$(-1)$ system implies the existence of disks whose boundary is divided into different portions lying either on the D3 or on the D$(-1)$ branes (see for example Fig. \[fig:md0\]). These disks, which we call mixed disks, are characterized by the insertion of at least two vertex operators associated to excitations of strings that stretch between a D3 and a D$(-1)$ brane (or viceversa), and clearly depend on the parameters ([*i.e.*]{} the moduli) that accompany these mixed vertex operators. Moreover, due to the change in the boundary conditions caused by the mixed operators, in general one can expect that $$\big\langle\,{\cal V}_{\phi_{\rm \,open}}\,\big\rangle_{\rm mixed~disk}
\not= 0~~.
\label{vev3}$$ In this paper we will confirm this expectation and in particular show that the massless fields of the ${\cal N}=4$ gauge vector multiplet propagating on the D3 branes have non-trivial tadpoles on the mixed disks; for example, for the gauge potential $A_\mu$, we will find that $$\big\langle\,{\cal V}_{A_\mu}\,\big\rangle_{\rm mixed~disk}
\not= 0~~.
\label{vev4}$$ Furthermore, by taking the Fourier transform of these massless tadpoles after including a propagator [@DiVecchia:1997pr; @DiVecchia:1999uf], we find that the corresponding space-time profile is precisely that of the classical instanton solution of the ${\mathrm{SU(N)}}$ gauge theory in the singular gauge [@Belavin:fg; @'tHooft:fv]. For simplicity we show this only in the case of the D3/D$(-1)$ brane system in flat space, [*i.e.*]{} for instantons of the ${\cal N}=4$ supersymmetry, but a similar analysis can be performed without difficulties also in orbifold backgrounds that reduce the supersymmetry to ${\cal N}=2$ or ${\cal N}=1$.
We can therefore assert that the mixed disks are the sources for gauge fields with an instanton profile, and thus, contrarily to the ordinary disks (see eq. (\[vev\])) they are the appropriate world-sheets one has to consider in order to compute instanton contributions to correlation functions within string theory. We believe that this fact helps to clarify the analysis and the prescriptions presented in Refs. [@Green:1997tv; @Green:2000ke] and also provides the conceptual bridge necessary to relate the D-instanton techniques of string theory with the standard instanton calculus in field theory.
This paper is organized as follows. In [section \[sec:review\]]{} we review the main properties of the D3/D$(-1)$ brane system, discuss its supersymmetries and the spectrum of its open string excitations. In [section \[sec:d3d-1\]]{} we derive the effective action for the D3/D$(-1)$ brane system by taking the field theory limit $\alpha'\to 0$ of string scattering amplitudes on (mixed) disks. In this derivation we introduce also a string representation for the auxiliary fields that linearize the supersymmetry transformation rules, and discuss how the effective action of the D3/D$(-1)$ system reduces to the ADHM measure on the instanton moduli space by taking a suitable scaling limit. In [section \[sec:instanton\]]{} we present one of the main result of this paper, namely that the gauge vector field emitted from a mixed disk with two boundary changing operators is exactly the leading term in the large distance expansion of the classical instanton solution in the singular gauge. We also discuss how the complete solution can be recovered by considering mixed disks with more boundary changing insertions. In [section \[sec:superinstanton\]]{} we complete our analysis by considering the other components of the ${\cal N}=4$ vector multiplet and obtain the full superinstanton solution from mixed disks. In the last section we show how instanton contributions to correlation functions in gauge theories can be computed using string theory methods, and also clarify the relation with the standard field theory approach. Finally, in the appendices we list our conventions, give some more technical details and briefly review the ADHM costruction of the superinstanton solution.
A review of the D3/D(-1) system {#sec:review}
===============================
The $k$ instanton sector of a four-dimensional ${\cal N}=4$ SYM theory with gauge group $\mathrm{SU}(N)$ can be described by a bound state of $N$ D$3$ and $k$ D$(-1)$ branes [@Witten:1995im; @Douglas]. In this section we review the main properties of this brane system, and in particular analyze its supersymmetries and the spectrum of its open string excitations.
In the D3/D$(-1)$ system the string coordinates $X^{M}(\tau,\sigma)$ and $\psi^{M}(\tau,\sigma)$ (${M}=1,\ldots,10$) obey different boundary conditions depending on the type of boundary. Specifically, on the D$(-1)$ brane we have Dirichlet boundary conditions in all directions, while on the D3 brane the longitudinal fields $X^\mu$ and $\psi^\mu$ ($\mu=1,2,3,4$) satisfy Neumann boundary conditions, and the transverse fields $X^a$ and $\psi^a$ ($a=5,\ldots,10$) obey Dirichlet boundary conditions. To fully define the system, it is necessary to specify also the reflection rules of the spin fields $S^{\dot{\cal A}}$, which transform as a Weyl spinor of $\mathrm{SO}(10)$ (say with negative chirality). As explained for example in Ref. [@Polchinski:1996na], these reflection rules must be determined consistently from the boundary conditions of the $\psi^{\cal
M}$’s. Introducing $z=\exp{(\tau+{\rm i}\sigma)}$ and $\bar z =
\exp{(\tau-{\rm i}\sigma)}$, and denoting with a $~\widetilde{}~$ the right-moving part, it turns out that on the D$(-1)$ boundary $$\label{spinbc1}
S^{\dot{\cal A}}(z) = \varepsilon\, \left.\widetilde S^{\dot{\cal A}}(\bar z)
\right|_{z=\bar z}~,$$ while on the D3 boundary $$\label{spinbc2}
S^{\dot{\cal A}}(z) = \varepsilon'\,\left.
(\Gamma^{0123}\widetilde S)^{\dot{\cal A}}(\bar z)
\right|_{z=\bar z}~.$$ Here, $\varepsilon$ and $\varepsilon'$ are signs that distinguish between branes and anti-branes. However, only the relative sign $\varepsilon\varepsilon'$ is relevant, and thus we loose no generality in setting $\varepsilon=1$ from now on.
Since the presence of the D3 branes breaks $\mathrm{SO}(10)$ to $\mathrm{SO}(4)\times \mathrm{SO}(6)$, we decompose the spin fields $S^{\dot{\cal A}}$ as follows $$\label{decospinor}
S^{\dot{\cal A}} \rightarrow \left(S_\alpha\,S_A,
S^{\dot\alpha}\,S^A\right)~~,$$ where $S_\alpha$ ($S^{\dot\alpha}$) are SO(4) Weyl spinors of positive (negative) chirality, and $S^A$ ($S_A$) are SO(6) Weyl spinors of positive (negative) chirality which transform in the fundamental (anti-fundamental) representation of $\mathrm{SU}(4)\sim \mathrm{SO}(6)$ (see appendix \[app:conventions\] for our conventions). Then, the D$(-1)$ boundary conditions (\[spinbc1\]) become $$\label{undottedbc}
S_\alpha(z) \,S_A(z)
=\left.\widetilde S_\alpha(\bar z) \,\widetilde S_A(\bar z)\right|_{z=\bar z}
~~~,~~~~
S^{\dot\alpha}(z) \,S^A(z)
=\left.\widetilde S^{\dot \alpha}(\bar z) \,\widetilde S^A(\bar z)
\right|_{z=\bar z}~~,$$ while the D3 boundary conditions (\[spinbc2\]) become $$\label{dottedbc}
S_\alpha(z) \,S_A(z) = \varepsilon'\,
\left.\widetilde S_\alpha(\bar z) \,\widetilde S_A(\bar z)\right|_{z=\bar z}
~~~,~~~~
S^{\dot\alpha}(z) \,S^A(z) = - \varepsilon'\,
\left.\widetilde S^{\dot \alpha}(\bar z) \,\widetilde S^A(\bar z)
\right|_{z=\bar z}~~.$$ These reflection rules are essential in determining which supersymmetries are preserved or broken by the different branes.
Broken and unbroken supersymmetries {#subsec:susy}
-----------------------------------
Let us recall that the charge $q$ corresponding to a [holomorphic]{} current can be written in terms of the left and right bulk charges $Q$ and $\widetilde Q$ as $$\label{uchargez}
q= Q - \widetilde Q = \frac{1}{2\pi\ii}\left(
\int dz~ j(z)
~- \int d\bar z~ \widetilde j(\bar z)\right)~,$$ where the $z$ ($\bar z$) integral is over a semicircle of constant radius in the upper (lower) half complex plane. The charge $q$ is conserved at the boundary if the following condition $$\label{bcj}
j(z) = \left.\widetilde j(\bar z)\right|_{\bar z = z}$$ holds. On the contrary, the other combination of bulk charges $$\label{bchargez}
q'= Q + \widetilde Q = \frac{1}{2\pi\ii}\left( \int dz~ j(z) ~+
\int d\bar z~ \widetilde j(\bar z)\right)$$ is broken by the boundary conditions (\[bcj\]). In this case, when the integration contours are deformed to real axis, the integrand does not vanish and thus it contributes to $q'$ with the following amount $$\label{bint} \left.\int_{\rm boundary} dx \, (j + \widetilde j)
\right|_{\bar z=z \equiv x}~.$$ This corresponds to the integrated insertion on the boundary of the massless vertex operator $(j + \widetilde j)(x)$ which describes the Goldstone field associated to the broken symmetry generated by $q'$.
Let us now return to the D3/D$(-1)$ system, and consider the bulk supercharges $$\label{defqa} Q^{\dot{\cal A}} = {1\over 2\pi\ii}\int dz~ j^{\dot{\cal A}}(z)
~~~,~~~{\widetilde Q}^{\dot{\cal A}} = {1\over 2\pi\ii}\int d{\bar
z}~ {\widetilde j}^{\dot{\cal A}}({\bar z})~~,$$ where $j^{\dot{\cal A}}$ (${\widetilde j}^{\dot{\cal A}}$) is the left (right) supersymmetry current. In the $(-1/2)$ picture, we simply have $$j^{\dot{\cal A}}(z) =S^{\dot{\cal A}}(z)~ {\rm e}^{-\frac{1}{2}\phi(z)}
\label{susycurr1}$$ (and similarly for the right moving current) where $\phi$ is the chiral boson of the superghost fermionization formulas [@FMS].
Decomposing the spin field as in (\[decospinor\]), and using the reflection rules (\[undottedbc\]) and (\[dottedbc\]), from the previous analysis it is easy to conclude that for $\varepsilon' = -1$
- the charge $Q^{\dot\alpha A} -\widetilde Q^{\dot\alpha A}$ is preserved both on the D3 and on the D$(-1)$ boundary. Adopting the same notation as in [@Green:2000ke], we denote by $\bar\xi_{\dot \alpha A}$ the fermionic parameters of the supersymmetry transformations generated by this charge;
- the charge $Q^{\dot\alpha A} + \widetilde Q^{\dot\alpha A}$ is broken on both types of boundaries. The corresponding parameter is denoted by $\rho_{\dot\alpha A}$;
- the charge $Q_{\alpha A} -\widetilde Q_{\alpha A}$ is preserved on the D$(-1)$ boundary but is broken on the D3 boundary. The corresponding parameter is denoted by $\xi^{\alpha A}$;
- the charge $Q_{\alpha A} +\widetilde Q_{\alpha A}$ is preserved on the D3 boundary but is broken by the D$(-1)$. The corresponding parameter is denoted by $\eta^{\alpha A}$.
If $\varepsilon'=1$, the chiralities get exchanged and the charges $Q_{\alpha A} -\widetilde Q_{\alpha A}$ and $Q_{\alpha A} +\widetilde Q_{\alpha A}$ are respectively preserved and broken on both boundaries, while the charges $Q^{\dot\alpha A} -\widetilde Q^{\dot\alpha A}$ and $Q^{\dot\alpha A} +\widetilde Q^{\dot\alpha A}$ are preserved only on the D$(-1)$ boundary and on the D3 boundary respectively. This exchange of chiralities is consistent with the fact that the two cases $\varepsilon'=\mp 1$ correspond to instanton and anti-instanton configurations in the four-dimensional gauge theory.
Massless spectrum
-----------------
In the D$3$/D$(-1)$ brane system there are four different kinds of open strings: those stretching between two D3-branes (3/3 strings in the following), those having both ends on a D$(-1)$-brane ($(-1)$/$(-1)$ strings), and finally those which start on a D$(-1)$ and end on a D$3$ brane or vice-versa ($(-1)$/3 or 3/$(-1)$ strings).
Let us first consider the 3/3 strings. In the NS sector at the massless level we find a gauge vector $A^\mu$ and six scalars $\varphi^a$ which can propagate in the four longitudinal directions of the D3 brane. The corresponding vertex operators (in the $(-1)$ superghost picture) are $$\begin{aligned}
V^{(-1)}_A(z) &= &{A}^\mu(p)~ {\cal V}^{(-1)}_{A^\mu}(z;p)~~,
\label{vertgaugevect}
\\
V^{(-1)}_\varphi(z)&=&{\varphi}^a(p)~ {\cal
V}^{(-1)}_{\varphi^a}(z;p)~~, \label{vertgaugescal}\end{aligned}$$ where $$\begin{aligned}
&&{\cal V}^{(-1)}_{A^\mu}(z;p) \,=\,\frac{1}{\sqrt 2} \,
\psi_{\mu}(z)\,\ee^{-\phi(z)} \,\ee^{\ii p_\nu X^\nu(z)}
\label{calverA}~~,\\ &&{\cal V}^{(-1)}_{\varphi^a}(z;p)
\,=\,\frac{1}{\sqrt 2}\, \psi_{a}(z)\,\ee^{-\phi(z)} \,\ee^{\ii
p_\nu X^\nu(z)} \label{calverPhi}\end{aligned}$$ with $p_\nu$ being the longitudinal incoming momentum. Here we have taken the convention that $2\pi\alpha'=1$; in the next section when we compute string scattering amplitudes we will reinstate the appropriate dimensional factors.
In the R sector at the massless level we find two gauginos, ${\Lambda}^{\alpha A}$ and ${\bar\Lambda_{\dot\alpha A}}$, that have opposite ${\mathrm{SO}}(4)$ chirality and transform respectively in the fundamental and anti-fundamental representation of ${\mathrm{SU}}(4)$. In the $(-1/2)$ picture, the gaugino vertex operators are $$\begin{aligned}
\label{vertgaugino} V^{(-1/2)}_\Lambda(z)&=& {\Lambda}^{\alpha
A}(p)~ {\cal V}_{\Lambda^{\alpha A}}^{(-1/2)}(z;p)~~,
\\
V^{(-1/2)}_{\bar\Lambda}(z)&=& {{\bar\Lambda_{\dot\alpha A}}}(p)~
{\cal V}_{\bar\Lambda_{\dot\alpha A}}^{(-1/2)}(z;p)~~,
\label{vertbargaugino}\end{aligned}$$ where $$\begin{aligned}
&&{\cal V}_{\Lambda^{\alpha A}}^{(-1/2)}(z;p) \,=\,
S_{\alpha}(z)\,S_A(z)\,\ee^{-\frac{1}{2}\phi(z)}\, \ee^{\ii p_\nu
X^\nu(z)}~~, \label{calverLambda1}
\\
\label{calverLambar2} &&{\cal V}_{\bar\Lambda_{\dot\alpha
A}}^{(-1/2)}(z;p) \,=\,
S^{\dot\alpha}(z)\,S^A(z)\,\ee^{-\frac{1}{2}\phi(z)}\, \ee^{\ii
p_\nu X^\nu(z)}~~.\end{aligned}$$ The massless fields introduced above form the ${\cal N}=4$ vector multiplet and are connected to each other by the sixteen supersymmetry transformations which are preserved on a D3 boundary and whose parameters are ${\bar \xi}_{\dot\alpha A}$ and $\eta^{\alpha A}$, namely $$\begin{aligned}
\label{susygauge} &&\delta A^\mu \, = \,
\ii\,\bar\xi_{\dot\alpha A}\, (\bar\sigma^\mu)^{\dot\alpha\beta}
\,\Lambda_{\beta}^{~A}\, + \,\ii\,\eta^{\alpha
A}\,(\sigma^\mu)_{\alpha\dot\beta}\, \bar\Lambda^{\dot\beta}_{~A}~~,
\nonumber\\ &&\delta \Lambda^{\alpha A} \, = \, {\ii\over 2}
\,\eta^{\beta A}\,(\sigma^{\mu\nu})_{\beta}^{~\alpha} \,F_{\mu\nu} \,+
\,\ii \,\bar\xi_{\dot\beta B}\,
(\bar\sigma^\mu)^{\dot\beta\alpha}\,(\Sigma^a)^{BA}\,
{\partial}_\mu\varphi_{a}~~,
\nonumber\\ &&
\delta \bar\Lambda_{\dot\alpha A} \, = \, {\ii\over 2}
\,\bar\xi_{\dot\beta A}\,(\bar\sigma^{\mu\nu})^{\dot\beta}_{~\dot\alpha}
\,F_{\mu\nu} \,-
\,\ii \,\eta^{\beta B}\,
(\sigma^\mu)_{\beta\dot\alpha}\,(\bar\Sigma^a)_{BA}\,
{\partial}_\mu\varphi_{a}~~,
\nonumber\\ &&
\delta\varphi^{a} \, = \,
-\,\ii\,\bar\xi_{\dot\alpha A}\,(\Sigma^a)^{AB}
\,\bar\Lambda^{\dot\alpha}_{~B}
\,+\,\ii\,\eta^{\alpha A}\, (\bar\Sigma^a)_{AB}
\,\Lambda_{\alpha }^{~B}~~,\end{aligned}$$ where $\sigma$ and $\bar\sigma$ are the Dirac matrices of ${\mathrm{SO(4)}}$, and $\Sigma$ and $\bar\Sigma$ are those of ${\mathrm{SO(6)}}$. (see appendix \[app:conventions\] for our conventions).
The transformation laws (\[susygauge\]) can be obtained by reducing to four dimensions the supersymmetry transformations of the ${\cal N}=1$ SYM theory in ten dimensions. However, they can also be obtained directly in the string formalism by using the vertex operators (\[vertgaugevect\])-(\[vertgaugescal\]) and computing their commutators with the supersymmetry charges that are preserved on the D3 brane. For instance, taking the vertex operator (\[vertgaugino\]) for the gaugino $\Lambda^{\alpha A}$ and the supersymmetry charge $q^{\dot\alpha A}\equiv Q^{\dot\alpha A}
- \widetilde Q^{\dot\alpha A}$, both in the $(-1/2)$ picture, we have $$\begin{aligned}
\label{susyA} &&\comm{\bar\xi_{\dot\alpha A}\,q^{\dot\alpha A}}
{V^{(-1/2)}_\Lambda(z)} = \bar\xi_{\dot\alpha A}\,\oint_z {dy\over
2\pi\ii}~ j^{\dot\alpha A}(y)\, V^{(-1/2)}_\Lambda(z) \nonumber\\
& & ~~~=~ -\,\bar\xi_{\dot\alpha A} \,\Lambda^{\beta B}\,\oint_z
{dy\over 2\pi\ii}
\left(S^{\dot\alpha}(y)\,S^A(y)\,\ee^{-\frac{1}{2}\phi(y)}\right)\,
\left(S_{\beta}(z)\,S_B(z)\,\ee^{-\frac{1}{2}\phi(z)}\, \ee^{\ii
p_\nu X^\nu(z)}\right) \nonumber\\ & & ~~~=
\left(-\,\ii\,\bar\xi_{\dot\alpha A}\,
(\bar\sigma^\mu)^{\dot\alpha}_{~\beta}\, \Lambda^{\beta
A}\right)\, \frac{1}{\sqrt
2}\,\psi_\mu(z)\,\ee^{-\phi(z)}\,\ee^{\ii p_\nu X^\nu(z)}\end{aligned}$$ where in the last step we have used the contraction formulas (\[spincorr\]). Comparing with (\[vertgaugevect\]), we recognize in the last line of (\[susyA\]) the vertex operator of a gauge boson with polarization $$\delta_{\bar\xi}\,A^\mu = \,\ii\,\bar\xi_{\dot\alpha A}\,
(\bar\sigma^\mu)^{\dot\alpha\beta}\, \Lambda_{\beta}^{~A}
\label{susyA1}$$ in agreement with the first of eqs. (\[susygauge\]). Thus, we can schematically write (\[susyA\]) as follows $$\comm{\bar \xi\, q}{V_\Lambda} = V_{\delta_{\bar\xi} A}
\label{susyschem1}~.$$ By proceeding in this way with all other vertex operators, we can reconstruct the entire transformation rules (\[susygauge\]). Since in this approach the supersymmetry generators act on the vertex operators, and not on their polarizations, in order to derive the transformation rule of a given field we have to work “backwards” and apply the supercharges to the vertices of the fields which appear in the right hand side of the supersymmetry transformations.
If one considers $N$ coincident D3-branes, all vertex operators for the 3/3 strings acquire $N\times N$ Chan-Paton factors $T^I$ and correspondingly all polarizations will transform in the adjoint representation of ${\rm U}(N)$ (or ${\rm SU}(N)$). In this case, the supersymmetry transformation rules (\[susygauge\]) must be modified accordingly, and in particular in the variation of the gauginos one must replace $F_{\mu\nu}$ with the full non-abelian field strength, the ordinary derivatives with the covariant ones and also add a term proportional to $\left[\varphi^{a},{\varphi}^{b}\right]$.
Let us now consider the $(-1)/(-1)$ strings. Since now there are no longitudinal Neumann directions, the states of these strings do not carry any momentum, and thus they correspond more to moduli rather than to dynamical fields. In the NS sector we find ten bosonic moduli. Even if they are all on the same footing, for later purposes it is convenient to distinguish them into four $a^\mu$ (corresponding to the longitudinal directions of the D3 branes) and six $\chi^a$ (corresponding to the transverse directions to the D3’s). Their vertex operators (in the $(-1)$ superghost picture) read $$\begin{aligned}
\label{vertA}
V^{(-1)}_a(z) &=& \frac{a^\mu}{\sqrt 2} ~\psi_{
\mu}(z)\,\ee^{-\phi(z)} ~~,
\\
V^{(-1)}_\chi(z)&=&\frac{\chi^a}{\sqrt 2} ~ \psi_{
a}(z)\,\ee^{-\phi(z)}~~. \label{vertchi}\end{aligned}$$
In the R sector of the $(-1)/(-1)$ strings we find sixteen fermionic moduli which are conventionally denoted by $M^{\alpha
A}$ and $\lambda_{\dot\alpha A}$, and correspond to the following vertex operators (in the $(-1/2)$ superghost picture) $$\begin{aligned}
\label{vertM'} V^{(-1/2)}_{M}(z)&=& M^{\alpha A}~
S_{\alpha}(z)\, S_A(z)\,\ee^{-\frac{1}{2}\phi(z)}~~,
\\
V^{(-1/2)}_{\lambda}(z)&=&
{{\lambda_{\dot\alpha A}}}~S^{\dot\alpha}(z)\,S^A(z)
\,\ee^{-\frac{1}{2}\phi(z)}
~~. \label{vertlambda}\end{aligned}$$ The moduli we have introduced so far are related to each other by the sixteen supersymmetry transformations which are preserved on a D$(-1)$ boundary. These can be obtained by reducing to zero dimensions the ${\cal N}=1$ supersymmetry transformations of the SYM theory in ten dimensions. However, since we will be ultimately interested in discussing the instanton properties of the four-dimensional gauge theory living on the D3 branes, we write only the moduli transformations which are preserved also by a D3 boundary and whose parameters have been denoted by $\bar\xi_{\dot\alpha A}$. They are $$\begin{aligned}
\label{susymoduli} &&\delta_{\bar\xi}\, a^\mu \, = \,\ii\,\bar
\xi_{\dot\alpha A}\,(\bar\sigma^\mu)^{\dot\alpha\beta}\,M_{\beta}^{~A}~~,
\nonumber\\
&&\delta_{\bar\xi}\,\chi^{a} \,=\,
-\,\ii\,\bar\xi_{\dot\alpha A}\,(\Sigma^a)^{AB}
\,\lambda^{\dot\alpha}_{~B}
~,\\
&&\delta_{\bar\xi}\, M^{\alpha A}\,=\, 0
~~~,~~~\delta_{\bar\xi}\,\lambda_{\dot\alpha A}=0\nonumber~~.\end{aligned}$$ Also these supersymmetry transformations can be obtained by commuting the charge $q^{\dot\alpha A}$ with the vertex operators of the various moduli, in complete analogy with what we have shown in (\[susyA\]). For example, we have $$\comm{\bar \xi\, q}{V_{M}} = V_{\delta_{\bar\xi}\, a}
\label{susyschem2}~~.$$
If we consider a superposition of $k$ D$(-1)$ branes, the vertex operators (\[vertchi\])-(\[vertlambda\]) acquire $k\times k$ Chan-Paton factors $t^U$ and the associated moduli an index in the adjoint representation of ${\rm U}(k)$. Moreover, the supersymmetry transformations of the fermionic moduli $M^{\alpha A}$ and $\lambda_{\dot\alpha A}$ get modified and become $$\begin{aligned}
&&\delta_{\bar\xi}\, M^{\alpha A}\,=\,
-\,\bar\xi_{\dot\beta B}\,(\bar\sigma^\mu)^{\dot\beta\alpha}\,
(\Sigma^a)^{BA}\,\comm{\chi_{a}}{a_\mu}~~,
\label{susymoduli10}\\
&&\delta_{\bar\xi}\,\lambda_{\dot\alpha A} \,=\,\frac{1}{2}\,
\bar\xi_{\dot\alpha B}
\,(\bar\Sigma^{ab})_{~A}^{B}\,
\comm{\chi_a}{\chi_b}\,+\,\frac{1}{2}
\,\bar\xi_{\dot\beta A}\,(\bar\sigma^{\mu\nu})^{\dot\beta}_{~\dot\alpha}\,
\comm{a_\mu}{a_\nu}
\label{susymoduli1}~~.\end{aligned}$$ Notice that these transformations being non linear in the moduli cannot be obtained using the vertex operator approach previously discussed. However, in the next section, we will show that this is actually possible after introducing suitable auxiliary fields.
Finally, let us consider the $3/(-1)$ and $(-1)/3$ strings which are characterized by the fact that four directions (those that are longitudinal to the D3 brane) have mixed boundary conditions. These conditions forbid any momentum and imply that in the NS sector the fields $\psi^\mu$ have integer-moded expansions with zero-modes that represent the ${\rm SO}(4)$ Clifford algebra. Therefore, the massless states of this sector are organized in two bosonic Weyl spinors of ${\rm SO}(4)$ which we denote by $w$ and $\bar w$ respectively. The chirality of these spinors is fixed by the GSO projection, and depends on whether the D$(-1)$ brane represents an instanton or an anti-instanton. In the instanton case, [*i.e.*]{} for $\varepsilon'=-1$ in (\[dottedbc\]), it turns out that $w$ and $\bar w$ must be anti-chiral, and thus the corresponding vertex operators (in the $(-1)$ superghost picture) are $$\begin{aligned}
\label{vertexw} V^{(-1)}_w(z) &=&{w}_{\dot\alpha}\, \Delta(z)\,
S^{\dot\alpha}(z) \,\ee^{-\phi(z)}~~, \nonumber\\ V^{(-1)}_{\bar
w}(z) &=&{\bar w}_{\dot\alpha}\, \bar\Delta(z)\,
S^{\dot\alpha}(z)\, \ee^{-\phi(z)}~~.\end{aligned}$$ Here $\Delta(z)$ and $\bar\Delta(z)$ are the bosonic twist and anti-twist fields with conformal dimension $1/4$, that change the boundary conditions of the $X^\mu$ coordinates from Neumann to Dirichlet and vice-versa by introducing a cut in the world-sheet [@orbifold] [^1].
In the R sector of the 3/$(-1)$ and $(-1)$/3 strings the fields $\psi^\mu$ have half-integer mode expansions so that there are fermionic zero-modes only in the six common transverse directions. Thus, the massless states of the R sector form two fermionic Weyl spinors of ${\rm SO}(6)$ which we denote by $\mu$ and $\bar \mu$ respectively. Again, it is the GSO projection, together with the requirement of locality with respect to the conserved supercurrent, that fixes the ${\rm SO}(6)$ chirality of $\mu$ and $\bar\mu$. The appropriate choice for instanton configurations is that they must transform in the fundamental representation of ${\rm SU}(4)$ so that their vertices (in the $(-1/2)$ picture) are $$\begin{aligned}
\label{vertexmu} V^{(-1/2)}_\mu(z) &=&{\mu}^{A}\,
\Delta(z)\,S_{A}(z)\, \ee^{-{1\over 2}\phi(z)}~~, \nonumber\\
V^{(-1/2)}_{\bar\mu}(z) &=&{{\bar \mu}}^{A}\,
\bar\Delta(z)\,S_{A}(z)\, \ee^{-{1\over 2}\phi(z)}~~.\end{aligned}$$ In the presence of $N$ D3 and $k$ D$(-1)$ branes, the vertices (\[vertexw\]) and (\[vertexmu\]) acquire Chan-Paton factors $\zeta_{ui}$ and $\bar\zeta^{ui}$ transforming, respectively, in the bifundamental representations $\mathbf{N}\times \mathbf{k}$ and $\mathbf{\bar N}\times \mathbf{\bar k}$ of the gauge groups.
The unbroken supersymmetries of the D$3$/D$(-1)$ system act on $w$ and $\mu$ by the following transformations $$\begin{aligned}
&&\delta_{\bar\xi}\,w_{\dot\alpha} \,=\,-\,\ii\,
{\bar \xi}_{\dot\alpha A}\,\mu^A~~,\label{susyw} \\
&&\delta_{\bar\xi}\,\mu^A \,=\,-\,\frac{1}{\sqrt 2}\,
{\bar\xi}_{\dot\alpha B}\,
(\Sigma^a)^{BA}\, w^{\dot\alpha}\,
{\chi}_{a}~~,
\label{susymu}\end{aligned}$$ and similarly for ${\bar w}_{\dot\alpha}$ and ${\bar \mu}^A$. The linear supersymmetry transformation (\[susyw\]) can be obtained in the string operator formalism by commuting the charge $q^{\dot\alpha A}$ with the vertex operator $V_\mu$; indeed we have $$\comm{\bar \xi\, q}{V_{\mu}} = V_{\delta_{\bar\xi}\, w}~~.
\label{susyschem3}$$ On the contrary, we have $$\comm{\bar \xi\, q}{V_{w}} = 0~~,
\label{susyschem4}$$ and to derive the non-linear transformation (\[susymu\]) from the string vertex operators suitable auxiliary fields are required. Furthermore, the presence of $w$ and $\bar w$ modifies the supersymmetry transformation of $\lambda_{\dot\alpha A}$ by a non-linear term $$\delta_{\bar \xi}\,\lambda_{\dot\alpha A} \sim
{\bar \xi}_{\dot\alpha A}\,\bar w w~~,
\label{newsusylambda}$$ which also requires auxiliary fields in order to be derived in the string operator formalism. We conclude by mentioning that under the eight supercharges $q'_{\alpha A}$ that are preserved by the D3 branes but are broken by the D-instantons, the moduli $w$, ${\bar w}$, $\mu$ and $\bar \mu$ are invariant and that $\comm{\eta\, q'}{V_{w}} = 0$.
Effective actions and ADHM measure on moduli space {#sec:d3d-1}
==================================================
In this section we compute the (tree-level) string amplitudes in the D3/D$(-1)$ system by using the vertex operators previously introduced, and discuss the field theory limit $\alpha'\to 0$ that yields the effective actions and the ADHM measure on the instanton moduli space.
As a first example, let us consider the (color ordered) amplitude among one gauge boson and two gauginos of the 3/3 strings. This is obtained by inserting the vertex operators (\[vertgaugevect\]), (\[vertgaugino\]) and (\[vertbargaugino\]) on a disk representing $N$ D3 branes and is given by $$\begin{aligned}
{\cal A}_{({\bar \Lambda}A\Lambda)} &=&
{\Big\langle\hskip -5pt\Big\langle}V_{\bar\Lambda}^{(-1/2)}\,
V_{A}^{(-1)}\,V_{\Lambda}^{(-1/2)}{\Big\rangle\hskip -5pt\Big\rangle}\nonumber
\\ &\equiv& C_4\,\int\frac{\prod_{i} dz_i}{dV_{123}}~
\left\langle V_{\bar\Lambda}^{(-1/2)}(z_1)\,
V_{A}^{(-1)}(z_2)\,V_{\Lambda}^{(-1/2)}(z_3) \right\rangle~~.
\label{ampl33}\end{aligned}$$ In this expression $dV_{abc}$ is the projective invariant volume element $$dV_{abc}= \frac{dz_a\,dz_b\,dz_c}{(z_a-z_b)(z_b-z_c)(z_c-z_a)}
\label{projvolume}$$ and the prefactor $C_4$ represents the topological normalization of a disk amplitude with the boundary conditions of a D3 brane. In general, the normalization $C_{p+1}$ for disk amplitudes on a D$p$ brane can be determined using for example the unitarity methods of Ref. [@DiVecchia:1996uq], and if we take $(2\pi\alpha')^{1/2}$ as the unit of length, it reads $$C_{p+1} = \frac{1}{2\pi^2\alpha'^2}\,\frac{1}{x_{p+1}\,g_{p+1}^2}
\label{cdp}$$ where $g_{p+1}$ is the coupling constant of the $(p+1)$-dimensional gauge theory living on the brane world-volume which is given by $$g_{p+1}^2 = 4\pi\left(4\pi^2\alpha'\right)^{\frac{p-3}{2}}\,g_s
\label{g(p+1)}$$ in terms of the string coupling constant $g_s$, and $x_{p+1}$ is the Casimir invariant of the fundamental representation of the gauge group of the D$p$ branes. Here we follow the standard conventions and normalize the $\mathrm{SU}(N)$ generators $T^I$ on the D3 branes with $x_4=1/2$ , [*i.e.*]{} $${\rm Tr}\,(T^I\,T^J)\,=\,\frac{1}{2}\,\delta^{IJ}
\label{norm}$$ and the $\mathrm{U}(k)$ generators $t^U$ on the D-instantons with $x_0=1$, [*i.e.*]{} [^2] $${\rm tr}\,(t^U\,t^V)\,=\,\delta^{UV}~~.
\label{norm1}$$ With this choice we have $$C_4 =
\frac{1}{\pi^2\alpha'^2}\,\frac{1}{g_{\rm YM}^2} \label{C4}$$ where $g_{\rm YM}^2\equiv g_4^2=4\pi g_s$ is the gauge coupling constant of the four-dimensional SYM theory, and $$C_0 = \frac{1}{2\pi^2\alpha'^2}\,\frac{1}{g_{0}^2}=\frac{2\pi}{g_s}=
\frac{8\pi^2}{g_{\rm YM}^2} ~~~\label{C0}$$ Notice that the normalization $C_4$ of a D3 amplitude is dimensionful, whereas the normalization $C_0$ of a D-instanton amplitude is dimensionless and equal to the action of a gauge instanton.
To compute the amplitude (\[ampl33\]), we must further remember that in [section \[sec:review\]]{} all vertex operators have been written with the convention that $2\pi\alpha'=1$, and thus suitable dimensional factors must be reinstated in the calculation. This can be systematically done by rescaling all bosonic fields of the NS sector by a factor of $(2\pi\alpha')^{1/2}$ so that they acquire the canonical dimension of $(\rm{length})^{-1}$, and by rescaling all fermionic fields of the R sector by a factor of $(2\pi\alpha')^{3/4}$ so that they acquire the canonical dimension of $(\rm{length})^{-3/2}$. Taking all these normalization factors into account and using the contraction formulas of appendix \[app:conventions\], we find $${\cal A}_{({\bar \Lambda}A\Lambda)} = -\,\frac{2\,\ii}{g_{\rm
YM}^2}\, {\rm Tr}\left(\bar
\Lambda_{\dot\alpha A}\,{\bar A\!\!\!/}^{\dot\alpha \beta}
\,\Lambda_{\beta}^{~A}\right) \label{ampl331}$$ where the $\delta$-function of momentum conservation is understood. The complete result is obtained by adding to (\[ampl331\]) all other inequivalent color orderings, and thus the total coupling among two gauginos and one gauge boson is given by $$-\,\frac{2\,\ii}{g_{\rm
YM}^2}\, {\rm Tr}\left(\bar
\Lambda_{\dot\alpha A}\,\left[{\bar A\!\!\!/}^{\dot\alpha \beta}
\,,\,\Lambda_{\beta}^{~A}\right]\right)~~.
\label{ampl3322}$$ All other interactions among the massless 3/3 string modes can be computed in a similar way. After taking the limit $\alpha'\to 0$ with $g_{\rm YM}$ held fixed in all string amplitudes and taking their Fourier transform, one finds that their 1PI parts are encoded in the (euclidean) action of the ${\cal N}=4$ SYM theory [^3] $$\begin{aligned}
\label{N4susy} {\cal S}_{\rm SYM}
&=&\frac{1}{g_{\rm YM}^2}\,\int d^4x~ {\rm Tr}
\Bigg\{ \frac{1}{2} F_{\mu\nu}^{\,2} -\,2\,\bar\Lambda_{\dot
\alpha A} \not\!\!{\bar{{\cal D}}}^{{\dot\alpha}\beta} \,\Lambda_{\beta }^{~A}
+ \, \left({{\cal D}}_\mu \varphi_{a}\right)^2
\,- \,\frac{1}{2}\,
\left[ \varphi_a, \varphi_b \right]^2
\nonumber
\\
&&~~-\,\ii\,(\Sigma^a)^{AB}\,\bar\Lambda_{\dot\alpha A}\!
\left[\varphi_a,\bar\Lambda^{\dot\alpha}_{~B}\right]
\,-\,\ii\,(\bar\Sigma^a)_{AB}\,\Lambda^{\alpha A}\!
\left[\varphi_a,\Lambda_{\alpha}^{~B}\right] \Bigg\}~~,\end{aligned}$$ which is invariant under the non-abelian version of the supersymmetry transformation rules (\[susygauge\]).
Let us now turn to the interactions among the $(-1)$/$(-1)$ strings which are obtained by evaluating correlation functions on disks representing $k$ D$(-1)$ branes. For example, the color ordered coupling among $\lambda_{\dot\alpha A}$, $a_\mu$ and $M^{\alpha A}$ corresponds to $${\cal A}_{({\lambda}aM)} = {\Big\langle\hskip -5pt\Big\langle}V_{\lambda}^{(-1/2)}\, V_{a}^{(-1)}\,V_{M}^{(-1/2)}
{\Big\rangle\hskip -5pt\Big\rangle}\label{ampl11}$$ where the vertex operators are given in (\[vertA\]), (\[vertM’\]) and (\[vertlambda\]) with suitable factors of $2\pi\alpha'$ inserted as discussed above in order to assign the canonical dimensions to the various fields. In (\[ampl11\]) the expectation value is computed in analogy with (\[ampl33\]) but now the overall normalization is $C_0$ given in (\[C0\]), as is appropriate for a disk with a D$(-1)$ boundary. After adding all color orderings, one finds that the total coupling under consideration is $$-\,\frac{\ii}{g_0^2}\, {\rm tr}\left(
\lambda_{\dot\alpha A}\,\left[{\bar a\!\!\!/}^{\dot\alpha \beta}
\,,\,M_{\beta}^{~A}\right]\right)
\label{ampl332}$$ where the trace is now taken on the indices labeling the $k$ D$(-1)$ branes. Interestingly, the various normalization coefficients have conspired to reproduce the (dimensionful) coupling constant $g_0$ with no other factors of $\alpha'$ left over. If we proceed in a similar way and take the field theory limit $\alpha'\to 0$ with $g_0$ held fixed, we find that all irreducible couplings of the $(-1)/(-1)$ strings are encoded in the effective action $${\cal S}_{(-1)}=
{\cal S}_{\rm cubic}+{\cal S}_{\rm quartic}
\label{s-1}$$ where $${\cal S}_{\rm cubic} =
\frac{\ii}{g_{0}^2}\, {\rm tr}
\Bigg\{\lambda_{\dot\alpha A}
\!\left[{\bar a\!\!\!/}^{\dot\alpha \beta},M_{\beta
}^{~A}\right]\,-\,
\frac{1}{2}\,(\Sigma^a)^{AB}\,\lambda_{\dot\alpha A}
\left[\chi_a,\lambda^{\dot\alpha}_{~B}\right]
\,-\,\frac{1}{2}\,(\bar\Sigma^a)_{AB}\,M^{\alpha A}\!
\left[\chi_a,M_{\alpha}^{~B}\right]\!\Bigg\}
\label{cubic}$$ and $${\cal S}_{\rm quartic} =
-\,\frac{1}{g_{0}^2}\, {\rm tr}
\Bigg\{\frac{1}{4} \left[ a_\mu, a_\nu \right]^2\,+\,
\frac{1}{2} \left[ a_\mu, \chi_a \right]^2\,+\,
\frac{1}{4} \left[ \chi_a, \chi_b \right]^2
\Bigg\}~~.
\label{quartic}$$ This action, which is the reduction to zero dimensions of the ${\cal N}=1$ SYM action in ten dimensions, vanishes in the abelian case of a single D$(-1)$ brane, [*i.e.*]{} for $k=1$. It is interesting to observe that the quartic interactions in (\[quartic\]) can be decoupled by means of auxiliary fields. In fact, ${\cal S}_{\rm quartic}$ is equivalent to $$\begin{aligned}
\label{s'}
{\cal S}\,' &=& \frac{1}{g_{0}^2}\, {\rm tr}\,
\Bigg\{\frac{1}{2}\,D_{c}^{\,2}\,+\,\frac{1}{2}\,D_{c}\,
\bar\eta_{\mu\nu}^c\,\left[a^\mu,a^\nu\right]
\,+\, \frac{1}{2}\,Y_{\mu a}^{\,2}\,+\,Y_{\mu a}\,\left[a^\mu,
\chi^a\right]
\nonumber \\
&&~~~~~~~~~~~~~~~
+\,\frac{1}{4}\,Z_{ab}^{\,2}\,+\,\frac{1}{2}\,Z_{ab}\,
\left[\chi^a,\chi^b\right]
\Bigg\}\end{aligned}$$ where $\bar\eta$ is the anti-self dual ’t Hooft symbol and $D$, $Y$ and $Z$ are auxiliary fields with dimensions of $(length)^{-2}$ which reproduce the quartic couplings of (\[quartic\]) after they are eliminated through their equations of motion. It is worth remarking that, in order to decouple the interaction ${\rm tr} \left[ a_\mu, a_\nu \right]^2$, it is enough to introduce three independent degrees of freedom which correspond to an antisymmetric tensor $D_{\mu\nu}$ of a given duality. For definiteness we have chosen this tensor to be anti-self dual and thus have written $D_{\mu\nu}=D_c\,\bar\eta^c_{\mu\nu}$.
The cubic couplings of ${\cal S}\,'$ can be obtained in the string operator formalism by introducing the following vertices for the auxiliary fields (in units of $2\pi\alpha'=1$) $$\begin{aligned}
\label{vertaux}
V_D^{(0)}(z) &=& \frac{1}{2}\,
D_c\,\bar\eta_{\mu\nu}^c\,\psi^\nu(z) \psi^\mu(z)
~~, \nonumber\\
V_Y^{(0)}(z)&=&{Y}_{\mu a}\,
\psi^a(z) \psi^\mu(z)~~,
\\
V_Z^{(0)}(z)&=&\frac{1}{2}\,{Z}_{ab}\,
\psi^b(z) \psi^a(z) ~~.
\nonumber\end{aligned}$$ These NS vertices are written in the 0-superghost picture and, even if they are not BRST invariant [^4], they provide the correct structures and interactions. Fox example, the (color-ordered) coupling among the auxiliary field $D$ and two $a$’s is reproduced by $${\cal A}_{(Daa)} = \frac{1}{2}\,{\Big\langle\hskip -5pt\Big\langle}V_{D}^{(0)}\, V_{a}^{(-1)}\,V_{a}^{(-1)}
{\Big\rangle\hskip -5pt\Big\rangle}= -\,\frac{1}{2g_0^2}\,{\rm
tr}\left(D_c\,\bar\eta^c_{\mu\nu} \,a^\mu\,a^\nu\right)
\label{ampl13}$$ where a symmetry factor of 1/2 has been inserted to account for the presence of two alike vertices, and the auxiliary field has been rescaled with $(2\pi\alpha')$ to make it of canonical dimension. All other cubic interactions of the action (\[s’\]) can be obtained in a similar way.
The vertex operators (\[vertaux\]) are useful also because they linearize the supersymmetry transformation rules of the various moduli which can therefore be obtained completely within the string operator formalism. In fact, using the method described in [section \[sec:review\]]{}, one can show for example that $$\comm{\bar \xi\, q}{V_D}
= V_{\delta_{\bar\xi}\,\lambda}$$ where $V_{\delta_{\bar\xi}\,\lambda}$ is the vertex (\[vertlambda\]) with polarization $$\delta_{\bar\xi}\,\lambda_{\dot\alpha A} \,=\,-\,
\frac{1}{4}\,\bar\xi_{\dot\beta A}
\,(\bar\sigma^{\mu\nu})^{\dot\beta}_{~\dot\alpha}
\,D_c\,\bar\eta^c_{\mu\nu}~~.
\label{deltaD}$$ If the auxiliary fields $D_c$ are eliminated through their equations of motion following from ${\cal S}\,'$, then (\[deltaD\]) reproduces exactly the last non-linear term in the supersymmetry transformation rule (\[susymoduli1\]). Similarly, the other terms in (\[susymoduli1\]) and (\[susymoduli10\]) can be obtained by computing $\comm{\bar \xi\, q}{V_Z}$ and $\comm{\bar \xi\, q}{V_Y}$.
Let us now analyze the interactions of the $(-1)/3$ and $3/(-1)$ strings. In this case the novelty is represented by the fact that the vertex operators (\[vertexw\]) and (\[vertexmu\]) contain the twist and anti-twist fields, $\Delta$ and $\bar\Delta$, which change the boundary conditions of the longitudinal coordinates $X^\mu$. Thus, for consistency in any correlation function a vertex operator of the $(-1)/3$ sector must always be accompanied by one of the $3/(-1)$ sector. This gives rise to mixed disks whose boundary is divided into an even number of portions with different boundary conditions [^5]. The simplest case is the mixed disk represented in Fig. \[fig:md0\] where a pair of twist/anti-twist operators divides its boundary in two portions with D3 and D$(-1)$ boundary conditions respectively. The topological normalization for the expectation value on such a mixed disk is $C_0$ given in (\[C0\]), [*i.e.*]{} the normalization of the lowest brane.
Let us now consider a 3-point amplitude originating from the insertion of a $(-1)/(-1)$ state on a mixed disk, like for example $${\cal A}_{(w\lambda\bar\mu)} = {\Big\langle\hskip -5pt\Big\langle}V_{w}^{(-1)}\,V_{\lambda}^{(-1/2)}\,V_{\bar\mu}^{(-1/2)}
{\Big\rangle\hskip -5pt\Big\rangle}~~. \label{amplmix1}$$ This correlation function can be computed in a straightforward manner by using the OPE’s of appendix \[app:conventions\], and the result is $${\cal A}_{(w\lambda\bar\mu)} \,= \,\frac{2\,\ii}{g_0^2}\,{\rm tr}\left(
w_{\dot\alpha}^{~u}\,\lambda^{\dot\alpha}_{~A}\,\bar\mu^{A}_{~u}\right)
\label{amplmix2}$$ where we have explicitly indicated also the index $u$ of the fundamental representation of $\mathrm{SU}(N)$ carried by the “twisted” moduli. Again all normalizations have conspired to reconstruct the coupling constant $g_0$ with no other factors of $\alpha'$ left over. Thus, this amplitude survives in the limit $\alpha'\to 0$ with $g_0$ fixed, and must be added to the zero-dimensional effective action ${\cal S}_{(-1)}$. Other terms of this effective action could arise from amplitudes involving the vertex operators (\[vertaux\]) of the auxiliary fields. For example, we have $$\begin{aligned}
{\cal A}_{( w D \bar w)} &=& {\Big\langle\hskip -5pt\Big\langle}V_{w}^{(-1)}\, V_{D}^{(0)}\,V_{\bar w}^{(-1)}
{\Big\rangle\hskip -5pt\Big\rangle}\nonumber \\ &=&
\frac{1}{2g_0^2}\,\bar\eta^c_{\mu\nu}\,{\rm tr}\left(
w_{\dot\alpha}^{~u}\,D_c\,\bar w^{\dot\beta}_{~u}\right)\,
(\bar\sigma^{\mu\nu})^{\dot \alpha}_{~\dot\beta }
=\frac{2\,\ii}{g_0^2}\,{\rm tr}\left(D_c\,W^c\right)
\label{amplmix3}\end{aligned}$$ where in the last step we have introduced the $k\times k$ matrices $$(W^c)_j^{~i} = w_{\dot\alpha}^{~ui}\,(\tau^c)^{\dot\alpha}_{~\dot\beta}
\, \bar w^{\dot\beta}_{~uj}
\label{Wc}$$ with $\tau^c$ being the Pauli matrices. We remark in passing that the coupling (\[amplmix3\]) modifies the field equations of $D_c$ by a term proportional to $W_c$. Thus, when the auxiliary fields are eliminated from the supersymmetry transformation rule (\[deltaD\]), the structure (\[newsusylambda\]) can be reproduced.
If we proceed systematically and compute all amplitudes on mixed disks which survive in the field theory limit, we can reconstruct the following effective action for $w$, $\bar w$, $\mu$ and $\bar\mu$ $${\cal S}\,''=\frac{2\,\ii}{g_0^2}\,{\rm tr}\,\Bigg\{\!
\Big(\bar\mu^{A}_{~u} w_{\dot\alpha}^{~u}
+\bar w_{\dot\alpha u}\mu^{Au}\Big)
\lambda_A^{\dot\alpha}-D_c\,W^c+
\frac{1}{2}(\bar\Sigma^a)_{AB}\,\bar\mu^{A}_{~u}\,\mu^{Bu}\chi_a
-\ii\,\chi_a\,{\bar w}_{\dot\alpha u}w^{\dot\alpha u}\chi^a
\Bigg\}~~.
\label{s''}$$ Notice that the auxiliary fields $Y$ and $Z$ do not appear in this action. In fact, all mixed amplitudes involving them vanish either at the string level, or in the field theory limit. We point out that in analogy with what we have done before, also the quartic interaction of (\[s”\]) can be decoupled by introducing a pair of auxiliary fields $X_{\dot\alpha a}$ and $\bar X_{\dot\alpha a}$. Their corresponding vertex operators, which are proportional to $S^{\dot\alpha}\psi^a\Delta$ and $S^{\dot\alpha}\psi^a
\bar\Delta$ respectively, can be used to derive the non-linear supersymmetry transformations rules (\[susymu\]) in the string operator formalism. However, since these auxiliary fields do not play any other role, we will not introduce them in our analysis.
We can summarize our findings by saying that the total effective action for the moduli produced by the D-instantons is given by $${\cal S}_{\rm moduli} = {\cal S}_{\rm cubic} + {\cal S}\,' +
{\cal S}\,''~~.
\label{smoduli}$$ As we have thoroughly discussed, the zero-dimensional action (\[smoduli\]) arises from string scattering amplitudes on D$(-1)$ branes in the limit $\alpha'\to 0$ with $g_0$ fixed, whereas the four-dimensional SYM action (\[N4susy\]) is obtained from string amplitudes on D3 branes in the limit $\alpha'\to 0$ with $g_{\rm YM}$ fixed. However, as is clear from (\[g(p+1)\]), $g_{\rm YM}$ and $g_0$ cannot be kept fixed at the same time: indeed, when $\alpha'\to 0$ either $g_{\rm YM}\to 0$ if $g_0$ is fixed, or $g_0\to\infty$ if $g_{\rm YM}$ is fixed. This simple fact shows that while a system made of D3 and D$(-1)$ branes is perfectly well-defined and stable at the string level, its field theory limit, instead, is more subtle and requires some care. Since we are interested in analyzing the four-dimensional SYM theory, we clearly must keep fixed $g_{\rm YM}$ when $\alpha'\to
0$, and hence we should consider the zero-dimensional moduli action in the strong coupling limit $g_0\to\infty$. If we take this limit in a naive way, we obtain a rather trivial result because the action (\[smoduli\]), which is inversely proportional to $g_0^2$, becomes negligible and all effects of the D-instantons inside the D3 branes disappear. However, there is another possibility that yields more interesting results: it consists in taking $g_0$ [and]{} (some of) the moduli to infinity. In particular, if we take $$\begin{aligned}
&&a = \sqrt{2}\,g_0\,a'~~~,~~~\chi= \chi'~~~,~~~ M=
\frac{g_0}{\sqrt{2}}\,M'~~~,~~~ \lambda= \lambda'~~, \nonumber \\
&&~~~~~~~~~~~~~D=D'~~~,~~~Y=\sqrt{2}\,g_0\,Y'~~~,~~~Z=g_0\,Z'~~,
\label{rescaling}\\ &&w=\frac{g_0}{\sqrt{2}}\,w'~~~,~~~ \bar
w=\frac{g_0}{\sqrt{2}}\,\bar w'~~~,~~~
\mu=\frac{g_0}{\sqrt{2}}\,\mu'~~~,~~~
\bar\mu=\frac{g_0}{\sqrt{2}}\,\bar\mu'~~, \nonumber\end{aligned}$$ and keep the primed variables fixed when $g_0\to\infty$, we can easily see that the moduli action (\[smoduli\]) survives in the field theory limit, and becomes $$\begin{aligned}
{S}_{\rm moduli}&=& {\rm tr}\Bigg\{ {Y'}_{\mu
a}^{\,2}\,+\,2\,{Y'}_{\mu a}\,\left[{a'}^\mu, \chi'^a\right]
\,+\,\frac{1}{4}\,{Z'}_{ab}^{\,2}\,+\,
{\chi'}_a\,\bar{w'}_{\dot\alpha u}\,{w'}^{\dot\alpha u}\,{\chi'}^a
\nonumber \\ &&+\,\frac{\ii}{2}\,(\bar\Sigma^a)_{AB}\,
\bar{\mu'}^{A}_{~u}\,{\mu'}^{Bu}\,{\chi'}_a\,
-\,\frac{\ii}{4}\,(\bar\Sigma^a)_{AB}\,{M'}^{\alpha A}\!
\left[{\chi'}_a,{M'}_{\alpha}^{~B}\right] \nonumber \\
&&+\,\ii\left(\bar{\mu'}^{A}_{~u}\,{w'}_{\dot\alpha}^{~u}\,+\,
\bar {w'}_{\dot\alpha u}\,{\mu'}^{A u} + \left[{M'}^{\beta
A}\,,\,a'_{\beta\dot\alpha}\right]
\right){\lambda'}^{\dot\alpha}_{~A} \nonumber \\
&&-\,\ii\,D'_c\Big({W'}^c +\,\ii\,
\bar\eta_{\mu\nu}^c\,\left[{a'}^\mu,{a'}^\nu\right]\Big)\Bigg\}~~.
\label{smoduli4}\end{aligned}$$ If we integrate out the auxiliary fields $Y'$ and $Z'$, the action (\[smoduli4\]) reduces exactly to the sum of the actions $S_K$ and $S_D$ defined in eqs. (10.70b) and (10.70c) of Ref. [@Dorey:2002ik] (up to a redefinition of $\chi'_a\to-\ii\,\chi'_a$). The action (\[smoduli4\]) provides the ADHM measure on the moduli space of the $k$-instanton sector of the ${\cal N}=4$ $\mathrm{SU}(N)$ SYM theory; in particular, the equations of motion for $D'_c$ are precisely the three non-linear ADHM constraints $${W'}^c +\,\ii\,
\bar\eta_{\mu\nu}^c\,\left[{a'}^\mu,{a'}^\nu\right] \,=\,0~~,
\label{adhm1}$$ while the equations of motion for ${\lambda'}^{\dot\alpha}_A$ are the fermionic constraints $$\bar{\mu'}^{A}_{~u}\,{w'}_{\dot\alpha}^{~u}\,+\,
\bar{w'}_{\dot\alpha u}\,{\mu'}^{Au} + \left[{M'}^{\beta
A}\,,\,a'_{\beta\dot\alpha}\right] \,=\,0 \label{adhm2}$$ of the ADHM construction. From now on, to avoid clutter we drop the $'$ from all moduli, but we keep the traditional notation for $a'$ and $M'$ [^6].
In this section we have explicitly reviewed that the D3/D$(-1)$ system accomodates all instanton moduli of a four-dimensional supersymmetric gauge theory. It is worth pointing out, however, that the ADHM measure on moduli space does not follow automatically from this construction. In fact, as we have shown, this measure emerges only by taking the field theory limit of the D3/D$(-1)$ system in a very specific way, which includes a rescaling of some of the string moduli with the dimensionful coupling $g_0$, as indicated in (\[rescaling\]), and the strong coupling limit $g_0\to\infty$.
The instanton solution from mixed disks {#sec:instanton}
=======================================
The disk diagrams considered in the previous section do not exhaust all possibilities, since there exhist also mixed disks with the emission of 3/3 strings. In this and the following sections we explicitly analyze such mixed diagrams and show that they are directly related to the classical instanton solutions of the four-dimensional SYM theory. In particular we show that the D$(-1)$ branes effectively act as a source for the various fields of the gauge supermultiplet and that the $(-1)/(-1)$ strings together with the boundary changing operators associated to the 3/$(-1)$ and $(-1)$/3 strings provide the correct dependence of the instanton profile on the ADHM moduli. For simplicity we will discuss in detail only the case of instanton number $k=1$ in a $\mathrm{SU}(N)$ gauge theory. However, no substantial changes occur in our analysis if one considers higher values of $k$. Moreover, in the following we will set again $2\pi\alpha'=1$ since all dimensional factors cancel out in the final results.
The gauge vector profile {#gaugevector}
------------------------
Let us begin by considering the emission of the gauge vector field $A_\mu^I$ from a mixed disk. The simplest diagram which can contribute to this process contains two bosonic boundary changing operators ($V_{\bar w}$ and $V_{w}$) and no D$(-1)$/D$(-1)$ moduli, as shown in Fig. \[fig:md2\].
The amplitude (in momentum space) associated to this diagram is $$\label{dia1} A^I_\mu(p;{\bar w, w}) =
{\Big\langle\hskip -5pt\Big\langle}V^{(-1)}_{\bar w}\,{\cal V}^{(0)}_{A^I_\mu}(-p)\,V^{(-1)}_{w}
{\Big\rangle\hskip -5pt\Big\rangle}$$ where, like for any mixed disk, the expectation value is normalized with $C_0$. Since we want to describe the source for the emission of a gauge boson, in the correlation function (\[dia1\]) we have inserted a gluon vertex operator with [*outgoing*]{} momentum and [*without*]{} polarization, so that the amplitude (\[dia1\]) carries the Lorentz structure and the quantum numbers that are appropriate for an emitted gauge vector field. Moreover, the gluon vertex is in the 0 superghost picture. This can be obtained by performing a picture changing on the vertex (\[calverA\]) and reads $$\label{vert2} {\cal V}^\ppz_{A^I_\mu}(z;-p) = 2\ii\, T^I\left(
\partial X_\mu \,-\, \ii \,p\cdot \psi\, \psi_\mu\right)\,
\ee^{-\ii p\cdot X(z)}$$ where $T^I$ is the adjoint $\mathrm{SU}(N)$ Chan-Paton factor [^7]. The vertices for the $w$ and $\bar w$ moduli are instead in the $(-1)$ superghost picture, and are given in (\[vertexw\]). However, due to the rescalings (\[rescaling\]), an overall factor of $(g_s/2\pi)^{1/2}$ must be incorporated in each of these vertices in order to interprete their polarizations as the $w$ and $\bar w$ moduli of the ADHM construction. Using the contraction formulas of appendix \[app:conventions\], and taking into account (see eq. (\[deltadelta\])) that $$\label{corr2} \left\langle \,\bar \Delta(z_1) \,\ee^{-\ii p\cdot
X(z_2)} \,\Delta(z_3)\,\right\rangle = -\,\ee^{-\ii p\cdot x_0}\,
(z_1-z_3)^{-1/2}$$ where $x_0$ denotes the location of the D-instanton inside the world-volume of the D3 branes (see also eq. (\[deltadelta\])), one easily finds that the amplitude (\[dia1\]) is given by $$\label{corr5} A^I_\mu(p;\bar w, w) = {\ii}\, (T^I)^{v}_{~u}\,p^\nu
\, \bar\eta^c_{\nu\mu}
\left(w_{\dot\alpha}^{~u}\,(\tau_c)^{\dot\alpha}_{~\dot\beta}\,
\bar w^{\dot \beta}_{~v}\right) \, \ee^{-\ii p\cdot x_0} \equiv
\ii\,p^\nu\,J^I_{\nu\mu}(\bar w, w)\,\ee^{-\ii p\cdot x_0}$$ where, in the last step, we have introduced the convenient notation $J^I_{\nu\mu}(\bar w, w)$ for the moduli dependence. Note that the various factors of $g_s$ and $\pi$’s coming from the rescalings and from the normalization $C_0$ of the mixed disk have canceled out completely in this calculation.
As we have discussed before, the mixed disk of Fig. \[fig:md2\] represents the source in momentum space for the emission of the gauge vector field in a non-trivial background. To obtain the space-time profile of this background, we simply have to take the Fourier transform of the amplitude $A^I_\mu(p;\bar w, w)$ after attaching to it the gluon propagator $\delta_{\mu\nu}/p^2$. Thus, the classical field associated to the mixed disk of Fig. \[fig:md2\] is $$\begin{aligned}
\label{gf1} A^I_\mu(x) &=& \int {d^4 p\over (2\pi)^2} \,
A^I_\mu(p; \bar w, w) \,{1\over p^2}\,\ee^{\ii p\cdot x} \nonumber
\\ &=& -\,2\,(T^I)^{v}_{~u} \, \left(w_{\dot\alpha
}^{~u}\,(\tau_c)^{\dot\alpha}_{~\dot\beta}\, \bar w^{\dot \beta
}_{~v} \right)\,\bar\eta^c_{\nu\mu} \, {(x-x_0)^\nu\over
(x-x_0)^4}~~.\end{aligned}$$ This result can also be rewritten in terms of the antisymmetric “source” tensor $J^I_{\nu\mu}$ as follows $$\label{gfsource} A^I_\mu(x) = J^I_{\nu\mu}(\bar w, w)\, \int {d^4
p \over (2\pi)^2}\, {\ii p^\nu\over p^2}\, \ee^{\ii p\cdot
(x-x_0)} = J^I_{\nu\mu}(\bar w, w) \,\partial^\nu G(x-x_0)$$ where $$G(x-x_0)=\int\frac{d^4 p}{(2\pi)^2}\,\frac{\ee^{\ii p\cdot(x-x_0)}
} {p^2} = \frac{1}{(x-x_0)^2} \label{prop}$$ is the scalar massless propagator in configuration space.
The gauge field $A_\mu^I(x)$ in (\[gf1\]) depends on the $4N$ moduli $w_{\dot\alpha}^{~u}$ and $\bar w_{\dot\alpha u}$, up to an overall phase redefinition $w\sim \ee^{\ii\theta}w$ and $\bar w
\sim \ee^{-\ii\theta}\bar w$, and on the position $x_0^\mu$ of the D-instanton inside the world-volume of the D3 branes. This amounts to $4N+3$ real parameters which are precisely those of the *unconstrained* instanton moduli space in the ADHM construction. In fact, upon enforcing the three bosonic ADHM constraints $W^c=0$ (see eq. (\[adhm1\]) for $k=1$), these parameters reduce exactly to the $4N$ moduli of the $\mathrm{SU}(N)$ instanton, namely the position of its center $x_0^\mu$, its size $\rho$ and the $4N-5$ varibles that parametrize the coset space $\mathrm{SU}(N)/\mathrm{S}[\mathrm{U}(N-2)\times \mathrm{U}(1)]$ and specify the orientation of a $\mathrm{SU}(2)$ subgroup inside $\mathrm{SU}(N)$. To see this explicitly, let us define $$2\rho^2 \equiv \bar w^{\dot\alpha}_{~u}\,
w_{\dot\alpha}^{~u}~~,
\label{rho}$$ and consider the three $N\times N$ matrices $$\label{gf2}
(t_c)^{u}_{~v} \equiv {1\over 2 \rho^2} \left(w_{\dot\alpha}^{~u}\,
(\tau_c)^{\dot\alpha}_{~\dot\beta}
\,\bar w^{\dot \beta }_{~v}\right)~~.$$ Then, it is not difficult to show that these matrices generate a SU$(2)$ subalgebra of SU$(N)$, [*i.e.*]{} $\comm{t_c}{t_d} = \ii \epsilon_{cde}\, \, t_e$, [provided]{} the ADHM constraints $W^c=0$ are satisfied. In conclusion, we can rewrite the gauge field (\[gf1\]) as follows $$\label{gf5}
A^I_\mu(x) = 4\rho^2\,\Tr\, (T^I\, t_c) \,
\bar\eta^c_{\mu\nu} \, {(x-x_0)^\nu\over (x-x_0)^4}~~.$$ In the case of $\mathrm{SU}(2)$ the indices $I$ and $c$ can be identified and, taking into account the trace normalization, we obtain $$\label{gf6}
A^c_\mu(x) = 2\rho^2\,
\bar\eta^c_{\mu\nu} \, {(x-x_0)^\nu\over (x-x_0)^4}~~.$$ In this expression we recognize precisely the leading term in the large distance expansion ([*i.e.*]{} $|x-x_0|>\!>\rho$) of the classical BPST $\mathrm{SU}(2)$ instanton [@Belavin:fg; @'tHooft:fv] with center $x_0$ and size $\rho$, in the so-called *singular gauge*, namely $$\begin{aligned}
{A}^c_\mu(x) &=& 2\rho^2 \,\bar\eta^c_{\mu\nu}\,\frac{(x - x_0)^\nu}{
(x - x_0)^2 \Big[(x-x_0)^2 + \rho^2\Big]} \nonumber \\
&\simeq&
2\rho^2 \,\bar\eta^c_{\mu\nu}\, \frac{(x - x_0)^\nu}{
(x - x_0)^4}\,\left(1 - {\rho^2\over (x-x_0)^2} + \ldots\right)~~.
\label{gf7}\end{aligned}$$ Notice that such a configuration has a self-dual field strength, despite the appearance of the anti self-dual ’t Hooft symbols $\bar\eta^c_{\mu\nu}$.
More generally, from the mixed disk amplitude (\[gf1\]) with the ADHM constraint (\[adhm1\]) enforced, we can reconstruct the following anti-hermitian $\mathrm{SU}(N)$ connection $$({\widehat A}_\mu(x))^{u}_{~v} \equiv -\,\ii\,A_\mu(x)^I\,(T^I)^{u}_{~v}
= w_{\dot\alpha}^{~u}\,
(\bar\sigma_{\nu\mu})^{\dot\alpha}_{~\dot\beta}\,{\bar w}^{\dot\beta}_{~v}\,
\frac{(x-x_0)^\nu}{(x-x_0)^4}~~,
\label{connection}$$ which is precisely the leading term in the large distance expansion of the one-instanton connection of the ADHM construction [@Atiyah:ri] in the singular gauge $$({\widehat A}_\mu(x))^{u}_{~v} = w_{\dot\alpha}^{~u}\,
(\bar\sigma_{\nu\mu})^{\dot\alpha}_{~\dot\beta}\,{\bar w}^{\dot\beta}_{~v}
\, \frac{(x-x_0)^\nu}{(x-x_0)^2\Big[(x-x_0)^2 + \rho^2\Big]}~~.
\label{connection1}$$ This analysis clarifies the interpretation of the string amplitude associated to the mixed disk of Fig. \[fig:md2\]. However, a few comments are in order. Firstly, we would like to remark that the amplitude (\[dia1\]) is a 3-point function from the point of view of the two dimensional conformal field theory on the string world sheet, but it should be regarded instead as a 1-point function from the point of view of the four-dimensional gauge theory on the D3 branes. Indeed, the two boundary changing operators $V_{\bar w}$ and $V_{w}$ in (\[dia1\]) just describe the non-dynamical parameters on which the background depends, [*i.e.*]{} the size of the instanton and its orientation inside the gauge group. To emphasize this point, we introduce the convenient notation $$A^I_\mu(p;{\bar w, w}) = {\Big\langle\hskip -5pt\Big\langle}{\cal V}_{A^I_\mu}(-p)
{\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}(\bar w,w)}
\label{vevnew}$$ where ${\cal D}(\bar w,w)$ is the mixed disk produced by the insertion of $V_{\bar w}$ and $V_{w}$. Secondly, the fact that the instanton connection is in the singular gauge should not come as a surprise, but on the contrary it should be expected in this D-brane set-up. In fact, as we have seen, the gauge instanton is produced by a D$(-1)$ brane which is a point-like object inside the D3 brane world-volume, and thus it is natural that the instanton connection arising in this way exhibits a singularity at the location $x_0$ of the D-instanton. We recall that in the singular gauge all non-trivial properties of the instanton profile come entirely from the region near the singularity through the embedding of a 3-sphere surrounding $x_0$ into a $\mathrm{SU}(2)$ subgroup of $\mathrm{SU}(N)$. This is to be contrasted with what happens in the regular gauge, where all non-trivial properties of the instanton come instead from the asymptotic 3-sphere at infinity. Furthermore, in the singular gauge the instanton field falls off as $1/x^3$ at large distances, thus guaranteeing the convergence of many integrals, like for example that of the topological charge.
An obvious question to ask at this point is whether also the subleading terms in the large distance expansion of the instanton solution can have a direct interpretation in string theory. Since these higher-order terms contain higher powers of $\rho^2$, they are naturally associated to mixed disks with more insertions of boundary changing operators. For example, the diagram one should consider to study the emission of the vector field at the next-to-leading order is a mixed disk with two more vertices $V_w$ and $V_{\bar w}$ as shown in Fig. \[fig:2ndorder0\]. However, extending the closed string analysis of Ref. [@Bertolini:2000jy] to the present case, one can argue that in the limit $\alpha'\to 0$ this diagram reduces to a simpler one in which two first-order diagrams are sewn with a 3-gluon vertex of the SYM theory, as shown in Fig. \[fig:2ndorder\]. In appendix \[app:subleading\] we will explicitly compute this diagram and find that, for example for $\mathrm{SU}(2)$, the corresponding emitted gauge field is $$\label{ft2ndorder0}
{A^{c}_\mu(x)}^{(2)} =
-2 \rho^4 \bar\eta^c_{\mu\nu} {(x - x_0)^\nu\over (x - x_0)^6}~,$$ that is exactly the second-order term in the large distance expansion of ${A}_\mu^c(x)$ in (\[gf7\]). The higher order terms in this expansion can be in principle computed in a similar manner and eventually the full instanton solution can be reconstructed. This analysis shows that the relevant building block for the complete solution is actually the leading term at large distance which corresponds to the “source” diagram of Fig. \[fig:md2\] whose evaluation, as we have seen, is extremely simple.
What we have described above is the open string analogue of the procedure introduced in Refs. [@DiVecchia:1997pr; @DiVecchia:1999uf] for closed strings. There, the so-called boundary states [@DiVecchia:1999rh; @Billo:1998vr] were recognized to be the sources for the various massless fields of the closed string spectrum in a D-brane background, and the classical supergravity D-brane solutions were obtained by taking the Fourier transform of the various tadpoles produced by the boundary states. Similarly here, the mixed disks have been shown to be the sources for the emission of open strings in a background whose profile is precisely that of the classical gauge instanton. Just like the boundary state approach has been very useful to obtain information on the classical geometry associated to complicated D-brane configurations, also the present method based on the use of mixed disks could play a very useful role in determining non-standard classical backgrounds of the gauge theory.
Insertions of the translational zero-modes {#a_insertions}
------------------------------------------
It is a familiar fact that in the instanton background there are collective coordinates associated to the presence of broken translational symmetries. From the string point of view, these zero-modes describe the motion of the D-instanton within the D3 branes and correspond to the vertex operators of $a'$ (see eq. (\[vertA\])) which, in the 0 superghost picture, are given by $$\label{verta}
V^{(0)}_{a'}=a'_\mu \,\partial_\sigma X^{\mu}~~.$$ These vertex operators can be used to establish in a stringy way a relation between $a'$ and the instanton collective coordinate $x_0$. Indeed, if one considers all disk diagrams obtained from that of Fig. \[fig:md2\] by inserting any number of vertices $V^{(0)}_{a'}$ along the D$(-1)$ part of the boundary, and then resums the corresponding perturbative series, one finds that all occurrences of $x_0$ are replaced by $x_0+a'$. This fact could be proved by adding to the action of the D$(-1)$ open strings the following marginal deformation along the boundary $$\label{coupledaction}
\delta S=
\frac{1}{2\pi\alpha'}\int\! d\tau \Big[V^{(0)}_{a'}(\sigma=\pi,\tau)
-V^{(0)}_{a'}(\sigma=0,\tau)\Big]~~.$$ However, it is quite difficult to treat this interaction in a non-perturbative way, since it is not easy to find an exact solution of the new equations of motion for the string coordinates with the required boundary conditions and regularity properties. For this reason it is convenient to exploit the open-closed string duality and translate the problem into the closed string language. This amounts to represent the D-instanton localized at $x_0$ with a boundary state $|\mathrm{D}(-1);x_0\rangle$ (see for example Ref. [@Billo:1998vr] for more details) and to perform a world-sheet modular transformation that interchanges the roles of $\sigma$ and $\tau$. Then, adding the marginal deformation (\[coupledaction\]) to the D$(-1)$ open strings is equivalent, in the closed string channel, to $$\label{aonboundary}
P\,\exp\left({-\,\frac{\ii}{2\pi\alpha'}\,
\int_0^\pi \!d\sigma \, a'_\mu \,\partial_\tau X^\mu}\right)
|\mathrm{D}(-1);x_0\rangle~~,$$ as one can easily see by generalizing the discussion of Ref. [@Callan:1988wz]. Notice that the path ordering is a consequence of the Chan-Paton factor that must be added to the vertex operator (\[verta\]) when $k>1$. For $k=1$ instead, the path ordering is trivial and the expression (\[aonboundary\]) can be easily evaluated. In particular, one finds that the relevant zero-more part is given by $$\ee^{-\ii\, a'_\mu p^\mu}\, \delta^4(x-x_0)\, {|{p=0}\rangle}
= \delta^4(x-x_0-a')\, {|{p=0}\rangle}~~,$$ which clearly shows that all occurrences of $x_0$ are to be replaced by $x_0 + a'$, as desired. For this reason in the following we will not distinguish any more between $x_0$ and $a'$.
The superinstanton profile {#sec:superinstanton}
==========================
The procedure we have discussed in the previous section can be easily extended to the other components of the ${\cal N}=4$ vector multiplet, thus allowing to recover the full superinstanton solution from mixed disks. Indeed, acting with the supersymmetry transformations that are preserved also by the D$(-1)$ branes, one can obtain from the diagram of Fig. \[fig:md2\] those that describe the emission of the gauginos and the scalar fields, and hence their classical profiles as function of the supermoduli. On the other hand, acting with the supersymmetries that are broken by the D$(-1)$ branes, one can shift the supermoduli in the classical solution and account in this way for the fermionic zero-modes of the superinstantons.
Unbroken supersymmetries
------------------------
The simplest diagrams which contribute to the emission of a gaugino are mixed disks with one bosonic and one fermionic boundary changing operators. The two possibilities are represented in Fig. \[fig:gauginoemission\]. The amplitude (in momentum space) associated to the diagram (a) is given by $$\bar\Lambda^{\dot\alpha A\,,\,I}(p;\bar w,{\mu})
\equiv {\Big\langle\hskip -5pt\Big\langle}{\cal
V}_{\bar\Lambda_{\dot\alpha A}^I}(-p)
{\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}(\bar w,{\mu})}=
{\Big\langle\hskip -5pt\Big\langle}V^{(-1)}_{\bar w}\, {\cal
V}_{\bar\Lambda_{\dot\alpha A}^I}^{(-1/2)}(-p) \,
V^{(-1/2)}_{\mu}{\Big\rangle\hskip -5pt\Big\rangle}\label{gauginoampl}$$ where ${\cal D}(\bar w,{\mu})$ is the mixed disk created by the insertion of $V_{\bar w}$ and $V_\mu$, and is easily evaluated to be $$\bar\Lambda^{\dot\alpha A\,,\,I}(p;\bar w,{\mu})
=\ii\,(T^I)^{v}_{~u}\,\mu^{A u}\,{\bar w}^{\dot\alpha}_{~v}\,
\ee^{-\ii p\cdot x_0}~~. \label{gauginiampl1}$$ Notice again that in the amplitude (\[gauginoampl\]) we have inserted a gaugino emission vertex with outgoing momentum. Similarly, the amplitude corresponding to the diagram (b) is $$\bar\Lambda^{\dot\alpha A\,,\,I}(p;\bar \mu,{w})
\equiv {\Big\langle\hskip -5pt\Big\langle}{\cal
V}_{\bar\Lambda_{\dot\alpha A}^I}(-p)
{\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}(w,{\bar\mu})}=
\ii\,(T^I)^{v}_{~u}\,{w}^{\dot\alpha u}\,\bar \mu^{A}_{~v}\,
\ee^{-\ii p\cdot x_0}~~. \label{gauginiampl2}$$ An alternative method to compute these amplitudes is based on the use of the supersymmetries which are preserved both on the D3 and on the D$(-1)$ boundary and have been denoted by $\bar\xi \,q$ in [section \[sec:review\]]{}. Exploiting the fact that these supersymmetries annihilate the vacuum, we have the following Ward identity $${\Big\langle\hskip -5pt\Big\langle}\Big[\,\bar\xi \,q\,,V_{\bar w}\,\Big]
\, {\cal V}_{A^I_\mu}(-p)\, V_\mu{\Big\rangle\hskip -5pt\Big\rangle}+
{\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,\Big[\bar\xi \,q\,,{\cal V}_{A^I_\mu}(-p)\Big] \, V_\mu{\Big\rangle\hskip -5pt\Big\rangle}+
{\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,{\cal V}_{A^I_\mu}(-p)\, \Big[\bar\xi\, q\,,V_\mu\Big]
{\Big\rangle\hskip -5pt\Big\rangle}=0~~,
\label{ward1}$$ where for simplicity we have understood the picture assignments [^8]. The only new ingredient appearing in (\[ward1\]) is the commutator in the second term; this can be computed from (\[vert2\]) and reads $$\label{atogaugino2} \comm{\bar\xi\, q\,}{{\cal V}_{A^I_\mu}(-p)} =
\bar\xi_{\dot\beta A}\, p_\nu\,
(\bar\sigma^{\nu\mu})^{\dot\beta}_{~\dot\alpha} \, {\cal
V}_{\bar\Lambda^I_{\dot\alpha A}}(-p)~~.$$ Then, using (\[susyschem3\]) and (\[susyschem4\]), we can rewrite the Ward identity (\[ward1\]) as follows $$\bar\xi_{\dot\beta A}\, p_\nu\,
(\bar\sigma^{\nu\mu})^{\dot\beta}_{~\dot\alpha}
\,{\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,{\cal
V}_{\bar\Lambda^I_{\dot\alpha A}}(-p) \,
V_{\mu}{\Big\rangle\hskip -5pt\Big\rangle}+ {\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,{\cal V}_{A^I_\mu}(-p) \, V_{\delta_{\bar \xi}w}
{\Big\rangle\hskip -5pt\Big\rangle}=0 \label{ward2}$$ which allows to obtain the gaugino amplitude in terms of the gauge boson amplitude (\[corr5\]) with $w$ replaced by its supersymmetry variation $\delta_{\bar \xi}w$ given in (\[susyw\]). In this way we can immediately get (\[gauginiampl1\]), and with a similar relation also (\[gauginiampl2\]) can be retrieved.
The space-time profile of the emitted gaugino is then obtained by taking the Fourier transform of the sum of the amplitudes (\[gauginiampl1\]) and (\[gauginiampl2\]) multiplied by the free fermion propagator $\ii\!\not \hskip -2pt
p^{\dot\beta\alpha}/p^2 \equiv \ii \,p^\nu
(\bar\sigma_\nu)^{\dot\beta\alpha}/p^2$, that is $$\begin{aligned}
\Lambda^{\alpha A\,,\,I}(x)&=&\int {d^4 p\over (2\pi)^2} \,
\left(\bar\Lambda_{\dot\beta}^{~A\,,\,I}(p;\bar w,{\mu})+
\bar\Lambda_{\dot\beta}^{~A\,,\,I}(p;\bar \mu,{w})\right)
\frac{\ii\!\not \hskip -2pt p^{\dot\beta\alpha}}{p^2}\,\ee^{\ii
p\cdot x} \nonumber \\ &=&-2\ii\, (T^I)^{v}_{~u} \,
\left(w_{\dot\beta}^{~u}\,{\bar \mu}^{A}_{~v}+ \mu^{Au}\,{\bar
w}_{\dot\beta v}\right) \,(\bar\sigma_\nu)^{\dot\beta\alpha}
\,\frac{(x - x_0)^\nu}{(x - x_0)^4}~~. \label{gauginosol2}\end{aligned}$$ Just as the gauge field (\[gf5\]), also the gaugino (\[gauginosol2\]) naturally arises in terms of *unconstrained* parameters which become the instanton moduli when they are restricted to satisfy the ADHM constraints (\[adhm1\]) and (\[adhm2\]). In particular, once the fermionic constraint (\[adhm2\]) is imposed, it is immediate to extract from (\[gauginosol2\]) the following matrix-valued gaugino profile $$({\widehat\Lambda}^{\alpha A}(x))^{u}_{~v}
\equiv -\,\ii\,\Lambda^{\alpha A\,,\,I}(x)\,
(T^I)^{u}_{~v} = (\sigma_\nu)^{\alpha}_{~\dot\beta}
\left(w^{\dot\beta u}\,{\bar\mu}^{A}_{~v}+
\mu^{Au}\,{\bar w}^{\dot\beta }_{~v}\right)
\frac{(x - x_0)^\nu}{(x - x_0)^4}~~.
\label{gauginosol3}$$ In this expression we recognize exactly the leading term in the large distance expansion of the gaugino instanton solution in the singular gauge (see for example appendix \[app:ADHM\]) $$({\widehat \Lambda}^{\alpha A}(x))^{u}_{~v}
=(\sigma_\nu)^{\alpha}_{~\dot\beta}
\left(w^{\dot\beta u}\,{\bar\mu}^{A}_{~v}+
\mu^{Au}\,{\bar w}^{\dot\beta }_{~v}\right)
\frac{(x - x_0)^\nu}{\sqrt{(x - x_0)^2\Big[(x-x_0)^2+\rho^2\Big]^3}}~~.
\label{gauginosol4}$$ The subleading terms can be obtained from diagrams with more sources, in complete analogy with what we did for the gauge field.
Let us now turn to the scalar components $\varphi_a^I$ of the ${\cal N}=4$ vector multiplet. The simplest diagram which can describe their emission is a mixed disk with two fermionic boundary changing operators, like the one represented in Fig. \[fig:scalaremission\]. The corresponding amplitude in momentum space is $$\begin{aligned}
\varphi_a^I(p;{\bar \mu},\mu) &
\equiv& {\Big\langle\hskip -5pt\Big\langle}{\cal
V}_{\varphi_{a}^I}(-p)
{\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}({\bar\mu},\mu)}
=
{\Big\langle\hskip -5pt\Big\langle}V^{(-1/2)}_{\bar\mu}\,{\cal
V}_{\varphi_{a}^I}^{(-1)}(-p) \, V^{(-1/2)}_{\mu}
{\Big\rangle\hskip -5pt\Big\rangle}\nonumber \\ &=&
-\,\frac{\ii}{2}\,(T^I)^{v}_{~u}\,(\bar\Sigma_a)_{AB}\,
{\mu}^{Bu}\,{\bar\mu}^{A}_{~v}\,\ee^{-\ii p\cdot x_0}
\label{scalarampl}\end{aligned}$$ where ${\cal D}({\bar\mu},\mu)$ is the mixed disk created by the insertion of $V_{\bar\mu}$ and $V_\mu$. Defining $$\varphi^{AB}= \frac{1}{2\sqrt 2}\,(\Sigma^a)^{AB}\,\varphi^a~~,
\label{phiAB}$$ we can rewrite (\[scalarampl\]) as $$\varphi^{AB\,,\,I}(p;{\bar \mu},\mu) =-\,\frac{\ii}{\sqrt 2}\,
(T^I)^{v}_{~u}\, \mu^{[Au}\,{\bar\mu}^{B]}_{~\,v}\,\ee^{-\ii
p\cdot x_0} \label{phiAB1}$$ where the square brackets mean antisymmetrization with weight one. Alternatively, this result can be obtained from the Ward identity $${\Big\langle\hskip -5pt\Big\langle}\Big[\,\bar\xi
\,q\,,V_{\bar\mu}\,\Big] \, {\cal V}_{{\bar\Lambda}_{\dot\alpha
A}^I}\!(-p)\,V_{\mu}{\Big\rangle\hskip -5pt\Big\rangle}+
{\Big\langle\hskip -5pt\Big\langle}V_{\bar\mu}\,\Big[\bar\xi \,q\,, {\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}\!(-p)\Big] \,
V_{\mu}{\Big\rangle\hskip -5pt\Big\rangle}+{\Big\langle\hskip -5pt\Big\langle}V_{\bar\mu}\,{\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}\!(-p)\, \Big[\bar\xi\,q\,,
V_{\mu}\Big] {\Big\rangle\hskip -5pt\Big\rangle}=0 \label{ward3}$$ which establishes a relation between the scalar and the gaugino amplitudes [^9]. Indeed, working out the commutators, we find $${\Big\langle\hskip -5pt\Big\langle}V_{\delta_{\bar\xi}\bar w} \, {\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}\!(-p)\,
V_{\mu}{\Big\rangle\hskip -5pt\Big\rangle}-\,\ii\,\bar\xi^{\dot\alpha}_B
\,(\Sigma^a)^{BA}\, {\Big\langle\hskip -5pt\Big\langle}V_{\bar\mu}\, {\cal
V}_{\varphi_a^I}(-p) \, V_{\mu}{\Big\rangle\hskip -5pt\Big\rangle}+{\Big\langle\hskip -5pt\Big\langle}V_{\bar\mu}\,{\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}\!(-p)\,
V_{\delta_{\bar\xi}w}{\Big\rangle\hskip -5pt\Big\rangle}=0~~,
\label{ward4}$$ from which (\[phiAB1\]) easily follows upon using (\[gauginiampl1\]), (\[gauginiampl2\]) and (\[susyw\]).
The space-time profile of the adjoint scalars is obtained by taking the Fourier transform of the amplitude (\[phiAB1\]) multiplied by the massless scalar propagator $1/p^2$, namely $$\begin{aligned}
\varphi^{AB\,,\,I}(x) &=&\int {d^4 p\over (2\pi)^2} \,
\varphi^{AB\,,\,I}(p;\bar\mu,{\mu})\,\frac{1}{p^2}\,\ee^{\ii
p\cdot x} \nonumber \\ &=&-\,\frac{\ii}{\sqrt
2}\,(T^I)^{v}_{~u}\,\mu^{[Au}\,{\bar\mu}^{B]}_{~\,v}
\,\frac{1}{(x-x_0)^2}~~. \label{scalarsol}\end{aligned}$$ When the parameters are restricted to satisfy the ADHM constraints, this expression represents the leading term of the adjoint scalars in the singular gauge. Moreover, from (\[scalarsol\]) one can see that $$({\widehat \varphi}^{AB}(x))^{u}_{~v} \equiv
-\,\ii\,\varphi^{AB\,,\,I}(x)\,(T^I)^{u}_{~v}
=-\,\frac{1}{2\sqrt 2}\left(\mu^{[Au}\,{\bar\mu}^{B]}_{~\,v}
-\frac{1}{2}\,\mu^{[Ap}\,{\bar\mu}^{B]}_{~\,p}\,\tilde\delta^{u}_{~v}\right)
\frac{1}{(x-x_0)^2}
\label{scalarsol1}$$ with $$\big|\big|\tilde\delta^u_{~v}\big|\big|
=\pmatrix{0_{[N-2]\times[N-2]}&
0_{[N-2]\times[2]}\cr
0_{[2]\times[N-2]}& 1_{[2]\times[2]}}~~,
\label{fizeromod30}$$ which is indeed the leading term at large distance of the exact instanton solution (see for example appendix \[app:ADHM\]). As before, the subleading terms are given by diagrams with more insertions of source terms.
We can summarize our findings by saying that the mixed disks with two boundary changing operators represented in Figs. \[fig:md2\], \[fig:gauginoemission\] and \[fig:scalaremission\] describe, respectively, the large distance behavior in the instanton background of the vector $A_\mu^I$, of the gaugino $\Lambda_{\alpha A}^I$ and of the scalars $\varphi_{AB}^I$ in the singular gauge, and that their space-time profiles can be written as $$\begin{aligned}
\label{AsymSol}
A^I_\mu (x) &=& J^I_{\nu\mu}\, \partial^\nu G(x-x_0)~~,
\nonumber\\
\Lambda^{\alpha A\,,\,I}(x) &=& J_{\dot\beta}^{A\,,\,I}
\,(\bar\sigma^\nu)^{\dot\beta\alpha}\, \partial_\nu G(x-x_0)~~,\\
\varphi^{AB\,,\,I}(x) &=& J^{AB\,,\,I}\,G(x-x_0)~~,
\nonumber\end{aligned}$$ where the scalar Green function $G(x-x_0)$ is defined in (\[prop\]) and the various source terms $J^I_{\nu\mu}$, $J_{\dot\beta}^{A\,,\,I}$ and $J^{AB\,,\,I}$ are bilinear expressions in the instanton moduli which can be read from (\[corr5\]), (\[gauginiampl1\]), (\[gauginiampl2\]) and (\[scalarampl\]) respectively. Moreover, taking into account the fall-off at infinity of the various fields, one can easily realize that the equations of motion that follow from the SYM action (\[N4susy\]) in the Lorentz gauge reduce at large distances simply to free equations [*i.e.*]{} $$\square A^I_\mu=0~~,~~
{\partial\!\!\!/}_{\alpha\dot\beta}\,\Lambda^{\alpha A\,,\,I}=0~~,~~
\square \varphi^{AB\,,\,I}=0~~,
\label{freeeqs}$$ which indeed admit a solution of the form (\[AsymSol\]) in the presence of source terms.
Broken supersymmetries
----------------------
Let us now consider the supersymmetries of the D3 branes which are broken by the D-instantons, namely those that are generated by the charges $q'_{\alpha A}\equiv\left(Q_{\alpha A}
+ \widetilde Q_{\alpha A}\right)$ (see section \[subsec:susy\]). As shown in (\[bint\]), when one pulls the integration contour of a charge operator to a boundary that does not preserve it, one obtains the integrated emission vertex for the Goldstone field corresponding to the broken charge. In our case, the goldstino associated to the breaking of $q'_{\alpha A}$ by the D$(-1)$ boundary is the modulus $M'^{\alpha A}$. Therefore, by acting with the broken supercharges $q'_{\alpha A}$ on a given instanton solution, one can modify it by shifting its supermoduli with $M'$ dependent terms. In particular, one can relate the “minimal” emission diagrams of Figs. \[fig:md2\], \[fig:gauginoemission\] and \[fig:scalaremission\], that contain no D$(-1)$/D$(-1)$ moduli, to diagrams which instead have additional insertions of $M'$ moduli [@Green:2000ke]. Thus, the use of the broken supersymmetries allows us to determine the $M'$ dependence and complete the full superinstanton solution.
Let us see how this works in a specific example and consider the following Ward identity $$\begin{aligned}
&&{\Big\langle\hskip -5pt\Big\langle}\Big[M'q'\,,V_{\bar w}\Big] \, {\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p)\,
V_{w}{\Big\rangle\hskip -5pt\Big\rangle}+{\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,\Big[M' q'\,,{\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p)\Big] \,
V_{w}{\Big\rangle\hskip -5pt\Big\rangle}\label{ward5}
\\&&~~~~~~~+\, {\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,{\cal
V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p)\, \Big[M' q'\,,V_{w}\Big]
{\Big\rangle\hskip -5pt\Big\rangle}= -\,{\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,{\cal V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p) \,V_{w}\!
\int \hskip -4pt V_{M'}
{\Big\rangle\hskip -5pt\Big\rangle}~~. \nonumber\end{aligned}$$ Differently from the identities (\[ward1\]) and (\[ward3\]) associated to the preserved supersymmetries, the right hand side of (\[ward5\]) is non-zero as a consequence of the fact that the supercharge $q'$ is broken on the D$(-1)$ boundary. A pictorial representation of this Ward identity is provided in Fig. \[fig:broken\]. Using the fact that the commutators of $q'$ with $V_w$ and $V_{\bar w}$ vanish (as we already noticed at the end of section \[sec:review\]), and that $$\label{q'barlambda}
\comm{M' q'}{{\cal V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p)}
= \ii\,M'^{\beta A}\,(\sigma_\mu)_{\beta}^{~\dot\alpha}\,
{\cal V}_{A_\mu^I}(-p)~~,$$ we can deduce from (\[ward5\]) the following relation $$\begin{aligned}
{\bar \Lambda}^{\dot\alpha A\,,\,I}(p;\bar w,w,M') &\equiv&
{\Big\langle\hskip -5pt\Big\langle}{\cal V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p)\,
{\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}(\bar w,w,M')}
=
{\Big\langle\hskip -5pt\Big\langle}V_{\bar w}\,{\cal V}_{{\bar\Lambda}_{\dot\alpha A}^I}(-p)\,
V_{w}\int \hskip -4pt V_{M'}
{\Big\rangle\hskip -5pt\Big\rangle}\nonumber \\\nonumber\\&=&
-\,\ii\,M'^{\beta A}\,(\sigma^\mu)_{\beta}^{~\dot\alpha}\,
A_\mu^I(p;\bar w, w)~~,
\label{LM'fromA}\end{aligned}$$ which reduces the calculation of the 4-point amplitude ${\bar \Lambda}^{\dot\alpha A\,,\,I}(p;\bar w,w,M')$ to an algebraic manipulation on the 3-point amplitude (\[dia1\]). Notice again that, despite the presence of many vertex operators, the amplitude (\[LM’fromA\]) is actually a 1-point function from the point of view of the four-dimensional gauge theory, since the only dynamical field is the emitted gaugino. To obtain its corresponding space-time profile we multiply ${\bar \Lambda}_{\dot\beta}^{~\dot\alpha A\,,\,I}(p;w,\bar w,M')$ by the propagator $\ii\!\not \hskip -2pt p^{\dot\beta\alpha}/p^2$ and take the Fourier transform, getting $$\begin{aligned}
\label{xspaceLM'}
\Lambda^{\alpha A\,,\,I} (x) & = & \int {d^4p\over (2\pi)^2}\,
{\bar \Lambda}_{\dot\beta}^{~ A\,,\,I}(p;\bar w,w,M')
\,\frac{\ii\!\not \hskip -2pt p^{\dot\beta\alpha}}{p^2}\,\ee^{\ii p\cdot x}
\nonumber \\
&=&M'^{\beta A}\,(\sigma^\mu\,\bar\sigma^\nu)_{\beta}^{~\alpha}
\int {d^4p\over (2\pi)^2}\,\frac{p_\nu \,A_\mu^I(p;w,\bar w)}{p^2}\,
\ee^{\ii p\cdot x}
\\
& = &
-\,\ii\,M'^{\beta A}\,(\sigma^\mu\,\bar\sigma^\nu)_{\beta}^{~\alpha}\,
\partial_\nu \,A^I_\mu(x)~
\stackrel{x\to\infty}{\simeq}~
{\ii\over 2}\, M'^{\beta A}\,(\sigma^{\mu\nu})_{\beta}^{~\alpha}\,
F^I_{\mu\nu}(x)~~.
\nonumber\end{aligned}$$ In the last step we have used the fact that in the instanton solution (\[corr5\]) the vector field $A_\mu^I$ is in the Lorenz gauge and that, due to the fall-off at infinity of the potential, the associated non-abelian field strength $F^I_{\mu\nu}$ simply reduces to $\partial_{\mu} A^I_{\nu} - \partial_{\nu} A^I_{\mu}$ in the large distance limit. Eq. (\[xspaceLM’\]) shows that a mixed disk with one $M'$ insertion and one emitted gaugino reproduces exactly the chiral fermionic profile that is created by acting with a broken supercharge on the instanton background according to the $\eta$-supersymmetry transformation rules (\[susygauge\]). Of course, with a repeated use of these supercharges, further insertions of $M'$ can be obtained and the entire structure of the superinstanton zero-modes can be reconstructed (see for example eq. (4.60) in the recent review [@Dorey:2002ik]). Our analysis, which for simplicity we have illustrated only in the simplest case, shows the precise relation between these zero-modes and the mixed disk amplitudes with insertions of $M'$ vertex operators. Finally, we recall that with the replacement $$M'^{\alpha A}~\to~- {\bar \zeta}_{\dot\alpha}^{~A}
\,(\bar\sigma^\mu)^{\dot\alpha\beta}\,a'_\mu
\label{replacement}$$ one can account for the superconformal zero-modes of the ${\cal N}=4$ instanton solution parametrized by the fermionic variables $\bar{\zeta}$.
String amplitudes and instanton calculus {#S1}
========================================
In this section we want to explain what is the stringy procedure to compute instanton corrections to scattering amplitudes in gauge theories and show its relation with the standard instanton calculus of field theory. The key ingredient will be the identification of the instanton solution with the string theory 1-point function on mixed disks that we have proven in the previous sections. Exploiting this fact, we will also be able to relate our approach to the analysis of the leading D-instantons effects on scattering amplitudes that has been presented in Ref. [@Green:2000ke]. Let us first recall a few basic facts on the relation between string theory correlators, effective actions and Green functions in field theory. As we have reviewed in [section \[sec:d3d-1\]]{}, the tree-level scattering amplitude among $n$ states of the 3/3 strings (which we denote generically by $\phi_i$) is given by [^10] $${\cal A}_{\phi_1\ldots\phi_n}= {\Big\langle\hskip -5pt\Big\langle}{V}_{\phi_1}(p_1)\ldots{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}\equiv \phi_n(p_n)\ldots\phi_1(p_1)\, {\Big\langle\hskip -5pt\Big\langle}{\cal V}_{\phi_1}(p_1)\ldots{\cal V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}\label{propervertex}$$ where the correlator among the vertex operators is computed on a disk with D3 boundary conditions (see for example eq. (\[ampl33\])). By taking the limit $\alpha'\to 0$ and extracting the 1PI part, we obtain the following contribution to the effective action $$-\int \!\frac{d^4p_1}{(2\pi)^2}\ldots\frac{d^4p_n}{(2\pi)^2}
~\phi_n(p_n)\ldots\phi_1(p_1)
\left.{\Big\langle\hskip -5pt\Big\langle}{\cal V}_{\phi_1}(p_1)\ldots{\cal V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}\right|_{\alpha'\to 0}^{\rm 1PI}~~,$$ which, in turn, induces the following [*amputated*]{} Green function [^11] $$\left.\Big\langle
\phi_1(p_1)\ldots\phi_n(p_n)\Big\rangle\right|_{\rm amput.}
= \left.{\Big\langle\hskip -5pt\Big\langle}{\cal V}_{\phi_1}(-p_1)\ldots{\cal V}_{\phi_n}(-p_n)
{\Big\rangle\hskip -5pt\Big\rangle}\right|_{\alpha'\to 0}^{\rm 1PI}~~.
\label{amputfunct}$$ If one computes the above correlators on world-sheets with more boundaries one obtains the perturbative loop corrections to the effective action and Green functions.
We now want to investigate how the previous relations get modified by the presence of $k$ D-instantons. In this case, as we have thoroughly explained, the correlators of vertex operators receive contributions also from world-sheets with a part of their boundary on the D-instantons, and specifically, at the lowest order in the string perturbation theory, from mixed disks. It is convenient to denote by ${\cal D}({\cal M})$ the sum of all disks with all possible insertions of the moduli ${\cal M}$ of the $k$ instantons, as represented in Fig. \[fig:amplitude0\]. Each term in this sum corresponds to an amplitude with no vertex operator of the 3/3 strings, and thus it represents a vacuum contribution from the point of view of the theory on the D3 branes. A noteworthy point is that also the first term in ${\cal D}({\cal M})$, [*i.e.*]{} the pure D$(-1)$ disk without insertions, contributes. Indeed, as shown in Ref. [@Polchinski:fq], it evaluates to minus $k$ times the topological normalization $C_0$ given in (\[C0\]). Collecting all terms and using the results of [section \[sec:d3d-1\]]{}, we obtain that the vacuum contribution of the “disk” ${\cal D}({\cal M})$ is such that $$\label{topo}
\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt\big\rangle_{{\cal D}({\cal M})}
\stackrel{\scriptstyle \alpha'\to 0}{\simeq}~ -\,S\big[{\cal M}\big]
\,\equiv \,-\,
\frac{8\pi^2k}{g_{\rm YM}^2}\,-\,S_{\rm moduli}$$ where the moduli action is defined in (\[smoduli4\]).
Let us now consider the correlators of 3/3 string vertex operators on ${\cal D}({\cal M})$, which are defined by $$\begin{aligned}
\label{cftcorr1}
&&\hskip -20pt {\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(p_1)\ldots \mathcal{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})} =
\\\nonumber \\
&&=\,C_0
\sum_m \!\int\!\frac{\prod_{i} dz_i\,\prod_{j} dy_j}{dV_{abc}}~
\Big\langle \mathcal{V}_{\phi_1}(z_1;p_1)\ldots \mathcal{V}_{\phi_n}(z_n;p_n)
V_{\mathcal{M}_1}(y_1)\ldots V_{\mathcal{M}_m}(y_m)
\Big\rangle~~.
\nonumber\end{aligned}$$ As is obvious from this definition, the string theory correlator depends on the $k$-instanton moduli ${\cal M}$, over which one has to integrate in order to account for all possible configurations. This fact is intuitively clear, since in our description all possible mixed boundary conditions are obtained by inserting the moduli.
The integration over ${\cal M}$ is the analogue of what one typically does in quantum field theory, where the path integral describing a specific correlator is split into the sum of path integrals restricted to the different topological sectors, namely $$\label{ftpi1}
\int \!\!{\cal D}\phi ~\phi_1(p_1)\ldots \phi_n(p_n)\, \ee^{-S[\phi]}=
\sum_k \!\int \!\!{\cal D}\delta\phi^{(k)} ~ \delta\phi_1^{(k)}(p_1)\ldots
\delta\phi_n^{(k)}(p_n)\,
\ee^{-S_k -S[\delta\phi^{(k)}]}$$ where $\delta\phi^{(k)}$ denotes the fluctuation of $\phi$ around a classical background with topological charge $k$ and action $S_k$. In this framework, the integration over all moduli of the non-trivial background arises directly from the path-integral, as a trade-off for the integration over the zero-mode fluctuations. However, from string theory we obtain a first-quantized description in which the string world-sheet gives rise for $\alpha'\to 0$ to the world-lines of a (super)particle description of the Feynman diagrams of the field theory. In this description, the different topological sectors can be described only by explicitly coupling the (super)particle to a non-trivial background field $A_\mu$ through the insertion of $$\label{ftwl1}
\Tr\, P\, \exp\left(\int_\gamma
A_\mu(x(\tau);{\cal M}) \dot x^\mu \, d\tau \right)$$ and then integrating over the background parameters ${\cal M}$. This procedure is pictorially illustrated in Fig. \[fig:amplitude01\] for a specific disk amplitude.
The integration over the moduli ${\cal M}$ has several important consequences. First of all, also world-sheets with disconnected components must be taken into account. For example, besides the correlator (\[cftcorr1\]), one should also consider the following one $${\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(p_1)\ldots \mathcal{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\,\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt\big\rangle_{{\cal D}({\cal M})}~~,
\label{cftcorr11}$$ which is disconnected from the two-dimensional point of view but connected from the point of view of the four-dimensional theory on the D3 branes. Obviously, we can add more disconnected components, and thus in general we have $$\frac{1}{\ell\,!}~{\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(p_1)\ldots \mathcal{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}({\cal M})}\, \left(\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt\big\rangle_{{\cal D}({\cal M})}\right)^\ell
\label{cftcorr1ell}$$ where the symmetry factor is due to the combinatorics of boundaries [@Polchinski:fq]. Summing over all these terms, we therefore get $${\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(p_1)\ldots \mathcal{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\, \ee^{\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt\big\rangle_{{\cal D}({\cal M})}}~~.
\label{cftcorr1exp}$$ However, this is not yet the full story. In fact, for the same arguments we should also take into account diagrams in which the $n$ vertex opertors $\mathcal{V}_{\phi_i}(p_i)$ are distributed among various disconnected components. For example, besides the correlator (\[cftcorr1exp\]) we should also consider the following one $${\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(p_1)\,\mathcal{V}_{\phi_2}(p_2){\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\,
{\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_3}(p_3)\ldots \mathcal{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\, \ee^{\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt\big\rangle_{{\cal D}({\cal M})}}~~.
\label{cftcorr2exp}$$ This contribution appears to be totally disconnected; however, it is connected with respect to the $\phi$’s because of the integration over the moduli ${\cal M}$ which all sit at the same point where the stack of $k$ D-instantons is located. Distributing the $\phi$’s in all possible ways, one generates various configurations which are compactly represented in Fig. \[fig:amplitude1\].
Since each expectation value on ${\cal D}({\cal M})$ is proportional to $C_0 \propto g_s^{-1}$ (see eqs. (\[cftcorr1\]) and (\[C0\])), the dominant contribution for small $g_s$ is the one in which a single vertex $\mathcal{V}_{\phi}$ is inserted in each disk [@Green:1997tv; @Green:2000ke], namely $${\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(p_1){\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\!\!\!\cdots~
{\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_n}(p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\, \ee^{\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt\big\rangle_{{\cal D}({\cal M})}}~~,
\label{cftcorrdom}$$ whereas other terms, like for example (\[cftcorr2exp\]), are subleading for small $g_s$ [^12]. Moreover, this correlator is clearly 1PI. Thus, we can conclude that in the field theory limit, the dominant contribution to the amputated Green function of $n$ fields of the 3/3 string sector in the presence of $k$ D-instanton is given by (see Fig. \[fig:amplitude2\]) $$\begin{aligned}
\label{corrinst}&&\hskip -15pt\left.\Big\langle
\phi_1(p_1)\ldots\phi_n(p_n)\Big\rangle\right|_{\rm amput.}^{\rm D-inst.}
=\\
&&\hskip 15pt=\int d{\cal M}\,\left.
{\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_1}(-p_1){\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\!\!\!\cdots~
{\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi_n}(-p_n)
{\Big\rangle\hskip -5pt\Big\rangle}_{\!{\cal D}({\cal M})}\, \ee^{\big\langle\hskip -2.5pt\big\langle \,1\,
\big\rangle\hskip -2.5pt
\big\rangle_{{\cal D}({\cal M})}}\right|_{\alpha'\to 0}~~.
\nonumber\end{aligned}$$ Reinstating the propagators (see footnote \[footnote\]) and Fourier transforming, we obtain the following Green function in configuration space $$\left.\Big\langle
\phi_1(x_1)\ldots\phi_n(x_n)\Big\rangle\right|_{\rm D-inst.}
=
\int d{\cal M}~\phi_1^{\rm disk}(x_1;{\cal M})
\cdots~\phi_n^{\rm disk}(x_n;{\cal M})
\, \ee^{-S[{\cal M}]}
\label{Green1}$$ where we have used (\[topo\]) and defined $$\phi^{\rm disk}(x;{\cal M})
= \int {d^4p\over (2\pi)^2}~\ee^{\ii p\cdot x} \, {1\over p^2}
\left.{\Big\langle\hskip -5pt\Big\langle}\mathcal{V}_{\phi}(-p){\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}({\cal M})}\right|_{\alpha'\to 0}
~~.
\label{onepoint}$$ Using the results of sections \[sec:instanton\] and \[sec:superinstanton\] we can identify the right hand side of (\[onepoint\]) with the classical profile $\phi^{\rm cl}(x;\CM)$ of the superinstanton solution for the field $\phi$. For example, the contributions from the simplest mixed disks, [*i.e.*]{} those with only two insertions of boundary changing operators, account for the leading terms in the large distance expansion of the superinstanton solution, as we have seen explicitly for $k=1$ in eqs. (\[gf1\]), (\[gauginosol2\]) and (\[scalarsol\]). The contributions from mixed disks with more boundary changing operators in the limit $\alpha'\to 0$ account instead for the sub-leading terms in the large distance expansion, as we have shown for the gauge field in [section \[sec:instanton\]]{} (see also appendix \[app:subleading\]). Thus, we can write $$\label{onepointclass}
\phi(x;\CM)^{\mathrm{disk}} =
\phi^{\rm cl}(x;\CM)$$ and conclude that the stringy prescription (\[Green1\]) of computing correlation functions in the presence of D-instantons is exactly equivalent to the standard field theory prescription of the instanton calculus $$\left.\Big\langle
\phi_1(x_1)\ldots\phi_n(x_n)\Big\rangle\right|_{\rm inst.}
=
\int d{\cal M}~\phi_1^{\rm cl}(x_1;{\cal M})
\cdots~\phi_n^{\rm cl}(x_n;{\cal M})
\, \ee^{-S[{\cal M}]}~~.
\label{Green10}$$
The effects of D-instantons on the scattering amplitudes of the gauge theory on the D3 branes can be encoded by introducing new effective vertices for the 3/3 fields $\phi_i$’s which suitably modify the SYM action (see also Ref. [@Green:2000ke]). These D-instanton induced vertices originate from the amputated Green functions (\[corrinst\]) upon including the polarization fields for the external legs, and are clearly moduli dependent. At [*fixed*]{} moduli, only the 1-point functions are irreducible and so the gauge effective action induced by the D-instantons on the D3 branes will be $$S_{(-1)/3}= - \,\sum_\phi\int\frac{d^4p}{(2\pi)^2}~\phi(p)\, \left.{\Big\langle\hskip -5pt\Big\langle}{\cal
V}_{\phi}(p){\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}({\cal M})}\right|_{\alpha'\to 0}
\label{insteffact}$$ where the sum is over all massless fields of the ${\cal N}=4$ vector multiplet. Since the tadpoles ${\Big\langle\hskip -5pt\Big\langle}{\cal
V}_{\phi}(p){\Big\rangle\hskip -5pt\Big\rangle}_{{\cal D}({\cal M})}$ are generically of the form $J_\phi({\cal M})\,\ee^{\ii p\cdot x_0}$ (see for instance eqs. (\[gauginiampl1\]) and (\[scalarampl\])) [^13], we can write this effective action simply as $$S_{(-1)/3}= - \,\sum_\phi\phi(x_0)\,J_\phi({\cal M})
\label{insteffact1}$$ which manifestly shows that the 1-point functions on the mixed disks are sources for the gauge fields at the instanton location. Using the expressions for the various tadpoles computed in sections \[sec:instanton\] and \[sec:superinstanton\], it is easy to realize that $$\label{effaclinear1}
S_{(-1)/3} = -\,\frac{1}{2}\,F^{I}_{\mu\nu}(x_0)\,J^{\mu\nu\,,\,I}(\CM)
- \bar\Lambda^I_{\dot\alpha A}(x_0)\,J^{\dot\alpha A\,,\,I}(\CM)
- \varphi^I_{AB}(x_0)\,J^{AB\,,\,I}(\CM)$$ where the various sources are defined in (\[AsymSol\]). This expression represents the non-abelian extension of the action given for example in Ref. [@Green:2000ke; @Dorey:2001ym].
We think that our analysis clarifies the role played by D-instantons on the scattering amplitudes of four-dimensional gauge theories already discussed in the literature. In particular we have shown that the stringy procedure to compute instanton corrections to correlation functions reproduces in the field theory limit the standard instanton calculus in virtue of the identification (\[onepointclass\]). We hope that these ideas and techniques can be useful also for practical calculations in the ${\cal N}=4$ SYM theory considered in this paper as well as in gauge theories with lower supersymmetries. 0.2cm We thank Rodolfo Russo for useful discussions and exchange of ideas.
Notations and conventions {#app:conventions}
=========================
#### Notations:
We use the following notations for indices:
- $d=10$ vector indices: $M,N,\dots\in\{1,\dots,10\}$;
- $d=4$ vector indices: $\mu,\nu,\dots\in\{1,\dots,4\}$;
- $d=6$ vector indices: $a,b,\dots\in\{5,\dots,10\}$;
- chiral and anti-chiral spinor indices in $d=10$: ${\cal A}$ and $\dot{\cal A}$;
- chiral and anti-chiral spinor indices in $d=4$: ${\alpha}$ and $\dot{\alpha}$;
- spinor indices in $d=6$: ${}^{A}$ and ${}_A$ in the fundamental and anti-fundamental of ${\mathrm{SU}}(4)\simeq {\mathrm{SO}}(6)$.
Our choice for the group indices is the following:
- $\mathrm{SU}(N)$ colour indices: $I,J,\dots\in\{1,\dots,N^2 - 1\}$;
- $U(k)$ colour indices: $U,V,\dots\in\{1,\dots,k^2\}$;
- D$3$ indices: $u,v,\dots\in\{1,\dots,N\}$;
- D$(-1)$ indices: $i,j,\dots\in\{1,\dots,k\}$;
- $\mathrm{SU}(2)$ adjoint indices: $c,d,\dots\in\{1,2,3\}$.
0.7cm
#### $\mathbf{d=4}$ Clifford algebra:
The Euclidean Lorentz group $\mathrm{SO}(4)\sim
\mathrm{SU}(2)_+\times \mathrm{SU}(2)_-$ is realized on spinors in terms of the matrices $(\sigma^\mu)_{\alpha\dot\beta}$ and $(\bar\sigma^{\mu})^{\dot\alpha\beta}$ with $$\label{sigmas}
\sigma^\mu =
(\mathbf{1},-\ii\vec\tau)~,\hskip 0.8cm
\bar\sigma^\mu =
\sigma_\mu^\dagger = (\mathbf{1},\ii\vec\tau)~~,$$ where $\tau^c$ are the ordinary Pauli matrices. They satisfy the Clifford algebra $$\label{cliff4}
\sigma_\mu\bar\sigma_\nu + \sigma_\nu\bar\sigma_\mu =
2\delta_{\mu\nu}\,\mathbf{1}~~,$$ and correspond to a Weyl representation of the $\gamma$-matrices, $$\label{gamma4def}
\gamma^\mu
=\left(\matrix{0 & \sigma^\mu \cr
\bar\sigma^\mu & 0}\right)$$ acting on the spinor $$\psi=\left(\matrix{ \psi_\alpha \cr \psi^{\dot\alpha} }\right) ~~~.$$ Out of these matrices, the $\mathrm{SO}(4)$ generators are defined by $$\label{sigmamunu}
\sigma_{\mu\nu}
={1\over 2}(\sigma_\mu\bar\sigma_\nu -
\sigma_\nu\bar\sigma_\mu)~,
\hskip 0.8cm
\bar\sigma_{\mu\nu}
={1\over 2}(\bar\sigma_\mu\sigma_\nu -
\bar\sigma_\nu\sigma_\mu)~~;$$ the matrices $\sigma_{\mu\nu}$ are self-dual and thus generate the $\mathrm{SU}(2)_+$ factor; the anti-self-dual matrices $\bar\sigma_{\mu\nu}$ generate instead the $\mathrm{SU}(2)_-$ factor. Notice that the indices in the $\mathbf{2}$ of $\mathrm{SU}(2)_+$ are denoted by $\alpha$ and those for the $\mathbf{2}$ of $\mathrm{SU}(2)_-$ by $\dot\alpha$. The charge conjugation matrix is block-diagonal in this Weyl basis: $$\label{charge4}
C_{(4)}
=\left(\matrix{C^{\alpha\beta} & 0 \cr
0 & C_{\dot\alpha\dot\beta} }\right)
=\left(\matrix{-\varepsilon^{\alpha\beta} & 0 \cr
0 & -\varepsilon_{\dot\alpha\dot\beta} }\right)$$ with $\varepsilon^{12}=\varepsilon_{12}
=-\varepsilon^{\dot 1\dot 2}=-\varepsilon_{\dot 1\dot 2}=+1$. Moreover we raise and lower spinor indices as follows $$\psi^\alpha=\varepsilon^{\alpha\beta}\,\psi_\beta
\,\,\,\,\, , \,\,\,\,\,
\psi_{\dot\alpha}= \varepsilon_{\dot\alpha\dot\beta}\,\psi^{\dot\beta}~~.$$
0.7cm
#### ’t Hooft symbols:
The explicit mapping of a self-dual $\mathrm{SO}(4)$ tensor into the adjoint representation of the $\mathrm{SU}(2)_+$ factor is realized by the ’t Hooft symbols $\eta^c_{\mu\nu}$; the analogous mapping of an anti-self dual tensor into the adjoint of the $\mathrm{SU}(2)_-$ subgroup is realized by $\bar\eta^c_{\mu\nu}$. One has $$(\sigma_{\mu\nu})_{\alpha}^{~\beta} =
\ii \,\eta^c_{\mu\nu}\, (\tau^c)_{\alpha}^{~\beta}
,\qquad
(\bar\sigma_{\mu\nu})^{\dot\alpha}_{~\dot\beta} = \ii \,\bar\eta^c_{\mu\nu}\,
(\tau^c)^{\dot\alpha}_{~\dot\beta}~~.$$ An explicit representation of the ’t Hooft symbols is given by $$\begin{aligned}
\label{etadef}
\eta^c_{\mu\nu} & = &\bar\eta^c_{\mu\nu}
= \varepsilon_{c\mu\nu},\qquad \mu,\nu\in \{1,2,3\},
\nonumber\\
\bar\eta^c_{4\nu}& =& -\eta^c_{4\nu} =\delta^c_{\nu},
\\
\eta^c_{\mu\nu} &= &- \eta^c_{\nu\mu},
\qquad \bar \eta^c_{\mu\nu} = - \bar \eta^c_{\nu\mu}~~.
\nonumber\end{aligned}$$ [F]{}rom it one can easily see that $$\begin{aligned}
\eta^c_{\mu\nu}\,\eta^{d\,\mu\nu} &=& 4\,\delta^{cd}~~,\\
\eta^{c}_{\mu\nu}\,\eta^{c}_{\rho\sigma}&=&
\delta_{\mu\rho}\,\delta_{\nu\sigma}
-\delta_{\mu\sigma}\,\delta_{\nu\rho}\,+\,
\varepsilon_{\mu\nu\rho\sigma}~~.
\label{etaeta}\end{aligned}$$ Analogous formulas hold for the contractions of two $\bar\eta$’s with a minus sign in the $\varepsilon$ term of (\[etaeta\]).
0.7cm
#### $\mathbf{d=6}$ Clifford algebra:
Taking advantage of the equivalence $\mathrm{SO}(6)\sim\mathrm{SU}(4)$, upon which a positive (negative) chirality spinor corresponds to a fundamental (anti-fundamental) $\mathrm{SU}(4)$ representation, we can represent the $\mathrm{SO}(6)$ spinor as $$\Lambda=\left(\matrix{ \Lambda^A \cr \Lambda_A }\right)$$ on which the following gamma matrices act $$\label{gamma6def}
\Gamma^a =
\left(\matrix{0 & \Sigma^{a} \cr
\bar\Sigma^a & 0}\right)~~~.$$ The matrices $\Sigma^a$ and $\bar\Sigma^a$ realize the six-dimensional Clifford algebra $$\label{gamma6}
(\Sigma^a)^{AB}(\bar\Sigma^b)_{BC} + (\Sigma^b)^{AB}(\bar\Sigma^a)_{BC} = 2\,
\delta^{ab}\,\delta^A_{\,\,\,C}~~,$$ (with $(\bar\Sigma^a)_{\, AB} = (\Sigma^{a\, BA})^*$). An explicit realization can be given in terms of ’t Hooft symbols $$\label{Gammadef}
\Sigma^a = \left(\eta^3,\ii \bar\eta^3,\eta^2,\ii\bar\eta^2,
\eta^1,\ii\eta^1\right)~~,
\hskip 0.8cm
\bar\Sigma^a = \left(-\eta^3,\ii \bar\eta^3,-\eta^2,\ii\bar\eta^2,
-\eta^1,\ii\eta^1\right)~~.$$ The charge conjugation matrix is off-diagonal in this chiral basis: $$\label{C6}
C_{(6)}
=\left(\matrix{ 0 & C_{A}^{~B} \cr
C^{A}_{~ B} & 0 }\right)
=\left(\matrix{ 0 & -\ii\,\delta_{A}^{~ B} \cr
-\ii\,\delta^{A}_{~ B} & 0 }\right) ~~.$$ 0.7cm
#### $\mathbf{d=10}$ Clifford algebra:
The ten-dimensional $\gamma$-matrices $\Gamma^{M}_{(10)}$ and the charge conjugation matrix $C_{(10)}$ are expressed in terms of the four- and six-dimensional matrices as $$\begin{aligned}
\Gamma^\mu_{(10)}=\gamma^\mu\otimes\mathbf{1}
\,\,& ,&\,\,
\Gamma^a_{(10)}=\gamma^5\otimes\Gamma^a~~,
\nonumber \\
\Gamma^{11}_{(10)}=\gamma^5\otimes\Gamma^7\,\,& ,&\,\,
C_{(10)}=C_{(4)}\otimes C_{(6)}~~,\end{aligned}$$ such that $$C_{(10)}\Gamma^{M}_{(10)}C_{(10)}^{-1}
=-\Gamma^{{M} \,\, T}_{(10)}~~.$$
0.7cm
#### Spin field correlators:
From the general formulae of [@Kostelecky:1986xg], by decomposing the ten-dimensional fields into four-dimensional and six-dimensional ones, we can derive the following “effective” OPE’s: $$\begin{aligned}
\label{spincorr}
S^{\dot\alpha}(z) \,S_\beta(w) \sim
\frac{1}{\sqrt{2}}\,
(\bar \sigma^\mu)^{\dot\alpha}_{~\beta}\, \psi_\mu(w)
~~& , &~~
S^A(z)\, S_B(w) \sim
\frac{\ii \,\delta^A_{~B}}{(z - w)^{3/4}}~~,
\\
\nonumber
S^{\dot\alpha}(z)\, S^{\dot\beta}(w) \sim
- \,\frac{\varepsilon^{\dot\alpha\,\dot\beta}}{(z - w)^{1/2}}
~~& , &~~
S^A(z) \,S^B(w) \sim
\frac{\ii}{\sqrt{2}}\,
\frac{(\Sigma^a)^{AB}\,\psi_a(w)}{(z - w)^{1/4}}~~,
\\
\nonumber
S_{\alpha}(z) \,S_{\beta}(w) \sim
\frac{\varepsilon_{\alpha\beta}}{(z - w)^{1/2}}
~~& ,&~~
\psi^a(z)\, S_A(w) \sim
\frac{1}{\sqrt{2}}\,
\frac{(\bar \Sigma^a)_{AB} \, S^B(w)}{(z - w)^{1/2}}~~,
\\
\nonumber
\psi^\mu(z) \,S^{\dot\alpha}(w) \sim
\frac{1}{\sqrt{2}}\,
\frac{(\bar \sigma^\mu)^{\dot\alpha\beta}\, S_\beta(w)}{(z - w)^{1/2}}
~~& , &~~
\psi^a(z) \,S^A(w) \sim
-\,\frac{1}{\sqrt{2}}\,
\frac{(\Sigma^a)^{AB} \, S_B(w)}{(z - w)^{1/2}}~~,
\\
\nonumber
\psi^\mu\psi^\nu(z)\, S^{\dot\alpha}(w) \sim
- \,\frac{1}{2}\,
\frac{(\bar\sigma^{\mu\nu})^{\dot\alpha}_{~\dot\beta}\,
S^{\dot\beta}(w)}{(z - w)}
~~& , &~~
\psi^a\psi^b(z)\, S^A(w) \sim
\frac{1}{2}\,
\frac{(\bar\Sigma^{ab})^A_{~B} \, S^B(w)}{(z - w)}~~.\end{aligned}$$ Other OPE’s which do not appear in (\[spincorr\]) can be simply obtained by a suitable change of the chiralities. From these OPE’s we can derive the following 3-point functions which have been used in the main text $$\begin{aligned}
\big\langle S^{\dot\alpha}(z_1)\, \psi_\mu(z_2)\,S_{\beta}(z_3)
\big\rangle &=&
\frac{1}{\sqrt{2}}\,
(\bar\sigma_{\mu})^{\dot\alpha}_{~\beta}
(z_1-z_2)^{-{1}/{2}}\,(z_2-z_3)^{-{1}/{2}}~~,
\\
\nonumber
\big\langle
S^{\dot\alpha}(z_1)\,\psi_\mu\psi_\nu(z_2)\,S^{\dot\beta}(z_3)
\big\rangle &=&
- \frac{1}{2} \,
(\bar\sigma_{\mu\nu})^{\dot\alpha\dot\beta}
(z_1-z_3)^{{1}/{2}}\,(z_1-z_2)^{-1}\,(z_2-z_3)^{-1}~~,
\\
\nonumber
\big\langle S^A(z_1)\,\psi^a(z_2)\,S^B(z_3)\big\rangle
&=&
\frac{\ii}{\sqrt{2}}\,(\Sigma^a)^{AB}\,(z_1-z_2)^{-1/2}\,
(z_1-z_3)^{-1/4}\,(z_2-z_3)^{-1/2}~~,
\\
\nonumber
\big\langle S_A(z_1)\,\psi^a(z_2)\,S_B(z_3)\big\rangle &=&
-\,\frac{\ii}{\sqrt{2}}\,(\bar\Sigma^a)_{AB}\,
(z_1-z_2)^{-1/2}\,
(z_1-z_3)^{-1/4}\,(z_2-z_3)^{-1/2}~~.\end{aligned}$$
0.7cm
#### Twist field correlators:
The $(-1)$/3 and the 3/$(-1)$ strings have four Neumann-Dirichlet directions, namely those along the world-volume of the D3 branes. Thus, the string fields $X^\mu$ have twisted boundary conditions; this fact can be seen as due to the presence of twist and anti-twist fields $\Delta(z)$ and $\bar\Delta(z)$ that change the boundary conditions from Neumann to Dirichlet and vice-versa by introducing a cut in the world-sheet (see for example Ref. [@orbifold]). The twist fields $\Delta(z)$ and $\bar\Delta(z)$ are bosonic operators with conformal dimension $1/4$ and their OPE’s are $$\Delta(z_1)\,\bar\Delta(z_2) \sim (z_1-z_2)^{1/2}~~~, ~~~
\bar\Delta(z_1)\,\Delta(z_2) \sim -\,(z_1-z_2)^{1/2} ~~,
\label{deltadelta}$$ where the minus sign in the second correlator is again an “effective” rule to correctly account for the space-time statistics in correlation functions.
A short review of the ADHM construction and of zero modes around an instanton background {#app:ADHM}
========================================================================================
Following the notation of Refs. [@Dorey:2002ik; @Vandoren_TO], we begin by introducing the basic objects in the ADHM construction of the $\mathrm{SU}(N)$ instanton solution in four dimensions, namely the $[N+2k]\times [2k]$ and $[2k]\times [N+2k]$ matrices $$\Delta(x)
\ =\ a \ +\
b \, x ~~~,~~~\bar{\Delta} (x)= \bar{a}+ \bar{x} \bar{b}
\label{del}$$ where $x_{\alpha\dot\beta}=x_\mu\,(\sigma^\mu)_{\alpha\dot\beta}$ and $\bar{x}^{\dot\alpha\beta}=x_\mu\,(\bar{\sigma}^\mu)^{\dot\alpha\beta}$ describe the position of the multi-instanton center of mass, and all the remaining moduli are collected in the matrix $a$ (see formula (\[aM\]) below). Finally, $b$ is a $[N+2k]\times [2k]$ matrix which can be conveniently chosen to be $$b= \pmatrix{0 \cr {\bf 1}_{[2k]\times[2k]}}
~~~,~~~
\bar{b}=\big(0 \ ,\ {\bf 1}_{[2k]\times[2k]}\big)~~.
\label{bs}$$ The moduli space of the solutions to the self-dual equations of motion is characterized in terms of the supercoordinates $$a \equiv \pmatrix{ w_{\dot\alpha}^{~ui}\cr
{a'}_{\alpha \dot\beta~li}}
~~~,~~~
{\cal M}^A \equiv
\pmatrix{ \mu^{Aui} \cr
{M'}^{\beta A}_{~~li}}~~,
\label{aM}$$ which satisfy the bosonic and fermionic ADHM constraints $$\begin{aligned}
\bar{\Delta}\,\Delta &=& f^{-1}_{k\times k} \,{\bf 1}_{[2]\times[2]}~~,
\label{aMa}\\
\bar{\Delta}{\cal M}^A &=& \bar{\cal M}^A \,\Delta
\label{constr}\end{aligned}$$ with $f_{k\times k}$ an invertible $k\times k$ matrix.
The solutions to the self-dual equations of motion for the various fields in the ${\cal N}=4$ vector multiplet are given by $$\begin{aligned}
{\widehat A}_\mu &=&
\bar{U} \, \partial_{\mu}
\, U~~, \nonumber\\
{\widehat \Lambda}^A &=& \bar{U}\left( {\cal M}^A f\, \bar{b}-
b\, f\, \bar{{\cal M}}^A\right)U ~~,\nonumber\\
{\widehat \varphi}^{AB} &=&
-\,{\ii\over2\sqrt{2}}\,\bar{U}\,\Big({\cal{M}}^B f\,\bar{\cal M}^A
-{\cal M}^A f\,\bar{\cal M}^B \Big)\,U\nonumber\\
&& -\,\ii \,\bar{U} \cdot
\pmatrix{
0_{[N]\times [N]}& 0_{[N]\times[2k]} \cr
0_{[2k]\times[N]}
&L^{-1}\Lambda^{AB}_{ [k]\times[k]}\otimes 1_{[2]\times[2]}}
\cdot U ~~,
\label{An}\end{aligned}$$ in terms of the kernels $U_{[N+2k]\times [N]}$ and $\bar{U}_{[N]\times [N+2k]}$ of the ADHM matrices $\bar{\Delta}$ and $\Delta$. In (\[An\]), the hatted gauge fields are taken to be anti-hermitian, $\Lambda^{AB}$ is the fermionic bilinear $$\Lambda^{AB}={1\over{2\sqrt{2}}}
\left(\bar{\cal M}^A{\cal M}^B - \bar{\cal M}^B{\cal M}^A\right)~~,
\label{lambda}$$ and the operator $L$ is defined as $$L\cdot \Omega = {1\over 2} \{W^0,\Omega\} + [a_\mu,[a^\mu,\Omega]] ~~,
\label{L}$$ with $(W^0)_{j}^{~i}=w_{\dot\alpha}^{~ui}\,{\bar w}^{\dot\alpha}_{~uj}$.
For simplicity, from now on we concentrate on solutions with winding number $k=1$, which for $\mathrm{SU}(N)$ can be found starting from those for $\mathrm{SU}(2)$. For $k=1$ the ADHM constraints drastically simplify; indeed, the bosonic constraint (\[aMa\]) simply reduces to $$\bar{w}^{\dot\alpha}_{~u}\, w_{\dot\beta}^{~u}=\rho^2\,
\delta^{\dot\alpha}_{~\dot\beta}
\label{vinc}$$ (see eq. (\[rho\])), which is solved by $$\left|\left|w_{\dot\alpha}^{~u}\right|\right| =
\left|\left|\bar w^{\dot\alpha}_{~u}\right|\right|
=\rho\, T\, \pmatrix{0_{[N-2]\times[2]}\cr 1_{[2]\times[2]}}
\label{solvinc}$$ where $T\in {\mathrm{SU}}(N)/{\mathrm{SU}}(N-2)$. This is just the standard $\mathrm{SU}(2)$ instanton solution embedded inside the $\mathrm{SU}(N)$ in the lower right corner. The matrices $T$ describe the orientation of the $\mathrm{SU}(2)$ instanton inside $\mathrm{SU}(N)$ with $\mathrm{SU}(N-2)$ being the stability group of the $\mathrm{SU}(2)$ instanton solution. If we temporarily set $T=1$, the vector field, which solves the equations of motion in the singular gauge, can be written as $$({\widehat A}_\mu)^u_{~v} ={\rho^2\over x^2\,(x^2+\rho^2)}\,
(\bar\sigma_{\nu\mu})^u_{~v}\,x^\nu~~,
\label{gfield}$$ where $$(\bar\sigma_{\nu\mu})_u^{~v}
=\pmatrix{0_{[N-2]\times[N-2]}&
0_{[N-2]\times[2]}\cr
0_{[2]\times[N-2]}& (\bar\sigma_{\nu\mu})_{\dot\alpha}^{~\dot\beta}}~~,$$ and the center of the instanton has been set at $x_0=0$ for simplicity. If we remove the $T=1$ constraint and shift the instanton center, we find the general $\mathrm{SU}(N)$ solution ${\widehat A}_\mu=T\,{\widehat A}_\mu \,T^{-1}$ which is given in (\[connection1\]). As we have also found in the main text, an explicit representation of our embedding is given by the matrices in (\[gf2\]) where the $w_{\dot\alpha}^{~u}$’s are chosen according to (\[solvinc\]).
We now turn to the fermionic the zero modes. Their number is $2kN{\cal N}$ and obviously depends on the number of supersymmetries. For compatibility with the rest of the paper we will discuss the ${\cal N}=4$ case. The ${\cal N}=2$ and ${\cal N}=1$ cases can easily be deduced from our discussion by restricting the range of the capital latin indices in the following to $A, B=1,2$ and $A, B=1$ respectively. It is well-known that in the ${\mathrm{SU}}(2)$ case the fermionic zero modes are in the adjoint representation and that their explicit form can be found by acting with the supersymmetry charges of the superconformal algebra on the instanton solution, leading to \^[A]{}=[2]{}(\^[A]{}- [|]{}\_\^[ A]{}(|\_)\^ x\^)(\^)\_\^[ ]{} F\_ . \[zeromodiagg\] These solutions can be singled out also for arbitrary winding numbers $k$, since they correspond to solutions of the constraint (\[constr\]) in which the fermionic matrix ${\cal M}^A$ is taken to be proportional to the matrices $a, b$ introduced in (\[del\]), namely $$\mu^{Aui} = 0~~,~~
{M'}^{\beta A}_{~~ij}= b_{ij}\,\eta^{\beta A}~~,
\label{zeromodahhadhm1}$$ and $$\mu^{Aui} = w_{\dot\alpha}^{~ui}\,\bar\zeta^{\dot\alpha A}
~~,~~
{M'}^{\beta A}_{~~ij} = - {\bar \zeta}_{\dot\alpha}^{~A}
\,(\bar\sigma^\mu)^{\dot\alpha\beta}\,{a'}_{\mu ij}~~,
\label{zeromodahhadhm2}$$ for the supersymmetric and superconformal zero-modes respectively.
Besides the zero-modes (\[zeromodiagg\]), in the ${\mathrm{SU}}(N)$ case we have other $4{\cal N}(N-2)$ fermionic zero-modes, which are the partners of the color rotations parametrized by $w_{{\dot{\alpha}}}^{~u}$’s. They transform in the fundamental representation of the embedded ${\mathrm{SU}}(2)$ and correspond to the $2(N-2)$ doublets in the decomposition of the adjoint representation of ${\mathrm{SU}}(N)$ with respect to ${\mathrm{SU}}(2)$. For example, for ${\mathrm{SU}}(3)$ we have ${\bf 8}={\bf 3}\oplus{\bf 2}\oplus\bar{\bf 2}\oplus {\bf 1}$. Since there are no solutions to the Dirac equation which are ${\mathrm{SU}}(2)$ singlets, and since we already know the form (\[zeromodiagg\]) of the solution in the adjoint representation, we simply have to recall the form of the ${\mathrm{SU}}(2)$ solutions in the fundamental. They are $$\psi_{\alpha s}
={\rho\,\epsilon_{\alpha s}\over \sqrt{(x^2+\rho^2)^3}}
\label{zeromodifun}$$ where $s=1,2$ is an index which runs in the fundamental. The solutions for $\bar{\bf 2}$ are obtained from those in (\[zeromodifun\]) by raising the indices $\alpha$ and $s$. Let us now turn to the $\mathrm{SU}(N)$ case and introduce the gauge invariant quantity $(W^{\dot\alpha}_{\dot\beta})^{j}_{~i}=\bar{w}^{{\dot{\alpha}}}_{~ui}\,
w_{{\dot{\beta}}}^{~uj}$. By definition, the infinitesimal gauge rotations which leave this quantity invariant are those which satisfy $$\delta\bar{w}^{{\dot{\alpha}}}_{~ui}\,w_{{\dot{\beta}}}^{~uj}+
\bar{w}^{{\dot{\alpha}}}_{~ui}\,
\delta w_{{\dot{\beta}}}^{~uj}=0~~.
\label{gaugetransf}$$ Using for $\delta w$ and $\delta \bar w$ the transformations (\[susymu\]), from (\[gaugetransf\]) we get $$\bar\xi^{{\dot{\alpha}}}_{~A}\,\bar{\mu}^{A}_{~ui}\,w_{{\dot{\beta}}}^{~uj}+
\bar\xi_{{\dot{\beta}}A}\,\bar{w}^{{\dot{\alpha}}}_{~ui}\,\mu^{Auj}=0~~,
\label{gaugetransffer}$$ from which we infer $$\bar{\mu}^{A}_{~ui}\,w_{{\dot{\beta}}}^{~uj}=0~~~,~~~
\bar{w}^{{\dot{\alpha}}}_{~ui}\mu^{Auj}=0~~.
\label{gaugefervinc}$$ For $k=1$, given the choice Eq.(\[vinc\]), this implies $\mu^{Au}=(\mu^A_1,\ldots,\mu^A_{N-2},0,0)$. Starting from (\[zeromodifun\]) we can now deduce the ${\mathrm{SU}}(N)$ formulae by replacing the index $s$ in the fundamental of ${\mathrm{SU}}(2)$ with an index $v$ in the fundamental of ${\mathrm{SU}}(N)$, and adding another index $u$ to label the $N-2$ different solutions. For convenience the range of $u$ will be extended to $N$. For consistency with our previous notation, we also substitute $\epsilon$ with $\mu$. Putting together doublets and anti-doublets, we finally find $$\left.({\widehat{\Lambda}^{{\dot{\alpha}}A}})^u_{~v}\right|_{\rm reg.}
={\rho\over\sqrt{(x^2+\rho^2)^3}}\,
\big(\mu^{Au}\,\delta^{\dot\alpha}_{~v}\,+\,
\varepsilon^{{\dot{\alpha}}u}\,\bar\mu^A_{~v}\big)
\label{zeromodifin}$$ where $\varepsilon^{{\dot{\alpha}}u}=(0,\ldots,0,\varepsilon^{{\dot{\alpha}}{\dot{\beta}}})$ is a natural extension of the Levi-Civita symbol to our case. To go to the singular gauge we perform a ${\mathrm{SU}}(N)$ gauge transformation extending the standard ${\mathrm{SU}}(2)$ one, [*i.e.*]{} $g={x_\mu\sigma^\mu}/\sqrt{x^2}$, to $g'=(0,\ldots,0,x_\mu{\sigma^\mu})/\sqrt{x^2}$, and get $$({{\widehat \Lambda}^{\alpha A}})^u_{~v}=
{\rho\over\sqrt{x^2(x^2+\rho^2)^3}}\,\big(\mu^{Au} \,x^\alpha_{~v}
+x^{\alpha u}\bar\mu^A_{~v}\big)~~,
\label{zeromodifinfun}$$ where $x^\alpha_{~v}=(0,\ldots,0,x_\mu\,(\sigma^\mu)^{\alpha}_{~{\dot{\beta}}})$.
At last we discuss the inhomogeneous solutions of the equations of motion for the adjoint scalars ${\widehat\varphi}^{AB}$. These equations follow from the SYM action (\[N4susy\]) [^14] and are $${\cal D}^2\,{\widehat \varphi}^{AB}
\,-\,\frac{1}{\sqrt 2}\,\big\{{\widehat \Lambda}^{\alpha A}\,,\,
{\widehat\Lambda}_{\alpha}^{~B}\big\}\,+\cdots = 0~~,
\label{EqOfMot}$$ where the ellipses stand for terms that contain $\bar\Lambda^{\alpha A}$ or are trilinear in the scalar fields, which are not relevant for our present analysis. A first part of the solution of (\[EqOfMot\]) is obtained by using for ${\widehat \Lambda}^{\alpha A}$ the supersymmetric zero-modes (\[zeromodiagg\]). This leads to $$({\widehat\varphi}^{AB})^u_{~v}
={4\sqrt{2}\over (x^2+\rho^2)^2}\eta^{[Au}\eta^{B]}_{~\,v}~~.
\label{fizeromod1}$$ In the ${\mathrm{SU}}(N)$ case there is an additional contribution to (\[fizeromod1\]) coming from the zero modes (\[zeromodifinfun\]). For $k=1$ and $T=1$, it is easy to see that $$\big\{{\widehat\Lambda}^{\alpha A}\,,\,{\widehat\Lambda}_\alpha^{~B}
\big\}^u_{~v}=
{4\rho^2\over (x^2+\rho^2)^3}\,\Big(\mu^{[Au}\bar\mu^{B]}_{~\,v}-
{1\over 2}\mu^{[Ap}\,\bar\mu^{B]}_{~\,p}\,\tilde\delta^u_{~v}\Big)
\label{fizeromod2}$$ where $\tilde\delta^u_{~v}$ is defined in (\[fizeromod30\]). Substituting the tentative solution $$({\widehat\varphi}^{AB})^u_{~v}
=f(x,\rho)\,\Big(\mu^{[Au}
\bar\mu^{B]}_{~\,v}-\frac{1}{2}\,\mu^{[Ap}\,\bar\mu^{B]}_{~\,p}\,
\tilde\delta^u_{~v}\Big)$$ in (\[EqOfMot\]) and solving the resulting differential equation for $f(x,\rho)$, one obtains $$f(x,\rho) = -{1\over 2\sqrt2(x^2+\rho^2)}~~.
\label{fizeromod1_0}$$
Subleading order of the instanton profile in the $\alpha' \rightarrow 0$ limit {#app:subleading}
==============================================================================
In section \[gaugevector\] we mentioned that the subleading terms in the large distance expansion of the instanton solution are naturally associated to mixed disks with more insertions of boundary changing operators (see Fig. \[fig:2ndorder0\]), and that in the limit $\alpha'\to 0$ they reduce to simple tree-level field theory diagrams, in complete analogy with the gravitational brane solutions as discussed in Ref. [@Bertolini:2000jy]. As an example, in this appendix we explicitly compute the second order contribution to the gauge field, which is represented by the diagram in Fig. \[fig:2ndorder\]. For simplicity we just consider the $\mathrm{SU}(2)$ case. The necessary ingredients to compute this diagram are:
- the ordinary 3-gluon vertex of YM theory $$V_{\mu\nu\lambda}^{cde}(p,q,k)= {\rm i}\,
\varepsilon^{cde}\, \Big[ (q-k)_\mu \,\delta_{\nu\lambda}\,+\,
(p-q)_\lambda \,\delta_{\mu\nu}\,+\, (k-p)_\nu\, \delta_{\lambda\mu} \Big]
\label{vertf}$$ where all momenta are incoming, and
- the source subdiagram representing the leading order expression of the gauge field in momentum space given in (\[corr5\]), namely $${A^c_\mu (p; \rho)}^{(1)}= \ii \rho^2 \,{\bar\eta^c_{\nu\mu}}\,p^\nu\,
\ee^{-\ii p\cdot x_0}~~.
\label{firstorder}$$
The amplitude in Fig. \[fig:2ndorder\] is then obtained by sewing two first-order diagrams to a 3-gluon vertex and reversing the sign of the momentum of the free gluon line to describe an [*outgoing*]{} field. Taking into account a simmetry factor of 1/2, we have $$\begin{aligned}
\label{2ndorderdia}
{A^c_\mu (p; \rho)}^{(2)} & = & \frac{1}{2}\,
\int \!\!{d^4 q\over (2\pi)^2}\,\Big[V_{\mu\nu\lambda}^{cde}(-p,q,p-q)\,\,
\frac{1}{q^2}\,\,{A^d_\nu (q; \rho)}^{(1)}\,
\frac{1}{(p-q)^2}\,\,{A^e_\lambda (p-q; \rho)}^{(1)}\Big]
\nonumber \\
&=&{\ii\over 2} \,\rho^4\,
\epsilon^{cde}\,\bar\eta^d_{\sigma\nu}\,\bar\eta^e_{\tau\lambda}\,
\ee^{-\ii p\cdot x_0} \,
\int \!\!{d^4 q\over (2\pi)^2}~ {1\over q^2(p-q)^2}\,\,
q^\sigma\,(p-q)^\tau~\times
\nonumber\\
&&~~~
\times \Big[(p-2q)_\mu \,\delta_{\nu\lambda}\, +\,
(q+p)_\lambda\, \delta_{\mu\nu}\,
+\, (q-2p)_\nu\, \delta_{\lambda\mu}\Big]\end{aligned}$$ where the momentum integral can be computed in dimensional regularization. To obtain the space-time profile, we take the Fourier transform of ${A^c_\mu (p; \rho)}^{(2)}$ multiplied by $1/p^2$, and after some standard manipulations we find $$\label{ft2ndorder}
(A^c_\mu (x))^{(2)} \equiv
\lim_{d\to 4} \int {d^d p\over (2 \pi)^{d/2}} ~
(A^c_\mu (p; \rho))^{(2)}\,\frac{1}{p^2} \,\ee^{\ii p\cdot x} =
-2 \rho^4 \bar\eta^c_{\mu\nu} {(x - x_0)^\nu\over (x - x_0)^6}~~,$$ which is exactly the second order term in eq. (\[gf7\]). The higher order terms in the large distance expansion can in principle be computed in a similar manner and thus the full instanton solution can eventually be reconstructed.
[99]{}
Z. Bern, L.J. Dixon and D.A. Kosower, “Progress in one-loop QCD computations”, Ann. Rev. Nucl. Part. Sci. [**46**]{} (1996) 109 \[arXiv:hep-ph/9602280\];\
Z. Bern, “Perturbative quantum gravity and its relation to gauge theory”, \[arXiv:gr-qc/0206071\], and references therein. J. Polchinski, “Dirichlet-branes and Ramond-Ramond charges,” Phys. Rev. Lett. [**75**]{} (1995) 4724 \[arXiv:hep-th/9510017\]. J. Polchinski, “TASI lectures on D-branes”, \[arXiv:hep-th/9611050\]; C. V. Johnson, “D-brane primer”, \[arXiv:hep-th/0007170\]. C.G. Callan, C. Lovelace, C.R. Nappi and S.A. Yost, “Loop corrections to superstring equations of motion”, Nucl. Phys. B [**308**]{} (1988) 221.\
J. Polchinski and Y. Cai, “Consistency of open superstring theories”, Nucl. Phys. B [**296**]{} (1988) 91. P. Di Vecchia and A. Liccardo, “D branes in string theory. I”, \[arXiv:hep-th/9912161\]; “D-branes in string theory. II”, \[arXiv:hep-th/9912275\]. P. Di Vecchia, M. Frau, I. Pesando, S. Sciuto, A. Lerda and R. Russo, “Classical p-branes from boundary state,” Nucl. Phys. B [**507**]{} (1997) 259 \[arXiv:hep-th/9707068\]. P. Di Vecchia, M. Frau, A. Lerda and A. Liccardo, “(F,Dp) bound states from the boundary state,” Nucl. Phys. B [**565**]{}, 397 (2000) \[arXiv:hep-th/9906214\]. E. Witten, “Bound states of strings and p-branes,” Nucl. Phys. B [**460**]{} (1996) 335 \[arXiv:hep-th/9510135\]. M. R. Douglas, “Gauge fields and D-branes,” J. Geom. Phys. [**28**]{}, 255 (1998) \[arXiv:hep-th/9604198\]; “Branes within branes,” arXiv:hep-th/9512077. J. Polchinski, “Combinatorics of boundaries in string theory,” Phys. Rev. D [**50**]{} (1994) 6041 \[arXiv:hep-th/9407031\]. M.B. Green and M. Gutperle, “Effects of D-instantons,” Nucl. Phys. B [**498**]{} (1997) 195, \[arXiv:hep-th/9701093\]; “D-particle bound states and the D-instanton measure”, JHEP [**9801**]{} (1998) 005, \[arXiv:hep-th/9711107\]; “D-instanton partition functions”, Phys. Rev. D [**58**]{} (1998) 046007, \[arXiv:hep-th/9804123\]. T. Banks and M. B. Green, “Non-perturbative effects in AdS(5) x S\*\*5 string theory and d = 4 SUSY Yang-Mills”, JHEP [**9805**]{} (1998) 002, \[arXiv:hep-th/9804170\]. C.S. Chu, P.M. Ho and Y.Y. Wu, “D-instanton in AdS(5) and instanton in SYM(4)”, Nucl. Phys. B [**541**]{} (1999) 179, \[arXiv:hep-th/9806103\]. I.I. Kogan and G. Luzon, “D-instantons on the boundary”, Nucl. Phys. B [**539**]{} (1999) 121, \[arXiv:hep-th/9806197\]. M. Bianchi, M.B. Green, S. Kovacs and G. Rossi, “Instantons in supersymmetric Yang-Mills and D-instantons in IIB superstring theory”, JHEP [**9808**]{}(1998) 013, \[arXiv:hep-th/9807033\]. N. Dorey, V.V. Khoze, M.P. Mattis and S. Vandoren, “Yang-Mills instantons in the large-N limit and the AdS/CFT correspondence”, Phys. Lett. B [**442**]{} (1998) 145, \[arXiv:hep-th/9808157\]; \[arXiv:hep-th/9808157\]. N. Dorey, T.J. Hollowood, V.V. Khoze, M.P. Mattis and S. Vandoren, “Multi-instantons and Maldacena’s conjecture”, JHEP [**9906**]{} (1999) 023, \[arXiv:hep-th/9810243\]. N. Dorey, T. J. Hollowood, V. V. Khoze, M. P. Mattis and S. Vandoren, “Multi-instanton calculus and the AdS/CFT correspondence in N = 4 superconformal field theory”, Nucl. Phys. B [**552**]{} (1999) 88 \[arXiv:hep-th/9901128\]. M.B. Green and M. Gutperle, “D-instanton induced interactions on a D3-brane,” JHEP [**0002**]{} (2000) 014 \[arXiv:hep-th/0002011\]. N. Dorey, T. J. Hollowood and V. V. Khoze, “Notes on soliton bound-state problems in gauge theory and string theory”, \[arXiv:hep-th/0105090\]. A.V. Belitsky, S. Vandoren and P. van Nieuwenhuizen, “Yang - Mills and D - instantons”, Class. Quant. Grav. [**17**]{} (2000) 3521 \[arXiv:hep-th/0004186\]. N. Dorey, T. J. Hollowood and V. V. Khoze, “A brief history of the stringy instanton,” \[arXiv:hep-th/0010015\]. N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “The calculus of many instantons,” \[arXiv:hep-th/0206063\]. M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, “Construction of instantons”, Phys. Lett. A [**65**]{} (1978) 185. A. A. Belavin, A. M. Polyakov, A. S. Schwarz and Y. S. Tyupkin, “Pseudoparticle solutions of the Yang-Mills equations,” Phys. Lett. B [**59**]{}, 85 (1975). G. ’t Hooft, “Computation of the quantum effects due to a four-dimensional pseudoparticle,” Phys. Rev. D [**14**]{}, 3432 (1976) \[Erratum-ibid. D [**18**]{}, 2199 (1978)\]. D. Friedan, E. J. Martinec and S.H. Shenker, “Conformal invariance, supersymmetry and string theory”, Nucl. Phys. B [**271**]{} (1986) 93. L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, “Strings on orbifolds”, Nucl. Phys. B [**261**]{} (1985) 678; S. Hamidi and C. Vafa, “Interactions on orbifolds”, Nucl. Phys. B [**279**]{} (1987) 465. P. Di Vecchia, L. Magnea, A. Lerda, R. Russo and R. Marotta, “String techniques for the calculation of renormalization constants in field theory”, Nucl. Phys. B [**469**]{} (1996) 235 \[arXiv:hep-th/9601143\]. D. Polyakov, “BRST properties of new superstring states,” \[arXiv:hep-th/0111227\]. A. Sen, “Type I D-particle and its interactions,” JHEP [**9810**]{}, 021 (1998) \[arXiv:hep-th/9809111\]. L. Gallot, A. Lerda and P. Strigazzi, “Gauge and gravitational interactions of non-BPS D-particles,” Nucl. Phys. B [**586**]{}, 206 (2000) \[arXiv:hep-th/0001049\]. M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta and R. Russo, “Is a classical description of stable non-BPS D-branes possible?”, Nucl. Phys. B [**590**]{} (2000) 471, \[arXiv:hep-th/0007097\]. M. Billo, P. Di Vecchia, M. Frau, A. Lerda, I. Pesando, R. Russo and S. Sciuto, “Microscopic string analysis of the D0-D8 brane system and dual R-R states”, Nucl. Phys. B [**526**]{} (1998) 199 \[arXiv:hep-th/9802088\]. V.A. Kostelecky, O. Lechtenfeld, W. Lerche, S. Samuel and S. Watamura, “Conformal techniques, bosonization and tree level string amplitudes”, Nucl. Phys. B [**288**]{} (1987) 173.
[^1]: The fact that $w$ and $\bar w$ must be anti-chiral can be understood by observing that the vertices (\[vertexw\]) are local with respect to the supercurrent $j^{\dot\alpha A}(z)$ associated to the only conserved supercharges $q^{\dot\alpha A}$ of the D3/D$(-1)$ strings. Indeed, using the OPE’s summarized in appendix \[app:conventions\], we have $$\begin{aligned}
\label{localw} j^{\dot\alpha A}(z)~ V^{(-1)}_w(y) & = & \left[
S^{\dot\alpha}(z)\,S^A(z)\,\ee^{-{1\over 2}\phi(z)}\right]\,
\left[{w}_{\dot\beta}\,\Delta(y)\,S^{\dot\beta}(y)\,
\ee^{-\phi(y)}\right]\nonumber\\ & \sim & {1\over (z - y)} \left[{
w}^{\dot\alpha} \,\Delta(y) \,S^A(y)\, \ee^{-{3\over
2}\phi(y)}\right] + \cdots
\nonumber\end{aligned}$$ where the ellipses stand for regular terms. If one had chosen the other chirality (corresponding to chiral moduli $w_\alpha$ and $\bar w_\alpha$), one would have obtained a branch cut in the OPE with the supercurrent $j^{\dot\alpha A}(z)$ and thus locality would have been spoiled. On the contrary, the chiral moduli would be local with respect to the supercurrent $j_{\alpha
A}(z)$ that is conserved for an anti-instanton ([*i.e.*]{} for $\varepsilon'=-1$ in (\[dottedbc\])).
[^2]: In this way the one-instanton case ($k=1$) can be simply obtained by removing the trace symbol from all formulas without extra numerical factors.
[^3]: Remember that in Euclidean space the 1PI part of a scattering amplitude is equal to [*minus*]{} the corresponding interaction term in the action. Moreover, the terms of higher order in $\alpha'$ in the scattering amplitudes represent string corrections to the standard field theory.
[^4]: The lack of BRST invariance of the vertices (\[vertaux\]) should not be regarded as a serious problem since, when dealing with auxiliary fields, one is effectively working off-shell. Vertices similar to those of (\[vertaux\]) (but in the $(-2)$ superghost picture) have been considered in Ref. [@Polyakov:2001zr].
[^5]: String amplitudes on mixed disks have been previously analyzed in Ref. [@Sen:1998ki; @Gallot:1999hs] to study the gauge interactions of the non-BPS D-particles of the type IIB theory.
[^6]: The procedure to obtain the ADHM measure that we have explained consists of two distinct steps: the first is the field theory limit on the D$(-1)$ branes, the second is the strong coupling limit accompanied by a rescaling of the D$(-1)$ fields which survive the first step. However, it is also possible to obtain the ADHM measure directly in a single step. This can be done by using always adimensional polarizations rescaled as follows $$\begin{aligned}
&&a = \left(\frac{2g_s}{\pi}\right)^{1/2}\!s^\alpha\,a' ~~,~~
\chi= s^{-\alpha}\,\chi' ~~,~~ M=
\left(\frac{g_s}{2\pi}\right)^{1/2}s^{\alpha/2}\,M' ~~,~~ \lambda=
\,s^{-3\alpha/2}\,\lambda' ~~,
\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
D=s^{-2\alpha}\,D'
~~~,~~~
Y=\sqrt{2}\,Y'
~~~,~~~
Z=\,Z'~~,
\\
&&
w=\left(\frac{g_s}{2\pi}\right)^{1/2}s^\alpha\,
w'
~~,~~
\bar w=\left(\frac{g_s}{2\pi}\right)^{1/2}s^\alpha\,\bar w'
~~,~~
\mu=\left(\frac{g_s}{2\pi}\right)^{1/2}s^{\alpha/2}\,\mu'
~~,~~
\bar\mu=\left(\frac{g_s}{2\pi}\right)^{1/2}s^{\alpha/2}\,\bar\mu'
~~,\end{aligned}$$ with $\alpha<0$, and then letting $s\rightarrow0$. It turns out that the action which survives in this limit is precisely given by eq. (\[smoduli4\]). The standard dimensions of the ADHM moduli can then be recovered by introducing suitable factors of $(2\pi\alpha')$.
[^7]: The overall factor of $2\ii$, which is not determined by the picture changing, is fixed by requiring the appropriate normalization of the three gluon amplitude.
[^8]: The latter are $(-1/2)$, 0 and $(-1)$ for $V_\mu$, ${\cal V}_{A^I_\mu}$ and $V_{\bar w}$ respectively, and $(-1/2)$ for the supercharges.
[^9]: In eq. (\[ward3\]) all vertex operators, as well as the supersymmetry charges, are in the $(-1/2)$ picture.
[^10]: Suitable symmetry factors must be included when not all field $\phi_i$ are different.
[^11]: For simplicity, we assume that the propagators are $\langle \phi_i(p)\phi_j(k)\rangle= (2\pi)^2\delta^{4}(p+k)\,
\frac{\delta_{ij}}{p^2}$; if this does not happen, like for instance for the gauginos, appropriate changes are required, but these can be straightforwardly implemented in our formulas.\[footnote\]
[^12]: Notice that world-sheets with higher Euler number can also give contributions to the sub-leading orders.
[^13]: For the gauge field $A^I_\mu$ there is also an explicit momentum factor, see eq. (\[corr5\]).
[^14]: We recall that the fields appearing in the SYM action (\[N4susy\]) are hermitian, while the hatted fields we are now considering are anti-hermitian; the precise relation between the two is given by ${\widehat{\varphi}}^{AB}=-\ii\,{\varphi^{AB}}$ and similarly for the other components of the supermultiplet.
|
---
author:
- |
\
Department of Physics, Keio University\
E-mail:
title: 'Domain-wall, overlap, and topological insulators'
---
Introduction
============
Construction of the doubler-free chiral fermion on a lattice has been a longstanding problem in lattice field theory. Nowadays it is known that the domain-wall fermion [@Kaplan1992] and the overlap fermion [@Neuberger1998] present a solution to this problem, and a lot of theoretical and numerical works are performed with these lattice fermions. In this report, we discuss a potential application of these formalisms to condensed-matter physics, especially topological insulators, where topology and quantum anomaly play an important role in the characterization of the system. We especially show that the topological insulator edge (surface) state is a physical realization of the domain-wall/overlap fermion.
Band topology and mass gap {#sec:band}
==========================
Let us explain how the mass term plays a role in the band topology. We start with the two-dimensional massive Dirac Hamiltonian, which is a toy model for the two-dimensional quantum Hall effect, classified into the class A system [@Schnyder2008; @Kitaev2009], $$\begin{aligned}
H_\text{2D}(p)
= p_x \sigma_x + p_y \sigma_y + m \sigma_z
=
\left(
\begin{array}{cc}
m & \Delta(p)^\dag \\ \Delta(p) & -m
\end{array}
\right)
\label{eq:2D_Ham}\end{aligned}$$ where $\Delta(p) = p_x + i p_y \in \mathbb{C}$. This Hamiltonian is actually given as the Bloch Hamiltonian of the real-space Hamiltonian. This is also seen as a Hermitian version of the Dirac operator through multiplication of $\sigma_z$-matrix, $\displaystyle H_\text{2D} = \sigma_z \, D_\text{2D}$. The Hamiltonian has two eigenvalues $$\begin{aligned}
\lambda^{(\pm)} = \pm \sqrt{p_x^2 + p_y^2 + m^2}
\, ,\end{aligned}$$ and we can define the topological charge associated with each eigenstate. For the positive eigenvalue state, it is given by $$\begin{aligned}
\nu_\text{2D} = \frac{i}{2\pi}
\int dp_x dp_y \ F
= \frac{1}{2} \operatorname{sgn} (m)
\, ,
\label{eq:2D_top}\end{aligned}$$ while an extra factor $(-1)$ is needed for the negative eigenvalue state. This is called the TKNN number, which computes the Hall conductivity of the model. Now the integral provides the first Chern number, which is topological, such that its value depends only on the sign of the mass parameter. The curvature $F$, called the Berry curvature, is defined with the Berry connection using the eigenfunction of the Hamiltonian . Let us show the explicit expressions for latter convenience, $$\begin{aligned}
\psi
& =
\frac{1}{\sqrt{1 + |\xi|^2}}
\left( \begin{array}{c} \xi \\ 1 \end{array} \right) \, ,
\qquad
A = \psi^\dag d \psi
= i {\operatorname{Im}}\frac{\xi^\dag d \xi}{1 + |\xi|^2} \, ,
\qquad
F = \frac{d\xi^\dag d\xi}{\left( 1 + |\xi|^2 \right)^2} \, ,
\label{eq:forms}\end{aligned}$$ where we apply the differential form notation to the connection and the curvature, and the derivative is with respect to momentum, $d = \partial_{p_i} dp^i$. The complex parameter $\xi$ is now given by $$\begin{aligned}
\xi
= \frac{\lambda^{(\pm)} + m}{\Delta}
= \frac{\Delta^\dag}{\lambda^{(\pm)} - m}
\, .
\label{eq:CP1_coord}\end{aligned}$$ This system enjoys the ${\mathrm{U}}(1)$ gauge symmetry, corresponding to the phase rotation of $\xi$. This ${\mathrm{U}}(1)$ symmetry reflects the particle number conservation for each energy band, which is well-defined as long as the mass parameter takes a non-zero value $m \neq 0$. This is nothing but a consequence of the adiabatic approximation in the original sense of the Berry phase.
Let us comment on the reason why the topological charge takes a half-integer value, while its difference must be integer $\delta \nu_\text{2D} = \pm 1$, which occurs at the massless point $m=0$ [@Oshikawa1994]. Usually the integral like is evaluated with the asymptotic behavior of the curvature and the connection at infinity $p \to \infty$, which plays a role of the boundary. In this case, however, the origin of the momentum space $p = 0$ also plays a similar role, because the parameter $\xi$ takes a non-trivial value at $p=0$. This also reflects the anomaly of the (2+1)-dimensional Dirac fermion, namely the parity anomaly. We also remark that the construction of the topological charge using the fermion, or more precisely, the Clifford algebra, e.g. Pauli matrices, is known as the Atiyah–Bott–Shapiro construction [@Atiyah1964], which is essentially related to the K-theoretical point of view.
The argument above is straightforwardly generalized to the four-dimensional case with the following Hamiltonian, which again belongs to the class A system (the four-dimensional quantum Hall system), $$\begin{aligned}
H_\text{4D}(p)
= p \cdot \gamma + m \gamma_5
=
\left(
\begin{array}{cc}
m & \Delta(p)^\dag \\ \Delta(p) & -m
\end{array}
\right)
\label{eq:4D_Ham}\end{aligned}$$ where each matrix element is 2-by-2, so that the Hamiltonian itself is a 4-by-4 matrix. This is again related to the Dirac operator, $\displaystyle H_\text{4D} = \gamma_5 \, D_\text{4D}$. The off-diagonal element is given by $\Delta(p) = p \cdot \sigma$ with the four-vector $\sigma = ({\mathbbm{1}},\vec{\sigma})$, which takes a quaternionic value $\Delta(p) \in \mathbb{H}$. We can formally apply the same expression as in two dimensions to the four-dimensional model . This Hamiltonian has two eigenvalues $$\begin{aligned}
\lambda^{(\pm)}
& =
\pm \sqrt{p^2 + m^2}
\, ,\end{aligned}$$ and each eigenstate is degenerated twice. Thus the Berry connection becomes a 2-by-2 matrix, and the corresponding topological charge is now given by $$\begin{aligned}
\nu_\text{4D}
& =
- \frac{1}{16\pi^2} \int d^4 p \, {\operatorname{Tr}}F * F
= \frac{1}{2} {\operatorname{sgn}}(m)
\, .
\label{eq:4D_top}\end{aligned}$$ This implies that the Berry connection realizes the four-dimensional instanton configuration in the momentum space.[^1] The topological number takes a half-integer value, due to the same reason as the two-dimensional case, and its difference is again given by $\delta \nu_\text{4D} = \pm 1$ at $m=0$.
We can discuss the topological charge also with the lattice system. We define the Hermitian operator, associated with the four-dimensional Wilson fermion $$\begin{aligned}
H_\text{W}(p) = \gamma_5 \, D_\text{W}(p)
\, , \qquad
D_\text{W}(p)
& =
m
+ \sum_{\mu=1}^4 i \gamma_\mu \sin p_\mu
+ r \sum_{\mu=1}^4 (1 - \cos p_\mu)
\, .\end{aligned}$$ Since it is a lattice system, the topological charge is then defined as an integral over the Brillouin zone [@Qi2008], $$\begin{aligned}
\nu_\text{4D}
& =
- \frac{1}{16\pi^2} \int_\text{BZ} d^4 p \, {\operatorname{Tr}}F * F
=
\begin{cases}
0 & (0 < m \ \& \ m < -8r) \\
1 & (-2r < m < 0) \\
-3 & (-4r < m < -2r) \\
3 & (-6r < m < -4r) \\
-1 & (-8r < m < -6r) \\
\end{cases}
\, .
\label{eq:4D_top_lat}\end{aligned}$$ In this case, the topological number itself takes an integral value, and its change $\delta \nu_\text{4D} = +1$, $-4$, $+6$, $-4$, $+1$ at $m=0$, $-2r$, $-4r$, $-6r$, $-8r$, corresponds to the number of massless modes appearing in the Wilson fermion formalism, and the sign of $\delta \nu_\text{4D}$ indeed reflects the chirality of each mode. This is in contrast to the continuum theory, directly associated with the anomaly, leading to the half-integer topological charge.
Topological phase, domain-wall, and overlap {#sec:top_phase}
===========================================
Domain-wall fermion at the topological insulator boundary
---------------------------------------------------------
As discussed in Sec. \[sec:band\], the topology associated with the band structure is well-defined for massive theory, and the topology change occurs at the critical point which is massless $m = 0$. This means that, according to the band theory, the band topology is only well-defined with an insulator since the mass term plays a role of the band gap. Nowadays an insulator involving non-trivial topology is called [*the topological insulator*]{}, in general, and the quantum Hall effect is known as the most fundamental example.
(0,0) rectangle ++(1.5,1.5); (1.5,0) rectangle ++(1.5,1.5); (0,1.5) – (.75,2) – (2.25,2) – (1.5,1.5); (1.5,1.5) – (2.25,2) – (3.75,2) – (3,1.5); (3,0) – (3.75,.5) – (3.75,2); (.75,.5) – (3.75,.5); (0,0) – (.75,.5) – (.75,2); (1.5,0) – (2.25,.5) – (2.25,2);
(1.5,0) – (2.25,.5) – (2.25,2) – (1.5,1.5) – cycle;
at (.75,-.5) [$\nu_1$]{}; at (2.25,-.5) [$\nu_2$]{};
(-.5,0) – (-.5,2) node \[left\] [$m(x)$]{}; (-.5,0) – (3.5,0) node \[right\] [$x$]{};
(0,.5) – (1,.5) to \[out=right,in=240\] (1.5,1) to \[out=60,in=left\] (2,1.5) – (3,1.5);
(.7,.7) – (1,.7) .. controls (1.3,.7) and (1.3,2) .. (1.5,2) node (loc) .. controls (1.7,2) and (1.7,.7) .. (2,.7) – (2.3,.7);
(.7,.7) – (1,.7) .. controls (1.3,.7) and (1.3,2) .. (1.5,2) node (loc+) .. controls (1.7,2) and (1.7,.7) .. (2,.7) – (2.3,.7);
(.7,.7) – (1,.7) .. controls (1.3,.7) and (1.3,2) .. (1.5,2) node (loc-) .. controls (1.7,2) and (1.7,.7) .. (2,.7) – (2.3,.7);
(m0) at (4.5,-1.5) [$m=0$]{};
(m0) – ++(3,0) – ++ (0,1.2); (m0) – ++(-3,0) – ++ (0,1.2);
(loc+) node \[above\] [\#modes $= \delta \nu$]{};
Typically the topological charge depends only on the sign of the mass term, e.g. and , which directly implies that the boundary of the topological insulator, where the topology change must occur, realizes the domain-wall configuration of the mass term. See Fig. \[fig:DW\]. This can be thought as the origin of the gapless surface (edge) state of the topological insulator. The number of edge states exactly corresponds to the difference of topological numbers. See the argument around . In this sense, we can compute the topological number in two ways: the characteristic class associated with the bulk system and the number of edge states. This principle is called the bulk/edge correspondence in the topological phases. The domain-wall formalism discussed here is a possible way to define a single massless chiral fermion on a lattice, and the existence of odd-number gapless excitations at the topological insulator boundary has been confirmed in several experiments. See a review article [@Hasan2010] for details.
Overlap formula for the topological edge state
----------------------------------------------
There are various interesting phenomena for the topological insulator surface state from both theoretical and experimental point of view: the Tomonaga–Luttinger liquid behavior of the quantum Hall edge current, the quantum Hall effect at the three-dimensional topological insulator surface, and so on. In order to study such phenomena peculiar to the topological insulator boundary, it is required to construct an effective theory of the topological surface state. Fig. \[fig:edge\] shows a band spectrum of the topological system with the open boundary condition so that the non-trivial edge state exists at the boundary. This is a typical situation for the domain-wall fermion. In addition to the gapless mode corresponding to the edge state, there are massive bulk spectra to be removed to obtain the edge effective theory. The prescription to obtain such an effective theory is nothing but the overlap formula [@Neuberger1998]: $$\begin{aligned}
\det D_\text{eff} =
\frac{\det D_\text{open}}{\det D_\text{period}}
\, .\end{aligned}$$ The meaning of this formula is as follows: The Dirac operator $D_\text{open}$ exhibits the spectrum with the gapless edge state due to the open boundary condition, as depicted in Fig. \[fig:edge\]. The operator in the denominator $D_\text{period}$ comes from the auxiliary (bosonic) degrees of freedom to remove the massive modes. The latter one is introduced with the periodic boundary condition, and thus it has no edge excitation. Since the boundary condition does not affect the bulk spectra except for the edge state, the remaining degrees of freedom in the effective operator $D_\text{eff}$ describes the gapless edge state. In this sense the topological edge state is a physical realization of the overlap fermion.
(.3,-1) – (1.7,-1) node \[right\] [$p$]{}; (-.5,1) – ++(0,.5) node \[left\] [$E$]{};
(0,0) to \[out=30,in=left\] (1,.3) to \[out=right,in=150\] (2,0);
(0,0) to \[out=30,in=left\] (1,.3) to \[out=right,in=150\] (2,0) – (2,-.3) – (0,-.3) – cycle;
(0,1.5) to \[out=-30,in=left\] (1,1.2) to \[out=right,in=210\] (2,1.5);
(0,1.5) to \[out=-30,in=left\] (1,1.2) to \[out=right,in=210\] (2,1.5) – (2,1.8) – (0,1.8) – cycle;
(.5,.25) – (1.5,1.25); (.5,1.25) – (1.5,.25);
(2.7,.75) – ++ (1,0);
\[top color=red!50!magenta,bottom color=red!10,middle color=red,shading=axis,opacity=0.25\] (0,0) circle (.5 and 0.1);
\[left color=red!50!magenta,right color=red!50!magenta,middle color=red!50, shading=axis, opacity=0.25\] (.5,0) – (0,.75) – (-.5,0) arc (180:360:.5 and 0.1);
\[top color=red!50!magenta,bottom color=red!50,middle color=red,shading=axis,opacity=0.25\] (0,1.5) circle (.5 and 0.1);
\[left color=red!50!magenta,right color=red!50!magenta,middle color=red!50, shading=axis, opacity=0.25\] (.5,1.5) – (0,0.75) – (-.5,1.5) arc (180:0:.5 and 0.1);
Once we identify the topological edge state as the overlap fermion, we can apply its several interesting properties to the topological insulator. The first example is the Ginsparg–Wilson (GW) relation [@Ginsparg1982]: $$\begin{aligned}
\gamma_5 D + D \gamma_5 = a \, D \gamma_5 D
\, .
\label{eq:GW_rel}\end{aligned}$$ A direct consequence of this GW relation is asymmetric chiral transformations of $\psi$ and $\bar{\psi}$ [@Luscher1998], $$\begin{aligned}
\psi \ \to \ e^{i \theta \Gamma_5} \psi \, ,
\qquad
\bar{\psi} \ \to \ \bar{\psi} e^{i \theta \gamma_5} \, ,
\qquad
\Gamma_5 = \gamma_5 \left( 1 - a D \right)
\, .\end{aligned}$$ This implies the particle-antiparticle symmetry is broken in the lattice spacing order $O(a)$. This is a natural realization of the chiral anomaly on a lattice.
In order to generalize this argument to other dimensions, we introduce another expression of the GW relation without using the chiral operator $\gamma_5$, $$\begin{aligned}
D + D^\dag = a \, D^\dag D
\label{eq:GW_rel2} \end{aligned}$$ where we just assume the $\gamma_5$-Hermiticity, $\gamma_5 D \gamma_5 = D^\dag$. Since this expression does not use the chiral operator $\gamma_5$ explicitly, it is applicable also in odd dimensions. A solution to this GW relation is obtained with a unitary operator $V$ as follows, $$\begin{aligned}
D = \frac{1}{a} \left( 1 - V \right) \, .\end{aligned}$$
To apply this relation to the three-dimensional topological insulator and its surface state, let us consider $(2+1)$-dimensional GW relation. In this case, we obtain a similar asymmetric behavior between $\psi$ and $\bar{\psi}$ under the parity transformation [@Bietenholz2001],$$\begin{aligned}
\psi \ \to \ i \, R \, V \psi \, , \qquad
\bar\psi \ \to \ i \bar{\psi} R
\label{eq:3D_asym}\end{aligned}$$ and vice versa. Here we introduced the reflection operator $R$: $(x,y,z) \to (-x,-y,-z)$ with ${R} \, D \, {R} = D^\dag$. This is a realization of the parity anomaly on a lattice, which is similar to the chiral anomaly discussed above. In this case we observe the asymmetry under the parity transformation, and in this way, we can apply this argument to arbitrary topological phases associated with the quantum anomaly with the corresponding symmetry and dimensions.
Let us comment on observation of such an asymmetric behavior in experiments. In the three-dimensional topological insulator surface, the quantum Hall effect can be induced by doping the magnetic impurity, which breaks the time-reversal symmetry. In this case, if we have the particle-antiparticle (hole) asymmetry, anomalous peak shift of the magneto-optical conductivity ${\operatorname{Re}}\sigma_{xx}(\omega)$ will be observed, which is peculiar to the zero energy state [@Tabert2015]. While the authors of Ref. [@Tabert2015] explicitly introduce the mass term, which leads to the asymmetry, our argument based on the GW relation could provide more systematic explanation to such an anomalous behavior, for example, as a result of the discretization error.
Topological phases with additional symmetry {#sec:top_add_symm}
===========================================
In addition to the topological phases, based on the well-known periodic table with respect to the dimension and symmetry [@Schnyder2008; @Kitaev2009], recently several topological systems associated with additional symmetry, e.g. symmetry of the lattice, gain a great interest in condensed-matter physics. In this section, we demonstrate that the argument discussed in Sec. \[sec:top\_phase\] is also applicable even in the topological phase with the additional symmetry.
Let us consider the three-dimensional system with the reflection symmetry in $x$-direction. We define the $x$-reflection operator ${R}_x$: $(x,y,z) \to (-x,y,z)$. In this case, the invariant plane $x = 0$ under this reflection plays an important role: The Dirac operator has a symmetry with this operation, $$\begin{aligned}
{R}_x \, D(x=0,y,z) \, {R}_x = D(x=0,y,z)
\, .\end{aligned}$$ Then, defining the three-dimensional chiral operator with the $x$-reflection, $\displaystyle \Gamma_x = i \gamma_x \, {R}_x$, we obtain the GW relation with respect to this operator, $$\begin{aligned}
\Gamma_x \, D + D \, \Gamma_x = a \, D \Gamma_x D
\, .\end{aligned}$$ Since we now have the chiral operator $\Gamma_x$, the topological classification is modified as $\mathbb{Z}_2$ (parity) $\to$ $\mathbb{Z}$ (chiral). This is known as the topological crystalline insulator [@Ando2015]. Naively speaking, the origin of this modification is understood as follows: Due to the additional reflection symmetry, a sort of dimensional reduction occurs at the invariant plane with respect to the reflection symmetry. In this case, therefore, the original three-dimensional system is reduced to the two-dimensional one at the $x=0$ plane, and the operator $\Gamma_x$ plays a role of the two-dimensional chiral operator.
Acknowledgements {#acknowledgements .unnumbered}
----------------
TK would like to thank T. Morimoto for collaboration at early stages of this work. The work of TK was supported in part by JSPS Grant-in-Aid for Scientific Research (No. 13J04302) from MEXT of Japan.
[99]{} D. B. Kaplan, [Phys. Lett. **B288** (1992) 342–347](http://dx.doi.org/10.1016/0370-2693(92)91112-M) \[[hep-lat/9206013](http://arxiv.org/abs/hep-lat/9206013)\].
H. Neuberger, [Phys. Lett. **B417** (1998) 141–144](http://dx.doi.org/10.1016/S0370-2693(97)01368-3) \[[hep-lat/9707022](http://arxiv.org/abs/hep-lat/9707022)\].
A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, [Phys. Rev. **B78** (2008) 195125](http://dx.doi.org/10.1103/PhysRevB.78.195125) \[[arXiv:0803.2786](http://arxiv.org/abs/0803.2786)\].
A. Kitaev, [AIP Conf. Proc. **1134** (2009) 22–30](http://dx.doi.org/10.1063/1.3149495) \[[arXiv:0901.2686](http://arxiv.org/abs/0901.2686)\].
M. Oshikawa, [Phys. Rev. **B50** (1994) 17357](http://dx.doi.org/10.1103/PhysRevB.50.17357) \[[cond-mat/9409079](http://arxiv.org/abs/cond-mat/9409079)\].
M. F. Atiyah, R. Bott, and A. Shapiro, [Topology **3** (1964) 3–38](http://dx.doi.org/10.1016/0040-9383(64)90003-5).
K. Hashimoto and T. Kimura, \[[arXiv:1509.04676](http://arxiv.org/abs/1509.04676)\].
X.-L. Qi, T. Hughes, and S.-C. Zhang, [Phys. Rev. **B78** (2008) 195424](http://dx.doi.org/10.1103/PhysRevB.78.195424) \[[arXiv:0802.3537](http://arxiv.org/abs/0802.3537)\].
M. Z. Hasan and C. L. Kane, [Rev. Mod. Phys. **82** (2008) 3045](http://dx.doi.org/10.1103/RevModPhys.82.3045) \[[arXiv:1002.3895](http://arxiv.org/abs/1002.3895)\].
P. H. Ginsparg and K. G. Wilson, [Phys. Rev. **D25** (1982) 2649](http://dx.doi.org/10.1103/PhysRevD.25.2649).
M. Lüscher, [Phys. Lett. **B428** (1998) 342–345](http://dx.doi.org/10.1016/S0370-2693(98)00423-7) \[[hep-lat/9802011](http://arxiv.org/abs/hep-lat/9802011)\].
W. Bietenholz and J. Nishimura, [JHEP **0107** (2001) 015](http://dx.doi.org/10.1088/1126-6708/2001/07/015) \[[hep-lat/0012020](http://arxiv.org/abs/hep-lat/0012020)\].
C. J. Tabert and J. P. Carbotte, [Phys. Rev. **B91** (2015) 235405](http://dx.doi.org/10.1103/PhysRevB.91.235405) \[[arXiv:1505.06660](http://arxiv.org/abs/1505.06660)\].
Y. Ando and L. Fu, [Ann. Rev. Cond. Mat. Phys. **6** (2015) 361–381](http;//dx.doi.org/10.1146/annurev-conmatphys-031214-014501) \[[arXiv:1501.00531](http://arxiv.org/abs/1501.00531)\].
[^1]: This construction can be actually discussed in parallel with the systematic construction of instantons, called the ADHM construction [@Hashimoto2015].
|
---
abstract: 'Fitting simultaneously SPX and VIX smiles is known to be one of the most challenging problems in volatility modeling. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate jointly these two quantities with a model with continuous sample-paths. We present the quadratic rough Heston model as a counterexample to this conjecture. The key idea is the combination of rough volatility together with a price-feedback (Zumbach) effect.'
author:
- 'Jim Gatheral[^1], Paul Jusselin[^2] and Mathieu Rosenbaum[^3]\'
bibliography:
- 'biblio20200106.bib'
title: |
The quadratic rough Heston model and\
the joint S&P $500$/VIX smile calibration problem
---
**Keywords:** SPX smiles, VIX smiles, rough Heston model, Zumbach effect, quadratic rough Heston model, Guyon’s conjecture.
Introduction {#sec:introduction}
============
The Volatility Index, or VIX, was introduced in 1993 by the Chicago Board Options Exchange (CBOE for short) and was originally designed according to [@vixwhite] to “measure the market’s expectation of 30-day volatility implied by at-the-money S&P 100 Index option price”. Since 2003, the VIX has been redefined as the square root of the price of a specific basket of options on the S&P 500 Index (SPX) with maturity $30$ days. The basket coefficients are chosen so that at any time $t$, the VIX represents the annualized square root of the price of a contract with payoff equal to $\log(S_{t+\Delta}/S_t)$ where $\Delta = 30$ days and $S$ denotes the value of the SPX. Consequently, it can be formally written via risk-neutral expectation under the form $$\label{eq:vix_def}
\text{VIX}_t = \sqrt{\mathbb{E}[\log(S_{t+\Delta}/S_t)|\mathcal{F}_t]} \times 100,$$ where $(\mathcal{F}_t)_{t\geq 0}$ is the natural filtration of the market.\
Since 2004, investors have been able to trade VIX futures. Quoting the CBOE white paper, they “provide market participants with a variety of opportunities to implement their view using volatility trading strategies, including risk management, alpha generation and portfolio diversification”[^4]. Subsequently in 2006, CBOE introduced VIX options “providing market participants with another tool to manage volatility. VIX options enable market participants to hedge portfolio volatility risk distinct from market price risk and trade based on their view of the future direction or movement of volatility”[^5]. Those products are now among the most liquid financial instruments in the world. There are indeed more than 500,000 VIX options traded each day, with most of the liquidity concentrated on the first three monthly contracts.\
Nevertheless, despite that more vega is now traded in the VIX market than in the SPX market, the wide bid-ask spreads in the VIX options market betray its lack of maturity. One of the reasons underlying these wide spreads is that the market lacks a reliable pricing methodology for VIX options; since the VIX is by definition a derivative of the SPX, any reasonable methodology must necessarily be consistent with the pricing of SPX options. Designing a model that jointly calibrates SPX and VIX options prices is known to be extremely challenging. Indeed, this problem is sometimes considered to be [*the holy grail of volatility modeling*]{}. We will simply refer to it as the *joint calibration problem.*\
The joint calibration problem has been extensively studied by Julien Guyon who provides a review of various approaches in [@guyon2019joint]. We can split the different attempts to solve it into three categories. In probably the most technical and original proposal, and the first to have succeeded in obtaining a perfect joint calibration, the joint calibration problem is interpreted as a model-free constrained martingale transport problem, as initially observed in [@de2015linking]. In his recent paper [@guyon2019joint], using this viewpoint, Guyon manages to get a perfect calibration of VIX options smile at time $T_1$ and SPX options smiles at dates $T_1$ and $T_2 = T_1+30$ days. As noticed by the author, although this methodology can theoretically be extended to any set of maturities, it is much more intricate in practice because of the computational complexity.\
This drawback is avoided in the second and third types of approach where models are in continuous-time. Continuous-time models have the advantage that they rely on observable properties of assets and so allow for practical intuition on their dynamics. The second approach is to attempt joint calibration with models where SPX trajectories are continuous, see in particular [@goutte2017regime]. Unfortunately, for now, continuous models have not been completely successful in this task. An interpretation for this failure is given in [@guyon2019joint] where the author explains that “the very large negative skew of short-term SPX options, which in continuous models implies a very large volatility of volatility, seems inconsistent with the comparatively low levels of VIX implied volatilities”. To circumvent this issue, it is then natural to think of rough volatility models as recently introduced in [@gatheral2018volatility]. However, these models also appear unsuccessful thus far, see [@guyon2018joint].\
The last approach is to allow for jumps in the dynamic of the SPX, see [@baldeaux2014consistent; @cont2013consistent; @kokholm2015joint; @pacati2018smiling; @papanicolaou2014regime]. Doing so, one can reconcile the skewness of SPX options with the level of VIX implied volatilities. Nevertheless, probably besides those in [@cont2013consistent] and [@pacati2018smiling], existing models with jumps do not really achieve a satisfying accuracy for the joint calibration problem. Specifically, most of them fail to reproduce VIX smiles for maturities shorter than one month.\
In summary, according to Guyon in [@guyon2019joint], despite the many efforts “so far all the attempts at solving the joint SPX/VIX smile calibration problem \[using a continuous time model\] only produced imperfect, approximate fits”. In particular, regarding continuous models, Guyon concludes that “joint calibration seems out of the reach of continuous-time models with continuous SPX paths”. In this paper, we provide a counterexample to Guyon’s conjecture, namely a model with continuous SPX and VIX paths that enables us to fit SPX and VIX options smiles simultaneously.
Rough volatility and the Zumbach effect
=======================================
Recently rough volatility models, where volatility trajectories, though continuous, are very irregular, have generated a lot of attention. The reason for this success is the ability of these very parsimonious models to reproduce all the main stylized facts of historical volatility time series and to fit SPX options smiles, see [@bayer2016pricing; @euch2019roughening; @gatheral2018volatility]. One particularly interesting rough volatility model is the rough Heston model introduced in [@el2019characteristic] which as its name suggests, is a rough version of the classical Heston model. This model arises as limit of natural Hawkes process-based models of price and order flow, see [@el2018microstructural; @jaisson2016rough; @jusselin2018no]. Moreover, there is a quasi-closed form formula for the characteristic function of the rough Heston model, just as in the classical case. So fast pricing of European options is possible, see [@callegaro2018rough; @gatheral2019rational]. In addition derivatives hedging is fully understood as shown in [@el2018perfect; @el2019characteristic], see also [@jaber2019affine; @cuchiero2018generalized].\
Despite these successes, a subtle question raised by Jean-Philippe Bouchaud remains: can a rough volatility model reproduce the so-called [*Zumbach effect*]{}, the observation originally due to Gilles Zumbach, see [@lynch2003market; @zumbach2009time; @zumbach2010volatility], that financial time series are not time-reversal invariant? To answer this question, we introduce two notions, each of which corresponds to different aspects of the Zumbach effect:
- The [*weak Zumbach effect*]{} (typically considered in the econophysics literature, see [@zumbach2009time]): Past squared returns forecast better future integrated volatilities than past integrated volatilities forecast future squared returns. This property is not satisfied in classical stochastic volatility models. However, rough stochastic volatility models are consistent with the weak Zumbach effect, see [@el2019zumbach] for explicit computations in the rough Heston model.
- The [*strong Zumbach effect*]{}: Conditional dynamics of volatility with respect to the past depend not only on the past volatility trajectory but also on the historical price path; specifically, price trends tend to increase volatility, see [@zumbach2010volatility]. Such feedback of the historical price path on volatility also occurs on implied volatility as illustrated in Figure 1 of [@foschi2008path] and in [@zumbach2010volatility]. Rough stochastic volatility models such as the rough Heston model are not consistent with the strong Zumbach effect, see [@el2018perfect].
The quest for a rough volatility model consistent with the strong Zumbach effect and the empirical success of quadratic Hawkes process-based models documented in [@blanc2017quadratic] led to the development of super-Heston rough volatility models in [@dandapani2019quadratic]. These extensions of the rough Heston model arise as limits of quadratic Hawkes process-based microstructural models just as the rough Heston model arises as the continuous-time limit of a linear Hawkes process-based microstructural model.\
The idea of using super-Heston rough volatility models to solve the joint calibration problem came after a presentation of Julien Guyon at École Polytechnique in March 2019. In this talk, he gave a necessary condition for a continuous model to fit simultaneously SPX and VIX smiles: The inversion of convex ordering between volatility and the local volatility implied by option prices, see [@acciaio2019inversion; @guyon2019inversion]. The intuition behind this condition could be reinterpreted as some kind of strong Zumbach effect. It was then natural for us to investigate the ability of super-Heston rough volatility models to solve the joint calibration problem.
The quadratic rough Heston model {#sec:model_definition}
================================
The quadratic rough Heston model that we consider is a special case of the super-Heston rough volatility models of [@dandapani2019quadratic]. The joint dynamics of the asset $S$ (here the SPX), and its spot variance $V$ satisfy $$\mathrm{d}S_t=S_t \sqrt{V_t}dW_t,~ V_t = a(Z_t - b)^2 +c,$$ where $W$ is a Brownian motion, $a$, $b$ and $c$ some positive constants and $Z_t$ follows a rough Heston model. More precisely, $$Z_t = \int_0^t (t-s)^{\alpha-1} \frac{\lambda}{\Gamma(\alpha)} (\theta_0(s) - Z_s) \mathrm{d}s + \int_0^t (t-s)^{\alpha-1} \frac{\lambda}{\Gamma(\alpha)} \eta \sqrt{V_s} \mathrm{d}W_s,$$ with $\alpha \in (1/2,1)$, $\lambda>0$, $\eta>0$ and $\theta_0$ a deterministic function. In this special case of a quadratic rough Heston model, the asset $S$ and its volatility depend on the history of only one Brownian motion. The model is thus a pure feedback model; volatility is driven only by the price dynamics, with no additional source of randomness. In general of course, the volatility process does not need to depend only on the Brownian motion driving the asset price $S$. For simplicity however, we will refer to this pure feedback version of the quadratic rough Heston model as the [*quadratic rough Heston model*]{}.
The quadratic rough Heston process
----------------------------------
The process $Z_t$ may be understood as a weighted moving average of past price log returns. Indeed from Lemma A.1 in [@el2018perfect], we have that $$Z_t=\int_0^t f^{\alpha, \lambda}(t-s) \theta_0(s)\mathrm{d}s + \int_0^t f^{\alpha, \lambda}(t-s) \eta \sqrt{V_s}\mathrm{d}W_s,$$ where $f^{\alpha, \lambda}(t)$ is the Mittag-Leffler density function defined for $t\geq 0$ as $$f^{\alpha, \lambda}(t)=\lambda t^{\alpha-1} E_{\alpha, \alpha}(-\lambda t^{\alpha}),$$ with $$E_{\alpha, \beta}(z) = \sum_{n\geq 0}\frac{z^n}{\Gamma(\alpha n+\beta)}.$$ The variable $Z_t$ is therefore [*path-dependent*]{}, a weighted average of past returns of the type typically considered in path-dependent volatility models, see [@hobson1998complete]. As explained in [@guyon2014path], modeling with path-dependent variables is a natural way to reproduce the fact that volatility depends on recent price changes. However the kernels used to model this dependency are typically exponential, see for example [@hobson1998complete]. Here a crucial idea, motivated by our previous work [@dandapani2019quadratic], is to use a rough kernel, more precisely the Mittag-Leffler density function. Thanks to this kernel, the “memory” of $Z$ decays as a power law and $Z$ is highly sensitive to recent returns since $$f^{\alpha, \lambda}(t) \underset{t\rightarrow +\infty}{\sim} \frac{\alpha}{\lambda \Gamma(1-\alpha)}t^{-\alpha - 1} \text{ and } f^{\alpha, \lambda}(t) \underset{t\rightarrow 0^+}{\sim} \frac{ \lambda }{\Gamma(\alpha)}t^{\alpha - 1}.$$ This essentially means that long periods of trends or sudden upwards or downwards moves of the price generate large values for $|Z|$ and so high volatility, in particular when $Z$ is negative. Such link is clearly observed on data, see Figure \[fig:historical\_spx\_vix\] where the VIX index spikes almost instantaneously after large negative returns of the SPX and then decreases slowly afterwards. We plot in Figure \[fig:simulation\_spx\_vix\] an example of sample paths of SPX and VIX indexes in our model. The feedback of negative price trends on volatility is very well reproduced. Finally the choice of $f^{\alpha, \lambda}$ as kernel ensures that the volatility process is rough, with Hurst parameter equal to $H = \alpha - 1/2$. As shown in [@gatheral2018volatility], this enables us to reproduce the behavior of historical volatility time series provided $H$ is taken of order $0.1$.\
As explained above, an immediate consequence of the feedback effect is that negative price trends generate high volatility levels. But such trends also impact the instantaneous variance of volatility in our model. To see this, consider the classical case with $\alpha = 1$. In that case, an application of Itô’s Formula gives that up to a drift term, $$\mathrm{d}V_t = 2a (Z_t-b) \lambda \eta \sqrt{V_t} \mathrm{d}W_t.$$ Thus the “variance of instantaneous variance" coefficient is proportional to $ a (Z_t - b)^2$ which, up to $c$, is equal to the variance of $\log S$. Thus when volatility is high, volatility of volatility is also high. In particular, conditional on a large downwards move in SPX, we would expect $V$ to be high and so also the volatility of $V$. This explains why our model generates upward sloping VIX smiles.\
We remark that incorporating the influence of price trends on volatility and instantaneous variance of volatility is the main motivation underlying the model of [@goutte2017regime]. That model, although not solving the joint calibration problem, is probably the best of the continuous models introduced so far. In this switching model, the price follows a classical Heston dynamic where the parameters can change depending on the value of an hidden Markov chain with three states. It is motivated by a $100$-days rolling calibration of the classical Heston model performed by the authors, see Figure 2 in [@goutte2017regime]. This rolling calibration suggests that volatility, leverage and volatility of volatility are higher in period of crisis. Hence they introduce a Markov chain to trigger crisis phases and switch the parameters of the Heston model depending on the situation. The three possible states of the chain can therefore be interpreted as corresponding to the following situations:
- Flat or increasing SPX.
- Transition phase between flat SPX and crisis.
- Crisis with dramatically decreasing SPX.
The Markov chain in [@goutte2017regime] can therefore somehow be seen as an [*ad hoc*]{} version of the process $Z$ in the quadratic rough Heston model.
Parameter interpretation
------------------------
The parameters $a$, $b$ and $c$ in the specification $$V_t = a(Z_t - b)^2 + c$$ can be interpreted in the following way.
- $c$ represents the minimal instantaneous variance. When calibrating the model, we use $c$ to shift upward or downward the smiles of SPX options.
- $b>0$ encodes the asymmetry of the feedback effect. Indeed for the same absolute value of $Z$, volatility is higher when $Z$ is negative than when it is positive. Such asymmetry aims at reproducing the empirical behavior of the VIX. This is illustrated in Figure \[fig:historical\_spx\_vix\] where we observe that the VIX spikes when the SPX tumbles down, but not after it goes up. From a calibration viewpoint, the higher $b$ the more SPX options smiles are shifted to the right.
- $a$ is the sensitivity of the volatility to the feedback of price returns. The greater $a$, the greater the role of feedback in the model and the higher is volatility of volatility. Consistent with this SPX smiles become more extreme as $a$ increases.
Infinite dimensional Markovian representation
---------------------------------------------
Though the quadratic rough Heston model is not Markovian in the variables $(S, V)$, it does admit an infinite dimensional Markovian representation. Inspired by the computations in [@el2018perfect], we obtain that for any $t$ and $t_0$ positive $$\label{eq:fwd_vol}
Z_{t_0+t} = \int_0^t (t-s)^{\alpha-1} \frac{\lambda}{\Gamma(\alpha)} (\theta_{t_0}(s) - Z_{t_0+s}) \mathrm{d}s + \int_0^t (t-s)^{\alpha-1} \frac{\lambda}{\Gamma(\alpha)} \eta \sqrt{V_{t_0+s}} \mathrm{d}W_{t_0+s},$$ with $\theta_{t_0}$ a $\mathcal{F}_{t_0}$-measurable function. More precisely $\theta_{t_0}$ is given by $$\theta_{t_0}(u) = \theta_0(t+u) + \frac{\alpha}{\lambda\Gamma(1-\alpha)}\int_0^{t_0} (t_0-v+u)^{-1-\alpha}(Z_v - Z_{t_0})\mathrm{d}v.$$ Equation implies that the law of $(S_t, V_t)_{t\geq t_0}$ only depends on $S_{t_0}$ and $\theta_{t_0}$. In view of and using the same methodology as in [@el2018perfect], it means that we can express the VIX at time $t$ as a function of $\theta_t$ and $S_t$. Consequently we can write the price of any European option with pay-off depending on SPX and VIX as a function of time, $S$ and $\theta$.
![SPX (in blue) and VIX (in red) indexes from $25$ November 2004 to $25$ November 2019.[]{data-label="fig:historical_spx_vix"}](historical_values.png){width="16cm" height="8cm"}
![SPX (in blue) and VIX (in red) indexes from simulation of the quadratic rough Heston model.[]{data-label="fig:simulation_spx_vix"}](path_simulation.png){width="16cm" height="8cm"}
Numerical results {#sec:numerical results}
=================
In this section, we illustrate how successfully we can fit both SPX and VIX smiles on May 19, 2017[^6], one of the dates considered in [@euch2019roughening], an otherwise randomly chosen date. We focus on short expirations, from $2$ to $5$ weeks, where the bulk of VIX liquidity is. Moreover, short-dated smiles are the ones that are typically poorly fitted by conventional models.\
In the quadratic rough Heston model, the function $\theta_0(\cdot)$ needs to be calibrated to market data. In the rough Heston model there is a simple bijection between $\theta_0(\cdot)$ and the forward variance curve. In the quadratic rough Heston model, this connection is more intricate and so for simplicity we choose the following restrictive parametric form for $Z$: $$Z_t = Z_0 - \int_0^t (t-s)^{\alpha - 1} \frac{\lambda}{\Gamma(\alpha)} Z_s \mathrm{d}s + \int_0^t (t-s)^{\alpha - 1} \frac{\lambda}{\Gamma(\alpha)} \eta \sqrt{V_s} \mathrm{d}W_s,$$ which is equivalent to taking $$\theta_0(t) = \frac{Z_0}{\lambda \Gamma(1-\alpha) } t^{-\alpha}.$$ Allowing $\theta_0(\cdot)$ to belong to a larger space would obviously lead to even better results, but would require a more complex calibration methodology. Thus we are left to calibrate the parameters $\nu = (\alpha, \lambda, a, b, c, Z_0)$. We use the following objective function:$$F(\nu) = \frac{1}{\# \mathcal{O}^{SPX}}\sum_{o \in \mathcal{O}^{SPX}} (\sigma^{o,mid} - \sigma^{o, \nu})^2 + \frac{1}{\# \mathcal{O}^{VIX}}\sum_{o\in \mathcal{O}^{VIX}}(\sigma^{o,mid} - \sigma^{o, \nu})^2,$$ where $\mathcal{O}^{SPX}$ is the set of SPX options, $\mathcal{O}^{VIX}$ the set of VIX options, $\sigma^{o, mid}$ denotes the market mid implied volatility for the option $o$ and $\sigma^{o,\nu}$ is the implied volatility of the option $o$ in the quadratic rough Heston model with parameter $\nu$ obtained by Monte-Carlo simulations. To calibrate the model, we minimize the function $F$ over a grid centered around an initial guess $\nu_0$.\
We obtain the following parameters: $$\alpha = 0.51;~ \lambda = 1.2;~a = 0.384;~b = 0.095;~c = 0.0025,~Z_0 = 0.1.
\label{eq:params}$$ The corresponding SPX and VIX options smiles are plotted in Figures \[fig:vol\_spx\] and \[fig:vol\_vix\].\
![Implied volatility on SPX options for $19$ May 2017. The blue and red points are respectively the bid and ask of market implied volatilities. The implied volatility smiles from the model are in green. The strikes are in log-moneyness and $T$ is time to expiry in years.[]{data-label="fig:vol_spx"}](SPX_20170519.png){width="16cm" height="8cm"}
![Implied volatility on VIX options for $19$ May 2017. The blue and red points are respectively the bid and ask of market implied volatilities. The implied volatility smiles from the model are in green. The strikes are in log-moneyness and $T$ is time to expiry in years.[]{data-label="fig:vol_vix"}](VIX_20170519.png){width="16cm" height="8cm"}
Despite that our calibration methodology is suboptimal and we only have six parameters, VIX smiles generated by the model with parameters fall systematically within market bid-ask spreads. The whole SPX volatility surface is also very well reproduced, in particular the overall shape of the SPX smiles, including extreme left tails. Obviously fits can be made even greater by reducing the range of strikes of interest or by fine tuning the calibration, notably through improving the $\theta_0(\cdot)$ function. We are currently working on a fast calibration approach, inspired by recent works on neural networks, see for example [@horvath2019deep].
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Julien Guyon for numerous inspiring discussions and Stefano de Marco for relevant comments. Paul Jusselin and Mathieu Rosenbaum gratefully acknowledge the financial support of the [*ERC Grant 679836 Staqamof*]{} and of the chair [*Analytics and Models for Regulation*]{}.
[^1]: Baruch College, CUNY, jim.gatheral@baruch.cuny.edu
[^2]: École Polytechnique, paul.jusselin@polytechnique.edu
[^3]: École Polytechnique, mathieu.rosenbaum@polytechnique.edu
[^4]: <https://cfe.cboe.com/cfe-products/vx-cboe-volatility-index-vix-futures>
[^5]: <http://www.cboe.com/products/vix-index-volatility/vix-options-and-futures/vix-options>
[^6]: Market data is from OptionMetrics via Wharton Data Research Services (WRDS).
|
---
author:
- |
Amir Hertz Rana HanockaRaja GiryesDaniel Cohen-Or\
Tel Aviv University
bibliography:
- 'references.bib'
title: 'PointGMM: a Neural GMM Network for Point Clouds'
---
|
---
author:
- 'Denis S. Goldobin$^{1,2,3}$, Kseniya V. Kovalevskaya$^1$'
- 'Dmitry V. Lyubimov$^2$'
title: 'Elastic and inelastic collisions of interfacial solitons and integrability of two-layer fluid system subject to horizontal vibrations'
---
Introduction
============
The experimental observations of the occurrence of steady wave patterns on the interface between immiscible fluids subject to horizontal vibrations were first reported by Wolf [@Wolf-1961; @Wolf-1970]. Wolf also noticed the opportunities for vibrational stabilization of the system states, which are gravitationally unstable in the absence of vibrations, and initiated exploration for these possibilities. The development of a rigorous theoretical basis for these experimental findings was contributed by the linear instability analysis of the flat state of the interface [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999] (in Fig.\[fig\_sketch\], one can see the sketch of the system considered in these works). It was found that in thin layers the instability is a long-wavelength one [@Lyubimov-Cherepanov-1987]. In [@Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999], the linear stability was analyzed for the case of arbitrary frequency of vibrations.
In Wolf’s experiments with horizontal vibrations [@Wolf-1961], the viscous boundary layer in the most viscous liquid was one order of magnitude smaller than the layer thickness, meaning the approximation of inviscid liquid to be appropriate. According to [@Lyubimov-Cherepanov-1987], the layer is thin enough for the marginal instability to be long-wavelength, when its half-thickness $h<\sqrt{3\alpha/[(\rho_2-\rho_1)g]}$, where $\alpha$ is the interface tension coefficient, $\rho_1$ and $\rho_2$ are the light and heavy liquid densities, respectively, and $g$ is the gravity. This critical layer thickness can be one or two orders of magnitude larger than the thickness of the viscous boundary layer, meaning the problem with long-wavelength instability remains physically relevant for the case of inviscid fluids. In the opposite limiting case, for a viscosity-dominated system, the problem of pattern formation was studied in [@Shklyaev-Alabuzhev-Khenner-2009; @Benilov-Chugunova-2010]. The case of dynamics of nearly-inviscid system is essentially different from the purely dissipative dynamics reported in [@Shklyaev-Alabuzhev-Khenner-2009] for extremely thin layers.
Until recently [@Goldobin-etal-Nonlinearity-2014], advances in theoretical studies of the relief of interface or free surface under high-frequency vibration fields were focused on the quasi-steady profiles (e.g., [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]). Within the approach of [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989], a kind of energy variational principle can be derived. This principle was employed for calculation of the average profile shape about which the interface trembles with small amplitude and high frequency. This approach, however, does not allow considering the pattern evolution and determining stability properties of the quasi-stationary relief. In [@Goldobin-etal-Nonlinearity-2014], the rigorous weakly nonlinear analysis was applied for derivation of the governing equations for large-scale (long-wavelength) patters below the instability threshold. With these equations the family of solitons can be found in the system. Remarkably, the standing solitons, which are the only patterns that could be derived with the variational principle, are always unstable. Thus, the governing equations we derived in [@Goldobin-etal-Nonlinearity-2014] provide for the first time opportunity for a reliable and physically informative theoretical analysis of nonlinear dynamics of the system.
With this letter, we will provide a comprehensive analysis of the dynamics resulting from the nonlinear evolution equations for long-wavelength patterns. For the soliton families we reported in [@Goldobin-etal-Nonlinearity-2014], we will analyze nonlinear stages of development of perturbations leading to either an ‘explosion’ or to falling-apart of an unstable soliton into pair of stable solitons; the latter kind of behavior was previously reported for soliton-bearing systems in [@Orlov-1983; @Falkovich-Spector-Turitsyn-1983; @Bogdanov-Zakharov-2002]. The self-similar explosion solution agrees remarkably well with the results of direct numerical simulation. Two integrals of motion will be derived for the equations, corresponding to the laws of conservation of mass and momentum in the virgin physical system. With these integrals of motion, we will see that unstable solitons can be represented by superpositions of pairs of stable ones, while stable solitons are elementary, in a sense that they can not be decomposed into any superpositions. The equivalence between unstable solitons and certain pairs of stable solitons suggests that stable soliton collisions can be ‘inelastic’. In agreement with the latter, we will find that soliton collisions can be either elastic or lead to an explosion; at the boundary between elastic and explosive collisions, colliding stable solitons coalesce into unstable ones. Finally, we will see that the system dynamics can be completely represented as a kinetics of a soliton gas and governed by the equation which is known from [@Bogdanov-Zakharov-2002] to be fully integrable beyond the vicinities of the explosion sites. Thus, we deal with the situation where the nonlinear dynamics of a real physical system—which pertains to one of the classic problems of fluid dynamics—can be fully integrated, and, even more intriguingly, demonstrates features which are not very common for soliton-bearing systems, such as decomposition of certain solitons, possibility of inelastic collisions, etc.
Governing equations for large-scale patterns
============================================
With the standard multiscale method one can derive, that large-scale ([*or*]{} long-wavelength) patterns in the system of inviscid liquids are governed by the equation system [@Goldobin-etal-Nonlinearity-2014] $$\left\{
\begin{array}{rcl}
\displaystyle
B_0\frac{\rho_2+\rho_1}{\rho_\ast}T^2\frac{\partial\varphi}{\partial t}\!\!\!&
\displaystyle =&\!\!\!
\displaystyle \left[L^2-\frac{h^2}{3}\right]\frac{\partial^2\eta}{\partial x^2}
\\[15pt]
&&\displaystyle
{}+\frac{3}{2h}\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\eta^2
+\frac{B_1}{B_0}\eta\,,\\[15pt]
\displaystyle
\frac{\partial\eta}{\partial t}\!\!\!&
\displaystyle =&\!\!\!
\displaystyle -h\frac{\partial^2\varphi}{\partial x^2}\,.
\end{array}
\right.
\label{eq:dimensional}$$ Here $\eta(x,t)$ is the non-pulsating part of the interface displacement from the flat state, $\varphi(x,t)$ is the non-pulsating part of the upper fluid flow ($\vec{v}_1=-\nabla\varphi+...$, $\vec{v}_2=\nabla\varphi+...$, where “$...$” stands for the pulsing part of the flow and smaller corrections); notice $\varphi(x,t)$ is independent of $z$, since $\vec{v}_{1,2}$ are nearly constant along $z$. Reference length $L=\sqrt{\alpha/[(\rho_2-\rho_1)g]}$, $\alpha$: surface tension, $\rho_1$ and $\rho_2$: density of the upper and lower fluids, respectively, $\rho_1<\rho_2$, $g$: gravity, $h$: unperturbed thickness of the layers, $T=L/b$: reference time, $\rho_\ast$: reference fluid density ($\rho_\ast$ can be chosen as convenient). Parameter $B$ is the dimensionless vibration parameter; $$B\equiv\rho_\ast b^2/\sqrt{\alpha(\rho_2-\rho_1)g}=B_0+B_1\,,
\label{eq:B}$$ where $b$ is the container vibration velocity amplitude; $B_0$ is the linear instability threshold $$B_0=\frac{\rho_\ast(\rho_2+\rho_1)^3h}{2\rho_1\rho_2(\rho_2-\rho_1)^2}
\sqrt{\frac{(\rho_2-\rho_1)g}{\alpha}}\,,
\label{eq:B0}$$ and $B_1$ is the deviation from the stability threshold.
We consider the system dynamics slightly below the linear instability threshold, i.e., for $B_1<0$ and $|B_1|\ll B_0$. With rescaling $$\begin{array}{c}
x\to x\,L\sqrt{\frac{B_0}{(-B_1)}\left[1-\frac{h^2}{3L^2}\right]}\,,
\quad
\eta\to\eta\,h\frac{\rho_2+\rho_1}{\rho_2-\rho_1}\frac{(-B_1)}{B_0}\,,
\\[10pt]
t\to t\sqrt{\frac{\rho_2-\rho_1}{\rho_\ast}\frac{L^3B_0^3}{h\,b^2B_1^2}
\left[1-\frac{h^2}{3L^2}\right]}\,,\mbox{ and}\\[10pt]
% \eta\to\eta\,h\frac{\rho_2+\rho_1}{\rho_2-\rho_1}\frac{(-B_1)}{B_0}\,,
% \\[10pt]
\varphi\to\varphi\sqrt{\frac{\rho_\ast(\rho_2+\rho_1)^2}{(\rho_2-\rho_1)^3}
\frac{L^3B_1^2}{h\,b^2B_0^3}\left[1-\frac{h^2}{3L^2}\right]}\,,
\end{array}
\label{eq:rescaling}$$ the governing equations (\[eq:dimensional\]) take zero-parametric dimensionless form; $$\begin{aligned}
\dot\varphi&=&\eta_{xx}+{\textstyle\frac{3}{2}}\eta^2-\eta\,,
\label{eq:dimensionless1}
\\
\dot\eta&=&-\varphi_{xx}\,.
\label{eq:dimensionless2}\end{aligned}$$ Here subscripts denote the partial derivative with respect to the specified coordinate.
The latter equation system can be recast as a ‘plus’ Boussinesq equation (BE); $$\ddot{\eta}-\eta_{xx}+\left({\textstyle\frac{3}{2}}\eta^2+\eta_{xx}\right)_{xx}=0\,.
\label{eq:plBe}$$ From the view point of dynamics, this equation essentially differs from original Boussinesq equation B (BE B) for waves in a shallow water layer [@Boussinesq-1872] or in a two-layer system without vibrations [@Choi-Camassa-1999], which is $$\ddot{\eta}-\eta_{xx}-\left({\textstyle\frac{3}{2}}\eta^2+\eta_{xx}\right)_{xx}=0\,.
\label{eq:BeB}$$ The equation system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]) is rigorously derived for the vicinity of the vibrational instability threshold (e.g., one cannot consider the case of vanishing vibrations, when departure from the threshold is finite, with this equation system). On the contrast, the Boussinesq equation A (BE A) [@Boussinesq-1872] with nonlinear term $[(\varphi_x)^2+(1/2)\eta^2]$ in place of $(3/2)\eta^2$ in Eq.(\[eq:BeB\]) is derived for small dispersion and nonlinear terms, i.e., only small-amplitude waves with velocity close to $1$ are quantitatively governed by BE A. With assumptions required by BE A, one needs further to restrict consideration to the case of the wave package moving in one direction for to set $(\varphi_x)^2=(\dot\varphi)^2\approx\eta^2$ and obtain BE B (Eq.(\[eq:BeB\])). Summarizing, the results on soliton waves derived with BE B are rigorous for waves in shallow water only for the edge of the soliton spectrum and never rigorous for collisions of contrpropagating solitons, while Eq.(\[eq:plBe\]) is rigorous for interfacial waves in the system subject to horizontal vibrations.
The integrability of the ‘plus’ BE was considered in [@Bogdanov-Zakharov-2002] where the $\partial$-dressing method was employed for deriving multisoliton solutions. Bogdanov and Zakharov [@Bogdanov-Zakharov-2002] reported existence of unstable solitons which can decay into pairs of stable solitons and thoroughly treated bounded states of two and more ‘singular’ solitons of the form $\eta=-4/(x-x_0)^2$. For our system the ‘singular’ solitons cannot be considered as the long-wavelength approximation is violated for them. As Ref. [@Bogdanov-Zakharov-2002] does not provide answers to some significant questions and omit certain important scenarios of the system dynamics, it will be more convenient to perform a comprehensive analysis of the system dynamics, without employment of a laborious $\partial$-dressing technique, and postpone comparison of this analysis to [@Bogdanov-Zakharov-2002] for the Discussion section.
Solitons
========
Equation system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]) admits solutions in the form of propagating wave with time-independent profile; $\eta(x,t)=\eta(x-ct)\equiv\eta(\xi)$, $\varphi(x,t)=\varphi(x-ct)\equiv\varphi(\xi)$, where $c$ is the wave propagation speed. For these waves $\partial_x=\partial_\xi$ and $\partial_t=-c\partial_\xi$, and Eqs.(\[eq:dimensionless1\])–(\[eq:dimensionless2\]) yield $$\textstyle
0=\eta_{\xi\xi}+\frac32\eta^2-(1-c^2)\eta\,.
\label{eq:s01}$$ (Here we used the condition $\eta(\xi=\pm\infty)=0$.) The latter equation admits the soliton solution $$\eta_0(\xi)=\frac{1-c^2}{\displaystyle\cosh^2\big[\sqrt{1-c^2}\xi/2\big]}\,;
\label{eq:s02}$$ for a given initial profile $\eta(x)$, the propagation direction ($+c$ or $-c$) is determined by the flow, $\varphi_\xi=\pm
c\eta(\xi)$ (cf. Eq.(\[eq:dimensionless2\])). The family of solitons is one-parametric, parameterized by the speed $c$ only. Speed $c$ varies within the range $[0,1]$; the standing soliton ($c=0$) is the sharpest and the highest one, while for the fastest solitons, $c\to1$, the width tends to infinity ($\propto
1/\sqrt{1-c^2}$) and the height tends to $0$ ($\propto [1-c^2]$).
Let us interpret these results in original dimensional space–time. The spatial and temporal scales for dynamic system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]) depend on the deviation from the threshold $(-B_1)$ (see rescaling (\[eq:rescaling\])). For a transparent interpretation of the dynamics of patterns in original dimensional space–time, one can consider solitons for the dimensional equation system (\[eq:dimensional\]) and find equation of form (\[eq:s01\]) with coefficient $$G:=\frac{(-B_1)}{B_0}-c_{\mathrm{dim}}^2\frac{(\rho_2+\rho_1)\,\alpha^{1/2}}{h\,[(\rho_2-\rho_1)g]^{3/2}}
\label{eq:s03}$$ ahead of the last term $(-\eta)$, where $c_{\mathrm{dim}}$ is the dimensional speed of the soliton, with all the other coefficients being physical parameters of the physical system under consideration. Considering a given physical system with vibration parameter $B$ as a control parameter, one can see that the shape of a soliton is controlled by expression (\[eq:s03\]). Hence, the same interface inflection soliton can exist for different deviation $B_1$, but the same $G$, which will be achieved by tuning $c_\mathrm{dim}$; for larger negative deviation $(-B_1)$ from the instability threshold, the speed $c_\mathrm{dim}$ is larger.
The family of solitons can be compared against the wave packages of linear waves—small perturbation of the flat-interface state. For small normal perturbations $(\eta,\varphi)\propto e^{i(kx-\Omega t)}$, Eqs.(\[eq:dimensionless1\])–(\[eq:dimensionless2\]) yield the dispersion relation $\Omega(k)=k\sqrt{1+k^2}$. The group velocity is $$v_\mathrm{gr}=\frac{d\Omega}{dk}=\frac{1+2k^2}{\sqrt{1+k^2}}\,,
\label{eq:s04}$$ which varies from $v_\mathrm{gr}=1$ (for $k\to0$) to infinity (for $k\to\infty$). Thus, any packages of linear waves travel with higher velocity than the fastest solitons.
Stability of solitons (initial perturbations)
=============================================
The linear stability analysis for solitons in dynamic system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]) reveals that the slow solitons, with $0\le c\le1/2$, are unstable (see Fig.\[fig\_stability\](a)), with one unstable degree of freedom (in Fig.\[fig\_stability\](b), one can see the instability mode for $c=0$). The fast solitons, with $1/2<c\le1$, are stable both linearly and to non-large finite perturbations. Detailed analysis of the linear stability can be found in [@Goldobin-etal-Nonlinearity-2014].
It is interesting to consider the nonlinear development of perturbations of unstable solitons. Two possible scenarios were encountered in direct numerical simulations:\
(i) Explosive growth of perturbation and formation of an infinitely sharp and high peak in a finite time;\
(ii) Falling-apart of the unstable soliton into exactly two stable ones.\
In Fig.\[fig\_nonlin\_instab\], one can see the development of these scenarios.
Obviously, for the scenario (i), after violation of the conditions of the long-wavelength approximation, the dynamics will deviate from the one dictated by Eqs.(\[eq:dimensionless1\])–(\[eq:dimensionless2\]); still, the formation of an sharpening of the interface with large deviation from the flat state is certain. In the following we will derive scaling laws for this explosion regime.
For the falling-apart of the unstable solution, we can observe in Fig.\[fig\_nonlin\_instab\](b) that two fast solitons can then collide and coalesce again into the same initial unstable soliton, which will exist for awhile. The smaller perturbation of this unstable soliton, the longer it exists before falling apart again. It suggests that collisions of solitons can be ‘inelastic’. In the text below, we will investigate the collisions of fast stable solitons numerically and reveal analytical conditions for coalescence of colliding solitons, elastic collisions and explosions (notice, in Fig.\[fig\_nonlin\_instab\](b) we observe collision not for an arbitrary pair of stable solitons but for the products of decay of the unstable soliton).
Noteworthy, one can predict wether the initial infinitesimal perturbation will lead to one or another scenario. It is determined by the projection of the initial perturbation onto the unstable mode. According to results of direct numerical simulation, if one normalizes the instability mode $(e^{\lambda t}\eta_1(\xi),e^{\lambda t}\varphi_1(\xi))$ so that $\eta_1(0)>0$—cf. Fig.\[fig\_stability\](b), where the instability mode for the soliton with $c=0$ is plotted with the blue dashed line—then the perturbation with a positive contribution of $(\eta_1(\xi),\varphi_1(\xi))$ will lead to explosion while the one with a negative contribution will lead to the splitting.\[[^1]\]
Explosions
==========
For the explosion solution, field $\eta$ becomes large and the term $(-\eta)$ can be neglected against the background of the term $(3/2)\eta^2$ in Eq.(\[eq:dimensionless1\]); therefore, equation system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]) can be rewritten as $$\ddot{\eta}\approx-\big(\eta_{xx}+{\textstyle\frac32}\eta^2\big)_{xx}\,.
\label{eq:e01}$$ The last differential equation is homogeneous: it admits self-similar solutions of the form $$\eta(x,t)=t^nf(s),\qquad s=x/t^m\,,
\label{eq:e02}$$ where $n$ and $m$ are to be determined from the condition that Eq.(\[eq:e01\]) yields a differential equation for $f(s)$ which is free from $t$ and $x$. After substitution (\[eq:e02\]), Eq.(\[eq:e01\]) reads $$\begin{array}{r}
n(n-1)f-m(2n-m-1)sf^\prime+m^2s^2f^{\prime\prime}
\qquad\\[10pt]
=-t^{2-4m}\big[f^{\prime\prime}
+\frac32 t^{n+2m}f^2\big]^{\prime\prime}\,,
\end{array}$$ i.e. requires $m=1/2$ and $n=-1$. With these values of $n$ and $m$ the equation for $f(s)$ reads $$\textstyle
f^{\prime\prime\prime\prime}+\frac32(f^2)^{\prime\prime}
+\frac14 s^2f^{\prime\prime}+\frac74 sf^\prime+2f=0\,.
\label{eq:e03}$$ The last equation has unique solution neither diverging at $s=0$ nor nonvanishing at $s\to\pm\infty$. This solution is $$f(s)=8\big(3\sqrt{2}-s^2\big)/\big(3\sqrt{2}+s^2\big)^2\,.
\label{eq:peak}$$ In Fig.\[fig\_explosion\](a), one can see that the rescaled profiles of the explosion solution from Fig.\[fig\_nonlin\_instab\](a) become nearly indistinguishable from the solution $f(s)$ to Eq.(\[eq:e03\]) quite quickly.
While the solitons are running, the explosive solution is standing. This can be seen as well in Fig.\[fig\_nonlin\_instab\](a), where the pattern stops when the explosion happens.
Integrals of motion
===================
As an integrable system, BE possesses infinite number of integrals of motion. In particular, one can show the dynamic system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]) to possess two following independent integrals of motion\[[^2]\]: $$\begin{aligned}
I_0&\textstyle=\int_{-\infty}^{+\infty}\eta(x,t)\,\mathrm{d}x\,,
\label{eq:i01}
\\
I_1&\textstyle=\int_{-\infty}^{+\infty}\eta(x,t)\,\varphi_x(x,t)\,\mathrm{d}x\,.
\label{eq:i02}\end{aligned}$$ The first integral is owned by the mass conservation law and the second integral represents the momentum conservation law—thus, the both conservation laws valid for the virgin fluid dynamical system have their reflections in presented integrals of motion of the system (\[eq:dimensionless1\])–(\[eq:dimensionless2\]). These integrals will be useful for our following consideration.
For the soliton (\[eq:s02\]), one can find $$\begin{aligned}
I_0[\eta_0(x-ct)]&\textstyle=\int_{-\infty}^{+\infty}\eta_0\,\mathrm{d}x
=\sqrt{1-c^2}\,,
\label{eq:i03}
\\
I_1[\eta_0(x-ct)]&\textstyle=c\int_{-\infty}^{+\infty}\eta_0^2\,\mathrm{d}x
=c\left[1-c^2\right]^{3/2}.
\label{eq:i04}\end{aligned}$$ Here it is important, that soliton velocity $c$ in Eq.(\[eq:i04\]) is negative for solitons propagating to the left.
Two-soliton collisions
======================
While the system dynamics is integrable, the possibility of explosions and the coalescence of decay products of an unstable soliton (as seen in Fig.\[fig\_nonlin\_instab\](b)) suggest that collision of stable solitons can be ‘inelastic’. The results of direct numerical simulation of the collisions of pairs of solitons are presented in Fig.\[fig\_collisions\](a). Collisions of copropagating solitons are always elastic and they exchange their velocities during this collisions. Collisions of contrpropagating solitons can be either elastic, when solitons are fast enough, or lead to an explosion. With elastic collision, they exchange velocities and effectively transpass through one another. At the explosion boundary, solitons are found to coalesce, forming an unstable soliton. It is interesting to consider this coalescence and the decay of unstable solitons into two stable products from the perspective of the motion integrals. Indeed, for solitons that are at the distance from each other the profiles are nearly mutually unaffected and $I_j=I_j[\eta_0(x-c_1t,c_1)]+I_j[\eta_0(x-c_2t,c_2)]$, where $c_2<0$, which should be the same as for the coalescence product soliton of velocity $c_0$. These integrals are best to be recast in terms of $Z_j\equiv\sqrt{1-c_j^2}$; $$\begin{aligned}
Z_1+Z_2&=Z_0\,,
\label{eq:c01}
\\
Z_1^3\sqrt{1-Z_1^2}-Z_2^3\sqrt{1-Z_2^2}
&=Z_0^3\sqrt{1-Z_0^2}\,.
\label{eq:c02}\end{aligned}$$ For any $Z_0\in(1/\sqrt{2};1]$, which corresponds to $c_0\in[0;1/2)$, i.e. an unstable soliton, a pair of solutions $Z_1$ and $Z_2$ exist, both of which are in the semi-open interval $(0;1/\sqrt{2}]$, i.e. correspond to stable solitons with $|c_j|\in[1/2;1]$. For $Z_0<1/\sqrt{2}$ the equation system (\[eq:c01\])–(\[eq:c02\]) has no solution. Thus, one obtains the soliton collision instability boundary semi-analytically, with the solution to algebraic equation system (\[eq:c01\])–(\[eq:c02\]). This solution is plotted in Fig.\[fig\_collisions\](a) with the solid blue line and one can see this result to match the results of direct numerical simulation plotted with triangles. In Fig.\[fig\_collisions\](b) one can see relations between $c_0$ and the pair $c_1$ and $|c_2|<c_1$.
This result is quite interesting: half of solitons, with $c\in[0,1/2)$, are unstable and can be represented as a superposition of two solitons from the other half of them, with $c\in[1/2,1)$. The solitons of the latter half are stable and cannot be decomposed into other solitons. Moreover, the unstable solitons are not merely the superposition of stable ones, they are also the boundary of the basin of the system trajectories leading to an explosive rapture of the upper layer.
Discussion
==========
In Fig.\[fig\_solitons\_ensemble\], a sample of the system dynamics from arbitrary initial conditions is presented in domain $x\in[0;200]$ with periodic boundary conditions. One can see this dynamics can be well treated as a kinetics of a gas of stable solitons.
Let us compare the big picture of the system dynamics constructed above with the results from [@Bogdanov-Zakharov-2002] where the dynamics of the ‘plus’ Boussinesq equation was considered [*on manifolds of superpositions of finite number of solitons*]{}. Two subfamilies of stable and unstable solitons were revealed in [@Bogdanov-Zakharov-2002] as well, however their stability was not considered with respect to arbitrary perturbations. While the decay of an unstable soliton into pair of stable solitons was reported with explicit analytical solutions, possibility of an ‘explosion’ of single unstable soliton was not reported. Thus, the picture of scenarios of instability development was incomplete. Although the formation of an explosion can be seen in [@Bogdanov-Zakharov-2002] for collisions of two unperturbed unstable solitons, its universal asymptotic shape (Eq.(\[eq:peak\])) and scaling properties (Eq.(\[eq:e02\])) were not considered. For the problem of two-soliton collisions, there were two general conclusions in [@Bogdanov-Zakharov-2002]: (1) stable copropagating solitons “do not form singularities as a result of two-soliton interaction” (which is important for us, as we observe no explosions and no coalescences for copropagating solitons) and (2) two-soliton interaction of unstable solitons necessarily leads to formation of a singularity. In the light of the results of the analysis of stability with respect to arbitrary perturbations, the latter statement becomes less informative. The case of collision of contrpropagating stable solitons, which yields us most important results, was not addressed previously. Ref. [@Bogdanov-Zakharov-2002] stands as a prominent work in the theory of solitons, presenting general multi-soliton solution for a paradigmatic ‘plus’ Boussinesq equation, the phenomenon of decay of unstable solitons into pairs of stable ones, and forth and back transformations of interacting solitons into bounded states of singularities.
Conclusion
==========
We presented a comprehensive analysis of the dynamics resulting from the nonlinear evolution equations for long-wavelength patterns in a system of two layers of immiscible inviscid fluids subject to horizontal vibrations. In this system the standing and slow solitons, $c<1/2$, are unstable, while the fast solitons, $c\ge1/2$, are stable. For unstable solitons, nonlinear stages of development of perturbations lead to either an ‘explosion’ or to falling-apart of an unstable soliton into pair of stable solitons. The self-similar explosion soliton was derived and found to agree well with the results of direct numerical simulation. Two integrals of motion were obtained and employed for demonstrating that unstable solitons can be represented by superpositions of pairs of stable ones, while stable solitons are elementary. We found that soliton collisions can be either elastic or lead to an explosion; at the boundary between elastic and explosive collisions, colliding stable solitons coalesce into unstable ones. To conclude, beyond the vicinities of explosions, the system dynamics is completely representable by a kinetics of a soliton gas and the system is fully integrable. Coexistence of explosion solutions and integrability for a real physical system is quite remarkable.
The work has been financially supported by the Russian Science Foundation (grant no. 14-21-00090).
[10]{} url\#1[`#1`]{}
.
.
.
.
.
.
.
. \[[*in Russian*]{}\]. (The results of this work related to the problem we consider also follow from [@Lyubimov-Cherepanov-1987])
.
.
.
.
.
[^1]: This can be intuitively expected from Fig.\[fig\_stability\](b) as well. Indeed, the addition of the dashed profile to the soliton (solid line) with a positive weight means shrinking of the interface embossment and increase of its height, which is the beginning of explosion, while the subtraction of the dashed profile corresponds to decrease of the middle peak and further formation of two peaks on the sides of the main one, which are ‘embryos’ of two splitting products.
[^2]: Indeed, $\dot{I}_0=\int\dot{\eta}\,\mathrm{d}x=-\int\varphi_{xx}\mathrm{d}x=0$, and $\dot{I}_1=\int(\dot{\eta}\varphi_x+\eta\dot{\varphi}_x)\mathrm{d}x
=\int(\dot{\eta}\varphi_x-\eta_x\dot{\varphi})\mathrm{d}x
=\int(-\varphi_x^2/2-\eta_x^2/2-\eta^3/2+\eta^2/2)_x\mathrm{d}x=0$; the integrals here vanish as integrals of $x$-derivatives of functions vanishing at infinity.
|
---
abstract: |
We analyze general aspects of exchangeable quantum stochastic processes, as well as some concrete cases relevant for several applications to Quantum Physics and Probability. We establish that there is a one–to–one correspondence between quantum stochastic processes, either preserving or not the identity, and states on free product $C^*$–algebras, unital or not unital respectively, where the exchangeable ones correspond precisely to the symmetric states. We also connect some algebraic properties of exchangeable processes, that is the fact that they satisfy the product state or the block–singleton conditions, to some natural ergodic ones. We then specialize the investigation for the $q$–deformed Commutation Relations, $q\in(-1,1)$ (the case $q=0$ corresponding to the reduced group $C^{*}$–algebra $C^*_r(\bbf_\infty)$ of the free group on infinitely many generators), and the Boolean ones. We also provide a generalization of De Finetti Theorem to the Fermi CAR algebra (corresponding to the $q$–deformed Commutation Relations with $q=-1$), by showing that any state is symmetric if and only if it is conditionally independent and identically distributed with respect to the tail algebra. The Boolean stochastic processes provide examples for which the condition to be independent and identically distributed w.r.t. the tail algebra, without mentioning the [*a–priori*]{} existence of a preserving conditional expectation, is in general meaningless in the quantum setting. Finally, we study the ergodic properties of a class of remarkable states on the group $C^{*}$–algebra $C^*(\bbf_\infty)$, that is the so–called Haagerup states.\
0.1cm[**Mathematics Subject Classification**]{}: 60G09, 46L53, 46L05, 46L30, 46N50.\
[**Key words**]{}: Exchangeability; Non commutative probability and statistics; $C^{*}$–algebras; States; Applications to Quantum Physics.
address:
- |
Vitonofrio Crismale\
Dipartimento di Matematica\
Università degli studi di Bari\
Via E. Orabona, 4, 70125 Bari, Italy
- |
Francesco Fidaleo\
Dipartimento di Matematica\
Università di Roma Tor Vergata\
Via della Ricerca Scientifica 1, Roma 00133, Italy
author:
- Vitonofrio Crismale
- Francesco Fidaleo
title: exchangeable stochastic processes and symmetric states in quantum probability
---
introduction {#sec1}
============
The study of random systems with distributional symmetries, started by De Finetti in [@DeF] for sequences of $2$–point valued exchangeable random variables, has known, throughout the years, an increasing attention in many branches of Mathematics and Physics, especially in Probability Theory, Operator Algebras, Quantum Information Theory and Entanglement. In particular, characterizing systems of exchangeable, or symmetrically dependent, random variables is a problem of major interest, since generally there appear strong and useful relations with independence. Some general answers to this problem were achieved in Probability Theory. For instance, in [@HS] it was shown the following generalization of De Finetti Theorem: infinite sequences of exchangeable random variables distributed on $X=E\times E\times\cdots$, $E$ being a compact Hausdorff space, are mixtures of independent identically distributed random variables. The case of finite sequences, useful for the applications, was considered in [@DF], whereas in [@DF2] the authors succeeded to characterize the so–called partially exchangeable sequences as mixtures of Markov chains, under the hypothesis of recurrence. These results were the source for many extensions to the non commutative setting. In particular, in [@St2] De Finetti–Hewitt–Savage Theorem was firstly generalized to the infinite tensor product $C^*$–algebras by showing that the symmetric states are mixture of extremal ones, the last consisting of infinite products of a single state. Moreover, some general properties of exchangeable stochastic processes based on a continuous index–set were studied, see e.g. [@AL]. We also mention the investigation the analogous of De Finetti Theorem in a setting involving quantum symmetries, see e.g. [@K] and the references cited therein.
Recently, it was shown in [@CrF] that de Finetti Theorem still holds for the Fermi $C^*$–algebra based on the Canonical Anticommutation Relations (CAR for short). More precisely, the convex compact set of symmetric states states on the CAR algebra, corresponding to exchangeable stochastic processes involving Fermions, is indeed a Choquet simplex, where the extremal (i.e. ergodic with respect to the action of all the finite permutations of indices) ones are precisely the Araki–Moriya products (cf. [@AM2]) of a single, necessarily even, state. Thus, any symmetric state is a mixture of product states by a unique barycentric measure.
As a consequence of such results, it appears now natural to address the systematic investigation of the structure of exchangeable stochastic processes in quantum setting. Our starting point in the present paper is to establish the perhaps expected following facts. We show that there is a one–to–one correspondence between quantum stochastic processes based on a $C^*$–algebra $\ga$ and states on the free product $C^*$–algebra of the same algebra $\ga$. When $\ga$ has the unity and the stochastic process under consideration has the identity, we have to consider the free product $C^*$–algebra in the category of the unital $C^*$–algebras. We also show that the exchangeable processes correspond precisely to the states on the free product which are invariant with respect the action of all the permutations moving only a finite number of indices. In addition, as the unital free product $C^*$–algebra is a quotient of the free product obtained, in a natural way, by forgetting the identity of $\ga$, such a one–to–one correspondence passes to the quotient. This approach, based on the universal property of free product $C^*$–algebra, can be applied to several remarkable examples. Thus, quite naturally in many cases of interest, the investigation of stochastic processes can be achieved directly on “concrete” $C^*$–algebras, seen as the quotient. This is the case of the infinite tensor product and the CAR algebras, both useful for applications in Quantum Statistical Mechanics, as well as the classical (i.e. commutative) case, covered by considering directly the free Abelian product (which corresponds to the infinite tensor product of a single Abelian $C^*$–algebra). We also mention the cases of interest in Free Probability, that is processes on the concrete $C^*$–algebras describing the $q$–deformed Commutation Relations, where $q=0$ corresponds to the reduced group $C^*$–algebra of the free group on infinitely many generators, or, more generally, processes on the whole free group $C^*$–algebra.
Those preliminary results clarify us the following considerations. Even if, for our purposes it appears completely natural to study symmetric states on the free product $C^*$–algebra, the wide generality of this structure makes almost impossible to provide too general results. For most of the cases relevant for applications, it will be enough and potentially more useful, to study the properties of invariance under the natural action of the group of permutations, of the stochastic processes directly on the quotient algebra. On the other hand, it would be nevertheless of some interest the investigation of the structure and/or the properties of some relevant classes of states naturally arising in Free Probability. Among them, we mention the class of the so–called Haagerup states as a pivotal example, see e.g. [@AHO; @Ha].
In the present paper we aim to cover these topics. More in detail, after a preliminary section containing the notations and results useful in the sequel, in Section \[sec3a\] we briefly describe the free product of a $C^*$–algebra in the category of non unital and unital $C^*$–algebras and prove that the action of the group of permutations can be really extended to both cases so that it is compatible with the passage to the quotient. Then we recall the definition of a quantum stochastic process, and in Theorem \[eqstost\] we establish that assigning a class of unitarily equivalent quantum stochastic processes on a $C^*$–algebra $\ga$ indexed by the set $J$, is equivalent to give a state on its free product $C^*$–algebra $\gb_{\ga,J}$. In this picture, the exchangeable stochastic processes are in one–to–one correspondence with symmetric states on such algebra. Passing to the unital case, it is also shown that exchangeable identity preserving stochastic processes on unital $C^*$–algebra $\ga$ indexed by $J$, correspond to symmetric states on the free product $C^*$–algebra $\gc_{\ga,J}$ in the category of unital $C^*$–algebras. Moreover, using the universal property of $\gb_\ga$, the ergodic (i.e. extremal) properties of symmetric states on it can be exploited for studying the structure of exchangeable stochastic processes in some “concrete” $C^*$–algebras (cf. Remark \[refact\]), as described below. We end this part relative to general properties of exchangeable stochastic processes with Theorem \[frecazo\] in Section \[secerg\]. Namely, we show that some algebraic properties, such as that to satisfy the product state or block–singleton conditions, are equivalent to some natural ergodic properties enjoyed by exchangeable stochastic processes. In addition, this result yields some general considerations (cf. the comments at the end of Sections \[secerg\], and \[sec4\]) about the boundary of symmetric states, whose structure does appear extremely complex. This circumstance changes if one takes special $C^*$–algebras as a starting point for the investigation of properties of (exchangeable) stochastic processes. Indeed in Section \[sec3\] we show that, for Fermion algebra, a state is symmetric if and only if the corresponding process is conditionally independent and identically distributed with respect to the tail algebra, known in Statistical Mechanics as the algebra at infinity. As in the commutative case, such a result entails, for a state on the CAR algebra, the equivalence among invariance under the action of the group of permutations, the fact that it is a mixture of independent and identically distributed product states (cf. Theorem 5.5 of [@CrF]), and the property to be conditionally independent and identically distributed with respect to the tail algebra. This appears as the first case in which de Finetti Theorem, in the form including also its conditional version, is fully extended to a non commutative $C^*$–algebra. Moreover the equivalences above are inherited in the case of infinite tensor product algebra (see Remark \[inften\]), whereas they fail in the general non commutative setting, see e.g. [@K] for details. Section \[sec4\] deals with the concrete $C^*$–algebra generated by $q$–Canonical Commutation Relations for $-1<q<1$. The CAR case corresponds to $q=-1$, and the reduced group $C^*$–algebra $C^*_r(\bbf_\infty)$ of the free group on infinitely many generators is described by the case $q=0$. We prove that the set of the symmetric states on all these algebras (i.e. for $-1<q<1$) including $C^*_r(\bbf_\infty)$, reduces to a singleton. We show in Section \[sec5\] that the same is essentially true for the case arising from the Boolean Commutation Relations. Contrarily to the classical situation, the Boolean case clarifies that the formulation of the condition to be independent and identically distributed w.r.t. the tail algebra without mentioning the [*a–priori*]{} existence of a preserving conditional expectation, is meaningless in quantum case. Section \[sec4\] ends with the study of the ergodic properties of a class of remarkable states in Free Probability, that is the Haagerup states (see e.g. Corollary 3.2 of [@Ha] for the definition). Such states, defined on the whole group algebra $C^*(\bbf_\infty)$ and symmetric by definition, are shown to be ergodic or, equivalently, extremal (even if it is expected that they do not fill all the extreme boundary), but their support in the bidual algebra $C^*(\bbf_\infty)^{**}$ does not belong to the center (i.e. they do not generate any natural KMS dynamics on the von Neumann algebra generated by the GNS representation), except for the tracial case. These results are achieved by using the above cited Theorem \[frecazo\].
preliminaries and notations {#sec2}
===========================
Throughout the section we will present and recall some known definitions and notations useful in the sequel.
Let $J$ be an arbitrary set, and $\ga$ a $C^*$–algebra. Take a family $\{\ga_j\}_{j\in J}\subset\ga$ of $C^*$–subalgebras. With $\alg\{\ga_j\mid j\in J\}$, we denote its $*$–algebraic span in the ambient algebra $\ga$. Let $I_k\subset J$, $k=1,2,3$ be finite subsets.
The state $\f\in\cs(\ga)$ is said to satisfy the *product state condition* (see e.g. [@AM2]) if $$\f(A_1A_2)=\f(A_1)\f(A_2)\,,$$ whenever $A_k\in\alg\{\ga_{j_k}\mid j_k\in I_k\}$, $k=1,2$, and $\quad I_1\cap I_2=\emptyset$.
The state $\f$ satisfies the [*block–singleton condition*]{} (cf. [@ABCL], Definition 2.2) if $$\f(A_1A_2A_3)=\f(A_1A_3)\f(A_2)\,,$$ whenever $A_k\in\alg\{\ga_{j_k}\mid j_k\in I_k\}$, $k=1,2,3$, and $(I_1\cup I_3)\cap I_2=\emptyset$.
Suppose that $\{M_j\mid j\in J\}$ are von Neumann algebras acting on the same Hilbert space $\ch$. We denote with $\bigvee_{j\in J} M_j:=\big(\bigcup_{j\in J} M_j\big)''$ the von Neumann algebra generated by the $M_j$.
A group $G$ is said to act as a group of automorphisms of $\ga$ if there exists a representation $\a: g\in G\rightarrow \a_g\in\aut(\ga)$. We denote by $(\ga, G)$ this circumstance. A state $\f\in\cs(\ga)$ is called $G$–invariant if $\f=\f\, \circ \a_g$ for each $g\in G$. The subset of the $G$–invariant states is denoted by $\cs_G(\ga)$. If $\ga$ is unital, it is $*$–weakly closed and its extremal points are called [*ergodic*]{} states. For $(\ga, G)$ as above, and an invariant state $\f$ on $\ga$, $(\pi_\f,\ch_\f,U_\f,\Omega_\f)$ is the GNS covariant quadruple canonically associated to $\f$ (see e.g. [@BR1; @T1]). If $(\pi_\f,\ch_\f,\Omega_\f)$ is the GNS triple associated to $\f$, the unitary representation $U_\f$ of $G$ on $\ch_\f$ is uniquely determined by $$\begin{aligned}
&\pi_\f(\a_g(A))=U_\f(g)\pi_\f(A)U_\f(g)^{-1}\,,\\
&U_\f(g)\Om_\f=\Om_\f\,,\quad A\in\ga\,, g\in G\,.\end{aligned}$$ If $\gz_\f:=\pi_\f(\ga)''\bigwedge\pi_\f(\ga)'$ is the center of $\pi_\f(\ga)''$, $\gb_G(\f)$ denotes its fixed point algebra under the adjoint action $\ad(U_\f)$ of $G$, i.e. $$\gb_G(\f):=\gz_\f\bigwedge \{U_\f(G)\}'$$ In addition, let $s(\f)$ be the support of $\f$ in the bidual $\ga^{**}$. Then $s(\f)\in Z(\ga^{**})$ if and only if $\Om_\f$ is separating for $\pi_\f(\ga)''$, $Z(\gb)$ being the center of any algebra $\gb$ (see [@SZ], Section 10.17). The invariant state $\f\in\cs_G(\ga)$ is said to be $G$–[*Abelian*]{} if all the operators $E_\f\pi_\f(\ga)E_\f$ mutually commute. The $C^*$–dynamical system $(\ga,G)$ is $G$–Abelian if $\f$ is $G$–abelian for each $\f\in\cs_G(\ga)$.
The group of permutations of $J$, $\bp_J:=\bigcup\{\bp_I|I\subseteq J\, \text{finite}\}$ is given by the permutations leaving fixed all the elements in $J$ but a finite number of them. If $J$ is countable, we sometimes denote $\bp_J$ simply as $\bp_\infty$. If $\bp_J$ acts as a group of automorphisms on the $C^*$–algebra $\ga$, $\f\in\cs(\ga)$ is called [*symmetric*]{} if it is $\bp_J$–invariant. Following the notation introduced above, in the unital case $\cs_{\bp_J}(\ga)$ and $\ce\left(\cs_{\bp_J}(\ga)\right)$ denote respectively the convex closed subset of all the symmetric states of $\ga$, and the ergodic ones. Let $M$ be the Cesaro Mean w.r.t. $\bp_J$, given for a generic object $f(g)$ by $$M\{f(g)\}:=\lim_{I\uparrow J}\frac1{|\bp_I|}\sum_{g\in \bp_I}f(g)\,,$$ provided the l.h.s. exists in the appropriate sense, and $I\subset J$ runs over all the finite parts of $J$. The state $\f\in \cs_{\bp_J}(\ga)$ is called *weakly clustering* if $$M\{\f(\a_g(A)B)\}=\f(A)\f(B)\,,\quad A,B\in\ga,\,g\in\bp_J\,.$$ In unital case, any weakly clustering state is ergodic and the converse holds true if $\f$ is $\bp_J$–Abelian, see e.g. Proposition 3.1.12 of [@S].
A [*conditional expectation*]{} $E: \ga\rightarrow \gb$ between $C^*$–algebras $\ga$, $\gb$ is a norm–one linear projection of $\ga$ onto $\gb$. If in addition it preserves the state $\f\in\cs(\ga)$: $\f\circ E=\f$, then $E$ is called a $\f$–[*preserving*]{} conditional expectation. The reader is referred to Chapter II of [@Sr] or Section IX of [@T1], and the references cited therein, for the main properties and the existence conditions of conditional expectations.
Finally, we report the following Lemma useful in the sequel. Its immediate proof is omitted.
\[permtra\] Consider a finite interval $J=[k,l]\subset\bz$. Then there exists a cycle $\g\in\bp_\bz$ such that $[k+1,l+1]=\g(J)$.
exchangeable stochastic processes {#sec3a}
=================================
In order to define exchangeable stochastic processes, we preliminary and briefly describe free products of a single $C^{*}$–algebra $\ga$ in the categories of $C^{*}$–algebras and unital $C^{*}$–algebras, provided that $\ga$ has a unit $\idd$ for the latter. Indeed, if $J$ is an index set, the algebraic free product $\gb^{(0)}_{\ga,J}\equiv\gb^{(0)}$ in the category of the ${*}$–algebras is given, as a vector space, by $$\gb^{(0)}:=\bigoplus_{n\geq1}\left(\bigoplus_{i_1\neq i_2\neq\cdots\neq i_n}
V_{i_1}\otimes V_{i_2}\otimes\cdots\otimes V_{i_n}\right)\,,$$ where $V_{i}=\ga$, $i\in J$. Notice that the indices $i_1, i_2,\dots,i_n$ will appear possibly more than once in the r.h.s.. The adjoint and the product in $\gb^{(0)}$ are defined in the usual way. In fact, let $v=A_{i_1}\otimes A_{i_2}\cdots\otimes A_{i_m}$, $w=B_{j_1}\otimes B_{j_2}\cdots\otimes B_{j_n}$ be two reduced words of length $m$ and $n$, respectively.[^1] Then $v^*:=A^*_{i_m}\otimes A^*_{i_2}\cdots\otimes A^*_{i_1}$, and $$vw:=\left\{
\begin{array}{ll}A_{i_1}\otimes A_{i_2}\cdots\otimes A_{i_m}
B_{j_1}\otimes B_{j_2}\cdots\otimes B_{j_n}\,,\,\qquad i_m=j_1\,,\\
A_{i_1}\otimes A_{i_2}\cdots\otimes A_{i_m}\otimes
B_{j_1}\otimes B_{j_2}\cdots\otimes B_{j_n}\,,\quad i_m\neq j_1\,.
\end{array} \right.$$ It is straightforwardly seen that the product is well defined. Indeed, let $B\in\ga\subset\ca$ where $\ca$ is the $C^*$–algebra obtained by $\ga$ by adding a unit $\idd$. We have, for $i_1\neq i_2\neq\cdots\neq i_n$, $$\begin{aligned}
\label{bendef}
&\sum_lA^{(l)}_{i_1}\otimes A^{(l)}_{i_2}\cdots\otimes A^{(l)}_{i_m}B\\
=\bigg(\sum_lA^{(l)}_{i_1}&\otimes A^{(l)}_{i_2}\cdots\otimes A^{(l)}_{i_m}\bigg)
(\idd\otimes\idd\otimes\cdots\otimes B)\,,{\nonumber}\end{aligned}$$ where $l$ runs over a finite set. Thus, the adjoint and product defined above extend by linearity on the whole $\gb^{(0)}$. Notice that $\gb^{(0)}$ does not have the unit even if $\ga$ has.
The algebraic free product $\gc_{\ga,J}^{(0)}\equiv\gc^{(0)}$ in the category of the unital ${*}$–algebras is given as a vector space, $$\gc^{(0)}:=\bc\idd\oplus\bigoplus_{n\geq1}\left(\bigoplus_{i_1\neq i_2\neq\cdots\neq i_n}
W_{i_1}\otimes W_{i_1}\otimes\cdots\otimes W_{i_n}\right)\,,$$ after writing $\ga=\bc\idd\oplus W$ as a vector space (cf. Lemma 4.21 of [@Rud]), and $W_i=W$, $i\in J$. Also in this situation the indices $i_1, i_2,\dots,i_n$ will appear possibly more than once in the r.h.s.. The $*$–operation works as in $\gb^{(0)}$, whereas the definition of the product has to be done more carefully. The identity (i.e. the zero degree word) is the neutral element for it. Concerning the other situation relative to homogeneous words $v$, $w$ of length $n,m>0$ as above (with elementary tensors belonging to $W$), we get the same result of the non unital case when $i_m\neq j_1$, i.e. $$vw:=A_{i_1}\otimes A_{i_2}\cdots\otimes A_{i_m}\otimes
B_{j_1}\otimes B_{j_2}\cdots\otimes B_{j_n}\,.$$ If $i_m= j_1$ we uniquely write $
A_{i_m}B_{j_1}=\a\idd+\b C$, where $\a,\b\in\bc$ and $C\in W$. Then $$\begin{aligned}
vw:=
&\a A_{i_1}\otimes A_{i_2}\cdots\otimes A_{i_{m-1}}\otimes B_{j_2}\cdots\otimes B_{j_n}\\
+&\b A_{i_1}\otimes A_{i_2}\cdots\otimes C
\otimes B_{j_2}\cdots\otimes B_{j_n}\,.\end{aligned}$$ In this case, holds true without adding any other identity to $\ga$. Then the product is again well defined.
The free product $*$–algebras $\gb^{(0)}$ and $\gc^{(0)}$ are the universal algebras making commutative the following diagrams $$\label{unni}
\xymatrix{ \ga \ar[r]^{i_j} \ar[d]_{\F_{j}} &
\gb^{(0)} \ar[dl]^\F \\
B}\qquad
\xymatrix{ \ga \ar[r]^{i_j} \ar[d]_{\F_{j}} &
\gc^{(0)} \ar[dl]^\F \\
C}\qquad j\in J\,.$$ Here, for $j\in J$, $i_j$ is the canonical embedding of $\ga$ into $\gb^{(0)}$, or into $\gc^{(0)}$ in unital case, $B$ and $C$ are arbitrary $*$–algebras with $C$ unital, and $\F$, $\F_j$ $*$–homomorphisms preserving the corresponding identities in the unital case. This simply means that $\F$ is uniquely determined by the $\F_j$. Thanks to the universal character of the free product algebra, the construction of $\gc^{(0)}$ does not depend on the splitting $\ga=\bc\idd\oplus W$, up to isomorphisms. >From now on, if $A_j\in V_{i_j}$ ($A_j\in W_{i_j}$ in the unital case), we will frequently use the identification $\ga_j\sim i_j(\ga)$ without further mention. It is not difficult to check that, if $\ga$ is unital, there is a natural quotient map (i.e. a $*$-epimorphism) $$\r^{(0)}:\gb^{(0)}\to\gc^{(0)}\,.$$ Indeed, since $$\ga=V_i=\bc\idd+W_i\sim\bc\idd\oplus W_i\,,\quad i\in J\,,$$ such a map is induced at all the levels of the tensor products by $$A\in\ga\longmapsto a\idd\oplus(A-a\idd)\in\bc\idd\oplus W$$ whereas $A=a\idd+(A- a\idd)$. For example, if $A\in\gb^{(0)}$ is a degree 1 element, then $A=a\idd+(A-a\idd)$, and $$\r^{(0)}(A)=a\idd\oplus(A-a\idd)\,.$$ At degree 2 level, $A=A_1\otimes A_2\in V_{i_1}\otimes V_{i_2}$, with $A_j=a_j\idd+(A_j-a_j\idd)$, $j=1,2$. Then $$\begin{aligned}
&\r^{(0)}(A_1\otimes A_2)=a_1a_2\idd\oplus a_2(A_1-a_1\idd)\oplus(a_1(A_2-a_2\idd)\\
\oplus&(A_1-a_1\idd)\otimes(A_2-a_2\idd)\in\bc\idd\oplus W_{i_1} \oplus W_{i_2}
\oplus W_{i_1}\otimes W_{i_2}\,.\end{aligned}$$ The highest level formulas can be obtained in similar way, and $\r^{(0)}$ extends on $\gb^{(0)}$ by linearity. By using the previous definitions of product and adjoint, it is also almost immediate to show that, for $v,w\in\gb^{(0)}$, $\r^{(0)}(v^*)=\r^{(0)}(v)^*$ and $\r^{(0)}(vw)=\r^{(0)}(v)\r^{(0)}(w)$. Namely, $\r^{(0)}$ is a surjective linear $*$–map preserving the algebraic structure, i.e. a $*$–epimorphism.[^2]
As $g(j_{1})\neq g(j_{2})\neq\cdots\neq g(j_{n})$ if $j_1\neq j_2\neq\cdots\neq j_n$, $j_h\in J$, $g\in\bp_J$, the permutation group $\bp_J$ acts in a natural way on $\gb^{(0)}$, and $\gc^{(0)}$ in the unital case, by means of the algebraic morphisms $g\in\bp_J\mapsto\b^{(0)}_g$ and $g\in\bp_J\mapsto\g^{(0)}_g$, respectively. When $\ga$ is unital, we easily get $$\label{ekv}
\r^{(0)}\circ\b^{(0)}_g=\g^{(0)}_g\circ\r^{(0)}\,,\quad g\in\bp_J\,.$$ Let $X$ be a generic element of $\gb^{(0)}$, or $\gc^{(0)}$ in the unital case. Define on $\gb^{(0)}$ and $\gc^{(0)}$, the extended–value seminorm $$\label{semm}
\|X\|:=\sup\{\|\pi(X)\|\mid \pi\,\text{is a $*$--representation}\,\}\,.$$ By definition, for $n=0$ in the unital case, $X$ is a multiple of the identity, whereas for $n\geq1$, it is a finite combinations of words $w_n$ of length $n$ which have the forms $w_n=A_{1}\otimes A_{2}\cdots\otimes A_{n}$, where $A_j\in V_{i_j}$ ($W_{i_j}$ respectively) for any choice of indices $i_1\neq i_2\neq\dots\neq i_n$. The elementary computation $\|w_n\|\leq \prod_{j}\|A_{j}\|$ yields that the extended–valued seminorm is effectively a seminorm. It is known that is indeed a norm (see pag. 286 of [@CES] for $\gb^{(0)}$, or Proposition 2.3 in [@Av] for $\gc^{(0)}$ when $\ga$ is $\s$–finite), but we do not use this fact in the sequel. It is seen (cf. [@Av; @CES]) that the enveloping $C^{*}$–algebras $\gb_{\ga,J}\equiv\gb$ and $\gc_{\ga,J}\equiv\gc$, of $\gb^{(0)}$ and $\gc^{(0)}$ respectively, are precisely the universal [*free product $C^{*}$–algebra*]{} and [*unital free product $C^{*}$–algebra*]{} $\gc$ making commutative the following diagrams, analogous to , $$\xymatrix{ \ga \ar[r]^{i_j} \ar[d]_{\F_{j}} &
\gb \ar[dl]^\F \\
B}\qquad
\xymatrix{ \ga \ar[r]^{i_j} \ar[d]_{\F_{j}} &
\gc \ar[dl]^\F \\
C}\qquad j\in J\,.$$ Here, $B$ is any $C^{*}$–algebra, $C$ any unital $C^{*}$–algebra, and the involved homomorphisms preserve the identities in the unital case.
The algebraic actions $\b^{(0)}_g$ and $\g^{(0)}_g$ uniquely extend to actions of the permutation group $$g\in\bp_J\mapsto\b_g\in\aut(\gb)\,,\quad g\in\bp_J\mapsto\g_g\in\aut(\gc)$$ on the free $C^{*}$–algebras $\gb$ and $\gc$, respectively.
In addition, the projection map $\r^{(0)}$ described above uniquely extends to a $C^{*}$–epimorphism $\r:\gb\to\gc$ from the free product $C^{*}$–algebra $\gb$ onto the unital free product $C^{*}$–algebra $\gc$ fulfilling, for $\ga$ unital, $$\label{ekv1}
\r\circ\b_g=\g_g\circ\r\,,\quad g\in\bp_J\,.$$
Fix $g\in\bp_J$ and define $\cb$ and $\cc$ the pre–$C^{*}$–algebras obtained by taking quotient of $\gb^{(0)}$ and $\gc^{(0)}$ respectively, with the ideal made of all the elements for which the seminorm vanishes. Since $\b^{(0)}_g$ and $\g^{(0)}_g$ are one–to–one norm preserving maps on the dense subsets $\cb$ and $\cc$ respectively, they uniquely extend to automorphisms of the corresponding the enveloping $C^*$–algebras. Moreover, by the universal property of $\gb$, $\r^{(0)}$ extends to a homomorphism $\r$ into $\gc$, whose range contains $\cc$. By Corollary I.8.2 of [@T1], it induces a $*$–isomorphism of the quotient $C^*$–algebra onto the range of $\r$. Thus $\r$ is a $C^{*}$–epimorphism and it satisfies , by taking into account the corresponding property .
Now we pass to recall the definition of a quantum stochastic process.
\[dqspi\] A [*stochastic process*]{} labelled by the index set $J$ is a quadruple $\big(\ga,\ch,\{\iota_j\}_{j\in J},\Om\big)$, where $\ga$ is a $C^{*}$–algebra, $\ch$ is an Hilbert space, the $\iota_j$’s are $*$–homomorphisms of $\ga$ in $\cb(\ch)$, and $\Om\in\ch$ is a unit vector, cyclic for the von Neumann algebra $M:=\bigvee_{j\in J}\iota_j(\ga)$ naturally acting on $\ch$.
The process is said to be [*exchangeable*]{} if, for each $g\in\bp_J$, $n\in\bn$, $j_1,\ldots j_n\in J$, $A_1,\ldots A_n\in\ga$ $$\langle\iota_{j_1}(A_1)\cdots\iota_{j_n}(A_n)\Om,\Om\rangle
=\langle\iota_{g(j_{1})}(A_1)\cdots\iota_{g(j_{n})}(A_n)\Om,\Om\rangle.$$ It is said to be [*unital*]{} if $\iota_{j}(\idd)=I$, $j\in J$, provided that $\ga$ has the unit $\idd$.
Two stochastic processes $\big(\ga,\ch_i,\{\iota^{(i)}_j\}_{j\in J},\Om_i\big)$, $i=1,2$ based on the same $C^{*}$–algebra $\ga$ and the same index set $J$, are said [*(unitarily) equivalent*]{} if there exists a unitary operator $V:\ch_1\to \ch_2$ such that $V\Om_1=\Om_2$, $V\iota^{(1)}_j(A)V^*=\iota^{(2)}_j(A)$, $a\in\ga$, $j\in J$.
It is worth noticing that the first definition of stochastic processes for the quantum case was given in [@AFL], where the reader is referred for further details and comparison with classical stochastic processes in the sense of Doob. Moreover, in literature one can find a slightly different definition (see, e.g. [@AFL]). Namely, the assignments of $\ch$ and $\Om$ are replaced by an arbitrary unital $C^{*}$–algebra $\gg$ for which $\bigvee_{j\in J}\iota_j(\ga)$ is dense, and a state $\f$ on $\gg$ (compare with Remark \[refact\] below). Then the equivalence condition for two processes on $\ga$ indexed by the same $J$ can be expressed in terms of mixed moments agreement of the two states (see [@AFL], Proposition 1.1).[^3] By using the GNS construction, it is straightforward to see the equivalence of the approaches (see. e.g. [@Sch], Section 1.2), and a quantum stochastic process defined as in Definition \[dqspi\] is said to be in the [*canonical form*]{}. Definition \[dqspi\] is given only for discrete index but it can be easily generalized to other situations (cf. [@AL]). Here, we deal only with stochastic processes where the index set $J$ is discrete, that is $J\sim\mathbb{N}$ or $\bz$ in countable situation.
Let a stochastic process be given. We introduce the linear forms $\f^{(0)}$ and $\psi^{(0)}$, and the $*$–representations $\pi^{(0)}$ and $\s^{(0)}$ on the Hilbert space $\ch$, respectively of $\gb^{(0)}$, and $\gc^{(0)}$ in the unital case. We firstly take $\psi^{(0)}(\idd):=1$, $\s^{(0)}(\idd):=I$ and define, on the linear generators of $\gb^{(0)}$ and $\gc^{(0)}$ respectively, $$\begin{aligned}
\f^{(0)}(A_{1}\otimes A_{2}\cdots\otimes A_{n})
:=&\langle\iota_{j_1}(A_1)\cdots\iota_{j_n}(A_n)\Om,\Om\rangle\,,\\
\psi^{(0)}(A_{1}\otimes A_{2}\cdots\otimes A_{n})
:=&\langle\iota_{j_1}(A_1)\cdots\iota_{j_n}(A_n)\Om,\Om\rangle\,,\end{aligned}$$ $$\begin{aligned}
\pi^{(0)}(A_{1}\otimes A_{2}\cdots\otimes A_{n})
:=&\iota_{j_1}(A_1)\cdots\iota_{j_n}(A_n)\,,\\
\s^{(0)}(A_{1}\otimes A_{2}\cdots\otimes A_{n})
:=&\iota_{j_1}(A_1)\cdots\iota_{j_n}(A_n)\,,\end{aligned}$$ where $n\in \mathbb{N}$, $A_k\in V_{j_k}$ ($W_{j_k}$ respectively), $j_k\in J$ and $j_1\neq j_2\neq\cdots\neq j_n$. The following relations hold, provided that the objects under consideration are well–defined (i.e. the process is unital), $$\f^{(0)}(B)=\langle\pi^{(0)}(B)\Om,\Om\rangle,\, B\in\gb^{(0)},\quad
\psi^{(0)}(C)=\langle\s^{(0)}(C)\Om,\Om\rangle,\, C\in\gc^{(0)}$$ In addition, in the unital case and/or when the process is exchangeable, $$\begin{aligned}
\label{untl}
&\f^{(0)}=\psi^{(0)}\circ\r^{(0)}\,,\quad \pi^{(0)}=\s^{(0)}\circ\r^{(0)}\,;\\
&\f^{(0)}=\f^{(0)}\circ\b_g^{(0)}\,,\quad \psi^{(0)}=\psi^{(0)}\circ\b_g^{(0)}\,,\,\,\,g\in\bp_J\,.{\nonumber}\end{aligned}$$
\[strep\] Fix a stochastic process $\big(\ga,\ch,\{\iota_j\}_{j\in J},\Om\big)$. Then the linear forms and the $*$-representations $\f^{(0)}$, $\pi^{(0)}$ of $\gb^{(0)}$, and $\psi^{(0)}$, $\s^{(0)}$ of $\gc^{(0)}$ in the unital case, uniquely extend to states and representations $\f$, $\pi$ of $\gb$, and $\psi$, $\s$ of $\gc$ in the unital case, satisfying the analogues of when the process is unital and/or exchangeable.
The representations $(\ch, \pi, \Om)$, and $(\ch, \s, \Om)$ in the unital case, are the GNS representations for the states $\f\in\cs(\gb)$ and $\psi\in\cs(\gc)$, respectively.
Consider the diagrams $$\xymatrix{ \ga \ar[r]^{i_j} \ar[d]_{\iota_{j}} &
\gb \ar[dl]^\F \\
M}\qquad
\xymatrix{ \ga \ar[r]^{i_j} \ar[d]_{\iota_{j}} &
\gc \ar[dl]^\F \\
M}\qquad j\in J\,,$$ where, for each $j\in J$, the $i_j$’s are the canonical embeddings of $\ga$ in $\gb$, or in $\gc$ in the unital case. By the universal property, the morphisms $\F$ making commutative the above diagrams are nothing but the extensions of $\pi^{(0)}$, and $\s^{(0)}$ in the unital case, to representations $$\pi:\gb\to M\subset\cb(\ch)\,,\quad \s:\gc\to M\subset\cb(\ch)\,.$$ Such representations are nondegenerate since $\Om$ is cyclic. Thanks to , it is almost immediate to verify that, when the process is unital, $\pi$ factors through $\s$ by the epimorphism $\r$, that is $\pi=\s\circ\r$.
Concerning the functionals $\f$ and $\psi$, if $A\in\gb$ or $A\in\gc$, we get $\f(A)=\langle\pi(A)\Om,\Om\rangle$, $\psi(A)=\langle\s(A)\Om,\Om\rangle$ in the unital case. Then they are positive and $\psi$ is automatically a state, as $\psi(\idd)=1$. Since $\Om$ is cyclic, $\s$ is nothing else than the GNS representation of $\psi$. Moreover, for $\f$, we immediately get $\|\f\|\leq1$. Concerning the non unital case, fix an approximate identity $\{u_a\}\subset\gb$ which always exists, see e.g. [@T1], Corollary I.7.5. As $\pi$ is nondegenerate, we have $\sup_a\pi(u_a)=I$. Thus, $$\|\f\|=\sup_a\f(u_a)=\sup_a\langle\pi(u_a)\Om,\Om\rangle
=\big\langle\big(\sup_a\pi(u_a)\big)\Om,\Om\big\rangle=1\,.$$ Thus, also in this case, $\f$ is a state with GNS representation given by $\pi$. It is matter of routine to check all the remaining properties.
The next crucial result gives that a stochastic process on a $C^*$–algebra and a state on its free product $C^{*}$–algebra are uniquely determined each other. In fact, we have the following
\[eqstost\] There is a one–to–one correspondence between equivalence classes of stochastic processes on $\ga$ and states on its free product $C^{*}$–algebra $\gb$ which factorize through the unital $C^{*}$–algebra $\gc$, provided that $\ga$ has the unit and the processes are unital. A class is given by exchangeable stochastic process if and only if the corresponding state is symmetric.
Fix a stochastic process $\big(\ga,\ch,\{\iota_j\}_{j\in J},\Om\big)$ on the $C^{*}$–algebra $\ga$. From Theorem \[strep\], it follows that there corresponds a unique state $\f$ on $\gb$ with GNS representation $(\ch, \pi, \Om)$ and, in the unital case, a unique state $\psi$ on $\gc$ with GNS representation $(\ch, \s, \Om)$, linked by the required factorization property. Moreover, a process unitarily equivalent to the given one gives rise to the same state, since it determines $*$–representations on $\ch$ which are unitarily equivalent to $\pi$ or $\s$. The proof of the last property is immediate.
Following an established tradition in literature, we generally refer the terms *exchangeable* and *symmetric*, to stochastic processes and states, respectively. We also denote symbolically by $\gf$ the free product universal $C^*$–algebras, that is $\gf$ stands for $\gb$, or $\gc$ in the unital case. Very often, it is more convenient to think of a (class of) stochastic process(es) as a state on a “concrete” $C^{*}$–algebra, rather than on a general free product $C^{*}$–algebra. The idea can be borrowed from the construction of a stochastic process not in canonical form. Indeed, take $\ga$ a $C^{*}$–algebra, equipped with a collection of $*$–homomorphisms $\iota_j:\ga\to\gg$ of $\ga$ into another $C^{*}$–algebra $\gg$ equipped with an action $\a$ of the permutation group $\bp_J$. Suppose further that the algebraic span $\alg\{\iota_j(\ga)\mid j\in J\}$ is dense in $\gg$, and $$\a_g\circ\iota_j=\iota_{g(j)}\,,\quad j\in J\,,\,\,\, g\in \bp_J\,.$$ Due to the universal property of $\gb_{\ga,J}\equiv\gb$, the $C^*$–free product of $\ga$, $\gg$ can be viewed as a quotient of $\gb_{\ga,J}$. Then we have the following
\[refact\] Any state on $\gg$ can be viewed as a class of unitarily equivalent stochastic processes for the algebra $\ga$ (i.e. as a state on $\gb_{\ga,J}$) factoring through $\gg$. The stochastic processes are exchangeable (i.e. the corresponding state on $\gb_{\ga,J}$ is symmetric) if and only if the corresponding state on $\gg$ is symmetric. Analogous considerations can be done for unital algebras, and for the corresponding unital processes.
As examples of “concrete” $\gg$, we mention the Boolean (cf. Section \[sec5\]) and the Canonical Anticommutation Relations (cf. Section \[sec3\]) algebras, for the non unital and unital cases, respectively. Other remarkable examples are those arising from the infinite tensor product, where the classical case is contained, and the concrete algebra corresponding to the $q$–deformed Commutation Relations, including the free group reduced $C^{*}$–algebra as a particular case.
ergodic properties of exchangeable stochastic processes {#secerg}
=======================================================
We start by reporting the noncommutative version of the definition of conditionally independent and identically distributed stochasticc processes, which is useful for our purposes.
Let a stochastic process $\big(\ga,\ch,\{\iota_j\}_{j\in J},\Om\big)$ be given, together with its corresponding state $\f$ on the free product $C^*$–algebra $\gf$ in unital or not unital case, of $\ga$. Define the [*tail algebra*]{} of the process under consideration as $$\label{ttai}
\gz^\perp_\f:=\bigcap_{\begin{subarray}{l}I\subset J,\,
I \text{finite} \end{subarray}}\bigg(\bigcup_{\begin{subarray}{l}K\bigcap I=\emptyset,
\\\,\,\,K \text{finite} \end{subarray}}\bigg(\bigvee_{k\in K}\iota_k(\ga)\bigg)\bigg)''\,.$$ For some applications in the sequel, we provide the definition of conditionally independent and identically distributed process w.r.t. $\gz^\perp_\f$.
\[cocaind\] The stochastic process described by the state $\f\in\cs(\gf)$, is [*conditionally independent and identically distributed*]{} w.r.t. the tail algebra if there exists a conditional expectation $E_\f:\bigvee_{j\in J}\iota_j(\ga)\to \gz^\perp_\f$ preserving the vector state $\langle\,{\bf\cdot}\,\Om_\f,\Om_\f\rangle$ such that,
- $E_\f(XY)=E_\f(X)E_\f(Y)$, for each finite subsets $I,K\subset J$, $I\cap K=\emptyset$, and $$X\in\bigg(\bigvee_{i\in I}\iota_i(\ga)\bigg)\bigvee\gz^\perp_\f\,,
Y\in\bigg(\bigvee_{k\in K}\iota_k(\ga)\bigg)\bigvee\gz^\perp_\f\,;$$
- $E_\f(\iota_i(A))=E_\f(\iota_k(A))$ for each $i,k\in J$ and $A\in\ga$.
The following results link together algebraic and ergodic properties for a symmetric state on the free product $C^*$–algebra. Notice that the second equivalence in Theorem \[frecazo\] below was achieved in Section 3 of [@ABCL], where firstly the relations between block–singleton condition and ergodicity w.r.t. the shift were explored for symmetric faithful states. Here, after considering directly the action of the permutations, we drop faithfulness since, if such condition holds true, or the state is asymptotically Abelian, block singleton and product state conditions are equivalent, see e.g. Proposition 5.1 in [@F22].
\[frecazo\] Consider a symmetric state $\f$ on the free product $C^*$–algebra $\gf$ . The following assertions hold true.
- $\f$ satisfies the product state condition if and only if it is weakly clustering,
- $\f$ is a block–singleton state if and only if $\pi_\f(\gf)''\bigwedge \{U_\f(\bp_J)\}'=\bc I$.
\(i) Suppose that $\f$ satisfies the product state condition. Consider two words $v,w\in\gf$ with support $I_v, I_w$ respectively. If $I$ is a finite part of $J$, define $A:=\{g\in\bp_I\mid I_v\cap I_{\a_g(w)}=\emptyset\}$, where hereafter $\a_g$ denotes $\b_g$, or $\gamma_g$ in the unital case. We get by applying the product state condition, $$\begin{aligned}
\quad\quad\bigg|\frac1{|\bp_I|}\sum_{g\in\bp_I}\f(v\a_g(w))&-\f(v)\f(w)\bigg|\\
\leq\bigg|\frac1{|\bp_I|}\sum_{g\in A}\f(v\a_g(w))- \f(v)\f(w)\bigg|
+&\bigg|\frac1{|\bp_I|}\sum_{g\in\bp_I\backslash A}\f(v\a_g(w))- \f(v)\f(w)\bigg|\\
=\bigg|\frac1{|\bp_I|}\sum_{g\in\bp_I\backslash A}\f(v\a_g(w))-&\f(v)\f(w)\bigg|
\leq2\|v\|\|w\|\frac{|A^c|}{|\bp_I|}\,.\end{aligned}$$ Taking the limit $I\uparrow J$, by Lemma 3.3 of [@CrF] one has that $\frac{|A|}{|\bp_I|}\to 1$, and $\frac{|A^c|}{|\bp_I|}\rightarrow 0$, where $A^c:= \bp_I\backslash A$. Thus $\f$ is weakly clustering.
Suppose now that $\f$ satisfies the weakly clustering condition. Let $v,w$ two words such that for their respective supports $I_v\cap I_w=\emptyset$. Define $B$ the subset in $\bp_J$ leaving pointwise fixed all the elements in $I_v$. We have $|B|=(|I|-|I_v|)!$. Since $\f$ is symmetric, one has $$\begin{aligned}
&\f(vw)=\frac1{(|I|-|I_v|)!}\sum_{g\in B}\f(\a_g(vw))=\frac1{(|I|-|I_v|)!}\sum_{g\in B}\f(v\a_g(w))\\
=&\frac{|I|!}{(|I|-|I_v|)!}\bigg(\frac1{|\bp_I|}\sum_{g\in\bp_I}\f(v\a_g(w))\bigg)
-\frac1{(|I|-|I_v|)!}\sum_{g\in\bp_I\backslash B}\f(v\a_g(w))\,.\end{aligned}$$ Taking the limit $I\uparrow J$, again Lemma 3.3 of [@CrF] gives $\frac{|B|}{|\bp_I|}=\frac{(|I|-|I_v|)!}{|I|!}\to1$, $\frac{|B^c|}{|\bp_I|}\to0$, and consequently $\frac{|B^c|}{|B|}=\frac{|B^c|}{|\bp_I|}\frac{|\bp_I|}{|B|}\to0$, whereas the weakly clustering condition ensures that $$\frac1{|\bp_I|}\sum_{g\in\bp_I}\f(v\a_g(w))\to \f(v)\f(w).$$ Then $|\f(vw)-\f(v)\f(w)|$ is infinitesimal and the product state condition follows.
\(ii) We firstly observe that $\pi_\f(\gf)''\bigwedge \{U_\f(\bp_J)\}'$ is generated by cluster points in the weak operator topology, of the Cauchy net of the Cesaro averages $\bigg\{\frac1{|\bp_I|}\sum_{g\in\bp_I}\pi_\f(\a_g(A))\mid A\in\gf\bigg\}$, where as usual, $I$ is any finite part of $J$. It is almost immediate to show that any of such a limit point is invariant, whereas the reverse inclusion easily follows by Kaplanski Density Theorem. Suppose now that $\f$ is a block–singleton state. Fix three words $u,v,w$ and take $\xi=\pi_\f(u)\Om_\f$, $\eta=\pi_\f(w)^*\Om_\f$. By counting the set $B:=\{g\in I\mid (I_u\cup I_w)\cap I_{\a_g(v)}=\emptyset\}$, and arguing as in the previous part, we get for $I\uparrow J$, $$\frac1{|\bp_I|}\sum_{g\in\bp_I}\langle U_\f(g)\pi_\f(v)U_\f(g)^{-1}\xi,\eta\rangle\longrightarrow
\f(v)\langle\xi,\eta\rangle\,,$$ as a consequence of the block–singleton condition and the fact that $\frac{|B|}{|\bp_I|}\to1$ and $\frac{|B^c|}{|\bp_I|}\to0$ as before. This implies that the Cesaro averages of $\{\pi_\f(\a_g(A))|A\in\gf\}$ converge to $\f(A)I$ in the weak operator topology. As a consequence, $\pi_\f(\gf)''\bigwedge \{U_\f(\bp_J)\}'$ is trivial, since the cluster points of all these possible averages generate the whole algebra.
Suppose now that $\pi_\f(\gf)''\bigwedge \{U_\f(\bp_J)\}'=\bc I$. Fix words $u,v,w$ such that their respective supports satisfy $I_v\cap(I_u\cup I_w)=\emptyset$, and consider the set $B\subset \bp_I$ made by permutations leaving $I_u\cup I_w$ pointwise fixed. Since $\f$ is symmetric, we get $$\begin{aligned}
&\f(uvw)=\frac1{|B|}\sum_{g\in B}\f(\a_g(uvw))=\frac1{|B|}\sum_{g\in B}\f(u\a_g(v)w)\\
=&\frac{|\bp_I|}{|B|}\bigg(\frac1{|\bp_I|}\sum_{g\in\bp_I}\f(u\a_g(v)w)\bigg)
-\frac1{|B|}\sum_{g\in\bp_I\backslash B}\f(u\a_g(v)w)\,.\end{aligned}$$ Consider now any cluster point in the weak operator topology $$\lim_\b\bigg(\frac1{|\bp_{I_\b}|}\sum_{g\in\bp_{I_\b}}U_\f(g)\pi_\f(v)U_\f(g)^{-1}\bigg)=:\G\,,$$ which exists by compactness. We have $\G\in\pi_\f(\gf)''\bigwedge \{U_\f(\bp_J)\}'$, as these cluster points generate such an algebra. Moreover, by assumption $\G=\g I$, for such number $\g\in\bc$ depending on the chosen net. Take $\xi=\pi_\f(u)\Om_\f$, $\eta=\pi_\f(w^{*})\Om_\f$. By usual arguments, $\frac{|B|}{|\bp_I|}\rightarrow 1$ and $\frac{|B^c|}{|B|}\rightarrow 0$, as $I\uparrow J$. Then $$\begin{aligned}
&\left|\f(uvw)-\f(v)\f(uw)\right|\\=&\left|\lim_\b\bigg(\frac1{|\bp_{I_\b}|}\sum_{g\in\bp_{I_\b}}\langle U_\f(g)\pi_\f(v)U_\f(g)^{-1}\xi,\eta\rangle\bigg)-\f(v)\f(uw)\right|\\
=&\left|\g\langle\xi,\eta\rangle-\f(v)\f(uw)\right|=\left|\g\f(uw)-\f(v)\f(uw)\right|.\end{aligned}$$ By choosing $u,w$ as the empty words, we also find $\g=\f(v)$, and the block–singleton condition follows.
We end the present section with the following considerations which distinguish the cases arising from Free Probability from all the remaining cases relevant for the applications.
Let $\f$ be a symmetric state on a free product $C^*$–algebra $\gf$, or equivalently an exchangeable stochastic process on a $C^*$–algebra. If it satisfies the block–singleton condition, then by (ii) of Theorem \[frecazo\] and Theorem 4.3.20 in [@BR1], it is extremal. But a priori, one cannot say that the converse is true. A sufficient condition is that the support of $\f$ in the bidual is central, that is $\Om_\f$ is a separating vector for $\pi_\f(\gf)''$. It is well–known that the last condition cannot be satisfied for many cases relevant for the applications like the so–called ground states (cf. [@BR1]). Suppose now $\f$ merely satisfies the product state condition. It is still extremal by (i) of Theorem \[frecazo\] and Proposition 3.1.10 of [@S]. But, again a priori, the converse is not automatically true. Indeed, a sufficient condition is given by the asymptotic Abelianess or merely the $G$–Abelianess with $G\equiv\bp_J$, of the state under consideration, see Proposition 3.1.12 of [@S]. It is expected that all such sufficient conditions, automatically true in the commutative setting, and true for some cases relevant for the applications like the CAR or the tensor product algebras, are not generally satisfied in Free Probability. As a consequence, the boundary of the convex of symmetric states in the free product $C^*$–algebra, or equivalently exchangeable stochastic processes on a fixed $C^*$–algebra, contains states satisfying the product state condition, but might not generally filled by them.
exchangeable processes on the CAR algebra {#sec3}
=========================================
The section deals with the conditional form of De Finetti–Hewitt–Savage Theorem on the Fermion algebra. Indeed we show that a state on such algebra is symmetric if and only if it is conditionally independent and identically distributed w.r.t the tail algebra. Since in [@CrF] the authors have already characterized the symmetric states on CAR algebra as mixture of products of a single even state, as a result we finally obtain that, being the three conditions above mutually equivalent, there is a perfect similarity with the classical setting, see e.g. [@Ka], Section 1.1.
Let $J$ be an arbitrary set. The [*Canonical Anticommutation Relations*]{} (CAR for short) algebra over $J$ is the $C^{*}$–algebra $\carf(J)$ with the identity $\idd$ generated by the set $\{a_j,
a^{\dagger}_j\mid j\in J\}$ (i.e. the Fermi annihilators and creators respectively), and the relations $$a_{j}^{*}=a^{\dagger}_{j}\,,\,\,\{a^{\dagger}_{j},a_{k}\}=\d_{jk}\idd\,,\,\,
\{a_{j},a_{k}\}=\{a^{\dagger}_{j},a^{\dagger}_{k}\}=0\,,\,\,j,k\in J\,$$ where $\{A,B\}:=AB+BA$ is the anticommutator between $A$ and $B$. The parity automorphism $\Th$ acts on the generators as $$\Th(a_{j})=-a_{j}\,,\,\,\Th(a^{\dagger}_{j})=-a^{\dagger}_{j}\,,\quad
j\in J\,$$ and induces on $\carf(J)$ a $\bz_{2}$–grading, which gives $\carf(J)=\carf(J)_{+} \oplus\carf(J)_{-}$, where $$\begin{aligned}
&\carf(J)_{+}:=\{A\in\carf(J) \ | \ \Th(A)=A\}\,,\\
&\carf(J)_{-}:=\{A\in\carf(J) \ | \ \Th(A)=-A\}\,.\end{aligned}$$ Elements in $\carf(J)_+$ and in $\carf(J)_-$ are called [*even*]{} and [*odd*]{}, respectively. Moreover, $\carf(J)=\overline{\carf{}_0(J)}$, where $$\carf{}_0(J):=\bigcup\{\carf(I)\mid\, I\subset
J\,\text{finite}\,\}$$ is the dense subalgebra of the [*localized elements*]{}. For the details relative to general properties of $\carf(J)$ and the symmetric states on it we refer the reader to [@CrF] and the literature cited therein. We only mention the fact that $\bp_J$ acts as a group of automorphisms on $\carf(J)$ (cf. [@CrF], Prop. 3.2), and each symmetric state is automatically even, that is $\f\circ \Theta=\f$ (cf. [@CrF], Th. 4.1). Moreover, one can see that at least for countable $J$, $$\carf(J)\sim\overline{\bigotimes_{J}\bm_{2}(\bc)}^{C^*}\,.$$ Such an isomorphism is established by a Jordan–Klein–Wigner transformation, as shown in [@T1], Exercise XIV. This transformation, contrarily to the action of the permutation group on the CAR and on tensor product algebras, does not intertwines the local structures. Thus, it cannot be directly used for our purposes.
Consider the state $\f\in\cs_{\bp_J}(\carf(J))$. As it is even, the parity $\Th$ is unitarily implemented on $\ch_\f$ by a unitary $V_\f$ satisfying $V_\f\Om_\f=\Om_\f$ and $\ad(V_\f)(\pi_\f(A))=\pi_\f(\Th(A))$. The [*even*]{} and the [*odd*]{} part of $\pi_\f(\carf(J))''$ are defined in a similar way as before by using $\ad(V_\f)$.
The tail algebra, known in Statistical Mechanics as the [*algebra at infinity*]{}, is defined as in : $$\gz^\perp_\f:=\bigwedge_{\begin{subarray}{l}I\subset J,\,
I \text{finite} \end{subarray}}\left(\bigcup_{\begin{subarray}{l}K\bigcap I=\emptyset,
\\\,\,\,K \text{finite} \end{subarray}}\pi_\f(\carf(K))\right)''\,.$$ In [@Rob] (see also pag. 156 of [@BR1]), it was proven that $$\gz^\perp_\f=\gz_\f\bigwedge\pi_\f(\carf(J)_+)''$$ where as usual, $\gz_\f$ denotes the center. It is easy to see that $\pi_\f(\carf(J)_+)''=\pi_\f(\carf(J))''_+$, so the tail algebra is automatically even: $$\label{veve}
\gz^\perp_\f=\gz_\f\bigwedge\pi_\f(\carf(J))''_+\,.$$ From now on, we take $J$ countable, say $J\equiv\mathbb{N}$ since we need the CAR algebra is separable. We symbolically denote with an abuse of notation $\ga:=\carf(\bn)$, and with $\a$ the action of $\bp_\bn$. We also report the following result (cf. Theorem 5.5 of [@CrF]) for the convenience of the reader, since it will be used in the sequel.
\[csyx\] $\cs_{\bp_{\bn}}(\ga)$ is a Choquet simplex. Then for each $\f\in\cs_{\bp_{\bn}}(\ga)$ there exists a unique Radon probability measure $\m$ on $\cs_{\bp_{\bn}}(\ga)$ such that $$\f(A)=\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\psi(A)\di\m(\psi)\,,\quad A\in\ga\,.$$
We recall that, in Probability Theory, one of the main ingredients to gain the equivalence between exchangeability and conditionally independent and identically distributed condition for infinite sequences of random variables, is the Hewitt–Savage Theorem [@HS]. It states that, for a given exchangeable stochastic process, the symmetric and the tail $\s$–algebras coincide. Here, before proving the main theorem, we firstly check that $$\gb_{\bp_\mathbb{N}}(\f)=\{\gz_\f\bigwedge \{U_\f(\bp_\mathbb{N})\}'\}_+=\gz^\perp_\f\,,$$ that is $\gb_{\bp_\mathbb{N}}(\f)$, the fixed point algebra of the center, is always Abelian and even and coincides with the tail algebra. This result can be seen as a generalization to the Fermi case of the above cited commutative statement, where $\gb_{\bp_\mathbb{N}}(\f)$ and $\gz^\perp_\f$ are the counterparts of the symmetric and tail $\s$–algebras, respectively. Furthermore, it is crucial for our purpose to show that the map $\F_\f:\pi_\f(\ga)''\to\gb_{\bp_\mathbb{N}}(\f)$ given in Theorem 3.1 of [@St1], is precisely the conditional expectation onto the tail algebra preserving the vector state $\langle\,{\bf\cdot}\,\Om_\f,\Om_\f\rangle$. Notice that the existence of such a conditional expectation is not a priori guaranteed if the corresponding state is not a trace.
\[invper\] For each $\f\in\cs_{\bp_\mathbb{N}}(\ga)$, we get $$\gz^\perp_\f\subset\pi_\f(\ga)''\bigwedge U_\f(\bp_\mathbb{N})'\,.$$
For the finite set $I\subset\mathbb{N}$, define $$M_I:=\left(\bigcup_{\begin{subarray}{l}K\bigcap I=\emptyset,
\\\,\,\,K \text{finite} \end{subarray}}\pi_\f(\carf(K))\right)''.$$ If $g\in\bp_I$, then $\ad(U_\f(g))(X)=X$ for each element $X\in M_I$. This implies $\gz^\perp_\f\subset U_\f(\bp_I)'$, for $I\subset \mathbb{N}$, $I$ finite. As the l.h.s. does not depend on $I$, we get $$\gz^\perp_\f\subset\bigg(\bigcup_{I\subset\mathbb{N},\, I \text{finite}}U_\f(\bp_I)\bigg)'=U_\f(\bp_\mathbb{N})'\,.$$
\[havest\] Let $\f\in\cs_{\bp_{\bn}}(\ga)$. Then $\gz_\f^\perp=\gb_{\bp_\mathbb{N}}(\f)$, and the map $\F_\f$ is the conditional expectation of $\pi_\f(\ga)''$ onto $\gz_\f^\perp$ preserving the vector state $\langle\,{\bf\cdot}\,\Om_\f,\Om_\f\rangle$. It assumes the form $$\F_\f(X)=\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle
X_\psi\Om_\psi,\Om_\psi\rangle\idd_{\ch_\psi}\di\m(\psi)\,, \quad
X\in\pi_\f(\ga)''\,.$$
Let $\f=\int_{{\ce(\cs_{\bp_{\bn}}(\ga))}}\psi\di\m(\psi)$ be the ergodic decomposition of $\f\in\cs_{\bp_{\bn}}(\ga)$ given in Theorem \[csyx\]. By Theorem 4.4.3 of [@BR1], Proposition 3.1.10 of [@S] and Theorem IV.8.25 of [@T1], we get $$\pi_\f(\ga)'\bigwedge\{U_\f(\bp_J)\}'=\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}\bc\idd_{\ch_\psi} \di\m(\psi)
\sim L^\infty(\ce(\cs_{\bp_{\bn}}(\ga)),\m)\,.$$ Let $x,y\in\ch_\f$ and $A\in\ga$. Then by Lebesgue Dominated Convergence Theorem and Propositions 4.3 and 5.4 of [@CrF], we obtain, for a sequence $\{I_n\}$ of finite subsets invading $\mathbb{N}$, $$\begin{aligned}
&\langle\F_\f(\pi_\f(A))x,y\rangle
=\lim_{I_n\uparrow\bn}
\frac1{|\bp_{I_n}|}\sum_{g\in\bp_{I_n}}\langle U_\f(g)\pi_\f(A)U_\f(g)^{-1}x,y\rangle\\
=&\lim_{I_n\uparrow\bn}
\frac1{|\bp_{I_n}|}\sum_{g\in\bp_{I_n}}\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle U_\psi(g)\pi_\psi(A)U_\psi(g)^{-1}
x_\psi,y_\psi\rangle\di\m(\psi)\\
=&\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\bigg(\lim_{I_n\uparrow\bn}
\frac1{|\bp_{I_n}|}\sum_{g\in\bp_{I_n}}\langle U_\psi(g)\pi_\psi(A)U_\psi(g)^{-1}
x_\psi,y_\psi\rangle\bigg)\di\m(\psi)\\
=&\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\psi(A)
\langle x_\psi,y_\psi\rangle\di\m(\psi)
=\left\langle\left(\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}\psi(A)\idd_{\ch_\psi} \di\m(\psi)\right)x,y\right\rangle\,.\end{aligned}$$ Then we immediately conclude that $$\label{dddn}
\F_\f(\pi_\f(\ga)'')\subset\pi_\f(\ga)'\bigwedge\{U_\f(\bp_\mathbb{N})\}'\,.$$ Consider the algebraic span $\ca:=\alg\{\pi_\f(\ga),\F(\pi_\f(\ga))\}$. If $$X:=\sum_k\pi_\f(A_k)\F(\pi_\f(B_k))\in\ca\,,$$ then $$\label{dddn1}
\F_\f(X)=\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}f_X(\psi)\idd_{\ch_\psi} \di\m(\psi)\,,$$ where $$f_X(\psi):=\sum_k\psi(A_k)\psi(B_k)\,.$$ It is then straightforward to see that the set of the functions $\{f_X\mid X\in\ca\}\subset C(\ce(\cs_{\bp_{\bn}}(\ga)))$ is an algebra of continuous functions containing the constants and separating the points of the weak–$*$ compact space $\ce(\cs_{\bp_{\bn}}(\ga))$, which is then dense by the Stone Theorem. As a consequence, by means of and one has, $$\gb_{\bp_\mathbb{N}}(\f)=\F_\f(\pi_\f(\ga)'')=\pi_\f(\ga)'\bigwedge\{U_\f(\bp_\mathbb{N})\}'\subset\gz_\f\,,$$ where the first equality and the last inclusion follow by taking into account Theorem 3.1 of [@St1]. Moreover, the inclusion $\pi_\f(\ga)'\bigwedge\{U_\f(\bp_\mathbb{N})\}'\subset\gz_\f$ gives (cf. Lemma 8.4.1 of [@DX0]), $$\pi_\f(\ga)''=\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}\pi_\psi(\ga)''\di\m(\psi)\,.$$ Thus, since $\ce(\cs_{\bp_{\bn}}(\ga))$ is made of factor states (cf. Proposition 5.7 of [@CrF]), the decomposition in Theorem \[csyx\] is indeed the factor decomposition of $\f$ and, by Corollary IV.8.20 of [@T1], $\gz_\f$ is the diagonal algebra, that is it coincides with $\pi_\f(\ga)'\bigwedge\{U_\f(\bp_\mathbb{N})\}'$. In addition, as $\f$ and all the $\psi\in\ce(\cs_{\bp_{\bn}}(\ga))$ are even, we can consider their covariant implementations $V_\f$, and $V_\psi$ respectively, in the GNS representation. The unitary $V_\f$ can be simultaneously diagonalized by using $\pi_\f(\ga)'\bigwedge\{U_\f(\bp_\mathbb{N})\}'$ as well. Then we get $$V_\f=\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}V_\psi\di\m(\psi)\,.$$ This implies that the center $\gz_\f$ coincides with its even part $\gz_\f\bigwedge\{V_\f\}'$. As in our situation $\gz^\perp_\f=\gz_\f\bigwedge\{V_\f\}'$ (cf. ), we obtain $$\label{prinvp}
\gz^\perp_\f=\gz_\f\bigwedge\{V_\f\}'=\gz_\f\,.$$ Collecting together Lemma \[invper\] and , we get $$\gz^\perp_\f=\gz^\perp_\f\bigwedge\{U_\f(\bp_\mathbb{N})\}'=\gz_\f\bigwedge\{U_\f(\bp_\mathbb{N})\}'=\gb_{\bp_\mathbb{N}}(\f)=
\F_\f(\pi_\f(\ga)'')\,.$$ Let now $X\in\pi_\f(\ga)''$ together with its direct integral decomposition $X=\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}X_\psi\di\m(\psi)$. By Kaplansky Density Theorem, there exists a sequence $\{A_n\}_{n\in\bn}\subset\ga$ such that $\pi_\f(A_n)\to X$ in the strong operator topology. This implies, by eventually passing to a subsequence, that $$\psi(A_n)=\langle\pi_\psi(A_n)\Om_\psi,\Om_\psi\rangle
\to\langle X_\psi\Om_\psi,\Om_\psi\rangle$$ $\m$–almost everywhere. Fix now $x,y\in\ch_\f$. By Lebesgue Dominated Convergence Theorem, we get $$\begin{aligned}
\langle\F_\f(X)&x,y\rangle=\lim_n
\langle\F_\f(\pi_\f(A_n))x,y\rangle\\
=&\lim_n\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle\pi_\psi(A_n)\Om_\psi,\Om_\psi\rangle
\langle x_\psi,y_\psi\rangle\di\m(\psi)\\
=&\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle X_\psi\Om_\psi,\Om_\psi\rangle
\langle x_\psi,y_\psi\rangle\di\m(\psi)\\
=&\left\langle\left(\int^\oplus_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle X_\psi\Om_\psi,\Om_\psi\rangle\idd_{\ch_\psi} \di\m(\psi)\right)x,y\right\rangle\,,\end{aligned}$$ which leads as a particular case, $$\begin{aligned}
&\langle X\Om_\f,\Om_\f\rangle=\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle X_\psi\Om_\psi,\Om_\psi\rangle\di\m(\psi)\\
=&\int_{\ce(\cs_{\bp_{\bn}}(\ga))}\langle X_\psi\Om_\psi,\Om_\psi\rangle
\langle \Om_\psi,\Om_\psi\rangle\di\m(\psi)=\langle\F_\f(X)\Om_\f,\Om_\f\rangle\,.\end{aligned}$$
Once having established in Proposition \[havest\] that $\gz^\perp_\f=\gb_{\bp_\mathbb{N}}(\f)$, and $\F_\f$ is the conditional expectation onto the tail algebra preserving the vector state $\langle\,{\bf\cdot}\,\Om_\f,\Om_\f\rangle$, it is meaningful to check that any state $\f\in\cs_{\bp_{\bn}}(\ga)$ is conditionally independent and identically distributed w.r.t. the tail algebra.
\[extdef\] A state $\f\in\cs(\ga)$ is symmetric if and only if the corresponding stochastic process is conditionally independent and identically distributed w.r.t. the tail algebra.
Fix any state $\f\in\ga$. Thanks to $\gz^\perp_\f\subset \gz_\f$ (cf. Theorem 2.6.5 of [@BR1]), by Proposition 3.1.1 of [@S] we can decompose $\f=\int_{\cs(\ga)}\psi\di\n(\psi)$ where $\n$ is the orthogonal measure associated to $\gz^\perp_\f$. By reasoning as in Proposition \[havest\] (cf. Lemma 8.4.1 of [@DX0]), we have $\pi_\f(\ga)''=\int^\oplus_{\cs(\ga)}\pi_\psi(\ga)''\di\n(\psi)$, and in addition for $X=\int^\oplus_{\cs(\ga)}X_\psi\di\g(\psi)\in\pi_\f(\ga)''$, the map $E_\f:\pi_\f(\ga)''\to\gz^\perp_\f$ given by $$E_\f(X):=\int^\oplus_{\cs(\ga)}\langle X_\psi\Om_\psi,\Om_\psi\rangle \idd_{\ch_\psi}\di\n(\psi)$$ defines a conditional expectation of $\pi_\f(\ga)''$ onto the tail algebra $\gz^\perp_\f$ preserving the vector state $\langle\,{\bf\cdot}\,\Om_\f,\Om_\f\rangle$. Suppose that the stochastic process $(\iota_j:\bm_2(\bc)\to\ga)_{j\in \mathbb{N}}$ corresponding to $\f$ is conditionally independent and identically distributed w.r.t $\gz^\perp_\f$. By the anticommutation relations, the dense $*$–algebra of the localized elements of $\ga$ coincides with the linear span of terms of the type $\iota_{i_1}(A_1)\iota_{i_2}(A_2)\cdots\iota_{i_n}(A_n)$, where $A_1,\ldots,A_n\in\bm_{2}(\bc)$ and the integers $i_j$ appear only once in the sequence. By a standard density argument we can reduce the matter to objects of this form. Put $A:=\iota_{i_1}(A_1)\iota_{i_2}(A_2)\cdots\iota_{i_n}(A_n)$. For $g\in\bp_\bn$, we get $$\a_g(A)=\iota_{g(i_1)}(A_1))\iota_{g(i_2)}(A_2)\cdots\iota_{g(i_n)}(A_n)\,,$$ where the $\a_g(i_j)$ again appear only once in the sequence. We compute (cf. Definition \[cocaind\]), $$\begin{aligned}
&\f\big(\iota_{i_1}(A_1)\iota_{i_2}(A_2)\cdots\iota_{i_n}(A_n)\big)\\
=&\big\langle E_\f\big(\pi_\f\big(\iota_{i_1}(A_1)\iota_{i_2}(A_2)\cdots\iota_{i_n}(A_n)\big)\big)
\Om_\f,\Om_\f\big\rangle\\
=&\big\langle E_\f\big(\pi_\f\big(\iota_{i_1}(A_1)\big)\big)E_\f\big(\pi_\f\big(\iota_{i_2}(A_2)\big)\big)\cdots
E_\f\big(\pi_\f\big(\iota_{i_n}(A_n)\big)\big)\Om_\f,\Om_\f\big\rangle\\
=&\big\langle E_\f\big(\pi_\f\big(\iota_{g(i_1)}(A_1)\big)\big)E_\f\big(\pi_\f\big(\iota_{g(i_2)}(A_2)\big)\big)\cdots
E_\f\big(\pi_\f\big(\iota_{g(i_n)}(A_n)\big)\big)\Om_\f,\Om_\f\big\rangle\\
=&\big\langle E_\f\big(\pi_\f\big(\iota_{g(i_1)}(A_1)\iota_{g(i_2)}(A_2)\cdots\iota_{g(i_n)}(A_n)\big)
\big)\Om_\f,\Om_\f\big\rangle\\
=&\f\big(\iota_{g(i_1)}(A_1)\iota_{g(i_2)}(A_2)\cdots\iota_{g(i_n)}(A_n)\big)\\
=&\f\circ\a_g\big(\iota_{i_1}(A_1)\iota_{i_2}(A_2)\cdots\iota_{i_n}(A_n)\big)\,,\end{aligned}$$ that is $\f$ is symmetric.
Let now $\f\in\cs_{\bp_{\bn}}(\ga)$. Then the conditional expectation $\F_\f$ onto the tail algebra $\gz^\perp_\f$ is invariant, i.e. $\F_\f=\F_\f\circ\ad U_\f(g)$, $g\in\bp_\bn$. Thus, the associated process is identically distributed. We now prove that it is conditionally independent w.r.t. $\gz^\perp_\f$. To this aim, for $I\subset\mathbb{N}$ finite denote by $\gt_{\f,I}$ the von Neumann algebra given by $\pi_\f(\carf(I))\bigvee \gz^\perp_\f$. Fix two finite subsets $I_1$, $I_2$ of $\mathbb{N}$, with $I_1\cap I_2=\emptyset$, and consider $X_j\in\gt_{\f,I_j}$, $j=1,2$. We then easily get for such elements, $$X_j=\sum_{\b_j }B^{(j)}_{\b_j}\eps^{(j)}_{\b_j}\,,\quad j=1,2\,,$$ where $\{B^{(j)}_{\b_j }\}\subset\gz^\perp_\f$, and $\{\eps^{(j)}_{\b_j}\}\subset\pi_\f(\carf(I_j))$ are any system of matrix–units (cf [@T1], Definition IV.1.7) for $\pi_\f(\carf(I_j))\sim\carf(I_j)$, $j=1,2$. To complete the proof we need to show that $\Phi_\f(X_1X_2)=\Phi_\f(X_1)\Phi_\f(X_2)$. As $I_1$, $I_2$ are disjoint, and $\ce(\cs_{\bp_{\bn}}(\ga))$ is made of product states, it easily follows $\F_\f(R_1R_2)=\F_\f(R_1)\F_\f(R_2)$, whenever $R_j\in\pi_\f(\carf(I_j))$, $j=1,2$. Concerning the general case, we compute, $$\begin{aligned}
\F_\f(X_1X_2)=&\F_\f\bigg(\sum_{\b_1,\b_2}
B^{(1)}_{\b_1 }
\eps^{(1)}_{\b_1}B^{(2)}_{\b_2}\eps^{(2)}_{\b_2}\bigg)
=\sum_{\b_1,\b_2}
\F_\f\big(B^{(1)}_{\b_1 }
B^{(2)}_{\b_2}\eps^{(1)}_{\b_1}\eps^{(2)}_{\b_2}\big)\\
=&\sum_{\b_1,\b_2}B^{(1)}_{\b_1 }B^{(2)}_{\b_2}
\F_\f\big(\eps^{(1)}_{\b_1}\eps^{(2)}_{\b_2}\big)
=\sum_{\b_1,\b_2}B^{(1)}_{\b_1}B^{(2)}_{\b_2}
\F_\f\big(\eps^{(1)}_{\b_1}\big)\F_\f\big(\eps^{(2)}_{\b_2}\big)\\
=&\bigg(\sum_{\b_1}B^{(1)}_{\b_1 }\F_\f\big(\eps^{(1)}_{\b_1}\big)\bigg)
\bigg(\sum_{\b_2}B^{(2)}_{\a_2,\b_2 }\F_\f\big(\eps^{(2)}_{\b_2}\big)\bigg)\\
=&\bigg(\sum_{\b_1}\F_\f\big(B^{(1)}_{\b_1 }\eps^{(1)}_{\b_1}\big)\bigg)
\bigg(\sum_{\b_2}\F_\f\big(B^{(2)}_{\b_2 }\eps^{(2)}_{\b_2}\big)\bigg)\\
=&\F_\f(X_1)\F_\f(X_2)\,.\end{aligned}$$
\[inften\] By following the same lines of the above proof, we can show that a stochastic process factoring through the countably infinite tensor product of a single separable $C^*$–algebra is conditionally independent and identically distributed w.r.t. the tail algebra if and only if the corresponding state is symmetric.
symmetric states in free and $q$–deformed probability {#sec4}
=====================================================
The $q$–deformed Commutation Relations were introduced in Quantum Physics in [@FB]. The reader is referred to [@DyF] and the references cited therein for further details. Suppose $-1<q<1$ and fix an Hilbert space $\ch$. The $q$–deformed Fock space $\G_q(\ch)$ is the completion of the algebraic linear span of the vacuum vector $\Om$, together with vectors $$f_1\otimes\cdots\otimes f_n\,,\quad
f_j\in\ch\,,j=1,\dots,n\,,n=1,2,\dots$$ with respect to the $q$–deformed inner product $$\langle f_1\otimes\cdots\otimes
f_n\,,g_1\otimes\cdots\otimes g_m\rangle_q
:=\d_{n,m}\sum_{\pi\in\bp_n}q^{i(\pi)}\langle
f_1\,,g_{\pi(1)}\rangle_\ch\cdots\langle f_n\,,g_{\pi(n)}\rangle_\ch\,,$$ $\d_{n,m}$ being the Kronecker symbol, $\bp_n$ the symmetric group of $n$ elements, and $i(\pi)$ the number of inversions of $\pi\in\bp_n$. Fix $f,f_1,\ldots,f_n\in\ch$. Define the creator $a_q^\dagger(f)$ as $$a_q^\dagger(f)\Omega=f\,,\quad a_q^\dagger(f)f_1\otimes\cdots\otimes f_n=f\otimes
f_1\otimes\cdots\otimes f_n\,,$$ and the annihilator $a(f)$ as $$\begin{aligned}
&a_q(f)\Omega=0\,,\quad
a_q(f)(f_1\otimes\cdots\otimes f_n)\\
=&\sum_{k=1}^nq^{k-1}\langle
f_k,f\rangle_\ch f_1\otimes\cdots f_{k-1}\otimes
f_{k+1}\otimes\cdots\otimes f_n\,.\end{aligned}$$ $a_q^\dagger(f)$ and $a_q(f)$ are mutually adjoint with respect to the $q$–deformed inner product, and satisfy the commutation relations $$a_q(f)a_q^\dagger(g)-qa_q^\dagger(g)a_q(f)=\langle
g,f\rangle_{\ch}\idd\,,\qquad f,g\in\ch\,.$$ The limiting cases are the Canonical Commutation Relations (Bosons) when $q=1$, and the Canonical Anticommutation Relations (Fermions) for $q=-1$, the latter treated exhaustively in [@CrF] and in Section \[sec3\]. The case $q=0$ corresponds to the free group reduced $C^*$–algebra (see below).
The concrete $C^*$–algebra $\gar_q$ and its subalgebra $\gg_q$, acting on $\G_q(\ch)$ are the unital $C^*$–algebras generated by the annihilators $\{a(f)\mid f\in\ch\}$, and the selfadjoint part of annihilators $\{s_q(f)\mid f\in\ch\}$, $$s_q(f):=a_q(f)+a_q^\dagger(f)\,,\quad f\in\ch\,,$$ respectively. The Fock vacuum expectation is the state on both the mentioned $C^*$–algebras defined as $\om_q:=\langle\,{\bf\cdot}\,\Om,\Om\rangle$. Notice that $\gar_0$ is an extension of the Cuntz algebra $\co_n$ by the compact operators if $\ch$ has dimension $2\leq n<\infty$, see e.g. see [@VDN], pag. 6.
\[cuntz\] The Cuntz algebra $\co_\infty$ coincides with $\gar_0$ with $\ch$ separable infinite dimensional. Its generators are $\big\{a_0^\dagger(e_j)\mid j\in\bn\big\}$, where $\{e_j\}_{j\in\bn}$ is any orthonormal basis of $\ch$.
In order to study the symmetric states on $\gar_q$ and $\gg_q$, we need to consider the action of the group of permutations $\bp_J$ on them. As in the previous sections, we get $J\sim \mathbb{Z}$. Then $\gar_q$ and $\gg_q$ are concrete $C^*$–algebras on $\G_q(\ell^2(\bz))$. If $i\in\mathbb{Z}$ and $e_i\in\ell^2(\bz)$ is the sequence taking value $1$ at $i$ and zero elsewhere, we denote $a_i:=a(e_i)$, $a_i^\dagger:=a^\dagger(e_i)$ and $s_i:=a_i+a^\dagger_i$. As usual, the group of permutations $\bp_\bz$ naturally acts on $\gar_q$ and $\gg_q$ by $$\a_g(a_i):=a_{g(i)}\,,\quad i\in\bz\,,\,g\in\bp_\bz\,,$$ and the Fock vacuum is invariant under such an action. The group $\bz$ also acts on both $\gar_q$ and $\gg_q$ as powers of the right shift $\b$, uniquely determined by $\b(a_i):=a_{i+1}$, $i\in\bz$, on the generators. In addition, the Fock vacuum $\om_q$ is shift–invariant, and it is shown in [@DyF] that it is the unique invariant state. The same state is moreover the unique symmetric one on both $\gar_q$ and $\gg_q$, as the following proposition shows.
\[frepp\] The set $\cs_{\bp_\bz}(\gar_q)$ and $\cs_{\bp_\bz}(\gg_q)$ of the invariant states under the action of $\bp_\bz$ on $\gar_q$ and $\gg_q$ respectively, consist of a singleton which is precisely the Fock vacuum expectation.
Fix the integers $k<l$ and take an arbitrary polynomial $A$ in $a_q(f)$ and $a_q^\dagger(f)$, or in $s_q(f)$ where the degree zero corresponds to the multiple of the identity, with $f\in\ell^2([k,l])\subset \ell^2(\bz)$. By Lemma \[permtra\], there is a cycle $g_A\in\bp_\bz$ such that $\b(A)=\a_{g_A}(A)$. Thus, $$\f(\b(A))=\f(\a_{g_A}(A))=\f(A)\,,$$ provided that $\f\in\cs_{\bp_\bz}(\gar_q)$ or $\f\in\cs_{\bp_\bz}(\gg_q)$, respectively. As the polynomials as above are dense in $\gar_q$ or $\gg_q$ respectively, we conclude that, if $\f$ is symmetric, then it is shift–invariant. But it is shown in Theorem 3.3 and Corollary 3.4 of [@DyF], that the Fock vacuum expectation is the unique shift invariant state.
An immediate application of Proposition \[frepp\] gives that the Fock vacuum $\om_0$ is the unique symmetric state on the Cuntz algebra $\co_\infty$ (see Remark \[cuntz\]), since the natural action of the permutation group $\bp_\infty$ on this algebra.
Let now $\bbf_\infty$ be the free group with countably many generators $\{g_i\mid i\in\mathbb{Z}\}$. For $i_1\neq i_2\neq\cdots\neq i_n$ where the indices can appear more than once in the string, and $k_1,k_2,\dots,k_n\in\bz\backslash\{0\}$, denote $w:=g_{i_1}^{k_1}\cdots g_{i_n}^{k_n}$ a (reduced) word. The length of $w\in\bbf_\infty$ is defined as $$w:=|k_1|+|k_2|+\cdots+|k_n|\,.$$ The empty word $w_{\emptyset}$, whose length is zero by definition, is the unity of $\bbf_\infty$. All the words of arbitrary length, together with the natural group operations, generate $\bbf_\infty$. The universal $C^*$–algebra generated by the linear combinations of $\{\l_w\mid w\in\bbf_\infty\}$, together with the relations $\l_w^*=\l_{w^{-1}}$, $w\in\bbf_\infty$, is the group $C^*$–algebra $C^*(\bbf_\infty)$. This is nothing but the unital free product $C^*$–algebra of the group $\bz$. The concrete $C^*$–algebra $C_r^*(\bbf_\infty)$ generated by (left) regular representation is precisely the reduced group $C^*$–algebra $C_r^*(\bbf_\infty)$. It is well–known that it differs from $C^*(\bbf_\infty)$ as $\bbf_\infty$ is not amenable. The GNS representation $\pi_\t$ associated to the tracial state $\t$ on $C^*(\bbf_\infty)$, uniquely defined as $$\t\bigg(\sum_{w\in\bbf_\infty}a_w\l_w\bigg):=a_{w_{\emptyset}}\,,$$ generates the left regular representation, so $\pi_\t\big(C^*(\bbf_\infty)\big)=C_r^*(\bbf_\infty)$. Finally the group $\bp_\infty$ acts in a natural way on $C^*(\bbf_\infty)$ and $C_r^*(\bbf_\infty)$.
\[cofr\] The tracial state $\t$ is the unique symmetric state on $C_r^*(\bbf_\infty)$.
The proof directly follows collecting together Theorem 2.6.2 of [@VDN], and Proposition \[frepp\].
The last part of the section is devoted to determine some remarkable ergodic properties of the Haagerup states. Recall that the Haagerup states on $C^*(\bbf_\infty)$, labelled by $\l\in(0,+\infty)$, are defined in [@Ha] as $$\f_\l(w):=e^{-\l|w|}\,.$$ The case $\l=+\infty$ corresponds to the tracial state and it is covered by Corollary \[cofr\]. It generates the regular representation of $\bbf_\infty$ and in addition, it is shown in Corollary 3.2 of [@Ha] that is the unique Haagerup state which is normal w.r.t. the regular representation. Namely, it is the unique state of the family whose corresponding stochastic process take values in the reduced free product $C^*$–algebra $C^*_r(\bbf_\infty)$. The Haagerup states are automatically symmetric by construction, and satisfy the product state condition $\f_\l(vw)=\f_\l(v)\f_\l(w)$ if $I_v\cap I_w=\emptyset$.[^4] But they do not fulfil the block–singleton condition for $\l\in(0,+\infty)$. In fact, if $i\neq j$, $$\f_\l(g_ig_jg_i^{-1})=e^{-3\l}\neq e^{-\l}=\f_\l(g_j)\f_\l(\idd)=\f_\l(g_j)\f_\l(g_ig_i^{-1})\,.$$ By (i) of Theorem \[frecazo\] they are weakly clustering, then extremal (i.e. ergodic under the action of $\bp_\infty$) thanks to Proposition 3.1.10. But (ii) of Theorem \[frecazo\] gives that $\pi_{\f_\l}\big(C^*(\bbf_\infty)\big)''\bigwedge \big\{U_{\f_\l}(\bp_\infty)\big\}'$ cannot be trivial, this property being crucial for the proof of the following
For $\l\in(0,+\infty)$, the support $s(\f_\l)\in C^*(\bbf_\infty)^{**}$ of $\f_\l$ does not belong to $Z(C^*(\bbf_\infty)^{**})$.
As we have shown, the Haagerup states are extremal symmetric. Suppose that for the support of $\f_\l$ in the bidual is central, then the cyclic vector $\Om_{\f_\l}$ is also separating for $\pi_{\f_\l}\big(C^*(\bbf_\infty)\big)''$. By Theorem 4.3.20 in [@BR1], it follows that $\pi_{\f_\l}\big(C^*(\bbf_\infty)\big)''\bigwedge \big\{U_{\f_\l}(\bp_\infty)\big\}'=\bc I$. As a consequence, by Theorem \[frecazo\], $\f_\l$ satisfies the block–singleton condition, This contradicts the above discussion.
In conclusion, as realized throughout the section, the structure of the convex set made by the symmetric states and its boundary, change radically when one considers $C^*_r(\bbf_\infty)$ or $C^*(\bbf_\infty)$, passing from a singleton to a richer structure which contains, perhaps properly, all the Haagerup states.
the boolean case {#sec5}
================
Let $\ch$ be a complex Hilbert space. Recall that the Boolean Fock space over $\ch$ (cf. [@BGS]) is given by $\G(\ch):=\mathbb{C}\oplus \ch$, where the vacuum vector $\Om$ is $(1,0)$. On $\Gamma(\ch)$ we define the creation and annihilation operators, respectively given for $f\in \ch$, by $$b^\dagger(f)(\alpha\oplus g):=0\oplus \alpha f,\,\,\,\, b(f)(\alpha\oplus g):=\langle g,f\rangle_\ch \oplus 0,\,\,\, \alpha\in\mathbb{C},\, g\in\ch.$$ They are mutually adjoint, and satisfy the following relations for $f,g\in \ch$, $$b(f)b^\dagger(g)=\langle g, f\rangle_\ch \langle\,{\bf\cdot}\,,\Om\rangle\Om\,,\quad
b^\dagger(f)b(g)=\langle\,{\bf\cdot}\,,0\oplus g\rangle 0\oplus f\,.$$ As in Section \[sec4\], we consider the unital $C^*$–algebras acting on $\Gamma(\ch)$, which are respectively generated by the annihilators $\{b(f)\mid f\in\ch\}$, and the selfadjoint part of annihilators $\{r(f)\mid f\in\ch\}$, where $$r(f):=b(f)+b^\dagger(f)\,,\quad f\in\ch\,.$$ Again, we are interested in the action of the group of permutations $\bp_J$ on these algebras. Then, as usual, we put $J\sim \mathbb{Z}$, and take $\ch=\ell^2(\mathbb{Z})$. From now on, we denote the vacuum vector by $e_\#$, and it can be seen as an element of $\ell^2(\{\#\}\cup\mathbb{Z})=\Gamma(l^2(\mathbb{Z}))$. As a consequence, $\{e_\#, e_i|i\in \mathbb{Z}\}$ is an orthonormal basis for this space, where for each $i\in \mathbb{Z}$, $e_i$ is the sequence taking value $1$ on $i$ and $0$ elsewhere. For each $j\in \mathbb{Z}$, define $b_j:=b(e_j)$, $b^\dagger_j:=b^\dagger(e_j)$, and $r_j:=r(e_j)$, $j\in\bz$. With $\cf(\ell^2(\{\#\}\cup\mathbb{Z}))$ and $\ck(\ell^2(\{\#\}\cup\mathbb{Z}))$ we will denote respectively, the finite rank operators and the compact linear operators acting on $\ell^2(\{\#\}\cup\mathbb{Z})$.
\[bolca\] The unital $C^*$–algebras acting on the Boolean Fock space $\Gamma(\ell^2(\bz))$ generated by the Boolean annihilators $\{b_j\mid j\in\bz\}$, or by their selfadjoint parts $\{r_j\mid j\in\bz\}$, are equal and coincide with $\ck(\ell^2(\{\#\}\cup\bz))+\bc\idd$.
First we note that $\alg\{b_j\mid j\in\bz\}$ generates all of $\cf(\ell^2(\{\#\}\cup\bz))$. In fact, consider in $\cb(\ell^2(\{\#\}\cup\bz))$, the canonical system of matrix–units $\{\varepsilon_{ij}\mid i,j\in(\{\#\}\cup\bz)\}$. It is easy to check that $$\varepsilon_{\#j}=b_j\,,\,\, \varepsilon_{j\#}=b^\dagger_j\,,\,\,
\varepsilon_{\#\#}=b_ib^\dagger_i\,,\quad \varepsilon_{ij}=b^\dagger_ib_j\,,\quad i,j\in\bz\,,$$ that is the assertion. The equality $\alg\{r_j\mid j\in\bz\}=\cf(\ell^2(\{\#\}\cup\bz))$ follows after noticing that $$r_ir_j=\d_{ij}\varepsilon_{\#\#}+\varepsilon_{ij}\,,
r^2_i-r^2_ir^2_j=\varepsilon_{ii}-\d_{ij}\varepsilon_{ij}\quad i,j\in\bz\,.$$ Since $\ck(\ell^2(\{\#\}\cup\bz))$ is the norm closure of $\cf(\ell^2(\{\#\}\cup\bz))$ in $\cb(\ell^2(\{\#\}\cup\bz))$, the thesis follows.
It is worth noticing that also the universal $C^*$–algebra generated by Boolean annihilators, or equivalently by their selfadjoint part, is isomorphic to the algebra of compact operator on $\ell^2(\{\#\}\cup\bz)$, see e.g. the arguments outlined in [@Sz].
Denote $\gpb=\ck(\ell^2(\{\#\}\cup\bz))+\bc I$ the unital Boolean $C^*$–algebra. The finite permutations and the shift naturally act on the indices of $\bz$, leaving invariant the vacuum index $\#$ corresponding to the vacuum vector $e_\#$. Thus, we have natural actions of the permutations and the shift on $\gpb$ and $\ck(\ell^2(\{\#\}\cup\bz))$, both denoted by an abuse of notations, by $\{\a_g\mid g\in\bp_\bz\}$ and $\b$, respectively.
\[shiv\] The unique invariant state under the shift for $\ck(\ell^2(\{\#\}\cup\bz))$ is the vacuum state.
Let $U$ be the canonical implementation of the shift on the Boolean Fock space: $$Ue_\#=e_\#\,,\quad Ue_k=e_{k+1}\,,\,\,\,k\in\bz\,.$$ Fix any state $\om_T\in\cs(\ck(\ell^2(\{\#\}\cup\bz)))$, together with its representation through a positive trace class operator $$T=\sum_{\l\in\s_{{\rm pp}}(T)}\l E_\l\,$$ where “$\s_{{\rm pp}}$” stands for pure points spectrum. The fact that $\om_T$ is invariant w.r.t. the shift leads to $T=UTU^*$, which turns out to be equivalent to $$U=\sum_{\l\in\s_{{\rm pp}}(T)}E_\l UE_\l\,.$$ As the unique eigenspace of $U$ is $\bc e_\#$ (see e.g. Section 6 of [@F]), the last condition is possible only if $T=\langle\,{\bf\cdot}\, e_\#,e_\#\rangle e_\#$.
We have for the compact convex set of the symmetric states, $$\cs_{\bp_{\bz}}(\gpb)=\{\g\om_\#+(1-\g)\om_\infty\mid\g\in[0,1]\}\,,$$ where $\om_\#=\langle\,{\bf\cdot}\,e_\#,e_\#\rangle$ is the Fock vacuum state, and $$\om_\infty(A+aI):=a\,,\quad A\in \ck(\ell^2(\{\#\}\cup\bz))\,, a\in\bc\,.$$
Fix $\om\in\cs_{\bp_{\bz}}(\gpb)$. Arguing as in Proposition \[frepp\], one finds that $\om$ is also shift–invariant. Its restriction $\om\lceil_{\ck(\ell^2(\{\#\}\cup\bz))}$ yields a positive functional which is also shift–invariant. Thus, by Proposition \[shiv\] we get, $$\om\lceil_{\ck(\ell^2(\{\#\}\cup\bz))}
=\big\|\om\lceil_{\ck(\ell^2(\{\#\}\cup\bz))}\big\|\om_\#\,.$$ This means that $\om=\g\om_\#+(1-\g)\om_\infty$, where $\g=\big\|\om\lceil_{\ck(\ell^2(\{\#\}\cup\bz))}\big\|$.
To end the present section, we show that there are plenty of Boolean processes for which the tail algebra is not expected, that is no conditional expectation onto such an algebra which preserves the state corresponding to the process under consideration. To simplify, we consider a pure state $\om_\xi:=\langle\,{\bf\cdot}\,\xi,\xi\rangle$, $\xi\in\ell^2(\{\#\}\cup\bz)$ being a unit vector. In this situation $\pi_{\om_\xi}(\gpb)''=\cb(\ell^2(\{\#\}\cup\bz))$. Concerning the tail algebra, we get with $I_n:=\{k\in\bz\mid |k|>n\}$ and $P_\#:=\langle\,{\bf\cdot}\, e_\#,e_\#\rangle e_\#$, $$\gz^\perp_{\om_\xi}=\bigcap_{n\in\bn}\cb(\ell^2(\{\#\}\cup I_n))\bigoplus\bc P_{\ell^2(\bz\backslash I_n)}
=\bc P_\#\oplus\bc P_\#^\perp\,,$$ where $\cb(\ell^2(\{\#\}\cup I_n))$ is considered as a non unital subalgebra of $\cb(\ell^2(\{\#\}\cup\bz))$ in a canonical way. Notice that each conditional expectation $F$ onto $\gz^\perp_{\om_\xi}$ satisfies $F=F\circ E$, with $$E(A)=\om_\#(A)P_\#+P_\#^\perp AP_\#^\perp\,,\quad A\in \cb(\ell^2(\{\#\}\cup \bz))\,.$$ By taking into account that any conditional expectation of a $C^*$–algebra with unity $\idd$, into $\bc\idd$ is just given by a state, we conclude that any conditional expectation $F=F_\f$ as above assumes the form $$F_\f(A)=\om_\#(A)P_\#+\f(P_\#^\perp AP_\#^\perp)P_\#^\perp\,,\quad A\in \cb(\ell^2(\{\#\}\cup \bz))\,,$$ where $\f$ is any state, not necessarily normal, on $\cb(\ell^2(\bz))$, the last viewed again as a non unital subalgebra of $\cb(\ell^2(\{\#\}\cup\bz))$. We now show that $\gz^\perp_{\om_\xi}$ cannot be expected if $0<|\langle e_\#,\xi\rangle|<1$. For the rank–one operator $A=\langle \,{\bf\cdot}\,,\xi\rangle e_\#$ and any $\f$ as above, we get $$\frac{\om_\xi(F_\f(A))}{\om_\xi(A)}=|\langle e_\#,\xi\rangle|^2\,.$$ The examples of Boolean stochastic processes described above explain that, contrarily to the classical case, the condition to be independent and identically distributed w.r.t. the tail algebra (cf. Definition \[cocaind\]), cannot be formulated in the general case, without mentioning the [*a–priori*]{} existence of a preserving conditional expectation.
[9999]{}
Accardi L., Ben Ghorbal A., Crismale V., Lu Y. G. [*Singleton conditions and quantum De Finetti’s theorem*]{}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. [**11**]{} (2008), 639–660.
Accardi L., Frigerio A., Lewis A. T. [*Quantum stochastic processes*]{}, Publ. Res. Inst. Math. Sci. [**18**]{} (1982), 97–133.
Accardi L, Hashimoto U., Obata N. [*Notions of independence related to the free group*]{}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. [**1**]{} (1998), 201–220.
Accardi L., Lu Y.G. [*A continuous version of De Finetti’s theorem*]{}, Ann. Probab. [**21**]{} (1993), 1478–1493.
Araki H., Moriya H. [*Joint extension of states of subsystems for a CAR system*]{}, Commun. Math. Phys. [**237**]{} (2003), 105–122.
Avitzour D. [*Free products of $C^*$–algebras*]{}, Trans. Amer. Math. Soc. [**271**]{} (1982), 423–435.
Ben Ghorbal A. Schurmann M. [*Quantum stochastic calculus on Boolean Fock space*]{}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. [**7**]{} (2004), 631–650.
Bratteli O., Robinson D. W. [*Operator algebras and quantum statistical mechanics I, II*]{}, Springer, Berlin–Heidelberg–New York, 1981.
Christensen E., Effros E. G., Sinclair A. [*Completely bounded multilinear maps and $C^*$–cohomology*]{}, Invent. Math. [**90**]{} (1987), 279–296.
Crismale V., Fidaleo F. [*De Finetti theorem on the CAR algebra*]{}, Commun. Math. Phys. [**315**]{} (2012), 135–152.
De Finetti B. [*Funzione caratteristica di un fenomeno aleatorio*]{}, Atti Accad. Naz. Lincei, VI Ser., Mem. Cl. Sci. Fis. Mat. Nat. [**4**]{} (1931), 251–259.
Diaconis P., Freedman D. A. [*De Finetti’s theorem for Markov chains*]{}, Ann. Probab. [**8**]{} (1980), 115–130.
Diaconis P., Freedman D. A. [*Finite exchangeable sequences*]{}, Ann. Probab. [**8**]{} (1980), 745–764.
Dixmier J. *$C^*$–algebras*, North Holland, Amsterdam–New York–Oxford, 1977.
Dykema K., Fidaleo F. [*Unique mixing of the shift on the $C^*$–algebras generated by the $q$–canonical commutation relations*]{}, Houston J. Math. [**36**]{} (2010), 275–281.
Fidaleo F. [*On strong ergodic properties of quantum dynamical systems*]{}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. [**12**]{} (2009), 551–564.
Fidaleo F. [*A Nonconventional Ergodic Theorem for quantum “diagonal measures”*]{}, arXiv:1203.5200v2 \[math.OA\] (2013).
Frisch U., Borret R., [*Parastochastics*]{}, J. Math. Phys. [**11**]{} (1970), 364–390.
Haagerup U. [*An example of non nuclear $C^*$–algebra, which has the metric approximation property*]{}, Invent. Math. [**50**]{} (1979), 279–293.
Hewitt E., Savage L. F. [*Symmetric measures on Cartesian products*]{}, Trans. Amer. Math. Soc. [**80**]{} (1955), 470–501.
Kallenberg O. [*Probabilistic symmetries and invariance principles*]{}, Springer, Berlin–Heidelberg–New York, 2005.
Köstler C. [*A noncommutative extended De Finetti theorem*]{}, J. Funct. Anal. [**258**]{} (2010), 1073–1120.
Robinson, D. W. [*Characterization of clustering states*]{}, Commun. Math. Phys. [**41**]{} (1975), 79–88.
Rudin W. [*Functional Analysis*]{}, McGraw–Hill Series in Higher Mathematics, New York–Düsseldorf–Johannesburg, 1973.
Sakai S. [*$C^*$ –algebras and $W^*$ –algebras*]{}, Springer, Berlin–Heidelberg–New York, 1971.
Schurmann M. [*White Noise on Bialgebras*]{}, Lecture Notes in Mathematics, 1544. Springer-Verlag, Berlin, 1993.
St[ø]{}rmer E. [*Large groups of automorphisms of $C^{*}$–algebras*]{}, Commun. Math. Phys. [**5**]{} (1967), 1–22.
St[ø]{}rmer E. [*Symmetric states of infinite tensor products of $C^{*}$–algebras*]{}, J. Funct. Anal. [**3**]{} (1969), 48–68.
Strǎtilǎ S. [*Modular theory in operator algebras*]{}, Abacus press, Tunbridge Wells, Kent, 1981.
Strǎtilǎ S., Zsidó, L. [*Lectures on von Neumann algebras*]{}, Abacus press, Tunbridge Wells, Kent, 1979.
Szabo G. [*Ideal spanned by matrix units isomorphic to compact operators*]{}, http://mathoverflow.net/questions/114780 (version: 2012-11-28).
Takesaki M. [*Theory of operator algebras I, II, III*]{}, Springer, Berlin–Heidelberg–New York, 2003.
Voiculescu D. V., Dykema K. J., Nica A. [*Free random variables*]{}, CRM Monogr. Ser., vol. 1, American Mathematical Society, Providence, 1992.
[^1]: As a word in the generators of an algebra or a group, we mean always a reduced word without any further specification.
[^2]: The existence of $\r^{(0)}$ directly follows also from the universality of $\gb^{(0)}$: in the l.h.s. of take $B=\gc^{(0)}$. Since the map “forgetting the identity” $\ga\to\ga$ is one–to–one, $\r^{(0)}$ is an epimorphism.
[^3]: Since many classical stochastic processes are defined in terms of their finite–dimensional distributions, irrespective of the probability spaces, this property is the transposition to the quantum case of the fact that two stochastic processes, defined on two different probability spaces but having the same state space, are identified if they have the same finite–dimensional distributions.
[^4]: The Haagerup states satisfy a stronger condition $\f_\l(vw)=\f_\l(v)\f_\l(w)$ if $|vw|=|v|+|w|$.
|
---
abstract: |
We introduce the notion of $T$-stability for torsion-free Higgs sheaves as a natural generalization of the notion of $T$-stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of torsion-free Higgs sheaves are important in the definition of $T$-stability. Using this, we show that this notion is preserved under dualization and tensor product with an arbitrary Higgs line bundle. Then, we prove that for a torsion-free Higgs sheaf over a compact Kähler manifold, $\omega$-stability implies $T$-stabilty. As a consequence of this we obtain the $T$-semistability of any reflexive Higgs sheaf with an admissible Hermitian-Yang-Mills metric. Finally, we prove that $T$-stability implies $\omega$-stability if, as in the classical case, some additional requirements on the base manifold are assumed. In that case, we obtain the existence of admissible Hermitian-Yang-Mills metrics on any $T$-stable reflexive sheaf.\
[*Keywords*]{}: Higgs sheaves; $T$-stability; Mumford-Takemoto stability and Hermitian-Yang-Mills metrics.\
[*MS Classification*]{}: 53C07, 53C55, 32C15.
author:
- |
S. A. H. Cardona[^1]\
CIMAT A.C. - Via Jalisco S/N - 36240, Gto. - México
title: '$T$-stability for Higgs sheaves over compact complex manifolds'
---
Introduction
============
The notion of $T$-stability was introduced by Bogomolov [@Bogomolov] in the case of coherent sheaves over projective algebraic manifolds, and it was studied latter by Kobayashi in [@Kobayashi; @0] and [@Kobayashi] for coherent sheaves over compact complex manifolds. In the Kähler case, the $T$-stability was related to Mumford-Takemoto stability (also called $\omega$-stability, where $\omega$ denotes the Kähler form of the base manifold). To be precise, it was shown by Bogomolov and Kobayashi that a $\omega$-stable (resp. $\omega$-semistable) torsion-free coherent sheaf over a compact Kähler manifold $X$, was $T$-stable (resp. $T$-semistable). They proved also the converse result if $H^{1,1}(X,{\mathbb C})$ was one dimensional, or if $\omega$ represented an integral class (so that $X$ was projective algebraic) and ${\rm Pic}(X)/{\rm Pic}^{0}(X)={\mathbb Z}$, where here ${\rm Pic}^{0}(X)$ denotes the subgroup of the Picard group ${\rm Pic}(X)$ consisting of holomorphic line bundles with vanishing first Chern class. In proving the connection between these two concepts of stability, it was important to consider a classical vanishing theorem for holomorphic line bundles. As we will see, the same result is important also to connect these two notions of stability for Higgs sheaves.\
Now, vanishing theorems are important in Complex Geometry. Indeed, some of these results, first proved by Bochner and Yano [@Yano-Bochner], have been used by Kobayashi [@Kobayashi] to prove one direction of the classical Hitchin-Kobayashi correspondence for holomorphic vector bundles over compact Kähler manifolds. As it is well known, this correspondence establishes an equivalence between the notion of $\omega$-polystabilty and the existence of Hermitian-Einstein metrics for such bundles. Kobayashi also proved that a holomorphic vector bundle admitting an approximate Hermitian-Einstein metric was $\omega$-semistable. As a consequence of this, it was followed that a holomorphic vector bundle over a compact Kähler manifold admitting a Hermitian-Einstein metric (resp. an approximate Hermitian-Einstein metric) was necessarily $T$-stable (resp. $T$-semistable). The Hitchin-Kobayashi correspondence has been extended to reflexive sheaves by Bando and Siu [@Bando-Siu] by introducing the notion of admissible metric on a sheaf.\
On the other hand, Higgs bundles and Higgs sheaves were introduced by Hitchin [@Hitchin] and Simpson [@Simpson], [@Simpson; @2] and they also introduced the corresponding notion of Mumford-Takemoto stability for these objects. As it is well known, several results on holomorphic vector bundles and coherent sheaves can be extended to Higgs bundles and Higgs sheaves. In particular, Vanishing theorems for Higgs bundles have been recently studied in [@Cardona; @3], and Simpson proved in [@Simpson] a Hitchin-Kobayashi correspondence for Higgs bundles over compact Kähler manifolds, i.e., an equivalence between the notion of Mumford-Takemoto polystability and the existence of Hermitian-Yang-Mills metrics (henceforth usually abbreviated $HYM$-metric). Now, Bruzzo and Graña Otero [@Bruzzo-Granha] proved that if a Higgs bundle admits an approximate Hermitian-Yang-Mills (henceforth abbreviated $apHYM$-metric), it is necessarily semistable in the sense of Mumford-Takemoto. Following the ideas of Bando and Siu [@Bando-Siu], Biswas and Schumacher proved in [@Biswas-Schumacher] that a reflexive Higgs sheaf over a compact Kähler manifold is $\omega$-polystable if and only if it has an admissible Hermitian-Yang-Mills metric, which is indeed a Hitchin-Kobayashi correspondence for Higgs sheaves.\
This article is organized as follows, in the second section we review some basic definitions concerning Higgs sheaves and using a classical isomorphism involving determinants, we show that the determinant bundle of certain quotient Higgs sheaves is indeed a Higgs line bundle. In the final part of the second section, we review some results on reflexive Higgs sheaves and Higgs bundles and we rewrite a classical vanishing theorem of holomorphic line bundles over compact Kähler manifolds in the context of Higgs line bundles. In the third section, we introduce the notion of a weighted flag for a torsion-free Higgs sheaf, and using this we define the $T$-stability for torsion-free Higgs sheaves, as a natural extension of the notion of $T$-stability for torsion-free coherent sheaves introduced in [@Kobayashi]. Then, we prove some basic properties that are indeed extensions of classical results; in particular, we prove that in the definition of $T$-stability it is enough to consider saturated flags (i.e., flags in which the quotients between the Higgs sheaf and all Higgs subsheaves of the flag are torsion-free). We prove also that $T$-stability is preserved under tensor products with Higgs line bundles and dualizations. Finally, in a four section we prove that in the Kähler case, a Mumford-Takemoto stable (resp. semistable) torsion-free Higgs sheaf is also $T$-stable (resp. $T$-semistable) and hence, as a consequence of the main result of Biswas and Schumacher in [@Biswas-Schumacher], we obtain that any reflexive Higgs sheaf with an admissible $HYM$-metric is necessarily $T$-semistable and it is in general a direct sum of $T$-stable Hermitian-Yang-Mills Higgs sheaves with equal slope; and that any locally free Higgs sheaf admitting a $HYM$-metric (resp. $apHYM$-metric) is $T$-stable (resp. $T$-semistable). At the end, we show that if either $H^{1,1}(X,{\mathbb C})$ is one dimensional, or if $\omega$ represents an integral class and ${\rm Pic}(X)/{\rm Pic}^{0}(X)={\mathbb Z}$, the classical proof of Kobayashi for the converse implication (i.e., $T$-stability as a sufficient condition of $\omega$-stability) can be easily adapted to the Higgs case.
Preliminaries
=============
We start with some basic definitions. Let $X$ be a compact complex manifold of complex dimension $n$ and let $\Omega_{X}^{1}$ be the cotangent sheaf to $X$. A Higgs sheaf over $X$ is a pair ${\mathfrak E}=(E,\phi)$ where $E$ is a coherent sheaf over $X$ and $\phi : E\rightarrow E\otimes\Omega_{X}^{1}$ is a morphism of ${\cal O}_{X}$-modules such that $\phi\wedge\phi : E\rightarrow E\otimes\Omega_{X}^{2}$ vanishes. The morphism $\phi$ is usually called the Higgs field. A section $s$ of $E$ is said to be a $\phi$-invariant section of ${\mathfrak E}$, if there exists a holomorphic 1-form $\lambda$ of $X$ such that $\phi(s)=s\otimes\lambda$. A Higgs sheaf ${\mathfrak E}$ is said to be torsion-free (resp. locally free, reflexive, normal, torsion) if the coherent sheaf $E$ is torsion-free (resp. locally free, reflexive, normal, torsion). The support of a Higgs sheaf is the support of the corresponding coherent sheaf, and hence ${\rm supp}(\mathfrak E)=\{x\in X; E_{x}\neq 0\}$. If ${\mathfrak T}=(T,\psi)$ is a torsion Higgs sheaf, from a classical result of Kobayashi [@Kobayashi], we know that ${\rm det}\,T$ admits a nonzero holomorphic section and, in particular, if ${\rm supp}(\mathfrak T)$ has codimension at least two, ${\rm det}\,{\mathfrak T}$ is a trivial holomorphic line bundle.\
As it is well known [@Biswas-Schumacher], on Higgs sheaves we can apply the same operations that we normaly apply to sheaves. For instance, the dual of a Higgs sheaf and its pullback are again Higgs sheaves, and tensor products and the direct sums of Higgs sheaves are Higgs sheaves. If ${\mathfrak E}$ is a Higgs sheaf we denote its dual by ${\mathfrak E}^{*}$, and if $f:Y\longrightarrow X$ is a map between compact complex manifolds, we denote its pullback by $f^{*}{\mathfrak E}$. If now ${\mathfrak E}_{1}$ and ${\mathfrak E}_{2}$ are Higgs sheaves, we denote its tensor product and direct sum by ${\mathfrak E}_{1}\otimes{\mathfrak E}_{2}$ and ${\mathfrak E}_{1}\oplus{\mathfrak E}_{2}$ respectively. Now, a Higgs subsheaf ${\mathfrak F}$ of ${\mathfrak E}$ is a subsheaf $F$ of $E$ such that $\phi(F)\subset F\otimes\Omega_{X}^{1}$, and hence the pair ${\mathfrak F}=(F,\phi|_{F})$ becomes itself a Higgs sheaf. A morphism $f:{\mathfrak E}_{1}\longrightarrow{\mathfrak E}_{2}$ between two Higgs sheaves over $X$, is a morphism $f:E_{1}\longrightarrow E_{2}$ of the corresponding coherent sheaves such that the diagram $$\xymatrix{
E_{1} \ar[r]^{\phi_{1}} \ar[d]^{f} & E_{1}\otimes\Omega_{X}^{1} \ar[d]^{f\otimes 1} \\
E_{2} \ar[r]^{\phi_{2}} & E_{2}\otimes\Omega_{X}^{1} \\
}$$ is commutative. If ${\mathfrak E}=(E,\phi)$ is a Higgs sheaf over $X$, the natural morphism $\sigma:E\rightarrow E^{**}$ is a first example of a Higgs morphism $\sigma:{\mathfrak E}\rightarrow {\mathfrak E}^{**}$. The kernel and the image of Higgs morphisms are Higgs sheaves and the torsion subsheaf of a Higgs sheaf is again a Higgs sheaf (see [@Cardona; @2] for details), these two results will be particularly important in the study of $T$-stability. An exact sequence of Higgs sheaves is an exact sequence of the corresponding coherent sheaves in which each morphism is a morphism of Higgs sheaves.\
Let ${\mathfrak E}$ be a torsion-free Higgs sheaf of rank $r$, from a classical result (see [@Kobayashi] for details of this and what follows) we know that ${\rm det\,}E\cong\left(\bigwedge^{r}E\right)^{**}$ and hence, by using this isomorphism, it is possible to induce a Higgs field on the determinant bundle. As a consequence of this, we see that the determinant bundle of a torsion-free Higgs sheaf is a Higgs line bundle, we denote this bundle by ${\rm det\,}{\mathfrak E}$. Clearly, from this definition we have canonically ${\rm det}\,{\mathfrak E}\cong\left(\bigwedge^{r}{\mathfrak E}\right)^{**}$ as an isomorphism of Higgs bundles. Now, let us consider the short exact sequence of Higgs sheaves $$\xymatrix{
0 \ar[r] & {\mathfrak F} \ar[r] & {\mathfrak E} \ar[r] & {\mathfrak G} \ar[r] & 0
} \nonumber$$ (also called a Higgs extension). If ${\mathfrak E}$ is torsion-free, then ${\mathfrak F}$ is torsion-free, but ${\mathfrak G}$ may have torsion. In this case, we induce a Higgs morphism on ${\rm det}\,G$ using the Higgs fields of ${\rm det}\,{\mathfrak F}$ and ${\rm det}\,{\mathfrak E}$, and the isomorphism ${\rm det}\,G\cong({\rm det}\,F)^{-1}\otimes{\rm det}\,E$. We denote by ${\rm det}\,{\mathfrak G}$ the Higgs line bundle defined by ${\rm det}\,G$ and this induced morphism. In this way we obtain ${\rm det}\,{\mathfrak E}\cong{\rm det}\,{\mathfrak F}\otimes{\rm det}\,{\mathfrak G}$ as an isomorphism of Higgs bundles.\
Suppose now that $X$ is a Kähler manifold with $\omega$ its Kähler form, then the first Chern class of $\mathfrak E$ is by definition the first Chern class of $E$, and hence following Kobayashi [@Kobayashi], $c_{1}(\mathfrak E)=c_{1}({\rm det}\,E)$ and the degree of $\mathfrak E$ is given by $${\rm deg\,}{\mathfrak E} = \int_{X}c_{1}(\mathfrak E)\wedge\omega^{n-1}\,. \label{deg}$$ It is important to note that the degree defined by (\[deg\]) depends on $\omega$ if the complex dimension of $X$ is greater than one. Now, if we denote the rank of $\mathfrak E$ by ${\rm rk\,}{\mathfrak E}$, and if this rank is positive, we introduce the quotient $\mu({\mathfrak E})={\rm deg\,}{\mathfrak E}/{\rm rk\,}{\mathfrak E}$, which is called the slope of the Higgs sheaf. A Higgs sheaf $\mathfrak E$ is said to be $\omega$-stable (resp. $\omega$-semistable), if it is torsion-free and for any Higgs subsheaf ${\mathfrak F}$ with $0<{\rm rk\,}{\mathfrak F}<{\rm rk\,}{\mathfrak E}$ we have the inequality $\mu({\mathfrak F})<\mu(\mathfrak E)$ (resp. $\le$). We say that a Higgs sheaf is $\omega$-polystable if it decomposes into a direct sum of two or more $\omega$-stable Higgs sheaves all these with the same slope.\
This notion of stability was introduced by Hitchin [@Hitchin] and Simpson [@Simpson] as an analog of the Mumford-Takemoto stability for coherent sheaves [@Kobayashi]. However, it is important to note that the coherent sheaf $E$ associated to a torsion-free Higgs sheaf ${\mathfrak E}$, is $\omega$-stable (resp. $\omega$-semistable) if and only if for any proper nontrivial subsheaf $F$ of $E$, we have $\mu(F)<\mu(E)$ (resp. $\le$). Therefore, if ${\mathfrak E}$ is $\omega$-stable (resp. $\omega$-semistable) in the classical sense, it is $\omega$-stable (resp. $\omega$-semistable) as a Higgs object, but the converse is not true in general (see [@Hitchin] for examples). In this sense, the notion of $\omega$-stability for Higgs sheaves is a generalization of the classical notion of $\omega$-stability for coherent sheaves.\
As it is well known (see for instance [@Cardona; @2] or [@Simpson]), for this notion of stability it is suffice to consider only Higgs subsheaves with torsion-free quotients (we will see in the next section that there exists a similar result for $T$-stability). As we said before, Biswas and Schumacher proved [@Biswas-Schumacher] the equivalence between $\omega$-stability and the existence of $HYM$-metrics for Higgs sheaves. To be precise, they proved the following result:
\[HK-corresp. sheaves\] Let ${\mathfrak E}$ be a reflexive Higgs sheaf over a compact Kähler manifold $X$ with Kähler form $\omega$. Then, there exists an admissible HYM-metric on ${\mathfrak E}$ if and only if it is $\omega$-polystable.
A Higgs bundle is by definition a locally free Higgs sheaf. We say that a Higgs bundle is Hermitian flat if there exists a Hermitian metric $h$ on it, such that the Hitchin-Simpson connection ${\cal D}_{h}=D_{h}+\phi+\bar\phi_{h}$ is flat, i.e., if the Hitchin-Simpson curvature ${\cal R}_{h}={\cal D}_{h}\wedge{\cal D}_{h}$ vanishes. Now, following [@Bruzzo-Granha] we know that $${\cal R}_{h} = R_{h} + D'_{h}(\phi) + D''(\bar\phi_{h}) + [\phi,\bar\phi_{h}] \label{R Hitchin-Simpson}$$ where $R_{h}$ is the Chern curvature, $D'_{h}$ and $D''$ are the holomorphic and anti-holomorphic parts of the Chern connection $D_{h}$ and the commutator is the usual abbreviation for $\phi\wedge\bar\phi_{h} + \bar\phi_{h}\wedge\phi$. Now, on the right hand side of (\[R Hitchin-Simpson\]) the third term is the adjoint of the second term and since for Higgs line bundles the commutator is zero, a Higgs line bundle ${\mathfrak L}=(L,\phi)$ is Hermitian flat if and only if $R_{h}=0$ (i.e., $L$ is Hermitian flat in the classical sense) and the Higgs field satisfies $D'_{h}\phi=0$. Notice that in the case of Higgs line bundles, any Higgs morphism is in essence a holomorphic 1-form, hence every holomorphic section of a Higgs line bundle is an invariant section.\
On the other hand, in Complex Geometry there is a well known vanishing theorem for holomorphic line bundles depending on its degree [@Kobayashi], since the degree of a Higgs bundle is the same degree of the corresponding vector bundle, this result can be applied to Higgs line bundles, and hence the classical vanishing theorem in the Higgs context becomes
\[Prop. Line bundles\] Let $\mathfrak L$ be a Higgs line bundle over a compact Kähler manifold $X$. Then\
[**(i)**]{} If ${\rm deg}\,{\mathfrak L}<0$, then ${\mathfrak L}$ admits no nonzero (invariant) holomorphic sections;\
[**(ii)**]{} If ${\rm deg}\,{\mathfrak L}=0$, then every nonzero (invariant) holomorphic section of ${\mathfrak L}$ has no zeros.
Finally, as it is well known, in the case of Higgs bundles over compact Kähler manifolds, Simpson [@Simpson] proved a Hitchin-Kobayashi correspondence, and Bruzzo and Graña Otero proved in [@Bruzzo-Granha] that Higgs bundles admitting $apHYM$-metrics are semistable in the sense of Mumford-Takemoto. The converse of this result has been proved in [@Cardona] in the one-dimensional case and by Li and Zhang [@Jiayu-Zhang] for compact Kähler manifolds of greater dimensions. These results can be summarized as follows:
\[HK-corresp.\] Let ${\mathfrak E}$ be a Higgs bundle over a compact Kähler manifold $X$ with Kähler form $\omega$. Then, there exists a HYM-metric (resp. apHYM-metric) on ${\mathfrak E}$ if and only if it is $\omega$-polystable (resp. $\omega$-semistable).
Notice that, since for compact Kähler manifolds an admissible $HYM$-metric is just a $HYM$-metric, part of Theorem \[HK-corresp.\] is indeed a particular case of Theorem \[HK-corresp. sheaves\]. However, there is known a differential geometric analog of $\omega$-semistability only for bundles[^2] and it is precisely the notion of $apHYM$-metric, a natural extension for Higgs bundles of an approximate Hermitian-Einstein structure for holomorphic vector bundles.
$T$-stability
=============
In order to define the notion of $T$-stability, we need to define first the notion of a weighted flag in the Higgs case. Let ${\mathfrak E}$ be a torsion-free Higgs sheaf over a compact complex manifold $X$, a weighted flag of ${\mathfrak E}$ is a sequence of pairs ${\cal F}=\{({\mathfrak E}_{i},n_{i})\}_{i=1}^{k}$ consisting of Higgs subsheaves $${\mathfrak E}_{1}\subset{\mathfrak E}_{2}\subset\cdots\subset{\mathfrak E}_{k}\subset{\mathfrak E} \nonumber$$ together with positive integers $n_{1},n_{2},...,n_{k}$ and such that $$0<{\rm rk}\,{\mathfrak E}_{1}<{\rm rk}\,{\mathfrak E}_{2}<\cdots<
{\rm rk}\,{\mathfrak E}_{k}<{\rm rk}\,{\mathfrak E}\,. \nonumber$$ Let $r_{i}={\rm rk}\,{\mathfrak E}_{i}$ and $r={\rm rk}\,{\mathfrak E}$. In analogy to the classical case, to each weighted flag ${\cal F}$ we associate the Higgs line bundle $${\mathfrak T}_{\cal F}=\prod_{i=1}^{k}(({\rm det}\,{\mathfrak E}_{i})^{r}\otimes({\rm det}\,{\mathfrak E})^{-r_{i}})^{n_{i}}. \label{def T}$$ We say that a weighted flag ${\cal F}$ is saturated if the quotients ${\mathfrak E}/{\mathfrak E}_{i}$, $i=1,2,...,k$, are all torsion-free. A torsion-free Higgs sheaf ${\mathfrak E}$ over $X$ is said to be $T$-stable (resp. $T$-semistable), if for every weighted flag ${\cal F}$ of ${\mathfrak E}$ and every Hermitian flat Higgs line bundle ${\mathfrak L}$ over $X$, the Higgs line bundle ${\mathfrak T}_{\cal F}\otimes{\mathfrak L}$ admits no nonzero holomorphic sections (resp. every nonzero holomorphic section of ${\mathfrak T}_{\cal F}\otimes{\mathfrak L}$, if any, vanishes nowhere on $X$). Notice that since weighted flags for Higgs sheaves consist of Higgs subsheaves, if a Higgs sheaf is $T$-stable (resp. $T$-semistable) in the classical sense, i.e., as a coherent sheaf, it is also $T$-stable (resp. $T$-semistable) in the Higgs sense. However, as we will see in the next section the converse is not true in general.\
From this definition of $T$-stability we have the following results, which are natural extensions to the Higgs case of classical results of Kobayashi [@Kobayashi].
\[prop. saturated\] Let ${\mathfrak E}$ be a torsion-free Higgs sheaf over a compact complex manifold $X$. Then it is $T$-stable (resp. $T$-semistable) if and only if for every saturated flag ${\cal F}$ of ${\mathfrak E}$ and every Hermitian flat Higgs line bundle $\mathfrak L$ over $X$, the bundle ${\mathfrak T}_{\cal F}\otimes{\mathfrak L}$ admits no nonzero holomorphic sections (resp. every holomorphic section of ${\mathfrak T}_{\cal F}\otimes{\mathfrak L}$, if any, vanishes nowehere on $X$).
[*Proof:*]{} There is nothing to prove in one direction[^3]. Now, in order to prove the other direction, let us assume that such conditions on existence or not of holomorphic sections are satisfied for any saturated flag and any Hermitian flat Higgs line bundle.\
Let ${\cal F}'=\{({\mathfrak E}_{i},n_{i})\}_{i=1}^{k}$ be an arbitrary flag of ${\mathfrak E}$ and ${\mathfrak L}$ a Hermitian flat Higgs line bundle. Let ${\mathfrak T}_{i}$ be the torsion of ${\mathfrak E}/{\mathfrak E}_{i}$. Then, if we define $\tilde{\mathfrak E}_{i}$ as the kernel of the morphism ${\mathfrak E}\rightarrow({\mathfrak E}/{\mathfrak E}_{i})/{\mathfrak T}_{i}$ we obtain the following commutative diagram $$\xymatrix{
& & & 0 \ar[d] & \\
& 0 \ar[d] & & {\mathfrak T}_{i} \ar[d] & \\
0 \ar[r] & {\mathfrak E}_{i} \ar[r]\ar[d] & {\mathfrak E} \ar[r]\ar[d]^{\rm Id} & {\mathfrak E}/{\mathfrak E}_{i} \ar[r]\ar[d] & 0 \\
0 \ar[r] & \tilde{\mathfrak E}_{i}\ar[r]\ar[d] & {\mathfrak E} \ar[r] & {\mathfrak E}/\tilde{\mathfrak E}_{i}
\ar[r]\ar[d] & 0 \\
& \tilde{\mathfrak E}_{i}/{\mathfrak E}_{i}\ar[d] & & 0 & \\
& 0 & & & \\
}$$ with ${\mathfrak T}_{i}\cong\tilde{\mathfrak E}_{i}/{\mathfrak E}_{i}$. Since $\tilde{\mathfrak E}_{i}$ and ${\mathfrak E}_{i}$ are torsion-free, from Section 2 we see that the determinant of $\tilde{\mathfrak E}_{i}/{\mathfrak E}_{i}$ is a Higgs bundle, and consequently also is the determinant of ${\mathfrak T}_{i}$ and we have ${\rm det}\,\tilde{\mathfrak E}_{i}\cong{\rm det}\,{\mathfrak E}_{i}\otimes{\rm det}\,{\mathfrak T}_{i}$. If we use this isomorphism and we consider now the saturated flag[^4] $\tilde{\cal F}=\{(\tilde{\mathfrak E}_{i},n_{
i})\}_{i=1}^{k}$ of ${\mathfrak E}$ we get $$\begin{aligned}
{\mathfrak T}_{\tilde{\cal F}} &=& \prod_{i=0}^{k}(({\rm det}\,\tilde{\mathfrak E}_{i})^{r}\otimes({\rm det}\,{\mathfrak E})^{-r_{i}})^{n_{i}}\\
&\cong& {\mathfrak T}_{{\cal F}'}\otimes\prod_{i=1}^{k}({\rm det}\,{\mathfrak T}_{i})^{rn_{i}}.\end{aligned}$$ Since each ${\mathfrak T}_{i}$ is torsion, from a classical result in [@Kobayashi] each ${\rm det}\,{\mathfrak T}_{i}$ admits a nonzero holomorphic section; Now, since $\tilde{\cal F}$ is saturated, these conditions on the existence or not of holomorphic sections are satisfied for ${\mathfrak T}_{\tilde{\cal F}}\otimes{\mathfrak L}$. At this point, by using the isomorphism above if follows that the same is true also for the Higgs line bundle ${\mathfrak T}_{{\cal F}'}\otimes{\mathfrak L}$. Q.E.D.\
Let ${\mathfrak E}$ be a torsion-free Higgs sheaf over a compact complex manifold $X$. Then\
[**(i)**]{} If ${\rm rk}\,{\mathfrak E}=1$, then ${\mathfrak E}$ is $T$-stable;\
[**(ii)**]{} If ${\mathfrak L}$ is a Higgs line bundle over $X$, then the tensor product ${\mathfrak E}\otimes{\mathfrak L}$ is $T$-stable (resp. $T$-semistable) if and only if ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable);\
[**(iii)**]{} ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable) if and only if its dual ${\mathfrak E}^{*}$ is $T$-stable (resp. $T$-semistable).
[*Proof:*]{} If ${\rm rk}\,{\mathfrak E}=1$, there are no flags to be considered, hence (i) is trivial. Suppose now that ${\mathfrak L}$ is a Higgs line bundle, then in analogy to the classical case, there exists a natural correspondence between flags ${\cal F}=\{({\mathfrak E}_{i},n_{i})\}$ of ${\mathfrak E}$ and flags ${\cal F}\otimes{\mathfrak L}=\{({\mathfrak E}_{i}\otimes{\mathfrak L},n_{i})\}$ of ${\mathfrak E}\otimes{\mathfrak L}$. Now, since ${\mathfrak E}$ and ${\mathfrak E}_{i}$ are torsion-free and we have the identities $$\bigwedge^{r_{i}}({\mathfrak E}_{i}\otimes{\mathfrak L})\cong(\bigwedge^{r_{i}}{\mathfrak E}_{i})\otimes{\mathfrak L}^{r_{i}}\,,
\quad\quad \bigwedge^{r}({\mathfrak E}\otimes{\mathfrak L})\cong(\bigwedge^{r}{\mathfrak E})\otimes{\mathfrak L}^{r} \nonumber$$ we have the following isomorphisms of determinant Higgs bundles $${\rm det}({\mathfrak E}_{i}\otimes{\mathfrak L})\cong{\rm det}\,{\mathfrak E}_{i}\otimes{\mathfrak L}^{r_{i}}\,,
\quad\quad {\rm det}({\mathfrak E}\otimes{\mathfrak L})\cong{\rm det}\,{\mathfrak E}\otimes{\mathfrak L}^r\,. \nonumber$$ Now, using these isomorphisms and the expression (\[def T\]) for the flag ${\cal F}\otimes{\mathfrak L}$ we obtain $$\begin{aligned}
{\mathfrak T}_{{\cal F}\otimes{\mathfrak L}} &=& \prod_{i=1}^{k}\left({\rm det}({\mathfrak E}_{i}\otimes{\mathfrak L})^{r}
\otimes{\rm det}({\mathfrak E}\otimes{\mathfrak L})^{-r_{i}}\right)^{n_{i}} \\
&\cong& \prod_{i=1}^{k}\left(({\rm det}\,{\mathfrak E}_{i})^{r}\otimes
{\mathfrak L}^{r_{i}r}\otimes({\rm det}\,{\mathfrak E})^{-r_{i}}\otimes{\mathfrak L}^{-rr_{i}}\right)^{n_{i}}
\cong {\mathfrak T}_{\cal F} \nonumber\end{aligned}$$ and (ii) follows. Finally, assume that ${\mathfrak E}^{*}$ is $T$-stable (resp. $T$-semistable) and let ${\cal F}=\{({\mathfrak E}_{i},n_{i})\}_{i=1}^{k}$ be a saturated flag of ${\mathfrak E}$. By dualizing the Higgs extension of ${\mathfrak E}$ associated to ${\mathfrak E}_{i}$ we get the exact sequence $$\xymatrix{
0 \ar[r] & ({\mathfrak E}/{\mathfrak E}_{i})^{*} \ar[r] & {\mathfrak E}^{*} \ar[r] & {\mathfrak E}_{i}^{*}
}$$ with ${\rm rk}({\mathfrak E}/{\mathfrak E}_{i})^{*}=r-r_{i}$ and we obtain from this a flag ${\cal F}^{*}=\{(({\mathfrak E}/{\mathfrak E}_{i})^{*},n_{i})\}$ of ${\mathfrak E}^{*}$ with $$({\mathfrak E}/{\mathfrak E}_{k})^{*}\subset\cdots\subset({\mathfrak E}/{\mathfrak E}_{1})^{*}\subset
{\mathfrak E}^{*}\,. \nonumber$$ Now, ${\mathfrak E}$ is torsion-free and ${\mathfrak E}/{\mathfrak E}_{i}$ is torsion-free because the flag ${\cal F}$ is saturated, then ${\rm det}\,{\mathfrak E}^{*}\cong({\rm det}\,{\mathfrak E})^{*}$ and ${\rm det}({\mathfrak E}/{\mathfrak E}_{i})^{*}\cong({\rm det}\,{\mathfrak E}/{\mathfrak E}_{i})^{*}$ and hence $$\begin{aligned}
{\mathfrak T}_{{\cal F}^{*}} &=& \prod_{i=1}^{k}\left(({\rm det}({\mathfrak E}/{\mathfrak E}_{i})^{*})^{r}\otimes({\rm det}\,{\mathfrak E}^{*})^{-(r-r_{i})}\right)^{n_{i}} \\
&\cong& \prod_{i=1}^{k}\left(({\rm det}\,{\mathfrak E}/{\mathfrak E}_{i})^{-r}\otimes({\rm det}\,{\mathfrak E})^{r-r_{i}}\right)^{n_{i}} \\
&\cong& \prod_{i=1}^{k}\left(({\rm det}\,{\mathfrak E}_{i})^{r}\otimes({\rm det}\,{\mathfrak E})^{-r_{i}}\right)^{n_{i}} = {\mathfrak T}_{\cal F}\end{aligned}$$ and it follows that ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable). Conversely, assume that ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable) and let ${\cal F}^{*}=\{({\mathfrak R}_{i},n_{i})\}_{i=1}^{k}$ be a saturated flag of ${\mathfrak E}^{*}$. Then for ${\mathfrak R}_{i}$ we have the short exact sequence $$\xymatrix{
0 \ar[r] & {\mathfrak R}_{i} \ar[r] & {\mathfrak E}^{*} \ar[r] & {\mathfrak H}_{i} \ar[r] & 0
} \label{Higgs ext. dual}$$ with ${\mathfrak H}_{i}={\mathfrak E}^{*}/{\mathfrak R}_{i}$ torsion-free. By dualizing the Higgs extension (\[Higgs ext. dual\]) we get the exact sequence $$\xymatrix{
0 \ar[r] & {\mathfrak H}_{i}^{*} \ar[r] & {\mathfrak E}^{**} \ar[r] & {\mathfrak R}_{i}^{*}\,.
}$$ Since ${\mathfrak E}$ is torsion-free, the natural morphism $\sigma:{\mathfrak E}\rightarrow{\mathfrak E}^{**}$ is injective and we can consider the Higgs sheaf ${\mathfrak E}_{i}=\sigma({\mathfrak E})\cap{\mathfrak H}_{i}^{*}$ as a Higgs subsheaf of ${\mathfrak E}$ with rank $r_{i}=r-{\rm rk}\,{\mathfrak R}_{i}$. From this we have a flag ${\cal F}=\{({\mathfrak E}_{i},n_{i})\}$ of ${\mathfrak E}$ with $${\mathfrak E}_{k}\subset{\mathfrak E}_{k-1}\subset\cdots\subset{\mathfrak E}_{1}\subset{\mathfrak E}\,. \nonumber$$ Now, we define torsion Higgs sheaves ${\mathfrak T}={\mathfrak E}^{**}/{\mathfrak E}$ and ${\mathfrak T}_{i}={\mathfrak H}_{i}^{*}/{\mathfrak E}_{i}\subset{\mathfrak T}$. Again, since ${\mathfrak E}$ is torsion-free, ${\rm det}\,{\mathfrak E}^{**}\cong{\rm det}\,{\mathfrak E}$ and consequently ${\rm det}\,{\mathfrak T}$ is trivial (as a classical bundle) and ${\rm det}\,{\mathfrak H}_{i}^{*}\cong{\rm det}\,{\mathfrak E}_{i}$. From this we get $${\rm det}\,{\mathfrak R}_{i} \cong {\rm det}\,{\mathfrak E}^{*}\otimes({\rm det}\,{\mathfrak H}_{i})^{-1}
\cong {\rm det}\,{\mathfrak E}^{*}\otimes{\rm det}\,{\mathfrak H}_{i}^{*}
\cong {\rm det}\,{\mathfrak E}^{*}\otimes{\rm det}\,{\mathfrak E}_{i}\,. \label{Iso dets}$$ Then, from (\[Iso dets\]) we get $$\begin{aligned}
{\mathfrak T}_{{\cal F}^{*}} &=& \prod_{i=1}^{k}\left(({\rm det}\,{\mathfrak R}_{i})^{r}\otimes({\rm det}\,{\mathfrak E}^{*})^{r_{i}-r}\right)^{n_{i}}\\
&\cong& \prod_{i=1}^{k}\left(({\rm det}\,{\mathfrak E}_{i})^{r}\otimes({\rm det}\,{\mathfrak E}^{*})^{r_{i}}\right)^{n_{i}}\\
&\cong& \prod_{i=1}^{k}\left(({\rm det}\,{\mathfrak E}_{i})^{r}\otimes
({\rm det}\,{\mathfrak E})^{-r_{i}}\right)^{n_{i}} = {\mathfrak T}_{\cal F}\\\end{aligned}$$ From this isomorphism it follows that ${\mathfrak E}^{*}$ is $T$-stable (resp. $T$-semistable) and hence we have proved (iii). Q.E.D.\
The Kähler case
===============
As it is well known [@Kobayashi], if $X$ is Kähler there exists a connection between the Mumford-Takemoto stability and $T$-stability for coherent sheaves. This result extends naturally to Higgs sheaves and can be written as:
\[mu-s–>T-s\] Let ${\mathfrak E}$ be a torsion-free Higgs sheaf over a compact Kähler manifold $X$ with Kähler form $\omega$. If ${\mathfrak E}$ is $\omega$-stable (resp. $\omega$-semistable), then it is $T$-stable (resp. $T$-semistable).
[*Proof:*]{} Assume that ${\mathfrak E}$ is $\omega$-stable (resp. $\omega$-semistable) and let ${\cal F}=\{({\mathfrak E}_{i},n_{i})\}_{i=1}^{k}$ be a flag (not necesarily saturated) of ${\mathfrak E}$. Then, as in the classical case we have $$\begin{aligned}
\int_{X}c_{1}({\mathfrak T}_{\cal F})\wedge\omega^{n-1} &=& \sum_{i=1}^{k}n_{i}\int_{X}c_{1}[({\rm det}\,{\mathfrak E}_{i})^{r}\otimes({\rm det}\,{\mathfrak E})^{-r_{i}}]\wedge\omega^{n-1}\\
&=& \sum_{i=1}^{k}n_{i}\int_{X}(rc_{1}({\mathfrak E}_{i}) - r_{i}
c_{1}({\mathfrak E}))\wedge\omega^{n-1} \\
&=& \sum_{i=1}^{k}n_{i}r_{i}r
(\mu({\mathfrak E}_{i})-\mu({\mathfrak E})) < 0 \end{aligned}$$ (resp. $\le 0$). If ${\mathfrak L}$ is a Hermitian flat Higgs line bundle, in particular $c_{1}({\mathfrak L})=0$ and we have $${\rm deg}({\mathfrak T}_{\cal F}\otimes{\cal L}) = \int_{X}c_{1}({\mathfrak T}_{\cal F}\otimes{\cal L})\wedge\omega^{n-1}
= \int_{X}c_{1}({\mathfrak T}_{\cal F})\wedge\omega^{n-1} < 0 \nonumber$$ (resp. $\le 0$). Therefore, by using Proposition \[Prop. Line bundles\] it follows that ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable). Q.E.D.\
At this point, as a direct consequence of Theorem \[mu-s–>T-s\] and Theorem \[HK-corresp. sheaves\] we obtain the following result for reflexive Higgs sheaves over compact Kähler manifolds.
\[HYM–>T-ps reflex\] Let ${\mathfrak E}$ be a reflexive Higgs sheaf over a compact Kähler manifold $X$. If ${\mathfrak E}$ has an admissible HYM-metric, then it is $T$-semistable and ${\mathfrak E} = \bigoplus_{i=1}^{s}{\mathfrak E}_{i}$, where each ${\mathfrak E}_{i}$ is a $T$-stable Hermitian-Yang-Mills Higgs sheaf with $\mu({\mathfrak E}_{i})=\mu({\mathfrak E})$.
If, on the other hand, we consider locally free Higgs sheaves over compact Kähler manifolds, then there exists a relation between the notion of $HYM$-metric and the concept of $T$-stability. In fact, from Theorem \[mu-s–>T-s\] and Theorem \[HK-corresp.\] we obtain
\[HYM–>T-s\] Let ${\mathfrak E}$ be a Higgs bundle over a compact Kähler manifold $X$. Then\
[**(i)**]{} If ${\mathfrak E}$ admits a HYM-metric, then it is $T$-semistable and ${\mathfrak E} = \bigoplus_{i=1}^{s}{\mathfrak E}_{i}$, where each ${\mathfrak E}_{i}$ is a $T$-stable Hermitian-Yang-Mills Higgs bundle with $\mu({\mathfrak E}_{i})=\mu({\mathfrak E})$;\
[**(ii)**]{} If ${\mathfrak E}$ admits an apHYM-metric, then it is $T$-semistable.
Notice that, the part (i) of Corollary \[HYM–>T-s\] can be seen as a particular case of Corollary \[HYM–>T-ps reflex\], however the part (ii) is new. Now, Kobayashi proved in [@Kobayashi] a partial converse of this Corollary for torsion-free sheaves. The proof of Kobayashi can be easily adapted to Higgs sheaves and gives a partial converse of Theorem \[mu-s–>T-s\]. So we have the following
\[T-s–>mu-s, proj case\] Let ${\mathfrak E}$ be a torsion-free Higgs sheaf over a compact Kähler manifold $X$ with Kähler form $\omega$ and assume either\
[**(a)**]{} The dimension of $H^{1,1}(X,{\mathbb C})$ is equal to one; or\
[**(b)**]{} $\omega$ represents an integral class and ${\rm Pic}(X)/{\rm Pic}^{0}(X)={\mathbb Z}$.\
If ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable), then it is $\omega$-stable (resp. $\omega$-semistable).
[*Proof:*]{} In analogy to the classical proof of Kobayashi, let ${\mathfrak E}'$ be any Higgs subsheaf of ${\mathfrak E}$ with nonzero rank $r'<r$, and consider the Higgs line bundle $${\mathfrak U}=({\rm det}\,{\mathfrak E}')^{r}\otimes({\rm det}\,{\mathfrak E})^{-r'}\, \label{def U}$$ with its degree given by $${\rm deg}\,{\mathfrak U} = \int_{X}(rc_{1}({\mathfrak E}') - r'c_{1}({\mathfrak E}))\wedge\omega^{n-1}
= rr'(\mu({\mathfrak E}') - \mu({\mathfrak E}))\,. \nonumber$$ If $[\omega]$ denotes the cohomology class of $\omega$, from either of hypothesis (a) or (b), we obtain $c_{1}({\mathfrak U})=a[\omega]$ for some $a\in{\mathbb R}$; hence by integrating this formula it follows that ${\rm deg}\,{\mathfrak U}=0$ (resp. $>0$) if and only if $a=0$ (resp. $>0$).\
If ${\mathfrak E}$ is not $\omega$-semistable, there exists a Higgs subsheaf ${\mathfrak E}'$ such that $\mu({\mathfrak E}')>\mu({\mathfrak E})$. Then, for the corresponding Higgs line bundle ${\mathfrak U}$ defined by (\[def U\]) we get ${\rm deg}\,{\mathfrak U}>0$ and hence $a>0$. From this we know there exists a positive integer $p$ such that ${\mathfrak U}^{p}$ admits a nonzero section, say $s$. Now, if $s$ vanishes nowhere on $X$, then ${\mathfrak U}^{p}$ is trivial as a classical bundle and $c_{1}({\mathfrak U}^{p})=0$ and therefore $a=0$, which is a contradiction. Therefore, $s$ necessarily vanishes at some point[^5]. In particular, ${\mathfrak U}^{p}\otimes{\mathfrak L}$ admits a nonzero section, vanishing at some point, for any Hermitian flat Higgs line bundle ${\mathfrak L}$ with $L$ trivial. This shows that ${\mathfrak E}$ is not $T$-semistable if it is not $\omega$-semistable.\
If ${\mathfrak E}$ is not $\omega$-stable, then there exists a Higgs subsheaf ${\mathfrak E}'$ such that $\mu({\mathfrak E}')\ge\mu({\mathfrak E})$. Now, if the inequality is strict it is also not $\omega$-semistable, and hence from the above analysis we conclude that it is not $T$-semistable and in particular it is not $T$-stable. If, on the other hand $\mu({\mathfrak E}')=\mu({\mathfrak E})$, it follows that ${\rm deg}\,{\mathfrak U}=0$ and $a=0$. Hence, the corresponding ${\mathfrak U}$ is flat as a classical bundle. Then, by defining $\tilde{\mathfrak L}=(U^{-1},0)$, with $U$ the holomorphic line bundle associated to ${\mathfrak U}$, it follows that ${\mathfrak U}\otimes\tilde{\mathfrak L}$ is trivial as a classical bundle and hence it admits a nonzero holomorphic section. This shows that ${\mathfrak E}$ is not $T$-stable if it is not $\omega$-stable. Q.E.D.\
As it is well known [@Hitchin], there are Higgs bundles over curves that are stable in the sense of Mumford-Takemoto, that are not stable as classical bundles. From Theorem \[mu-s–>T-s\] these bundles are $T$-stable as Higgs bundles. Now, from Kobayashi [@Kobayashi] it is known that for holomorphic bundles over curves, the notions of Mumford-Takemoto stability and $T$-stability are equivalent; hence, such Higgs bundles are not $T$-stable in the classical sense. This fact shows that $T$-stability is indeed an extension of the classical notion of $T$-stability. Finally, as a direct consequence of Theorem \[T-s–>mu-s, proj case\] and Theorem \[HK-corresp. sheaves\] we get a partial converse of Corollary \[HYM–>T-ps reflex\]. To be precise we have the following result
\[T-s–>admissible HYM\] Let ${\mathfrak E}$ be a reflexive Higgs sheaf over a compact Kähler manifold $X$ and assume that either (a) or (b) of Theorem \[T-s–>mu-s, proj case\] holds. If ${\mathfrak E}$ is $T$-stable, then it has an admissible $HYM$-metric.
This Corollary can be extended to $T$-semistable Higgs sheaves in a very special case. Indeed, if ${\mathfrak E}$ is $T$-semistable and ${\mathfrak E} = \bigoplus_{i=1}^{s}{\mathfrak E}_{i}$, where each ${\mathfrak E}_{i}$ is a $T$-stable Higgs sheaf with $\mu({\mathfrak E}_{i})=\mu({\mathfrak E})$, then from Corollary \[T-s–>admissible HYM\] each ${\mathfrak E}_{i}$ has an admissible $HYM$-metric and (see [@Biswas-Schumacher] or [@Cardona; @2] for details) we get also an admissible $HYM$-metric on ${\mathfrak E}$.\
[**Acknowledgements**]{}\
This paper was mostly done during a stay of the author at the International School for Advanced Studies (SISSA) in Trieste, Italy. The author wants to thank SISSA for the hospitality and support. Finally, the author would like to thank U. Bruzzo for some useful comments and suggestions.
[99]{}
*Stable sheaves and Einstein-Hermitian metrics*, Geometry and analysis on complex manifolds, (World Scientific Publishing, River Edge, NJ, 1994), pp. 39-50.
, International Journal of Mathematics, Vol. 20, No 5 (2009), pp. 541-556.
, Math USSR Izv., Vol. 13, (1978) pp. 499-555.
, J. reine ang. Math., [**612**]{} (2007), pp. 59-79.
*Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case*, Ann. Glob. Anal. Geom., Vol. 42, Number 3 (2012), pp. 349-370 (DOI 10.1007/s10455-012-9316-2).
*Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures*, Ann. Glob. Anal. Geom., Vol. 44, Number 4 (2013), pp. 455-469 (DOI 10.1007/s10455-013-9376-y).
*On vanishing theorems for Higgs bundles*, Differential Geometry and its Applications, Vol. 35, (2014), pp. 95-102 (DOI 10.1016/j.difgeo.2014.06.005).
, Proc. London. Math., [**55**]{} (1987), pp. 59-126.
*On two concepts of stability for vector bundles and sheaves*, Aspects of Math. and its Applications, Elsevier Sci. Publ. B. V., (1986), pp. 477-484.
*Differential geometry of complex vector bundles*, Iwanami Shoten Publishers and Princeton Univ. Press (1987).
*Existence of approximate Hermitian-Einstein structures on semistable Higgs bundles*, Calculus of Variations and Partial Differential Equations (2014), (DOI 10.1007/s00526-014-0733-x).
*Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization*, J. Amer. Math. Soc. [**1**]{} (1988), pp. 867-918.
, Publ. Math. I.H.E.S., [**75**]{} (1992), pp. 5-92.
*Curvature and Betti numbers*, Annals of Math. Studies [**32**]{} (Princeton Univ. Press, 1953).
[^1]: Electronic addresses: s.a.h.cardona@gmail.com, andres.holguin@cimat.mx
[^2]: Indeed, even in the classical case of reflexive sheaves, there is no yet an equivalence of $\omega$-semistability (see [@Bando-Siu] for more details).
[^3]: If ${\mathfrak E}$ is $T$-stable (resp. $T$-semistable), the conditions on existence or not of holomorphic sections hold for any flag and any Hermitian flat Higgs line bundle; in particular this is true if the flag is saturated.
[^4]: Since ${\mathfrak E}_{i}\subset{\mathfrak E}_{i+1}$, there exists a map ${\mathfrak E}/{\mathfrak E}_{i}\rightarrow{\mathfrak E}/{\mathfrak E}_{i+1}$ such that the obvious diagram commutes. Now, any element in $\tilde{\mathfrak E}_{i}$ can be projected on $\tilde{\mathfrak E}_{i}/{\mathfrak E}_{i}\cong{\mathfrak T}_{i}\subset{\mathfrak T}_{i+1}$, so it is zero in ${\mathfrak E}/\tilde{\mathfrak E}_{i+1}$ and hence $\tilde{\mathfrak E}_{i}\subset\tilde{\mathfrak E}_{i+1}$ and $\tilde{\cal F}$ is a flag, which is obviously saturated.
[^5]: Notice that non $T$-semistability means that there exists a nonzero section of $T_{\cal F}\otimes{\cal L}$ for some flag ${\cal F}$ and some ${\mathfrak L}$ Hermitian flat Higgs line bundle, and such a section vanishes at least at some point.
|
---
abstract: |
We measure the obscured star formation in $z\sim 1$ early-type galaxies. This constrains the influence of star formation on their optical/near-IR colors, which, we found, are redder than predicted by the model by Bruzual & Charlot (2003). From deep ACS imaging we construct a sample of 95 morphologically selected early-type galaxies in the HDF-N and CDF-S with spectroscopic redshifts in the range $0.85<z<1.15$. We measure their $24~\mu\rm{m}$ fluxes from the deep GOODS-MIPS imaging and derive the IR luminosities and star formation rates. The fraction of galaxies with $>2\sigma$ detections ($\sim 25~\mu\rm{Jy}$) is $17_{-4}^{+9}\%$. Of the 15 galaxies with significant detections at least six have an AGN. Stacking the MIPS images of the galaxies without significant detections and adding the detected galaxies without AGN we find an upper limit on the mean star formation rate (SFR) of $5.2\pm 3.0~M_{\odot}~\rm{yr^{-1}}$, and on the mean specific SFR of $4.6\pm
2.2\times10^{-11}~\rm{yr}^{-1}$. Under the assumption that the average SFR will decline at the same rate as the cosmic average, the *in situ* growth in stellar mass of the early-type galaxy population is less than $14\pm 7\%$ between $z=1$ and the present. We show that the typically low IR luminosity and SFR imply that the effect of obscured star formation (or AGN) on their rest-frame optical/near-IR SEDs is negligible for $\sim 90\%$ of the galaxies in our sample. Hence, their optical/near-IR colors are most likely dominated by evolved stellar populations. This implies that the colors predicted by the Bruzual & Charlot (2003) model for stellar populations with ages similar to those of $z\sim 1$ early-type galaxies ($\sim 1-3~\rm{Gyr}$) are most likely too blue, and that stellar masses of evolved, high-redshift galaxies can be overestimated by up to a factor of $\sim 2$.
author:
- 'A. van der Wel, M. Franx, G.D. Illingworth P.G. van Dokkum'
title: 'Low Star Formation Rates for $z=1$ Early-Type Galaxies in the Very Deep GOODS-MIPS Imaging: Implications for their Optical/Near-Infrared Spectral Energy Distributions'
---
INTRODUCTION
============
A convenient way to estimate stellar masses of galaxies is through modeling their spectral energy distributions (SEDs) with stellar population models [e.g., @worthey94; @vazdekis96; @bruzual03; @maraston05]. This is the most efficient method to quantify the stellar mass function at high redshift [e.g., @kauffmann04; @drory04; @forster04; @rudnick06; @borch06]. The robustness of this method relies on the reliability of the assumed model parameters, such as the star formation history. The correctness of the models, given the star formation history and other parameters, is an obvious requirement as well.
With the advent of the Infrared Array Camera [IRAC, @fazio04] on the *Spitzer Space Telescope* [@werner04] the rest-frame near-infrared (near-IR) has become a commonly used part of the SED to infer stellar masses of $z\gtrsim 1$ galaxies. However, especially in the near-IR different stellar population models differ from each other [@maraston05], which indicates that there might be systematic biases in such mass estimates. In @vanderwel06a we used IRAC imaging of a sample of early-type galaxies at $z\sim 1$ in the GOODS-South field with dynamically measured masses in order to infer the evolution of the rest-frame near-IR $M/L$ of early-type galaxies between $z=1$ and the present. We compared this with the evolution of the rest-frame optical $M/L$ and the predictions from several stellar population models. We found that the near-IR $M/L$ of the $z\sim 1$ galaxies are lower (or, the rest-frame $B-K$ colors redder) than predicted by, for example, the model by @bruzual03 for a range of model parameters. This indicates that stellar mass estimates inferred from near-IR photometry suffer from significant systematic uncertainties [@vanderwel06b].
{width="18cm"}
The most straightforward interpretation of this result is that the model colors are too blue, but this depends on the assumption that the stellar populations of early-type galaxies are simple and can be well described by a single burst stellar population. If we add a significant population of young, heavily obscured stars ($\gtrsim
10\%$ in mass), then we would reconcile the observations with the Bruzual & Charlot model [@vanderwel06b]. Moreover, such a high level of obscured star formation could account for the observed increase of the stellar mass density of red sequence galaxies between $z=1$ and the present [@bell04b; @faber05; @borch06; @brown06]. Hence, determination of the star formation rate is interesting and relevant in its own right. Recently, @rodighiero07 showed through pan-chromatic SED modeling that, indeed, a significant fraction of the early-type galaxy population at $z<1$ shows signs of hidden activity.
In this paper we construct a sample of morphologically selected early-type galaxies with spectroscopic redshifts at $z\sim 1$, and measure (upper limits of) their $24~\mu\rm{m}$ fluxes with the Multiband Imaging Photometer for Spitzer [MIPS, @rieke04] on the *Spitzer Space Telescope* (Sec. 2). We use these to constrain their star formation rates (SFRs) and the increase of their stellar masses with cosmic time in Sec. 3. Then we proceed, in Sec. 4, to test the hypothesis that the rest-frame optical/near-IR colors of $z\sim 1$ early types are significantly affected by obscured star formation or Active Galactic Nuclei (AGN), and can explain the observed red colors. We summarize our conclusions in Sec. 5. Throughout the paper we adopt the concordance cosmology, $(\Omega_{\rm{M}},~\Omega_{\Lambda},~h) = (0.3,~0.7,~0.7)$.
SAMPLE SELECTION AND MULTI-WAVELENGTH PHOTOMETRY
================================================
We select galaxies with spectroscopic redshifts and early-type morphologies from *Hubble Space Telescope*/Advanced Camera for Surveys (HST/ACS) imaging from the Great Observatories Origin Deep Survey [GOODS, @giavalisco04] in the Chandra Deep Field-South (CDF-S) and the Hubble Deep Field-North (HDF-N). Several spectroscopic surveys conducted in the CDF-S [@lefevre04; @mignoli05; @vanderwel05; @vanzella06] are combined to give 381 spectroscopic redshifts in the range $0.85< z< 1.15$. Similarly, the compilation from [@wirth04] and the fundamental plane study by @treu05b are combined to give 404 redshifts in the same redshift range in the HDF-N.
Early-type galaxies are morphologically distinguished from late-type galaxies by fitting Sersic models to the F850LP (hereafter, $z_{850}$) GOODS images of all galaxies using the technique developed by @vandokkumfranx96. The best fitting $n$-parameter was determined for every galaxy, with $n$ an integer, 1, 2, 3 or 4. The asymmetry $A$, the fraction of the total $z_{850}$ flux not situated in the point-symmetric part of the light distribution, was also determined for each object. The criteria used to select early-type galaxies are $n\geq 3$, $A<0.15$, $\chi^2<3$, and $r_{\rm{eff}}>0\farcs 09$. The latter three criteria are chosen upon visual inspection of the images and the residuals of the fits. The limit $\chi^2<3$ excludes galaxies that are poorly fit by any model, and the requirement $r_{\rm{eff}}>0\farcs 09$ excludes galaxies that are very compact and for which the shape of the light distribution cannot be reliably inferred. The majority of the galaxies satisfying these criteria have early-type morphologies as visually determined, but a small number of Sa galaxies with clear spiral structure and compact galaxies with small-scale irregularities, most likely mergers, are rejected upon visual inspection. The sample contains 95 early-type galaxies with E+S0+S0/a morphologies, with $T\leq 0$ in the classification scheme of @postman05. Total $z_{850}$-band magnitudes are derived from the fitted profiles. PSF-matched ACS and GOODS IRAC images are used to measured rest-frame $B-K$ colors within $5''$ diameter apertures, as described by @vanderwel06a. Even though the used spectroscopic surveys are neither complete nor necessarily representative for the $z\sim 1$ galaxy population, there is no reason to assume that our morphologically selected sample has a bias in favor of or against early-type galaxies with obscured star formation or AGN. Therefore, we can assume that our sample is representative for $z\sim 1$ early types as far as their IR properties are concerned.
We use the GOODS MIPS images[^1] to obtain $24~\mu\rm{m}$ photometry for our sample of 95 early-type galaxies. Six galaxies turn out to be located in areas with exposure times less than 10% of the deepest parts of the images. We henceforth exclude these objects from the analysis, such that we have a final sample of 89 early-type galaxies. The publicly available catalogs from the GOODS team[^2] are used as a reference to verify our own photometry, which we push deeper than than the $83~\mu\rm{Jy}$ flux limit from the GOODS catalogs. This limit is a trade-off between completeness and the number of spurious detections, but since we already know the positions of our objects, deeper photometry is justified. Deeper photometry is necessary because @bell05 already concluded that the vast majority of early-type galaxies at $z\sim 0.7$ are not detected down to $83~\mu\rm{Jy}$, an upper limit that is too conservative for our goals.
We determined the two-dimensional background across the image with the Sextractor software [@bertin96], using the global background setting. We subtracted this background to obtain an image with background set to zero. Following, e.g., @papovich04 and @zheng06, we produce composite PSF-images for both fields, comprised of isolated stars (identified in the ACS images), which we use to create model images of MIPS sources at the positions of the galaxies in our sample and objects in their vicinity. The $3.6~\mu
\rm{m}$ IRAC images are used as a reference to identify these sources and pinpoint their positions[^3] The total flux of an object is inferred from the PSF fitting model. By default, the positions of the $24~\mu m$ sources are left as free parameters, however, in case of obviously erroneous results, we fix the positions at the IRAC positions. This is necessary for very low $S/N$ sources. In all cases, IRAC and ACS images are used to visually identify the counterpart of $24~\mu\rm{m}$ sources. This procedure gives robust $24~\mu\rm{m}$ flux measurements for 83 out of 89 galaxies in our sample, of which 15 have a significant $24~\mu\rm{m}$ fluxes, with a signal-to-noise ratio of at least two. For six galaxies we cannot produce reliable measurements because the centers of the low $S/N$ $24~\mu\rm{m}$ objects cannot be defined sufficiently accurate to identify their counterparts with any confidence. We label these objects as ’possibly misidentified’.
The photometric error is dominated by noise and the uncertainty in the background level. In addition we include a 5% error due to the uncertainty in the aperture correction and a 2% error due to the uncertainty in the absolute photometric calibration.
10 of the galaxies with the brightest MIPS counterparts are also found in the GOODS catalogs. The total fluxes as listed in the GOODS catalogs agree within $\sim 5\%$ with the values that we derive, except for one confused object for which we determined an accurate flux measurement with the deconvolution method described above. As an *a posteriori* verification of our morphological classification methodology we show ACS $z_{850}$-band cutout images of nine of these galaxies in Fig. \[stamps\], together with nine randomly selected galaxies without significant MIPS counterparts: even the IR-bright galaxies in our sample are genuine early types, as far as their optical morphologies are concerned. We conclude that our morphological selection criteria described above are sufficiently stringent to exclude all late-type galaxies.
We list the measured fluxes in Table 1, and in Fig. \[z\_histM\]a we show the redshift distribution of our sample. The mean redshift is $z=0.984$. The shaded regions in Fig. \[z\_histM\]a show the galaxies with MIPS counterparts. The lightly shaded regions indicate the six galaxies with possibly misidentified MIPS counterparts. The fraction of galaxies in our sample with MIPS counterparts (typically $\gtrsim 25~\mu\rm{Jy}$) is $f=0.17_{-0.04}^{+0.09}$. If we adopt the brighter flux limit of $83~\mu\rm{Jy}$, the limit used for the GOODS MIPS catalogs, we find $f=0.11\pm 0.03$. In Fig. \[z\_histM\]b we show the $24~\mu\rm{m}$ flux distribution. The typical flux of the objects with significant detections is $S_{24}\sim 100~\mu\rm{Jy}$, and ranges from $\sim 25~\mu\rm{Jy}$ up to almost $1~\rm{mJy}$. As mentioned above, most galaxies in the sample have no significant $24~\mu\rm{m}$ counterparts. For those galaxies the $2\sigma$ flux levels, i.e., $2\times$ the photometric error, are shown.
CONSTRAINTS ON THE STAR FORMATION RATE
======================================
We use $S_{24}$ to constrain the bolometric infrared luminosity and SFR. @papovich06 have shown that this is feasible with reasonable accuracy. First, given $S_{24}$, we compute $L_{12}=\nu
L_{\nu,12~\mu\rm{m}}$ for $z=1.0$ and $L_{15}=\nu L_{\nu,15~\mu\rm{m}}$ for $z=0.6$. With the conversions from @chary01 (Equations 4 and 5) we estimate the associated values for the bolometric infrared luminosity $L_{\rm{IR}}$. We introduce a $K$-correction by interpolating between the values for $L_{\rm{IR}}$ inferred from $L_{12}$ and $L_{15}$ to obtain $L_{\rm{IR}}$ at the observed redshift $z$ of each object. The $K$-correction is generally small (typically 10%) since all redshifts are in the range $0.85<z<1.15$. Finally, assuming a Salpeter IMF, $L_{\rm{IR}}$ is converted into a SFR as $(1.71\times10^{-10}~L_{\rm{IR}}/L_{\odot})~M_{\odot}~\rm{yr}^{-1}$ [@kennicutt98].
The systematic uncertainties in the derived $L_{\rm{IR}}$ and SFR are considerable. According to @chary01 the uncertainty in the transformation of $L_{12}$ into $L_{\rm{IR}}$ is of order 50%. Furthermore, as noted by @papovich06, the models by @dale02 yield $L_{\rm{IR}}$ that are lower by a factor of 2-3 for the most luminous objects ($L_{\rm{IR}}>10^{12.5}~L_{\odot}$). For less luminous objects, like the objects in our sample, the differences are smaller, therefore we adopt a systematic uncertainty of 50%. Finally, the conversion of $L_{\rm{IR}}$ into SFR is uncertain by about 30%, such that the total uncertainty in the derived SFR is a factor of two.
We have 15 galaxies with significant and secure detections, six of which have X-ray counterparts [@alexander03], with total X-ray luminosities in the range $L_{\rm{X}}=1-25\times10^{42}~\rm{erg~s^{-1}}$, which most certainly means that these galaxies harbor type 2 AGN. This is corroborated by the fact that in all cases at least half of $L_{\rm{X}}$ is due to hard X-rays.
Assuming that the $24~\mu\rm{m}$ flux of the other nine galaxies is due to dust heated by star formation, we find SFRs ranging from 5 to 80 $M_{\odot}~\rm{yr^{-1}}$. The IRAC colors of the two galaxies with SFR exceeding 50 $M_{\odot}~\rm{yr^{-1}}$ are consistent with the colors of a star-forming galaxy, and, moreover, in the rest-frame UV F435W and F606W ACS filters these galaxies clearly show irregular morphologies. We stress, however, that, according to their rest-frame optical morphologies, these galaxies are genuine early-type galaxies with $\sim 90\%$ of the $z_{850}$-band flux accounted for by a smooth De Vaucouleurs profile (see Fig. \[stamps\]).
In order to constrain the SFR of the galaxies without significant individual detections we stack their MIPS images. We omit the galaxies with X-ray counterparts. The stacked image shown in Figure \[stack\] is created by co-adding the individual images, clipping the pixels (outside a 6” radius aperture centered on the fiducial position of the co-added object) at $2.3\sigma$ to mask surrounding objects [see also, e.g., @burgarella06]. Then we proceed to determine the total flux within a $12"$ diameter aperture. To compute the total flux, we subtract the background (the median in a concentric annulus between $30"$ and $40"$) and multiply by the aperture correction 1.697. The uncertainty is determined by the noise properties of the stacked image, the systematic uncertainties in the zero point calibration and aperture correction, and an additional uncertainty of 5% due to sub-pixel variations in the source positions of the individual images [@zheng06]. The measured flux is $6.8\pm 1.2~\mu\rm{Jy}$ per galaxy. This corresponds to a SFR of $1.2~M_{\odot}~\rm{yr}^{-1}$ for a galaxy at the average redshift of the sample ($z=1$), which should be regarded an upper limit to the true SFR. Some of the flux will be due low-luminosity AGN, and, in addition, low levels of silicate emission can be expected from circumstellar dust around Asymptotic Giant Branch stars [see, e.g., @bressan98; @piovan03; @bressan06]. It is beyond the scope of this paper to constrain the various contributions of the measured flux.
The average SFR of the galaxies without significant detections (those used to create the stacked image) and the nine galaxies with significant detections but without X-ray counterparts is $5.2\pm3.0~M_{\odot}~\rm{yr}^{-1}$. This is an order of magnitude lower than the SFR of the typical MIPS source at $z\sim 1$ [@perez05]. Also, $\sim 80\%$ of the star formation is accounted for by $\sim 10\%$ of the galaxies, which suggests either interlopers or that such obscured star-formation events in early-type galaxies are short lived.
Next, we estimate an upper limit on the specific SFR. We assume that $M/M_{\odot}=2 \times L_{\rm{B}}/L_{\odot,\rm{B}}$, which is the typical $M/L$ as was determined by recent $z\sim 1$ fundamental plane studies [@vanderwel05; @treu05b]. The inferred masses are typically in the range of $5\times 10^{10} - 2\times 10^{11}
M_{\odot}$. We find an upper limit for the average specific SFR of $4.6\pm 2.2\times10^{-11}~\rm{yr}^{-1}$, that is, a growth in stellar mass of 5% per Gyr. This is two orders of magnitude lower than the specific SFR of a typical MIPS source at $z\sim 1$, which has a stellar mass of $M/M_{\odot}=10^{10} M_{\odot}$ [@perez05]. We can use the specific SFR to crudely constrain the growth in stellar mass of the early-type galaxy population between $z\sim 1$ and the present. In the case that the SFR remains constant for the population as a whole, i.e., that obscured bursts of star formation are as prevalent in the local universe as they are at $z=1$, the stellar mass would increase by $35\pm 17\%$. It is quite unlikely that the average SFR in early types has remained constant over the past 7.5 Gyr, as many studies have shown that the SFR has decreased by an order of magnitude since $z\sim 1$ [e.g., @madau96; @wolf03; @lefloch05; @bell05; @perez05; @zheng06]. If we assume that the SFR declines exponentially and by a factor of 10 between $z=1$ and the present, the growth in stellar mass is $14\pm
7\%$. We stress that these numbers are upper limits due to the various other potential contributors to the measured IR flux. Most likely, the true *in situ* growth of the stellar mass of the early-type galaxy population is still lower.
These upper limits are consistent with the residual stellar mass growth of $\sim 7\%$ estimated by @gebhardt03 and the residual star formation of $\sim 2~M_{\odot}~\rm{yr^{-1}}$ derived by @koo05 for early-type galaxies and bulges in the Groth Strip Survey. @treu05b find significant young stellar populations in low-mass early-type galaxies ($M<10^{11}~M_{\odot}$) at redshifts $0.2<z<1$ in the HDF-N, which suggests a considerable growth in stellar mass ($20-40\%$) between $z=1.2$ and the present. For more massive galaxies, they find that the growth in stellar mass is negligible. The upper limits that we derive here are marginally consistent with a mass increase of more than $20\%$, but then we have to assume that all the observed $24~\mu\rm{m}$-flux is due to star formation, which is probably unrealistic. The specific SFRs for galaxies more and less massive than $10^{11}~M_{\odot}$ (the median mass) are $5.1\pm 2.5\times10^{-11}~\rm{yr}^{-1}$ and $3.3\pm
1.6\times10^{-11}~\rm{yr}^{-1}$, respectively. This difference is not statistically significant since the co-added fluxes only differ from each other on the $1.2\sigma$ level. Still, if anything, the specific SFR of high-mass early types is higher than that of low-mass early types, but, in addition to the low significance of the measurement, we should keep in mind that high mass galaxies are more likely to have AGN that might contribute to the $24~\mu\rm{m}$ flux. To reconcile these results with the large fraction of young stars in low-mass early types [@treu05b], these young stellar populations must have formed in other galaxies that later became part of an early type, or at a time when the galaxies had not yet attained their early-type morphologies.
We conclude that the *in situ* SFR of the early-type galaxy population at $z\sim 1$ is low, and can only account for an increase in the stellar mass density of early-type galaxies by $\lesssim 20\%$ between $z=1$ and the present. Additional mergers and/or morphological transformations of galaxies are required to explain the observed increase in stellar mass density of red galaxies by a factor of two [@bell04b; @faber05; @borch06; @brown06].
THE EFFECT OF STAR FORMATION ON THE OPTICAL/NEAR-IR COLOR
=========================================================
Now we explore the question whether star formation affects the rest-frame optical/near-IR colors of the galaxies in our sample, or whether light from evolved stellar populations dominates that part of their SEDs, as is usually assumed. In order to do so, we use the (upper limits on the) specific star-formation rate derived in the previous section, and compare this with the rest-frame $B-K$ color (see Fig. \[ssfr\_bk\]). The median $B-K$ color of the galaxies without significant $24~\mu\rm{m}$ fluxes is $B-K=3.52$. The median $B-K$ of the 15 galaxies with significant $24~\mu\rm{m}$ fluxes is 0.2 mag redder, whereas the uncertainty in the $B-K$ color is only $\lesssim 0.05$ mag. This implies that obscured activity can indeed affect the $B-K$ colors of galaxies.
We compare the colors of the galaxies in our sample with the expected color of a $z=1$ galaxy with an evolved stellar populations. We estimate this expected color as follows. Given the measured evolution of $M/L$ [@treu05b; @vanderwel05], the @bruzual03 model (with solar metallicity and a Salpeter IMF) predicts a certain amount of evolution in $B-K$. Therefore, from the $B-K$ color of local early types [@vanderwel06a], we can derive the expected color for $z\sim 1$ early types. We find $B-K=3.29$. We note that the $B-K$ colors of the galaxies with measured dynamical $M/L$ [@vanderwel06a] do not deviate from those of the larger sample presented in this paper.
The median observed color is 0.2 mag redder than the expected color $B-K=3.29$ (see Fig. \[ssfr\_bk\]). In particular, the galaxies without significant $24~\mu\rm{m}$ counterparts are redder than $B-K=3.29$. The question is how much room the upper limits on their $24~\mu\rm{m}$ fluxes leave for attenuation of the $B-K$ color by obscured star formation. With the star-formation rates derived in Sec. 3 we can constrain this scenario.
In Fig. \[ssfr\_bk\] we show two-component Bruzual-Charlot models, consisting of an evolved stellar population with $B-K=3.29$ (see above) and a 200 Myr old stellar population with a constant star formation rate (solar metallicity, Salpeter IMF). Varying the age of the young population with constant star formation between 50 and 500 Myr does not significantly change the models shows in Fig. \[ssfr\_bk\]. Besides the specific SFR, the attenuation $A_{\rm{V}}$ (increasing from bottom to top) is the only other variable. Only the young component is attenuated: $A_{\rm{V}}$ of the evolved component with $B-K=3.29$ is assumed to be zero. We assume the @calzetti00 extinction law.
As expected, models with low $A_{\rm{V}}$ predict blue colors for high SFRs, whereas models with high $A_{\rm{V}}$ predict red colors. Models with highly obscured star formation can reach $B-K$ colors that match those of the galaxies in our sample. However, the associated SFRs of those models are only observed for a handful of objects that have significant MIPS detections. For the majority of the galaxies in our sample, the observed SFRs are much lower than expected on the basis of these models. This implies these models are inconsistent with the red colors of the galaxies in our sample. As indicated in Sec. 3, the measured SFRs are systematically uncertain on the level of a factor of two or so. The discrepancy with the model SFRs is much larger than this (an order of magnitude for most galaxies).
Besides obscured SFR, obscured AGN could potentially also redden the $B-K$ color of a galaxy. However, the vast majority of the galaxies in our sample do not show evidence for nuclear activity in the IR or in X-ray [see also @rodighiero07]. It is highly unlikely that AGN affect the $B-K$ colors of the majority of the galaxies without leaving a trace at other wavelengths that are commonly used to identify AGN. The most straightforward conclusion is that the @bruzual03 model predicts colors that are too blue for stellar populations in the age range of those of early-type galaxies at $z\sim
1$ ($1-3~\rm{Gyr}$, assuming solar metallicity).
SUMMARY
=======
From MIPS $24~\mu\rm{m}$ imaging we derived constraints on the IR luminosities of a sample of 89 morphologically selected early-type galaxies at $z\sim 1$ with the purpose to identify obscured star formation or AGN activity. We find that 15 ($17^{+9}_{-4}\%$) have $>2\sigma$ ($\sim 25~\mu\rm{Jy}$) $24~\mu\rm{m}$ counterparts. The X-ray luminosities of six of these indicate that an obscured AGN is responsible for the IR emission. These findings are consistent with recent work by @rodighiero07 who study obscured activity in a $z<1$ sample of morphologically selected early-type galaxies.
We derive an upper limit on the $24~\mu\rm{m}$ flux of the galaxies without significant individual detections by stacking their MIPS images. When we add this sample to the galaxies with significant detections but without luminous AGN we find $5.2\pm
3.0~M_{\odot}~\rm{yr^{-1}}$ as the upper limit of the average star formation rate. If we assume that the SFR of the early-type population as a whole is constant between $z=1$ and the present, this implies that the increase in stellar mass density of the early-type galaxy population through *in situ* star formation is at most $35\pm 17\%$. More realistically, if we assume that the average SFR declines by an order of magnitude between $z=1$ and the present, i.e., if we assume that it follows the evolution of the average cosmic SFR, we find $14\pm 7\%$. This amount is too small to explain the observed increase by a factor of $\sim 2$ [@bell04b; @faber05; @borch06; @brown06].
$10\%$ of the galaxies account for as much as $\sim 80\%$ of the integrated IR luminosity, i.e., the majority of the galaxies are quiescent in terms of star formation and AGN activity [see also, @bell05]. The low IR luminosities of the galaxies imply that the optical/near-IR colors of $z\sim 1$ early-type galaxies are dominated by their evolved stellar populations, and are most likely not significantly affected by obscured star formation or AGN (see Sec. 4). Hence, the conclusions from @vanderwel06a and @vanderwel06b hold: the rest-frame $B-K$ colors of the stellar populations of $z\sim 1$ early-type galaxies are most likely redder than predicted by the stellar population model from @bruzual03, and, consequently, stellar masses of evolved galaxies at high redshift that are inferred from optical/near-IR photometry are overestimates by up to a factor of two relative to stellar mass estimates at low redshift.
[47]{} natexlab\#1[\#1]{}
, D. M. et al. 2003, , 126, 539
, E. F. et al. 2004, , 608, 752
, E. F. et al. 2005, , 625, 23
, E., & [Arnouts]{}, S. 1996, , 117, 393
, A. et al. 2006, , 453, 869
, A., [Granato]{}, G. L., & [Silva]{}, L. 1998, , 332, 135
, A. et al. , 639, L55
Brown, M. et al. 2006, , in press, astro-ph/0609584
, G., & [Charlot]{}, S. 2003, , 344, 1000
, D. et al. 2006, , 450, 69
, D., [Armus]{}, L., [Bohlin]{}, R. C., [Kinney]{}, A. L., [Koornneef]{}, J., & [Storchi-Bergmann]{}, T. 2000, , 533, 682
, R., & [Elbaz]{}, D. 2001, , 556, 562
, D. A., & [Helou]{}, G. 2002, , 576, 159
, N., [Bender]{}, R., [Feulner]{}, G., [Hopp]{}, U., [Maraston]{}, C., [Snigula]{}, J., & [Hill]{}, G. J. 2004, , 608, 742
, N. M. et al. 2004, , 616, 40
, S. M. et al. 2005, , submitted, astro-ph/0506044
, G. G. et al. 2004, , 154, 10
, K. et al. 2003, , 597, 239
, M. et al. 2004, , 600, L93
, G. et al. 2004, , 353, 713
, R. C. 1998, , 36, 189
, D. C. et al. 2005, , 157, 175
, O. et al. 2004, , 428, 1043
, E. et al. 2005, , 632, 169
, P., [Ferguson]{}, H. C., [Dickinson]{}, M. E., [Giavalisco]{}, M., [Steidel]{}, C. C., & [Fruchter]{}, A. 1996, , 283, 1388
, C. 2005, , 362, 799
, M. et al. 2005, , 437, 883
, C. et al. 2004, , 154, 70
, C. et al. 2006, , 640, 92
, P. G. et al. 2005, , 630, 82
, L., [Tantalo]{}, R., & [Chiosi]{}, C. 2003, , 408, 559
, M. et al. 2005, , 623, 721
, G. H. et al. 2004, , 154, 25
, G. et al. 2007, , accepted, astro-ph/0701178
2006, , accepted, astro-ph/0606536
, T. et al. 2005, , 633, 174
, A., [Franx]{}, M., [van Dokkum]{}, P. G., [Huang]{}, J., [Rix]{}, H.-W., & [Illingworth]{}, G. 2006, , 636, L21
, A., [Franx]{}, M., [van Dokkum]{}, P. G., [Rix]{}, H.-W., [Illingworth]{}, G. D., & [Rosati]{}, P. 2005, , 631, 145
, A., [Franx]{}, M., [Wuyts]{}, S., [van Dokkum]{}, P. G., [Huang]{}, J., [Rix]{}, H.-W., & [Illingworth]{}, G. 2006, , 652, 97
, P. G., & [Franx]{}, M. 1996, , 281, 985
, E. et al. 2006, , 454, 423
, A., [Casuso]{}, E., [Peletier]{}, R. F., & [Beckman]{}, J. E. 1996, , 106, 307
, M. W. et al. 2004, , 154, 1
, G. D. et al. 2004, , 127, 3121
, C., [Meisenheimer]{}, K., [Rix]{}, H.-W., [Borch]{}, A., [Dye]{}, S., & [Kleinheinrich]{}, M. 2003, , 401, 73
, G. 1994, , 95, 107
, X. Z., [Bell]{}, E. F., [Rix]{}, H.-W., [Papovich]{}, C., [Le Floc’h]{}, E., [Rieke]{}, G. H., & [P[é]{}rez-Gonz[á]{}lez]{}, P. G. 2006, , 640, 784
[^1]: http://data.spitzer.caltech.edu/popular/goods/Documents/goods\_dr3.html
[^2]: http://www.stsci.edu/science/goods/
[^3]: The GOODS ACS, IRAC and MIPS images are registered to the same world coordinate system with high accuracy, with virtually no systematic offset ($<0.1"$) and a rms of $\sim
0.4''$ in the difference between the centroid positions of bright MIPS sources and their IRAC counterparts, such that uncertainties therein do not affect our measurements.
|
---
abstract: |
We present stellar evolutionary models covering the mass range from 0.4 to 1 M$_{\odot}$ calculated for metallicities Z=0.020 and 0.001 with the MHD equation of state (Hummer & Mihalas, 1988; Mihalas et al. 1988; Däppen et al. 1988). A parallel calculation using the OPAL (Rogers et al. 1996) equation of state has been made to demonstrate the adequacy of the MHD equation of state in the range of 1.0 to 0.8 M$_{\odot}$ (the lower end of the OPAL tables). Below, down to 0.4 M$_{\odot}$, we have justified the use of the MHD equation of state by theoretical arguments and the findings of Chabrier & Baraffe (1997).
We use the radiative opacities by Iglesias & Rogers (1996), completed with the atomic and molecular opacities by Alexander & Fergusson (1994). We follow the evolution from the Hayashi fully convective configuration up to the red giant tip for the most massive stars, and up to an age of 20 Gyr for the less massive ones. We compare our solar-metallicity models with recent models computed by other groups and with observations.
The present stellar models complete the set of grids computed with the same up-to-date input physics by the Geneva group \[Z=0.020 and 0.001, Schaller et al. (1992), Bernasconi (1996), and Charbonnel et al. (1996); Z=0.008, Schaerer et al. (1992); Z=0.004, Charbonnel et al. (1993); Z=0.040, Schaerer et al. (1993); Z=0.10, Mowlavi et al. (1998); enhanced mass loss rate evolutionary tracks, Meynet et al. (1994)\].
author:
- 'C.Charbonnel'
- 'W.Däppen'
- 'D.Schaerer'
- 'P.A.Bernasconi'
- 'A.Maeder'
- 'G.Meynet'
- 'N.Mowlavi'
date: 'Received, accepted October 23, 1998'
subtitle: 'VIII. From 0.4 to 1.0 ${\rm M_{\odot}}$ at Z=0.020 and Z=0.001, with the MHD equation of state [^1] '
title: 'Grids of stellar models.'
---
\#1\#2\#3[=\#3cm ]{}
Introduction
============
In stellar evolution computations, and in particular in the case of stars more massive than the Sun, it is generally sufficient to use a simple equation of state. The plasma of the stellar interior is treated as a mixture of perfect gases of all species (atoms, ions, nuclei and electrons), and the Saha equation is solved to yield the degrees of ionization or molecular formation. In the case of low mass stars however, non ideal effects, such as Coulomb interactions become important. It is then necessary to use a more adequate equation of state than the one employed in the Geneva code for more massive stars. This simple equation of state essentially contains a mixture of ideal gases, ionization of the chemicals is dealt with by the Saha equation, excited states and molecules are neglected, complete pressure ionization is artificially imposed above certain temperatures and pressures, and no Coulomb-pressure correction is included (see Schaller et al. 1992).
For the present grids of models of 0.4 to 1.0 M$_{\odot}$ stars these assumptions are obviously inadequate. For such stars, the most useful equations of state, as far as their smooth realization and versatility are concerned, are (i) the so-called Mihalas-Hummer-Däppen (MHD) equation of state (Hummer and Mihalas 1988; Mihalas et al. 1988; Däppen et al. 1988), and (ii) the OPAL equation of state, the major alternative approach developed at Livermore (Rogers 1986, and references therein; Rogers et al. 1996). A brief description of these two equations of state is given in the next section.
Here, we chose the MHD equation of state. First, we were able to compute very smooth tables specifically for our cases of chemical composition, instead of relying on pre-computed, relatively coarse tables that would require interpolation in the chemical composition. Second, our choice was forced by the fact that the currently available OPAL equation of state tables do not allow to go below stars less massive than $\sim$0.8 M$_{\odot}$. Third, we validated our choice by a comparative calculation with OPAL at its low-mass end. We found results that are virtually indistinguishable from MHD. Fourth, we examined in a parallel theoretical study (Trampedach & Däppen 1998) the arguments about the validity of the MHD equation of state down to the limit of our calculation of 0.4 M$_{\odot}$ (see below). Therefore we do not have to include a harder excluded-volume term such as the one included in the Saumon-Chabrier (SC) equation of state (Saumon & Chabrier 1991, 1992).
Although the MHD equation of state was originally designed to provide the level populations for opacity calculations of stellar [*envelopes*]{}, the associated [*thermodynamic quantities*]{} of MHD can none the less be reliably used also for stellar cores. This is due to the fact that in the deeper interior the plasma becomes virtually fully ionized. Therefore, in practice, it does not matter that the condition to apply the detailed Hummer-Mihalas (1988) occupation formalism for bound species is not fulfilled, because essentially there are no bound species. Other than that, the MHD equation of state includes the usual Coulomb pressure and electron degeneracy, and can therefore be used for low-mass stars and, in principle, even for envelopes of white dwarfs (W. Stolzmann, [*private communication*]{}). The present paper, with its MHD-OPAL comparison (see §3) corroborates this assertion.
This broad applicability of the MHD equation of state for entire stars was specifically demonstrated by its successes in solar modeling and helioseismology (Christensen-Dalsgaard et al. 1988, Charbonnel & Lebreton 1993, Richard et al. 1996, Christensen-Dalsgaard et al. 1996). A solar model that is based on the MHD equation of state from the surface to the center is in all respects very similar to one based on the OPAL equation of state, the major alternative approach developed at Livermore (Rogers 1986, and references therein; Rogers et al. 1996). This similarity even pertains to the theoretical oscillation frequencies that are used in comparisons with the observed helioseismic data.
Although the difference between the MHD and OPAL equations of state is of helioseismological relevance, it has no importance for the lower-mass stellar modeling of the present analysis. This is explicitly validated in the present paper. For the much finer helioseismological analyses, it turned out that in some respect the OPAL model seems to be closer than the MHD model to the one inferred from helioseismological observations (Christensen-Dalsgaard et al. 1996, Basu & Christensen-Dalsgaard 1997). However, we stress that for the present stellar modelling these subtle differences are no compelling reason to abandon the convenience of our ability to compute MHD equation of state tables ourselves, and to go below the range of the available OPAL tables ($\sim$0.8 M$_{\odot}$).
Not only helioseismology, but also fine features in the Hertzsprung-Russell diagram of low- and very low-mass stars impose strong constraints on stellar models (Lebreton & Däppen 1988, D’Antona & Mazzitelli 1994, 1996, Baraffe et al. 1995, Saumon et al. 1995). They have all confirmed the validity of the principal equation-of-state ingredients employed in MHD (Coulomb pressure, partial degeneracy of electrons, pressure ionization). Finally, we have checked that even at the low-mass end of our calculations the physical mechanism for pressure ionization in the MHD equation of state is still achieved by the primary pressure ionization effect of MHD (the reduction of bound-state occupation probabilities due to the electrical microfield; see Hummer & Mihalas 1988). Such a verification was necessary to be sure that our results are not contaminated by the secondary, artificial pressure-ionization device included in MHD for the very low-temperature high-density regime (the so-called $\Psi$ term of Mihalas et al. 1988). A parallel calculation has confirmed that in our models a contamination by this $\Psi$ term can be ruled out (Trampedach & Däppen 1998).
In the present paper, we expand the current mass range of the Geneva evolution models from 0.8 down to 0.4 M$_{\odot}$, by using a specifically calculated set of tables of the MHD equation of state. This work aims to complete the base of extensive grids of stellar models computed by the Geneva group with up-to-date input physics \[Z=0.020 and 0.001, Schaller et al. (1992), Bernasconi (1996), and Charbonnel et al. (1996); Z=0.008, Schaerer et al. (1992); Z=0.004, Charbonnel et al. (1993); Z=0.040, Schaerer et al. (1993); Z=0.10, Mowlavi et al. (1998); enhanced mass loss rate evolutionary tracks, Meynet et al. (1994) \]. In Sect. 2, we present the characteristics of our equation of state and recall the physical ingredients used in our computations. In Sect. 3, we summarize the main characteristics of the present models and discuss the influence of the equation of state on the properties of low mass stars. Finally, we compare our solar-metallicity models with recent models computed by other groups and with observations in §4.
Input physics
=============
The basic physical ingredients used for the complete set of grids of the Geneva group are extensively described in previous papers (see Schaller et al. 1992, hereafter Paper I). With the exception of the equation of state, described in detail in the following subsection, we therefore just mention the main points.
Equation of state
-----------------
We have already justified our choice of the MHD equation of state in the introduction. As mentioned there, MHD is one of the two recent equations of state that have been especially successful in modeling the Sun under the strong constraint of helioseismological data. Historically, the MHD equation of state was developed as part of the international “Opacity Project” \[OP, see Seaton (1987, 1992)\]. It was realized in the so-called [*chemical picture*]{}, where plasma interactions are treated with modifications of atomic states, [*i.e.*]{} the quantum mechanical problem is solved before statistical mechanics is applied. It is based on the so-called free-energy minimization method. This method uses approximate statistical mechanical models (for example the nonrelativistic electron gas, Debye-Hückel theory for ionic species, hard-core atoms to simulate pressure ionization via configurational terms, quantum mechanical models of atoms in perturbed fields, etc). From these models a macroscopic free energy is constructed as a function of temperature $T$, volume $V$, and the concentrations $N_1, \ldots, N_m$ of the $m$ components of the plasma. The free energy is minimized subject to the stoichiometric constraint. The solution of this minimum problem then gives both the equilibrium concentrations and, if inserted in the free energy and its derivatives, the equation of state and the thermodynamic quantities.
The other of these two equations of state is the one underlying the OPAL opacity project (see §2.2). The OPAL equation of state is realized in the so-called [*physical picture*]{}. It starts out from the grand canonical ensemble of a system of the basic constituents (electrons and nuclei), interacting through the Coulomb potential. Configurations corresponding to bound combinations of electrons and nuclei, such as ions, atoms, and molecules, arise in this ensemble naturally as terms in cluster expansions. Any effects of the plasma environment on the internal states are obtained directly from the statistical-mechanical analysis, rather than by assertion as in the chemical picture.
More specifically, in the chemical picture, perturbed atoms must be introduced on a more-or-less [*ad-hoc*]{} basis to avoid the familiar divergence of internal partition functions (see [*e.g.*]{} Ebeling et al. 1976). In other words, the approximation of unperturbed atoms precludes the application of standard statistical mechanics, [*i.e.*]{} the attribution of a Boltzmann-factor to each atomic state. The conventional remedy of the chemical picture against this is a modification of the atomic states, [*e.g.*]{} by cutting off the highly excited states in function of density and temperature of the plasma. Such cut-offs, however, have in general dire consequences due to the discrete nature of the atomic spectrum, [*i.e.*]{} jumps in the number of excited states (and thus in the partition functions and in the free energy) despite smoothly varying external parameters (temperature and density). However, the occupation probability formalism employed by the MHD equation of state avoids these jumps and delivers very smooth thermodynamic quantities. Specifically, the essence of the MHD equation of state is the Hummer-Mihalas (1988) occupation probability formalism, which describes the reduced availability of bound systems immersed in a plasma. Perturbations by charged and neutral particles are taken into account. The neutral contribution is evaluated in a first-order approximation, which is good for stars in which most of the ionization in the interior is achieved by temperature \[the aforementioned study (Trampedach & Däppen 1998) has verified the validity of this assumption down to the lowest mass of our calculation\]. For colder objects (brown dwarfs, giant planets), higher-order excluded-volume effects become very important (Saumon & Chabrier 1991, 1992; Saumon et al. 1995). In the common domain of application of the Saumon et al. (1995) and MHD equations of state, Chabrier & Baraffe (1997) showed that both developments yield very similar results, which strongly validates the use of the MHD equation of state for our mass range of 0.4 to 1.0 M$_{\odot}$.
Despite undeniable advantages of the physical picture, the chemical picture approach leads to smoother thermodynamic quantities, because they can be written as analytical (albeit complicated) expressions of temperature, density and particle abundances. In contrast, the physical picture is normally realized with the unwieldy chemical potential as independent variable, from which density and number abundance follow as dependent quantities. The physical-picture approach involves therefore a numerical inversion before the thermodynamic quantities can be expressed in their “natural” variables temperature, density and particle numbers. This increases computing time greatly, and that is the reason why so far only a limited number of OPAL tables have been produced, only suitable for stars more massive than $\sim$0.8 M$_{\odot}$. Therefore we chose MHD for its smoothness, availability, and the possibility to customize it directly for our calculation, despite the – in principle – sounder conceptual foundation of OPAL.
Opacity tables, treatment of convection, atmosphere and mass loss
-----------------------------------------------------------------
- The OPAL radiative opacities from Iglesias & Rogers (1996) including the spin-orbit interactions in Fe and relative metal abundances based on Grevesse & Noels (1993) are used. These tables are completed at low temperatures below 10000 K with the atomic and molecular opacities by Alexander & Fergusson (1994).
- We use a value of 1.6 for the mixing length parameter $\alpha$. Various observational comparison support this choice. $\alpha = 1.6 \pm 0.1$ leads to the best fit of the red giant branch for a wide range of clusters (see Paper I). It is also the value we get for the calibration of solar models including the same input physics (Richard et al. 1996).
- A grey atmosphere in the Eddington approximation is adopted as boundary condition. Below $\tau = 2/3$, full integration of the structure equations is performed. We discuss the implications of such an approximation in §4.
- Evolution on the pre-main sequence and on the main sequence are calculated at constant mass. On the red giant branch, we take mass loss into account by using the expression by Reimers (1975) : $\dot{M} = 4 \times 10^{-13} \eta L R / M$ (in M$_{\odot}$yr$^{-1}$) where L, M and R are the stellar luminosity, mass and radius respectively (in solar units). At solar metallicity, $\eta=$0.5 is chosen (see Maeder & Meynet 1989). At Z=0.001, the mass loss is lowered by a factor (0.001/0.020)$^{0.5}$ with respect to the models at Z=0.020 for the same stellar parameters.
Nuclear reactions
-----------------
- Nuclear reaction rates are due to Caughlan & Fowler (1988). The screening factors are included according to the prescription of Graboske et al. (1973).
- Deuterium is destroyed on the pre-main sequence at temperatures higher than 10$^6$ K by D(p,$\gamma)^3$He and, to a lower extent, by D(D,p)$^3$H(e$^- \nu)^3$He and D(D,n)$^3$He. We take into account these three reactions. In order to avoid the follow-up of tritium, we consider the last two reactions as a single process, D(D,nucl)$^3$He. The $\beta$ desintegration is considered as instantaneous, which is justified in view of the lifetime of tritium ($\tau _{1/2}$=12.26 yr) compared with the evolutionary timescale. The rate of the D(D,nucl)$^3$He reaction is written as $$<DD>_{nucl} = (1 + {{<DD>_p}\over{<DD>_n}} ) <DD>_n$$ where we take for ${{<DD>_p}\over{<DD>_n}}$ a mean value of 1.065, in agreement with the rates given by Caughlan & Fowler (1988). The corresponding mean branch ratios are I$_p$ = 0.5157 and I$_n$ = 0.4843.
Initial abundances
------------------
- The initial helium content is determined by Y=0.24+($\Delta$Y/$\Delta$Z)Z, where 0.24 corresponds to the current value of the cosmological helium (Audouze 1987). We use the value of 3 for the average relative ratio of helium to metal enrichment ($\Delta$Y/$\Delta$Z) during galactic evolution. This leads to (Y,Z) = (0.300,0.020) and (0.243,0.001). In addition, computations were also performed with (Y,Z) = (0.280,0.020).
- The relative ratios for the heavy elements correspond to the mixture by Grevesse & Noels (1993) used in the opacity computations by Iglesias & Rogers (1996).
- Choosing initial abundance values for D and $^3$He is more complex. Pre-solar abundances for both elements have been reviewed in Geiss (1993), however galactic chemical models face serious problems to describe their evolution (see e.g. Tosi 1996), and no reliable prescription exists to extrapolate their values in time. Deuterium is only destroyed by stellar processing since the Big Bang Nucleosynthesis, so that its abundance decreases with time (i.e. with increasing metallicity). On the other hand, the actual contribution of stars of different masses is still subject to a large debate (Hogan 1995, Charbonnel 1995, Charbonnel & Dias 1998), and observations of ($^3$He/H) in the proto-solar nebulae (Geiss 1993) and in galactic HII regions present a large dispersion. We adopt the same (D/H) and ($^3$He/H) initial values for both metallicities, namely 2.4$\times 10^{-5}$ and 2.0 $\times 10^{-5}$ respectively.
Initial models
--------------
At the low mass range considered in this paper, the observed quasi-static contraction begins rather close to the theoretical deuterium main sequence, once the stars emerge from their parental dense gas and dust. For the present purposes then, we take as starting models polytropic configurations on the Hayashi boundary, neglecting the corrections brought to isochrones and upper tracks by the modern accretion paradigm of star formation (Palla & Stahler 1993, Bernasconi & Maeder 1996). For the mass range considered here, these corrections are likely not to exceed 3$\%$ of the Kelvin-Helmholtz timescale for the pre-main sequence contraction times (Bernasconi 1996). We note, however, that the predicted upper locus for the optical appearance of T Tauri stars in the HR diagram can be as much as half less luminous than the deuterium ignition luminosity on the convective tracks ($\Delta \log L \approx$ 0.3).
Short discussion of the main results
====================================
HR diagram and lifetimes
------------------------
The HR diagrams for pre-main sequence evolution and for the following phases are given in Fig. 1 and 2 respectively for both metallicities. For each stellar mass, Table 1 displays the lifetimes in the contraction phase and in the deuterium- and hydrogen-burning phases. Note that we did not complete the main sequence evolution computations for the less massive stars which have a H-burning phase longer than the age of the universe; for these stars, our last computed model corresponds to an age of 20 Gyr.
- In the stellar mass range we consider, the ignition of deuterium burning (indicated in Fig.1) takes place in a fully convective interior. During pre-main sequence evolution, a radiative core develops. However a proper radiative branch is absent, since these stars maintain a convective envelope all along contraction until the ZAMS has been reached, and further on.
- From Table 1 one can see that the contraction time lasts less than 4 thousandths of the hydrogen-burning stage.
- Due to opacity effects, the entire evolution (pre-main sequence, main sequence and red giant branch) occurs at higher luminosity and effective temperature for a given stellar mass when the initial metallicity is smaller, or when the hydrogen content is lower for the same initial metallicity.
- As a consequence, the contraction phase and the deuterium- and hydrogen-burning phases for a given stellar mass are shorter at lower metallicity (see Table 1), and at lower hydrogen content for the same value of Z.
Influence of the equation of state
----------------------------------
$${\begin{array}{r@{.}lr@{.}lr@{.}lr@{.}lr@{.}lr@{.}l} \hline \\[1mm]
\multicolumn{2}{c}{\mbox{Z}}&
\multicolumn{2}{c}{\mbox{Initial}}&
\multicolumn{2}{c}{\mbox{Contraction}} &
\multicolumn{2}{c}{\mbox{D-burning}} &
\multicolumn{2}{c}{\mbox{H-burning}}&
\multicolumn{2}{c}{\mbox{ {t$_c$ / t$_H$} }} \\
\multicolumn{2}{c}{\mbox{Y}}&
\multicolumn{2}{c}{\mbox{mass}} &
\multicolumn{2}{c}{\mbox{phase}} &
\multicolumn{2}{c}{\mbox{phase}} &
\multicolumn{2}{c}{\mbox{phase}} &
\multicolumn{2}{c}{} \\[1mm]
\hline
\\[0.5mm]
0&001 & 0&4 & 160&10 & 0&417 & \tiret & \tiret \\
0&243 & 0&5 & 81&11 & 0&321 & \tiret & \tiret \\
& & 0&6 & 51&89 & 0&277 & \tiret & \tiret \\
& & 0&7 & 36&44 & 0&240 & \tiret & \tiret \\
& & 0&8 & 31&20 & 0&214 & 14338&03 & 0&0022 \\
& & 0&9 & 22&82 & 0&196 & 9051&01 & 0&0025 \\
& & 1&0 & 19&34 & 0&187 & 6847&20 & 0&0028 \\[1mm]
\hline
0&020 & 0&4 & 153&84 & 0&605 & \tiret & \tiret \\
0&300 & 0&5 & 117&34 & 0&476 & \tiret & \tiret \\
& & 0&6 & 88&26 & 0&389 & \tiret & \tiret \\
& & 0&7 & 68&80 & 0&330 & \tiret & \tiret \\
& & 0&8 & 58&15 & 0&302 & 22713&11 & 0&0026 \\
& & 0&9 & 44&30 & 0&276 & 14070&96 & 0&0031 \\
& & 1&0 & 32&80 & 0&252 & 9059&52 & 0&0036 \\[1mm]
\hline
0&020 & 0&4 & 161&79 & 0&640 & \tiret & \tiret \\
0&280 & 0&5 & 124&50 & 0&484 & \tiret & \tiret \\
& & 0&6 & 93&75 & 0&407 & \tiret & \tiret \\
& & 0&7 & 72&86 & 0&344 & \tiret & \tiret \\
& & 0&8 & 62&34 & 0&309 & 26372&72 & 0&0024 \\
& & 0&9 & 48&26 & 0&286 & 16421&94 & 0&00002 \\
& & 1&0 & 37&98 & 0&263 & 10662&62 & 0&00002 \\[1mm]
\hline
\end{array}}$$
[ccccc]{}\
& & & &\
\
0.020 & 1.0 & 9059.5 & 8968.1 & 9970.1\
0.020 & 0.8 &22713.3 & 22361.4 & 25151.3\
0.001 & 0.8 &14338.0 & 13986.0 &\
When we first compare the results obtained with the MHD and with the simple Geneva equations of state (see Fig. 3 and 4, and Table 2), we obtain essentially the same results than Lebreton & Däppen (1988). Firstly, the fact that MHD contains ${\rm H}_2$ molecules, and the simple Geneva code does not, is reflected in a shift essentially [*along*]{} the ZAMS. On the other hand, the Coulomb pressure correction, also contained in MHD, causes a slight shift of the ZAMS, clearly visible for higher masses, where there are no hydrogen molecules in the photosphere. This Coulomb effect has been well discussed in the case of helioseismology ([*e.g.*]{} Christensen-Dalsgaard et al. 1996). Conformal to the effect of the MHD equation of state to push the apparent position on the ZAMS upward, it also decreases the lifetime on the ZAMS (see Table 2).
For comparison, we have computed with the OPAL equation of state two 0.8M$_{\odot}$ models (the lowest mass that can be computed with the current OPAL tables), for both metallicities. As can be seen in Fig.3, the corresponding tracks are very close to those obtained with the MHD equation of state, the use of the OPAL equation of state leading to slightly higher effective temperature on the ZAMS. As far as their internal structure is concerned, the models computed with MHD equation of state have slightly deeper convection zones. The main sequence lifetime obtained with the MHD equation of state is slightly higher than the one obtained with the OPAL equation of state (Table 2). The comparison shows that down to 0.8M$_{\odot}$ all is fine with the MHD pressure ionization. As mentioned in the introduction, Trampedach & Däppen (1998) predict a correct functioning of pressure ionization in MHD even for much smaller masses. With the present comparison, we have validated their prediction at least to 0.8M$_{\odot}$.
Comparisons with other sets of models and with observations
===========================================================
We now compare our models with recent models computed by other groups and with various observations.
The strongest approximation in our models lies in the treatment of the atmosphere and of the surface boundary conditions, which are specified by the Eddington approximation. In Figs. 5 and 6 we compare our results with the models of D’Antona & Mazzitelli (1994) and Tout et al. (1996) which also rely on a simple treatment of the boundary conditions. For the mass range considered in our work we obtain predictions very similar to D’Antona & Mazzitelli.
Sophisticated model atmospheres for the computation of very low mass stars have been developed recently (Baraffe et al. 1995, 1998; Brett 1995, Allard et al. 1997). Their use becomes crucial for very low mass stars ($M \la 0.4$ M$_{\odot}$) down to the brown dwarf limit. We refer to Chabrier & Baraffe (1997) and Baraffe et al. (1998) for a discussion of the physical basis of the differences. As can be seen in Fig.5, the Eddington approximation we use results in higher T${\rm eff}$ (from 2 to 5 $\%$ depending on the metallicity) compared to the Chabrier & Baraffe (1997) models at our low mass end; our models also have slightly smaller radius (see Fig.6). In this mass range the predictions agree well with the observations of Popper (1980) and Leggett et al. (1996) and do not allow to disregard one model with respect to the other.
In Fig.7 we show the predicted mass-luminosity relations for the V and K band for different metallicities and ages. The magnitudes were derived from the standard stellar library for evolutionary synthesis of Lejeune et al.(1998), which provides empirically calibrated colours for solar metallicity and a semi-empirical correction for non-solar metallicities. The comparison with the observations of Andersen (1991) and Henry & McCarthy (1993) shows a good agreement with our predictions for both the V and K band (Fig.7). Again a similar agreement is obtained with the models of Brocato et al. (1998) and Baraffe et al. (1998).
From the comparisons in Figs.5 to 8, we conclude (in agreement with Alexander et al. 1997) that for the mass range considered in this work an approximate treatment of the stellar atmosphere leads to a satisphactory agreement between theoretical predictions and observations. Independently ab initio stellar interior and atmosphere models allowing the detailed predictions of all observational properties are of fundamental importance for our understanding of very low mass stars which are out of the scope of the present grids.
Appendix: How to obtain the tables by file-transfer
===================================================
The results of this work and of our previous grids (Papers I to VI) are published by Astronomy and Astrophysics at the Centre de Données Spatiales (CDS at Strasbourg) where the corresponding tables are available in electronic form:
[http://cdsweb.u-strasbg.fr]{}. These data can also be obtained from the Geneva Observatory
[http://obswww.unige.ch/]{} (contact Corinne.Charbonnel@obs-mip.fr).
An ensemble of models were selected to describe each evolutionary track. For each model the tables display the age, actual mass, log $L/L_\odot$, log $T_{eff}$, the surface abundances in mass fraction of H, $^4$He, $^{12}$C, $^{13}$C, $^{14}$N, $^{16}$O, $^{17}$O, $^{18}$O, $^{20}$Ne, $^{22}$Ne, the core mass fraction Qcc, log$(-\dot{M})$ (where $\dot{M}$ is the mass loss rate on the red giant branch), log $\rho _c$ (where $\rho _c$ is the central density), log T$_c$ (where T$_c$ is the central temperature), and the central abundances in mass fraction of the above elements. A detailed description of the models selection and of the tables contents is given in Paper I. We now also provide photometric data in verious systems for all tracks and isochrones (see Schaerer & Lejeune 1998).
Alexander D.R., Fergusson J.W., 1994, ApJ 437, 879 Alexander D.R., Brocato E., Cassisi S., Castellani V., Ciacio F., Degl’Innocenti S., 1997, A&A 317, 90 Allard F., Hauschildt P.H., Alexander D.R., Starrfield S., 1997, ARA&A 35, 137 Andersen J., 1991, A&A Rev 3, 91 Audouze J., 1987, in Observational Cosmology, IAU Symp. 124, Eds. A.Hewitt et al., Reidel Publ. p.89 Baraffe I., Chabrier G., Allard F., Hauschildt P.H., 1995, ApJ 446, L35 Baraffe I., Chabrier G., Allard F., Hauschildt P.H., 1998, A&A, submitted (astro-ph/9805009) Basu S., Christensen-Dalsgaard J., 1997, A&A, 322, L5 Bernasconi P.A. 1996, A&AS, 120, 57 Bernasconi P.A., Maeder A., 1996, A&A, 307, 829 Brett J.M., 1995, A&A 295, 736 Brocato E., Cassisi S., Castellani V., 1998, MNRAS 295, 711 Caughlan G.R., Fowler W.A., 1988, Atomic Data Nuc. Data Tables 40,283 Chabrier G., Baraffe I., 1997, A&A 327, 1039 Charbonnel C., 1995, ApJ 453, L41 Charbonnel C., Dias do Nascimento J., 1998, A&A, 336, 915 Charbonnel C., Lebreton Y., 1993, A&A 280, 666 Charbonnel C., Meynet G., Maeder A., Schaller G., Schaerer D., 1993, A&A 102, 339 Charbonnel C., Meynet G., Maeder A., Schaerer D., 1996, A&AS 115, 339 Christensen-Dalsgaard, J., Däppen, W., Lebreton, Y., 1988, Nature, 336, 634 Christensen-Dalsgaard J., Däppen, W., and the GONG team, 1996, Science, 272, 1286 D’Antona F., Mazzitelli I., 1994, ApJS 90, 467 D’Antona F., Mazzitelli I., 1996, ApJ 456, 329 Däppen, W., Mihalas, D., Hummer, D.G. & Mihalas, B.W.: 1988, Astrophys. J. 332, 261 Ebeling, W., Kraeft, W.D. & Kremp, D., 1976, [*Theory of Bound States and Ionization Equilibrium in Plasmas and Solids*]{}, Akademie Verlag, Berlin, DDR Geiss J., 1993, Origin and Evolution of the Elements, Eds. N.Prantzos, E.Vangioni-Flam & M.Cassé, Cambridge University Press, 89 Graboske H.C., de Witt H.E., Grossman A.S., Cooper M.S., 1973, ApJ 181, 457 Grevesse N., Noels A., 1993, in “Origin and Evolution of the Elements", Eds Prantzos N., Vangioni-Flam E., Cassé M., CUP Henry T.J., McCarthy D.W., 1993 AJ 106, 773 Hogan C., 1995, ApJ 441, L17 Hummer, D.G. & Mihalas, D.: 1988, ApJ, 331, 794 Iglesias C.A., Rogers F.J., 1996, ApJ 464, 943 Lebreton Y., Däppen W., 1988, Seismology of the Sun and the Sun-like Stars, Eds.V.Domingo, E.J.Rolfe, ESA SP-286, 661 Leggett S.K., Allard F., Berriman G., Dahn C.C., Hauschildt P.H., 1996, ApJS 104, 117 Lejeune T., Cuisinier F., Buser R., 1997, A&AS, in press (astro-ph/9710350) Maeder A., Meynet G., 1989, A&A 210, 155 Meynet G., Maeder A., Schaller G., Schaerer D., Charbonnel C., 1994, A&AS 103, 97 Mihalas D., Hummer D.G., Däppen W., 1988, ApJ 331, 815 Mowlavi N., Schaerer D., Meynet G., Bernasconi P.A., Maeder A., Charbonnel C., 1998, A&AS 128, 1 Palla F., Stahler S.W. 1993, ApJ, 418, 414 Popper D.M., Ann.Rev.Astr.Ap. 18, 115 Reimers D., 1975, Mem. Soc. Roy. Sci. Liège 8, 369 Richard O., Vauclair S., Charbonnel C., Dziembowski W.A., 1996, A&A 312, 1000 Rogers, F.J.: 1986, ApJ. 310, 723 Rogers, F.J., Swenson, F.J. and Iglesias, C.A., 1996, ApJ, 456, 902 Saumon D., Chabrier G., 1991, [*Phys. Rev*]{}, A44, 5122 Saumon D., Chabrier G., 1992, [*Phys. Rev*]{}, A46, Saumon D., Chabrier G., VanHorn H.M., 1995, ApJS 99, 713 Schaerer D., Lejeune T., 1998, in preparation Schaerer D., Meynet G., Maeder A., Schaller G., 1992, A&AS 98, 523 Schaerer D., Charbonnel C., Meynet G., Maeder A., Schaller G., 1993, A&AS 102, 339 Schaller G., Schaerer D., Meynet G., Maeder A., 1992, A&AS 96, 269 Seaton, M.: 1987, J. Phys. B: Atom. Molec. Phys. 20, 6363 Seaton, M.J.: 1992, in [*Astrophysical Opacities*]{}, eds. C. Mendoza & C. Zeippen ([*Revista Mexicana de Astronomía y Astrofísica*]{}) 180 Tosi M., 1996, From Stars to Galaxies: The Impact of Stellar Physics on Galaxy Evolution”, ASP Conf. Series, Vol. 78, Eds. C. Leitherer, U. Fritze - von Alvensleben, J. Huchra, p. 299 Tout C.A., Pols O., Eggleton P.P., Han Z., 1996, MNRAS 281, 257 Trampedach R., Däppen W., 1998, A&A, submitted
[^1]: Data available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/Abstract.html
|
---
abstract: 'We study the diversity of complex spatio-temporal patterns in the behavior of random synchronous asymmetric neural networks (RSANNs). Special attention is given to the impact of disordered threshold values on limit-cycle diversity and limit-cycle complexity in RSANNs which have ‘normal’ thresholds by default. Surprisingly, RSANNs exhibit only a small repertoire of rather complex limit-cycle patterns when all parameters are fixed. This repertoire of complex patterns is also rather stable with respect to small parameter changes. These two unexpected results may generalize to the study of other complex systems. In order to reach beyond this seemingly-disabling ‘stable and small’ aspect of the limit-cycle repertoire of RSANNs, we have found that if an RSANN has threshold disorder above a critical level, then there is a rapid increase of the size of the repertoire of patterns. The repertoire size initially follows a power-law function of the magnitude of the threshold disorder. As the disorder increases further, the limit-cycle patterns themselves become simpler until at a second critical level most of the limit cycles become simple fixed points. Nonetheless, for moderate changes in the threshold parameters, RSANNs are found to display specific features of behavior desired for rapidly-responding processing systems: accessibility to a large set of complex patterns.'
author:
- |
Patrick C. McGuire [^1] [^2] [^3] [^4] [^5] , Henrik Bohr [^6] , John W. Clark [^7] ,\
Robert Haschke , Chris L. Pershing [^8] , Johann Rafelski\
title: |
Threshold Disorder as a Source of Diverse and\
Complex Behavior in Random Nets
---
Introduction
============
Random Synchronous Asymmetric Neural Networks (RSANNs) with fixed synaptic coupling strengths and fixed neuronal thresholds/inputs tend to have access to a very limited set of different limit cycles (Amari (1974), Clark, Kürten & Rafelski (1988), Littlewort, Clark & Rafelski (1988), Hasan (1989), Rand, Cohen & Holmes (1988), Clark (1990), Schreckenberg (1992)). We will show here, however, that when we add a moderate amount of randomly quenched noise or disorder, by choosing the neural thresholds or inputs to vary within a prescribed gaussian distribution, we can gain controllable, and we believe biologically relevant, access to a wide variety of limit cycles, each displaying dynamical participation by many neurons.
The appearance of limit-cycle behavior in central pattern generators is evidence for cyclic temporal behavior in biological systems (Hasan (1989), Marder & Hooper (1985)). Previous computational models, as discussed above, do not exhibit a diverse repertoire of limit-cycle behaviors, as biological systems often demonstrate (e.g., the different gaits of a horse, or the different rhythmic steps of a good human dancer). Additonally, it is our belief that the biologically-interesting networks are those in which a significant fraction of the neurons can (and often do) participate in the local dynamics. In principle, spatially-sparse neuronal firing patterns can be constructed from a large network of strongly-participatory neurons by self-organized architectural inhibition of selected neuronal assemblies. This can leave the uninhibited neuronal assemblies able to freely participate in the neural dynamics (for some time), though these uninhibited neurons or neuronal assemblies may be isolated in space from each other, only connected to other active neurons through non-local or indirect connections (Gray & Singer (1990)). Therefore, in this paper, we explore the problem of how to produce a computational neural model which possesses a diverse repertoire of strongly-participatory limit-cycle behaviors.
A system which can access [*many*]{} limit cycles should always be able to access a novel mode; hence the system would have the potential to be a ‘[*creative*]{}’ system. Herein, we demonstrate conditions sufficient to allow a simple computational neural system to access creative dynamical behavior.
In Section \[sec:RSANN\] we introduce RSANNs along with the concept of threshold disorder, as well as a measure to distinguish different limit cycles. In our quantitative investigations we need to introduce, with some precision, concepts which intuitively are easy to grasp, but which mathematically are somewhat difficult to quantify. We define ‘[*eligibility*]{}’ in Section \[sec:Elig\] as an entropy-like measure of the fraction of neurons which actively participate in the dynamics of a limit cycle. In order to quantify the RSANN’s accessibility to multiple limit-cycle attractors, we define ‘[*diversity*]{}’ in Section \[sec:Div\] as another entropy-like measure, calculated from the probabilities that the RSANN converges to each of the different limit cycles. The difference between eligibility and diversity is that the former applies to a limit cycle observed in a specific network, while the latter applies to the collection of limit cycles that the network can exhibit. To measure the creative potential of a system, we introduce the concept of ‘[*volatility*]{}’ as the ability to access a huge number of highly-eligible cyclic modes. We find that in terms of these diagnostic variables, as the neuronal threshold disorder ${\ensuremath{\varepsilon}\xspace}$ increases, our RSANN exhibits a phase transformation at ${\ensuremath{\varepsilon}\xspace}={\ensuremath{\varepsilon}\xspace}_1$ from a small number to a large number of different [*accessible*]{} limit-cycle attractors (Section \[sec:Div\]), and another phase transformation at ${\ensuremath{\varepsilon}\xspace}={\ensuremath{\varepsilon}\xspace}_2 > {\ensuremath{\varepsilon}\xspace}_1$ from high eligibility to low eligibility (Section \[sec:Elig\]). Our main result is that the volatility is high only in the presence of threshold disorder of suitably chosen strength between ${\ensuremath{\varepsilon}\xspace}_1 \leq {\ensuremath{\varepsilon}\xspace}\leq {\ensuremath{\varepsilon}\xspace}_2$, thereby allowing access to a diversity of eligible limit-cycle attractors (Section \[sec:Vol\]).
Random asymmetric neural networks with threshold or input noise {#sec:RSANN}
===============================================================
Symmetric neural networks (SNNs) (Hopfield (1982)) became widely used in associative memory applications due to their ability to store a large number of patterns as fixed points of their dynamics; however, their dynamical behaviour is restricted to fixed points or limit cycles of period 2. In contrast, asymmetric neural networks (RSANNs) show a complicated dynamical behaviour, including limit cycles of large periods or even chaos [^9] (Amari (1974), Clark, Rafelski & Winston (1985), Clark, Kürten & Rafelski (1988), Littlewort, Clark & Rafelski (1988), Kürten (1988), Bressloff & Taylor (1989), Clark (1990, 1991), McGuire, Littlewort & Rafelski (1991), McGuire [*et al.*]{} (1992), Bastolla & Parisi (1997)). Moreover, they offer considerably more biological realism, since real neuronal connections tend to be unidirectional.
We investigate a network of $N$ threshold elements, i.e. their firing states have binary values $x_i \in \{0, 1\}$. Each neuron $i$ is connected to $M < N$ presynaptic neurons by unidirectional weights $w_{ij}$, with $w_{ij} \ne w_{ji}$ and $w_{ii} = 0$. All weights are independent random variables, drawn from a uniform distribution within $[-1, 1]$. A neuron fires if its post-synaptic-potential (PSP) is greater than its specific threshold $V_i$. Therefore the network is described by the following system of equations for ‘sum-and-fire’ McCullough-Pitts neurons: $$\begin{aligned}
x_i (t+1) &= \Theta \bigl(\sum_{j=1}^M w_{ij} x_j(t) - V_i\bigr)
\quad \forall i \in 1, \dots, N \, ,
\label{eq:Dynamics}\end{aligned}$$ where $\Theta$ is the Heaviside function. Supposing that all neurons should actively participate in the dynamics, with a mean firing rate $\langle x_i \rangle = \frac{1}{2}$, the mean thresholds $V^0_i$ are adjusted so that the mean overall input $$\begin{aligned}
\langle u_i \rangle =
\langle \sum w_{ij} x_j - V^0_i \rangle \approx \sum w_{ij} \langle x
\rangle - V^0_i\,
\label{eq:approx}\end{aligned}$$ to a generic neuron $i$ becomes zero (so that it is poised on the boundary between firing and not firing). Thus, we have:
$$\begin{aligned}
V^0_i &= \frac{1}{2} \sum_j w_{ij} , \\
V_i &= \eta_i V_i^0 ,
\label{eq:ThreshNoise1}\end{aligned}$$
where the parameter $\eta_i$ is chosen to modulate the threshold. The case $\eta_i \equiv 1$ for all $i$ corresponds to the choice known as ‘normal’ thresholds (Clark (1991)). The mean firing rate of 0.5 is quite high biologically, but computationally, it is a reasonable point to begin our research; it is not too difficult to adapt the treatment to lower mean firing rates. In order for a given amount of threshold disorder to affect all neurons more-or-less equally, we have chosen here a multiplicative scaling of the thresholds relative to the normal thresholds rather than an additive scaling. We do not consider synaptic noise or modulation; hence the weights $w_{ij}$ are kept fixed for a given network.
In considering the living neural networks in the brain, some researchers treat the neuronal thresholds as constant and noiseless (as in the Hodgkin-Huxley and Fitzhugh-Nagumo models; see Murray (1989) for a summary); others are convinced that neurons live in a very noisy environment, both chemically and electrically, with nontrivial consequences for neuronal and network function (see Zador (1997), Chow & White (1996), Clark (1988), Buhmann & Schulten (1987), Shaw & Vasudevan (1974), Little (1974), Taylor (1972), and Lecar & Nossal (1971)). Examining the issue more closely, we may note that Mainen & Sejnowski (1995) have presented data suggesting a low intrinsic noise level for neurons, which does not seriously affect the precision of spike timing in the case of stimuli with fluctations resembling synaptic activity. On the other hand, Pei, Wilkens & Moss (1996) have presented evidence that noise can exert beneficial effects on neural processing through the phenomenon of stochastic resonance.
![ Qualitative sketch showing the varying threshold for neuron \#7 ($V_7^\alpha$) and the spatially-averaged firing rate $\bar{x}^\alpha(t)$, as a function of trial-number $\alpha$, and in the inset, a magnified view of $\bar{x}^\alpha(t)$, all as functions of time, $t$.[]{data-label="fig:clock"}](timing8.eps){width="13.0cm"}
Mathematically, the external inputs to a neuron from sensory organs or from other areas of the neural system can also be treated as a modulation of the threshold of that neuron. This suggests that the results obtained on threshold modulation might be easily generalized to the situation of external modulation.
Taken together, the noisiness of thresholds and the variability of inputs can be viewed as a changing environment. We simply model this complex changing environment by varying the normal thresholds using multiplicative gaussian noise $\eta_i$ with mean $\mu=1$ and standard deviation ${\ensuremath{\varepsilon}\xspace}$, leading to Eq. \[eq:ThreshNoise1\]. The components $\eta_i$ are chosen independently for all neurons $i$. It may be much more reasonable to consider spatially-correlated noise amplitudes, but such a study exceeds the scope of our present effort.
Limit-Cycle Search {#sec:Updating}
------------------
Since we wish to study the diversity of different limit cycles accessible with small changes of the thresholds, we need a robust criterion for detecting limit cycles. Even in the presence of small-amplitude noise effective on a shorter time scale than the cycle length, the neural net will never stabilize into a detectable perfect cycle. Rather, it will either converge to an [*approximate*]{} limit cycle with occasional misfirings or never converge at all. Such approximate limit-cycle behavior is more relevant to neurobiological systems than is its perfect realization, due to the inherent destabilizing noise (from membrane-potential or synaptic noise) and additional complicating factors, notably (1) the complexity of biological neurons, (2) the continuum of signal transmission times between neurons, and (3) the apparent lack of a clock to synchronously update all neurons.
However, although approximate limit-cycle behavior might be more common in volatile systems, it is not ideal for computer simulation and computer characterization. Therefore, for the sake of the computational tractability, we restrict our search to perfect limit cycles. In order to achieve this, we fix the neural thresholds $V_i$ until a limit cycle is found during network evolution via Eq. \[eq:Dynamics\]. Since the noise is frozen-in (‘quenched’) for a long period, it is more properly considered as disorder. Before the next trial, the thresholds are varied according to Eq. \[eq:ThreshNoise1\], changing the underlying network dynamics; they are then fixed again during limit cycle search. Each trial step starts with the activity pattern ${\ensuremath{{\boldsymbol{x}}}\xspace}(0)$ with which the previous trial was terminated. To gain a qualitative understanding of our approach, see Figure \[fig:clock\].
Fixed points (with limit-cycle period $L = 1$) are generally of less interest than cyclic modes in view of the spontaneous oscillatory behavior displayed by real neural systems (e.g walking or singing). Effectively chaotic or non-cycling behavior (with $L \sim 2^N$) is not predictable enough to be of much use for most applications in real neural systems. Limit cycles of intermediate period are consequently our paramount concern.
We update all neuron firing states in parallel, or ‘synchronously’, as opposed to either serial or random updating in which only one neuron is updated at a given time step.
In our simulations, we used $M = N/10$ incoming connections per neuron, where self-connections were not allowed, and we studied networks with $N \in \{10,20,30,40,50,100\}$ neurons. Self-connections tend to have a stabilizing effect on the network, often driving the behavior towards a fixed point with only very brief transients. The different behavior of networks with and without self-connections might be a worthy subject of future investigation, but we chose not to emphasize that direction here. For practical computational reasons, the network sizes investigated are primarily constrained by the existence of extremely long limit cycles and transients of large networks, occurring especially when the thresholds are near-normal, (see Clark (1990, 1991), Clark, Kürten & Rafelski (1988), and Littlewort, Clark & Rafelski (1988) for discussions of normal thresholds and the correlation between transient length and limit-cycle period). A network of $N$ threshold units can assume $2^N$ states, placing an upper limit on the length of a limit cycle. This upper limit for the cycle length is due to the facts that there are only a finite number of states and that the time development of the system is deterministic and depends only on the initial network state ${\ensuremath{{\boldsymbol{x}}}\xspace}(0)$.
Since the detection of limit cycles at the microstate level ${\ensuremath{{\boldsymbol{x}}}\xspace}$ is too time consuming (it requires ${\cal O}(N L^2)$ comparisons), we use the system-averaged firing rate $\bar{x}(t)$ in order to test for periodicity: $$\begin{aligned}
\label{eq:Activity}
\bar{x}(t) &\equiv \frac{1}{N} \sum_{i=1}^{N} x_i(t) \, , \\
\label{eq:Activity2}
\bar{x}(t+L) &= \bar{x}(t) \quad \forall t \in [0, 4 L] \, ,\end{aligned}$$ where the limit-cycle period is is identified as $L$. Though satisfying Equation \[eq:Activity2\] is only a necessary condition for an exact limit cycle of period $L$ at the microstate-level, in practice we observed no differences between exact and average comparison methods on a small test set. We used a window of $4 L$ time steps to ensure that the cycle does indeed repeat itself in $\bar{x}$ four times; without explicitly tracking the microstate ${\ensuremath{{\boldsymbol{x}}}\xspace}$, such care is necessary in order to avoid false limit-cycle detection and measurement.
Limit-cycle Comparison
----------------------
Since diversity and volatility (which we define in sections \[sec:Div\] & \[sec:Vol\]) require an abundance of [*different*]{} limit cycles, we need to introduce a high-contrast, direct, neuron-by-neuron measure to decide whether a given limit cycle is different from or similar to another limit cycle. One could, of course, just compare the full neuron-by-neuron time-dependence of the activity patterns of the cycles themselves, but that would require a vast amount of memory to store all observed cycles. However, if as above, we choose to compare the time-dependence of the system-averaged firing rates instead of the full firing vectors, different limit cycles may be remarkably similar, possibly distinguished by only a small numerical difference, which we elucidate here with a specific example. Given that:
- a network of $N$ neurons has two cycles: cycle [**A**]{} with period $L$ and cycle [**B**]{} with period $2 L$,
- cycle [**A**]{} has the same firing pattern at each time step as cycle [**B**]{} with the exception of two neurons at each time step in cycle [**A**]{} which differ from the corresponding two neurons in cycle [**B**]{},
- and the total number of firing neurons at each time step in cycle [**A**]{} is the same at each time step in cycle [**B**]{} (due to the two neurons cancelling each other),
then with system-averaged firing rates, the two cycles would be deemed identical, with the same period; though a comparison of the time-dependence of full firing vectors would show the different period of the two cycles.
Reliable discrimination clearly requires a compact measure, i.e. a fingerprint of a cycle, which is:
- independent of cycle length,
- capable of discriminating between a wide array of limit cycles, and
- easily computable.
For this purpose we use the vector $\bar{{\ensuremath{{\boldsymbol{x}}}\xspace}} = [\bar{x}_1, \dots, \bar{x}_N]^t$ formed by the time-averaged firing rates within $T=4 L$ time steps, $$\bar{x}_i = \frac{1}{T} \sum_{t=1}^{T} x_i(t).
\label{eq:TimeAvFiringRate}$$ Since we base our limit-cycle detection upon a temporal quantity (the time-dependent, system-averaged firing rate), our additional reference to a ‘spatial’ quantity (the time-independent, time-averaged firing vector) serves as a good cross-check. Obviously, use of limit-cycle period alone as our measure of similarity might have commonly led to misclassified cycles.
We do not want to distinguish between very similar limit cycles, separated by only a small number of misfirings. Therefore, for two cycles to be considered similar, we allow small non-zero values of the distance $$d(\bar{{\ensuremath{{\boldsymbol{x}}}\xspace}},\bar{{\ensuremath{{\boldsymbol{x}}}\xspace}}') =
\frac{1}{N} \sum_{i=1}^{N} |\bar{x}_i - \bar{x}_i'|$$ between their fingerprints $\bar{{\ensuremath{{\boldsymbol{x}}}\xspace}}$ and $\bar{{\ensuremath{{\boldsymbol{x}}}\xspace}}'$.
In Figure \[fig:DiffHisto\], we show histograms of the distances between limit cycles with (a) equal and (b) different lengths, accumulated over a large number of different cycles and all investigated networks and thresholds. Due to their high frequency, we have excluded pairs of cycles with $d=0.0$ from this figure. If the two cycles have the same period $L$, the overall number of misfirings during $T$ timesteps is given by $N T d$. The chosen value of the threshold $d_{\text{max}}=0.02$ for cycles to be regarded as similar is indicated in Figure \[fig:DiffHisto\] as well. Evidently, this choice is low enough to prevent misclassification of most of the different limit cycles, as it avoids the large peaks for larger $d$, and it is sufficiently larger than zero to tolerate small deviations in similar cycles. It corresponds to one ‘misfiring’ per time step for $N=50$, on average.
As observed in Figure \[fig:DiffHisto\].a (center column), cycle pairs with equal and long periods ($L>50$) produce a curious clustering of distances within $0.02< d < 0.1$. A closer inspection of this peak reveals that only three cycle pairs contribute to this clustering. Therefore, in order to be conservative, we raised the threshold for limit-cycle difference to $d_{\text{max}} = 0.1$ in this case. Since the number of cycle pairs within this peak is only a small fraction of the total number of cycle pairs observed, this change in $d_{\text{max}}$ changes the ‘diversity’ and ‘volatility’ (defined later) only by a small amount.
For cycle pairs with short periods ($L < 50$), the non-zero distances approximately follow a gaussian distribution with mean $\bar{d} =
0.45$ and standard deviation $\sigma_d\sim 0.15$ (see Figure \[fig:DiffHisto\], left column). Cycles with larger periods display a non-gaussian distribution (see Figure \[fig:DiffHisto\], center column) with positive skew, mean $\bar{d} \sim 0.18$, and standard deviation $\sigma_d \sim 0.12$.
Splitting the range of compared periods into smaller intervals reveals that within these smaller intervals the distribution is gaussian as well, but with decreasing mean for increasing periods (not shown). Long-period limit cycles often tend to have a large fraction of their neurons firing very close to 50% of the time, reducing the average distance.
Performance of the random asymmetric neural network
===================================================
The firing rate of random asymmetric neural networks tends to converge to a fixed point where the neural firing vector ${\ensuremath{{\boldsymbol{x}}}\xspace}(t)$ does not change in time, if the deviation from normal thresholds is large. If the mean threshold value is much greater or much less than normal, the neurons are less or more likely to fire and the RSANN will tend to have a fixed point with very few or very many neurons firing each time step. These extreme conditions are called network ‘death’ and ‘epilepsy’, respectively. By contrast, for normal thresholds $(\mu=1)$, limit cycles with very long periods are possible, as seen in Figs. \[fig:Period\_Mean\], \[fig:FireFrac\_Mean\] (cf., Clark, Kürten & Rafelski (1988), Kürten (1988), Clark (1990,1991), McGuire, Littlewort & Rafelski (1991), McGuire [*et al.*]{} (1992)).
![The mean neural firing rate $\bar{x}$ (averaged over time and neuron index) for each observed limit cycle as a function of $\mu$, for ${\ensuremath{\varepsilon}\xspace}=0$ and $N=40$.[]{data-label="fig:FireFrac_Mean"}](period_bias.eps){width="\linewidth"}
![The mean neural firing rate $\bar{x}$ (averaged over time and neuron index) for each observed limit cycle as a function of $\mu$, for ${\ensuremath{\varepsilon}\xspace}=0$ and $N=40$.[]{data-label="fig:FireFrac_Mean"}](MeanFireFrac_Bias.eps){width="\linewidth"}
When the mean threshold value is normal ($\mu = 1)$, but the threshold fluctuations from normality are large (${\ensuremath{\varepsilon}\xspace}>
{\ensuremath{\varepsilon}\xspace}_2$), then there also exist [*many*]{} different mixed death/epilepsy fixed points in which a fraction of the neurons are firing at each time step and the remaining neurons never fire. With growing ${\ensuremath{\varepsilon}\xspace}$, this fraction of neurons with constant firing state ($x_i(t)=0$ or $x_i(t)=1$ for all $t$) grows, due to the fact that a larger fraction of the neuron thresholds differ significantly from their normal values. Therefore, as ${\ensuremath{\varepsilon}\xspace}$ increases, a smaller fraction of neurons actively participate in the dynamics, making the effective network size smaller and the limit cycles shorter. As can be seen in Figure \[fig:Period\_Eps\], the average period and the maximal period both decrease roughly exponentially with growing ${\ensuremath{\varepsilon}\xspace}$.
Conversely, as ${\ensuremath{\varepsilon}\xspace}$ increases, the number of different limit cycles increases as well (as observed during many different trials, each with a different random realization of thresholds $V_i \equiv \eta_i
V_i^0$). This increase in the observed number of different cycles is caused by each network realization eventually producing a (short) limit cycle in a different portion of the network, as more and more different neurons drop out of the picture as dynamical participants (Figure \[fig:NewCycles\]). The saturation for some of the nets in the ensemble of 10 nets is artificial, since we limited the maximum number of trials to 1000 to constrain computational costs. [^10]
In the following three sections, we limit our discussion to a small ensemble of 10 networks with different connection-strength matrices, and we compute the mean and variation of the quantities-of-interest, as a function of the disorder amplitude, ${\ensuremath{\varepsilon}\xspace}$. In section \[sec:Ensemble\], we explore a much larger ensemble of 300 networks, but for only a few values of ${\ensuremath{\varepsilon}\xspace}$.
![The period of the observed limit cycles, averaged over 100 trials, decreases roughly exponentially as the noise amplitude ${\ensuremath{\varepsilon}\xspace}$ increases (for $\mu = 1$, $N = 50$; note the semi-logarithmic scale.) The average period for 10 networks is plotted as a solid line, and the average of the maximum periods for these 10 networks is plotted as a thick dashed line, with $1 \sigma$ deviation limits plotted as dotted lines.[]{data-label="fig:Period_Eps"}](period_eps.eps){width="\linewidth"}
![The total number of different limit cycles observed increases as the noise amplitude ${\ensuremath{\varepsilon}\xspace}$ increases (for $N=50$, and for a maximal number of trials $= 1000$, in linear and log-log plots). For ${\ensuremath{\varepsilon}\xspace}>10^{-3}$, the ensemble-average scaling is approximately as a power law, ${\cal N}_{\text{cycles}} \sim {\ensuremath{\varepsilon}\xspace}^{\alpha}$, with $\alpha \sim 1.0$, making it roughly a linear function as well.[]{data-label="fig:NewCycles"}](cycles_eps.eps "fig:"){width="0.65\linewidth"} ![The total number of different limit cycles observed increases as the noise amplitude ${\ensuremath{\varepsilon}\xspace}$ increases (for $N=50$, and for a maximal number of trials $= 1000$, in linear and log-log plots). For ${\ensuremath{\varepsilon}\xspace}>10^{-3}$, the ensemble-average scaling is approximately as a power law, ${\cal N}_{\text{cycles}} \sim {\ensuremath{\varepsilon}\xspace}^{\alpha}$, with $\alpha \sim 1.0$, making it roughly a linear function as well.[]{data-label="fig:NewCycles"}](cycles_eps.logxy.eps "fig:"){width="0.65\linewidth"}
Eligibility {#sec:Elig}
-----------
Since we are interested in complex dynamical behaviour, a large fraction of neurons should participate non-trivially in the dynamical collective activity of the network – such a network is said to have a high degree of eligibility. A limit cycle $\alpha$ will have a maximally eligible time-averaged firing pattern $\bar{{\ensuremath{{\boldsymbol{x}}}\xspace}}^{\alpha}$, if $\bar{x}_i^{\alpha}=\frac{1}{2}$ for all neurons $i$. There will be minimal eligibility if $\bar{x}_i^{\alpha} \in
\{0, 1\}$ for all neurons $i$. The Shannon information (or entropy) has these properties, so we will adopt an entropy function as our measure of the eligibility of a given limit-cycle attractor $\alpha$: $$e(\alpha) \equiv
-\frac{1}{N} \sum_{i=1}^N \bar{x}_i^{\alpha} \ln \bar{x}_i^{\alpha} .$$ The mean eligibility ${\cal E}$, averaged over all ${\cal N}_{\text{trials}}$ trials, is $${\cal E} \equiv \frac{1}{{\cal N}_{\text{trials}}}
\sum_{\alpha=1}^{{\cal N}_{\text{trials}}}
e(\alpha) \leq \frac{1}{2} \ln 2 \equiv {\cal E}_{\text{max}}$$ for fixed network connectivity and fixed ${\ensuremath{\varepsilon}\xspace}$. Despite its utility, we do not have a rigorous dynamical motivation for quantifying eligibility by entropy. As discussed at the beginning of this section, the fraction of actively participating neurons decreases with growing ${\ensuremath{\varepsilon}\xspace}$; thus eligibility is decreasing as well (Figure \[fig:Elig\_Eps\]). In other words, when the thresholds become grossly ‘out-of-tune’ with the mean membrane potential, the RSANN attractors become more trivial, with each neuron tending toward its own independent fixed point $x_i(t)=1$ or $x_i(t)=0$.
Diversity {#sec:Div}
---------
We measure the accessibility of a given attractor by estimating the probability $P(\alpha)$ that a given attractor (first observed at trial $\alpha$) is observed during all ${\cal N}_{\text{trials}}$ trials, identified with its relative frequency of occurence $$P(\alpha) \equiv \frac{{\cal N}(\alpha)}{{\cal N}_{\text{trials}}} \; ,$$ where ${\cal N}(\alpha)$ is the number of observations of limit cycle $\alpha$. Note that if a given attractor is only observed [*once*]{}, then $P(\alpha) = 1/{\cal
N}_{\text{trials}}$, while if the same attractor is observed in every trial, then $P(\alpha)=1$.
Given that each different limit-cycle attractor is accessed by the network with probability $P(\alpha)$, we can define the diversity ${\cal D}$ as the attractor occupation entropy: $${\cal D} = -\sum_{\alpha=1}^{{\cal N}_{\text{cycles}}}
P(\alpha)\ln P(\alpha)
\; \text{ , where} \quad
\sum_{\alpha=1}^{{\cal N}_{\text{cycles}}} P(\alpha) = 1
\label{eq:Div}$$ and ${{\cal N}_{\text{cycles}}}$ is the total number of different observed cycles. It is easily seen that a large ${\cal D}$ corresponds to the ability to occupy many different cyclic modes with nearly equal probability; the diversity will reach a maximum value of ${\cal
D}_{\text{max}} = \ln {{\cal N}_{\text{trials}}}$ when $P(\alpha) =
1 / {{\cal N}_{\text{trials}}}$ for all limit cycles $\alpha$, i.e. if in each trial a different cycle is observed. A small value of ${\cal
D}$ corresponds to a strong stability (or inflexibility) of the system – very few different cyclic modes are available. As can be seen from Figure \[fig:Div\_Eps\] the diversity grows rapidly with increasing disorder amplitude ${\ensuremath{\varepsilon}\xspace}$, and with a relatively small disorder value of ${\ensuremath{\varepsilon}\xspace}= {\ensuremath{\varepsilon}\xspace}_1 \sim 10^{-2}$, the diversity is already half of its maximal value. Since it takes into account the accessibility of all the detected cycles, this technique of quantifying diversity by an entropy function is considerably more robust and meaningful than simply counting the cycles.
Volatility {#sec:Vol}
----------
Volatility is defined as the ability to access a large number of highly eligible limit cycles, or a ‘mixture’ of high eligibility and high diversity. Having defined both eligibility and diversity, we can now combine them to define volatility as an entropy-weighted entropy: $${\cal V} = -\sum_{\alpha=1}^{{\cal N}_{\text{cycles}}}
e(\alpha) P(\alpha) \ln P(\alpha) \; .$$ Since the volatility curve in Figure \[fig:Vol\_Eps\] is roughly the product of the eligibility and the diversity curve in Figures \[fig:Elig\_Eps\] and \[fig:Div\_Eps\], there exists an intermediate regime ${\ensuremath{\varepsilon}\xspace}_1 < {\ensuremath{\varepsilon}\xspace}< {\ensuremath{\varepsilon}\xspace}_2$ of high volatility. At ${\ensuremath{\varepsilon}\xspace}= {\ensuremath{\varepsilon}\xspace}_1 \sim 10^{-2}$ the growing disorder amplitude causes a transformation to a condition of diversity, entailing many different limit cycles, whereas at ${\ensuremath{\varepsilon}\xspace}= {\ensuremath{\varepsilon}\xspace}_2 \approx 0.5$ the disorder amplitude has become so large that most limit cycles become fixed points. We accordingly identify three different regimes for the RSANN with disorder:
1. Stable Regime: ${\ensuremath{\varepsilon}\xspace}< 10^{-2}$
2. Volatile Regime: $10^{-2} \leq {\ensuremath{\varepsilon}\xspace}\leq 0.5$
3. Trivially Random Regime: ${\ensuremath{\varepsilon}\xspace}> 0.5$.
Larger Ensemble study {#sec:Ensemble}
---------------------
![ For 300 networks of $N=50$ neurons, each with different connection strength matrices, we form histograms of the number of limit cycles found in each different network. Histograms for four different values of ${\ensuremath{\varepsilon}\xspace}$ are presented. Note the rebinned histogram in the inset of the ${\ensuremath{\varepsilon}\xspace}= 0.0$ histogram. []{data-label="fig:histonumdiff"}](newCycL_0.0.comb.eps "fig:"){width="49.00000%"} ![ For 300 networks of $N=50$ neurons, each with different connection strength matrices, we form histograms of the number of limit cycles found in each different network. Histograms for four different values of ${\ensuremath{\varepsilon}\xspace}$ are presented. Note the rebinned histogram in the inset of the ${\ensuremath{\varepsilon}\xspace}= 0.0$ histogram. []{data-label="fig:histonumdiff"}](newCycL_0.1.eps "fig:"){width="50.00000%"} ![ For 300 networks of $N=50$ neurons, each with different connection strength matrices, we form histograms of the number of limit cycles found in each different network. Histograms for four different values of ${\ensuremath{\varepsilon}\xspace}$ are presented. Note the rebinned histogram in the inset of the ${\ensuremath{\varepsilon}\xspace}= 0.0$ histogram. []{data-label="fig:histonumdiff"}](newCycL_0.2.eps "fig:"){width="49.00000%"} ![ For 300 networks of $N=50$ neurons, each with different connection strength matrices, we form histograms of the number of limit cycles found in each different network. Histograms for four different values of ${\ensuremath{\varepsilon}\xspace}$ are presented. Note the rebinned histogram in the inset of the ${\ensuremath{\varepsilon}\xspace}= 0.0$ histogram. []{data-label="fig:histonumdiff"}](newCycL_0.4.eps "fig:"){width="50.00000%"}
The results obtained in Sections \[sec:Elig\] - \[sec:Vol\] were obtained from ten RSANNs with random weight matrices drawn from the uniform distribution $w_{ij} \in [-1,1]$. As can be seen from the standard deviation curves in figures \[fig:Elig\_Eps\]- \[fig:Vol\_Eps\], which are in close proximity to the average curves, all networks exhibit the same qualitative behaviour. To further confirm this finding, we have tabulated in tables \[tab:numfig\] & \[tab:periodlength\] and Figure \[fig:histonumdiff\] the statistics of the number of different limit cycles, their periods, and the diversity & volatility, for 300 networks with different connection strength matrices using 500 different disorder vectors ${\ensuremath{{\boldsymbol{\eta}}}\xspace}$ for each network. The ${\ensuremath{\varepsilon}\xspace}=0.0$ results in Table \[tab:numfig\] and Figure \[fig:histonumdiff\] confirm Amari’s theoretical result (1974) and the empirical results of Clark, Kürten & Rafelski (1988), that random networks tend to possess only a small number of different cyclic modes. For larger ${\ensuremath{\varepsilon}\xspace}$, the ensemble-average number of cycles, the diversity and volatility are all much larger than for ${\ensuremath{\varepsilon}\xspace}=0$.
In particular, the results shown in Table \[tab:periodlength\] suggest that with increasing ${\ensuremath{\varepsilon}\xspace}$, the minimum period rapidly decreases, whereas the maximum period rapidly increases, with the average period staying roughly constant and closer to the minimum than the maximum. This suggests that at least a few limit cycles having long periods exist with non-normal thresholds, and that there are many more fixed points than long-period limit cycles when the thresholds are far from normal, than when the thresholds are close to normal.
------------------------------------- ------------------- -------------------- ------------------------ ----------------- -----------------
${\ensuremath{\varepsilon}\xspace}$ max. \# of cycles ave. \# of cycles ave. \# of long cycles diversity volatility
0.0 6 2.11 $\pm$ 1.17 0.04 $\pm$ 0.19 0.06 $\pm$ 0.08 0.03 $\pm$ 0.04
0.1 176 45.58 $\pm$ 26.54 1.59 $\pm$ 2.12 0.33 $\pm$ 0.14 0.26 $\pm$ 0.10
0.2 422 179.98 $\pm$ 69.87 12.66 $\pm$ 12.89 0.70 $\pm$ 0.12 0.55 $\pm$ 0.12
0.4 500 461.34 $\pm$ 44.24 18.40 $\pm$ 19.89 0.98 $\pm$ 0.04 0.64 $\pm$ 0.14
------------------------------------- ------------------- -------------------- ------------------------ ----------------- -----------------
: For 300 networks of $N=50$ neurons, each with a different matrix of connection strengths, we tabulate the statistics of the number of observed limit cycles (maximum number, average number of cycles, and average number of long cycles), as well as the average diversity and the average volatility. The results are given for 4 different values of threshold disorder ${\ensuremath{\varepsilon}\xspace}$.[]{data-label="tab:numfig"}
------------------------------------- ------------------ ------------------ ------------------
${\ensuremath{\varepsilon}\xspace}$ ave. min. period ave. max. period ave. mean period
0.0 64.98 111.97 85.63
0.1 2.10 796.63 89.18
0.2 1.27 1233.89 146.66
0.4 1.02 1144.64 110.15
------------------------------------- ------------------ ------------------ ------------------
: For 300 networks of $N=50$ neurons, each a different matrix of connection strengths, we tabulate the averages of the statistics of the observed periods of the limit cycles for each network (minimum period, maximum period, mean period), for four different values of the threshold disorder ${\ensuremath{\varepsilon}\xspace}$.[]{data-label="tab:periodlength"}
Stability and attractor basins of observed cycles {#sec:Stability}
-------------------------------------------------
We have shown in Sections \[sec:Elig\] - \[sec:Ensemble\] that when we change the threshold parameters of a network sufficiently far from their default values, then we get new, non-trivial behavior for nearly each parameter realization. Since the level of threshold disorder which is needed to obtain new and complex behavior is not too high, the ensemble of networks can exhibit diverse and complex behavior with only slight changes in the network parameters.
![ The attractor occupation entropy (Eq. \[eq:Div\]) of the small number of observed cycles (compare Fig. \[fig:Stab\_Cycles\_Eps\]) allows us to estimate the relative sizes of the basins of attraction. Since the entropy reaches only half of its maximum, some cycles dominate over the others. The solid curve is a smoothed average curve; the dashed curves are smoothed $1\sigma$ error curves.[]{data-label="fig:Stab_Entropy_Eps"}](stab_cycles_eps){width="\linewidth"}
![ The attractor occupation entropy (Eq. \[eq:Div\]) of the small number of observed cycles (compare Fig. \[fig:Stab\_Cycles\_Eps\]) allows us to estimate the relative sizes of the basins of attraction. Since the entropy reaches only half of its maximum, some cycles dominate over the others. The solid curve is a smoothed average curve; the dashed curves are smoothed $1\sigma$ error curves.[]{data-label="fig:Stab_Entropy_Eps"}](stab_entropy_eps.eps){width="\linewidth"}
In order to demonstrate that this diversity originates only from ensemble diversity and is not intrinsic to the specific network realizations, we must show that each specific network of the ensemble possesses only a limited set of limit cycles. For this reason we counted the number of different observed cycles ${\cal N}_{\text{cycles}}^{\text{fixed}} (w_{ij},
{\ensuremath{{\boldsymbol{\eta}}}\xspace}({\ensuremath{\varepsilon}\xspace}))$ of each network $(w_{ij}, {\ensuremath{{\boldsymbol{\eta}}}\xspace}({\ensuremath{\varepsilon}\xspace}))$ in the ensemble starting with 100 randomly-chosen initial activity patterns ${\ensuremath{{\boldsymbol{x}}}\xspace}(0)$. Then we performed the ensemble average $${\cal N}_{\text{cycles}}^{\text{fixed}}({\ensuremath{\varepsilon}\xspace}) =
\langle
{\cal N}_{\text{cycles}}^{\text{fixed}} (w_{ij}, {\ensuremath{{\boldsymbol{\eta}}}\xspace}({\ensuremath{\varepsilon}\xspace}))
\rangle_{{\ensuremath{{\boldsymbol{\eta}}}\xspace}({\ensuremath{\varepsilon}\xspace})}\,.$$ As can be seen from Figure \[fig:Stab\_Cycles\_Eps\], the number of different observed cycles is rather small, [*and*]{} does not depend upon the disorder amplitude ${\ensuremath{\varepsilon}\xspace}$. Of course, the [*actual*]{} size of the repertoire does depend on the instantiation of ${\ensuremath{{\boldsymbol{\eta}}}\xspace}({\ensuremath{\varepsilon}\xspace})$, as evidenced by the deviations (indicated by dashed curves), but the degree of variation in ${\cal
N}_{\text{cycles}}^{\text{fixed}}({\ensuremath{\varepsilon}\xspace})$ in no way matches the substantial secular increase of ${\cal
N}_{\text{cycles}}^{\text{fixed}}({\ensuremath{\varepsilon}\xspace})$ observed in Figure \[fig:NewCycles\].
Using a similar diversity measure like in Equation \[eq:Div\], but now using the occurence probabilities of observed cycles with fixed network parameters, it is possible to estimate the relative sizes of the basins of attraction of these cycles. The diversity becomes maximal when all cycles are observed with equal probability, corresponding to basins of attractions of equal size. As can be seen from Figure \[fig:Stab\_Entropy\_Eps\], the diversity reaches only the half of its maximum, indicating basins of attraction of different sizes. The figure suggests that $-\langle\ln P\rangle \sim
\frac{1}{2}$, implying that $\langle P \rangle \sim 0.6$. In other words, for fixed thresholds, as we vary the initial firing vector, we observe the same cycle in greater than $\sim 60\%$ of the trials; as evident in Figure \[fig:Stab\_Entropy\_Eps\], there is little dependence in the fixed-threshold cycle-diversity upon the frozen-disorder amplitude ${\ensuremath{\varepsilon}\xspace}$.
Distribution of Limit-Cycle Periods and their Dependence upon Network Size {#sec:NetSize}
--------------------------------------------------------------------------
For the choice of threshold and connectivity parameters made here, the average cycle length $\langle L \rangle$ grows exponentially with the network size $N$ (see Fig. \[fig:PeriodLen\_NetSize\]). This exponential scaling of the cycle length puts it in the ‘chaotic’ regime of Kürten’s (1988) classification of dynamical phases, where the motion shows high sensitivity to initial conditions. Kürten also found a ‘frozen’ regime where the limit-cycle period scales as a power law in $N$, and where there is little sensitivity of the attracting limit cycle upon initial conditions. When ${\ensuremath{\varepsilon}\xspace}=0.02$, which is in the volatile region, a broad, non-gaussian distribution of cycle lengths is found (see Fig. \[fig:PeriodHisto\]). Furthermore, since the cycle-length distribution shown in Fig. \[fig:PeriodHisto\] does not exhibit peaks at regularly spaced intervals, the possibility that we have employed an errant limit-cycle comparison algorithm is unlikely. This distribution of cycle-lengths for RSANNs differs significantly from the $\exp(-L^2/\tau^2)/L$ distribution of cycle-lengths predicted for Kauffman’s Boolean nets by Bastolla and Parisi (1997). This difference in distributions implies that there may be some significant differences between these two types of nets.
![Distribution of different limit-cycle periods observed in the volatile regime ($N=50$, ${\ensuremath{\varepsilon}\xspace}=0.02$). We overlay a fit to the distribution predicted by Bastolla & Parisi (1997, 1997b) for Kauffman nets.[]{data-label="fig:PeriodHisto"}](meanperiod_N.eps){width="\linewidth"}
![Distribution of different limit-cycle periods observed in the volatile regime ($N=50$, ${\ensuremath{\varepsilon}\xspace}=0.02$). We overlay a fit to the distribution predicted by Bastolla & Parisi (1997, 1997b) for Kauffman nets.[]{data-label="fig:PeriodHisto"}](periodhist_fine.eps){width="\textwidth"}
Dependence of Attractor Count on Observation-Period Length {#sec:Count}
----------------------------------------------------------
In Figure \[fig:NewCycles\], we showed that the number of different attractors observed, ${\cal N}_{\text{cycles}}$, is a significant fraction of the total number of trials ${\cal N}_{\text{trials}}$ when ${\ensuremath{\varepsilon}\xspace}> 0.1$. For disorder between ${\ensuremath{\varepsilon}\xspace}=10^{-2}$ and ${\ensuremath{\varepsilon}\xspace}=0.1$, the number of cycles observed is much larger than one (see Figure \[fig:Div\_Eps\].b), but less than the total number of trials. In the stable regime (${\ensuremath{\varepsilon}\xspace}<
10^{-2}$), ${\cal N}_{\text{cycles}}$ is quite small and largely independent of ${\cal N}_{\text{trials}}$. These three results are complementary: the first result (at high ${\ensuremath{\varepsilon}\xspace}$) implying a nearly inexhaustible source of different limit-cycle attractors accompanied by a steady decrease of eligibility with increasing ${\ensuremath{\varepsilon}\xspace}$; the second result (at moderate ${\ensuremath{\varepsilon}\xspace}$) implying a large, but limited repertoire of limit cycles of high eligibility; and the third result (at very low ${\ensuremath{\varepsilon}\xspace}$) implying that we can exhaustively access a small group of high-eligibility limit-cycle attractors with a high degree of robustness.
In the stable regime, ${\cal N}_{\text{cycles}}$ is often greater than $1$ (though small), which means that the stable phase cannot be used to access a particular attractor upon demand, but we can demand reliable access to one of a small number of different attractors. Frequently, however, there is a single dominant attractor, as indicated by the low entropy seen in Fig. \[fig:Stab\_Entropy\_Eps\]. We conclude that, in practice, for a given set of network parameters or external inputs to the network, but starting with different initial conditions, the network will converge to the same attractor most of the time.
Discussion
==========
Ensemble-scanner, Multistability, ‘Creativity/Madness’, and Control Algorithms
------------------------------------------------------------------------------
In the ‘noisy’ runs at ${\ensuremath{\varepsilon}\xspace}>0$, epitomized by Fig. \[fig:clock\], we are actually sampling a significant fraction of an entire ensemble of closely-related networks over the course of time, as the noise or disorder or an external input slowly varies. Based upon the results for our RSANN model, quenched noise (disorder) can give a dynamical system access to a whole ensemble of different behaviors at different times during the lifetime of the dynamical system. In other words, slowly-varying threshold noise or disorder or external input can act as a ‘scanner’ for a host of dynamic modes.
The posed existence (Skarda & Freeman (1987), Yao & Freeman (1990), Freeman [*et al.*]{} (1997)) of a chaotic ground-state attractor for the olfactory system and the existence of ‘multistable’ limit-cycle excited-state attractor lobes provides a striking exemplification of our volatility concept, and potentially an [*in vivo*]{} demonstration of this phenomenon.
The RSANN networks studied here have a small repertoire of behaviors when there is no noise or disorder in the thresholds (${\ensuremath{\varepsilon}\xspace}=0$); hence, there is little multistability. The size of the repertoire becomes tremendously large (as does the extent of multistability) for ‘small’ changes in the threshold parameters (${\ensuremath{\varepsilon}\xspace}> 10^{-2}$). One is therefore tempted to call this behavior ‘chaotic’ with respect to the parameter changes, since it has one of the hallmarks of chaos (sensitive dependence upon small changes of the parameters). However, for smaller changes in the threshold parameters ($0 < {\ensuremath{\varepsilon}\xspace}< 10^{-3}$), the repertoire of ${\ensuremath{\varepsilon}\xspace}= 0$ behaviors is ‘stable’ – no new cycles are observed. For the purpose of discussion , we refer to this [*delimited*]{} sensitivity to parameter changes, as ‘quasi-chaos’ or ‘quasi-multistability’. Additionally, the stability is augmented by the fact that very frequently, the repertoire of a network in the stable regime is dominated by a single cycle (for fixed connections and fixed disorder), a phenomenon known as ‘canalization’ (Kauffman (1993)). Hence, for the sake of argument, we will also assume that in the stable regime only one cycle is accessible. By taking advantage of the quasi-chaotic/quasi-multistable threshold parameters (noting that the connection strength parameters are probably also quasi-chaotic), we can access a large number of different RSANN attractors, each with a small neighborhood of stability in threshold-parameter space. With a suitable feedback algorithm, one might be able to control this quasi-chaos (cf. Ott, Grebogi & Yorke (1990)) and access and stabilize a given attractor upon demand. Due to the proximity of other limit cycles just beyond the local neighborhood of stability of the given attractor, novel attractors are always within a stone’s throw of the given attractor, while maintaining a respectable distance so as not to be destabilizing. Such an approach to controlled creativity has been developed into the adaptive resonance formalism (Carpenter & Grossberg (1987)).
The volatile regime within ${\ensuremath{\varepsilon}\xspace}=0.001-0.5$ can be subdivided into two sub-regimes. The lower end of the range, ${\ensuremath{\varepsilon}\xspace}=0.001-0.1$, which corresponds to the upward-sloping part of the volatility curve in Figure \[fig:Vol\_Eps\].b, could suggestively be named the ‘creative’ regime, wherein new cycles are observed with some rarity, so as to provide truly new behavioral modes for the RSANN. These new cycles can be taken together with the more commonly-observed cycles in the net’s repertoire, perhaps to produce new and ‘interesting’ sequences of behavior (if these cyclic modes can be logically sequenced). The upper end of the range, ${\ensuremath{\varepsilon}\xspace}=0.1-0.5$, might be regarded as the ‘overly-creative’ or ‘slightly-mad’ regime of the RSANN. New, rather complex modes are being found with almost every trial, which could overwhelm the ‘bookkeeping’ resources necessary for the RSANN to implement or utilize the new mode to its full potential. Clearly, this abstract and simple RSANN model is insufficent to be a true neurobiological model of creativity/madness, but it could be a good starting point for a more detailed model.
Generalization to other Complex Systems
---------------------------------------
It may well be a common feature of a broad class of complex, non-linear systems without adaptability or noise that the diversity of non-trivial behaviors is limited. We have confirmed the lack of diverse behavior for a non-linear system with truly simple elements (McCullough-Pitts neurons); we have also seen similar non-volatile behavior for a slightly more general, discretized integrate-and-fire neuron model. This canalization result may be generalizable to other complex systems of either simple or complex units. Indeed, one of the first observations of canalization was in Kauffman’s Boolean immunological networks, which have some significant differences from RSANNs. The canalization property might have been more difficult to generalize if we had started with more complex units like Hodgkin-Huxley neurons. Furthermore, it is plausible that the introduction of a moderate amount of noise or disorder will [*generally*]{} increase the diversity of complex behaviors, as we have seen in RSANNs.
Our volatility-producing model might be applicable in more abstract situations. One might imagine that the states of our simple Boolean neuron reflect in some manner the ‘on or off’ state of complex subunits of a modular system. Such modular complex systems could be probed to determine whether dynamical diversity can or cannot be enhanced by small changes in network parameters. Examples might include:
1. A random neural network composed of subnetworks;
2. A network of complex, real neurons;
3. The brain of an organism with its different subsystems;
4. The geoeconomic or political structure of a large country composed of smaller states, regions, or cities; and
5. The ecological network of the world composed of different regions or of different subcommunities of animals or plants.
In models with subunits that are composed of many sub-subunits, the stability or canalizing ability of the system itself may be significantly enhanced either by a law-of-large-numbers decrease in the noise/disorder susceptibility of an individual subunit, or by self-stabilizing internal feedback loops which may be present by design within the subunits.
Conclusions & Prospects
-----------------------
Based on combinatorics and statistical arguments, one expects to find many limit cycles in a random synchronous asymmetric neural network (RSANN). Experience has shown otherwise. After much of this paper was completed, we found an analytical argument by Amari (1974, 1989) to the effect that RSANNs have only one attractor, in the thermodynamic limit of a large number of neurons, thus explaining our ${\ensuremath{\varepsilon}\xspace}=0$ results.
The main objective of our study has been to construct a volatile neural network which exhibits a large set of easily-accessible highly-eligible limit-cycle attractors, as has been achieved already in a non-neural system (Poon & Grebogi (1995)). First, we have demonstrated that in the absence of noise and in the absence of random, long-term imposition of threshold disorder, a random asymmetric neural network can reach only a small number of different limit-cycle attractors. Second, by imposing and freezing neuronal threshold disorder within a well-defined range (${\ensuremath{\varepsilon}\xspace}_1<{\ensuremath{\varepsilon}\xspace}<{\ensuremath{\varepsilon}\xspace}_2$), we show that RSANNs can access a diversity of highly-eligible limit-cycle attractors. RSANNs exhibit a phase transformation from a small number of distinct limit-cycle attractors to a large number at a disorder amplitude of ${\ensuremath{\varepsilon}\xspace}= {\ensuremath{\varepsilon}\xspace}_1\sim 10^{-2}$. Likewise, RSANNs exhibit an eligibility phase transformation at a threshold disorder amplitude of ${\ensuremath{\varepsilon}\xspace}={\ensuremath{\varepsilon}\xspace}_2\sim 0.5$.
Potentially, Amari’s argument can be extended to gain an understanding of how slight changes of threshold parameters beyond some minimal level can substantially increase the diversity of accessible cyclic modes. This extension is beyond the scope of the current work. Another very interesting question is how the diversity and volatility curves scale with the size of the network.
While the addition of threshold disorder seems to be a trivial mechanism for enhancing the volatility or diversity by constantly changing the parameters of the RSANN, we believe that since some biological systems (Neiman [*et al.*]{} (1999)) may use threshold, synapse and/or externally-generated noise or disorder to enhance their abilities, we have discovered a simple feature which could have some importance in modeling biological systems. We fully expect that other volatility-enhancing mechanisms are available beyond the particular one proposed here.
In summary, our key result is that a random neural network can be driven easily from one to another stable recurrent mode. While such behavior can be always accomplished by radical modifications of some of the network properties, the interesting result we have here presented is that plausibly small (e.g., RMS in the neighborhood of 0.1-1%) and random changes imposed [*simultaneously*]{} upon all of the neural threshold parameters suffices to access new dynamical behavior. Indeed, due to the combinatorics of changing many parameters simultaneously, an immense number of interesting modes become available to the system. We are aware that this does not yet create a network that can self-sequence a series of modes, though some authors have already made considerable progress in this direction (e.g. Daucé and Quoy (2000), Tani (1998)). The development of autonomous control algorithms that provide access to mode sequences is a natural but challenging objective that can potentially lead to a deeper understanding of information processing in recurrent neural networks.
Acknowledgements {#acknowledgements .unnumbered}
----------------
H. Bohr would like to acknowledge the hospitality of P. Carruthers (now deceased), J. Rafelski, and the U. Arizona Department of Physics during several visits when much of this work was carried out. During his graduate studies, P. McGuire was partially supported by an NSF/U.S. Department of Education/State of Arizona doctoral fellowship. J.W. Clark acknowledges research support from the U.S. National Science Foundation under Grant No. PHY-9900713. McGuire, Bohr and Clark were participants in the Research Year on the “The Sciences of Complexity: From Mathematics to Technology to a Sustainable World” at the Center for Interdisciplinary Studies (ZiF) at the University of Bielefeld, in Germany. We all thank many individuals who have provided different perspectives to our work, including the following: G. Sonnenberg, D. Harley, Z. Hasan, H. Ritter, R. Vilela-Mendes, and G. Littlewort.
[99]{}
S. Amari, “A Method of Statistical Neurodynamics,” [*Kybernetik*]{}, [**14**]{} (1974) 201.
S. Amari, “Mathematical Foundations of Neurocomputing,” [*Proceedings of the IEEE*]{}, [**78**]{} (1990) 1443; University of Tokyo Mathematical Engineering Technical Report 89-06.
U. Bastolla and G. Parisi, “Attractors in Fully Asymmetric Neural Networks,” [*J. Phys. A: Math. Gen.*]{} [**30**]{} (1997) 5613.
U. Bastolla and G. Parisi, “Attraction Basins in Discretized Maps,” [*J. Phys. A: Math. Gen.*]{}, [**30**]{} (1997b) 3757.
P. C. Bressloff and J. G. Taylor, “Random Iterative Networks,” [*Phys. Rev.*]{} [**A41**]{} (1990) 1126.
J. Buhmann and K. Schulten, “Influence of Noise on the Function of a ‘Physiological’ Neural Network,” [*Biol. Cybern.*]{} [**56**]{} (1987) 313.
G.A. Carpenter and S. Grossberg, “ART 2: Self-organization of Stable Category Recognition Codes for Analog Input Patterns,” [*Applied Optics*]{} [**26**]{} (1987) 4919.
C.C. Chow and J.A. White, “Spontaneous Action Potentials due to Channel Fluctuations,” [*Biophysical J.*]{} [**71**]{} (1996) 3013.
J.W. Clark, “Statistical Mechanics of Neural Networks,” [*Phys. Rep.*]{} [**158**]{} (1988) 9.
J.W. Clark, “Long-term Behavior of Neural Networks,” [*Relaxation Complex Systems and Related Topics*]{}, eds. I. Campbell and C. Giovannella (Plenum: New York, 1990) 205.
J.W. Clark, “Neural Network Modelling,” [*Phys. Med. Biol.*]{}[** 36**]{} (1991) 1259.
J.W. Clark, K.E. Kürten and J. Rafelski, “Access and Stability of Cyclic Modes in Quasirandom Networks of Threshold Neurons Obeying a Determinisitic Synchronous Dynamics,” [*Computer Simulation in Brain Science*]{}, edited by R.M.J Cotterill (Cambridge Univ. Press:Cambridge,1988) 316.
J.W. Clark, J. Rafelski and J.V. Winston, “Brain Without Mind: Computer Simulation of Neural Networks with Modifiable Neuronal Interactions,” [*Phys. Rep.*]{} [**123**]{} (1985) 215.
E. Daucé and M. Quoy, “Resonant Spatio-temporal Learning in Sparse Random Recurrent Networks,” submitted to [*Biol. Cybernetics*]{} (2000).
W.J. Freeman, H.J. Chang, B.C. Burke, P.A. Rose, and J. Badler, “Taming Chaos: Stabilization of Aperiodic Attractors by Noise,” [*IEEE Trans. on Circuits and Systems*]{}, [**44**]{} (1997) 989.
C.M. Gray & W. Singer, “Stimulus-specific Neuronal Oscillations in Orientation Columns of a Cat Visual Cortex” [*Proceedings of the National Acadamey of Sciences, USA*]{} [**86**]{} (1989) 1698.
Z. Hasan, “Biomechanical Complexity and the Control of Movement,” [*Lectures in the Sciences of Complexity*]{}, edited by D.L. Stein (Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley) (1989) 841.
J.J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” [*Proc. Natl. Acad. Sci. USA*]{} [**79** ]{} (1982) 2554.
Stuart A. Kauffman, [*The Origins of Order: Self-Organization and Selection in Evolution*]{}, Oxford University Press (1993).
K.E. Kürten, “Critical Phenomena in Model Neural Networks,” [*Phys. Lett.* ]{}[**A129**]{} (1988) 157.
H. Lecar and R. Nossal, “Theory of Threshold Fluctuations in Nerves. [**I.**]{} Relations between Electrical Noise and Fluctuations in Axon Firing,” [*Biophysical J.*]{} [**11**]{} (1971) 1048.
H. Lecar and R. Nossal, “Theory of Threshold Fluctuations in Nerves. [**II.**]{} Analysis of Various Sources of Membrane Noise,” [*Biophysical J.*]{} [**11**]{} (1971) 1068.
W.A. Little, “The Existence of Persistent States in the Brain,” [*Math. Biosci.*]{} [**19**]{} (1974) 101.
G.C. Littlewort, J.W. Clark and J. Rafelski, “Transition to Cycling in Neural Networks,” [*Computer Simulation in Brain Science*]{}, edited by R.M.J Cotterill (Cambridge Univ. Press:Cambridge,1988), 345
Z.F. Mainen and T.J. Sejnowski, “Reliability of Spike Timing in Neocortical Neurons,” [*Science*]{} [**268**]{} (1995) 1503.
E. Marder and S.L. Hooper, “Neurotransmitter Modulation of the Stomatogastric Ganglion of Decapod Crustaceans,” [*Model Neural Networks and Behavior*]{}, editted by A.I. Selverston (Plenum Press: New York,1985), 319.
P.C. McGuire, G.C. Littlewort and J.Rafelski, “Brainwashing Random Asymmetric ‘Neural’ Networks,” [*Phys. Lett.*]{} [**A160**]{} (1991) 255.
P.C. McGuire, G.C. Littlewort, C. Pershing and J. Rafelski, “Training Random Asymmetric ‘Neural’ Networks Towards Chaos – A Progress Report,” [*Proceedings of the Workshop on Complex Dynamics in Neural Networks*]{},edited by J.G Taylor, E.R. Caianiello, R.M.J. Cotterill and J.W. Clark ( Springer-Verlag, London, 1992) pp. 90-102.
J.D. Murray, [*Mathematical Biology*]{}, Springer-Verlag (1989), pp. 161-166.
A. Neiman, X. Pei, D. Russell, W. Wojtenek, L. Wilkens, F. Moss, H. Braun, M. Huber and K. Voight, “Synchronization of the Electrosensitive Noisy Cells in the Paddlefish,” [*Phys. Rev. Lett.*]{} [**82**]{} (1999) 660.
E. Ott, C. Grebogi and J.A. Yorke, “Controlling Chaos,” [*Phys. Rev. Lett.* ]{}[**64**]{} (1990) 1196.
X. Pei, L.A. Wilkens and F. Moss, “Noise-mediated Spike Timing Precision from Aperiodic Stimuli in an Array of Hodgkin-Huxley-type Neurons,” [*Phys. Rev. Lett.*]{} [**77**]{} (1996) 4679.
L. Poon and C. Grebogi, “Controlling Complexity,” [*Phys. Rev. Lett.*]{} [**75**]{} (1995) 4023-4026.
R.H. Rand, A.H. Cohen and P.J. Holmes, “Systems of Coupled Oscillators as Models of Central Pattern Generators,” [*Neural Control of Rhythmic Movements in Vertebrates*]{}, Eds. A.H. Cohen, S. Rossignol, and S. Grillner (John Wiley: New York, 1988), 333.
M. Schreckenberg, “Attractors in the Fully Asymmetric SK-model,” [*Z. Phys. B - Condensed Matter*]{} [**86**]{}, (1992) 453.
G. L. Shaw and R. Vasudevan, “Persistent States of Neural Networks and the Random Nature of Synaptic Transmission,” [*Math. Biosci.*]{}, [**21**]{} (1974) 207.
C.A. Skarda and W.J. Freeman, “How Brains Make Chaos in Order to Make Sense of the World,” [*Behavioral and Brain Sciences*]{}, [**10**]{} (1987) 161.
J. Tani, “An Interpretation of the ‘Self’ from the Dynamical Systems Perspective: a Constructivist Approach”, [*J. Consciousness Studies*]{} [**5**]{} (1998) 516.
J. G. Taylor, “Spontaneous Behaviour in Neural Networks,” [*J. Theor. Biol.*]{}, [**36**]{} (1972) 513.
Y. Yao and W.J. Freeman, “Model of Biological Pattern Recognition with Spatially Chaotic Dynamics,” [*Neural Networks*]{}, [**3**]{} (1990) 153.
A. Zador, “Impact of Synaptic Unreliability on the Information Transmitted by Spiking Neurons,” [*J. Neurophysiol.*]{} [**79**]{} (1997) 1219.
[^1]: Department of Physics; University of Arizona; Tucson, AZ 85721, USA
[^2]: Neuroinformatics Group: Computer Science Department: Technische Fakultät (Engineering College); University of Bielefeld; 33501 Bielefeld, Germany
[^3]: Center for Interdisciplinary Studies (ZiF): Complexity Program; University of Bielefeld, 33501 Bielefeld, Germany
[^4]: Before March 15, 2002, please correspond with this author at his current address in the University of Bielefeld Neuroinformatics Group; or by Phone (0049) 521-106-6059 or -6060; FAX (0049) 521-106-6011; or at his email address: mcguire@techfak.uni-bielefeld.de
[^5]: After March 15, 2002, please correspond with this author at his next address in the Centre de Astrobiología (CSIC/INTA); Instituto Nacional de Técnica Aeroespacial; Ctra de Torrejón a Ajalvir, km 4; 28850 Torrejón de Ardoz; Madrid, Spain; or by Phone (0034) 91-520-21-07; FAX (0034) 91-520-10-74, or at his email address: mcguire@cab.inta.es
[^6]: Department of Physics; DTU, The Technical University of Denmark, B. 307; DK-2800 Lyngby, Denmark
[^7]: Department of Physics and the McDonnell Center for the Space Sciences; Washington University; St. Louis, MO 63130, USA
[^8]: Biomedical Engineering Program; University of Arizona; Tucson AZ 85721, USA
[^9]: for networks of binary-valued neurons, the dynamical behavior can simulate chaos, but for networks of real-valued neurons, true chaos is observable, with the prerequisite for chaos: ‘sensitive dependence upon initial conditions’.
[^10]: In the accompanying log-log version of Figure \[fig:NewCycles\], the behavior seems much more regular, with the large deviations from the ensemble-average behavior for large ${\ensuremath{\varepsilon}\xspace}$ becoming less important. The general shape for ${\ensuremath{\varepsilon}\xspace}> 10^{-3}$ is of a power law with exponent near $1.0$ and positive coefficient (${\cal N}_{\text{cycles}}
\sim A {\ensuremath{\varepsilon}\xspace}^{\alpha}$, with $\alpha \sim 1.0$), the near-unity exponent of the power law making it roughly a linear dependence as well. For particular examples from this 10-network ensemble, the dependence of the ${\cal N}_{\text{cycles}}$ curve is sometimes not a power-law; and for the cases in which the behavior is similar to that of a power law, the coefficient $A$ and exponent $\alpha$ of the power law both differ by as much as a factor of 2 from the coefficient and exponent for the average behavior. The fact that the average behavior is a power law or even a linear function (rather than irregular behavior) means that it might be worthwhile and interesting in the future to perform a theoretical analysis of cycle diversity for RSANNs as a function of threshold disorder.
|
---
abstract: |
We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $\lambda$, called the activity or fugacity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $\omega$ which is the positive solution to $\lambda=\omega(1+\omega)^b$, and vertices are occupied with probability $\omega/(1+\omega)$ when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and non-reconstruction regions at $\omega_r\approx \ln{b}/b$. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs.
In this paper we show that the relaxation time of the Glauber dynamics on regular trees $T_h$ of height $h$ with branching factor $b$ and $n$ vertices, undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any $\omega \le \ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is $\Omega(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction threshold we show that for every $\delta>0$, for $\omega=(1+\delta)\ln{b}/b$, the relaxation time on $T_h$ with any boundary condition is $O(n^{1+\delta + o_b(1)})$, and we construct a boundary condition where the relaxation time is $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.
author:
- 'Ricardo Restrepo[^1] [^2]'
- 'Daniel Stefankovic[^3]'
- 'Juan C. Vera[^4]'
- 'Eric Vigoda[^5]'
- 'Linji Yang$^\P$'
date: 'July 11, 2010'
title: Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees
---
Introduction {#sec:introduction}
============
There has been much recent interest in possible connections between equilibrium properties of statistical physics models and efficiency of local Markov chains for studying these models (see, e.g., [@BKMP; @DSVW; @Martinelli-lecturenotes; @MSW-ising; @MSW-soda; @TVVY]). In this paper we study the hard-core model and establish new connections between the so-called reconstruction threshold in statistical physics with the convergence time of the single-site Markov chain known as the Glauber dynamics. The hard-core model was studied in statistical physics as model of a lattice gas (see, e.g., Sokal [@Sokal]), and in operations research as a model of communication network (see Kelly [@Kelly]). It is a natural combinatorial problem, corresponding to counting and randomly sampling weighted independent sets of an input graph $G=(V,E)$. Let $\Omega=\Omega(G)$ denote the set of independent sets of $G$. Each set is weighted by an activity (or fugacity) $\lambda>0$. For $\sigma\in\Omega$, its weight is $\wt(\sigma) = \lambda^{|\sigma|}$ where $|\sigma|$ is the number of vertices in the set $\sigma$. The Gibbs measure is defined over $\Omega$ as $\mu(\sigma) = \wt(\sigma)/Z$ where $Z=\sum_{\sigma\in\Omega} \wt(\sigma)$ is the partition function.
This paper studies the hard-core model on trees, in some cases with a boundary condition. Let $T_h$ denote the complete tree of height $h$ with branching factor $b$. For concreteness we are assuming the root has $b$ children, but our results, of course, easily extend to allow $b+1$ children for the root, the so-called Bethe lattice. Let $n$ denote the number of vertices in $T_h$, and let $L$ denote the leaves of the tree. A boundary condition is an assignment $\BD$ to the leaves, where in the case of the hard-core model, $\BD$ specifies a subset of the leaves $L$ that are in the independent set. Then let $\Omega_\BD = \{\sigma\in\Omega: \sigma(L)=\BD\}$ be the set of independent sets of $T_h$ that are consistent with $\BD$, and the Gibbs measure ${{\mu_{h,\BD}}}$ is defined with respect to $\Omega_\BD$, i.e., it is the projection of $\mu$ onto $\Omega_\BD$.
The (heat bath) Glauber dynamics is a discrete time Markov chain $(X_t)$ for sampling from the Gibbs distribution $\mu$ for a given graph $G=(V,E)$ and activity $\lambda$. We view $\Omega\subset\{0,1\}^V$ where for $X_t\in\Omega$, $X_t(v)=1$ iff $v$ is in the independent set. The transitions $X_t\rightarrow X_{t+1}$ of the Glauber dynamics are defined as:
- Choose a vertex $v$ uniformly at random;
- For all $w\neq v$ set $X_{t+1}(w) = X_t(w)$;
- If all of the neighbors of $v$ are unoccupied, set $X_{t+1}(v) = 1$ with probability $\lambda/(1+\lambda)$, otherwise set $X_{t+1}(v) = 0$.
When a boundary condition $\BD$ is specified, the state space is restricted to $\Omega_\BD$. For the case of the complete tree $T_h$ (possibly with a boundary condition $\BD$) it is straightforward to verify that the Glauber dynamics is ergodic with unique stationary distribution $\mu_h$ (or ${{\mu_{h,\BD}}}$ when a boundary condition is specified). Thus, the Glauber dynamics is a natural algorithmic process for sampling from the Gibbs distribution. We study the relaxation time of the dynamics, which is defined as the inverse of the spectral gap of the transition matrix. See Section \[sec:background\] for a more detailed definition of the relaxation time.
The Gibbs distribution describes the equilibrium state of the system, and the Glauber dynamics is a model of how the physical system reaches equilibrium [@Martinelli-lecturenotes]. Thus, it is interesting to understand connections between properties of the equilibrium state (i.e., the Gibbs distribution) and properties of how the system reaches equilibrium (i.e., the Glauber dynamics). Models from statistical physics are designed to study phase transitions in the equilibrium state. A phase transition is said to occur when a small change in the microscopic parameters of the system (in the case of the hard-core model that corresponds to $\lambda$) causes a dramatic change in the macroscopic properties of the system.
A well-studied phase transition is uniqueness/non-uniqueness of infinite volume Gibbs distributions, these are obtained as a limit of Gibbs measures for a sequence of boundary conditions as $h\rightarrow\infty$. For the hard-core model on the complete tree, Kelly [@Kelly] showed that the uniqueness threshold is at $\lambda_u =b^b/(b-1)^{b+1}$ (namely, uniqueness holds iff $\lambda<\lambda_u$). There are interesting connections between the uniqueness threshold $\lambda_u$ and the efficiency of algorithms on general graphs. In particular, Weitz [@Weitz] showed a deterministic fully-polynomial approximation scheme to estimate the partition function for any graph with constant maximum degree $b$ for activities $\lambda<\lambda_u$. Recently, Sly [@Sly:hardcore-negative] showed that it is NP-hard (unless $NP=RP$) to approximate the partition function for activities $\lambda$ satisfying $\lambda_u<\lambda<\lambda_u+\eps_b$ for some small constant $\eps_b$.
We are interested in the phase transition for reconstruction/non-reconstruction. This corresponds to extremality of the infinite-volume measure obtained by the “free” boundary condition where free means no boundary condition [@Georgii]. This measure can be generated by the following broadcast process which constructs an independent set $\sigma$. Let $\omega$ be the real positive solution of $\lambda=\omega(1+\omega)^b$. Consider the infinite complete tree with branching factor $b$, and construct $\sigma$ as follows. We first include the root $r$ in $\sigma$ with probability $\omega/(1+\omega)$ and leave it out with probability $1/(1+\omega)$. Then for each vertex $v$, once the state of its parent $p(v)$ is determined, if $p(v)\notin\sigma$ then we add $v$ into $\sigma$ with probability $\omega/(1+\omega)$ and leave it out with probability $1/(1+\omega)$; if $p(v)\in\sigma$ then we leave $v$ out of $\sigma$. Let $\sigma_h$ denote the configuration of $\sigma$ on level $h$, and let $\nu_h$ denote the broadcast measure on $T_h$.
Reconstruction addresses whether $\sigma_h$ “influences” the configuration at the root $r$. In words, we first generate $\sigma$ using the broadcasting measure, then we fix $\sigma_h$ and resample a configuration $\tau$ on $T_h$ from the Gibbs distribution ${{\mu_{h,\BD}}}$ with boundary condition $\BD=\sigma_h$. Of course, for finite $h$, the configuration at the root $r$ in $\tau$ has a bias to the initial configuration $\sigma(r)$. Non-reconstruction is said to hold if the root is unbiased in expectation in the limit $h\rightarrow\infty$. More precisely, reconstruction holds if and only if: $$\label{eq:reconstruction-defn}
\lim\limits_{h\rightarrow\infty} {{{\mathrm{E}}_{\sigma\sim\nu_h}\left[{\left| {{\mu_{h,\sigma_h}}}(r\in\tau) - \frac{\omega}{1+\omega}\right|}\right]}} > 0.$$ There are many other equivalent conditions to the above definition of reconstruction, see Mossel [@MosselSurvey] for a more extensive survey. We refer to the reconstruction threshold as the critical $\omega_r$ such that for all $\omega<\omega_r$ non-reconstruction holds and for all $\omega>\omega_r$ reconstruction holds. The existence of the reconstruction threshold follows from Mossel [@Mossel-2nd-eigenvalue Proposition 20], and, by recent work of Bhatnagar et al [@BST] and Brightwell and Winkler [@BW], it is known that $\omega_r=(\ln{b} + (1+o(1))\ln{\ln{b}})/b$.
Reconstruction for the Ising and Potts models has applications in phylogenetics [@DMR] and for random constraint satisfaction problems is connected to the geometry of the space of solutions on sparse random graphs [@ACO; @GMON; @KMRSZ; @MRT]. Our interest in this paper is on establishing more detailed connections between the reconstruction threshold and the relaxation time of the Glauber dynamics for trees. Berger et al [@BKMP] proved that for the tree $T_h$ with boundary condition $\BD$ such that ${{\mu_{h,\BD}}}=\nu_h$, $O(n)$ relaxation time for all $h$ implies non-reconstruction. For the Ising model and colorings the boundary condition is empty, i.e., $\nu_h$ corresponds to the free boundary condition. That is not the case for the hard-core model as discussed further in Section \[sec:lower-bound-approach\].
It was recently established for the Ising model [@BKMP; @MSW-ising; @DLP] and for k-colorings [@TVVY] that on the tree $T_h$ with free boundary condition, the relaxation is $O(n)$ in the non-reconstruction region and there is a slow down in the reconstruction region. Our starting point was addressing whether a similar phenomenon occurs in the hard-core model. Martinelli et al [@MSW-soda] showed that for the hard-core model on $T_h$ with free boundary condition the relaxation time is $O(n)$ for all $\lambda$ (and the mixing time is $O(n\log{n}))$. However, it is unclear whether the reconstruction threshold has any connection to the relaxation time of the Glauber dynamics on trees for the hard-core model. (In fact, we vacillated between proving that there is fast mixing for all boundary conditions and proving the following result.)
We prove there is a connection by constructing a boundary condition for which the relaxation time slows down at the reconstruction threshold. Here is the formal statement of our results.
\[thm:main\] For the Glauber dynamics on the hard-core model with activity $\lambda= \omega (1+\omega)^b$ on the complete tree $T_h$ with $n$ vertices, height $h$ and branching factor $b$, the following hold:
1. [**For all $\omega\le\ln{b}/b$:**]{} \[thm:below\] $$\Omega(n) \le \ {{T_{\rm{relax}}}}\ \le O(n^{1+o_b(1)}).$$
2. [**For all $\delta>0$ and $\omega=(1+\delta)\ln{b}/b$:**]{}
1. \[thm:above-upper\] For every boundary condition, $${{T_{\rm{relax}}}}\ \le O(n^{1+\delta+o_b(1)}).$$
2. \[thm:above-lower\] There exists a sequence of boundary conditions for all $h\rightarrow\infty$ such that, $${{T_{\rm{relax}}}}= \Omega(n^{1+\delta/2-o_b(1)}).$$
More precisely, we show that there is a function $g(b)=O(\ln\ln{b}/\ln{b})=o(1)$ such that for every $b$, the lower bound in Part \[thm:above-lower\] is $\Omega(n^{1+\delta/2-g(b)})$, and there is a function $f(b) = O((\ln\ln{b})^2/\ln{b})=o(1)$ such that for every $b$, the upper bound in Part \[thm:below\] is $O(n^{1+f(b)})$ and in Part \[thm:above-upper\] is $O(n^{1+\delta+f(b)})$.
The upper bound improves upon Martinelli et al [@MSW-soda] who showed $O(n)$ relaxation time (and $O(n\log{n})$ mixing time) for $\lambda < 1/(\sqrt{b}-1)$ for all boundary conditions. Note, $\lambda = 1/\sqrt{b}$ is roughly equivalent to $\omega \approx \frac{1}{2}\ln{b}/b$ which is below the reconstruction threshold. Our main result extends the fast mixing up to the reconstruction threshold, and shows the slow-down beyond the reconstruction threshold. Our lower bound in the reconstruction region uses a general approach that transforms an algorithm showing reconstruction into a set with poor conductance, which implies the lower bound on the relaxation time. This framework captures the proof approach used in [@TVVY]. In Section \[sec:background\] we formally define various terms and present the basic tools used in our proofs. The lower bound (Part \[thm:above-lower\] of Theorem \[thm:main\]) is presented in Sections \[sec:lower-bound-approach\], \[sec:lower-bound-broadcasting\] and \[sec:lower-bound-hardcore\]. Section \[sec:lower-bound-approach\] outlines the approach. We then prove an analogue of Theorem \[thm:main\] in Section \[sec:lower-bound-broadcasting\] for the broadcasting model and use it in Section \[sec:lower-bound-hardcore\] to prove Part \[thm:above-lower\] of Theorem \[thm:main\]. The argument for the upper bounds stated in Theorem \[thm:main\] is presented in Section \[sec:upper-bound\].
Background {#sec:background}
==========
Let $P(\cdot,\cdot)$ denote the transition matrix of the Glauber dynamics. Let $\gamma_1 \ge \gamma_2 \ge \dots\ge \gamma_{|\Omega|}$ be the eigenvalues of the transition matrix $P$. The spectral gap $\gap$ is defined as $1-\gamma$ where $\gamma=\max\{\gamma_2,|\gamma_{|\Omega|}|\}$ denotes the second largest eigenvalue in absolute value. The relaxation time ${{T_{\rm{relax}}}}$ of the Markov chain is then defined as $\gap^{-1}$, the inverse of the spectral gap. Relaxation time is an important measure of the convergence rate of a Markov chain (see, e.g., Chapter 12 in [@LPW]).
To lower bound the relaxation time we analyze conductance. The conductance of a Markov chain with state space $\Omega$ and transition matrix $P$ is given by $
\Phi=\min_{S\subseteq\Omega} \{\Phi_S\}
$, where $\Phi_S$ is the conductance of a specific set $S \subseteq \Omega$ defined as $$\Phi_S=\frac{\sum_{\sigma\in S}\sum_{\eta\in\bar{S}}\pi(\sigma)P(\sigma,\eta)}
{\pi(S)\pi(\bar{S})}.$$
Thus, a general way to find a good upper bound on the conductance is to find a set $S$ such that the probability of “escaping” from $S$ is relatively small. The well-known relationship between the relaxation time and the conductance was established in [@LawlerSokal] and [@SinclairJerrum], and we will use the form ${{T_{\rm{relax}}}}= \Omega(1/\Phi)$ for proving the lower bounds.
Lower Bound Approach {#sec:lower-bound-approach}
====================
First note that the lower bound stated in Part \[thm:below\] of Theorem \[thm:main\], namely, ${{T_{\rm{relax}}}}=\Omega(n)$, is trivial for all $\omega$. For example, by considering the set $S=\{\sigma\in\Omega:r\notin\sigma\}$ of independent sets which do not contain the root, $\Phi(S) = \Omega(1/n)$ since we need to update $r$ to leave $S$.
We begin by explaining the high level idea of the non-trivial lower bound in Part \[thm:above-lower\] of Theorem \[thm:main\]. To that end, we first analyze a variant of the hard-core model in which there are two different activities, the internal vertices have activity $\lambda$ and the leaves have activity $\omega$. The resulting Gibbs distribution is identical to the measure $\nu_h$ defined in Section \[sec:introduction\] for the broadcasting process. Thus we refer to the following model as the broadcasting model.
For the tree $T_h=(V,E)$, we look at the following equivalent definition of the distribution $\nu_h$ over the set $\Omega$ of independent sets of $T_h$. For $\sigma\in\Omega$, let $$\wt'(\sigma) = \lambda^{|\sigma\cap V\setminus L|}\omega^{|\sigma\cap L|},$$ where $L$ are the leaves of $T_h$ and $\omega$ is, as before, the positive solution to $\omega(1+\omega)^b=\lambda$. Let $\nu_h(\sigma)=\wt'(\sigma)/Z'$ where $Z'=\sum_{\sigma\in\Omega} \wt'(\sigma)$ is the partition function. By simple calculations, the following proposition holds.
\[pro:broadcasting-alpha\] The measure $\nu_h$ defined by the hard-core model with activity $\lambda$ for internal vertices and $\omega$ for leaves is identical to the measure defined by the broadcasting process.
In fact, we just need to verify that in the hard-core model with activity $\lambda$ for internal vertices and $\omega$ for leaves, the probability $p_v$ of a vertex $v$ being occupied conditioning on its parent is unoccupied is $\omega/(1+\omega)$. This can be proved by induction. The base case is $v$ being a leaf, which is obviously true by the Markovian property of the Gibbs measure. If $v$ is not a leaf, by induction, the probability $p_v$ has to satisfy the following equation $$p_v = (1-p_v)\frac{\lambda}{(1+\omega)^b},$$ which solves to $p_v = \omega/(1+\omega)$.
The result of Berger et al [@BKMP] mentioned in Section \[sec:introduction\] implies that the relaxation time of the Glauber dynamics on the broadcasting model is $\omega(n)$. We will prove a stronger result, analogous to the desired lower bound for Part \[thm:above-lower\] of Theorem \[thm:main\].
\[thm:broadcasting\] For all $\delta > 0$, the Glauber dynamics for the broadcasting model on the complete tree $T_h$ with $n$ vertices, branching factor $b$ and $w=(1+\delta)\ln{b}/b$ satisfies the following: $${{T_{\rm{relax}}}}= \Omega(n^{1+\delta/2-o_b(1)}),$$ where the $o_b(1)$ function is $O(\ln{\ln {b}}/\ln{b})$.
We can show a similar upper bound on the relaxation time for the Glauber dynamics in this setting as in Theorem \[thm:main\]. Moreover, we can show the same upper bound for the mixing time by establishing a tight bound between the inverse log-Sobolev constant and the relaxation time as was done for colorings in Tetali et al [@TVVY].
We will prove Theorem \[thm:broadcasting\] via a general method that relates any reconstruction algorithm (or function) with the conductance of the Glauber dynamics. A *reconstruction algorithm* is a function $A:\Omega(L)\rightarrow\{0,1\}$ (ideally efficiently computable) such that $A(\sigma_h)$ and $\sigma(r)$ are positively correlated. Basically, the algorithm $A$ takes the configurations at the leaves $L$ as the input and tries to compute the configuration at the root. When the context is clear, we write $A(\sigma)$ instead of $A(\sigma_h)$. Under the Gibbs measure $\nu_h$, the [*effectiveness*]{} of $A$ is the following measure of the covariance between the algorithm $A$’s output and the marginal at the root of the actual measure: $$r_{h,A}=\min_{x\in\{0,1\}}\left[
\nu_h(A(\sigma)=\sigma(r)=x) -\nu_h(A(\sigma)=x) \nu_h(\sigma(r)=x)\right].$$ If it is the case that $\liminf_{h\rightarrow\infty}r_{h,A} = c_0 >0$ for some positive constant $c_0$ depending only on $\omega$ and $b$, then we say that it is an [*effective reconstruction algorithm*]{}. In words, an effective algorithm, is able to recover the spin at the root, from the information at the leaves, with a nontrivial success, when $h\rightarrow \infty$. Notice that reconstruction (defined in ) is a necessary condition for any reconstruction algorithm to be effective, since $${{{\mathrm{E}}_{\sigma\sim\nu_h}\left[{\left| {{\mu_{h,\sigma_h}}}(r\in\tau) - \frac{\omega}{1+\omega}\right|}\right]}}
\ge {{{\mathrm{E}}_{\sigma\sim\nu_h}\left[{\big( {{\mu_{h,\sigma_h}}}\left(r\in\tau\right) - \nu_h\left(r\in \sigma\right)\big){{\mathbf{1}\!\left({A\left(\sigma\right)=1}\right)}}}\right]}}
\ge r_{h,A},$$ where ${{\mathbf{1}\!\left({}\right)}}$ is the indicator function. We define the *sensitivity* of $A$, for the configuration $\sigma\in \Omega(T_{h})$, as the fraction of vertices $v$ such that switching the spin at $v$ in $\sigma$ changes the final result of $A$. More precisely, let $\sigma^{v}$ be the configuration obtained from changing $\sigma$ at $v$. Define the sensitivity as: $$S_A(\sigma) = \frac{1}{n}\#\{v\in L: A(\sigma^v)\neq A(\sigma)\}.$$ The [*average sensitivity*]{} (with respect to the root being occupied) $\bar{S}_A$ is hence defined as $$\bar{S}_A = {{{\mathrm{E}}_{\sigma\sim\nu_h}\left[{S_A(\sigma){{\mathbf{1}\!\left({A(\sigma) = 1}\right)}}}\right]}}.$$ It is fine to define the average sensitivity without the indicator function, which only affects a constant factor in the analysis. We are doing so to simplify some of the results’ statements and proofs.
Typically when one proves reconstruction, it is done by presenting an effective reconstruction algorithm. Using the following theorem, by further analyzing the sensitivity of the reconstruction algorithm, one obtains a lower bound on the relaxation time or mixing time of the Glauber dynamics.
\[thm:sensitivity\] Suppose that $A$ is an effective reconstruction algorithm. Then, the relaxation time ${{T_{\rm{relax}}}}$ of the Glauber dynamics satisfies ${{T_{\rm{relax}}}}= \Omega\left((\bar{S}_A)^{-1}\right)$.
The above theorem can be generalized to any spin system. To illustrate the usefulness of this theorem, we note that the lower bound on the mixing time of the Glauber dynamics for $k$-colorings in the reconstruction region proved in [@TVVY] fits this conceptually appealing framework.
Throughout the proof let $\nu := \nu_h$. Consider the set $U=\left\{ \sigma:A(\sigma)=1\right\}$. Then, $$\Phi_{U} =\frac{\textstyle\sum\nolimits_{\sigma \in U}\nu(\sigma)\sum\nolimits_{w\in L}\sum\nolimits_{\tau:\tau(w)\neq \sigma(w)}P(\sigma,\tau)}{\nu(U)(1-\nu(U))} \le
\frac{\sum_{\sigma\in U}\nu(\sigma) S_A(\sigma)}{\nu(U)(1-\nu(U))}.$$ From the definition of $r_{h,A}$, we have that $\nu(U) \geq \nu(A(\sigma) = \sigma(r) = 1) \geq r_{h,A}$, and similarly $(1-\nu(U))\geq \nu(A(\sigma) = \sigma(r) = 0) \geq r_{h,A}$. Now, because the algorithm is effective, we have $\liminf_{h\rightarrow\infty}(r_{h,A})= c_0 >0$ and hence for all $h$ big enough, $r_{h,A}>0$. Therefore, $\Phi_{U} \leq (r_{h,A})^{-2} \bar{S}_A$, which concludes that $
{{T_{\rm{relax}}}}= \gap^{-1} \ge 1/\Phi_{U} =\Omega((\bar{S}_A)^{-1})
$.
To prove Theorem \[thm:broadcasting\], we analyze the sensitivity of the reconstruction algorithm by Brightwell and Winkler [@BW Section 5] which yields the best known upper bounds on the reconstruction threshold. Our goal is to show that the average sensitivity of this algorithm is small. The analysis of the sensitivity of the Brightwell-Winkler (BW) algorithm, which then proves Theorem \[thm:broadcasting\], is presented in Section \[sec:lower-bound-broadcasting\].
Our main objective remains of constructing a sequence of “bad” boundary conditions under which the Glauber dynamics for the hard-core model slows down in the reconstruction region. An initial approach is that if we can find a complete tree $T'$ with some boundary condition such that the marginal of the root being occupied exactly equals $\omega/(1+\omega)$, then by attaching the same tree $T'$ with the corresponding boundary conditions to all of the leaves of a complete tree $T$, we are able to simulate the nonuniform hard-core model on $T$, (i.e., the resulting measure projected onto $T$ is the same as the one in the broadcasting model) and hence we can do the same approach to upper-bound the conductance of the dynamics on this new tree. However, from a cardinality argument, not for every $\omega$ there exists a complete tree of finite height with some boundary condition such that the marginal probability of the root being occupied equals $\omega / (1+\omega)$. Alternatively, we give a constructive way to find boundary conditions that approximate the desired marginal probability relatively accurately. This is done in Section \[sec:lower-bound-hardcore\].
Finally, at the end of Section \[sec:lower-bound-hardcore\] we argue that since the error is shrinking very fast from the bottom level under our construction of boundary conditions, we can again analyze the sensitivity of the Brightwell-Winkler algorithm starting from just a few levels above the leaves. This approach yields the lower bound stated in Part \[thm:above-lower\] of Theorem \[thm:main\].
Lower Bound for Broadcasting: Proof of Theorem \[thm:broadcasting\] {#sec:lower-bound-broadcasting}
===================================================================
Throughout this section we work on the broadcasting model. To prove Theorem \[thm:broadcasting\] we analyze the average sensitivity of the reconstruction algorithm used by Brightwell and Winkler [@BW], which we refer to as the BW algorithm. For any configuration $\sigma$ as the input, the algorithm works in a bottom up manner labeling each vertex from the leaves: a parent is labeled to occupied if all of its children are labeled to unoccupied; otherwise, it is labeled to unoccupied. The algorithm will output the labeling of the root as the final result. Formally, it can be described by the following deterministic recursion deciding the labeling of every vertex: $$R_{\sigma}(v) =\left\{
\begin{array}
[c]{cc}\sigma(v) \text{ } & \text{if }\operatorname* v\in L\\
1 - \max\{R_{\sigma}(w_1),R_{\sigma}(w_2),\ldots,R_{\sigma}(w_b)\} \text{ } & \text{otherwise}
\end{array}
\right.$$ where $w_{1},\ldots,w_{b}$ are the children of $v$. Finally, let $\A(\sigma) =\A(\sigma_h) =R_{\sigma}(r)$. Note that, $\A(\sigma)$ only depends on the configuration $\sigma_h$ on the leaves. The algorithm is proved to be effective in [@BW] when $\delta>0$. Therefore, it can be used in our case to lower bound the relaxation time. In this algorithm, by definition we have $$\label{eq:s31}
\bar{S}_\A = O\big(n^{-1}{{{\mathrm{E}}_{\sigma\sim\nu_h}\left[{\#\{v\in L: \A(\sigma) = 1 \textrm{ and } \A(\sigma^{v}) = 0\}}\right]}}\big),$$ Due to the symmetry of the function $R_{\sigma}(v)$ and the measure $\nu_h$, the expectation can be further simplified as $$\label{eq:s32}
{{{\mathrm{E}}_{\sigma}\left[{\#\{v\in L: \A(\sigma) = 1 \textrm{ and } \A(\sigma^{v}) = 0\}}\right]}}=b^h\nu_h(\A(\sigma) = 1 \textrm{ and } \A(\sigma^{\hv}) = 0),$$ where $\hv$ is now a fixed leaf. To bound the right hand side of Eq., let $\kappa\in\Omega(T_h)$ be a fixed configuration such that $\A(\kappa) = 1$. Let the path $\mathcal{P}$ from $\hv$ to the root $r$ be $u_{0} = \hv \leadsto u_{1}\leadsto\cdots\leadsto u_{h} = r$, and for any $i > 0$, let $w_{i,j}$ be the children of $u_i$ so that the labeling is such that for $j =1$, $w_{i,1} = u_{i-1}$ and for $j\neq 1$, $w_{i,j}$ is not on the path $\Pa$. An important observation is that, in order to make $\A(\kappa)$ change to $0$ by changing only the configuration at $\hv$ of $\kappa$, a necessary condition for $\kappa$ is $R_\kappa(u_{i}) = 1 - R_\kappa(u_{i-1})$ for all $i \ge 1$. Then for all $i\ge 1$ and $j\in\{2,\ldots,b\}$, we have $R_{\kappa}(w_{i,j}) = 0$. To calculate the probability that a random $\kappa\sim\nu_h$ satisfies such conditions, it would be easier if we expose the configurations along the path $\Pa$. Since then, conditioning on the configurations on the path, the events $R_{\kappa}(w_{i,j}) = 0$ are independent for all $i,j$. And if $\kappa(u_i) = 0$, we have for all $j>1$, the conditional probability of $R_{\kappa}(w_{i,j}) = 0$ equals ${{{\mathrm{Pr}}_{\eta\sim\nu_{i-1}}\left[{\A(\eta) = 0}\right]}}$, the probability $\A$ algorithm outputs a $0$ over a random configuration $\eta$ of the leaves of the complete tree $T_{i-1}$ with height $i-1$. The analysis above leads to the following lemma, which bounds the probability $\nu_h(\A(\sigma) = 1 \textrm{ and } \A(\sigma^{\hv}) = 0)$.
\[lem:nonuni1\] For every $i > 0$, let $\eta \in \Omega(T_{i-1})$ be a configuration chosen randomly according to measure $\nu_{i-1}$, then $$\nu_{h}(\A(\sigma) = 1 \textrm{ and } \A(\sigma^{\hv}) = 0) \le {{{\mathrm{E}}_{\kappa\sim\nu_h}\left[{\prod\limits_{i>0:\kappa(u_i)=0} {{{\mathrm{Pr}}_{\eta\sim\nu_{i-1}}\left[{\A(\eta) = 0}\right]}}^{(b-1)}}\right]}}.$$
Complete proofs of lemmas in this section are deferred to Section \[App:broadcasting\]. To use Lemma \[lem:nonuni1\], we derive the following uniform bound on the probability ${{{\mathrm{Pr}}_{\eta\sim\nu_{i}}\left[{\A(\eta) = 0}\right]}}$, for all $i$. Here and through out the paper, $b_0(\delta)$ is a function explicitly defined in Lemma \[lem:b0h\] in Section \[App:tlemmas\]. This function is of order $\exp(\delta^{-1}\ln(\delta^{-1}))$ as $\delta\to 0$ and remains bounded as $\delta \to \infty$.
\[lem:nonuni2\] Let $\delta>0$, and let $\omega=(1+\delta)\ln b/{b}$. For all $b\geq \BW$ and $i \geq 1$, $${{{\mathrm{Pr}}_{\eta\sim\nu_i}\left[{\A(\eta) = 0}\right]}}\leq \frac{(1.01)^{1/b}}{1+\omega}.$$
Combining Equations , , Lemma \[lem:nonuni1\] and Lemma \[lem:nonuni2\], we are able to upper bound the average sensitivity of the BW algorithm: $$\bar{S}_\A = O\big( \nu_h(\A(\sigma) = 1 \textrm{ and } \A(\sigma^{\hv}) = 0) \big)
= O\left({{{\mathrm{E}}_{\kappa\sim\nu_h}\left[{ \left( \frac{1.01\omega(1+\omega)}{\lambda}\right)^{\#\left\{ i:\kappa(u_{i})=0\right\}}}\right]}}\right) \text{.}$$
In this expectation, the number of unoccupied vertices in the path $\mathcal{P}$ can be trivially lower bounded by $h/2$, since it is impossible that there exists $i > 0$, $\kappa(u_i) = \kappa(u_{i-1}) = 1$. Therefore, the above expectation can be easily bounded by $O^*(n^{-(1+\delta)/2})$. This is not good enough in our case. We sharpen the bound using Lemma \[lem:asympunn\] in Section \[App:tlemmas\], leading to the following theorem, whose complete proof is contained in Section \[App:broadcasting\].
\[th:slownonunif\] Let $\delta > 0$, and let $\omega=(1+\delta)\ln b /{b}$. For all $b\geq \BW$, $${{T_{\rm{relax}}}}= \Omega\big(n^{d}\big)\text{,\quad where }d=\left( 1+\frac{\ln\left(
\lambda/(1.01\omega b)^2\right)}{2\ln b}\right).$$
Theorem \[thm:broadcasting\] is a simple corollary of Theorem \[th:slownonunif\] by noticing that $d=1+\delta/2-O\left(\frac{\ln\ln b}{\ln b}\right)$. Furthermore, we can “hide” the fact that $b\geq b_0(\delta)$ in this residual term, by using the trivial lower bound $\Omega(n)$ for all $b<b_0(\delta)$ and $b_0(\delta)\approx \exp(\delta^{-1}\ln(\delta^{-1}))$ as $\delta \to 0$.
Proofs. {#App:broadcasting}
-------
Note that throughout this paper, we will use the following notations for the relationships between two functions $f(x)$ and $g(x)$ for simplicity. If $\lim_{x\rightarrow \infty} f(x)/g(x) = 1$, we write $f(x) \approx g(x)$; if $f(x) = O(g(x))$, we write $f(x) \lesssim g(x)$ and if $f(x) = \Omega(g(x))$, we write $f(x) \gtrsim g(x)$.
Let $\mathbf{x} = \{0,1\}^h$ be a valid configuration on the path $\mathcal{P}$. Conditioning on $\kappa(u_i) = \mathbf{x}(i)$ for all $i$, we know that the events $R_{\kappa}(w_{i,j}) = 0$ are independent for all $i$ and $j$. Given $\kappa(u_i) = 0$, the probability of the event $R_{\kappa}(w_{i,j}) = 0$ equals ${{{\mathrm{Pr}}_{\eta\sim\nu_{i-1}}\left[{\A(\eta) = 0}\right]}}$; and given $\kappa(u_i) = 1$, the probability of this event can be trivially upper bounded by $1$ (this bound is “safe”, in the sense that the actual quantity is close to $1$ for big $\lambda$). By this, we can conclude that $$\begin{aligned}
\nu_{h}(\A(\kappa) & =1\text{ and }\A(\kappa^{\hv})=0)\\
& \leq \sum\limits_{\mathbf{x}}\nu_{h}(\kappa:\forall i,\kappa
(u_{i})=\mathbf{x}(u_{i}))\prod\limits_{i>0:\kappa(u_{i})=0}{{{\mathrm{Pr}}_{\eta\sim\nu_{i-1}}\left[{\A(\eta) = 0}\right]}}^{(b-1)}\\
&
={{{\mathrm{E}}_{\kappa}\left[{\prod\limits_{i>0:\kappa(u_i)=0} {{{\mathrm{Pr}}_{\eta\sim\nu_{i-1}}\left[{\A(\eta) = 0}\right]}}^{(b-1)}}\right]}}.\end{aligned}$$
In the proof, we will use the fact that $\exp\left(\frac{2(1.01)(\omega b)^2 }{\lambda}\right)\leq 1.01$, whenever $b\geq \BW$ (Lemma \[lem:b0h\]). Now, for simplicity, denote $\tg_i = {{{\mathrm{Pr}}_{\eta\sim\nu_{i-1}}\left[{A(\eta) = 0}\right]}}$. First of all, notice the recurrences$$\begin{aligned}
\tg_{i+1} & =\frac{\omega}{1+\omega}\left(
1-\left( 1- \tg_{i-1}^b\right)
^b \right)
+\frac{1}{1+\omega}\left(1-\tg_{i}^b\right) \text{,}\\
\tg_{1} & =\frac{1}{1+\omega}\quad\text{,\quad}
\tg_{2}=
\frac{1}{1+\omega}\left(1-\left(\frac{1}{1+\omega}\right)^{b}\right) \text{.}$$ The result follows by an easy induction: For $h=1,2$, the result is clear. On the other hand, from the previous recurrences, if it is the case that $\tg_{i}\leq \frac{(1.01)^{1/b}}{1+\omega}$, then$$\begin{aligned}
\tg_{i+1} ^{b} &
\leq\left[ \frac{\omega}{1+\omega}\left( 1-\left( 1-
\tg_{i-1} ^{b}\right) ^{b}\right) +\frac{1}{1+\omega}\right] ^{b}\\
& \leq\left[ \frac{\omega}{1+\omega}\left( 1-\left( 1-\frac{1.01\omega }{\lambda}\right) ^{b}\right) +\frac{1}{1+\omega
}\right] ^{b}\\
& \leq\left( \frac{1+\frac{1.01\omega^2b }{\lambda}}{1+\omega}\right) ^{b}\leq\frac{\exp(1.01(\omega b)^2/\lambda)}{(1+\omega)^b}\leq\frac{1.01}{(1+\omega)^b}\text{,}$$ where the third inequality follows from the fact that $(1-u)^{b}\geq1-ub$ for $u<1$, the fourth inequality follows from $(1+u)\leq e^{u}$, and the last inequality follows from the fact that $\exp\left(\frac{2(1.01)(\omega b)^2 }{\lambda}\right)\leq 1.01$ for $b\geq \BW$.
From Lemma \[lem:asympunn\] in Section \[App:tlemmas\], we have that $${{{\mathrm{E}}\left[{ \left( \frac{1.01\omega(1+\omega) }{\lambda}\right)
^{\#\{ i:\sigma(u_{i})=0\}}}\right]}}
\approx
\left( 1+\frac{ 1-\epsilon }{2\epsilon (1+\omega)}\right)
\frac{\left( 1+\epsilon\right) }{2}
\left(
\frac{1.01 \omega}{2\lambda}
\left[
1+\sqrt{1+4\lambda/1.01}
\right]
\right)^{h}$$ where $\eps =\left[\sqrt{1+4\lambda/1.01}\right]^{-1}$. The previous term is asymptotically dominated by $\left(1.01\frac{\omega}{\lambda^{1/2}}\right)^h$. Therefore, $$\bar{S}_A =O \left(\left[\frac{1.01\omega}{\lambda^{1/2}}\right]^h\right)
=O\left(n^{-\left[
1+\frac{\ln\left(\lambda/(1.01\omega b)^2\right)}{2\ln b}
\right]}\right)\text{.}$$
Now, from [@BW Section 5], it is known that the BW algorithm is effective for $\omega> (1+\delta)\ln b/{b}$ and $b>\BW$. (This can also be deduced from Lemma \[lem:nonuni2\] by following the same steps as in Proposition \[prop:eff2\]), therefore Theorem \[thm:sensitivity\] applies. The conclusion follows.
Some Technical Lemmas. {#App:tlemmas}
----------------------
\[lem:b0h\] Define $b_0(\delta)=\min\{ b_0:\exp\left( \frac{2(1.01)(\omega b)^2 }{\lambda}\right)\leq 1.01 \text{ for all } b\geq b_0 \}$. Then $b_0(\delta)$ is a continuous function such that
1. $b_0(\delta)<\infty$ for all $\delta>0$ (that is, it is well defined).
2. $b_0(\delta)\approx\exp\left((1+o(1))\delta^{-1}\ln(\delta^{-1})\right)$ as $\delta \to 0$.
3. $b_0(\delta)\approx b_0(\infty)$ as $\delta \to \infty$, where $ b_0(\infty)$ is a fixed constant $\leq 2$.
For (1), just notice that for $\delta>0$ fixed, we have that, as $b\to \infty$, $$\exp\left( \frac{2(1.01)(\omega b)^2 }{\lambda}\right) \approx 1<1.01.$$
For (2), notice that if we let $b=\exp\left(\beta\delta^{-1}\ln(\delta^{-1})\right)$, then as $\delta\to 0$, we have that $$\exp\left( \frac{2(1.01)(\omega b)^2 }{\lambda}\right) \approx \exp\left(2(1.01)\beta\delta^{\beta-1}\ln(\delta^{-1})\right).$$ Therefore, if $\beta\leq 1$, it is the case that $\exp\left( \frac{2(1.01)(\omega b)^2 }{\lambda}\right) \to \infty$ as $\delta \to 0$, while if $\beta>1$ the same expression goes to $1$.
For (3), notice that for $b$ fixed, $\exp\left( \frac{2(1.01)(\omega b)^2 }{\lambda}\right) \approx \exp\left( \frac{\Theta(1) }{\delta^{b-1}}\right)$, as $\delta\to \infty$.
\[lem:asympunn\]Let $\zeta_{0},\zeta_{1},\ldots$ be a Markov process with state space $\left\{ 0,1\right\} $, such that $\zeta_{0}=0$ and with transition rates $p_{0\rightarrow0}=p$, $p_{0\rightarrow1}=q$, $p_{1\rightarrow0}=1$, $p_{1\rightarrow1}=0$. Let $N_{h}=\#\left\{ 1\leq
i\leq h:\zeta_{i}=0\right\} $, then $${{{\mathrm{E}}\left[{ a^{N_{h}}}\right]}} \approx\left( 1+\frac{p\left(
1-\epsilon\right) }{2\epsilon}\right) \frac{\left( 1+\epsilon\right) }{2}\left( \frac{pa}{2}\left[ 1+\sqrt{1+4q/\left( ap^{2}\right) }\right]
\right) ^{h}\text{.}$$ where $\epsilon=\frac{1}{\sqrt{1+4q/\left( ap^{2}\right) }}$. Moreover, if the transition rate $p_{0\rightarrow0}$ is inhomogeneous but such that $\left\vert p-p_{0\rightarrow0}^{\left( i\right) }\right\vert
\leq\delta$, then $${{{\mathrm{E}}\left[{ a^{N_{h}}}\right]}}
\lesssim
\left(1+\frac{\left(p+\delta\right)\left( 1-\bar{\epsilon}\right)}{2\bar{\epsilon}}\right)
\frac{\left( 1+\bar{\epsilon}\right)}{2}
\left(\frac{\left( p+\delta\right)a}{2}
\left[1+\sqrt{1+4\left(q+\delta\right)/\left(a\left(p+\delta\right)^{2}\right) }
\right]
\right)^{h}
\text{,}$$ where $\bar{\epsilon}=\frac{1}{\sqrt{1+4\left( q+\delta\right) /\left(
a\left( p+\delta\right) ^{2}\right) }}$.
Straightforward combinatorics leads to the expression $${{{\mathrm{E}}\left[{ a^{N_{h}}}\right]}} =
{\textstyle\sum\nolimits_{k=0}^{\left\lfloor h/2\right\rfloor }}
\tbinom{h-k}{k}p^{h-2k}q^{k}a^{h-k}+
{\textstyle\sum\nolimits_{k=1}^{\left\lfloor (h+1)/2\right\rfloor }}
\tbinom{h-k}{k-1}p^{h-2k+1}q^{k}a^{h-k}$$ Now, for the first term, we have that $${\textstyle\sum\nolimits_{k=0}^{\lfloor h/2\rfloor}}
\tbinom{h-k}{k}p^{h-2k}q^{k}a^{h-k}
= (pa)^{h}
{\textstyle\sum\nolimits_{k=0}^{ \lfloor h/2 \rfloor }}
\tbinom{h-k}{k}x^{k},$$ where $x=\frac{q}{ap^{2}}$, which by standard saddle point methods, noticing that the function $\displaystyle\phi(t)=\lim_{h \to \infty} h^{-1} \ln \left[\tbinom{h-th}{th}x^{th}\right]$ reaches its maximum at the point $t^{\ast}=\frac{1}{2}(1-\eps)$, where $\eps=1/{\sqrt{1+4x}}$ and $\phi^{\prime\prime}(t^{\ast})
=\sqrt{\frac{4}{\eps(1-\eps)(1+\eps)}}$, implying that $$\frac{1}{(pa)^{h}}
{\textstyle\sum\nolimits_{k=0}^{ \lfloor h/2 \rfloor }}
\tbinom{h-k}{k}p^{h-2k}q^{k}a^{h-k}
\approx
\frac{(1+\eps)}{2}\left(\frac{1+\sqrt{1+4x}}{2}\right)^{h}.$$ Similarly, $$\frac{1}{(pa)^{h}}
{\textstyle\sum\nolimits_{k=1}^{ \lfloor (h+1)/2 \rfloor }}
\tbinom{h-k}{k-1}p^{h-2k+1}q^{k}a^{h-k}
\approx
\frac{p(1-\eps)}{2\eps}
\frac{(1+\eps)}{2}\left(\frac{1+\sqrt{1+4q/(ap^{2})}}{2}\right)^{h},$$ from where the result follows. In the inhomogeneous case, we have that $${{{\mathrm{E}}\left[{ a^{N_{h}}}\right]}} \leq{\textstyle\sum\nolimits_{k=0}^{\left\lfloor h/2\right\rfloor }}
\tbinom{h-k}{k}\left( p+\delta\right) ^{h-2k}\left( q+\delta\right)
^{k}a^{h-k}+{\textstyle\sum\nolimits_{k=1}^{\left\lfloor (h+1)/2\right\rfloor }}
\tbinom{h-k}{k-1}\left( p+\delta\right) ^{h-2k+1}\left( q+\delta\right)
^{k}a^{h-k}\text{,}$$ from where the result follows using the same asymptotic.
“Bad" Boundary Conditions: Proof of Theorem \[thm:main\].\[thm:above-lower\] {#sec:lower-bound-hardcore}
=============================================================================
First, we will show that for any $\omega$, there exists a sequence of boundary conditions, denoted as $\GammaN := \{\BD_i\}_{i>0}$, one for each complete tree of height $i>0$, such that if $i\rightarrow \infty$, the probability of the root being occupied converges to $\frac{\omega}{1+\omega}$. Later in this section we will exploit such a construction to attain in full the conclusion of Part \[thm:above-lower\] of Theorem \[thm:main\].
As a first observation, note that, the Gibbs measure for the hard-core model on $T_i$ with boundary condition $\BD$ is the same as the Gibbs measure for the hard-core model (with the same activity $\lambda$) on the tree $T$ obtained from $T_i$ by deleting all of the leaves as well as the parent of each (occupied) leaf $v\in\BD$. It will be convenient to work directly with such “trimmed” trees, rather than the complete tree with boundary condition. Having this in mind, our construction will be inductive in the following way. We will define a sequence of (trimmed) trees $\{(L_i,U_i)\}_{i\geq 0}$ such that $L_{i+1}$ is comprised of $s_{i+1}$ copies of $U_i$ and $b-s_{i+1}$ copies of $L_i$ with $\{s_i\}_{i \geq 1}$ properly chosen. Similarly, $U_{i+1}$ is comprised of $t_{i+1}$ copies of $U_i$ and $b-t_{i+1}$ copies of $U_i$, with $\{t_i\}_{i \geq 1}$ properly chosen.
We will show that, for either $T^*_i=L_i$, or $T^*_i=U_i$, it is the case that the $Q$-value, defined as: $$Q(T^*_i) = \frac{ \mu_{T^*_i}\left(\sigma(r)=1\right)}{\omega \mu_{T^*_i}\left( \sigma(r)=0\right)},$$ where $\mu_{T^*_i}(\cdot)$ is the hard-core measure on the trimmed tree $T^*_i$, satisfies $Q(T^*_i) \rightarrow 1$. Note that if $Q(T^*_i) = 1$, then the probability of the root being occupied is $\omega/(1+\omega)$ as desired. To attain this, we will construct $L_i$ and $U_i$ in such a way that $Q(U_i) \ge 1$ and $Q(L_i) \le 1$.
The recursion for $Q(L_{i+1})$ can be derived easily as $$Q(L_{i+1}) = \frac{(1+\omega)^b}{(1+\omega Q(U_i))^{s_{i+1}}(a+\omega Q(L_i))^{b-s_{i+1}}},$$ and a similar equation holds for $Q(U_{i+1})$ by replacing $s_{i+1}$ with $t_{i+1}$.
To keep the construction simple, we inductively define the appropriate $t_i$ and $s_i$, so that once $L_i$ and $U_i$ are given, we let $t_{i+1}$ be the minimum choice so that the resulting $Q$-value is $\geq 1$, more precisely, we let: $$t_{i+1}=\arg\min_{\ell}\{Q = \frac{(1+\omega)^b}{(1+Q(U_{i}))^\ell(1+\omega Q(L_{i}))^{b-\ell}}: Q \ge 1\}.$$ $$\hbox{And similarly, we let:}~~s_{i+1}=\arg\max_{\ell}\{Q = \frac{(1+\omega)^b}{(1+Q(U_{i}))^\ell(1+Q(L_{i}))^{b-\ell}}: Q \le 1\}.$$
The recursion starts with $U_1$ being the graph of a single node and $L_1$ being the empty set, so that $Q(U_1) = \lambda/\omega$ and $Q(L_1) = 0$. Observe that, by definition, $s_{i+1}~\in~\{t_{i+1},t_{i+1}+1\}$ and that the construction guarantees that the values $Q(L_i)$ are at most $1$, and the values $Q(U_i)$ are at least $1$. The following simple lemma justifies the correctness of our construction.
\[lem:err1\] $$\lim_{i\rightarrow \infty} {Q(U_i)}/{Q(L_i)} = 1.$$
It is easy to see that either $t_i=s_i$ (meaning that $Q(L_i)=Q(U_i)=1$), or $t_i=s_i-1$, which implies that $$\frac{Q(U_i)}{Q(L_i)} = \frac{1+\omega Q(U_{i-1})}{1+\omega Q(L_{i-1})} < \frac{Q(U_{i-1})}{Q(L_{i-1})}.$$ So the ratio is shrinking. Suppose the limit is not $1$ but some value $q>1$. Then, $$\frac{Q(U_{i-1})}{Q(L_{i-1})}-\frac{Q(U_{i})}{Q(L_{i})} = \frac{Q(U_{i-1})-Q(L_{i-1})}{(1+\omega Q(L_{i-1}))Q(L_{i-1})}.$$ Since $Q(U_i)/Q(L_i) > q$ and $Q(L_i)\le 1$, we have $$\frac{Q(U_{i-1})-Q(L_{i-1})}{(1+\omega Q(L_{i-1}))Q(L_{i-1})}\ge
\frac{(q-1)Q(L_{i-1})}{Q(L_{i-1})(1+\omega)}=\frac{q-1}{1+\omega},$$ which is a constant.
Therefore as long as $q>1$, we show that the difference between the ratios for each step $i$ is at least some constant which is impossible. Hence the assumption is false.
By this lemma, it is easy to check that if we let $T^*_i$ to be equal to either $U_i$ or $L_i$, then $Q(T^*_i)\rightarrow 1$. Indeed, we can show that the additive error decreases exponentially fast. The following lemma indicates that this is the case for $\omega<1$ (although a similar result holds for any $\omega$).
\[lem:err2\] Let $\eps^+_i$ be the value of $Q(U_i)-1$ and let $\eps^-_i$ be the value of $1-Q(L_i)$, then $$\eps^+_{i+1}+\eps^-_{i+1} \le \omega(\eps^+_i + \eps^-_i).$$
We can rewrite the expression $$(1+\omega)^b / (1+\omega Q(U_i))^j (1+\omega Q(L_i))^{b-j}$$ as $$\frac{1}{(1+\frac{\omega}{1+\omega}\eps^+_i)^j(1-\frac{\omega}{1+\omega}\eps^-_i)^{b-j}}.$$ Now, let $k$ be the biggest index over $[b]$ such that the denominator of the previous expression is less than $1$ (thus, $k+1$ will be the least index such that the denominator is greater than $1$). Then,
$$\begin{aligned}
\eps^+_{i+1}+\eps^-_{i+1} & = & \frac{1}{(1+\frac{\omega}{1+\omega}\eps^+_i)^k(1-\frac{\omega}{1+\omega}\eps^-_i)^{b-k}}-
\frac{1}{(1+\frac{\omega}{1+\omega}\eps^+_i)^{k+1}(1-\frac{\omega}{1+\omega}\eps^-_i)^{b-k-1}}\\
&=&
\frac{\frac{\omega}{1+\omega}(\eps^+_i+\eps^-_i)}{(1+\frac{\omega}{1+\omega}\eps^+_i)^{k+1}(1-\frac{\omega}{1+\omega}\eps^-_i)^{b-k}}\\
&\le&
\frac{\frac{\omega}{1+\omega}(\eps^+_i+\eps^-_i)}{1-\frac{\omega}{1+\omega}\eps^-_i}~~~~\textrm{(by the property of $k+1$)}\\
&\le&
\omega(\eps^+_i + \eps^-_i).\end{aligned}$$
Coming back to the original tree-boundary notation, let ${{\BD^{\textrm{1}}_{h}}}$ be the boundary corresponding to the trimming of the tree $U_h$ and let ${{\BD^{\textrm{2}}_{h}}}$ be the boundary corresponding to the trimming of the tree $L_h$. By our construction, for any vertex $v$ on the tree of height $h$, the measure from ${{\mu_{h,{{\BD^{\textrm{1}}_{h}}}}}}$ (or ${{\mu_{h,{{\BD^{\textrm{2}}_{h}}}}}}$) projected onto the space of the independent sets of the subtree rooted at $v$ with the boundary condition corresponding to the correct part of $\BD$ and the parent of $v$ being unoccupied is either ${{\mu_{i,{{\BD^{\textrm{1}}_{i}}}}}}$ or ${{\mu_{i,{{\BD^{\textrm{2}}_{i}}}}}}$, where $i$ is the distance of $v$ away from the leaves on $T_h$. Conditioning on the parent of $v$ being unoccupied, in the broadcast process defined in the Introduction, we would occupy $v$ with probability $\omega/(1+\omega)$. Therefore, in the above construction, the probability $v$ is occupied (or rather unoccupied) is close to the desired probability, and the error will decay exponentially fast with the distance from the leaves. This is formally stated in the following corollary of Lemma \[lem:err2\].
\[cor:err1\] Given any $\omega<1$ and the complete tree of height $i$, for $\BD$ equal to ${{\BD^{\textrm{1}}_{i}}}$ or ${{\BD^{\textrm{2}}_{i}}}$ inductively constructed above, we have $$\left\vert {{\mu_{i,\BD}}}(\sigma(r)=0)-\frac{1}{1+\omega}\right\vert \leq \omega^{i-1}\lambda/b.$$
Throughout the rest of this section it is assumed that we are dealing with the boundary conditions $\{{{\BD^{\textrm{1}}_{h}}}\}_{h\in \mathbf{N}}$ and $\{{{\BD^{\textrm{2}}_{h}}}\}_{h\in \mathbf{N}}$ constructed above. We will then show that for every $\omega = (1+\delta)\ln b/b$ under these two boundary conditions, the Glauber dynamics on the hard-core model slows down, whenever $\delta > 0$. As we know from Corollary \[cor:err1\], the error of the marginal goes down very fast, so that roughly we can think of the marginal distribution of the configurations on the tree from the root to the vertices a few levels above the leaves as being close to the broadcasting measure. In fact, by following the same proof outline as we did in Section \[sec:lower-bound-broadcasting\], we are able to prove the same lower bound in the hard-core model for these boundaries. To do that we need a slight generalization of the reconstruction algorithm and extensions of the corresponding lemmas used in that section to handle the errors in the marginal probabilities.
To generalize the notion of a reconstruction algorithm to the case of a boundary condition we need to add an extra parameter $\hp$ depending only on $\omega$ and $b$. We will essentially ignore the bottom $\hp$ levels in the analysis, and we will use that for the top $h-\hp$ levels the marginal probabilities are close to those on the broadcasting tree. We define a *reconstruction algorithm with a parameter $\hp$* for the tree $T_h$ with boundary condition $\BD$ as a function $A_\hp:\Omega(L_{h-\hp})\rightarrow\{0,1\}$. The algorithm $A_\hp$ takes the configurations of the vertices at height $h-\hp$ as the input and tries to compute the configuration at the root. For any $\sigma \in\Omega(T_{h,\BD})$, the sensitivity is defined as: $
S_{\hp,A}(\sigma) = \frac{1}{n}\#\left\{v \in L_{h-\hp}: A_\hp(\sigma^v_{h-\hp}) \neq A_\hp(\sigma_{h-\hp}) \right\}
$. The average sensitivity of the algorithm at height $h-\hp$ with respect to the boundary $\BD$ is defined as: $\bar{S}_{\hp,A}^{\BD} = {{{\mathrm{E}}_{\sigma}\left[{S_{\hp,A}(\sigma){{\mathbf{1}\!\left({A_{\hp}(\sigma_{h-\hp}) = 1}\right)}}}\right]}}
$. And the effectiveness is defined as: $$r_{\hp,A}^{\BD} = \min_{x\in \{0,1\}} [\mu_{h,\BD}(A_\hp(\sigma_{h-\hp}) = x \textrm{ and } \sigma(r) = x) - \mu_{h,\BD}(A_\hp(\sigma_{h-\hp}) = x)\mu_{h,\BD}(\sigma(r) = x)].$$ We can show the analog of Theorem \[thm:sensitivity2\] in this setting.
\[thm:sensitivity2\] Suppose that $A_\hp$ is an effective reconstruction algorithm. Then, it is the case that the spectral gap $\gap$ of the Glauber dynamics for the hard-core model on the tree of height $h$ with boundary condition $\BD$, satisfies $\gap = O(\bar{S}_{\hp,A}^{\BD})$, and hence the relaxation time of this Glauber dynamics satisfies ${{T_{\rm{relax}}}}= \Omega(1/\bar{S}_{\hp,A}^{\BD})$.
To bound the average sensitivity for the boundary conditions ${{\BD^{\textrm{1}}_{h}}}$ and ${{\BD^{\textrm{2}}_{h}}}$ constructed above, we again use the same BW algorithm as we analyzed for the broadcasting tree. As in Eq. and , it is again enough to bound the probability $$P_{\hp,\A}^\BD := {{\mu_{h,\BD_h}}}(\A_{\hp}(\sigma_{h-\hp})~=~1 \textrm{ and } \A_{\hp}(\sigma^{\hv}_{h-\hp})~=~0)$$ for a fixed vertex $\hv$ at a distance $\hp$ from the leaves, although in this case, this probability will not be the same for all $\hv$. Let the path $\Pa$ from $\hv$ to the root $r$ be $u_{0} = \hv \leadsto u_{1}\leadsto\cdots\leadsto u_{h-\hp} = r$, and for each $i > 0$ and $j\in\{1,\dots,b\}$, let $w_{i,j}$ be defined similarly as in Section \[sec:lower-bound-broadcasting\]. Further, let $\BD_{i,j}$ be the boundary condition $\BD_h$ restricted to the subtree $T_{w_{i,j}}$ of $T_h$ rooted at the vertex $w_{i,j}$. These subtrees are of height $i+\ell-1$ for each $i$. Note that, by our construction of the boundary conditions, for each fixed $i$, $\BD_{i,j} = {{\BD^{\textrm{1}}_{i+\hp-1}}}$ or $\BD_{i,j} = {{\BD^{\textrm{2}}_{i+\hp-1}}}$. The probability $P_{\hp,\A}^\BD$ can be calculated by the following lemma, which is the analog of Lemma \[lem:nonuni1\] for the broadcasting tree.
\[lem:uni1\] $$\ P_{\hp,\A}^\BD \le {{{\mathrm{E}}_{\sigma}\left[{\prod\limits_{i>0:\sigma(u_i)=0} \prod_{j=2}^{b}
{{{\mathrm{Pr}}_{\eta\sim{{\mu_{i+\hp-1,\BD_{i,j}}}}}\left[{A_{\hp}(\eta) = 0}\right]}}
}\right]}},$$ where the expectation is over the measure ${{\mu_{h,\BD_h}}}$, and for each $i,j$, $\eta$ is a random configuration on the subtree rooted at $w_{i,j}$ with the probability measure ${{\mu_{i+\hp-1,\BD_{i,j}}}}$.
The proofs of Theorem \[thm:sensitivity2\] and Lemma \[lem:uni1\] use the same proof approach as for Theorem \[thm:sensitivity\] and Lemma \[lem:nonuni1\] respectively. However, to bound ${{{\mathrm{Pr}}_{\eta}\left[{A_{\hp}(\eta) = 0}\right]}}$ for every $i>0$, in spite of going along the lines of Lemma \[lem:nonuni2\], the proof does require extra care to deal with the errors in the marginal probabilities which were bounded in Corollary \[cor:err1\]. In particular, we will establish the following lemma to upper bound ${{{\mathrm{Pr}}_{\eta\sim{{\mu_{i+\hp-1,\BD_{i,j}}}}}\left[{ A_{\hp}(\eta) = 0}\right]}}$ for each $i>0$. Here and throughout the text, we define $\hp(\lambda,b)$ to be the minimum $\hp$ such that for all $i\geq \hp$, $${{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{2}}_{i}}}}}}}\left[{\eta(r) = 0}\right]}}\leq \frac{1}{1+\omega}\exp\left(\frac{1.01(\omega b)^2}{\lambda}\right).$$ The existence of such constant $\hp(\lambda,b)$ is guaranteed by Lemma \[lem:err1\], also from Corollary \[cor:err1\] we can deduce a explicit value for $\hp(\lambda,b)$, provided that $\omega <1$.
\[lem:uni2\] Given any $\delta>0$, and $i \geq \hp(\lambda,b) = \hp$, then both ${{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{1}}_{i}}}}}}}\left[{A_{\hp}(\eta) = 0}\right]}}$ and ${{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{2}}_{i}}}}}}}\left[{A_{\hp}(\eta) = 0}\right]}}$ are upper bounded by $\frac{1.01^{1/b} }{1+\omega}$ for any $b\geq \BW$.
And also it is not hard to show that the BW algorithm under this setting is effective.
\[prop:eff2\] The BW reconstruction algorithm is effective to recover the configuration at the root from the configurations at distance $\hp(\lambda,b)$ from the leaves.
Then, we are able to again bound $\bar{S}_{\hp,\textrm{BW}}^{\BD}$ for $\BD = {{\BD^{\textrm{1}}_{h}}}$ or ${{\BD^{\textrm{2}}_{h}}}$, proving the following theorem, which completes the proof of Part \[thm:above-lower\] in Theorem \[thm:main\]. Interested readers can look up Section \[App:part2b\] for the complete proofs.
\[thm:lowerbounderr\] Let $\delta > 0$, and let $\omega=(1+\delta)\ln b/{b}$. For all $b\geq \BW$, it is the case that$${{T_{\rm{relax}}}}= \Omega\big(n^{d}\big)\text{,\quad where }d=\left( 1+\frac{\ln\left(
\lambda/(1.01\omega b)^2\right)}{2\ln b}\right).$$
Proofs. {#App:part2b}
-------
Let $\tg_{i,1}$ denote ${{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{1}}_{i}}}}}}}\left[{A_{\hp}(\eta) = 0}\right]}}$ and let $\tg_{i,2}$ denote ${{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{2}}_{i}}}}}}}\left[{A_{\hp}(\eta) = 0}\right]}}$. Let $\bar{t}_{i} = b - t_{i}$ and $\bar{s}_{i} = b - s_{i}$ for simplicity. Now, recall the recurrences given in Lemma \[lem:nonuni2\], which in this case take the form $$\begin{aligned}
\tg_{i+1,1} & ={{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{1}}_{i}}}}}}}\left[{\eta(r) = 1}\right]}}
\left( 1-
\left( 1-
\tg_{i-1,1}^{t_{i}}
\tg_{i-1,2}^{\bar{t}_{i}}
\right)^{t_{i+1}}
\left( 1-
\tg_{i-1,1}^{s_{i}}
\tg_{i-1,2}^{\bar{s}_{i}}
\right)^{\bar{t}_{i+1}}
\right)\\
&+ {{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{1}}_{i}}}}}}}\left[{\eta(r) = 0}\right]}}
\left( 1-
\tg_{i,1}^{t_{i+1}}
\tg_{i,2}^{\bar{t}_{i+1}}
\right) \text{,}\\
\tg_{i+1,2} & = {{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{2}}_{i}}}}}}}\left[{\eta(r) = 1}\right]}}
\left( 1-
\left( 1-
\tg_{i-1,1}^{t_{i}}
\tg_{i-1,2}^{\bar{t}_{i}}
\right)^{s_{i+1}}
\left( 1-
\tg_{i-1,1}^{s_{i}}
\tg_{i-1,2}^{\bar{s}_{i}}
\right)^{\bar{s}_{i+1}}
\right)\\
&+ {{{\mathrm{Pr}}_{\eta\sim{{\mu_{i,{{\BD^{\textrm{2}}_{i}}}}}}}\left[{\eta(r) = 0}\right]}}
\left( 1-
\tg_{i,1}^{s_{i+1}}
\tg_{i,2}^{\bar{s}_{i+1}}
\right).\end{aligned}$$ And the base cases are: $$\begin{aligned}
\tg_{\hp,1} &= {{{\mathrm{Pr}}_{\eta\sim{{\mu_{\hp,{{\BD^{\textrm{1}}_{\hp}}}}}}}\left[{\eta(r) = 0}\right]}} \text{,}\quad \tg_{\hp,2} = {{{\mathrm{Pr}}_{\eta\sim{{\mu_{\hp,{{\BD^{\textrm{2}}_{\hp}}}}}}}\left[{\eta(r) = 0}\right]}} \text{,}\\
\tg_{\hp+1,1} & ={{{\mathrm{Pr}}_{\eta\sim{{\mu_{\hp+1,{{\BD^{\textrm{1}}_{\hp+1}}}}}}}\left[{\eta(r) = 0}\right]}}
\left( 1-
\tg_{\hp,1}^{t_{\hp+1}}
\tg_{\hp,2}^{\bar{t}_{\hp+1}}
\right) \text{,}\\
\tg_{\hp+1,2} & ={{{\mathrm{Pr}}_{\eta\sim{{\mu_{\hp+1,{{\BD^{\textrm{2}}_{\hp+1}}}}}}}\left[{\eta(r) = 0}\right]}}
\left( 1-
\tg_{\hp,1}^{s_{\hp+1}}
\tg_{\hp,2}^{\bar{s}_{\hp+1}}
\right) \text{.}\\\end{aligned}$$ Our purpose now, is to show by induction that $\tg_{i,1}\text{, } \tg_{i,2} \leq \fh$ for all $i \geq \hp$. The base case is simple: $$\tg_{\hp+1,1} \leq \tg_{\hp,1} \leq {{{\mathrm{Pr}}_{\eta\sim{{\mu_{\hp,{{\BD^{\textrm{1}}_{\hp}}}}}}}\left[{\eta(r) = 0}\right]}} \leq \frac{1}{1+\omega} \exp\left(\frac{1.01(\omega b)^2}{\lambda}\right),$$ and the last term is less or equal to $\fh$ for $b \geq \BW$ \[Lemma \[lem:b0h\]\]. Similarly, it is the case that $\tg_{\hp+1,2} \leq \tg_{\hp,2} \leq \fh$. Now, in general, assuming the inductive hypothesis, we get from the above recurrence, that $$\begin{aligned}
\tg_{i+1,1}^{b}
& \leq \left[ \frac{\omega}{1+\omega}\left( 1-\left( 1-\frac{1.01 \omega}{\lambda}\right) ^{b}\right)
+\frac{1}{1+\omega} \right] ^{b}\exp\left(\frac{1.01(\omega b)^2}{\lambda}\right) \text{,}\end{aligned}$$ where the first term, just as deducted in the proof of Lemma \[lem:nonuni2\], is dominated by $\frac{\exp(1.01(\omega b)^2/\lambda)}{(1+\omega)^b}$. Now, the hypothesis follows for $\tg_{i+1,1}$ (and seemingly for $\tg_{i+1,2}$), due to the fact that $\exp(2(1.01)(\omega b)^2/\lambda)\leq 1.01 $ for $b\geq\BW$, proving the induction.
Notice the formulas, $$\begin{aligned}
{{\mu_{h,{{\BD^{\textrm{1}}_{h}}}}}}(A(\sigma)=0:\sigma(r)=0) & =1-\left[
{{\mu_{h-1,{{\BD^{\textrm{1}}_{h-1}}}}}}(A(\sigma)=0)\right] ^{t_{h}}\left[
{{\mu_{h-1,{{\BD^{\textrm{2}}_{h-1}}}}}}(A(\sigma)=0)\right] ^{\bar{t}_{h}}\\
{{\mu_{h,{{\BD^{\textrm{1}}_{h}}}}}}(A(\sigma)=1:\sigma(r)=1) & =\left[ 1-\left(
{{\mu_{h-2,{{\BD^{\textrm{1}}_{h-2}}}}}}(A(\sigma)=0)\right) ^{t_{h-1}}\left(
{{\mu_{h-2,{{\BD^{\textrm{2}}_{h-2}}}}}}(A(\sigma)=0)\right) ^{\bar{t}_{h-1}}\right] ^{t_{h}}\\
& \cdot\left[ 1-\left( {{\mu_{h-2,{{\BD^{\textrm{1}}_{h-2}}}}}}(A(\sigma)=0)\right)
^{s_{h-1}}\left( {{\mu_{h-2,{{\BD^{\textrm{2}}_{h-2}}}}}}(A(\sigma)=0)\right) ^{\bar{s}_{h-1}}\right] ^{\bar{t}_{h}},\end{aligned}$$ whith similar expressions for ${{\mu_{h,{{\BD^{\textrm{2}}_{h}}}}}}\left( \cdot\right)$. Now, from these recurrences and the bounds stated in Lemma \[lem:uni2\], we deduce that $$\begin{aligned}
{{\mu_{h,{{\BD^{\textrm{1}}_{h}}}}}}(A(\sigma)=0,\sigma(r)=0)&-{{\mu_{h,{{\BD^{\textrm{1}}_{h}}}}}}(A(\sigma)=0){{\mu_{h,{{\BD^{\textrm{1}}_{h}}}}}}(\sigma(r)=0) \\
& = \Omega\left( 1-\frac
{1.01}{(1+\omega)^b}-\frac{1.01^{1/b}}{1+\omega}
\right) \gg0\end{aligned}$$ With the same result holding for ${{\mu_{h,{{\BD^{\textrm{2}}_{h}}}}}}( \cdot) $.
It goes along the lines of the proof of Theorem \[th:slownonunif\]. Here, we take $\hp$ as $\hp(\lambda,b)$, as in Lemma \[lem:uni2\]. Now, due to Lemma \[lem:uni1\], we have that $$\bar{S}^{\BD}_{\hp,\textrm{BW}} = O\left( {{{\mathrm{E}}\left[{ \left( \frac{1.01\omega(1+\omega)
}{\lambda}\right)^ { \#\left\{ i:\sigma(u_{i}) =0\right\} } }\right]}} \right),$$ and using the second statement of Lemma \[lem:asympunn\], we have that $${{{\mathrm{E}}\left[{\left( \frac{1.01\omega(1+\omega)
}{\lambda}\right)^ { \#\left\{ i:\sigma(u_{i}) =0\right\} } }\right]}} \lesssim \left(
\frac{1.01\omega\exp\left(\frac{1.01(\omega b)^2}{\lambda}\right) }{2\lambda}
\left[
1+\sqrt{1+\frac{4\lambda}{1.01\exp\left(\frac{1.01(\omega b)^2}{\lambda}\right)}}
\right]
\right)^{h-\hp},$$ where the previous term is dominated by $\left(1.01\frac{\omega}{\lambda^{1/2}}\right)^{h-\hp}$. And just as in the proof of Theorem \[th:slownonunif\], $$\bar{S}_A =O \left(\left[\frac{1.01\omega}{\lambda^{1/2}}\right]^h\right)
=O\left(n^{-\left[
1+\frac{\ln\left(\lambda/(1.01\omega b)^2\right)}{2\ln b}
\right]}\right)\text{.}$$
Now, from Proposition \[prop:eff2\], the BW algorithm is effective for $\omega>(1+\delta)\ln{b}/{b}$, $b>b_0(\delta)$, therefore Theorem \[thm:sensitivity2\] applies. The conclusion follows.
[Upper Bounds of the Relaxation Time via the Coupling Method]{} {#sec:upper-bound}
===============================================================
We will use the coupling technique in some of our proofs which upper bound the relaxation time and the mixing time. The mixing time ${{T_{\rm{mix}}}}$ for the Glauber dynamics is defined as the number of steps, from the worst initial state, to reach within variation distance $\le 1/2{{\mathrm{e}}}$ of the stationary distribution. It is an elementary fact that the mixing time gives a good upper bound on the relaxation time (see, e.g., [@LPW] for the following bound), we will use this fact in our upper bound proofs: $$\label{eqn:TmixTrel}
{{T_{\rm{relax}}}}\le {{T_{\rm{mix}}}}+ 1.$$
Given two copies $(X_t)$ and $(Y_t)$ of the Glauber dynamics, a coupling is a joint process $(X_t,Y_t)$ such that the evolution of each component viewed in isolation is identical to the Glauber dynamics, see [@LPW] for more background on the coupling technique. The Coupling Lemma [@Aldous] (c.f., [@LPW Theorem 5.2]) guarantees that if, there is a coupling and time $t >0$, so that for every pair $(X_0, Y_0)$ of initial states, ${{{\mathrm{Pr}}\left[{X_t \neq Y_t} \mid {X_0,Y_0} \right]}} \le 1/2{{\mathrm{e}}}$ under the coupling, then ${{T_{\rm{mix}}}}\le t$.
Before we show the main idea of proving the upper bound, let us introduction some notations for this section first. Let $\tau_{\rho}$ be the relaxation time of the following Glauber dynamics of the hard-core model on the star graph $G$, where $\rho$ is a $b$ dimensional vector. The dynamics is defined as: if the root is chosen, it will be occupied with probability $\lambda/(1+\lambda)$ if no leaf is occupied; if the leaf $i$ is chosen, it will be occupied with probability $\rho_{i}$ if the root is unoccupied. Therefore $\tau := \max_{\rho}\{\tau_{\rho}\}$ is defined as the worst case relaxation time over all the possible choice of $\rho$. Following the same block dynamics strategy [@Martinelli-lecturenotes] as in Section 2.3 of the paper by [@BKMP] (See also [@Molloy] and [@TVVY]), it is not hard to show that the relaxation time of the above Glauber dynamics is exactly the same as that of the natural block dynamics which updates the configurations of a whole subtree of the root in one step, and hence the following lemma holds.
\[lem:block\] The relaxation time $T_{rel}$ of the Glauber dynamics of the hard-core model on the complete tree of height $H$ is upper bounded by $\tau^{H}$ for any boundary condition on the leaves.
Note that, the relaxation time is quite sensitive with respect to the boundary conditions. Especially, in the paper by [@MSW-soda], they show that when the boundary conditions are even (or odd) meaning that occupied all the leaves when the height is even (odd) and unoccupied all the leaves when the height is odd (even), the mixing time is actually $O(n\ln n)$. In this paper we are dealing with any kind of boundary condition, and in the lower bound part, we show the boundary conditions that slow down the Glauber dynamics actually exist. The lower bound on the relaxation time for the Glauber dynamics under that boundary conditions roughly matches up with the upper bound here.
We need to bound the relaxation time $\tau$ of the Glauber dynamics on star graphs for different cases with respect to $\rho$. To do this, we first use the so-called maximum one step coupling to bound the mixing time and then by Eq. we get an upper bound on the relaxation time $\tau$. The maximal one-step coupling, originally studied for colorings by Jerrum [@Jerrum] gives a way to upper bound the mixing time of the Glauber dynamics on general graphs. The coupling $(X_t,Y_t)$ of the two chains is done by choosing the same random vertex $v$ for changing the states at step $t$ and maximizing the probability of the two chains choosing the same update for the state of $v$. Thus, if none of neighbors of $v$ is occupied in neither $X_t$ and $Y_t$, $v$ will be occupied/unoccupied in both chains at time $t+1$ with the correct marginal probability. In all other cases, the update choices for $X_{t+1}(v)$ and $Y_{t+1}(v)$ are coupled arbitrarily. By analyzing the maximum one-step coupling, we will show a series of lemmas for different $\rho$, which give the upper bound on the relaxation time $\tau$.
\[lemma:up1\] If $\sum \rho_{i} \le 4\ln\ln b$ and for all $i$, $\rho_{i} \le 1-1/\ln b$, then $\tau = O(b^{1+o(1)})$, where $o(1)$ is a term that goes to zero as $b$ goes to infinity.
\[lemma:up2\] If $\sum \rho_{i} \ge 4\ln\ln b$, then $\tau = O( (\lambda+1) b\ln b)$.
\[lemma:up3\] If there exists an $i$ such that $\rho_i > 1-1/\ln b$, then $\tau = O((\lambda+1) b\ln b)$.
The intuition behind is that the typical behaviors of the coupling chain change with respect to the leaves marginal probability $\rho$. When $\rho$ are all tiny (Lemma \[lemma:up1\]), the coupling chain has a large chance to have all the leaves unoccupied in both $(X_t)$ and $(Y_t)$ and hence can be coupled at the state where the root is occupied in both chains with a good probability; While when the sum of $\rho$ is big (Lemma \[lemma:up2\] and \[lemma:up3\]) suggesting that there is a good chance that one of the leaves is occupied always, the coupling chain is easier to get coupled once the root is not occupied in both chains. We delay the proofs of these lemmas in Section \[section:up\_proofs\]. By applying Lemma \[lem:block\] to Lemma \[lemma:up1\], \[lemma:up2\] and \[lemma:up3\], we get the conclusion that the relaxation time is upper bounded by $O(n^{1+\ln(\lambda+1)/\ln b+o_b(1)})$ for any $\lambda > 0$. Recall that the relationship between $\omega$ and $\lambda$ is $\lambda = \omega (1+\omega)^b$ and in this paper we are mainly interested in the cases when $\omega = (1+\delta)\ln b/b$ for any constant $\delta>-1$. Hence, in terms of $\omega$, if $-1 < \delta \le 0$ the relaxation time is upper bounded by $n^{1+o_b(1)}$ and if $\delta > 0$ the relaxation time is upper bounded by $n^{1+\delta+o_b(1)}$, which proves Theorem \[thm:main\].
Proofs. {#section:up_proofs}
-------
With out lose of generality, we assume that the root is unoccupied in $X_0$ and occupied in $Y_0$. Readers will see later in the proofs that these are indeed the worst cases scenarios. We prove the lemmas by analyzing the coupling in rounds, where each round consists of $T:=20 b\ln{b}$ steps. The following analysis says that if in each round we have a good probability of coalescing (i.e., achieving $X_t=Y_t$), then we will have a good upper bound on the mixing time (and hence the relaxation time). Suppose that in the coupling process, starting from any pair of initial states, after time $T$, with probability $P_T$ we have $X_T = Y_T$. By repeatedly applying this assumption $i$ many times we have, for all $(x_0,y_0)$, $${{{\mathrm{Pr}}\left[{X_{2iT} \neq Y_{2iT}} \mid {X_0 = x_0, Y_0 = y_0} \right]}} \leq
(1-P_T)^{2i}\leq 1/2{{\mathrm{e}}}$$ for $i=1/P_T$. Therefore, by applying the Coupling Lemma, mentioned in Section \[sec:background\], the mixing time is $O(b\ln b /P_T)$. Now, the only thing left is to lower bound $P_T$ for different $\rho$ assuming that the root $r\notin X_0$ and $r\in Y_0$. Let us define for any configuration $X_t \in \Omega(G)$, $\|X_t\|:= |\{ \ell \in V(G): deg(\ell)=1 \textrm{ and } \ell \in X_t\}|$, the number of leaves of the star graph $G$ that are in the independent set.
In this case, we will try to couple all of the leaves to unoccupied and then couple the root. Let $T_0$ be the last time the root is chosen to change the state in the $T$ steps running of the coupling chain. It is a simple fact from the coupon collector problem that with high probability, $T_0 > 10b\ln b$ and all the leaves have been chosen at least once before $T_0$. Note that throughout this section “with high probability” means the probability goes to $1$ as $b$ goes to infinity. Conditioning on this, consider the following process $(B_t)$:
- $B_t \subseteq V(G)$;
- For each time $t$ the process $(B_t)$ chooses a leaf $v_i$ to update: $B_{t+1}= B_{t}\cup {v_i}$ with probability $\rho_i$ and $B_{t+1}= B_{t}\setminus {\{v_i\}}$ with probability $1-\rho_i$;
- The process will ignore any update of the root and the state of the root at any time.
By the conditioning we assume, we know that after $10b\ln b$ steps, every leaf has been updated at least once. Therefore the chance that all of the leaves are not in $B_t$ (i.e. $(B_t)$ is an empty set) is $$\prod_{i=1}^{b} (1-\rho_i) \ge \prod_{i=1}^{b}e^{-\rho_i \cdot \ln\ln b}\ge e^{-4\ln^2\ln b},$$ where $1-x > e^{-x \cdot \ln\ln b}$ holds when $x < 1 - 1/\ln b$. There is a straightforward coupling of $(X_t)$ and $(B_t)$ such that for any $t\ge 0$, $\|X_t\| \le |B_t|$. And also, by the natural of the maximum one-step couplings with the assumption $r\notin X_0$ and $r\in Y_0$, for any $t\ge 0$, $\|X_t\| \ge \|Y_t\|$. Therefore, we can conclude that with probability at least $ e^{-4\ln^2\ln b}$, at time $T_0 - 1$, all the leaves are unoccupied in both $X$ and $Y$, and hence with probability $1$, when the coupling chain chooses the root to update at time $T_0$, we have $X_{T_0} = Y_{T_0}$. In conclusion, we show that with probability at least $e^{-4\ln^2\ln b}$, we have $X_T = Y_T$, which implies that the mixing time and hence the relaxation time is $O(b\ln b \cdot e^{4\ln^2\ln b}) = O(b^{1+o_b(1)})$, where $o_b(1)$ is a $O(\ln^2\ln b)/\ln b$ function.
In this case, we want to first unoccupied the root in both chains and then couple the leaves. The key observation is that once the root is not occupied, it will not be occupied again since there is a good chance that one of the leaves is always occupied and hence “blocking” the root from being occupied. First of all, it is easy to see that in the first $b$ steps, the coupling chain has a positive probability of choosing the root to change the state and at that moment, the chance of the root being unoccupied in both chains is at least $1/(\lambda+1)$. We denote this as event $\E_1$. Next, we want that in the following $b$ steps, with positive probability, the root will not be chosen to change the state, therefore the chain chooses $b$ leaves to change the states during these $b$ steps. Let the set of leaves that are chosen during these $b$ steps be $S$. We want to argue that with positive probability, $\sum_{i\in S} \rho_i > c_0 \ln\ln b$ for some constant $c_0 > 1$ so that $S$ is a set of good representatives of all the leaves. Let this be the event $\E_2$. By using the fact that for each leaf, the indicator random variable of whether the leaf is chosen or not during these $b$ steps are negatively associated to each other (c.f., Theorem 14 in [@DR]), we can still use the following Hoeffding bound (Theorem 2 in [@HF]) by Proposition 7 in [@DR].
Let $X_1,\dots,X_n$ be independent (or negatively associated) random variables with $a_i \le X_i \le b_i$. Let $X=\sum_i X_i$ and $\mu = E[X]$. Then the following inequality holds $${{{\mathrm{Pr}}\left[{|X-\mu|\ge t}\right]}}\le 2\exp(-\frac{2t^2}{\sum_{i=1}^{n}(a_i - b_i)^2}).$$
In our case, $X_i = \rho_i$ if the leaf $i$ is chosen in $b$ steps and otherwise zero. Therefore, since in our case $\sum_{i=1}^{n}(a_i - b_i)^2\le (\sum_{i=1}^{n}(b_i - a_i))^2 = (4\ln\ln b)^2$, $\mu \ge 4(1-1/e)\ln\ln b$ and $t = \delta \mu$ for some properly chosen $\delta > 0$, it is straightforward to show that with positive probability, the event $\E_2$ happens. Note that the events $\E_1$ and $\E_2$ are actually independent. And from now on, we are conditioning on $\E_1$ and $\E_2$.
For each $t > 2b$, let $A_t:= \{\ell \in S: \ell \in X_t \textrm{ and } \ell \in Y_t\}$ be the set of leaves in $S$ such that they are in the independent set at time $t$ in both $X_t$ and $Y_t$. Let the stopping time $T_s$ be the first time $t > 2b$ such that $A_t = \emptyset$ and the root is chosen. If we can show that ${{{\mathrm{Pr}}\left[{T_s < 2b\ln b}\right]}} < 1/2$, then with positive probability, the root is always blocked from changing to occupied since $A_t \neq \emptyset$. Hence once the chain chooses all the leaves during $2b\ln b-2b$ steps, we will have $X_T = Y_T$. Finally, because the events $\E_1$ and $\E_2$ happen with probability $\Omega(1/(\lambda+1))$, we can conclude that with probability $\Omega(1/(\lambda+1))$, $X_T = Y_T$.
In order to show that ${{{\mathrm{Pr}}\left[{T_s < 2b\ln b}\right]}} < 1/2$, we will use the same coupling strategy as in the last proof. Let $(C_t)$ be a stochastic process that update the states of $|S|$ many independent vertices in the following way:
- The process starts at time $t = 2b+1$ with a random starting configuration;
- For any $t$, $C_t \subseteq S$;
- For each time $t$ the process $(C_t)$ chooses a vertex $v\in V(G)$ randomly;
- If $v = v_i \in S$ then apply the following update rule: $C_{t+1}= C_{t}\cup {v_i}$ with probability $\rho_i$ and $B_{t+1}= B_{t}\setminus {\{v_i\}}$ with probability $1-\rho_i$.
Let the stopping time $T_c$ be the first time $t>2b$ such that $C_t = \emptyset$ and the root is chosen. We want to couple $(C_t)$ and $(X_t,Y_t)$ in such a way that $T_s > T_c$ for all runs and then if we can show that ${{{\mathrm{Pr}}\left[{T_c < 2b\ln b}\right]}} < 1/2$, by the coupling, we are done.
In fact, it is easy to show that ${{{\mathrm{Pr}}\left[{T_c < 2b\ln b}\right]}} < 1/2$ (any positive constant will do) by using union bound combining with the following two facts: for any fix time $t > 2b$, the probability that $C_t = \emptyset$ is upper bounded by $(\ln b)^{-c_0}$ with $c_0 > 1$; with high probability the root will be chosen at most $O(\ln b)$ times during $T$ steps. Now we just need a valid coupling.
We couple $(C_t)$ with the coupling chain $(X_t,Y_t)$ from time $t = 2b+1$ in the following way:
- Initially, for $t=2b$, $v\in C_t$ if and only if $v\in A_t$.
- For any time $t$, both of them pick the same vertex $v$ to update and if $v\in S$, $v\in C_t$ if and only if $v\in A_t$.
It is easy to verify that conditioning on the events $\E_1$ and $\E_2$, this coupling is valid and $T_s > T_c$, which completes the proof.
This case is the same as the last one. However, we can just use the leaf $i$ to block the root since it has a very high probability of being occupied, i.e., for the probability bound concerning $T_s$, it is enough to bound the probability that at a given time $t$, the leaf $i$ is occupied.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Prasad Tetali for many helpful discussions.
[99]{}
D. Achlioptas and A. Coja-Oghlan. Algorithmic barriers from phase transitions. In [*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science*]{} (FOCS), 793-802, 2008.
D. Aldous. Random walks on finite groups and rapidly mixing [M]{}arkov chains. , 243-297, 1983.
N. Berger, C. Kenyon, E. Mossel, and Y. Peres. Glauber dynamics on trees and hyperbolic graphs. , 131(3):311–340, 2005.
N. Bhatnagar, A. Sly, and P. Tetali. Reconstruction Threshold for the Hardcore Model. Preprint, 2010. Available from arXiv at: <http://arxiv.org/abs/1004.3531>
N. Bhatnagar, J. Vera, E. Vigoda, and D. Weitz. Reconstruction for colorings on tree. reprint, 2008. Available from arXiv at: <http://arxiv.org/abs/0711.3664>
G. R. Brightwell and P. Winkler. A second threshold for the hard-core model on a Bethe lattice. [*Random Struct. Algorithms*]{}, 24(3):303-314, 2004.
J. Ding, E. Lubetzky, and Y. Peres. Mixing time of critical [I]{}sing model on trees is polynomial in the height. [*Communications in Mathematical Physics*]{}, 295(1): 161-207, 2010.
C. Daskalakis, E. Mossel and S. Roch. Optimal Phylogenetic Reconstruction, in [*Proceedings of the 38th Annual ACM Symposium on Theory of Computing*]{} (STOC), 159-168, 2006.
D. Dubhashi and D. Ranjan. Balls and bins: A study in negative dependence. , 13(2):99–124, 1998.
M. E. Dyer, A. Sinclair, E. Vigoda and D. Weitz. Mixing in time and space for lattice spin systems: A combinatorial view. [*Random Struct. Algorithms*]{}, 24(4):461-479, 2004.
H. O. Georgii. [*Gibbs Measures and Phase Transitions*]{}, de Gruyter Studies in Mathematics, vol. 9, 1988.
A. Gerschenfeld, A. Montanari. Reconstruction for models on random graphs, In [*Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science*]{} (FOCS), 194-204, 2007.
W. Hoeffding. Probability inequalities for sums of bounded random variables. [*Journal of the American Statistical Association*]{}, 58(301):13-30, 1963.
M. Jerrum. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. , 7(2):157–166, 1995.
F. Kelly. Loss networks. [*Annals of Applied Probability*]{}. 1(3):319-378, 1991.
F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian and L. Zdeborová. Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems, [*Proc. Natl. Acad. Sci. USA*]{}, 104(25):10318-10323, 2007.
G. Lawler and A. Sokal. Bounds on the $L^2$ spectrum for [M]{}arkov chains and [M]{}arkov processes: [a]{} generalization of [C]{}heeger’s inequality. , 309:557–580, 1988.
D.A. Levin, Y. Peres, and E.L. Wilmer. . American Mathematical Society, 2009.
B. Lucier and M. Molloy. The [G]{}lauber dynamics for colourings of bounded degree trees. , 2008. Available at: [http://www.cs.toronto.edu/$\sim$molloy/webpapers/wb-jour.pdf](http://www.cs.toronto.edu/~molloy/webpapers/wb-jour.pdf)
F. Martinelli. . Springer Lecture Notes in Mathematics (Vol. 1717), 2000.
F. Martinelli, A. Sinclair, and D. Weitz. The [I]{}sing model on trees: [B]{}oundary conditions and mixing time. , 250(2):301-334, 2004.
F. Martinelli, A. Sinclair, and D. Weitz. Fast mixing for independent sets, colorings, and other models on trees. , 31(2):134-172, 2007.
A. Montanari, R. Restrepo, P. Tetali. Reconstruction and Clustering in Random Constraint Satisfaction Problems. , 2009. Available from arXiv at: <http://arxiv.org/abs/0904.2751>
E. Mossel. Survey: Information flow on trees. [*Graphs, [M]{}orphisms and [S]{}tatistical [P]{}hysics.*]{} [DIMACS series in discrete mathematics and theoretical computer science]{}, 155–170, 2004.
E. Mossel. Reconstruction on trees: Beating the second eigenvalue. [*Ann. Appl. Probab*]{}, 11(1):285-300, 2001.
A. Sinclair and M. Jerrum. Approximate counting, uniform generation and rapidly mixing [M]{}arkov chains. , 82:93-133, 1989.
A. Sly. Reconstruction of random colourings. , 288(3):943-961, 2009.
A. Sly. Computational Transition at the Uniqueness Threshold. , 2010. Available from arXiv at: <http://arxiv.org/abs/1005.5584>
A.D. Sokal. A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models. [*Markov Process. Related Fields*]{}, 7(1):21-38, 2001.
P. Tetali, J. C. Vera, E. Vigoda, and L. Yang. Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees. In [*Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms*]{} (SODA), 1646-1657, 2010.
D. Weitz. Counting independent sets up to the tree threshold. In [*Proceedings of the 38th Annual ACM Symposium on Theory of Computing*]{} (STOC), 140-149, 2006.
[^1]: School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332. Email: restrepo@math.gatech.edu. Research supported in part by NSF grant CCF-0910584.
[^2]: Universidad de Antioquia, Departamento de Matematicas, Medellin, Colombia. Scholarship ‘200 years’.
[^3]: Department of Computer Science, University of Rochester, Rochester, NY 14627. Email: stefanko@cs.rochester.edu. Research supported in part by NSF grant CCF-0910415.
[^4]: Department of Management Sciences, University of Waterloo, Waterloo ON. Email: jvera@waterloo.edu.
[^5]: School of Computer Science, Georgia Institute of Technology, Atlanta GA 30332. Email: {vigoda,ljyang}@gatech.edu. Research supported in part by NSF grant CCF-0830298 and CCF-0910584.
|
---
abstract: 'We say a graph is exactly $m$-coloured if we have a surjective map from the edges to some set of $m$ colours. The question of finding exactly $m$-coloured complete subgraphs was first considered by Erickson in 1994; in 1999, Stacey and Weidl partially settled a conjecture made by Erickson and raised some further questions. In this paper, we shall study, for a colouring of the edges of the complete graph on $\mathbb{N}$ with exactly $k$ colours, how small the set of natural numbers $m$ for which there exists an exactly $m$-coloured complete infinite subgraph can be. We prove that this set must have size at least $\sqrt{2k}$; this bound is tight for infinitely many values of $k$. We also obtain a version of this result for colourings that use infinitely many colours.'
address: 'Department of Pure Mathematics and Mathematical Statistics, University of Cambridge'
author:
- 'Bhargav P. Narayanan'
bibliography:
- 'exactrefs.bib'
date: 22 February 2013
title: 'Exactly $m$-Coloured Complete Infinite Subgraphs'
---
Introduction
============
A classical result of Ramsey [@Ramsey1930] says that when the edges of a complete graph on a countably infinite vertex set are finitely coloured, one can always find a complete infinite subgraph all of whose edges have the same colour.
Ramsey’s Theorem has since been generalised in many ways; most of these generalisations are concerned with finding other monochromatic structures. For a survey of many of these generalisations, see the book of Graham, Rothschild and Spencer [@Graham1990]. Ramsey theory has witnessed many developments over the last fifty years and continues to be an area of active research today; see for example [@Leader2012], [@Hindman2008], [@Thomason1988], [@Conlon2009].
Alternatively, anti-Ramsey theory, which originates in a paper of Erd[ő]{}s, Simonovits and S[ó]{}s [@Erdos1975], is concerned with finding large “rainbow coloured" or “totally multicoloured" structures. Between these two ends of the spectrum, one could consider the question of finding structures which are coloured with exactly $m$ different colours as was first done by Erickson [@Erickson1994]; this is the line of enquiry that we pursue here.
Our Results
===========
For a set $X$, denote by $X^{(2)}$ the set of all unordered pairs of elements of $X$; equivalently, $X^{(2)}$ is the complete graph on the vertex set $X$. As always, $[n]$ will denote $\{1,...,n\}$, the set of the first $n$ natural numbers. By a colouring of a graph $G$, we will always mean a colouring of the edges of $G$.
Let $\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]$ be a surjective $k$-colouring of the edges of the complete graph on the natural numbers with $k\geq2$ colours. We say that a subset $X\subset\mathbb{N}$ is (*exactly*) $m$*-coloured* if $\Delta \left( X^{(2)}\right)$, the set of values attained by $\Delta$ on the edges with both endpoints in $X$, has size exactly $m$. Our aim in this paper is to study the set $$\mathcal{F}_{\Delta}{\mathrel{:\mkern-0.25mu=}}\left\{ m\in[k]\mbox{ : }\exists X\subset\mathbb{N}\mbox{ such that }X\mbox{ is infinite and }m\mbox{-coloured}\right\} .$$
Clearly, $k\in\mathcal{F}_{\Delta}$ as $\Delta$ is surjective. Ramsey’s theorem tells us that $1\in\mathcal{F}_{\Delta}$. Furthermore, Erickson [@Erickson1994] noted that a fairly straightforward application of Ramsey’s theorem enables one to show that $2\in\mathcal{F}_{\Delta}$ for any surjective $k$-colouring $\Delta$ with $k\geq 2$. He also conjectured that with the exception of $1,2\mbox{ and }k$, no other elements are guaranteed to be in $\mathcal{F}_{\Delta}$ and that if $k>k'>2$, then there is a surjective $k$-colouring $\Delta$ such that $k'\notin\mathcal{F}_{\Delta}$. Stacey and Weidl [@Stacey1999] settled this conjecture in the case where $k$ is much bigger than $k'$. More precisely, for any $k'>2$, they showed that there is a constant $C_{k'}$ such that if $k>C_{k'}$, then there is a surjective $k$-colouring $\Delta$ such that $k'\notin\mathcal{F}_{\Delta}$.
In this note, we shall be interested in the set of possible sizes of $\mathcal{F}_{\Delta}$. Since $\mathcal{F}_{\Delta}\subset[k]$, we have $|\mathcal{F}_{\Delta}|\leq k$ and it is easy to see that equality is in fact possible. Things are not so clear when we turn to the question of lower bounds. Let us define $$\mathbb{\psi}(k){\mathrel{:\mkern-0.25mu=}}\min_{\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]}|\mathcal{F}_{\Delta}|.$$ We are able to prove the following lower bound for $\psi(k)$.
\[mainresult\] Let $n \geq 2$ be the largest natural number such that $k\geq{n \choose 2}+1$. Then $\psi(k)\geq n$.
It is not hard to check that Theorem \[mainresult\] is tight when $k={n \choose 2}+1$ for some $n\geq2$. To this end, we consider the “small-rainbow colouring" $\Delta$ which colours all the edges with both endpoints in $[n]$ with ${n \choose 2}$ distinct colours and all the remaining edges with the one colour that has not been used so far. Clearly $|\mathcal{F}_{\Delta}|=n$, and so Theorem \[mainresult\] is best possible for infinitely many values of $k$.
Turning to the question of upper bounds for $\psi$, the small-rainbow colouring demonstrates that $\psi (k) = O(\sqrt{k})$ for infinitely many values of $k$. However, the obvious generalisations of the small-rainbow colouring described above fail to give us any good upper bounds on $\psi (k)$ for general $k$; in particular, we are unable to decide if $\psi(k)=o(k)$ for all $k \in \mathbb{N}$. However, by considering colourings that colour all the edges of a small complete bipartite graph with distinct colours (as opposed to a small complete graph) and making use of some number theoretic estimates of Tenenbaum [@Tenenbaum1984] and Ford [@Ford2008], we get reasonably close to such a statement.
\[upperbound\] There exists a subset $A$ of the natural numbers of asymptotic density one such that for all $k\in A$, we have $$\psi(k) = O \left(\frac{k}{(\log{\log{k}})^{\delta}(\log{\log{\log{k}}})^{3/2}}\right)$$ where $\delta = 1 - \frac{1 + \log{\log{2}} }{\log{2}} \approx 0.086 > 0$.
Canonical Ramsey theory, which originates in a paper of Erdős and Rado [@Erdos1950], involves the study of colourings that use infinitely many colours. We are able to prove a result similar to Theorem \[mainresult\] for such colourings.
We say that $\Delta:\mathbb{N}^{(2)}\rightarrow\mathbb{N}$ is an an *infinite colouring* if it is a colouring that uses infinitely many colours; in other words, if the image of $\Delta$ is infinite. For an infinite colouring $\Delta$, the analogue of $\mathcal{F}_{\Delta}$ that is of interest to us is the set $$\mathcal{G}_{\Delta}{\mathrel{:\mkern-0.25mu=}}\left\{ m\in\mathbb{N}\mbox{ : }\exists X\subset\mathbb{N}\mbox{ such that }X\mbox{ is }m\mbox{-coloured}\right\} .$$
The difference between $\mathcal{G}_{\Delta}$ and $\mathcal{F}_{\Delta}$ is that we also consider finite complete subgraphs in the definition of $\mathcal{G}_{\Delta}$. Since the set of colours is no longer finite, it might be the case that for each infinite subset $X$ of $\mathbb{N}$, $\Delta\left(X^{(2)}\right)$ is infinite; this motivates our definition.
By considering the unique injective colouring $\Delta$ that colours each edge of the complete graph on $\mathbb{N}$ with a distinct colour, we see that unless $m={n \choose 2}$ for some $n\geq2$, $m$ is not guaranteed to be a member of $\mathcal{G}_{\Delta}$. In the other direction, since an edge is a $1$-coloured complete graph, ${2 \choose 2}=1$ is always an element of $\mathcal{G}_{\Delta}$. With a little work, one can prove that ${3 \choose 2}=3$ is also always an element of $\mathcal{G}_{\Delta}$. But for $n\geq4$, we are unable to decide whether or not there exists a colouring $\Delta$ with infinitely many colours such that ${n \choose 2}\notin\mathcal{G}_{\Delta}$.
However, we can prove the following analogue of Theorem \[mainresult\] for infinite colourings.
\[canonical\] Let $\Delta:\mathbb{N}^{(2)}\rightarrow\mathbb{N}$ be an infinite colouring and suppose $n\geq2$ is a natural number. Then, $|\mathcal{G}_{\Delta}\cap\left[{n \choose 2}\right]|\geq n-1$.
Again, by considering the injective colouring that colours each edge with a distinct colour, it is easy to see that Theorem \[canonical\] is best possible.
The paper is organised as follows. In the next section, we prove our lower bounds, namely Theorems \[mainresult\] and \[canonical\]. We remark that we do not prove Theorem \[mainresult\] and \[canonical\] as stated. Instead, we prove two stronger structural results that in turn imply the theorems. We postpone the statements of these results since they depend on a certain notion of homogeneity that we will introduce in the next section. In Section 3, we describe how Theorem \[upperbound\] follows from certain divisor estimates. We conclude by mentioning some open problems in Section 4.
Lower Bounds
============
In this section, we prove Theorem \[mainresult\] by proving a stronger structural result, namely Theorem \[nhomog\]. The proof of Theorem \[canonical\] via Theorem \[nwhomog\] is very similar and we shall only highlight the main differences in the proofs.
We first introduce a notational convenience. Given a colouring $\Delta$ of $\mathbb{N}^{(2)}$, a vertex $v\in\mathbb{N}$, and a subset $X\subset\mathbb{N}\backslash\{v\}$, we say that a colour $c$ is a *new colour from* $v$ *into* $X$ if some edge from $v$ to $X$ is coloured $c$ by $\Delta$ and also, no edge in $X^{(2)}$ is coloured $c$ by $\Delta$. We write $N_{\Delta}(v,X)$ for the set of new colours from $v$ into $X$.
Proof of Theorem \[mainresult\]
-------------------------------
Before we prove Theorem \[mainresult\], we note that Erickson’s argument showing that $2\in\mathcal{F}_{\Delta}$ can be generalised to give a quick proof of $\psi(k)=\Omega(\log k)$.
\[erickson\] Let $\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]$ and suppose $l\in\mathcal{F}_{\Delta}$ and $l<k$. Then there is an $l'\in\mathcal{F}_{\Delta}$ such that $l+1\leq l'\leq2l$.
Note that Lemma \[erickson\], coupled with the fact that we always have $1\in\mathcal{F}_{\Delta}$, implies that $\psi(k)\geq1+\log_{2}k$.
Let $X\subset\mathbb{N}$ be a maximal $l$-coloured set. As $l<k$, $X\neq\mathbb{N}$. Pick $v\in\mathbb{N}\backslash X$. Note that $N_{\Delta}(v,X)\neq\emptyset$ since otherwise $X\cup\{v\}$ is $l$-coloured, which contradicts the maximality of $X$.
If $|N_{\Delta}(v,X)|\leq l$, then $X\cup\{v\}$ is $l'$-coloured for some $l+1\leq l'\leq2l$. So suppose $|N_{\Delta}(v,X)|\geq l+1$. By the pigeonhole principle, there is an infinite subset $X'$ of $X$ such that all the vertices in $X'$ are connected to $v$ by edges of a single colour, say $c$.
We consider two cases. If $c\in N_{\Delta}(v,X)$, we pick $l-1$ vertices from $X$ which are joined to $v$ by edges coloured with $l-1$ distinct colours from $N_{\Delta}(v,X)\backslash\{c\}$. If on the other hand $c\notin N_{\Delta}(v,X)$, we pick $l$ vertices from $X$ which are joined to $v$ by edges coloured with $l$ distinct colours from $N_{\Delta}(v,X)$. Call this set of $l-1$ or $l$ vertices $X''$.
In both cases, it is easy to check that $X'\cup X''\cup\{v\}$ is $l'$-coloured with $l+1\leq l'\leq2l$.
Consequently, we have the following corollary.
\[powersof2\] Let $\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]$ and suppose $n$ is a natural number such that $k\geq2^{n}+1$. Then $\mathcal{F}_{\Delta}\cap\left(\left[2^{n+1}\right]\backslash\left[2^{n}\right]\right)\neq\emptyset$.
We shall show that for any $\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]$ with $k\geq{n \choose 2}+1$ for some $n$, we can find $n$ nested subsets $A_1 \subsetneq A_2 \subsetneq ... \subsetneq A_n$ of $\mathbb{N}$ such that $\Delta(A_{1}^{(2)})\subsetneq\Delta(A_{2}^{(2)})\subsetneq...\subsetneq\Delta(A_{n}^{(2)})$. To do this, we introduce the notion of $n$-homogeneity on which our first structural result, Theorem \[nhomog\], hinges.
For an ordered $n$-tuple $\mathbf{X}=(X_{1},X_{2},...,X_{n})$, write $\bar{X}_{i}$ for the set $X_{1}\cup X_{2}...\cup X_{i}$. Given a colouring $\Delta$, we call $\mathbf{X}=(X_{1},X_{2},...,X_{n})$, with each $X_{i}$ a non-empty subset of $\mathbb{N}$, $n$-*homogeneous with respect to* $\Delta$ if the following conditions are met:
1. $X_{i}\cap X_{j}=\emptyset$ for $i\neq j$,
2. $X_{1}$ is infinite and $1$-coloured,
3. $\Delta\left(\bar{X}_{1}^{(2)}\right)\subsetneq\Delta\left(\bar{X}_{2}^{(2)}\right)\subsetneq...\subsetneq\Delta\left(\bar{X}_{n}^{(2)}\right)$,
4. for each $X_{i}$ with $2\leq i\leq n$, every $v\in X_{i}$ satisfies $$N_{\Delta}(v,\bar{X}_{i-1})=\Delta\left(\bar{X}_{i}^{(2)}\right)\Big\backslash\Delta\left(\bar{X}_{i-1}^{(2)}\right)\mbox{,}$$
5. $\left|\Delta\left(\bar{X}_{n}^{(2)}\right)\right|\leq{n \choose 2}+1$.
Rather than proving Theorem \[mainresult\], we prove the following stronger statement.
\[nhomog\] Let $\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]$ and suppose $n$ is a natural number such that $k\geq{n \choose 2}+1$. Then there exists an $n$-homogeneous tuple with respect to $\Delta$.
Before we prove Theorem \[nhomog\], let us first recall the lexicographic order on $\mathbb{N}^{r}$: We say that $(a_{1},a_{2}...,a_{r})<(b_{1},b_{2}..,b_{r})$ if for some $r'\leq r-1$ we have $a_{i}=b_{i}$ for $1\leq i\leq r'$ and $a_{r'+1}<b_{r'+1}$.
Note that if $\mathbf{X}=(X_{1},X_{2},...,X_{n})$ is $n$-homogeneous, then by Condition 4, the set $N_{\Delta}(v,\bar{X}_{i-1})$ is identical for all $v\in X_{i}$ for $2\leq i\leq n$. For $n\geq2$, define the *rank* of an $n$-homogeneous tuple $\mathbf{X}$ to be the $(n-1)$-tuple $(x_{1},x_{2},...,x_{n-1})$, where $x_{i}$ is the number of new colours from any vertex in $X_{i+1}$ into the set $\bar{X}_{i}$. Note that the rank of an $n$-homogeneous tuple is an $(n-1)$-tuple of natural numbers and so we can compare ranks by the lexicographic order on $\mathbb{N}^{n-1}$.
We proceed by induction on $n$. The case $n=1$ is Ramsey’s theorem. Suppose that $k\geq{n+1 \choose 2}+1$ and assume inductively that at least one $n$-homogeneous tuple exists.
From the set of all $n$-homogeneous tuples, pick one with minimal rank in the lexicographic order, say $\mathbf{X}=(X_{1},X_{2},...,X_{n})$. If $n=1$, the rank is immaterial; it suffices to pick $\mathbf{X}=(X_{1})$ such that $X_{1}$ is an infinite $1$-coloured set. We will build an $(n+1)$-homogeneous tuple from $\mathbf{X}$.
Note that $k\geq{n+1 \choose 2}+1>{n \choose 2}+1$. Since $\Delta$ is surjective and attains at most ${n \choose 2}+1$ different values inside $\bar{X}_{n}$, clearly $\mathbb{N}\backslash\bar{X}_{n}\neq\emptyset$. We consider two cases.
<span style="font-variant:small-caps;">Case 1</span>: *$N_{\Delta}(v,\bar{X}_{n})\neq\emptyset$ for some $v\in\mathbb{N}\backslash\bar{X}_{n}$*.
If $|N_{\Delta}(v,\bar{X}_{n})|\leq n$, then it is easy to check that $(X_{1},X_{2},...,X_{n},\{v\})$ is an $(n+1)$-homogeneous tuple and we are done. So, assume without loss of generality that $|N_{\Delta}(v,\bar{X}_{n})|\geq n+1$.
Let $j$ be the smallest index such that $N_{\Delta}(v,\bar{X}_{j})\neq\emptyset$. Since $N_{\Delta}(v,\bar{X}_{n})\neq\emptyset$, this minimal index $j$ exists. We now build our $(n+1)$-homogeneous tuple $\mathbf{Y}=(Y_{1},Y_{2},...,Y_{n+1})$ as follows.
Set $Y_{1}=X_{1},Y_{2}=X_{2},...,Y_{j-1}=X_{j-1}$. We define $Y_{j}$ as follows. Choose $c\in N_{\Delta}(v,\bar{X}_{j})$. Note that by the minimality of $j$, $N_{\Delta}(v,\bar{X}_{j-1})=\emptyset$ and so all the edges between $v$ and $\bar{X}_{j}$ coloured $c$ are actually edges between $v$ and $X_{j}$. Take $Y_{j}\subset X_{j}$ to be that (non-empty) subset of vertices $v'$ of $X_{j}$ such that the edge between $v$ and $v'$ is either coloured $c$ or with a colour from $\Delta(\bar{X}_{j}^{(2)})$ (and hence a colour not in $N_{\Delta}(v,\bar{X}_{j})$). Note that if $j=1$, we can always choose $c$ such that $Y_{1}$ is an infinite subset of $X_{1}$.
Next, set $Y_{j+1}=\{v\}$. Now, note that the only colour from $\Delta(\bar{Y}_{j+1}^{(2)})$ that might possibly occur in $N_{\Delta}(v,\bar{X}_{n})$ is $c$. So we can choose $v_{1},v_{2},...,v_{n-j}$ from $X_{n}\cup X_{n-1}...\cup X_{j+1}\cup(X_{j}\backslash Y_{j})$ such that these $n-j$ vertices are joined to $v$ by edges which are all coloured by distinct elements of $N_{\Delta}(v,\bar{X}_{n})\backslash\{c\}$. Set $Y_{j+2}=\{v_{1}\},Y_{j+3}=\{v_{2}\},...,Y_{n+1}=\{v_{n-j}\}$.
We claim that $\mathbf{Y}$ is an $(n+1)$-homogeneous tuple. Indeed, Conditions 1 and 2 are obviously satisfied.
To check Condition 3, first note that $\Delta(\bar{Y}_{1}^{(2)})\subsetneq\Delta(\bar{Y}_{2}^{(2)})\subsetneq...\subsetneq\Delta(\bar{Y}_{j}^{(2)})$ follows from the $n$-homogeneity of $\mathbf{X}$. Next, $\Delta(\bar{Y}_{j}^{(2)})\subsetneq\Delta(\bar{Y}_{j+1}^{(2)})$ since $v$ is joined to at least one vertex in $Y_{j}$ by an edge coloured with $c$ and we know that $c$ is a new colour from $v$ into $\bar{Y}_{j}$. Finally, $\Delta(\bar{Y}_{j+1}^{(2)})\subsetneq\Delta(\bar{Y}_{j+2}^{(2)})\subsetneq...\subsetneq\Delta(\bar{Y}_{j+1}^{(2)})$ follows from the choice of $v_{1},v_{2},...,v_{n-j}$. So Condition 3 is also satisfied.
Condition 4 for each of $Y_{1},Y_{2},..,Y_{j}$ is equivalent to the same condition for $X_{1},X_{2},...,X_{j}$ respectively. Furthermore, Condition 4 is also satisfied by each of $Y_{j+1},Y_{j+2},...,Y_{n+1}$ since they each contain exactly one vertex.
Finally, we check Condition 5. Clearly, $\Delta(\bar{Y}_{n+1}^{(2)})$ is a subset of $\Delta(\bar{X}_{n}^{(2)})\cup T$ for some subset $T$ of $N_{\Delta}(v,\bar{X}_{n})$ of size at most $n$. Hence, we see that $|\Delta(\bar{Y}_{n+1}^{(2)})|\leq{n \choose 2}+1+n={n+1 \choose 2}+1$.
<span style="font-variant:small-caps;">Case</span> 2: *$N_{\Delta}(v,\bar{X}_{n})=\emptyset$ for every $v\in\mathbb{N}\backslash\bar{X}_{n}$*.
It is here that we use the fact that $\mathbf{X}$ has minimal lexicographic rank. To deal with this case, we need the following lemma.
\[lexlem\] Let $\mathbf{X}$ be an $n$-homogeneous tuple of minimal lexicographic rank and suppose $N_{\Delta}(v,\bar{X}_{n})=\emptyset$ for some $v\in\mathbb{N}\backslash\bar{X}_{n}$. Then there is an $n$-homogeneous tuple $\mathbf{X'}$ such that $X'_{j}=X_{j}\cup\{v\}$ for some $j\in[n]$, and $X'_{i}=X_{i}$ for $1\leq i\leq n$, $i\neq j$.
If $N_{\Delta}(v,\bar{X}_{i})=\emptyset$ for $1\leq i\leq n$, then $(X_{1}\cup\{v\},X_{2},...,X_{n})$ is $n$-homogeneous and we have $\mathbf{X'}$ as required.
Hence, let $ j<n$ be largest index such that $N_{\Delta}(v,\bar{X}_{j})\neq\emptyset$. So by the definition of $j$, $N_{\Delta}(v,\bar{X}_{i})=\emptyset$ for $j<i\leq n$. We claim that $\mathbf{X'}=(X_{1},X_{2},...,X_{j},X_{j+1}\cup\{v\},X_{j+2},...,X_{n})$ is $n$-homogeneous.
Consider a colour $c$ that belongs to $N_{\Delta}(v,\bar{X}_{j})$. Since $N_{\Delta}(v,\bar{X}_{j+1})=\emptyset$, this means that $c$ must occur in $\Delta(\bar{X}_{j+1}^{(2)})\backslash\Delta(\bar{X}_{j}^{(2)})$. But, by Condition 4, for each $v'\in X_{j+1}$, $N_{\Delta}(v',\bar{X}_{j})=\Delta(\bar{X}_{j+1}^{(2)})\backslash\Delta(\bar{X}_{j}^{(2)})$. Hence, $N_{\Delta}(v,\bar{X}_{j})\subset N_{\Delta}(v',\bar{X}_{j})$ for $v'\in X_{j+1}$.
Observe that since $N_{\Delta}(v,\bar{X}_{i})=\emptyset$ for $j<i\leq n$, we have $N_{\Delta}(v' \bar{X}_{i-1})=N_{\Delta}(v',\bar{X}_{i-1}\cup\{v\})$ for each $v'\in X_{i}$ with $j<i\leq n$. From this, it is easy to see that $\mathbf{X'}$ is $n$-homogeneous if $N_{\Delta}(v,\bar{X}_{j})=N_{\Delta}(v',\bar{X}_{j})$ for $v'\in X_{j+1}$.
So suppose that $N_{\Delta}(v,\bar{X}_{j})\subsetneq N_{\Delta}(v',\bar{X}_{j})$ for $v'\in X_{j+1}$. Consider then the $n$-tuple $\mathbf{Z}=(X_{1},X_{2},...,X_{j},\{v\},X_{j+1},X_{j+2},...,X_{n-1})$. We claim that $\mathbf{Z}$ is $n$-homogeneous and has strictly smaller lexicographic rank than $\mathbf{X}$, which is a contradiction.
We first check the $n$-homogeneity of $\mathbf{Z}$. Clearly, Conditions 1 and 2 are satisfied by $\mathbf{Z}$.
To check Condition 3, first note that $\Delta(\bar{Z}_{1}^{(2)})\subsetneq\Delta(\bar{Z}_{2}^{(2)})\subsetneq...\subsetneq\Delta(\bar{Z}_{j+1}^{(2)})$ follows from the $n$-homogeneity of $\mathbf{X}$ and the fact that $N_{\Delta}(v,\bar{X}_{j})\neq\emptyset$. Next, $\Delta(\bar{Z}_{j+1}^{(2)})\subsetneq\Delta(\bar{Z}_{j+2}^{(2)})$ since $N_{\Delta}(v,\bar{X}_{j})\subsetneq N_{\Delta}(v',\bar{X}_{j})$ for $v'\in X_{j+1}$. Finally, $\Delta(\bar{Z}_{j+2}^{(2)})\subsetneq\Delta(\bar{Z}_{j+3}^{(2)})\subsetneq...\subsetneq\Delta(\bar{Z}_{n}^{(2)})$ since we know that $N_{\Delta}(v',\bar{X}_{i-1}\cup\{v\})=N_{\Delta}(v',\bar{X}_{i-1})\neq\emptyset$ for each $v'\in X_{i}$ with $j<i\leq n$. So $\mathbf{Z}$ satisfies Condition 3.
Condition 4 is satisfied trivially by each of $Z{}_{1},Z_{2},...,Z_{j}$. Condition $4$ holds for $Z_{j+1}$ since $v$ is the only element in $Z_{j+1}$. We know that $N_{\Delta}(v,\bar{X}_{j+1})=\emptyset$. Hence, Condition 4 holds for $Z_{j+2}$ since for any vertex $v'\in Z_{j+2}=X_{j+1}$, we see that $N_{\Delta}(v',\bar{Z}_{j+1})=N_{\Delta}(v',\bar{X}_{j})\backslash N_{\Delta}(v \bar{X}_{j})=\Delta(\bar{Z}_{j+2}^{(2)})\backslash\Delta(\bar{Z}_{j+1}^{(2)})$. Finally, Condition 4 holds for each $Z_{i}$ with $j+2<i\leq n$ by the fact that $N_{\Delta}(v',\bar{X}_{i-1}\cup\{v\})=N_{\Delta}(v',\bar{X}_{i-1})$ for each $v'\in X_{i}$. It is easy to see that Condition 5 holds since $N_{\Delta}(v,\bar{X}_{n})=\emptyset$.
That $\mathbf{Z}$ has smaller lexicographic rank than $\mathbf{X}$ is clear from the fact that $N_{\Delta}(v,\bar{X}_{j})\subsetneq N_{\Delta}(v',\bar{X}_{j})$ for $v'\in X_{j+1}$. This completes the proof of the lemma.
We have assumed that $N_{\Delta}(v,\bar{X}_{n})=\emptyset$ for each $v\in\mathbb{N}\backslash\bar{X}_{n}$. Now, $\Delta$ is surjective, so there must exist two vertices $v_{1}$ and $v_{2}$ in $\mathbb{N}\backslash\bar{X}_{n}$ such that edge joining $v_{1}$ and $v_{2}$ is coloured with a colour $c$ not in $\Delta(\bar{X}_{n}^{(2)})$.
Let $\mathbf{X'}$ be the $n$-homogeneous tuple that we get by applying Lemma \[lexlem\] to $\mathbf{X}$ and $v_{1}$. Then, clearly $N_{\Delta}(v_{2},\bar{X'}_{n})=\{c\}$. Thus, $(X'_{1},X'_{2},...,X'_{n},\{v_{2}\})$ is an $(n+1)$-homogeneous tuple. This completes the proof of the theorem.
Proof of Theorem \[canonical\]
------------------------------
As we mentioned earlier, the proof of Theorem \[canonical\] is very similar to that of Theorem \[mainresult\] and, also goes via a stronger structural result. We only highlight the main differences.
To prove Theorem \[canonical\], we need to alter the definition of $n$-homogeneity slightly. We relax Condition 2; instead of demanding that our first set $X_{1}$ be infinite and $1$-coloured, we only require that $|X_{1}|=1$.
More precisely, given a colouring $\Delta$, we call an $n$-tuple $\mathbf{X}=(X_{1},X_{2},...,X_{n})$, with each $X_{i}$ a non-empty subset of $\mathbb{N}$, *weakly homogeneous with respect to* $\Delta$ if the following conditions are met:
1. $X_{i}\cap X_{j}=\emptyset$ for $i\neq j$,
2. $|X_{1}|=1$,
3. $\emptyset=\Delta\left(\bar{X}_{1}^{(2)}\right)\subsetneq\Delta\left(\bar{X}_{2}^{(2)}\right)\subsetneq...\subsetneq\Delta\left(\bar{X}_{n}^{(2)}\right)$,
4. for each $X_{i}$ with $2\leq i\leq n$, every $v\in X_{i}$ satisfies $$N_{\Delta}(v,\bar{X}_{i-1})=\Delta\left(\bar{X}_{i}^{(2)}\right)\Big\backslash\Delta\left(\bar{X}_{i-1}^{(2)}\right)\mbox{,}$$
5. $\left|\Delta\left(\bar{X}_{n}^{(2)}\right)\right|\leq{n \choose 2}$.
Theorem \[canonical\] is an easy consequence of the following stronger statement.
\[nwhomog\] Let $\Delta:\mathbb{N}^{(2)}\rightarrow\mathbb{N}$ be an infinite colouring and suppose $n\geq2$ is a natural number. Then there exists an $n$-weakly homogeneous tuple with respect to $\Delta$.
The proof is essentially identical to that of Theorem \[nhomog\]. Note that we only use the finiteness of the set of colours in two places in the proof of Theorem \[nhomog\]. First, to produce an infinite $1$-coloured set for the base case of the induction and second, to ensure that the subset $Y_{1}$ of $X_{1}$ that we construct in the inductive step (in Case 1) is infinite. The definition of weak homogeneity gets around both these difficulties.
Upper Bounds
============
Erdős proved in [@Erdos1955] that for a natural number $n$, the set $P_n=\lbrace ab : a,b \leq n \rbrace$ has size $o(n^2)$. We base the proof of Theorem \[upperbound\] on the observation that $P_n$ is exactly the set of sizes of all induced subgraphs of a complete bipartite graph between two equal vertex classes of size $n$.
Let $H(x,y,z)$ be the number of natural numbers $n\leq x$ having a divisor in the interval $(y, z]$. Tenenbaum [@Tenenbaum1984] showed that $$\label{Tenenbaum}
H(x,y,z) = (1+ o(1))x \mbox{ if }\log{y}=o(\log{z}), z\leq\sqrt{x}. \tag{4.1}$$ Ford [@Ford2008] proved that we have $$\label{Ford}
H(x,y,2y) = \Theta \left( \frac{x}{(\log{y})^{\delta}(\log{\log{y}})^{3/2}}\right) \mbox{ if } 3\leq y \leq \sqrt{x}, \tag{4.2}$$ where $\delta = 1 - \frac{1 + \log{\log{2}} }{\log{2}}$. Armed with these two facts, we can now prove Theorem \[upperbound\].
We shall take $$A = \lbrace k : \exists a,b\in \mathbb{N} \mbox{ with } k-1 = ab \mbox{ and } \log{k} \leq a \leq b\rbrace.$$ It follows from that $H(x,\log{x},\sqrt{x})=(1+ o(1))x$; as an easy consequence, we have that $A$ has asymptotic density one. Now, for a fixed $k\in A$ with $k-1 = ab$, consider a surjective $k$-colouring $\Delta$ of the complete graph on $\mathbb{N}$ which colours all the edges of the complete bipartite graph between $[a]$ and $[b]\backslash[a]$ with $ab$ distinct colours and all other edges with the one unused colour. It is easy to then see that $$\mathcal{F}_\Delta = \lbrace a'b' + 1 : a' \leq a, b'\leq b \rbrace.$$
Now, for any element $a'b'+1 \in \mathcal{F}_\Delta$, we have $a/2^{i+1} < a' \leq a/2^i$ for some $i\geq 0$ and so $a'b' \leq ab/2^i$. Thus, $$|\mathcal{F}_\Delta| \leq \sum_{i\geq 0}{H(\frac{ab}{2^i},\frac{a}{2^{i+1}},\frac{a}{2^i})}.$$ Using Ford’s estimate for $H(x,y,2y)$ and the fact that $a\geq \log{k}$, we obtain that $$\psi(k) = O \left(\frac{k}{(\log{\log{k}})^{\delta}(\log{\log{\log{k}}})^{3/2}}\right)$$ for all $k \in A$.
Concluding Remarks
==================
Our results raise many questions that we cannot yet answer. We suspect that something much stronger than Corollary \[powersof2\] is true.
\[triangconj\] Let $\Delta:\mathbb{N}^{(2)}\twoheadrightarrow[k]$ and suppose $n\geq2$ is a natural number such that $k\geq{n \choose 2}+2$. Then $\mathcal{F}_{\Delta}\cap\left(\left[{n+1 \choose 2}+1\right]\backslash\left[{n \choose 2}+1\right]\right)\neq\emptyset$.
If true, note that this statement would imply Theorem \[mainresult\]. When $n=2$, the conjecture is implied by Corollary \[powersof2\]. We are able to prove the first non-trivial instance of Conjecture \[triangconj\], namely that when $k\geq5$, $\mathcal{F}_{\Delta}\cap\{5,6,7\}\neq\emptyset$, but the proof we possess sheds no light on how to prove the conjecture in general.
We strongly suspect that the function $\psi$ is quite far from being monotone. We have shown that $\psi\left({n \choose 2}+1\right)=n$ and that $\psi\left({n+1 \choose 2}+1\right)=n+1$, and it is an easy consequence of our results $\psi\left({n \choose 2}+2\right)=n+1$. It appears to be true that even $\psi\left({n \choose 2}+3\right)$ is much bigger than $n$, though we cannot even prove $\psi\left({n \choose 2}+3\right)>n+1$.
There is an absolute constant $\epsilon>0$ such that $\psi\left({n \choose 2}+3\right)>(1+\epsilon)n$ for all natural numbers $n\geq2$.
The problem of determining $\psi$ completely is of course still open. We do not know the answer to even the following question.
Is $\psi(k)=o(k)$ for all $k\in\mathbb{N}$?
If we restrict our attention to colourings which use every colour but one exactly once, we are led to the following question about induced subgraphs, a positive answer to which would immediately imply that $\psi(k)=o(k)$ for all $k\in\mathbb{N}$.
Given $m\in\mathbb{N}$, let $G(m)$ be a graph on exactly $m$ edges for which $S(m)$, the set of sizes of all the induced subgraphs of $G(m)$, is smallest. Is $|S(m)|=o(m)$?
**Acknowledgements.** I would like to thank my supervisor Béla Bollobás for bringing this question to my attention and for carefully reading an earlier version of this paper.
|
---
abstract: 'Statistical methods relating tensor predictors to scalar outcomes in a regression model generally vectorize the tensor predictor and estimate the coefficients of its entries employing some form of regularization, use summaries of the tensor covariate, or use a low dimensional approximation of the coefficient tensor. However, low rank approximations of the coefficient tensor can suffer if the true rank is not small. We propose a tensor regression framework which assumes a *soft* version of the parallel factors (PARAFAC) approximation. In contrast to classic [PARAFAC]{}, where each entry of the coefficient tensor is the sum of products of row-specific contributions across the tensor modes, the soft tensor regression ([[Softer]{}]{}) framework allows the row-specific contributions to vary around an overall mean. We follow a Bayesian approach to inference, and show that softening the [PARAFAC]{} increases model flexibility, leads to more precise predictions, improved estimation of coefficient tensors, and more accurate identification of important predictor entries, even for a low approximation rank. From a theoretical perspective, we show that the posterior distribution of the coefficient tensor based on [[Softer]{}]{} is weakly consistent irrespective of the true tensor or approximation rank. In the context of our motivating application, we adapt [[Softer]{}]{} to symmetric and semi-symmetric tensor predictors and analyze the relationship between brain network characteristics and human traits.'
author:
- 'Georgia Papadogeorgou$^1$, Zhengwu Zhang$^2$, David B. Dunson$^1$'
bibliography:
- 'Softer.bib'
- 'Clustering-Networks.bib'
title: Soft Tensor Regression
---
$^1$Department of Statistical Science, Duke University, Durham NC 27708\
$^2$Department of Biostatistics and Computational Biology, University of Rochester, Rochester NY 14642
*keywords:* adjacency matrix, Bayesian, brain connectomics, graph data, low rank, network data, parafac, tensor regression
Introduction
============
In many applications, data naturally have an array or tensor structure. When the tensor includes the same variable across two of its modes, it is often referred to as a network. Graph or network dependence is often summarized via an adjacency matrix or tensor. For example, data might correspond to an $R \times R \times p$ array containing features measuring the strength of connections between an individual’s $R$ brain regions. In tensor data analysis interest often lies in characterizing the relationship between a tensor predictor and a scalar outcome within a regression framework. Estimation of such regression models most often requires some type of parameter regularization or dimensionality reduction since the number of entries of the tensor predictor is larger than the sample size.
In this paper, we propose a soft tensor regression ([[Softer]{}]{}) framework for estimating a high-dimensional regression model with a tensor predictor and scalar outcome. [[Softer]{}]{} directly accommodates the predictor’s structure by basing the coefficient tensor estimation on the parallel factors approximation, similarly to other approaches in the literature. However, in contrast to previously developed methodology, [[Softer]{}]{} adaptively expands away from its low-rank mean to adequately capture and flexibly estimate more complex coefficient tensors. [[Softer]{}]{}’s deviations from the underlying low-rank, tensor-based structure are interpretable as variability in the tensor’s row-specific contributions.
Tensor regression in the literature
-----------------------------------
Generally, statistical approaches to tensor regression fall in the following categories: they estimate the coefficients corresponding to each tensor entry with entry-specific penalization, regress the scalar outcome on low-dimensional summaries of the tensor predictor, or estimate a coefficient tensor assuming a low-rank approximation.
A simple approach to tensor regression considers vectorizing the tensor predictor and fitting a regression model of the outcome on the tensor’s entries while performing some form of variable selection or regularization. Examples include [@Cox2003functional] and [@Craddock2009disease] who employed support vector classifiers to predict categorical outcomes based on participants’ brain activation or connectivity patterns. Other examples in neuroscience include [@Mitchell2004learning; @Haynes2005predicting; @OToole2005partially; @Polyn2005category] and [@Richiardi2011decoding] (see [@Norman2006beyond] for a review). However, this regression approach to handle tensor predictors is, at the least, unattractive, since it fails to account for the intrinsic array structure of the predictor, effectively flattening it prior to analysis.
Alternatively, dimensionality reduction can be performed directly on the tensor predictor reducing it to low dimensional summaries. In such approaches, the expectation is that these summaries capture all essential information effectively decreasing the number of parameters to be estimated. For example, [@Zhang2019tensor] and [@Zhai2019predicting] use principal component analysis to extract information on the participants’ structural and functional brain connectivity, and use these principal components to study the relationship between brain network connections and outcomes within a classic regression framework. However, this approach could suffer due to its unsupervised nature which selects principal components without examining their relationship to the outcome. Moreover, the performance of the low-dimensional summaries is highly dependent on the number and choice of those summaries, and the interpretation of the estimated coefficients might not be straightforward.
[@Ginestet2017hypothesis] and [@Durante2018bayesian] developed hypothesis tests for differences in the brain connectivity distribution among subgroups of individuals, employed in understanding the relationship between categorical outcomes and binary network measurements. Even though related, such approaches do not address our interest in building regression models with tensor predictors.
An attractive approach to tensor regression performs dimension reduction on the coefficient tensor. Generally, these approaches exploit a tensor’s Tucker decomposition [@Tucker1966] and its restriction known as the parallel factors ([PARAFAC]{}) or canonical decomposition. According to the [PARAFAC]{}, a tensor is the sum of $D$ rank-1 tensors, and each entry can be written as the sum of $D$ products of row-specific elements. The minimum value of $D$ for which that holds is referred to as the tensor’s rank. Note that the word “row” along a tensor mode is used here to represent rows in the classic matrix sense (slice of the tensor along the first mode), columns (slice of the tensor along the second mode), or slices along higher modes.
Within the frequentist paradigm, [@Zhou2013tensor] and [@Li2018tucker] exploited the [PARAFAC]{} and Tucker decompositions respectively, and proposed low rank approximations to the coefficient tensor. [@Guhaniyogi2017bayesian] proposed a related Bayesian approach for estimating the coefficient tensor. Even though these approaches perform well for prediction in these high-dimensional tensor settings, they are bound by the approximation rank in the sense that they cannot capture any true coefficient tensor, and are not directly applicable for identifying important connections. In this direction, [@Guha2018bayesian] assume a [PARAFAC]{} decomposition of the mean of the coefficient tensor and use a spike-and-slab prior distribution to identify brain *regions* whose connections are predictive of an individual’s creativity index. Relatedly, [@Wang2018symmetric] used a penalization approach of the [PARAFAC]{} contributions to identify small brain subgraphs that are predictive of an individual’s cognitive abilities.
Low-rank approximations to the coefficient tensor provide a supervised approach to estimating the relationship between a tensor predictor and a scalar outcome. However, such approximations can lead to inaccurate predictions, a poorly estimated coefficient tensor, and misidentification of important connections, if the true rank of the coefficient tensor is not small. As we will illustrate in \[sec:parafac\], this performance issue arises due to the inflexibility of the [PARAFAC]{} approximation which specifies that each row has a *fixed* contribution to all coefficient entries that involve it, leading to an overly rectangular or block structure of the estimated coefficient tensor. Therefore, if the true coefficient tensor does not exhibit such a block structure, a large number of components $D$ might be necessary in order to adequately approximate it. Due to this strict structure, we refer to the [PARAFAC]{} approximation to estimating the coefficient tensor as the *hard* [PARAFAC]{}.
Our contribution
----------------
We address the inflexibility of the hard [PARAFAC]{} by proposing a hierarchical modeling approach to estimate the coefficient tensor. Similarly to the hard [PARAFAC]{}, each entry of the coefficient tensor is the sum of products of row-specific contributions. However, our model specification allows a row’s contribution to the coefficients that involve it to be entry-specific and to vary around a row-specific mean. This row-specific mean resembles the row-specific contribution in the hard [PARAFAC]{} approximation, and the entry-specific row contributions can be conceived as random effects. Conceptually, the row-specific mean can be thought of as a row’s overall importance, with entry-specific deviations representing small variations in the row’s importance when interacting with the rows of other tensor modes. Allowing for the row contributions to vary by entry leads to the softening of the hard structure in the [PARAFAC]{} approximation, and for this reason, we refer to it as the *soft* [PARAFAC]{}. We refer to the tensor regression model that utilizes the soft [PARAFAC]{} for estimation of the coefficient tensor as *Soft Tensor Regression* ([[Softer]{}]{}). In contrast to models that strictly utilize low-rank approximations of the coefficient tensor, the soft tensor regression model can capture any true coefficient tensor without increasing the base rank. We explicitly show this by showing that for any true coefficient tensor and any base rank, the posterior distribution is consistent. Further, due to its increased flexibility, [[Softer]{}]{} performs better than strictly low-rank models in identifying important entries of the tensor predictor. We follow a fully Bayesian approach to inference which allows for straightforward uncertainty quantification in the coefficient estimates and predictions. We apply the soft tensor regression framework in a study of the relationship between brain structural connectomics and human traits for participants in the Human Connectome Project (HCP; [@VanEssen2013]). In the last few years, HCP has played a very important role in expanding our understanding of the human brain by providing a database of anatomical and functional connections and individual demographics and traits on over a thousand healthy subjects. Data availability and increased sample sizes have allowed researchers across various fields to develop and implement new tools in order to analyze these complex and rich data (see [@Cole2014article; @McDonough2014network; @Smith2015positive; @Riccelli2017surface; @Croxson2018structural] among many others). Using data from the HCP, exploiting state-of-the-art connectomics processing pipelines [@Zhang2018mapping], and within an adaptation of the supervised [[Softer]{}]{} framework for symmetric tensor predictors, we investigate the relationship between structural brain connection characteristics and a collection of continuous and binary human traits.
Tensor regression {#sec:tensor_regression}
=================
Some useful notation
--------------------
Let $a \in \Rp[1]$ and $b \in \Rp[2]$. Then $a \otimes b \in \mathbb{R}^{p_1 \times p_2}$ is used to represent the outer product of $a$ and $b$ with dimension $p_1 \times p_2$ and entries $[a\otimes b]_{ij} = a_i b_j$. Similarly, for vectors $a_k \in \Rp$, $k = 1,2, \dots, K$, the outer product $a_1 \otimes a_2 \otimes \dots \otimes a_K$ is a $K$-mode tensor $\bm A$ of dimensions $p_1, p_2, \dots, p_K$ and entries $\bm A_{\subs} = \prod_{k = 1}^K a_{k, j_k}$. For two tensors $\bm A_1, \bm A_2$ of the same dimensions, we use $\bm A_1 {\circ}\bm A_2$ to represent the Hadamard product, defined as the element-wise product of the two tensors. Further, we use $\langle \bm A_1, \bm A_2 \rangle_F$ to represent the Frobenius inner product, which is the sum of the elements of $\bm A_1 {\circ}\bm A_2$. When the tensors are vectors (1-mode), the Frobenius inner product is the classic dot product.
For a $K$-mode tensor $\bm A$ of dimensions $p_1, p_2, \dots, p_K$, we use “$j^{th}$ slice of $\bm A$ along mode $k$” to refer to the $(K-1)$-mode tensor $\bm G$ with dimensions $p_1, p_2, \dots, p_{k-1}, p_{k + 1}, \dots, p_K$ and $[\bm G]_{j_1 j_2 \dots j_{k-1} j_{k +1} \dots j_K} = [\bm A]_{j_1 j_2 \dots j_{k-1} j j_{k +1} \dots j_K}$. For example, the $j^{th}$ slice of a $p_1 \times p_2$ matrix along mode 1 is the matrix’s $j^{th}$ row. As a result, we refer to “slice-specific” quantities as “row-specific”even when that slice is not along mode 1. For example, the $j^{th}$ row mean of a $p_1 \times p_2$ matrix along mode 2 is the mean of the $j^{th}$ column. Remembering that we use “row” to refer to slices (and not necessarily to rows in the classic matrix sense) will be useful when discussing the hard [PARAFAC]{} in \[sec:parafac\] and introducing the soft [PARAFAC]{} in \[sec:soft\_tensor\_regression\].
Regression of scalar outcome on tensor predictor
------------------------------------------------
Let $Y_i$ be a continuous outcome, $\C_i = (C_{i1}, C_{i2}, \dots, C_{ip})^T$ scalar covariates, and $\tensor_i$ a $K$-mode tensor of dimensions $p_1, p_2, \dots, p_K$ with entries $[\tensor_i]_\subs = X_{i,\subs}$, for unit $i = 1, 2, \dots N$. Even though our development is presented here for continuous outcomes, the relationship between tensor predictors and binary or categorical outcomes can be similarly evaluated by considering an appropriate link function as we do in \[sec:application\]. We study the relationship between the outcome and the scalar and tensor predictors by assuming a model $$Y_i = \mu + \C_i^T \bm \delta + \sum_{j_1 = 1}^{p_1} \sum_{j_2 = 1}^{p_2} \dots \sum_{j_K = 1}^{p_K} X_{i,\subs} \beta_{\subs} + \epsilon_i, \ \epsilon_i \sim N(0, \tau^2),
\label{eq:tensor_regression}$$ where $\bm \delta \in \Rp[]$ and $\beta_{\subs} \in \mathbb{R}$. Alternatively, organizing all coefficients $\beta_{\subs}$ in a tensor $\B$ of equal dimensions to $\tensor$ and $\subs$ entry equal to $\beta_{\subs}$, the same model can be written as $$Y_i = \mu + \C_i^T\bm \delta + \langle \tensor_i, \B \rangle_F + \epsilon_i.
\label{eq:tensor_regression2}$$
Since the coefficient tensor $\B$ includes $\prod_{k = 1}^K p_k$ coefficients, it is infeasible to estimate it without some form of regularization or additional structure. Penalization or variable selection approaches based on the vectorization of the tensor predictor are implemented directly on model \[eq:tensor\_regression\], ignoring the predictor’s tensor structure. Alternatively, one approach to account for the predictor’s inherent structure is to assume a low-rank approximation to $\B$ based on the hard [PARAFAC]{} decomposition.
Tensor regression using the hard [PARAFAC]{} approximation {#sec:parafac}
==========================================================
Under the [PARAFAC]{} decomposition, a tensor $\B \in \mathbb{R}^{p_1 \times p_2 \times \dots p_K}$ can be written as $$\B = \sum_{d = 1}^D \beta_1^{(d)} \otimes \beta_2^{(d)} \otimes \dots \otimes \beta_{K}^{(d)}
\label{eq:parafac}$$ for some integer $D$ and $\beta_k^{(d)} \in \Rp$. The minimum value of $D$ for which $\B$ equals its representation \[eq:parafac\] is referred to as its rank. For matrices ($2-$mode tensors), this decomposition is equivalent to the singular value decomposition, and $D$ is the matrix rank.
The tensor [PARAFAC]{} decomposition leads to a natural approximation of the coefficient tensor in \[eq:tensor\_regression2\] by assuming that the coefficient tensor is in the form \[eq:parafac\] for some *small* value of $D$, potentially much smaller than its true rank. Therefore, the $\prod_{k = 1}^K p_k$ coefficients in $\B$ are approximated using $D\sum_{k = 1}^K p_k$ parameters leading to a large decrease in the number of quantities to be estimated.
However, this reduction in the number of parameters might come at a substantial price if the rank $D$ used in the approximation is smaller than the tensor’s true rank. According to \[eq:parafac\], the $(\subs)$ entry of $\B$ is equal to $$\B_{\subs} = \sum_{d = 1}^D \beta_{1j_1}^{(d)}\beta_{2j_2}^{(d)}\dots \beta_{Kj_K}^{(d)}.
\label{eq:hardB_entry}$$ According to \[eq:hardB\_entry\], row $j_k$ along mode $k$ has *fixed* importance, expressed as fixed row contributions $\beta_{kj_k}\tod$, to all coefficient entries $\B_{\subs}$ that include it, irrespective of the remaining indices. We refer to $\beta_{kj_k}\tod$ as the $d^{th}$ $j_k$-row contribution along mode $k$. This is best illustrated by considering a rank-1 2-mode tensor (matrix) $\B = \beta_1 \otimes \beta_2$ for vectors $\beta_1 \in \Rp[1], \beta_2 \in \Rp[2]$. Then, $\B_{j_1j_2} = \beta_{1j_1} \beta_{2j_2}$, and the same entry $\beta_{1j_1}$ is used in $\B_{j_1j_2}$ irrespective of $j_2$. This gives rise to a *rectangular structure* in $\B$ in which a row’s importance, $\beta_{1j_1}$, is fixed across all columns (and similarly for $\beta_{2j_2}$).
We further illustrate this in \[fig:ordered\_parafac\_1a\] where we plot $\beta_1 \otimes \beta_2$ for randomly generated vectors $\beta_1, \beta_2 \in \{0, 1\}^{100}$. It is evident from \[fig:ordered\_parafac\_1a\] that rank-1 matrices are organized in a rectangular structure where rows and columns are either uniformly important or not. Even though the generated vectors are binary for ease of illustration, the rectangular structure persists even when $\beta_1, \beta_2$ include non-binary entries. The rectangular structure observed in rank-1 tensors indicates that a rank-1 ($D=1$) approximation to the coefficient tensor could be quite limiting. Generally, a rank-$D$ approximation for $D > 1$ is employed to estimate the coefficient tensor. \[fig:ordered\_parafac\_3\] shows a matrix $\B$ of rank $D = 3$, summing over three rank-1 tensors like the one in \[fig:ordered\_parafac\_1a\]. The rank-3 tensor alleviates but does not annihilate the rectangular structure observed previously. This is most obvious in \[fig:ordered\_parafac\_3b\] where the rows and columns of \[fig:ordered\_parafac\_3\] are re-ordered according to their mean entry. In \[app\_sec:hard\_demonstration\] we further demonstrate the inflexibility of the hard [PARAFAC]{}’s block structure.
The said block structure is also evident in the work by [@Zhou2013tensor; @Guhaniyogi2017bayesian] and [@Li2018tucker] where they simulated data based on binary coefficient matrices. When these matrices represent combinations of rectangles (such as squares or crosses), the approximation performed well in estimating the true coefficient tensor. However, in situations where the true coefficient tensor was irregular, an increase in the rank was necessary in order to vaguely approximate the truth.
Soft tensor regression {#sec:soft_tensor_regression}
======================
Our development proceeds by further increasing the number of parameters in the regression model \[eq:tensor\_regression2\] and subsequently imposing sufficient structure to ensure model regularization and adaptability, simultaneously. We introduce tensors $\B_k\tod$ of equal dimensions to $\B$ and write $$\B = \sum_{d = 1}^D \B_1\tod {\circ}\B_2\tod {\circ}\dots {\circ}\B_K\tod.
\label{eq:softB}$$ From \[eq:softB\], the coefficient with indices $\subss = (\subs)$ is written as the sum of $D$ products of $K$ parameters $$\B_{\subss} = \sum_{d = 1}^D \beta_{1\subss}\tod \beta_{2\subss}\tod \dots \beta_{K\subss}\tod,
\label{eq:softB_entry}$$ where $\beta_{k\subss}\tod$ is the $\subss$ entry of the tensor $\B_k\tod$. For reasons that will become apparent later, the parameters $\beta_{k\subss}\tod$ are referred to as the $j_k^{th}$ row-specific contributions along mode $k$ to the coefficient $\B_{\subss}$. Note that, for now, these row-specific contributions are allowed to depend on all indices $\subss$. For unrestricted $\B_k\tod$s, \[eq:softB\] does not impose any restrictions on the coefficient tensor and any tensor $\B$ can be written in this form (for example, take $D = 1$, $\B_1^{(1)} = \B$ and $\B_k^{(1)} = \mathbf{1}$, for all $k > 1$).
This implies that the already high-dimensional problem of estimating the $\prod_{k = 1}^K p_k$ parameters in $\B$ has been translated to an even higher-dimensional problem in \[eq:softB\]. We achieve dimensionality reduction by imposing structure on the tensors $\B_k\tod$ in a careful manner that allows flexible and low-dimensional estimation of $\B$. We refer to the resulting characterization as the soft [PARAFAC]{} of $\B$. Before introducing the soft [PARAFAC]{}, we demonstrate that the hard [PARAFAC]{} fits within the framework of \[eq:softB\]-\[eq:softB\_entry\] by assuming a specific structure on the $\B_k\tod$s.
Representation of the hard [PARAFAC]{} motivating the soft [PARAFAC]{} {#subsec:parafac_representation}
----------------------------------------------------------------------
As shown in \[eq:hardB\_entry\], the hard [PARAFAC]{} row-specific contributions to each entry of the coefficient tensor are fixed across the remaining indices. Hence, the hard [PARAFAC]{} can be written in the form \[eq:softB\] by specifying tensors $\B_k\tod$ that are constant within the rows of mode $k$, $$\big[\B_k\tod \big]_{j_1 j_2 \dots j_k \dots j_K} = \big[\B_k\tod \big]_{j_1' \dots j_{k-1}' j_k j_{k + 1}' \dots j_K'}.$$ This structure on the tensors $\B_k\tod$ can be visualized as $p_k$ *constant* slices along mode $k$ representing the fixed row-specific contributions to all coefficient entries that involve it. This structure is illustrated in \[fig:hard\_parafac\_visual\] for a 4-by-3 coefficient matrix. As an example, the contribution of row 2 along mode 1 is constant ($\beta_{1, (2, 1)}= \beta_{1, (2, 2)}= \beta_{1, (2, 3)}$), and the same is true for the contribution of row 1 along mode 2 ($\beta_{2, (1, 1)} = \beta_{2, (2, 1)} = \beta_{2, (3, 1)} = \beta_{2, (4, 1)}$). The connection between \[eq:softB\] and the hard [PARAFAC]{} is the reason why we refer to $\beta_{k\subss}\tod$ as row-specific contributions along mode $k$.
This demonstrates that the hard [PARAFAC]{} is one example of structure that can be imposed on the $\B_k\tod$s in order to approximate $\B$. However, the hard [PARAFAC]{} structure is quite strict, in that it imposes equalities across the $p_k$ slices of $\B_k\tod$. Furthermore, since the hard [PARAFAC]{} can only capture coefficient tensors of rank up to $D$, it is evident that this strict structure assumed on the tensors $\B_k\tod$ can limit the flexibility of the model in capturing a true coefficient tensor $\B$ of higher rank.
The soft [PARAFAC]{} {#subsec:soft_representation}
--------------------
The soft [PARAFAC]{} builds upon the hard [PARAFAC]{}’s low-rank structure, while providing additional flexibility by introducing entry-specific variability in the row contributions. Specifically, for all $k = 1, 2, \dots, K$, $j_k = 1, 2, \dots, p_k$, and $d = 1, 2, \dots D$, we specify $$\betakj\tod \sim N(\gammakj\tod, \sigma^2_k \zeta\tod),
\label{eq:soft_beta_dist}$$ for some $\gammakj\tod \in \mathbb{R}, \sigma^2_k, \zeta\tod > 0$. Then, ${\mathbb{E}}[\B_\subss | \Gamma, S, Z] = \sum_{d = 1}^D \gamma_{1j_1}\tod \gamma_{2j_2}\tod \dots \gamma_{Kj_K}\tod$ indicating that the tensor entries are centered around a $\gamma$-based rank-D hard [PARAFAC]{}, where $\Gamma, S, Z$ are the collections of the $\gamma, \sigma, \zeta$ parameters respectively. At the same time, \[eq:soft\_beta\_dist\] allows variation within the mode$-k$ slices of $\B_k\tod$ by considering them as random effects centered around an overall mean. This implies that row $j_k$’s importance is allowed to be entry-specific leading to a softening in the hard [PARAFAC]{} structure. The soft [PARAFAC]{} is illustrated in \[fig:soft\_parafac\_visual\]. Here, the row-contributions are centered around a common value (a value resembling the row-contribution according to the hard [PARAFAC]{}) but are entry-specific. For example, $\beta_{1, (2,1)}$ is similar but not equal to $\beta_{1, (2, 2)}, \beta_{1, (2, 3)}$.
The entry-specific contributions deviate from the baseline according to a mode-specific parameter, $\sigma^2_k$, and a parameter that depends on $d$. As we will discuss later, the inclusion of $\zeta\tod$ in the variance forces a larger amount of shrinkage on the entry-specific importance for components $d$ that have limited overall importance. For $\sigma^2_k \zeta\tod = 0$ the soft [PARAFAC]{} reverts back to the hard [PARAFAC]{}, with row-specific contributions fixed at $\gammakj\tod$. However, larger values of $\sigma^2_k \zeta\tod$ allow for a [PARAFAC]{}-based approximation that deviates from its hard underlying structure and can be used to represent any true tensor $\B$. This is further illustrated in \[fig:parafac\_plus\_sd\] where $\gamma_1, \gamma_2 \in \{0, 1\}^{64}$, and entry-specific contributions are generated according to \[eq:soft\_beta\_dist\] with $\sigma^2_k \zeta\tod \in \{0, 0.05, 0.1, 0.2\}$. The soft [PARAFAC]{} resembles a structured matrix with higher values of the conditional variance leading to further deviations from a low-rank structure.
The structure imposed by the soft [PARAFAC]{} has interesting interpretation. The parameters $\gammakj\tod$ represent the baseline importance of row $j_k$ along the tensor’s $k^{th}$ mode. In contrast to the hard [PARAFAC]{}, row $j_k$’s importance might manifest differently based on the rows of the other modes that participate with it in a coefficient entry, $\subss \setminus \{j_k\}$, through $\betakj \tod$. This interpretation of the soft [PARAFAC]{} structure is coherent in network settings like the one in our brain connectomics study, where we expect a brain region to have some baseline value for its connections, but the magnitude of this importance might slightly vary depending on the other region with which these connections are made. In this sense, defining deviations from the hard [PARAFAC]{} through deviations in the row-specific contributions as specified in \[eq:soft\_beta\_dist\] represents a tensor-based relaxation of the hard [PARAFAC]{} structure.
Bayesian inference in the soft tensor regression framework {#subsec:bayesian}
----------------------------------------------------------
[[Softer]{}]{} is placed within the Bayesian paradigm, which allows for straightforward uncertainty quantification. We consider the structure on $\B_k\tod$ expressed in \[eq:soft\_beta\_dist\] as part of the prior specification on the model parameters of \[eq:tensor\_regression2\]. Since $\gammakj\tod$ are the key building blocks for the mean of $\B$ representing the central hard [PARAFAC]{}, we borrow from [@Guhaniyogi2017bayesian] and specify $$\begin{aligned}
\gammakj \tod & \sim N(0, \tau_\gamma \zeta\tod \wkj \tod) \\
\tau_\gamma & \sim \Gamma(a_\tau, b_\tau) \\
\wkj \tod & \sim Exp((\lambda_k\tod)^2 / 2), \\
\lambda_k \tod & \sim \Gamma(a_\lambda, b_\lambda) \\
\bm \zeta & \sim \text{Dirichlet}(\alpha/D, \alpha/D, \dots, \alpha/D),\end{aligned}$$ where $\bm \zeta = (\zeta^{(1)}, \zeta^{(2)}, \dots, \zeta^{(D)})$. Therefore, the parameters $\gammakj\tod$ vary around 0 with variance that depends on an overall parameter $\tau_\gamma$, and component and row-specific parameters $\zeta\tod$ and $w_{k,j_k}\tod$. As discussed in [@Guhaniyogi2017bayesian], the row-specific components $\wkj\tod$ lead to an adaptive Lasso type penalty on $\gammakj\tod$ [@Armagan2013], and $\gammakj\tod | \tau_\gamma, \zeta\tod, \lambda_k\tod$ follows a double exponential (Laplace) distribution centered at 0 with scale $\tau_\gamma \zeta\tod / \lambda_k\tod$ [@Park2008].
The component-specific variance parameter $\zeta\tod$ is included in the prior of $\gammakj\tod$ to encourage only a subset of the $D$ components to contribute substantially in the tensor’s low-rank [PARAFAC]{} approximation. This is because parameters $\gammakj\tod$ for $d$ with small $\zeta\tod$ are shrunk closer to zero. For the same reason, we include $\zeta\tod$ in the conditional variance of $\betakj\tod$ in \[eq:soft\_beta\_dist\] to ensure that penalization of the baseline row contributions $\gammakj\tod$ is accompanied with penalization of the row contributions $\betakj\tod$, and that a reduction in the variance of $\gammakj\tod$ is not overcompensated by an increase in the variance of $\betakj\tod$.
We assume normal prior distributions on the intercept and scalar covariates’ coefficients $(\mu, \bm \delta) \sim N(0, \Sigma_0)$, and inverse gamma priors on the residual variance $\tau^2 \sim IG(a_\tau, b_\tau)$ and the mode-specific variance components $\sigma^2_k \sim \Gamma(a_\sigma, b_\sigma)$. Specific choices for the hyperparameter values are discussed in \[subsec:hyperparameters\].
Choosing hyperparameters to achieve desirable characteristics of the induced prior {#subsec:hyperparameters}
----------------------------------------------------------------------------------
The prior distribution on the coefficient tensor $\B \sim \pi_{\B}$ is induced by our prior specification on the remaining parameters. The choice of hyperparameters can have a large effect on model performance, and the use of diffuse, non-informative priors can perform poorly in some situations [@Gelman2008]. For that reason and in order to assist default choices of hyperparameters leading to weakly informative prior distributions, we study the properties of the induced prior on $\B$.
We do so in the following way. Firstly, we provide expressions for the induced prior expectation, variance and covariance for the entries in $\B$ in \[theory:variance\_B\]. We use these expressions to understand the importance of certain hyperparameters in how the soft [PARAFAC]{} transitions away from its low-rank, hard version. Then, in \[theory:prior\_targets\] we provide default values for hyperparameters for a standardized 2-mode tensor predictor such that, a priori, ${\mathrm{Var}}(\B_\subss) = V^*$, and the proportion of the variance that arises due to [PARAFAC]{} softening is equal to $AV^*$. Studying the proportion of prior variability due to the softening is motivated by \[fig:parafac\_plus\_sd\] in that hyperparameters should be chosen such that most of coefficient tensor’s *prior* variability arises from a low-rank tensor structure. All proofs are in \[app\_sec:proofs\].
For $\subss, \subss' \in \otimes_{k = 1}^K \{1, 2, \dots, p_k\}$ such that $\subss \neq \subss'$, we have that ${\mathbb{E}}(\B_\subss) = 0$, ${\mathrm{Cov}}(\B_\subss, \B_{\subss'}) = 0$, and for $a_\lambda > 2$, $${\mathrm{Var}}(\B_\subss) = \Big\{ D \prod_{r = 0}^{K - 1} \frac{\alpha/D + r}{\alpha + r} \Big\}
\Big[ \sum_{l = 0}^K \rho_l \binom{K}{l} \Big\{ \lambdafrac[T] \Big\}^l
\Big( {\frac{a_\sigma}{b_\sigma }}\Big)^{K -l} \Big],
$$ where $\rho_0 = 1$ and $\rho_l = a_\tau (a_\tau + 1) \dots (a_\tau + l - 1)$ for $l \geq 1$. \[theory:variance\_B\]
*The hyperparameters of the softening variance, $a_\sigma, b_\sigma$*. \[remark:add\_var\] Remember that $\sigma^2_k$ is the parameter driving the [PARAFAC]{} softening by allowing row-specific contributions to vary. From \[theory:variance\_B\], it is evident that the prior of $\sigma^2_k$ is only influential on the first two moments of $\B_\subss$ through its mean, ${\frac{a_\sigma}{b_\sigma }}$, with higher prior expectation of $\sigma^2_k$ leading to higher prior variance of $\B_\subss$. Therefore, prior elicitation for $a_\sigma, b_\sigma$ could be decided based on the ratio ${\frac{a_\sigma}{b_\sigma }}$.
*Variance of coefficient entries for the hard [PARAFAC]{}.* For ${\mathbb{E}}(\sigma^2_k) = 0$, the prior variance of the coefficient tensor entries is equal to the prior variance of the hard [PARAFAC]{}, $${\mathrm{Var}}^{hard}(\B_\subss) = \Big\{ D \prod_{r = 0}^{K - 1} \frac{\alpha/D + r}{\alpha + r} \Big\} \frac{\rho_K}{b_\tau^K} \Big\{ \lambdafrac \Big\}^K.$$
Comparing the variance of $\B_\subss$ based on the soft and hard [PARAFAC]{} allows us to quantify the amount of additional flexibility that is provided by the [PARAFAC]{} softening, expressed as $$AV = \frac{{\mathrm{Var}}(\B_\subss) - {\mathrm{Var}}^{hard}(\B_\subss)}{{\mathrm{Var}}(\B_\subss)} \in [0, 1).$$ We refer to this quantity as the *additional variance*. Motivated by \[fig:parafac\_plus\_sd\], we would like to ensure that chosen hyperparameters assign more prior weight to coefficient matrices that resemble low-rank factorizations. At the same time, choice of hyperparameters should ensure a sufficiently but not overly large prior variance of the regression coefficients. \[theory:prior\_targets\] provides conditions on the hyperparameters for matrix predictors ($K = 2$), for which ${\mathrm{Var}}(\B_\subss) = V^*$, and $AV = AV^*$, for values $V^*>0$ and $AV^* \in [0, 1)$. Conditions on the hyperparameters to ensure that the target variance and target additional variance are achieved can be acquired for a tensor predictor with $K > 2$ by following steps very similar to the ones in the proof of \[theory:prior\_targets\].
For a matrix predictor, target variance $V^* \in (0, \infty)$, target additional variance $AV^* \in [0,1)$, and hyperparameters satisfying $a_\lambda > 2$, $$\lambdafrac = \frac{b_\tau}{a_\tau} \sqrt{\frac{V^*(1 - AV^*)a_\tau}{C (a_\tau + 1)}}
\label{eq:condition_lambdafrac}$$ and $${\frac{a_\sigma}{b_\sigma }}= \sqrt{\frac{V^*(1 - AV^*)a_\tau}{C(a_\tau + 1)}} \bigg\{ \sqrt{ 1 - \frac{a_\tau + 1}{a_\tau} \big\{1 - \big(1 - AV^*)^{-1} \big\}} - 1 \bigg\},$$ where $C = (\alpha / D + 1)/(\alpha + 1)$, we have that a priori ${\mathrm{Var}}(\B_\subss) = V^*$, and $AV = AV^*$. \[theory:prior\_targets\]
\[theory:prior\_targets\] is used in our simulations and study of brain connectomics to choose hyperparameters such that, a priori, ${\mathrm{Var}}(\B_\subss) = 1$ and $AV = 10\%$, assuming a tensor predictor with standardized entries. Specifically, we set $a_\tau = 3$, $a_\sigma = 0.5$ and calculate the values of $b_\tau, b_\sigma$ for which $V^* = 1$ and $AV^* = 10\%$. These values correspond to $b_\tau \approx 6.3 \sqrt{C} $ and $b_\sigma \approx 8.5 \sqrt{C} $. We specify $\alpha_\sigma = 0.5 < 1$ to encourage, a priori, smaller values of $\sigma^2_k$. Throughout, we use $\alpha = 1$ and $D = 3$. Following [@Guhaniyogi2017bayesian] for the hyperparameters controlling the underlying hard [PARAFAC]{}, we specify $a_\lambda = 3$ and $b_\lambda = \sqrt[2K]{a_\lambda}$. Lastly, assuming centered and scaled outcome and scalar covariates, we specify $(\mu, \bm \delta^T)^T \sim N(0, \mathbb{I}_{p +1})$, and residual variance $\tau^2 \sim IG(2, 0.35)$ which specifies $P(\tau^2 < 1) \approx 0.99$.
*Interplay between variance hyperparameters*. The prior mean of $\sigma^2_k$, the variance component in the [PARAFAC]{} softening, depends on the target variance and the proportion of that variance that is attributable to the [PARAFAC]{} softening, and does not depend on the remaining hyperparameters (considering that $a_\tau / (a_\tau + 1) \approx 1$ for large $a_\tau$). This expresses a desirable separation between the hard and soft [PARAFAC]{} variance hyperparameters. Furthermore, since $\lambdafrac$ is the prior mean of $\wkj\tod$, \[eq:condition\_lambdafrac\] expresses the interplay between two components in the variance of $\gammakj\tod$. When the prior mean of $\tau_\gamma$ increases, the prior mean of $\wkj$ has to decrease in order to maintain the target variance at level $V^*$. Note that \[eq:condition\_lambdafrac\] depends on $V^*$ and $AV^*$ only through $V^*(1 - AV^*)$ expressing the prior variability in $\B_\subss$ due to the hard underlying [PARAFAC]{}.
[[Softer]{}]{}’s dependence on the rank of the underlying hard [PARAFAC]{}
--------------------------------------------------------------------------
As mentioned previously, the prior on $\bm \zeta$ allows for some form of sparsity in the components that contribute to the coefficient matrix approximation. In a sense, if the matrix can be well-approximated by a rank lower than $D$, the prior leads to a reduction in the approximation’s effective rank. However, if all $D$ components are useful in estimation, then all of them acquire sufficient weight. For that reason, [@Guhaniyogi2017bayesian] recommended using $D = 10$ for a predictor of dimension $64 \times 64$.
However, [[Softer]{}]{} is more robust to the choice of rank $D$ than hard [PARAFAC]{}. That is because [[Softer]{}]{} allows for deviations from the underlying [PARAFAC]{} structure when the true coefficient tensor is not of low rank, and these deviations can effectively capture components corresponding to singular values of any magnitude. In \[fig:rank\_choice\] we illustrate the range of singular values that would be accounted for when expanding away from a rank-$D_1$ hard [PARAFAC]{} approximation by (1) increasing the hard [PARAFAC]{} rank, and (2) softening the [PARAFAC]{}. Increasing the hard [PARAFAC]{} rank would include components corresponding to some small singular values, but softening the [PARAFAC]{} would accommodate deviations from the underlying $D_1$-rank structure across all singular values.
[[Softer]{}]{}’s ability to capture any true coefficient tensor is evident in the following results where we show that the posterior distribution of the coefficient tensor is consistent irrespective of the true coefficient tensor’s rank, or the rank used in the underlying hard [PARAFAC]{}. First, in \[prop:prior\_support\] we show that the prior on $\B$, $\pi_{\B}$, assigns positive prior weight to a neighborhood of any true coefficient tensor $\B^0$:
\[prop:prior\_support\] Let $\epsilon > 0$. Then, $\pi_{\B}\big(\neigh\big) > 0$, where $\neigh = \{ \B: \max_{\subss} | \B_\subss^0 - \B_\subss| < \epsilon \}.$
We assume that the true data generating model is \[eq:tensor\_regression2\] with true coefficient tensor $\B^0$. Since our interest is in estimating $\B^0$, we assume that $\tau^2 = 1$, $\mu = 0$ and $\bm \delta = \bm 0$ are known. The following result shows that our prior formulation assigns sufficient prior weight to distributions resembling the truth measured via the Kullback–Leibler divergence:
\[prop:KL\_diverge\] Assume that the tensor predictor $\tensor$ has bounded entries. Then, for any $\epsilon > 0$, there exists $\epsilon ^* > 0$ such that $ \Big\{\B: \max_{\subss} | \B_\subss^0 - \B_\subss| < \epsilon^* \Big\} \subseteq
\Big\{\B: KL(\B^0, \B) < \epsilon \Big\}, $ where $$KL(\B_0, \B) = \int \log \frac{\phi(y;\B^0)}{\phi(y;\B)} \phi(y;\B^0) \mathrm{d} y,$$ and $\phi(y;\B)$ is the density of a normal distribution with coefficient tensor $\B$ and variance 1.
From \[prop:KL\_diverge\] and using \[prop:prior\_support\] we see that any Kullback–Leibler neighborhood of the true distribution has positive probability, implying weak consistency of $\B$ [@Schwartz1965bayes]. Importantly, these results do *not* depend on assuming that the true coefficient tensor is of low rank, and hold irrespective of the rank used in the underlying hard [PARAFAC]{} structure.
[[Softer]{}]{}’s robustness to the choice of $D$ is further illustrated in simulated examples in \[sec:simulations\].
Approximating the posterior distribution of the coefficient tensor {#subsec:MCMC}
------------------------------------------------------------------
Since there is no closed-form for the posterior distribution of $\B$, we approximate it using Markov Chain Monte Carlo (MCMC). An MCMC scheme where most parameters are updated using Gibbs sampling is shown in \[app\_sec:MCMC\]. We found this approach to be sufficiently efficient when the sample size is of similar order to the number of parameters. However, in very high-dimensional settings, mixing and convergence was slow under reasonable time constraints. For that reason, and in order to provide a sampling approach that performs well across $n,p$ situations, we instead rely on Hamiltonian Monte Carlo (HMC) implemented in Stan [@Carpenter2017stan] and on the R interface [@Rstan] to acquire samples from the posterior distribution. HMC is designed to improve mixing relative to Gibbs sampling by employing simultaneous updates, and relying on gradients calculated with automatic differentiation to obtain efficient proposals.
MCMC convergence was assessed based on visual inspection of traceplots across chains with different starting values and the potential scale reduction factor [@Gelman1992] for the regression coefficients $\mu, \bm \delta, \B$ and the residual variance $\tau^2$. Note that the remaining parameters are not individually identifiable.
Simulations {#sec:simulations}
===========
To illustrate the performance of [[Softer]{}]{} and compare it against alternatives, we simulated data under various scenarios. In one set of simulations, we considered a tensor predictor of dimension $32 \times 32$ and corresponding coefficient tensors that were not necessarily of low-rank form. Sample size was set to 400. In another set of simulations, we considered a tensor predictor of dimension $20 \times 20$ and corresponding coefficient tensor of rank $3, 5, 7, 10$ and $20$ in order to investigate the performance of [[Softer]{}]{} relative to the hard [PARAFAC]{} for a true coefficient tensor that increasingly deviates from low rank form. The sample size in this situation was 200. In all situations, the predictor’s entries were drawn independently from a $N(0, 1)$ distribution, and the outcome was generated from a model in the form \[eq:tensor\_regression2\] with true residual variance $\tau ^ 2 = 0.5$.
In addition to [[Softer]{}]{}, we also considered (a) the Bayesian hard [PARAFAC]{} approach of [@Guhaniyogi2017bayesian] (for $D = 3$, same as in [[Softer]{}]{}), and (b) estimating the coefficient tensor by vectorizing the predictor and performing Lasso. We choose these two approaches because they represent the two extremes of how much prioritization is given to the predictor’s array structure (the hard [PARAFAC]{} directly depends on it, the Lasso completely ignores it), whereas [[Softer]{}]{} is designed to exploit the predictor’s structure while allowing deviations from it. Additional simulation results (including hard [PARAFAC]{} with higher rank) are shown in \[app\_sec:more\_sims\] and are summarized in the main text where appropriate.
Methods were evaluated in terms of how well they estimated the true coefficient tensor $\B$ by calculating (1) the entry-specific bias and mean squared error of the posterior mean (for the Bayesian approaches) and the penalized likelihood estimate (for the Lasso), and (2) the frequentist coverage of the 95% credible intervals. In order to evaluate the methods’ performance in accurately identifying important entries (entries with non-zero coefficients), we calculated methods’ (3a) sensitivity (percentage of important entries that were identified), (3b) specificity (percentage of non-important entries that were correctly deemed non-important), (3c) false positive rate (percentage of identified entries that are truly not important), and (3d) false negative rates (percentage of non-identified entries that are important). For the Bayesian methods, an entry was flagged as important if its corresponding 95% credible interval did not overlap with zero. Hierarchical Bayesian models have been shown to automatically perform adjustment for multiple testing error [@Scott2010bayes; @muller2006fdr]. Confidence intervals and entry selection for the Lasso were not considered. Further, we evaluated the models’ predictive performance by estimating (4) the predictive mean square error defined as the mean of the squared difference between the true outcome and the predictive mean over 1,000 new data points.
Simulation results for tensor predictor of dimensions 32$\times$32 {#subsec:sims_400}
------------------------------------------------------------------
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_true_squares2_small.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_true_feet_small.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_true_dog_small.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_true_diagonal_small.pdf "fig:"){width="\textwidth"}
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_s15_squares2_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_s15_feet_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_s15_dog_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_s15_diagonal_mean.pdf "fig:"){width="\textwidth"}
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_p2_squares2_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_p2_feet_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_p2_dog_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_p2_diagonal_mean.pdf "fig:"){width="\textwidth"}
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_l2_squares2_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_l2_feet_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_l2_dog_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso.[]{data-label="fig:sim_pictures"}](f_sim_l2_diagonal_mean.pdf "fig:"){width="\textwidth"}
The first column of \[fig:sim\_pictures\] shows the true coefficient tensors (squares, feet, dog, diagonal) which are of varying complexity and sparsity. The squares coefficient matrix is used as a scenario where the true coefficient matrix is rectangular, but not low rank. In another case, the diagonal coefficient matrix is used to represent a sparse and full-rank coefficient tensor that is expected to be hard to estimate using a block structure. The other two scenarios represent situations where the underlying structure is not of low-rank form, but could be potentially approximated by a low rank matrix up to a certain degree.
[[Softer]{}]{} [PARAFAC]{} Lasso
---------- ---------------- ---------- ---------------- ------------- --------------
squares Truly zero bias **0.003** 0.005 0.012
rMSE **0.033** 0.05 0.143
coverage 99.7% 98.3% –
\[5pt\] Truly non-zero bias **0.084** 0.106 0.501
rMSE **0.11** 0.148 0.601
coverage **80.1%** 68.4% –
\[5pt\] Prediction MSE **5.05** 8.99 111.8
feet Truly zero bias 0.037 0.046 **0.016**
rMSE **0.092** 0.109 0.198
coverage 96.9% 94.2% -
\[5pt\] Truly non-zero bias **0.116** 0.138 0.43
rMSE **0.184** 0.21 0.558
coverage **89.2%** 80% -
\[5pt\] Prediction MSE **31.9** 41.8 264.6
dog Truly zero bias 0.066 0.085 **0.013**
rMSE **0.125** 0.151 0.153
coverage 97.1% 87.7% -
\[5pt\] Truly non-zero bias **0.09** 0.112 0.197
rMSE **0.16** 0.182 0.3
coverage **92.9%** 87.7% -
\[5pt\] Prediction MSE **35.1** 45.4 138.4
diagonal Truly zero bias 0.002 0.004 **$<$0.001**
rMSE 0.019 0.051 **0.009**
coverage 100% 100% -
\[5pt\] Truly non-zero bias 0.112 0.899 **0.07**
rMSE 0.127 0.906 **0.084**
coverage **94.7%** 3% -
\[5pt\] Prediction MSE 1.39 29.7 **0.81**
: Average bias, root mean squared error, frequentist coverage of 95% credible intervals among truly zero and truly non-zero coefficient entries, and predictive mean squared error for [[Softer]{}]{}, hard [PARAFAC]{} and Lasso for the simulation scenario with tensor predictor of dimensions $32 \times 32$ and sample size $n = 400$. Bold text is used for the approach performing best in each scenario and for each metric.[]{data-label="tab:sims_n400"}
The remaining columns of \[fig:sim\_pictures\] show the average posterior mean or penalized estimate across simulated data sets. In the squares, feet and dog scenarios, the hard [PARAFAC]{} performs decently in providing a low-rank approximation to the true coefficient matrix. However, certain coefficient entries are estimated poorly to fit its rectangular structure. In the diagonal scenario, the hard [PARAFAC]{} almost totally misses the diagonal structure and estimates (on average) a coefficient matrix that is very close to zero. In contrast, the Lasso approach performs best in the sparse, diagonal scenario and identifies on average the correct coefficient matrix structure. However, in the squares, dog and feet settings, it underestimates the coefficient matrix since it is based on assumed sparsity and does not borrow any information across coefficient matrix entries. In all situations, [[Softer]{}]{} closely identifies the structure of the underlying coefficient matrix. The results shown in \[fig:sim\_pictures\] indicate that [[Softer]{}]{} provides a good compromise between tensor-based and unstructured estimation, with small biases across simulated scenarios (average bias also reported in \[tab:sims\_n400\]). Even though the Lasso approach shows the smallest shrinkage in the diagonal scenario, the strength of [[Softer]{}]{} is found in its ability to use the low-rank structure of the [PARAFAC]{} when necessary, and diverge from it when needed.
In \[tab:sims\_n400\], we report the root mean squared error (rMSE) of the three approaches, predictive mean squared error, and frequentist coverage of the 95% credible intervals for [[Softer]{}]{} and hard [PARAFAC]{}. Conclusions remain unchanged, with [[Softer]{}]{} performing similarly to the hard [PARAFAC]{} when its underlying structure is close to true, and has the ability to diverge from it and estimate a coefficient tensor that is not low-rank in other scenarios. This is evident by an average coverage of 95% posterior credible intervals that is 94.7% in the diagonal scenario. In terms of their predictive ability, the pattern observed for the bias and mean squared error persists, with smallest mean squared predictive error observed for [[Softer]{}]{} in the squares, feet and dog scenarios, and the Lasso for the diagonal case.
\[tab:sims\_significance\] shows the performance of [[Softer]{}]{} and hard [PARAFAC]{} approaches for identifying entries of the tensor predictor with non-zero coefficients. Perfect performance would imply specificity and sensitivity equal to 100, and false positive and negative rates (FPR, FNR) equal to 0. The methods perform comparably in terms of specificity, sensitivity, and FNR, except for the diagonal scenario where hard [PARAFAC]{}’s sensitivity is dramatically lower. However, the big difference between the two approaches is in the FPR. Even though [[Softer]{}]{}’s FPR is at times higher than 5%, it remains at much lower levels than hard [PARAFAC]{}’s which reaches an average of over 10% in the dog and almost 30% in the diagonal scenario. In \[app\_sec:significance\_disagree\] we investigate the cases where [[Softer]{}]{} and hard [PARAFAC]{} return contradicting results related to an entry’s importance. We illustrate that, when [[Softer]{}]{} disagrees with [PARAFAC]{}, it identifies entries as significant uniformly over the range of entries’ true coefficients. On the other hand, when [PARAFAC]{} identifies entries as important and [[Softer]{}]{} does not, it is most likely to be for entries with small (or zero) coefficient values.
Sensitivity Specificity FPR FNR
------------ ---------------- ------------- ------------- ------------------ ------
squares [[Softer]{}]{} 100 99.7 0.9 (0, 2.5) 0
[PARAFAC]{} 100 98.3 4.7 (1.2, 8.7) 0
feet [[Softer]{}]{} 64.5 96.9 2.9 (1.7, 4.1) 36.3
[PARAFAC]{} 68.4 94.1 5.2 (3.3, 7.1) 34.3
dog$^{**}$ [[Softer]{}]{} 52.9 96.7 5.2 (2.7, 8.1) 34.7
[PARAFAC]{} 63.1 90.1 12.4 (8.4, 15.9) 30.9
diagonal [[Softer]{}]{} 100 100 0 (0, 0) 0
[PARAFAC]{} 3 100 28.8 (0, 70) 3
: Methods performance in identifying important entries. For sensitivity, specificity and false negative rate (FNR), results are shown as average across simulated data sets ($\times 100$), and for false positive rate$^*$ (FPR) as average (10$^{th}$, 90$^{th}$ percentile) ($\times 100$).[]{data-label="tab:sims_significance"}
$\ $\
[$^*$The average FPR is taken over simulated data sets for which at least one entry was identified as important.]{}\
[$^{**}$Most coefficients in the dog simulation were non-zero. Results are presented considering coefficients smaller than 0.05 as effectively zero.]{}
\[app\_sec:more\_sims\] shows additional simulation results. \[app\_subsec:sims\_altern\] shows results for alternative coefficient matrices, including a coefficient matrix of rank 3. There, we see that [[Softer]{}]{} collapses to the underlying hard [PARAFAC]{} structure when such a structure is true. \[app\_sec:sims\_n200\] shows results for a subset of the coefficient matrices and for sample size $n = 200$. Simulations with a smaller $n$ to $p$ ratio show that [[Softer]{}]{} performs comparably to the hard [PARAFAC]{} for the dog and feet scenarios and has substantially smaller bias and rMSE for the truly non-zero coefficients in the diagonal scenario. The most notable conclusion is that [[Softer]{}]{} results are closer to the hard [PARAFAC]{} results when the sample size is small. This indicates that the data inform the variance components controlling the degree of departure from the underlying hard [PARAFAC]{}, and the amount of [PARAFAC]{} softening and shrinkage of [PARAFAC]{} deviations depends on the sample size. Lastly, \[app\_subsec:D\_7\] shows results for [[Softer]{}]{} and hard [PARAFAC]{} when $D = 7$. In short, [[Softer]{}]{} performs almost identically for rank 3 or 7, whereas the hard [PARAFAC]{} shows substantial improvements using the higher rank in certain scenarios. This indicates that [[Softer]{}]{} is very robust to the specification of the rank $D$.
Simulation results for coefficient tensor of increasing rank {#subsec:sims_rank}
------------------------------------------------------------
The simulation results presented here aim to evaluate the performance of the hard and soft [PARAFAC]{} approaches for fixed rank, when the rank of the true coefficient matrix increases and is equal to $3, 5, 7, 10$ and $20$ for a tensor predictor of dimensions $20 \times 20$. For every value of the true rank, we generated 100 data sets, and we estimated the regression model using the hard [PARAFAC]{} approach for $D = 3$ and $5$ and [[Softer]{}]{} for $D = 3$.
\[fig:sims\_increasing\_rank\] shows the average across entries of the coefficient matrix of the absolute bias and mean squared error, and the predictive mean squared error of [[Softer]{}]{} and the hard [PARAFAC]{}. When the true rank of the coefficient matrix is 3, all approaches perform similarly. This indicates that both the hard [PARAFAC]{} with $D = 5$ and [[Softer]{}]{} are able to convert back to low ranks when this is true. For true rank $D = 5, 7$, the hard [PARAFAC]{} with $D = 5$ slightly outperforms [[Softer]{}]{}. However, for $D > 7$, [[Softer]{}]{} based on a rank-3 underlying structure performs best both in estimation and in prediction. These results indicate that, in realistic situations where the coefficient tensor is not of low-rank form, [[Softer]{}]{} with a low rank has the ability to capture the coefficient tensor’s complex structure more accurately than the hard [PARAFAC]{}.
Estimating the relationship between brain connectomics and human traits {#sec:application}
=======================================================================
Data from the Human Connectome Project (HCP) contain information on about 1,200 healthy young adults including age, gender, various brain imaging data, and a collection of measures assessing cognition, personality, substance intake and so on (referred to as traits here). We are interested in studying the brain structural connectome, referring to anatomical connections of brain regions via white matter fibers tracts. The white matter fiber tracts can be indirectly inferred from diffusion MRI data. Two brain regions are considered connected if there is at least one fiber tract running between them. However, there can be thousands of fiber tracts connecting a pair of regions. Properties of the white matter tracts in a connection, such as number of tracts, and patterns of entering the regions, might be informative about an individual’s traits. Using data from the HCP and based on the soft tensor regression framework, we investigate the relationship between different connectome descriptors and human traits.
Structural connectivity data were extracted using state-of-the-art pipelines in [@Zhang2018mapping]. In total, about 20 connectome descriptors (adjacency matrices) describing different aspects of white matter fiber tract connections were generated (see [@Zhang2018mapping] for more information on the extracted descriptors). Each adjacency matrix has a dimension of $68 \times 68$, representing $R = 68$ regions’ connection pattern. The $68$ regions were defined using the Desikan-Killiany atlas [@Desikan2006automated]. Of the 20 extracted connectome features, we consider two in this analysis: (a) count, describing the number of streamlines, and (b) connected surface area (CSA), describing the area covered by small circles at the interactions of fiber tracts and brain regions, since they are the most predictive features according to results in [@Zhang2019tensor].
We examine the relationship between these descriptors of structural brain connections and [15]{} traits, covering domains such as cognition, motor, substance use, psychiatric and life function, emotion, personality and health. The full list of outcomes we analyze is presented in \[app\_tab:app\_outcomes\] and includes both binary and continuous traits. For binary traits, a logistic link function is assumed.
Adapting [[Softer]{}]{} for (semi-)symmetric brain connectomics analysis
------------------------------------------------------------------------
The nature of the brain connectivity data implies that the $R \times R$-dimensional tensor predictor including a specific connectivity feature among $R$ ROIs is symmetric and the diagonal elements can be ignored since self-loops are not considered. Further, considering $p$ features simultaneously would lead to an $R \times R \times p$ tensor predictor which is semi-symmetric (symmetric along its first two modes). The (semi-)symmetry encountered in the predictor allows us to slightly modify [[Softer]{}]{} and reduce the number of parameters by imposing that the estimated coefficient matrix $\B$ is also (semi-)symmetric. We provide the technical details for the (semi-)symmetric [[Softer]{}]{} in \[app\_sec:symmetric\_softer\].
Analyses of the brain connectomics data
---------------------------------------
For the purpose of this paper, we investigate the relationship between features of brain connections and human traits by regressing each outcome on each of the two predictors (count and CSA) separately. Even though analyzing the relationship between the traits and multiple features simultaneously is possible, we avoid doing so here for simplicity. We analyze the data employing the following methods: (1) symmetric [[Softer]{}]{} with $D = 6$, (2) the hard [PARAFAC]{} approach of [@Guhaniyogi2017bayesian] which does not impose symmetry of the coefficient tensor with $D = 10$, and (3) Lasso on the vectorized lower triangular part of the tensor predictor. Since publicly available code for non-continuous outcomes is not available for the hard [PARAFAC]{} approach, we only consider it when predicting continuous outcomes.
We compare methods relative to their predictive performance using cross-validation. In the case of [[Softer]{}]{} and hard [PARAFAC]{} we also investigate the presence of specific brain connections that are important in predicting any of the outcomes by checking whether their coefficients’ 95% posterior credible intervals include zero. Additional results based on [[Softer]{}]{} for a different choice of baseline rank or when symmetry is ignored are included in \[app\_sec:additional\_application\] and are summarized below.
Using features of brain connections for predicting human traits
---------------------------------------------------------------
\
For each approach, we estimate the out-of-sample prediction error by performing 15-fold cross validation, fitting the method on 90% of the data and predicting the outcome on the remaining 10%. For continuous outcomes, methods’ predictive performance was evaluated by calculating the percentage of the marginal variance explained by the model defined as $1 - (\text{CV MSE}) / (\text{marginal variance})$. For binary outcomes, we used the model’s estimated linear predictor to estimate the optimal cutoff for classification based on Youden’s index [@youden1950index] and calculated the average percentage of correctly classified observations in the held-out data.
\[fig:app\_predict\] shows these results for the three approaches considered, and for each feature separately. For most outcomes, one of the two features appeared to be most predictive of the outcome across approaches. For example, the count of streamlines was more predictive than CSA of an individual’s anger level (`AngHostil_Unadj`), independent of the modeling approach used. By examining the methods’ predictive performance, it is evident that features of brain connectomics are, in some cases, highly predictive of outcomes. Specifically, over 30% of the variance in an individual’s strength level, and over 10% of the variance in endurance, reading comprehension, and picture vocabulary ability can be explained by the count or CSA of streamlines of their brain connections.
Not one approach outperformed the others in prediction across all features and outcomes. However, approaches that accommodate the network structure perform better than Lasso in most situations. One example is [[Softer]{}]{}’s performance relative to Lasso when predicting individuals’ previous depressive episode. Here, Lasso performs worse than the random classifier, whereas [[Softer]{}]{} has over 90% accuracy. Even when the number of observations is less than 300 (indicator of having difficulty quitting tobacco), [[Softer]{}]{} performs only slightly worse than Lasso. For continuous outcomes, [[Softer]{}]{} and hard [PARAFAC]{} perform comparably. As we saw in the simulations in \[sec:simulations\] and in \[app\_sec:sims\_n200\], the similar predictive performance of [[Softer]{}]{} and hard [PARAFAC]{} could be due to the limited sample size that forces [[Softer]{}]{} to heavily rely on the underlying low-rank structure for estimation, essentially reverting back to the hard [PARAFAC]{}.
The low signal in predicting some outcomes implies low power in identifying pairs of brain regions whose connection’s features are important. In fact, 95% credible intervals for all coefficients using the hard [PARAFAC]{} overlapped with zero. In contrast, [[Softer]{}]{} identified seven important connections: five of them were for predicting whether an individual has had a depressive episode (three using count of streamlines as the predictor, and two using CSA), one in predicting an individual’s strength, and one in predicting the variable short penn line orientation (VSPLOT) using CSA. The identified connections are listed in \[tab:app\_connections\] and agree with the literature in neuroscience. All identified connections in predicting a depressive episode involve the parahippocampal, which is the posterior limit of the amygdala and hippocampus and is located in the temporal lobe, and ROIs located in the frontal lobe (paracentral, lateral orbitofrontal, pars orbitalis). Dysfunction of the parahippocampal (as well as the amygdala and hippocampus) has been identified in various studies as an important factor in major depression and emotion-related memory observed in depression [@Mayberg2003modulating; @Seminowicz2004limbic; @LaBar2006cognitive; @Zeng2012identifying]. Further, dysregulation of the pathways between the frontal and temporal lobes has been identified as predictive of depression [@mayberg1994frontal; @Steingard2002smaller], even when explicitly focusing on the cortical regions [[Softer]{}]{} identified as important [@Liao2013depression]. The identified connection in predicting strength involves the precuneus and superior parietal regions in the parietal lobe. Precuneus’ connectivity has been associated with a variety of human functions, including motor-related traits [@Cavanna2006precuneus; @Wenderoth2005role; @Simon2002topographical], and the parietal lobe in general is believed to control humans’ motor system [@Fogassi2005motor].
Outcome Feature ROI 1 ROI 2
------------ --------- ---------------------------------------- ----------------------------
Depressive Count (lh) Parahippocampal (lh) Paracentral
Episode (rh) Lateral Orbitofrontal
(lh) Pars Orbitalis
CSA (lh) Paracentral
(rh) Lateral Orbitofrontal
VSPLOT CSA (rh) Banks of Superior Temporal Sulcus (lh) Superior Frontal
Strength Count (rh) Precuneus (rh) Superior Parietal
: Brain connections with important features in predicting human traits.[]{data-label="tab:app_connections"}
\[app\_sec:application\_additional\_results\] includes additional study results, including results from the symmetric [[Softer]{}]{} using a smaller rank ($D=3$), and results from [[Softer]{}]{} using the same rank as the results in this section ($D=6$) but ignoring the known symmetry of the predictor. All three versions of [[Softer]{}]{} perform similarly in terms of prediction, with potentially slightly lower predictive power for symmetric [[Softer]{}]{} with rank 3. Importantly, when symmetry is not accounted for, [[Softer]{}]{} does not identify any important connections, indicating that incorporating symmetry directly in estimation leads to a reduction in the number of parameters and a subsequent increase in power to identify important entries.
Discussion
==========
In this paper, we considered modeling a scalar outcome as a function of a tensor predictor within a regression framework. Estimation of regression models in high dimensions is generally based on some type of assumed sparsity of the true underlying model: sparsity directly on covariates, or “latent sparsity” by assuming a low-dimensional structure. When the assumed sparsity is not true, the model’s predictive ability and estimation can suffer. Our approach is positioned within the class of latent sparsity, since it exploits a low-dimensional underlying structure. However, we explicitly focused on adequately relaxing the assumed structure by softening the low-dimensional [PARAFAC]{} approximation and allowing for interpretable deviations of row-specific contributions. We show that softening the [PARAFAC]{} leads to improved estimation of coefficient tensors, more accurate predictions, better performance in identifying important entries, and consistent estimation irrespective of the rank of the underlying structure used in estimation. The approach is applicable to both continuous and binary outcomes, and was adapted to (semi-)symmetric tensor predictors, which is common in settings where the predictor is measured on a network of nodes. [[Softer]{}]{} was used to study the relationship between brain connectomics and human traits, and identified several important connections in predicting depression.
Combining the two types of assumed sparsity for low-rank *and* sparse matrix estimation has received some attention in the literature, especially in machine learning with matrix data. [@Candes2011robust], [@Waters2011recovering] and [@Zhou2011randomized] decomposed matrices as the sum of a low-rank and a sparse component. [@Zhang2016lowrank] employed such decomposition in anomaly detection by studying the Mahalanobis distance between the observed data and the low-rank component. [@Richard2012estimation] developed a penalization-based approach to estimating matrices that are simultaneously sparse and low-rank by adopting one type of penalty for sparsity and one for rank. All these approaches have been formulated as optimization problems and algorithms for estimation are generally based on iterative procedures.
Within a Bayesian regression framework, [@Guha2018bayesian] combined the two types of sparsity and proposed a network-based spike and slab prior on the nodes’ importance. Under that model, a node is either active or inactive, and active nodes are expected to contribute to the outcome based on a low-rank coefficient tensor. In that sense, this approach has important commonalities to estimating a coefficient tensor that is simultaneously sparse and low-rank. Even though we find that approach to be promising, we find that node selection in itself can be too restrictive in some settings, and future work could incorporate hierarchical or parallel node and entry selection.
[[Softer]{}]{} has similarities but also differences from the methods discussed above. On one hand, [[Softer]{}]{} provides a relaxation of an assumed low-rank form. However, this relaxation (or softening) is not sparse in any way, and *every* element of the tensor is allowed to deviate from the low-rank structure. We find that an exciting line of research would combine low-rank and sparse approaches while allowing for sufficient flexibility to deviate from both of them.
Important questions remain on the interplay between entry selection and the assumed structure on the coefficient tensor. In this paper we showed that the hard [PARAFAC]{} employed for estimation in settings with tensor data directly affects and deteriorates the method’s performance for entry selection. Future work could focus on studying the properties of multiplicity control in structured settings, and forming principled variable selection approaches with desirable properties within the context of structured data.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by grant R01MH118927 from the National Institutes of Health (NIH) and R01-ES027498-01A1 from the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH).
Appendices {#appendices .unnumbered}
==========
Proofs {#app_sec:proofs}
======
$\ $\
**Expectation.** We use $S, Z$ and $W$ to denote the collection of $\sigma^2_k, \zeta\tod$ and $\wkj\tod$, over $k$, $d$, and $(k,j_k,d)$ accordingly. We start by noting that $$\betakj\tod | \sigma^2_k, \zeta\tod, \tau_\gamma, \wkj\tod \sim N(0, \sigma^2_k\zeta\tod + \tau_\gamma\zeta\tod\wkj\tod),$$ and, if $(k,j_k,d) \neq (k',j_k', d')$ $$\betakj\tod {\perp\!\!\!\perp}\betakj[F][F]\tod[F] | S, Z, W, \tau_\gamma.
\label{app_eq:cond_independece}$$ Note that $\betakj\tod$ is *not* independent of $\betakj[T][F]\tod$ conditional on $(S,Z,W,\tau_\gamma)$ when $j_k = j_k'$ due to their shared dependence on $\gammakj\tod$. Then, $$\begin{aligned}
{\mathbb{E}}(\B_\subss | S, Z, W, \tau_\gamma) &=
{\mathbb{E}}\Big(\sum_{d = 1}^D \prod_{k = 1}^K \betakj\tod | S, Z, W, \tau_\gamma \Big)
= \sum_{d = 1}^D \prod_{k = 1}^K {\mathbb{E}}\Big(\betakj\tod | S, Z, W, \tau_\gamma \Big).
= 0\end{aligned}$$ So, a priori, all elements of the coefficient tensor have mean 0, ${\mathbb{E}}(\B_\subss) = 0$.\
**Variance.**\
Furthermore, we have $${\mathrm{Var}}(\B_\subss) = {\mathbb{E}}\Big\{ {\mathrm{Var}}\Big( \B_\subss | S, Z, W, \tau_\gamma \Big)\Big\}
= {\mathbb{E}}\Big\{ {\mathrm{Var}}\Big( \sum_{d = 1}^D \prod_{k = 1}^K \betakj\tod | S, Z, W, \tau_\gamma \Big) \Big\}.$$ Since the $\betakj\tod$ are conditionally independent across $d$, $\prod_{k = 1}^K \betakj\tod$ are also conditionally independent across $d$. Moreover, the terms of the product $\betakj\tod$ are independent across $k$ and are mean-zero random variables, implying that $\prod_{k = 1}^K \betakj\tod$ are mean zero variables. Note here that two independent mean-zero random variables $A, B$ satisfy that ${\mathrm{Var}}(AB) = {\mathrm{Var}}(A){\mathrm{Var}}(B) $. Then, $$\begin{aligned}
{\mathrm{Var}}(\B_\subss) &= {\mathbb{E}}\Big\{ \sum_{d = 1}^D \prod_{k = 1}^K {\mathrm{Var}}\Big( \betakj\tod | S, Z, W, \tau_\gamma \Big) \Big\} \\
&= {\mathbb{E}}\Big\{ \sum_{d = 1}^D \prod_{k = 1}^K \zeta\tod (\sigma^2_k + \tau_\gamma \wkj\tod)\Big\} \\
&= {\mathbb{E}}_Z \Big\{ \sum_{d = 1}^D (\zeta\tod)^K \Big\} {\mathbb{E}}_{S,W, \tau_\gamma} \Big\{ \prod_{k = 1}^K (\sigma^2_k + \tau_\gamma \wkj\tod)\Big\},\end{aligned}$$ where in the last equation we used that, a priori, $Z {\perp\!\!\!\perp}(S, W, \tau_\gamma)$ to write ${\mathbb{E}}_{S,W,\tau_\gamma|Z}$ as ${\mathbb{E}}_{S,W,\tau_\gamma}$, and separate the two expectations.
However, $\sigma^2_k + \tau_\gamma \wkj\tod$ are not independent of each other for different values of $k$ since they all involve the same parameter $\tau_\gamma$. We overcome this difficulty in calculating the expectation of the product by writing $\prod_{k = 1}^K (\sigma^2_k + \tau_\gamma \wkj\tod) = \sum_{l = 0}^K c_l \tau_\gamma^l$, where $$c_l = \sum_{\mathcal{K} \subset \{1, 2, \dots, K\} : |\mathcal{K}| = l} \left( \prod_{k \in \mathcal{K}} \wkj\tod \prod_{k \not\in \mathcal{K}} \sigma^2_k \right).$$ So, for every power of $\tau_\gamma$, $\tau_\gamma^l$, $l \in \{0, 1, \dots, K\}$, the corresponding coefficient is a sum of all terms involving $l$ distinct $w$’s and $K-l$ distinct $\sigma^2$’s. For example, for $K = 2$, $c_1 = w_{1,j_1}\tod \sigma^2_2 + \sigma^2_1 w_{2,j_2}\tod$. Writing the product in this way, separates the terms $(\wkj\tod, \sigma^2_k)$ from $\tau_\gamma$, which are a priori independent. Then, $${\mathrm{Var}}(\B_\subss) = {\mathbb{E}}_Z \Big\{ \sum_{d = 1}^D (\zeta\tod)^K \Big\}
{\mathbb{E}}_{S,W, \tau_\gamma} \Big( \sum_{l = 0}^K c_l \tau_\gamma^l \Big)
= {\mathbb{E}}_Z \Big\{ \sum_{d = 1}^D (\zeta\tod)^K \Big\} \Big\{ \sum_{l = 0}^K {\mathbb{E}}(\tau_\gamma^l){\mathbb{E}}_{S,W} (c_l) \Big\}.$$
We continue by studying ${\mathbb{E}}_{S,W}(c_l) = \sum_{\mathcal{K}: |\mathcal{K}| = l} {\mathbb{E}}_{S,W} \Big( \prod_{k\in\mathcal{K}} \wkj\tod \prod_{k \not\in\mathcal{K}} \sigma^2_k \Big) $. Note that since all parameters $\{\wkj\tod,\sigma^2_k\}_k$ for fixed $j_k$ are a priori independent (any dependence in the $\wkj\tod$ exists across $j_k$ of the same mode due to the common value $\lambda_k\tod$), ${\mathbb{E}}_{S,W}(c_l) = \sum_{\mathcal{K}: |\mathcal{K}| = l} \Big( \prod_{k\in\mathcal{K}} {\mathbb{E}}_W (\wkj\tod) \prod_{k \not\in\mathcal{K}} {\mathbb{E}}_S (\sigma^2_k) \Big) $. Now, note that ${\mathbb{E}}(\sigma^2_k) = {\frac{a_\sigma}{b_\sigma }}$, and $${\mathbb{E}}_W (\wkj\tod) = {\mathbb{E}}_\Lambda\{{\mathbb{E}}_{W|\Lambda}[\wkj\tod]\} = 2 {\mathbb{E}}_{\Lambda}\{ (\lambda_k\tod)^{-2} \}.$$ Since $\lambda_k\tod \sim \Gamma(a_\lambda, b_\lambda)$, $1 / \lambda_k\tod \sim IG(a_\lambda, b_\lambda)$, we have that $${\mathbb{E}}\{(1 / \lambda_k\tod)^2\} =
{\mathrm{Var}}(1 / \lambda_k\tod) + {\mathbb{E}}^2(1 / \lambda_k\tod) =
\frac{b_\lambda^2}{(a_\lambda - 1)(a_\lambda - 2)}, \ a_\lambda > 2.$$ Putting this together, we have that, for $a_\lambda > 2$, $${\mathbb{E}}_{S,W}(c_l) = {K \choose l}
\Big\{ \lambdafrac \Big\}^l
\Big( {\frac{a_\sigma}{b_\sigma }}\Big)^{K -l}.$$ Further, since $\tau_\gamma \sim \Gamma(a_\tau, b_\tau)$, we have that $${\mathbb{E}}(\tau_\gamma^l) = \frac{b_\tau^{a_\tau}}{\Gamma(a_\tau)} \int \tau_\gamma^{a_\tau + l - 1} \exp\{ - b_\tau \tau_\gamma \}\ \mathrm{d}\tau_\gamma = \frac{\Gamma(a_\tau + l)}{\Gamma(a_\tau)b_\tau^l} = \frac{\rho_l}{b_\tau^l},$$ for $\rho_l = 1$ if $l = 0$, and $\rho_l = a_\tau (a_\tau + 1) \dots (a_\tau + l - 1)$ if $l \geq 1$. Lastly, since $\bm \zeta \sim Dir(\alpha/D, \alpha/D, \dots, \alpha/D)$, we have that $\zeta\tod \sim Beta(\alpha/D, (D-1)\alpha/D)$, and $$E\{(\zeta\tod)^K\} = \prod_{r = 0}^{K - 1} \frac{\alpha/D + r}{\alpha + r}$$
Combining all of these, we can write the prior variance for entries $\B_\subss$ of the coefficient tensor as $$\begin{aligned}
{\mathrm{Var}}(\B_\subss) &=
{\mathbb{E}}_Z \Big[ \sum_{d = 1}^D (\zeta\tod)^K \Big]
\sum_{l = 0}^K \rho_l {K \choose l} \Big\{ \lambdafrac[T] \Big\}^l
\Big( {\frac{a_\sigma}{b_\sigma }}\Big)^{K -l} \\
&= \Big\{ D \prod_{r = 0}^{K - 1} \frac{\alpha/D + r}{\alpha + r} \Big\}
\Big[ \sum_{l = 0}^K \rho_l {K \choose l} \Big\{ \lambdafrac[T] \Big\}^l
\Big( {\frac{a_\sigma}{b_\sigma }}\Big)^{K -l} \Big].\end{aligned}$$ **Covariance.**\
Since ${\mathbb{E}}(\B_\subss|S,Z,W,\tau_\gamma) = 0$, we have that ${\mathrm{Cov}}(\B_\subss, \B_{\subss'}) = {\mathbb{E}}\Big\{ {\mathrm{Cov}}(\B_\subss, \B_{\subss'} | S, Z, W, \tau_\gamma ) \Big\}$. Remember from \[app\_eq:cond\_independece\] that, when at least one of $k,j_k,d$ are different, $\betakj\tod {\perp\!\!\!\perp}\betakj[F][F]\tod[F] | S, Z, W, \tau_\gamma$. However, that is not true when $(k,j_k,d) = (k',j_k',d')$, even if $\subss \neq \subss$. We write $$\begin{aligned}
{\mathbb{E}}\Big\{ {\mathrm{Cov}}(\B_\subss, \B_{\subss'} | S, Z, W, \tau_\gamma ) \Big\}
&= {\mathbb{E}}\Big\{ {\mathrm{Cov}}\Big(\sum_{d = 1}^D \prod_{k = 1}^K \betakj\tod, \sum_{d = 1}^D \prod_{k = 1}^K \betakj[T][F]\tod \Big | S, Z, W, \tau_\gamma) \Big\} \\
&= \sum_{d,d'=1}^D {\mathbb{E}}\Big\{ {\mathrm{Cov}}\Big( \prod_{k = 1}^K \betakj\tod, \prod_{k = 1}^K \betakj[T][F]\tod[F] | S, Z, W, \tau_\gamma \Big) \Big\}.\end{aligned}$$ However, $$\begin{aligned}
{\mathrm{Cov}}\Big(\prod_{k = 1}^K & \betakj\tod, \prod_{k = 1}^K \betakj[T][F]\tod[F] | S,Z,W,\tau_\gamma \Big) = \\ & {\mathbb{E}}\Big(\prod_{k = 1}^K \betakj\tod \betakj[T][F]\tod[F] | S,Z,W,\tau_\gamma \Big) -
{\mathbb{E}}\Big(\prod_{k = 1}^K \betakj\tod | S,Z,W,\tau_\gamma \Big)
{\mathbb{E}}\Big(\prod_{k = 1}^K \betakj[T][F]\tod[F] | S,Z,W,\tau_\gamma \Big) = \\
& {\mathbb{E}}\Big(\prod_{k = 1}^K \betakj\tod \betakj[T][F]\tod[F] | S,Z,W,\tau_\gamma \Big),\end{aligned}$$ where the last equation holds because the $\betakj\tod$ are independent of each other across $k$ conditional on $S, Z, W, \tau_\gamma$ and have mean zero. Furthermore, since the $\betakj\tod$ are conditionally independent across $d$, we have that for $d \neq d'$, ${\mathrm{Cov}}\Big(\prod_{k = 1}^K \betakj\tod, \prod_{k = 1}^K \betakj[T][F]\tod[F] | S,Z,W,\tau_\gamma \Big) = 0$. So we only need to study the conditional covariance for $d = d'$. For $\Gamma$ representing the set of all $\gammakj\tod$, we write $${\mathbb{E}}\Big(\prod_{k = 1}^K \betakj\tod \betakj[T][F]\tod | S,Z,W,\tau_\gamma \Big)
= {\mathbb{E}}\Big\{ {\mathbb{E}}\Big( \prod_{k = 1}^K \betakj\tod \betakj[T][F]\tod | \Gamma, S,Z,W,\tau_\gamma \Big) | S,Z,W,\tau_\gamma \Big\}.$$ Conditional on $\Gamma, S,Z,W,\tau_\gamma$, and as long as $\subss \neq \subss'$, the $\betakj\tod$ are independent across all indices, even if they have all of $k,j_k,d$ common, leading to $$\begin{aligned}
{\mathbb{E}}\Big(\prod_{k = 1}^K \betakj\tod \betakj[T][F]\tod | S,Z,W,\tau_\gamma \Big)
&= {\mathbb{E}}\Big\{\prod_{k = 1}^K {\mathbb{E}}\Big( \betakj\tod | \Gamma, S,Z,W,\tau_\gamma \Big) {\mathbb{E}}\Big( \betakj[T][F]\tod | \Gamma, S,Z,W,\tau_\gamma \Big) | S,Z,W,\tau_\gamma \Big\} \\
&= {\mathbb{E}}\Big( \prod_{k = 1}^K \gammakj\tod \gammakj[T][F]\tod | S,Z,W,\tau_\gamma \Big) \\
&= \prod_{k = 1}^K {\mathbb{E}}\big( \gammakj\tod \gammakj[T][F]\tod | S,Z,W,\tau_\gamma \big) \\
&= \prod_{k: j_k = j_k'} {\mathbb{E}}\Big( \big(\gammakj\tod \big)^2 | S,Z,W,\tau_\gamma \Big) \\
& \hspace{40pt} \times \prod_{k:j_k \neq j_k'} {\mathbb{E}}\big( \gammakj\tod | S,Z,W,\tau_\gamma \big) {\mathbb{E}}\big( \gammakj[T][F]\tod | S,Z,W,\tau_\gamma \big) \\
&= 0\end{aligned}$$ where the first equality holds because $\subss \neq \subss'$, the third equality holds because the $\gammakj\tod$ are conditionally independent across $k$, and the fourth equality holds because they are conditionally independent across $j_k$.
We want ${\mathrm{Var}}(B_\subss) = V^*$ and $AV = AV^*$. The second target is achieved if ${\mathrm{Var}}(\B_\subss) / {\mathrm{Var}}^{hard}(\B_\subss) = (1 - AV^*)^{-1}$. Since ${\frac{a_\sigma}{b_\sigma }}$ is the quantity driving the soft [PARAFAC]{}’s additional variability we use this condition to acquire a form for ${\frac{a_\sigma}{b_\sigma }}$ as a function of the remaining hyperparameters. $$\begin{aligned}
\frac{{\mathrm{Var}}(\B_\subss)}{{\mathrm{Var}}^{hard}(\B_\subss)} &=
\frac{\sum_{l = 0}^2 \frac{\rho_l}{b_\tau^l} \binom{2}{l} \big\{ \lambdafrac \big\}^l \big( {\frac{a_\sigma}{b_\sigma }}\big)^{2 -l}}
{ \frac{\rho_2}{b_\tau^2} \big\{ \lambdafrac \big\}^2} \\
&= \sum_{l = 0}^2 \binom{2}{l} \frac{\rho_l}{\rho_2} \Big\{ \lambdafrac[T] \Big\}^{l - 2} \Big( {\frac{a_\sigma}{b_\sigma }}\Big)^{2 - l} \\
&= \frac1{\rho_2} \Big\{ \lambdafrac[T] \Big\}^{- 2} \Big( {\frac{a_\sigma}{b_\sigma }}\Big)^2 + 2 \frac{\rho_1}{\rho_2} \Big\{ \lambdafrac[T] \Big\}^{-1} {\frac{a_\sigma}{b_\sigma }}+ 1 \\
&= \frac1{a_\tau(a_\tau + 1)} \Big\{ \lambdafrac[T] \Big\}^{- 2} \Big( {\frac{a_\sigma}{b_\sigma }}\Big)^2 + \frac2{a_\tau + 1} \Big\{ \lambdafrac[T] \Big\}^{-1} {\frac{a_\sigma}{b_\sigma }}+ 1\end{aligned}$$ Therefore, in order for ${\mathrm{Var}}(\B_\subss) / {\mathrm{Var}}^{hard}(\B_\subss) = (1 - AV^*)^{-1}$, ${\frac{a_\sigma}{b_\sigma }}$ is the solution to a second degree polynomial. We calculate the *positive* root of this polynomial. $$\begin{aligned}
\Delta & = \frac{4}{(a_\tau + 1)^2} \Big\{ \lambdafrac[T] \Big\}^{-2} - \frac4{a_\tau(a_\tau + 1)} \Big\{ \lambdafrac[T] \Big\}^{- 2} \big(1 - (1 - AV^*)^{-1}\big) \\
& = \frac{4}{(a_\tau + 1)^2} \Big\{ \lambdafrac[T] \Big\}^{-2} \Big[ 1 - \frac{a_\tau + 1}{a_\tau}\Big\{1 - \big(1 - AV^*)^{-1} \Big\} \Big] > 0.\end{aligned}$$ Since ${\frac{a_\sigma}{b_\sigma }}$ is positive, we have that $$\begin{aligned}
{\frac{a_\sigma}{b_\sigma }}&= \frac{-\frac2{a_\tau + 1} \Big\{ \lambdafrac[T] \Big\}^{-1} + \sqrt{\frac{4}{(a_\tau + 1)^2} \Big\{ \lambdafrac[T] \Big\}^{-2} \Big[ 1 - \frac{a_\tau + 1}{a_\tau}\Big\{1 - \big(1 - AV^*)^{-1} \Big\} \Big]}}
{\frac2{a_\tau(a_\tau + 1)} \Big\{ \lambdafrac[T] \Big\}^{- 2}} \\
&= \frac{- 1 + \sqrt{ 1 - \frac{a_\tau + 1}{a_\tau}\Big\{1 - \big(1 - AV^*)^{-1} \Big\}}}
{\frac1{a_\tau} \Big\{ \lambdafrac[T] \Big\}^{- 1}} \\
&= \frac{a_\tau}{b_\tau} \lambdafrac \bigg\{
\sqrt{ 1 - \frac{a_\tau + 1}{a_\tau} \big\{1 - \big(1 - AV^*)^{-1} \big\}} - 1 \bigg\} {\addtocounter{equation}{1}\tag{{\thesection.\arabic{equation}}}}{} \label{proof_eq:add_var}.\end{aligned}$$ Denoting $ \displaystyle \xi = 1 - \frac{a_\tau + 1}{a_\tau} \big\{1 - \big(1 - AV^*)^{-1} \big\} $ and substituting the form of ${\frac{a_\sigma}{b_\sigma }}$ in ${\mathrm{Var}}(\B_\subss)$ we have that $$\begin{aligned}
{\mathrm{Var}}(\B_\subss) &= C \sum_{l = 0}^2 \binom{2}{l} \frac{\rho_l}{b_\tau^l} \Big\{ \lambdafrac \Big\}^2 \Big(\frac{a_\tau}{b_\tau} \Big)^{2 - l} \Big( \sqrt{ \xi } - 1 \Big) ^{2 - l} \\
&= C \Big\{ \lambdafrac \Big\}^2 \Big(\frac{a_\tau}{b_\tau} \Big)^2 \sum_{l = 0}^2 \binom{2}{l} \frac{\rho_l}{a_\tau^l} \Big( \sqrt{\xi} - 1 \Big) ^{2 - l}\\
&= C \Big\{ \lambdafrac \Big\}^2 \Big(\frac{a_\tau}{b_\tau} \Big)^2
\Big\{ \Big( \sqrt{\xi} - 1 \Big)^2 + 2 \Big(\sqrt{\xi} - 1 \Big) + 1 + \frac1{a_\tau} \Big\} \\
&= C \Big\{ \lambdafrac \Big\}^2 \Big(\frac{a_\tau}{b_\tau} \Big)^2
\Big\{ \xi + \frac1{a_\tau} \Big\}\end{aligned}$$ Also, $$\begin{aligned}
\xi + \frac1{a_\tau} &= 1 - \frac{a_\tau + 1}{a_\tau} \big\{1 - \big(1 - AV^* \big)^{-1} \big\} + \frac1{a_\tau} =
\frac{a_\tau + 1}{a_\tau} \big(1 - AV^*)^{-1} \end{aligned}$$ leading to $$\begin{aligned}
{\mathrm{Var}}(\B_\subss) = V^* \iff \lambdafrac = \frac{b_\tau}{a_\tau} \sqrt{\frac{V^*(1 - AV^*)a_\tau}{C(a_\tau + 1)}}.
{\addtocounter{equation}{1}\tag{{\thesection.\arabic{equation}}}}{} \label{proof_eq:lambda_frac}\end{aligned}$$ Substituting \[proof\_eq:lambda\_frac\] back into \[proof\_eq:add\_var\], we have that $${\frac{a_\sigma}{b_\sigma }}= \sqrt{\frac{V^*(1 - AV^*)a_\tau}{C(a_\tau + 1)}} \bigg\{
\sqrt{ 1 - \frac{a_\tau + 1}{a_\tau} \big\{1 - \big(1 - AV^*)^{-1} \big\}} - 1 \bigg\}.$$
Start by noting that $$\pi_{\B}(\neigh) = {\mathbb{E}}_{\Gamma, S, Z} \Big[ p \Big(\B : \max_{\subss} |\B_\subss^0 - \B_\subss| < \epsilon \Big| \Gamma, S, Z \Big) \Big],$$ where $\Gamma, S, Z$ are as defined in the proof of \[theory:variance\_B\]. Then, take $\epsilon^* = \sqrt[K]{\epsilon / (2(D - 1))}$ and write $$\begin{aligned}
p \Big(\B : \max_{\subss} |\B_\subss^0 - \B_\subss| < \epsilon \Big| \Gamma, S, Z \Big) & \geq
p \Big(\B : \max_{\subss} |\B_\subss^0 - \B_\subss| < \epsilon \Big| \Gamma, S, Z \big\{|\betakj\tod| <\epsilon^*, \text{all } k, \subss, \text{ and } d \geq 2 \big\} \Big) \\
& \hspace{40pt} \times p \Big( |\betakj\tod| <\epsilon^*, \text{all } k, \subss, \text{ and } d \geq 2 \big\} | \Gamma, S, Z \Big).\end{aligned}$$ Conditional on $\Gamma,S,Z$, $\betakj\tod$ are independent normal variables with positive weight in an $\epsilon^*$-neighborhood of 0, implying that $ p \Big( |\betakj\tod| <\epsilon^*, \text{all } k, \subss, \text{ and } d \geq 2 \big\} | \Gamma, S, Z \Big) > 0. $
Remember from \[eq:softB\] that $\B = \sum_{d = 1}^D \B_1\tod {\circ}\B_2\tod {\circ}\dots {\circ}\B_K\tod$, and denote $\B\tod = \B_1\tod {\circ}\B_2\tod {\circ}\dots {\circ}\B_K\tod$. Then, $\B_\subss = \B_\subss^{(1)} + \B_\subss^{(2)} + \dots + \B_\subss^{(D)}$. Note that $$\begin{aligned}
p \Big(\B : & \max_{\subss} |\B_\subss^0 - \B_\subss| < \epsilon \Big| \Gamma, S, Z \big\{|\betakj\tod| <\epsilon^*, \text{all } k, \subss, \text{ and } d \geq 2 \big\} \Big) \\
& \geq p \Big(\B : \max_{\subss} |\B_\subss^0 - \B_\subss^{(1)}| < \epsilon / 2 \Big| \Gamma, S, Z \big\{|\betakj\tod| <\epsilon^*, \text{all } k, \subss, \text{ and } d \geq 2 \big\} \Big) \\
& = p \Big(\B : \max_{\subss} |\B_\subss^0 - \B_\subss^{(1)}| < \epsilon / 2 \Big| \Gamma, S, Z \Big),\end{aligned}$$ where the equality holds because the entries of $\B^{(1)}$ are independent of all $\betakj\tod$ for $d \geq 2$ conditional on $\Gamma,S,Z$, and the inequality holds because $|\B_\subss^0 - \B_\subss^{(1)}| < \epsilon / 2$ and $|\betakj\tod| < \epsilon^*$ for $d \geq 2$ implies that $$\begin{aligned}
|\B_\subss^0 - \B_\subss| &= |\B_\subss^0 - \B_\subss^{(1)} - \B_\subss^{(2)} - \dots - \B_\subss^{(D)}| \\
&\leq |\B_\subss^0 - \B_\subss^{(1)}| + |\B_\subss^{(2)}| + \dots + |\B_\subss^{(D)}| \\
& < \epsilon / 2 + (D-1)(\epsilon^*)^K = \epsilon.\end{aligned}$$
Since all of $\betakj^{(1)}$ are independent conditional on $\Gamma, S,Z$, we have that $$p \Big(\B : \max_{\subss} |\B_\subss^0 - \B_\subss^{(1)}| < \epsilon / 2 \Big| \Gamma, S, Z \Big) =
\prod_{\subss} p \Big(|\B_\subss^0 - \B_\subss^{(1)}| < \epsilon / 2 \Big| \Gamma, S, Z \Big) > 0,$$ since all entries $\B_\subss^{(1)}$ are products of draws from $K$ normal distributions and therefore assign positive weight in all $\mathbb{R}$, including the $\epsilon /2 -$neighborhood of $\B_\subss^0$.
Putting all of this together, we have the desired result that $\pi_{\B}(\neigh) > 0$.
Assume that there exists $M$ such that $|\tensor_\subss| < M$ for all $\subss$ with probability 1. For two normal distributions with mean $\langle \tensor, \B^0\rangle_F$ and $\langle \tensor, \B \rangle_F$ respectively, we have that $$\begin{aligned}
KL(\B^0, \B) &= \frac12 \Big(\langle \tensor, \B^0\rangle_F - \langle \tensor, \B^0\rangle_F \Big)^2 \\
&= \frac12 \Big[ \sum_\subss \big( \B^0_\subss - \B_\subss \big) \tensor_\subss \Big]^2 \\
& \leq \frac12 \Big[ \sum_\subss \big( \B^0_\subss - \B_\subss \big)^2 \Big] \Big[ \sum_\subss \tensor_\subss^2 \Big]\end{aligned}$$ Take $\epsilon^* = \sqrt{2\epsilon}/ \big(M (p_1p_2\dots p_K)^2 \big)$ and consider $\B \in \neigh[*]$. We will show that $\B$ satisfies $KL(\B^0, \B) < \epsilon$ completing the proof. Note first that $$\begin{aligned}
\sum_\subss \big( \B^0_\subss - \B_\subss \big)^2
\leq p_1p_2\dots p_k \max_\subss \big( \B^0_\subss - \B_\subss \big)^2 < p_1p_2\dots p_k (\epsilon^*)^2 = \frac{2\epsilon}{M p_1p_2\dots p_K},\end{aligned}$$ and since $|\tensor_\subss| \leq M$ we have that $\sum_\subss \tensor_\subss^2 \leq M p_1p_2\dots p_K$. Putting these results together $$KL(\B^0, \B) < \frac12 \frac{2\epsilon}{M p_1p_2\dots p_K} M p_1p_2\dots p_K = \epsilon.$$
Hard [PARAFAC]{} error in estimating the true matrix for an increasing rank {#app_sec:hard_demonstration}
===========================================================================
Due to the block structure and subsequent “inflexibility” of the hard [PARAFAC]{} approximation, a large number of components $D$ might be required in order to adequately approximate a coefficient tensor $\B$. To further demonstrate this, we considered a coefficient matrix $\B$ whose entries $\B_{ij}$ are centered around (but are not equal to) the entries of a rank-1 matrix of the form $\beta_1 \otimes \beta_2$. Therefore, even though the matrix has a somewhat rectangular structure, it is not exactly in that form. Using the singular value decomposition (which is the [PARAFAC]{} analog for matrices), we considered the quality of the approximation based on $D$ factors, for various values of $D$. \[fig:random\_matrix\_parafac\] shows histograms of the difference of the true entries in $\B$ from the estimated ones. Even for $D = 20$, substantial error remains in estimating the matrix $\B$.
![Histogram of errors in estimating the entries of $\B$ when $\B$ resembles but is not exactly equal to a rank-1 tensor, and estimation is based on the singular value decomposition using $D\in \{3, 10, 20, 50\}$ factors.[]{data-label="fig:random_matrix_parafac"}](f_ill_estimation_error.jpg){width="90.00000%"}
Alternative sampling from the posterior distribution {#app_sec:MCMC}
====================================================
The full set of parameters is $\bm \theta = \{\mu, \bm \delta, \tau^2, \betakj\tod, \gammakj\tod, \sigma^2_k, \zeta\tod, \wkj\tod, \lambda_k\tod, \tau^2_\gamma, \text{ for all } d, k, j_k, \subss\}$. We use the notation $|\cdot$ and $|\cdot,-y$ to denote conditioning on the data and all parameters, and the data and all parameters but $y$, accordingly. Then, our MCMC updates are:
- $(\mu, \bm \delta) | \cdot \sim N(\bm \mu^*, \Sigma^*)$, for $\Sigma^* = (\Sigma_0^{-1} + \widetilde{\bm C}^T \widetilde{\bm C} / \tau^2)^{-1}$, and $\bm \mu^* = \Sigma^* \widetilde{\bm C}^T \bm R_B / \tau^2$, where $\widetilde{\bm C}$ is the $N\times(p+1)$ matrix with $i^{th}$ row equal to $(1, \bm C_i)$, and $\bm R_B = (Y_1 - \langle \tensor_1, \B\rangle_F, \dots, Y_N - \langle \tensor_N, \B\rangle_F)^T$ is the vector of residuals of the outcome on the tensor predictor.
- $\tau^2 | \cdot \sim IG(a_\tau + N / 2, b_\tau + \sum_{i = 1}^N (Y_i - \mu - \bm C_i^T \bm \delta - \langle \tensor_i, \B\rangle_F))$.
- $ \displaystyle \sigma^2_k | \cdot \sim giG(p^*, a^*, b^*)$, for $p^* = a_\sigma - D\prod_{k = 1}^K p_k / 2$, $a^* = 2 b_\sigma$, and $b^* = \sum_{d, \subss} (\betakj\tod - \gammakj\tod)^2 / \zeta\tod$. As a reminder, $X \sim giG(p, a, b)$ if $p(x) \propto x^{p-1}\exp\{- (ax + b/x) / 2 \} $.
- $\gammakj\tod | \cdot \sim N(\mu^*, \sigma^{2*})$, for [$ \displaystyle \sigma^{2*} = \Big\{(\tau_\gamma \wkj\tod \zeta\tod)^{-1} + \big(\sum_{l \neq k} p_l \big) / (\sigma^2_k\zeta\tod) \Big\}^{-1}, \mu^* = \sigma^{2*} \Big\{ \sum_{\subss:\subss_k = j_k} \betakj\tod / (\sigma^2_k\zeta\tod)\Big\}$]{}.
- $\tau_\gamma|\cdot \sim giG \big(a_\tau - D\sum_k p_k / 2, 2 b_\tau, \sum_{d,k,j_k} (\gammakj\tod)^2 / (\zeta\tod \wkj\tod) \big)$.
- $\wkj\tod |\cdot \sim giG(1/2, \lambda_k^2, (\gammakj\tod)^2 / (\tau_\gamma \zeta\tod))$.
- $[ \lambda_k\tod |\cdot, - \wkj\tod, \text{all } j_k ] \sim \Gamma(a_\lambda + p_k, b_\lambda + \sum_{j_k} |\gammakj\tod| / (\tau_\gamma\zeta\tod))$. Therefore, $\lambda_k\tod$ is updated conditional on all parameters excluding *all* $\wkj\tod$, $j_k = 1, 2, \dots, p_k$. Its distribution can be acquired by noting that $\gammakj\tod | \tau_\gamma, \zeta\tod, \lambda_k\tod \sim DE(\mu = 0, b = \tau_\gamma\zeta\tod / \lambda_k\tod)$ [@Park2008], where $DE$ stands for double exponential or Laplace distribution.
- for each $k = 1, 2, \dots, K$, $d = 1, 2, \dots, D$ and $j_k = 1, 2, \dots, K$, we use $\B_{kj_k}\tod$ to denote the $j_k^{th}$ slice of tensor $\B_k\tod$ along mode $k$, which is a $(K-1)$-mode tensor. Then, $\vect[\B_{kj_k}\tod] |\cdot \sim N(\bm \mu^*, \Sigma^*)$, for $\Sigma^* = \big(\Sigma_\pi ^{-1} + \big(\sum_{i = 1}^N \Psi_i\Psi_i^T \big) / \tau^2\big)^{-1}$, and $\bm \mu^* = \Sigma^* \big(\Sigma_\pi^{-1}\bm \mu_\pi + \big(\sum_{i = 1}^N \Psi_i R_{i,\Psi}\big) / \tau^2\big)$, where
- $\Sigma_\pi$ is a diagonal matrix of dimension $\big(\prod_{k = 1}^K p_k\big)/p_k$ with repeated entry $\sigma^2_k\zeta\tod$,
- $\bm \mu_\pi$ is a constant vector of length $\big(\prod_{k = 1}^K p_k\big)/p_k$ with entry $\gammakj\tod$,
- $\Psi_i = \vect[\B_1\tod {\circ}\dots {\circ}\B_{k-1}\tod {\circ}\B_{k+1}\tod \dots \B_K\tod {\circ}\tensor_i]$, and
- $R_{i,\Psi} = Y_i - \alpha - \bm C_i^T\delta - \langle \tensor_i, \sum_{r \neq d} \B_1^{(r)} {\circ}\B_2^{(r)} {\circ}\dots {\circ}\B_K^{(r)} \rangle_F$ is the residual excluding component $d$ of the coefficient tensor.
- for each $d = 1, 2, \dots, D$, we update $\zeta\tod$ from its full conditional, and then ensure that $\bm \zeta$ sums to 1, by dividing all its entries with $\sum_{d = 1}^D\zeta\tod$. The $\zeta\tod$ update is from $\zeta\tod |\cdot \sim giG(p^*, a^*, b^*)$, where $p^* = \alpha/D - K(\prod p_k + \sum p_k) / 2$, $a^* = 0$, and $b^* = \sum_{k,\subss} (\betakj\tod - \gammakj\tod)^2 / \sigma^2_k + \sum_{k,j_k} \gammakj^2 / (\tau_\gamma \wkj\tod)$.
Additional simulation results {#app_sec:more_sims}
=============================
Comparing [[Softer]{}]{} and [PARAFAC]{} in identifying important entries of tensor predictor {#app_sec:significance_disagree}
---------------------------------------------------------------------------------------------
In \[sec:simulations\] we presented an evaluation of the relative performance of [[Softer]{}]{} and hard [PARAFAC]{} in identifying important entries. There, we shows that [[Softer]{}]{} has significant lower FPR indicating that the two methods systematically disagree in the entries of the tensor predictor they identify as important. In order to study their disagreement, for each entry of the tensor predictor we calculate the percentage of data sets for which [[Softer]{}]{} or hard [PARAFAC]{} identifies the entry as important while the other does not. We plot the results in Figure \[app\_fig:disagreement\] as a function of the entry’s true coefficient. We see that the entries that [[Softer]{}]{} identifies as important and hard [PARAFAC]{} does not happen uniformly over the entries’ true coefficient. In contrast, when hard [PARAFAC]{} identifies an entry as important and [[Softer]{}]{} does not, it is more likely that the coefficient of this entry will be in reality small or zero.
When further investigating this feature of [PARAFAC]{}, we identified that the entries that it identifies as significant in disagreement to [[Softer]{}]{} are most often the ones that attribute to the coefficient tensor’s block structure. This is evident in Figure \[app\_fig:parafac\_disagree\] where we see that the entries with high identification by [PARAFAC]{} in contrast to [[Softer]{}]{} are the ones at the boundary of the truly non-zero entries.
Simulation results with alternative coefficient tensors {#app_subsec:sims_altern}
-------------------------------------------------------
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_true_squares_small.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_true_feet2_small.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_true_dog2_small.pdf "fig:"){width="\textwidth"}
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_s15_squares_mean.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_s15_feet2_mean.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_s15_dog2_mean.pdf "fig:"){width="\textwidth"}
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_p2_squares_mean.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_p2_feet2_mean.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_p2_dog2_mean.pdf "fig:"){width="\textwidth"}
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_l2_squares_mean.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_l2_feet2_mean.pdf "fig:"){width="\textwidth"}\
![Simulation results for alternative coefficient matrices. True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{} and the penalized estimator for the Lasso. Note that the color scale is the same for the true, [[Softer]{}]{} and hard [PARAFAC]{} approaches, but different for the Lasso because the order of estimated coefficients for the Lasso is much smaller.[]{data-label="app_fig:sim_pictures"}](f_sim_l2_dog2_mean.pdf "fig:"){width="\textwidth"}
For a tensor predictor of dimension $32\times 32$ we also considered three alternative coefficient matrices. The constant squares represent a rank-3, sparse scenario. Both the hard [PARAFAC]{} and the Lasso are expected to perform well in this situation, and we are interested in investigating [[Softer]{}]{}’s performance when softening is not necessary. The varying feet and dog scenarios are scenarios similarly to the dog and feet of the main text but for non-zero entries varying between 0.5 and 1.
[[Softer]{}]{} [PARAFAC]{} Lasso
---------- ---------------- ---------- ---------------- ------------- ------------
constant Truly zero bias 0.0016 0.0014 **0.0013**
squares rMSE 0.016 **0.015** 0.016
coverage 99.2% 99.7% -
\[5pt\] Truly non-zero bias 0.0211 **0.017** 0.073
rMSE **0.06** 0.076 0.093
coverage **90.9%** 79.6% -
\[5pt\] Prediction MSE **0.79** 1.25 1.32
varying Truly zero bias 0.06 0.076 **0.014**
feet rMSE **0.123** 0.145 0.169
coverage 96.4% 90.7% –
\[5pt\] Truly non-zero bias **0.165** 0.205 0.569
rMSE **0.244** 0.279 0.661
coverage **87.7**% 73.8% –
\[5pt\] Prediction MSE **40.79** 55.19 194
varying Truly zero bias 0.071 0.091 **0.013**
dog rMSE 0.159 0.176 **0.152**
coverage 98.3% 92.7 –
\[5pt\] Truly non-zero bias **0.263** 0.321 0.554
rMSE **0.358** 0.398 0.644
coverage **81.2** 63.2 –
\[5pt\] Prediction MSE **67.6** 85.3 159.2
: Average bias, root mean squared error, frequentist coverage of 95% credible intervals among truly zero and truly non-zero coefficient entries, and predictive mean squared error for [[Softer]{}]{}, hard [PARAFAC]{} and Lasso for the simulation scenario with tensor predictor of dimensions $32 \times 32$ and sample size $n = 400$. Bold text is used for the approach performing best in each scenario and for each metric.[]{data-label="app_tab:sims_n400"}
Figure \[app\_fig:sim\_pictures\] shows the true and average estimated coefficient matrices in these additional simulations. Even though we plot the true, [[Softer]{}]{}, and hard [PARAFAC]{} matrices using a common scale, we plot the expected coefficients employing Lasso using a different color scale. That is because, we want to show that Lasso gets the correct structure, on average, but largely underestimates coefficients due to the assumption of sparsity. Further, \[app\_tab:sims\_n400\] reports the average absolute bias, root mean squared error and 95% coverage of the truly zero, and truly non-zero coefficients, and the prediction mean squared error. When the true underlying hard [PARAFAC]{} structure is correct, [[Softer]{}]{} is able to revert back to its hard version, as is evident by the simulation results for the constant squares coefficient matrix. Further, [[Softer]{}]{} performs better than the hard [PARAFAC]{} for the varying feet and varying dog scenarios. In all three scenarios, [[Softer]{}]{} has the best out-of-sample predictive ability.
Simulation results for $32 \times 32$ tensor predictor and sample size $n = 200$ {#app_sec:sims_n200}
--------------------------------------------------------------------------------
Simulation results in this section represent a subset of the scenarios (dog, feet, diagonal) in \[subsec:sims\_400\] but for sample size $n = 200$. The general conclusions from \[subsec:sims\_400\] remain even when considering a smaller sample size. Figure \[app\_fig:sims\_n200\] shows a plot similar to the one in Figure \[fig:sim\_pictures\] including the true coefficient matrices and average posterior mean or penalized estimate across data sets. Again, the color scale of the results is the same for [[Softer]{}]{} and hard [PARAFAC]{}, but is different for the Lasso. Using different scales facilitates illustration of the underlying structure the methods estimate, even though different methods estimate different magnitude of coefficients. For example, Lasso estimates the feet structure, but non-zero coefficients are greatly underestimated around 0.1 (instead of 1). In contrast, in the truly sparse, diagonal scenario, Lasso estimates non-zero coefficients at about 0.8 whereas [[Softer]{}]{} estimates them to be close to 0.3, and hard [PARAFAC]{} near 0.06.
One of the main conclusions is that [[Softer]{}]{} deviates less from hard [PARAFAC]{} when the $n-p$ ratio is small, which is evident by the more rectangular structure in the mean coefficient matrix for the dog scenario, and the stronger shrinkage of the non-zero coefficients in the diagonal scenario. Further, \[app\_tab:sims\_n200\] show the average absolute bias and root mean squared error for estimating the coefficient matrix entries, and the prediction mean squared error.
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_true_feet_small.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_true_dog_small.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_true_diagonal_small.pdf "fig:"){width="\textwidth"}
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_s14_feet_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_s14_dog_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_s14_diagonal_mean.pdf "fig:"){width="\textwidth"}
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_p1_feet_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_p1_dog_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_p1_diagonal_mean.pdf "fig:"){width="\textwidth"}
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_l1_feet_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_l1_dog_mean.pdf "fig:"){width="\textwidth"}\
![True coefficient matrix and average across simulated data sets of the coefficient matrix posterior mean for [[Softer]{}]{} and hard [PARAFAC]{}. Note that the true matrices might be shown at a different color scale than the estimated ones.[]{data-label="app_fig:sims_n200"}](f_sim_l1_diagonal_mean.pdf "fig:"){width="\textwidth"}
------------------- ---------------- ---------------------- ----------------------- ----------------------
[[Softer]{}]{} [PARAFAC]{} Lasso
\[10pt\] feet Truly zero 0.06 (0.14) 0.065 (0.15) **0.01 (0.118)**
Truly non-zero **0.216** (0.332) 0.229 **(0.328)** 0.535 (0.585)
Prediction 95.4 (76.6, 111.1) **94 (78.8, 107.5)** 312 (299, 323)
\[10pt\] dog Truly zero 0.042 (0.155) 0.087 (0.164) **0.017 (0.155)**
Truly non-zero **0.171** (0.264) 0.174 **(0.254)** 0.248 (0.305)
Prediction 110.6 (99.6, 121.1) **101.2 (94, 107.8)** 177.3 (170.9, 183.4)
\[10pt\] diagonal Truly zero 0.005 (0.054) 0.003 (0.038) **0.002 (0.02)**
Truly non-zero 0.753 (0.773) 0.945 (0.948) **0.149 (0.181)**
Prediction 22.76 (21.04, 24.29) 31.08 (30.1, 31.78) **2.00 (1.57 2.32)**
------------------- ---------------- ---------------------- ----------------------- ----------------------
: Mean bias and rMSE among truly zero and truly non-zero coefficient entries (presented as bias (rMSE)), and average and IQR of the predictive mean squared error (presented as average (IQR)) for tensor predictor of dimensions $32 \times 32$ and $n = 200$. Bold text is used for the approach minimizing these quantities in each scenario.[]{data-label="app_tab:sims_n200"}
Simulation results for alternative rank of hard [PARAFAC]{} {#app_subsec:D_7}
-----------------------------------------------------------
[[Softer]{}]{} [PARAFAC]{}
---------- ---------------- ---------- ---------------- -------------
squares Truly zero bias 0.003 0.005
rMSE 0.034 0.049
coverage 99.5% 98.6%
\[5pt\] Truly non-zero bias 0.085 0.104
rMSE 0.111 0.146
coverage 79.6% 70.5%
\[5pt\] Prediction MSE 5.17 8.76
feet Truly zero bias 0.035 0.041
rMSE 0.092 0.104
coverage 97.3% 96.7%
\[5pt\] Truly non-zero bias 0.112 0.128
rMSE 0.181 0.199
coverage 89.2% 85.2%
\[5pt\] Prediction MSE 30.69 37.15
dog Truly zero bias 0.079 0.078
rMSE 0.138 0.138
coverage 92.8% 94.7%
\[5pt\] Truly non-zero bias 0.084 0.095
rMSE 0.157 0.166
coverage 92.4% 90.3%
\[5pt\] Prediction MSE 32.49 36.68
diagonal Truly zero bias 0.002 0.005
rMSE 0.02 0.06
coverage 100% 99.9%
\[5pt\] Truly non-zero bias 0.113 0.852
rMSE 0.128 0.861
coverage 93\. 3% 4.1%
\[5pt\] Prediction MSE 1.43 28.09
: Simulation results for [PARAFAC]{} rank $D = 7$.[]{data-label="tab:sims_n400_D7"}
In order to investigate the reliance of hard [PARAFAC]{} and [[Softer]{}]{} on the rank of the [PARAFAC]{} approximation, we considered the simulation scenarios of \[sec:simulations\] for $D = 7$. Results are shown in \[tab:sims\_n400\_D7\].
Comparing \[tab:sims\_n400\_D7\] to the results in \[sec:simulations\] it is evident that the performance of [[Softer]{}]{} is remarkably unaltered when $D = 3$ or $D = 7$. Perhaps the only difference is in the dog simulations where bias and mean squared error for the truly zero coefficients is slightly increased when $D = 7$. This indicates that [[Softer]{}]{} is robust to the specification of the rank of the underlying hard [PARAFAC]{}.
In contrast, the hard [PARAFAC]{} approach shows some improvements in performance when $D=7$ compared to $D = 3$. This is evident when examining the bias and rMSE for the coefficient entries in the feet, dog, and diagonal scenarios. Specifically, the hard [PARAFAC]{} shows up to 15% decreases in absolute bias and up to 10% in rMSE. When examining the mean estimated coefficient matrices (not shown) we see the improvements in estimation are in picking up the toes (in the feet scenario) and the eyes (in the dog scenario). However, the improvements in performance are quite minor. We suspect that the reason is that the decrease in singular values of the true coefficient matrices is slow after the first three, indicating that adding a few additional components does not drive much of the approximation. Relatedly, it is likely that the Dirichlet prior on $\bm \zeta$ in [@Guhaniyogi2017bayesian], along with a prior on Dirichlet parameter $\alpha$, effectively reduces the approximation rank to values smaller than 7.
Despite the improvements in performance for the hard [PARAFAC]{} when the rank is increased, [[Softer]{}]{} with either rank ($D = 3$ or 7) outperforms the hard [PARAFAC]{}. We suspect that the reason is that [[Softer]{}]{} allows for unstructured deviations from the underlying [PARAFAC]{} with $D = 3$, compared to structured increases in rank like the ones in the hard [PARAFAC]{}.
Additional information on our brain connectomics study {#app_sec:additional_application}
======================================================
Outcome information for our brain connectomics study
----------------------------------------------------
\[app\_tab:app\_outcomes\] shows the full list of outcomes we considered in our analysis. Both binary and continuous outcomes are considered. Information includes the category of trait. C/B represents continuous and binary outcomes.
[ll L[5.5cm]{} ccc]{}
\
Category & Name & Description & \#obs & Type & Mean (SD)\
Cognition & ReadEng\_AgeAdj & Age adjusted reading score & 1,065 & C & 107.1 (14.8)\
& PicVocab\_AgeAdj & Age adjusted vocabulary comprehension & 1,065 & C & 109.4 (15.2)\
& VSPLOT\_TC & Spatial Orientation (Variable Short Penn Line Orientation Test) & 1,062 & C & 15 (4.44)\
Motor & Endurance\_AgeAdj & Age Adjusted Endurance & 1,063 & C & 108.1 (13.9)\
& Strength\_AgeAdj & Age Adjusted Strength & 1,064 & C & 103.6 (20.1)\
Substance Use & Alc\_12\_Drinks\_Per\_Day & Drinks per day & 1,010 & C & 2.3 (1.57)\
& Alc\_Hvy\_Frq\_5plus & Frequency of $5+$ drinks & 1,011 & C & 3 (1.44)\
& TB\_DSM\_Difficulty\_Quitting & Tobacco difficulty quitting & 280 & B & 74.6%\
& Mj\_Ab\_Dep & Marijuana Dependence & 1,064 & B & 9.3%\
Psychiatric and & ASR\_Intn\_T & Achenbach Adult Self-Report (Internalizing Symptoms) & 1,062 & C & 48.5 (10.7)\
Life function & ASR\_Oth\_Raw & Achenbach Adult Self-Report (Other problems) & 1,062 & C & 9.1 (4.6)\
& Depressive\_Ep & Has the participant experienced a diagnosed DSMIV major depressive episode over his/her lifetime & 1,035 & B & 9.2%\
Emotion & AngHostil\_Unadj & NIH Toolbox Anger and Affect Survey (Attitudes of Hostility) & 1,064 & C & 50.5 (8.58)\
Personality & NEOFAC Agreeableness & “Big Five” trait: Agreeableness & 1,063 & C & 32 (4.93)\
Health & BMI & Body Mass Index & 1,064 & C & 26.4 (5.1)\
Additional Results {#app_sec:application_additional_results}
------------------
\[app\_fig:results\_all\] shows predictive power of (1) symmetric [[Softer]{}]{} with $D = 3$ and (2) $D=6$, (3) standard [[Softer]{}]{} ignoring symmetry with $D = 6$, (4) hard [PARAFAC]{} and (5) Lasso. Results from (2), (4), and (5) are the same as those shown in \[fig:app\_predict\]. Comparing results from (1) and (2) we see that increasing the rank from $D = 3$ to $D = 6$ improved predictions for a subset of outcomes. Ignoring the symmetry of the tensor predictor performed sometimes better and sometimes worse than accounting for symmetry directly into [[Softer]{}]{}, showing that the two approaches perform comparably for prediction (comparing (2) and (3)).
The gains of accounting for symmetry are evident when examining the approaches’ performance in identifying important entries. In fact, when symmetry is not accounted for, standard [[Softer]{}]{} did not identify any important connections. This indicates that accounting for symmetry leads to a reduction in the number of parameters, increased efficiency, and identification of important entries. When comparing the identification of important entries between symmetric [[Softer]{}]{} with rank 3 and 6, we see that the connections for strength and VSPLOT in \[tab:app\_connections\] are also identified by [[Softer]{}]{} with $D=3$. However, symmetric [[Softer]{}]{} with $D = 3$ identifies a larger set of important connections of the precuneus for strength and endurance, and no important connections for predicting depression. These results indicate the interplay between increasing the number of parameters to be estimated when increasing the rank, with the increased flexibility that increasing the rank provides to estimation.
\
Symmetric and semi-symmetric soft tensor regression {#app_sec:symmetric_softer}
===================================================
[[Softer]{}]{} for symmetric 2-mode predictor
---------------------------------------------
We start again from model \[eq:tensor\_regression2\] with $ Y_i = \mu + \C_i^T\bm \delta + \langle \tensor_i, \B \rangle_F + \epsilon_i. $ However, now $X_i$ is an $R \times R$ symmetric matrix with ignorable diagonal elements. This means that we can think of $\B$ as a real symmetric matrix with ignorable diagonal elements.
### Eigenvalue decomposition of real symmetric matrices
We can still approximate $\B$ in the same way as in [PARAFAC]{} (SVD) by writing $\B = \sum_{d = 1}^D \gamma_1\tod \otimes \gamma_2\tod$ for some $D$ large enough and $\gamma_1\tod, \gamma_2\tod \in \mathbb{R}^R$. However, this would not enforce that $\B$ is symmetric, since $\B_{j_1j_2} = \sum_{d = 1}^D \gamma_{1j_1}\tod \gamma_{2j_2}\tod \neq \sum_{d = 1}^D \gamma_{1j_2}\tod \gamma_{2j_1}\tod = \B_{j_2j_1}$. This implies that the entries of $\B$ would only be identifiable up to $\B_{j_1j_2} + \B_{j_2j_1}$.
Since $\B$ is a real symmetric matrix, it is diagonalizable and it has an eigenvalue decomposition. Therefore, we can think of approximating $\B$ using $$\B = \sum_{d = 1}^D \xi\tod \gamma\tod \otimes \gamma\tod,
\label{app_eq:eigen_decomp}$$ for sufficiently large $D$, where $\gamma\tod \in \mathbb{R}^R$ and $\xi\tod \in \mathbb{R}$. Note that the vectors $\gamma\tod$ here resemble the ones in the [PARAFAC]{} decomposition, but they are the same across the two tensor modes (matrix rows and columns).
The main difference between using the eigenvalue-based approximation in \[app\_eq:eigen\_decomp\], compared to the [PARAFAC]{}-based approximation is the inclusion of the parameters $\xi\tod$. Here, $\xi\tod$ are necessary in order to have a eigenvalue decomposition employing vectors with real entries. In fact, excluding $\xi\tod$ from \[app\_eq:eigen\_decomp\] can only be used to approximate positive definite symmetric matrices. To see this, take vector $\bm v \in \mathbb{R}^R$. Then, $$\bm v^T \Big(\sum_{d=1}^D \gamma\tod \otimes \gamma\tod \Big) \bm v =
\bm v^T \Big(\sum_{d=1}^D \gamma\tod \gamma\tod{}^T \Big) \bm v =
\sum_{d=1}^D \bm v^T \gamma\tod (\bm v^T \gamma\tod)^T =
\sum_{d=1}^D (\bm v^T \gamma\tod)^2 \geq 0$$
### Soft eigenvalue-based tensor regression
In the case of tensor predictors without symmetries, [[Softer]{}]{} was built based on the [PARAFAC]{} (multimodal equivalent to SVD) approximation of the coefficient tensor. Instead, for symmetric matrices, [[Softer]{}]{} is based on the eigenvalue decomposition while still allowing for deviations in row (and column)-specific contributions. However, these deviations also have to account for the symmetric nature of the tensor predictor.
Write $\B = \sum_{d = 1}^D \xi\tod \B_1\tod {\circ}\B_2\tod$ similarly to \[eq:softB\], and assume prior distributions on all parameters as in \[subsec:bayesian\]. Note that the parameters $\gamma\tod$ are not forced to be of norm 1 (as in classic eigenvalue decomposition), and they can have any magnitude. This allows us to restrict the parameters $\xi\tod$ to be in $\{-1, 1\}$, and base shrinkage of unnecessary ranks on shrinkage of the vectors $\gamma\tod$. Therefore, we assume a Bernoulli$(0.5)$ distribution over $\{-1, 1\}$ for parameters $\xi\tod$.
However, even though the symmetry of the underlying decomposition is enforced based on $\gamma\tod$, we need to ensure that it is also enforced when writing $\B = \sum_{d = 1}^D \xi\tod \B_1\tod {\circ}\B_2\tod$ using $\B_1\tod, \B_2\tod$. Note that the [[Softer]{}]{} framework assumes that entries $\beta_{1,j_1j_2}\tod$ are centered around $\gamma_{j_1}\tod$, and similarly entries $\beta_{2, j_1j_2}\tod$ are centered around $\gamma_{j_2}\tod$. However, doing so does not necessarily lead to symmetric matrices $\B$ since $$\B_{j_1j_2} = \sum_{d = 1}^D \beta_{1,j_1j_2}\tod \beta_{2, j_1j_2}\tod \neq \sum_{d = 1}^D \beta_{1,j_2j_1}\tod \beta_{2, j_2j_1}\tod = \B_{j_2j_1}.$$
We enforce symmetry of $\B$ by focusing only on the lower-triangular part. [[Softer]{}]{} for symmetric matrix predictor specifies row $i$’s contributions to entries $\B_{ij}$, $\beta_{1,ij}\tod$, as centered around $\gamma_i\tod$ only for $i > j$. Then, for $j > i$ we set $\beta_{1,ij}\tod = \beta_{1,ji}\tod$. Similarly, column $i$’s contributions to entries $\B_{ji}$, $\beta_{2, ji}\tod$ are centered around $\gamma_i\tod$ only for $i < j$, and for $j > i$ we set $\beta_{2, ji}\tod = \beta_{2, ij}\tod$. An equivalent way to enforce symmetry on $\B$ is to allow all entries in $\B_1\tod$ to have the same form as in [[Softer]{}]{}, and force $\B_2\tod = (\B_1\tod)^T$.
### Note on implementation using RStan
Note that RStan cannot directly handle discrete parameters as $\xi\tod$. The most common approach to discrete parameters is to specify the likelihood integrating these parameters out. However, this approach is not easily applicable in our setting since $\xi\tod$ are entangled in the likelihood through their role in the coefficient matrix $\B$. For that reason, we take an alternative approach, and assume that $\xi\tod$ are continuous and specify a mixture of normals distribution on each of them: $\xi\tod \sim 0.5 N(-1, 0.001) + 0.5 N(1, 0.001)$. Since the parameters $\xi\tod$ are not directly of interest, and shrinkage of the contributions of component $d$ in a rank$-D$ decomposition is achieved through the prior on $\gamma\tod$, we expect that this approach will closely resemble results from a specification that defines $\xi\tod$ to be binary taking values in $\{-1, 1\}$ from a Bernoulli$(0.5)$ distribution.
[[Softer]{}]{} for semi-symmetric 3-mode tensor
-----------------------------------------------
In brain connectomics, and specifically in our study of features of brain connections and their relationship to traits, tensor predictors are often of dimensions $R \times R \times p$ and are semi-symmetric. Semi-symmetry means that the predictor $\tensor$ is symmetric along its first two modes and $\tensor_{j_1j_2j_3} = \tensor_{j_2j_1j_3}$. An example of such tensor includes $R$ brain regions along the first two modes and $p$ features of brain connection characteristics along its third mode. When these features are symmetric (feature of connection from region $i$ to region $j$ is the same as the feature of connection from region $j$ to region $i$), the tensor predictor is semi-symmetric. In such cases, the standard [[Softer]{}]{} approach could be applied, but entries of $\B$ would be identifiable only up to $\B_{j_1j_2j_3} + \B_{j_2j_1j_3}$. In order to account for the semi-symmetry in $\tensor$ we can enforce the same type of semi-symmetry in $\B$ by adopting a [PARAFAC]{}-eigenvalue decomposition hybrid.
Specifically, assume that $\B$ is a 3-mode semi-symmetric coefficient tensor corresponding to the semi-symmetric predictor $\tensor$. Then, for sufficiently large $D$, $\gamma\tod \in \mathbb{R}^R$ and $\rho\tod \in \mathbb{R}^p$, we can write $$\B = \sum_{d = 1}^D \gamma\tod \otimes \gamma\tod \otimes \rho \tod.
\label{app_eq:semi-parafac}$$ This leads to a natural approximation for $\B$ for some value $D$ potentially smaller than the true one. [[Softer]{}]{} for semi-symmetric tensor predictor builds on \[app\_eq:semi-parafac\] while allowing for deviations in the row-specific contributions along the three modes.
We achieve that by specifying $\B = \sum_{d = 1}^D \B_1\tod {\circ}\B_2\tod {\circ}\B_3\tod$ for $\B_k\tod$ are tensors of dimensions $R \times R \times p$. The structure and specification of $\B_k\tod$ are as in \[subsec:bayesian\] with small changes to account for the semi-symmetric structure in $\tensor$ and ensure that the estimated coefficient tensor is also semi-symmetric. Note that the $(j_1, j_2, j_3)$ entry of $\B$ is equal to $$\B_{j_1j_2j_3} = \sum_{d = 1}^D \beta_{1,j_1j_2j_3}\tod \beta_{2,j_1j_2j_3}\tod \beta_{3,j_1j_2j_3}\tod,$$ and we want $\B_{j_1j_2j_3} = \B_{j_2j_1j_3}$. Borrowing from the symmetric case, we allow all row-specific contributions along mode 1, $\beta_{1, j_1j_2j_3}\tod$, to vary around the corresponding entry in the decomposition \[app\_eq:semi-parafac\], $\gamma_{j_1}\tod$, and set $\B_{2,..j_3} = \B_{1, ..j_3}^T$. Further, we allow entries $\beta_{3,j_1j_2j_3}\tod$ to vary around $\rho_{j_3}\tod$ for $j_1 < j_2$, and set $\beta_{3,j_1j_2j_3}\tod = \beta_{3,j_2j_1j_3}\tod$ when $j_1 > j_2$. Doing so, ensures that $\B_{j_1j_2j_3} = \B_{j_2j_1j_3}$.
|
---
author:
- |
Matteo Poggi, Fabio Tosi, Konstantinos Batsos, \
Philippos Mordohai, and Stefano Mattoccia,
bibliography:
- 'ref.bib'
title: 'On the Synergies between Machine Learning and Stereo: a Survey'
---
\[[{width="1in" height="1.25in"}]{}\][Matteo Poggi]{} received his PhD degree in Computer Science and Engineering from University of Bologna 2018. Currently, he is a Post-doc researcher at Department of Computer Science and Engineering, University of Bologna. His research interests include deep learning for depth estimation and embedded computer vision. He is the author of more than 30 papers about these topics.
\[[{width="1in" height="1.25in"}]{}\][Fabio Tosi]{} earned his MS degree from the University of Bologna in 2017. Currently he is a PhD student in Computer Science and Engineering at the University of Bologna, working on deep learning for stereo and monocular depth estimation.
\[[{width="1in" height="1.25in"}]{}\][Konstantinos Batsos]{} earned his MS from Stevens Institute of Technology in 2011 and returned to pursue his PhD in 2016. His research interests include binocular and multi-view stereo, deep learning and real-time computer vision.
\[[{width="1in" height="1.25in"}]{}\][Philippos Mordohai]{} is an associate professor of Computer Science at Stevens Institute of Technology. He earned his PhD from the University of Southern California and held postdoctoral appointments at the University of North Carolina and the University of Pennsylvania. His research interests span 3D reconstruction from images and video, range data analysis, perceptual organization and active vision. He has served as area chair for CVPR, ICCV and ECCV and program co-chair of the International Conference on 3D Vision (3DV), 2019.
\[[{width="1in" height="1.25in"}]{}\][Stefano Mattoccia]{} received a Ph.D. degree in Computer Science Engineering from the University of Bologna in 2002. Currently he is an associate professor at the Department of Computer Science and Engineering of the University of Bologna. His research interest is mainly focused on computer vision, depth perception from images, deep learning and embedded computer vision.
|
---
author:
- 'C. Lester'
- 'S. Ramos'
- 'R. S. Perry'
- 'T. P. Croft'
- 'R. I. Bewley'
- 'T. Guidi'
- 'P. Manuel'
- 'D. D. Khalyavin'
- 'E. M. Forgan'
- 'S. M. Hayden'
title: 'Field tunable spin density wave phases in Sr$_3$Ru$_2$O$_7$'
---
**The conduction electrons in a metal experience competing interactions with each other and the atomic nuclei. This competition can lead to many types of magnetic order in metals [@White2007_Whit]. For example, in chromium [@Fawcett1988_Fawc] the electrons order to form a spin-density-wave (SDW) antiferromagnetic state. A magnetic field may be used to perturb or tune materials with delicately balanced electronic interactions. Here we show that the application of a magnetic field can induce SDW magnetic order in a metal, where none exists in the absence of the field. We use magnetic neutron scattering to show that the application of a large ($\approx 8$ T) magnetic field to the metamagnetic perovskite metal Sr$_3$Ru$_2$O$_7$ (refs ) can be used to tune the material through two magnetically-ordered SDW states. The ordered states exist over relatively small ranges in field ($\lesssim 0.4$ T) suggesting that their origin is due to a new mechanism related to the electronic fine structure near the Fermi energy, possibly combined with the stabilising effect of magnetic fluctuations [@Berridge2009_BGGS; @Berridge2010_BGSG]. The magnetic field direction is shown to control the SDW domain populations which naturally explains the strong resistivity anisotropy or “electronic nematic” behaviour observed [@Grigera2004_GGBW; @Borzi2007_BGFP] in this material.**
An itinerant metamagnet [@Wohlfarth1962_WoRh; @Shimizu1982_Shim] is a metal which undergoes a sudden change from a low- to high- magnetisation state as a function of magnetic field. We investigated the layered perovskite Sr$_3$Ru$_2$O$_7$ (Fig. \[fig:WISH\_Tdep\]a). In this material, conduction takes place in the RuO$_2$ bilayers. Under a magnetic field applied parallel to the $\mathbf{c}$-axis and temperatures below $T \approx 1$ K, Sr$_3$Ru$_2$O$_7$ shows a rapid increase of magnetisation [@Rost2009_RPMM] (See Supplementary Fig. 2c) from 0.2 to 0.35 $\mu_B$ Ru$^{-1}$ over a field range of about 1 T near the metamagnetic field, $B_c \approx 7.95$ T. The metamagnetic behaviour is believed to be caused [@Wohlfarth1962_WoRh; @Shimizu1982_Shim; @Millis2002_MSLG; @Binz2004_BiSi] by proximity to ferromagnetism and the band structure having a local minimum in the density of states at the Fermi energy ($\varepsilon_F$) and/or a maximum near $\varepsilon_F$. Such features may result from a van-Hove singularity near $\varepsilon_F$, as observed [@Tamai2008_TAMM] by angle-resolved photo-emission spectroscopy.
A unique feature of Sr$_3$Ru$_2$O$_7$ is the observation of an unusual phase (denoted as “A”, see Fig. \[fig:WISH\_phase\_diagram\]a) near $B_c$ for $\mathbf{B} \parallel \mathbf{c}$ and for $T \lesssim 1$ K. The A-phase is a region of higher resistivity (see Fig. \[fig:LET\_resistivity\]b) whose boundaries can be identified from anomalies in a.c. susceptibility [@Grigera2004_GGBW], resistivity [@Grigera2004_GGBW], NMR (ref. ) and magnetostriction [@Stingl2011_SPMG]. Tilting the field $\mathbf{B}$ away from $\mathbf{c}$ to give a component along $\mathbf{a}$ or $\mathbf{b}$ induces a large anisotropy (“electron nematic” behaviour) in the in-plane resistivity both in the A-phase [@Borzi2007_BGFP] and the adjacent region [@Bruin2013_BBGR] in the $B-T$ plane. For example, a $\mathbf{B}$ component along $\mathbf{a}$ causes $\mathbf{b}$ to become the easy direction for current flow (see Fig. \[fig:LET\_resistivity\]a-c).
Motivated by previous reports [@Capogna2003_CFHW; @Ramos2008_RFBH] of strong low-energy spin fluctuations, we searched for static SDW order with higher energy resolution and lower temperatures. In SDW antiferromagnets the ordered moment is modulated in space with a wavevector $\mathbf{q}_\mathrm{SDW}$. This results in satellite peaks at reciprocal space positions $\mathbf{Q}=\boldsymbol{\tau}+\mathbf{q}_{\mathrm{SDW}}$, where $\boldsymbol{\tau}$ is a reciprocal lattice point (including $\boldsymbol{\tau}=0$) of the crystal structure. Fig. \[fig:WISH\_Tdep\]c shows Bragg scans along $\mathbf{Q}=(h,0,0)$ for a magnetic field $B_c$=7.95 T and a series of temperatures in the range $0.05 < T < 1.2$ K which traverse the A-phase (see Fig. \[fig:WISH\_phase\_diagram\]a). Below $T \approx 1.0$ K a magnetic Bragg peak develops at $\mathbf{q}_{\mathrm{SDW}}^{A}=(0.233 \pm 0.002,0,0)$. Scans along other directions parallel to $\mathbf{b}^{\star}$ and $\mathbf{c}^{\star}$ (see Supplementary Fig. 4) show that the peak is sharp in all three directions indicating 3D magnetic ordering with in-plane correlation lengths greater than 350 Å. Energy-dependent scans through the ordering position (see Supplementary Fig. 3) show that the peak is resolution limited in energy. This implies that the inverse lifetime of any magnetic fluctuations is less than $\tau^{-1}=$4 $\mu$eV$\approx$40 mK$\approx$1 GHz. From the intensity of the Bragg peak we estimate the magnitude of the ordered moment (for $T=50$ mK and $B=7.95$ T) to be $\langle m_q \rangle =0.10 \pm 0.02$ $\mu_B$ Ru$^{-1}$ assuming the structure in Fig. \[fig:LET\_spin\_structure\]a. Fig. \[fig:WISH\_phase\_diagram\]b shows the intensity of the $\mathbf{q}_{SDW}^{A}$ Bragg peak measured as a function of magnetic field. We find that the onset of the Bragg peak coincides with the boundaries of the A-phase indicating that it is associated with SDW order.
The susceptibility and resistivity (see Fig. \[fig:LET\_resistivity\]b) of Sr$_3$Ru$_2$O$_7$ show anomalous behaviour for fields above the A-phase boundary [@Borzi2007_BGFP; @Bruin2013_BBGR; @Grigera2004_GGBW]. We therefore also searched for SDW order in this region. For fields greater than the upper boundary of the A-phase, we observe (see Fig. \[fig:WISH\_Tdep\]d) an incommensurate peak at a different wavevector $\mathbf{q}_{\mathrm{SDW}}^{B}=(0.218 \pm 0.002,0,0)$ to that observed in the A-phase. We denote this second SDW-ordered region the “B-phase”.
The temperature and field dependencies of the $\mathbf{q}_{\mathrm{SDW}}^{A}$ and $\mathbf{q}_{\mathrm{SDW}}^{B}$ Bragg peak intensities are shown in Fig. \[fig:WISH\_phase\_diagram\]b,c. For the $\mathbf{q}_{\mathrm{SDW}}^{A}$ Bragg peak, we find that the fields and temperatures at which the magnetic order disappears coincide closely with the boundaries of the A-phase determined from a.c. susceptibility and resistivity [@Grigera2004_GGBW]. The boundaries of the B-phase for $\mathbf{B} \parallel \mathbf{c}$ are less well characterised. Borzi *et al*. [@Borzi2007_BGFP] observe a high field tail to the resistivity anomaly for $8.1 \lesssim B \lesssim 8.5$ T and $T=50$ mK, which defines the width of the B-phase in field. For a small tilt of the magnetic field away from the $c$-axis, Bruin *et al*. [@Bruin2013_BBGR] identify a region of anisotropic resistance which persists up to 0.4 K and appears to correspond to the B-phase. The SDW modulation of the ordered moment \[$\mathbf{m(r)}$\] results in satellite peaks. For the bct lattice of Sr$_3$Ru$_2$O$_7$, we expect SDW peaks around the $\boldsymbol{\tau}=$(0,0,0), (1,0,1), (0,1,1) and (1,1,0) reciprocal lattice points (Fig. \[fig:LET\_spin\_structure\]b). We observe satellite peaks along $(h,0,1)$, but not along $(0,k,1)$. The simplest structure consistent with this observation is the linear transverse SDW shown in Fig. \[fig:LET\_spin\_structure\]a. Other structures such as a cycloid [@Berridge2009_BGGS; @Berridge2010_BGSG] with $\mathbf{m(r)}$ in the $ab$ plane or a modulation of the moment $\mathbf{m(r)}$ parallel to $\mathbf{c}$ would give satellite peaks along $(0,k,1)$ with similar intensity to those along $(h,0,1)$.
One of the most fascinating properties of the A-phase is its electron nematic behaviour [@Borzi2007_BGFP]. For magnetic fields parallel to the $c$-axis, the A-phase is associated with a dramatic increase in the resistivity $\rho$ which is isotropic with respect to the direction of charge transport within the RuO$_2$ planes (see Fig. \[fig:LET\_resistivity\]b). By tilting the magnetic field away from the $c$-axis we can introduce an in-plane (IP) component of magnetic field $\mathbf{B}_{\mathrm{IP}}$ along the $a$-axis. Under these conditions, charge transport in the A-phase becomes strongly anisotropic. For current parallel to the in-plane field and the $a$-axis, the resistance anomaly in $\rho_a(B)$ associated with the A-phase remains (Fig. \[fig:LET\_resistivity\]b), while for currents perpendicular to $\mathbf{B}_{\mathrm{IP}}$, the anomaly in $\rho_b(B)$ is completely suppressed (Fig. \[fig:LET\_resistivity\]c).
We investigated the effect of similar tilted fields on the SDW order of the A-phase. For fields parallel to the $c$-axis (see Fig. \[fig:LET\_resistivity\]d) we observe two domains of the transverse SDW, one with a propagation vector $\mathbf{q}$ parallel to $\mathbf{a}^{\star}$ giving peaks at $(\pm 0.233,0,0)$ and the other with $\mathbf{q}$ parallel to $\mathbf{b}^{\star}$ giving peaks at $(0,\pm 0.233,0)$. When the field is tilted to give a component parallel to $\mathbf{a}$, the domain propagating along $\mathbf{b}$ is completely suppressed (Fig. \[fig:LET\_resistivity\]e). Thus, the applied magnetic field allows fine control of domain populations and the presence of the SDW domain modulated along $\mathbf{a}$ (Fig. \[fig:LET\_resistivity\]e) is associated with the resistance anomaly in $\rho_a$ (Fig. \[fig:LET\_resistivity\]c).
The existence of a SDW provides a natural explanation for the resistivity anomalies observed in Sr$_3$Ru$_2$O$_7$. SDW order in metals may increase the resistivity by gapping out the Fermi surface or by introducing additional scattering mechanisms due to the excitations associated with the magnetic order [@Fawcett1988_Fawc]. For Sr$_3$Ru$_2$O$_7$, the field-dependent resistivity shown in Fig. \[fig:LET\_resistivity\]b closely tracks the sum of the two magnetic order parameters (as measured by the SDW Bragg peak intensities) for the A and B phases (Fig. \[fig:WISH\_phase\_diagram\]b). The gapping of the Fermi surface will be closely related to the SDW order parameter, hence we believe the removal of electronic states from near the Fermi surface is the most likely cause of the resistivity anomalies.
There are other examples of inhomogeneous magnetically modulated states induced by a magnetic field. For example, the application of a magnetic field to certain insulating quantum magnets such as TlCuCl$_3$ causes a Bose-Einstein condensation of magnons and antiferromagnetic order [@Zapf2014_ZaJB]. In the heavy fermion superconductor CeCoIn$_5$ (ref. ), a “Q-phase” with spatially modulated superconducting and magnetic order parameters is created abutting $B_{c2}$. Since Sr$_3$Ru$_2$O$_7$ is neither a superconducting metal nor an insulator, the field-induced order must have a different mechanism to these two cases.
The formation of SDWs in metals (e.g. Cr) is usually described in terms of a Stoner theory including a wavevector-dependent-susceptibility $\chi_0(\mathbf{q})$ and an exchange interaction parameter $I$ (ref. ). SDW order occurs when the generalised Stoner criterion $\chi_0(\mathbf{q}) I \geq 1$ is satisfied. The ordering wavevector $\mathbf{q}_{\mathrm{SDW}}$ is determined from the peak in $\chi_0(\mathbf{q})$ and ultimately by Fermi surface nesting. In the vicinity of a metamagnetic transition, the Fermi surface changes rapidly with field as the Fermi energy of one of the spin species passes through a peak in the density of states. Such a rapid change may lead to a SDW phase that is only favoured over a narrow range in field [@Rice1970_Rice]. In Sr$_3$Ru$_2$O$_7$, two slightly different SDW states can be favoured. We note that the $\alpha_1$ and $\gamma_3$ sheets [@Tamai2008_TAMM] provide approximately the right nesting vectors to match $\mathbf{q}_{\mathrm{SDW}}^{A}$ and $\mathbf{q}_{\mathrm{SDW}}^{B}$ (refs ). It has recently been proposed [@Berridge2009_BGGS; @Berridge2010_BGSG] - based on a microscopic Stoner theory and a Landau-Ginzburg expansion - that transverse spin modulated states can be further stabilized by soft transverse magnetic fluctuations [@Wohlfarth1962_WoRh; @Shimizu1982_Shim].
The linear transverse nature of the SDW is not obviously predicted by the Landau-Ginzburg theory [@Berridge2009_BGGS; @Berridge2010_BGSG]. However, linearly polarised SDWs such as the one observed here (also in metals such as Cr, ref. ) may be favoured by additional contributions [@Overhauser1962_Over; @Walker1980_Walk] to the free energy. In the absence of strong anisotropy, antiferromagnets usually favour $\mathbf{m}_\mathbf{q} \perp \mathbf{B}$ and hence favour the structure for the A-phase shown in Fig. \[fig:LET\_spin\_structure\]a. This structure would naturally host single-$\mathbf{q}$ domains which would be enhanced or suppressed by tilting the field away from $\mathbf{c}$, leading to the population imbalance shown in Fig. \[fig:LET\_resistivity\]e. The imbalance naturally explains the observed “nematic” (anisotropic) transport properties of the SDW-A phase.
One can ask whether other examples of field-induced SDWs can be found in metals and what conditions are required for their existence. The relevant special properties of Sr$_3$Ru$_2$O$_7$ may include its two-dimensional electronic structure, nested Fermi surface, the existence of a van Hove singularity near the Fermi energy and concomitant metamagnetic transition, and its strongly enhanced magnetic susceptibility.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge helpful discussions with A.P. Mackenzie, R. Coldea, Y. Maeno, R. Evans, and R. M. Richardson. We are grateful to S. A. Grigera and A.P. Mackenzie for providing resistivity data from ref which is reproduced in Figs. \[fig:WISH\_phase\_diagram\] and \[fig:LET\_resistivity\]. Our work was supported by the UK EPSRC (Grant No. EP/J015423/1).
Additional Information {#additional-information .unnumbered}
======================
Correspondence and requests for materials should be addressed to S.M.H. (s.hayden@bristol.ac.uk).
[10]{} \[1\][`#1`]{} \[2\]\[\][[\#2](#2)]{}
White, R. M. *Quantum Theory of Magnetism* (Springer, 2007).
Fawcett, E. Spin-density-wave antiferromagnetism in chromium. *Rev. Mod. Phys.* **60**, 209–283 (1988).
Ikeda, S.-I., Maeno, Y., Nakatsuji, S., Kosaka, M. & Uwatoko, Y. Ground state in [Sr$_3$Ru$_2$O$_7$]{}: Fermi liquid close to a ferromagnetic instability. *Phys. Rev. B* **62**, R6089–R6092 (2000).
Perry, R. S. *et al.* Metamagnetism and critical fluctuations in high quality single crystals of the bilayer ruthenate [Sr$_3$Ru$_2$O$_7$]{}. *Phys. Rev. Lett.* **86**, 2661–2664 (2001).
Grigera, S. A. *et al.* Disorder-sensitive phase formation linked to metamagnetic quantum criticality. *Science* **306**, 1154–1157 (2004).
Borzi, R. A. *et al.* Formation of a nematic fluid at high fields in [Sr$_3$Ru$_2$O$_7$]{}. *Science* **315**, 214–217 (2007).
Rost, A. W., Perry, R. S., Mercure, J. F., Mackenzie, A. P. & Grigera, S. A. Entropy landscape of phase formation associated with quantum criticality in [Sr$_3$Ru$_2$O$_7$]{}. *Science* **325**, 1360–1363 (2009).
Berridge, A. M., Green, A. G., Grigera, S. A. & Simons, B. D. Inhomogeneous magnetic phases: A fulde-ferrell-larkin-ovchinnikov-like phase in [Sr$_3$Ru$_2$O$_7$]{}. *Phys. Rev. Lett.* **102**, 136404 (2009).
Berridge, A. M., Grigera, S. A., Simons, B. D. & Green, A. G. Magnetic analog of the [Fulde-Ferrell-Larkin-Ovchinnikov]{} phase in [Sr$_{3}$Ru$_{2}$O$_{7}$]{}. *Phys. Rev. B* **81**, 054429 (2010).
Wohlfarth, E. P. & Rhodes, P. Collective electron metamagnetism. *Phil. Mag.* **7**, 1817–1824 (1962).
Shimizu, M. Itinerant electron matamagnetism. *J. Phys. (Paris)* **43**, 155 (1982).
Millis, A. J., Schofield, A. J., Lonzarich, G. G. & Grigera, S. A. Metamagnetic quantum criticality in metals. *Phys. Rev. Lett.* **88**, 217204 (2002).
Binz, B. & Sigrist, M. Metamagnetism of itinerant electrons in multi-layer ruthenates. *Europhys. Lett.* **65**, 816 (2004).
Tamai, A. *et al.* Fermi surface and van hove singularities in the itinerant metamagnet [Sr$_3$Ru$_2$O$_7$]{}. *Phys. Rev. Lett.* **101**, 026407 (2008).
Kitagawa, K. *et al.* Metamagnetic quantum criticality revealed by [${}^{17}$O-NMR]{} in the itinerant metamagnet [Sr$_{3}$Ru$_{2}$O$_{7}$]{}. *Phys. Rev. Lett.* **95**, 127001 (2005).
Stingl, C., Perry, R. S., Maeno, Y. & Gegenwart, P. Symmetry-breaking lattice distortion in [Sr$_3$Ru$_2$O$_7$]{}. *Phys. Rev. Lett.* **107**, 026404 (2011).
Bruin, J. A. N. *et al.* Study of the electronic nematic phase of [Sr$_3$Ru$_2$O$_7$]{} with precise control of the applied magnetic field vector. *Phys. Rev. B* **87**, 161106 (2013).
Capogna, L. *et al.* Observation of two-dimensional spin fluctuations in the bilayer ruthenate [Sr$_3$Ru$_2$O$_7$]{} by inelastic neutron scattering. *Phys. Rev. B* **67**, 012504 (2003).
Ramos, S. *et al.* Spin dynamics in [Sr$_3$Ru$_2$O$_7$]{} near the metamagnetic transition by inelastic neutron scattering. *Physica B* **403**, 1270–1272 (2008).
Zapf, V., Jaime, M. & Batista, C. D. Bose-einstein condensation in quantum magnets. *Rev. Mod. Phys.* **86**, 563–614 (2014).
Kenzelmann, M. *et al.* Coupled superconducting and magnetic order in [CeCoIn$_5$]{}. *Science* **321**, 1652–1654 (2008).
Rice, T. M. Band-structure effects in itinerant antiferromagnetism. *Phys. Rev. B* **2**, 3619–3630 (1970).
Singh, D. J. & Mazin, I. I. Electronic structure and magnetism of [Sr$_3$Ru$_2$O$_7$]{}. *Phys. Rev. B* **63**, 165101 (2001).
Overhauser, A. W. Spin density waves in an electron gas. *Phys. Rev.* **128**, 1437 (1962).
Walker, M. B. Phenomenological theory of the spin-density-wave state of chromium. *Phys. Rev. B* **22**, 1338–1347 (1980).
Shaked, H., Jorgensen, J., Chmaissem, O., Ikeda, S. & Maeno, Y. Neutron diffraction study of the structural distortions in [Sr$_3$Ru$_2$O$_7$]{}. *J. Solid State Chem.* **154**, 361–367 (2000).
{width="0.7\linewidth"}
{width="0.95\linewidth"}
{width="0.95\linewidth"}
{width="0.6\linewidth"}
|
---
abstract: |
Abstract
We investigate some processes of the associated production of a neutral top-pion $\Pi^{0}_{t}$ with a pair of fermions($e^{+}e^{-}\rightarrow f\overline{f}\Pi^{0}_{t}$) in the context of top-color-assisted technicolor(TC2) theory at future $e^{+}e^{-}$ colliders. The studies show that the largest cross sections of the processes $e^{+}e^{-}\rightarrow f'\overline{f'}\Pi^{0}_{t}(f'=u,d,c,s,\mu,\tau)$ could only reach the level of 0.01fb, we can hardly detect a neutral top-pion through these processes. For the processes $e^{+}e^{-}\rightarrow
e^{+}e^{-}\Pi^{0}_{t}$, $e^{+}e^{-}\rightarrow
t\overline{t}\Pi^{0}_{t}$ and $e^{+}e^{-}\rightarrow
b\overline{b}\Pi^{0}_{t}$, the cross sections of these processes are at the level of a few fb for the favorable parameters and a few tens, even hundreds, of neutral top-pion events can be produced at future $e^{+}e^{-}$ colliders each year through these processes. With the clean background of the flavor-changing $t\bar{c}$ channel, the top-pion events can possibly be detected at the planned high luminosity $e^{+}e^{-}$ colliders. Therefore, such neutral top-pion production processes provide a useful way to detect a neutral top-pion and test the TC2 model directly.
address: ' College of Physics and Information Engineering, Henan Normal University, Henan 453002, China'
author:
- 'Xuelei Wang , Yueling Yang, and Bingzhong Li'
title: 'Detecting the neutral top-pion at $e^{+}e^{-}$ colliders '
---
Introduction
=============
Essential elements of fundamental constituents of matter and their interactions have been discovered in the past three decades by operating $e^{+}e^{-}$ colliders. A coherent picture of the structure of matter has emerged, that is adequately described by the standard model(SM), in many of its facets at a level of very high accuracy. However, the SM does not provide a comprehensive theory of matter. Neither the fundamental parameters, masses and couplings, nor the symmetry pattern can be explained. Therefore, some new theories beyond the SM(new physics models) have been studied to solve the problems of the SM. On the other hand, there exist new particles in the new physics models, probing these particles at future high-energy colliders can provide a direct way to test these models.
The top-color-assisted technicolor model
----------------------------------------
The top quark is the heaviest particle yet experimentally discovered and its mass of 174 GeV [@cdf] is close to the electroweak symmetry breaking(EWSB) scale. Much theoretical work has been carried out in connection to the top quark and EWSB. The top-color-assisted technicolor (TC2)[@TC2] model, the top seesaw model [@seesaw] and the flavor universal TC2 model [@utc2] are three such examples. The TC2 model is a more realistic one, which generates the large top quark mass through a dynamical $<t\bar{t}>$ condensation and provides a possible dynamical mechanism for breaking electroweak symmetry. In the TC2 model, the new strong dynamics is assumed to be chiral-critically strong but spontaneously broken by TC at the scale $\backsim$ 1 TeV and the EWSB is driven mainly by TC interaction. The extended technicolor(ETC) interaction gives the contribution to all ordinary quark and lepton masses including a very small portion of the top quark mass: $m^{'}_{t}=\varepsilon m_{t}$ $(0.03\leq
\varepsilon \leq 0.1)$[@Burdman]. The top-color interaction also makes a small contribution to the EWSB and gives rise to the main part of the top quark mass: $(1-\varepsilon)m_{t}$. The TC2 model also predicts the existence of a CP-even scalar $(h^{0}_{t})$ called the top-Higgs and three CP-odd pseudo Goldstone bosons(PGB’s) called top-pions $(\Pi^{0}_{t}$,$\Pi^{\pm}_{t})$ in a few hundreds GeV region. The physical particle top-pions can be regarded as a typical feature of the TC2 model. Thus, the study of the possible signatures of top-pions and top-pion contribution to some processes at high- energy colliders can be regarded as a good method to test the TC2 model and further to probe the EWSB mechanism.
At the energy scale $\Lambda\backsim$ 1 TeV, the new strong dynamics is coupled preferentially to the third generation. The dynamics of a general TC2 model involves the following structure[@TC2; @Burdman]: $$\begin{aligned}
SU(3)_{1}\bigotimes SU(3)_{2}\bigotimes U(1)_{Y_{1}}\bigotimes
U(1)_{Y_{2}}& &\bigotimes SU(2)_{L}\\
\rightarrow SU(3)_{QCD}\bigotimes U(1)_{EM}\end{aligned}$$ where $SU(3)_{1}\bigotimes U(1)_{Y_{1}}(SU(3)_{2}\bigotimes
U(1)_{Y_{2}})$ couples preferentially to the third generation(the first and the second generations). The $U(1)_{Y_{i}}$ is just strongly rescaled versions of electroweak $U(1)_{Y}$. $SU(3)_{1}\bigotimes U(1)_{Y_{1}}$ is assumed strong enough to produce a large top condensate which is responsible for the main part of the top quark mass. The b-quark mass is an interesting issue, involving a combination of ETC effects and instanton effects in $SU(3)_{1}$. The instanton induced b-quark mass can then be estimated as [@Simmons]: $$\begin{aligned}
m_{b}^{*}\approx \frac{3km_{t}}{8\pi^{2}}\thicksim 6.6k ~~GeV\end{aligned}$$ where we generally expect $k\thicksim 1$ to $10^{-1}$ as in QCD. In the TC2 model, the top-color gauge bosons include the color-octet colorons $B^{A}_{\mu}$ and color-singlet extra $U(1)$ gauge boson $Z^{'}$. These gauge bosons have very large masses which can be up to several TeV. Such large masses will depress the contribution to the cross sections. So, in our calculation, we can neglect the contributions of the gauge bosons.
Search for the new particles at planned $e^{+}e^{-}$ colliders
--------------------------------------------------------------
The planned linear $e^{+}e^{-}$ colliders(LC) with energy in the range from a few hundred GeV up to several TeV are under intense studies around the world. These studies are being done at the Next Linear Collider(NLC)(USA)[@NLC], the Japan Linear Collider(JLC)(Japan)[@JLC] and the DESY TeV Energy Superconducting Linear Accelerator(TESLA)(Europe)[@TESLA]. One task of these high-energy $e^{+}e^{-}$ colliders is to search for Higgs particle in the SM or some new particles predicted in the models beyond the SM\[such as Higgs bosons $A^{0},H^{0},h^{0},H^{\pm}$ in the minimal supersymmetric standard model(MSSM) and PGB’s in the TC2 model\]. So, the study of some new particle production processes can provide a theoretical instruction to search for these particles experimentally.
As it is known, top-pions are the typical particles in the TC2 model. Some neutral top-pion production processes have been studied in Ref.[@wang]. On the other hand, Ref.[@cao] has studied a top-charm associated production process at LHC to probe the top-pion. The above studies provide us with some useful information to detect top-pion events and test the TC2 model. In this paper, we study the neutral top-pion production processes $e^{+}e^{-}\rightarrow f\bar{f}\Pi^{0}_{t}$ in the framework of the TC2 model, where $f$ represents $u,d,c,s,t,b$ quarks and $e,\mu,\tau$ leptons. Our results show that the cross sections of the processes $e^{+}e^{-}\rightarrow f'\bar{f'}\Pi^{0}_{t}(f'=u,d,c,s,\mu,\tau)$ are very small. The largest cross section could only reach an order of magnitude $0.01fb$. With such small cross sections, the neutral top-pion can be hardly detected via these processes. So, we pay attention to the processes $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}$, $e^{+}e^{-}\rightarrow t\bar{t}\Pi^{0}_{t}$ and $e^{+}e^{-}\rightarrow b\bar{b}\Pi^{0}_{t}$. These cross sections are about two orders or even three orders of magnitude larger than that of processes $e^{+}e^{-}\rightarrow f'\bar{f'}\Pi^{0}_{t}$. The study in this paper is a useful addition to the previous studies. Some similar processes in the context of the SM and MSSM have also been studied quite extensively in the LC[@LC], Tevatron and LHC[@LHC]. Reference[@Leibovich] has investigated the production of the neutral scalar with a pair of top quarks at the hadron collider. They find that the neutral scalar may be observed at the LHC via the process $e^{+}e^{-}\rightarrow t\bar{t}\phi$. The $q\bar{q}H$ production mode is extremely interesting for physicists because this production mode provides a direct way to measure the Yukawa couplings of the quarks with scalar particles, on the other hand, we can detect these scalar particles at LC, LHC and Tevatron II with high luminosity through these processes.
The calculation of the production cross sections of the processes $e^{+}e^{-}\rightarrow f\bar{f}\Pi^{0}_{t}$ is presented in Sec II.
Cross sections of these processes
=================================
It is noticeable that the TC2 model may have rich top-quark phenomenology since it treats the top quark differently from other quarks. The couplings of top-pions to three family fermions are nonuniversal and the top-pions have large Yukawa couplings to the third generation. The Yukawa interactions of top-pions to $t,b,c$ quarks can be written as[@TC2; @He] $$\begin{aligned}
\nonumber
\frac{m_{t}\tan\beta} {\upsilon_{\omega}}&
&[iK^{tt}_{UR}K^{tt^{*}}_{UL}\overline{t_{L}}t_{R}
\Pi^{0}_{t}+\sqrt{2}K^{tt}_{UR}K^{bb^{*}}_{DL}\overline{b_{L}}t_{R}\Pi^{-}_{t}\\
\nonumber &
&+iK^{tc}_{UR}K^{tt^{*}}_{UL}\overline{t_{L}}c_{R}\Pi^{0}_{t}+\sqrt{2}K^{tc}_{UR}
K^{bb^{*}}_{DL}\overline{b_{L}}c_{R}\Pi^{-}_{t}\\ &
&+i\frac{m^{*}_{b}}{m_{t}}\overline{b_{L}}b_{R}\Pi^{0}_{t}+h.c.]\end{aligned}$$ where $\tan\beta=\sqrt{(\frac{\upsilon_{\omega}}{\upsilon_{t}})-1}$, $\upsilon_{t}\thickapprox 60-100$ GeV is the top-pion decay constant and $\upsilon_{\omega}=246$ GeV is the electroweak symmetry-breaking scale. $m_{b}^{*}$ is the part of b-quark mass induced by instanton. $K^{tt}_{UL}, K^{bb}_{DL},
K^{tt}_{UR},K^{tc}_{UR}$ are the elements of the rotation matrices $K_{UL,R}$ and $K_{DL,R}$. The rotation matrices $K_{UL,R},$ and $K_{DL,R}$ are needed for diagonalizing the up- and down-quark mass matrices $M_{U}$ and $M_{D}$, i.e., $K_{UL}^{+}M_{U}K_{UR}=M^{dia}_{U}$ and $K_{DL}^{+}M_{D}K_{DR}=M^{dia}_{D}$, from which the Cabibbo-Kobayashi-Maskawa(CKM) matrix is defined as $V=K^{+}_{UL}K_{DL}$. The matrix elements are given as $$\begin{aligned}
K^{tt}_{UL}\backsimeq K^{bb}_{DL}\approx 1 \hspace{1.5cm}
K^{tt}_{UR}=1-\varepsilon\end{aligned}$$ Here, we take $\varepsilon$ as a free parameter changing from 0.03 to 0.1.
With $t\bar{t}\Pi^{0}_{t}$ coupling, the neutral top-pion $\Pi^{0}_{t}$ can couple to a pair of gauge bosons through the top quark triangle loops in an isospin violating way. Calculating the top quark triangle loops, we can explicitly obtain the couplings of $\Pi^{0}_{t}-\gamma-\gamma$ , $\Pi^{0}_{t}-\gamma-Z$ and $\Pi^{0}_{t}-Z-Z$ $$\begin{aligned}
\nonumber
& & \Pi^{0}_{t}-\gamma-\gamma :\\
& & iN_{c}\frac{8}{9\pi}
\frac{tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)\alpha_{e}
\varepsilon_{\mu\nu\rho\delta}p_{in}^{\rho}p_{out}^{\delta}C_{0}
\end{aligned}$$ $$\begin{aligned}
\nonumber
& & \Pi^{0}_{t}-\gamma-Z: \\ \nonumber
& &iN_{c}\frac{\alpha_e}{3\pi c_w s_w}
\frac{tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon_{\mu\nu\rho\delta}
p_{in}^{\rho}p_{out}^{\delta}\\
& & (1-\frac{8}{3}s_{w}^{2})C_{0}\end{aligned}$$ $$\begin{aligned}
\nonumber
& & \Pi^{0}_{t}-Z-Z: \\ \nonumber
& &iN_{c}\frac{\alpha_e}{8\pi c_{w}^{2} s_{w}^{2}}
\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon_{\mu\nu\rho\delta}
p_{in}^{\rho}p_{out}^{\delta}\\
& & \{[(1-\frac{8}{3}s_{w}^{2})^{2}-1]C_{0}-2C_{11}\}
\end{aligned}$$ where $N_{c}$ is the color index with $N_{c}=3$, $s_{w}=\sin\theta_{w}$, $c_{w}=\cos\theta_{w}$ ($\theta_{w}$ is the Weinberg angle), $C_{0}=C_{0}(-p_{in},p_{out},m_{t},m_{t},m_{t})$ and $C_{11}=
C_{11}(-p_{in},p_{out},m_{t},m_{t},m_{t})$ are standard three-point scalar integrals with $p_{in}$ and $p_{out}$ denoting the momenta of the incoming gauge boson and the outcoming top-pion, respectively.
With the couplings of $\Pi^0_t\gamma\gamma$, $\Pi^0_tZ\gamma$,$\Pi^0_tZZ$, the neutral top-pion can be produced via the processes $e^{+}e^{-}\rightarrow
f\bar{f}\Pi^{0}_{t}$. The Feynman diagrams of these processes are shown in Fig.1.
From the diagrams, we can see that the process $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}$ can take place through t-channel and s-channel as shown in Fig. 1(a),(b). The light quark pairs$(u\bar{u}, d\bar{d}, c\bar{c},s\bar{s})$ or lepton pairs $(\mu\bar{\mu}, \tau\bar{\tau})$ can only be produced via Fig.1(b). The heavy quark pair production processes $e^{+}e^{-}\rightarrow
t\bar{t}\Pi^{0}_{t}$ and $e^{+}e^{-}\rightarrow
b\bar{b}\Pi^{0}_{t}$ are shown in Fig.1(b),(c),(d),(e). $h^{0}_{t}$ shown in Fig.1(e) is a CP even particle in TC2 model, the couplings $h^0_tq\bar{q}$ and $Zh^0_t\Pi^0_t$ are given in Ref.[@lugongru].
The explicit expressions of the amplitudes for different diagrams can be directly written as: $$\begin{aligned}
M^{a}_{\gamma\gamma\Pi^{0}_{t}}&=&iN_{c}\alpha_{e}^{2}\frac{32}{9}
\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}(p_{2}-p_{4})_{\rho}p_{5\delta}
C_{0}\\
& & G(p_{2}-p_{4},0)G(p_{2}-p_{4}-p_{5},0)\\
& & \overline{u}_{e^{-}}(p_{4})\gamma_{\mu}u_{e^{-}}(p_{2})
\overline{v}_{e^{+}}(p_{1})\gamma_{\nu}v_{e^{+}}(p_{3})\\
M^{a}_{\gamma Z\Pi^{0}_{t}}&=&-iN_{c}\frac{2^{\frac{5}{4}}}{3}\frac{\alpha_{e}^{2}}
{c^{2}_{\omega}s^{2}_{\omega}}\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}(p_{2}-p_{4})_{\rho}p_{5\delta}\\
& &(1-\frac{8}{3}s^{2}_{\omega})C_{0}G(p_{2}-p_{4},0)G(p_{2}-p_{4}-p_{5},M_{Z})\\
& &\overline{u}_{e^{-}}(p_{4})\gamma_{\mu}u_{e^{-}}(p_{2})
\overline{v}_{e^{+}}(p_{1})\gamma_{\nu}(-\frac{1}{2}L+s^{2}_{\omega})
v_{e^{+}}(p_{3})\\
M^{a}_{Z\gamma\Pi^{0}_{t}}&=&-iN_{c}\frac{2^{\frac{5}{4}}}{3}\frac{\alpha_{e}^{2}}
{c^{2}_{\omega}s^{2}_{\omega}}\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}(p_{2}-p_{4})_{\rho}p_{5\delta}\\
& &(1-\frac{8}{3}s^{2}_{\omega})C_{0}G(p_{2}-p_{4},M_{Z})G(p_{2}-p_{4}-p_{5},0)\\
& &\overline{u}_{e^{-}}(p_{4})\gamma_{\mu}(-\frac{1}{2}L+s^{2}_{\omega})u_{e^{-}}
(p_{2})\overline{v}_{e^{+}}(p_{1})\gamma_{\nu}v_{e^{+}}(p_{3})\\
M^{a}_{ZZ\Pi^{0}_{t}}&=&iN_{c}\frac{\alpha_{e}^{2}}{2c^{4}_{\omega}s^{4}_{\omega}}
\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}
(p_{2}-p_{4})_{\rho}p_{5\delta}\\
& &\{[(1-\frac{8}{3}s^{2}_{\omega})^{2}-1]C_{0}-2C_{11}\}G(p_{2}-p_{4},M_{Z})\\
& &G(p_{2}-p_{4}-p_{5},M_{Z})
\overline{u}_{e^{-}}(p_{4})\gamma_{\mu}(-\frac{1}{2}L+s^{2}_{\omega})\\
& &u_{e^{-}}(p_{2})
\overline{v}_{e^{+}}(p_{1})\gamma_{\nu}(-\frac{1}{2}L+s^{2}_{\omega})v_{e^{+}}(p_{3})\\
M^{b}_{\gamma\gamma\Pi^{0}_{t}}&=&-iQ_{f}N_{c}\alpha_{e}^{2}\frac{32}{9}
\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}(p_{1}+p_{2})_{\rho}\\
& &p_{5\delta} C'_{0}
G(p_{1}+p_{2},0)G(p_{1}+p_{2}-p_{5},0)\\
& & \overline{v}_{e^{+}}(p_{1})\gamma_{\mu}u_{e^{-}}(p_{2})
\overline{u}(p_{4})\gamma_{\nu}v(p_{3})\\
M^{b}_{\gamma Z\Pi^{0}_{t}}&=&-iN_{c}\frac{2^{\frac{5}{4}}}{3}\frac{\alpha_{e}^{2}}
{c^{2}_{\omega}s^{2}_{\omega}}\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}(p_{1}+p_{2})_{\rho}\\
& &p_{5\delta}(1-\frac{8}{3}s^{2}_{\omega})C'_{0}G(p_{1}+p_{2}-p_{5},M_{Z})\\
& &G(p_{1}+p_{2},0)
\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}u_{e^{-}}(p_{2})\\
& & \overline{u}(p_{4})\gamma_{\nu}(aL+b)v(p_{3})\\
M^{b}_{Z\gamma\Pi^{0}_{t}}&=&iQ_{f}N_{c}\frac{2^{\frac{5}{4}}}{3}\frac{\alpha_{e}^{2}}
{c^{2}_{\omega}s^{2}_{\omega}}\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}(p_{1}+p_{2})_{\rho}\\
& &p_{5\delta}(1-\frac{8}{3}s^{2}_{\omega})C'_{0}G(p_{1}+p_{2}-p_{5},0)G(p_{1}+p_{2},\\
& & M_{Z})\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}
(-\frac{1}{2}L+s^{2}_{\omega})u_{e^{-}}(p_{2})\\
& &\overline{u}(p_{4})\gamma_{\nu}v(p_{3})\\
M^{b}_{ZZ\Pi^{0}_{t}}&=&iN_{c}\frac{\alpha_{e}^{2}}{2c^{4}_{\omega}s^{4}_{\omega}}
\frac{\tan\beta}{\upsilon_{w}}m_{t}^{2}(1-\varepsilon)
\varepsilon^{\mu\nu\rho\delta}
(p_{1}+p_{2})_{\rho}p_{5\delta}\\
& &\{[(1-\frac{8}{3}s^{2}_{\omega})^{2}-1]C'_{0}-2C'_{11}\}G(p_{1}+p_{2},M_{Z})\\
& &G(p_{1}+p_{2}-p_{5},M_{Z})
\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}(-\frac{1}{2}L+s^{2}_{\omega})\\
& &u_{e^{-}}(p_{2})
\overline{u}(p_{4})\gamma_{\nu}(aL+b)v(p_{3})\\
M^{c}_{\gamma}&=&4Q_{f}\pi\alpha_{e}
\frac{\tan\beta}{\upsilon_{\omega}}m^{*}_{q_{3}}
G(p_{4}+p_{5},m_{q_{3}})G(p_{1}+p_{2},0)\\
& &\overline{u}_{q_{3}}(p_{4})\gamma_{5}(\pslash_{4}+\pslash_{5}+m_{q_3})\gamma^{\mu}
v_{\bar{q}_{3}}(p_{3})\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}\\
& &u_{e^{-}}(p_{2})\\
M^{c}_{Z}&=&-4\pi\frac{\alpha_{e}}{s^{2}_{\omega}c^{2}_{\omega}}
\frac{\tan\beta}{\upsilon_{\omega}}m^{*}_{q_{3}}
G(p_{4}+p_{5},m_{q_{3}})G(p_{1}+p_{2},\\& &M_{Z})
\overline{u}_{q_{3}}(p_{4})\gamma_{5}(\pslash_{4}+\pslash_{5}+m_{q_{3}})\gamma^{\mu}
(aL+b)\\ & &v_{\bar{q}_{3}}(p_{3})
\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}(-\frac{1}{2}L+s^{2}_{\omega})
u_{e^{-}}(p_{2})\\
M^{d}_{\gamma}&=&-4Q_{f}\pi\alpha_{e}
\frac{\tan\beta}{\upsilon_{\omega}}m^{*}_{q_{3}}
G(p_{3}+p_{5},m_{q_{3}})\\
& &G(p_{1}+p_{2},0)\overline{u}_{q_{3}}(p_{4})\gamma_{5}(\pslash_{3}+\pslash_{5}+m_{q_{3}})
\\ & &\gamma^{\mu}v_{\bar{q}_{3}}(p_{3})
\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}u_{e^{-}}(p_{2})\\
M^{d}_{Z}&=&4\pi\frac{\alpha_{e}}{s^{2}_{\omega}c^{2}_{\omega}}
\frac{\tan\beta}{\upsilon_{\omega}}m^{*}_{q_{3}}
G(p_{3}+p_{5},m_{q_{3}})G(p_{1}+p_{2},\\
& &M_{Z})\overline{u}_{q_{3}}(p_{4})\gamma^{\mu}(aL+b)(\pslash_{3}+\pslash_{5}+m_{q_{3}})\\
& &\gamma_{5}v_{\bar{q}_{3}}(p_{3})\overline{v}_{e^{+}}(p_{1})\gamma_{\mu}
(-\frac{1}{2}L+s^{2}_{\omega})u_{e^{-}}(p_{2})\\
M^{e}&=&-2\pi\frac{\alpha_{e}}{c_{\omega}s_{\omega}}\frac{\tan\beta}{\upsilon_{\omega}}
m^{*}_{q_{3}} G(p_{3}+p_{4},m_{h})G(p_{1}+p_{2},\\
& &M_{Z})\overline{v}_{e^{+}}(p_{1}) (\pslash_{5}-\pslash_{3}-\pslash_{4})
u_{e^{-}}(p_{2})\\
& &\overline{u}_{q_{3}}(p_{4})v_{\bar{q}_{3}}(p_{3})
\end{aligned}$$ where $L=\frac{1}{2}(1-\gamma_{5})$, $G(p,m)=\frac{1}{p^{2}-m^{2}}$ denotes the propagator of the particle and $C_{0}=C_{0}(-p_{2}+p_{4},p_{5},m_{t},m_{t},m_{t})$, $C'_{0}=C_{0}(-p_{1}-p_{2},p_{5},m_{t},m_{t},m_{t})$. $m_{q_{3}}$ represents the masses of the third generation quarks and $m_{q_{3}}^{*}$ denotes $m_{t}^{*}$ and $ m_{b}^{*}$, $m_{t}^{*}=(1-\varepsilon)m_{t}$ is induced by topcolor interaction and $m_{b}^{*}$ is the b quark mass produced by instanton. The values of $Q_{f},a,b$ are taken as following:\
0.8pt 0.12in
$ f $ $Q_{f}$ $ a $ $ b$
----------------------- ---------------- ---------------- ------------------------------
up-quarks$(u,c,t)$ $\frac{2}{3}$ $\frac{1}{2}$ $-\frac{2}{3}s^{2}_{\omega}$
down-quarks$(d,s,b)$ $-\frac{1}{3}$ $-\frac{1}{2}$ $\frac{1}{3}s^{2}_{\omega}$
leptons$(e,\mu,\tau)$ $-1$ $-\frac{1}{2}$ $s^{2}_{\omega}$
The production amplitudes for different processes are: $$\begin{aligned}
M_{e^{+}e^{-}\Pi^{0}_{t}} & = &M^{a}+M^{b} \hspace{2.2cm}
(e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t})\\
M_{q_{3}\overline{q_{3}}\Pi^{0}_{t}}&=&M^{b}+M^{c}+M^{d}+M^{e}\hspace{0.3cm}
(e^{+}e^{-}\rightarrow q_{3}\overline{q_{3}}\Pi^{0}_{t})\\
M_{f'\overline{f'}\Pi^{0}_{t}}&=&M^{b}(f'=u,d,c,s,\mu , \tau)
\hspace{0.2cm} (e^{+}e^{-}\rightarrow f'\overline{f'}\Pi^{0}_{t})
\end{aligned}$$
With the above production amplitudes, we can obtain the production cross sections directly.
Numerical results and conclusions
=================================
In our calculation, we take $m_{e}=0$, $m_{\mu}=0.105$ GeV, $m_{\tau}=1.784$ GeV, $m_{u}=0.005$ GeV, $m_{d}=0.009$ GeV, $m_{c}=1.4$ GeV, $m_{s}=0.15$ GeV, $m_{t}=174$ GeV, $m_{b}=4.9$ GeV, $M_{Z}=91.187$ GeV, $v_{t}=60$ GeV and $s^{2}_{\omega}=0.23$. The electromagnetic fine-structure constant $\alpha_{e}$ at a certain energy scale is calculated from the simple QED one-loop evolution with the boundary value $\alpha_{e}=\frac{1}{137.04}$. There are three free parameters $\varepsilon,M_{\Pi},s $ in the cross sections. To see the influence of these parameters on the cross sections, we take the mass of the top-pion $M_{\Pi}$ to vary in the range of 150 GeV$\leq M_{\Pi}\leq$ 450 GeV and $\varepsilon=0.03,0.06,0.1$, respectively. Considering the center-of-mass energy $\sqrt{s}$ in the planned $e^{+}e^{-}$ linear colliders(for example,TESLA), we take $\sqrt{s}=800$ GeV and 1600 GeV, respectively.
For the process $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}$, we can see that there exists a t-channel resonance effect induced by the photon propagator in fig.1(a) and an s-channel resonance effect induced by the Z boson propagator in fig.1(b). So we should take into account the effect of the width of the $Z$ boson in the calculations, i.e. we should take the complex mass term $M_{Z}^{2}-iM_{Z}\Gamma_{Z}$ instead of the simple $Z$ boson mass term $M_{Z}^{2}$ in the $Z$ boson propagator. Here, we take $\Gamma_{Z}=2.49$ GeV. All the resonance effects will enhance the cross section significantly. We find that the contribution to the cross section of $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}$ arises mainly from Fig.1(a). The final numerical results of the cross section are summarized in Figs 2. The figure is the plots of the cross section as a function of $M_{\Pi}$. One can see that there is a peak in the plot when $M_{\Pi}$ is close to 350 GeV, which arises from the top quark triangle loop. The largest cross sections are 1.78 fb and 4.15fb, when we take $\varepsilon=0.03$ and $\sqrt{s}=800$ and 1600 GeV, respectively. The cross section increases with $\sqrt{s}$. With a luminosity of $100fb^{-1}/$yr, there are several tens or even hundreds of $\Pi^{0}_{t}$ events to be produced via the process $e^{+}e^{-}\rightarrow
e^{+}e^{-}\Pi^{0}_{t}$. For $M_{\Pi}\leq$ 350 GeV, the dominate decay channel of $\Pi^{0}_{t}$ is $\Pi^{0}_{t}\rightarrow
t\overline{c}$. As has been investigated in Ref. [@yue], the branching ratio Br$(\Pi^{0}_{t}\rightarrow
t\overline{c})$ can reach about 60%. Because there is no tree level flavor-changing neutral current in the SM, the background of $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}\rightarrow
e^{+}e^{-}t\bar{c}$ is very clean. Therefore, $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}\rightarrow
e^{+}e^{-}t\bar{c}$ is an ideal channel to detect the neutral top-pion with small top-pion mass. For $ M_{\Pi}\geq$ 350 GeV, $\Pi^{0}_{t}\rightarrow t\overline{t}$ is permitted and the total width of $\Pi^{0}_{t}$ increases significantly. The branching ratio of the decay channel $\Pi^{0}_{t}\rightarrow t\overline{t}$ is close to 100%, all other decay modes may be ignored. In this case, we should detect $\Pi^{0}_{t}$ through the $t\bar{t}$ channel.
The numerical results of the process $e^{+}e^{-}\rightarrow
t\overline{t}\Pi^{0}_{t}$ are shown in Fig.3. The contributions arise from Fig.1(b)(c)(d)(e). There is no resonance effect for this process, but the large $\Pi_t^0t\bar{t}$ coupling in Fig.1(c)(d) can enhance the cross section significantly. We can see that the cross section increases with $M_{\Pi}$ decreasing and $\sqrt{s}$ increasing. In the all parameter space we consider, the cross section is larger than 2.5fb for $\sqrt{s}=1600$ GeV.
The same plots as Fig.2 for the process $e^{+}e^{-}\rightarrow
b\bar{b}\Pi^{0}_{t}$ are shown in Fig.4. The contributions to the cross section come mainly from Fig.1(e) due to the resonance effect of the $h^{0}_{t}$ propagator. In this case, the effect of the decay width of $h^0_t$ should be considered. The possible decay modes of $h^{0}_{t}$ are $b\bar{b},t\bar{c},gg,W^+W^-,ZZ,$ $t\bar{t}$(if $M_{h^0_t}\geq 2m_{t}$). The results show that although the coupling of $\Pi_t^0b\bar{b}$ is much smaller than the coupling of $\Pi_t^0t\bar{t}$, the resonance effect can enhance the cross section to the level of a few fb. When $\sqrt{s}=1600$ GeV, the cross section will be much smaller than that for $\sqrt{s}=800$ GeV and it hardly varies with $M_{\Pi}$. So we only draw up the plots for the case of $\sqrt{s}=800$ GeV. The plots show that the cross section is not sensitive to $M_{\Pi}$. The neutral top-pion can be more easily detected via the process $e^{+}e^{-}\rightarrow b\bar{b}\Pi^{0}_{t}$ than via the process $e^{+}e^{-}\rightarrow t\bar{t}\Pi^{0}_{t}$ with b-tagging. Therefore, the process $e^{+}e^{-}\rightarrow
b\bar{b}\Pi^{0}_{t}$ provides us with another useful way to search for a neutral top-pion.
For the other processes $e^{+}e^{-}\rightarrow
f'\bar{f'}\Pi^{0}_{t}(f'=u,d,c,s,\mu,\tau)$, the largest cross section could only be up to 0.01 fb. Therefore, we could hardly detect the neutral top-pion through these processes. So we do not discuss these processes in detail.
In conclusion, we have studied some neutral top-pion production processes $e^{+}e^{-}\rightarrow f\bar{f}\Pi^{0}_{t}(f=u,d,c,s,t,b,e,\mu,\tau)$ at the future $e^{+}e^{-}$ colliders in the framework of the TC2 model. We find that there are the following features for these processes: (i)The cross sections of the processes $e^{+}e^{-}\rightarrow
f'\bar{f'}\Pi^{0}_{t}(f'=u,d,c,s,\mu,\tau)$ are too small to detect a neutral top-pion. (ii)Due to the effect of the top quark triangle loops, there exists a narrow peak in the cross section plots of the process $e^{+}e^{-}\rightarrow
e^{+}e^{-}\Pi^{0}_{t}$. Because of the resonance effect, the cross section of $e^{+}e^{-}\rightarrow e^{+}e^{-}\Pi^{0}_{t}$ could be up to a few fb and a few tens or even hundreds of $\Pi^{0}_{t}$ events can be produced, which causes the neutral top-pion to become experimentally detectable through the flavor-changing $t\bar{c}$ channel with the clean background. (iii)The cross sections of both $e^{+}e^{-}\rightarrow b\overline{b}\Pi^{0}_{t}$ and $e^{+}e^{-}\rightarrow t\overline{t}\Pi^{0}_{t}$ can reach the level of a few fb, the strong coupling $t\bar{t}\Pi_t^0$ in the process $e^{+}e^{-}\rightarrow t\overline{t}\Pi^{0}_{t}$ and the resonance effect of the $h^0_t$ propagator in the process $e^{+}e^{-}\rightarrow b\overline{b}\Pi^{0}_{t}$ could enhance the cross sections significantly. These two processes provide us with another way to search for a neutral top-pion. Therefore, our studies could provide a direct way to test the TC2 model by detecting top-pion signals.
[**Acknowledgments**]{}
This work is supported by the National Natural Science Foundation of China(10175017 and 10375017), the Excellent Youth Foundation of Henan Scientific Committee(02120000300), and the Henan Innovation Project for University Prominent Research Talents(2002KYCX009).
CDF Collaboration, F. Abe,[*et al*]{}.,[*Phys. Rev. Lett.*]{} [**74**]{}, 2626(1995);D0 collaboration, S. Abachi,[*et al*]{}.,ibid.[**74**]{}, 2632(1995). C. T. Hill, [*Phys. Lett.*]{} B[**345**]{}, 483(1995); K. Lane and E. Eichten, [*Phys. Lett.*]{} B[**352**]{}, 383(1995); K. Lane, [*Phys. Lett.*]{} B[**433**]{}, 96(1998); G. Cvetic, [*Rev. Mod. Phy.*]{}[**71**]{}, 513(1999). B. A. Dobrescu and C. T. Hill, [*Phys. Rev. Lett.*]{} B[**81**]{}, 2634(1998); R. S. Chivukula, B. A. Dobrescu and C. T. Hill, [*Phys. Rev.*]{} D[**59**]{}, 075003(1999). M. B. Popovic and E. H. Simmons,[*Phys.Rev.*]{} D[ **58**]{}, 095007(1998). G. Buchalla, G. Burdman, C. T. Hill, D. Kominis, [*Phys. Rev.*]{} D[**53**]{}, 5185(1996). C. H. Hill, E. H. Simmons, hep-ph/0203079 The NLC Collaboration, 2001 report on the Next linear collider: A report submitted to snowmass’01, SLAC-R-571. S. Iwata, The JLC project, Proceedings of the worldwide, Study on Physics and Experiments with Future Linear $e^+e^-$ Colliders, Sitges, Vol.2, 611. R. Brinkmann et.al., “TESLA Technical Design Report Part II: The accelerator.” DESY-01-011B. Xuelei Wang, Yueling Yang, Bingzhong Li, Chongxing Yue, Jinyu Zhang, [*Phys.Rev.*]{} D[**66**]{}, 075009(2002); Xuelei Wang, Yueling Yang, Bingzhong Li, Lingde Wan, [*Phys.Rev.*]{} D[**66**]{}, 075013(2002); Xuelei Wang, Bingzhong Li, Yueling Yang, [*Phys.Rev.*]{} D[**67**]{}, 035005(2003). Junjie Cao, Zhaohua Xiong, Jin Min Yang, [*Phys. Rev.*]{} D[**67**]{},071701(2003). A. Djouadi, J. Kalinowski, P. M. Zerwas, [*Particles and Fields*]{} [**54**]{}, 255(1992); S. Dittmaier, M. Kr$\ddot{a}$mer, Y. Liao, M. Spira, P. M. Zerwas, [*Phys. Lett.*]{} B[**441**]{}, 383(1998); [*Phys. Lett.*]{} B[**478**]{}, 247(2000); S. Dawson and L.Reina, [*Phys. Rev.*]{} D[**59**]{}, 054012(1999); [*Phys. Rev.*]{} D[**60**]{}, 015003(1999). J. Goldstein, et.al., [*Phys. Rev. Lett.*]{} [**86**]{}, 1694(2001); L. Reina, S. Dawson, [*Phys. Rev. Lett.*]{}[**87**]{}, 201804(2001); W. Beenakker, et al., [*Phys. Rev. Lett.*]{}[**87**]{}, 201805(2001). A. K. Leibovich, D. Rainwater,[*Phys. Rev.*]{} D[**65**]{}, 055012(2002). H.-J He and C.-P Yuan, [*Phys. Rev. Lett.*]{}[83]{}, 28(1999);G. Burdman,[*Phys. Rev. Lett.*]{} [**83**]{}, 2888(1999). Gongru Lu, Furong Yin, Xuelei Wang, Lingde Wan, [*Phys. Rev.*]{} D[**68**]{}, 015002(2003). Chonbgxing Yue, et al.,[*Phys. Rev.*]{} D[**63**]{}, 115002(2001).
|
---
abstract: 'Thanks to improvements in machine learning techniques, including deep learning, speech synthesis is becoming a machine learning task. To accelerate speech synthesis research, we are developing Japanese voice corpora reasonably accessible from not only academic institutions but also commercial companies. In 2017, we released the JSUT corpus, which contains 10 hours of reading-style speech uttered by a single speaker, for end-to-end text-to-speech synthesis. For more general use in speech synthesis research, e.g., voice conversion and multi-speaker modeling, in this paper, we construct the JVS corpus, which contains voice data of $100$ speakers in three styles (normal, whisper, and falsetto). The corpus contains 30 hours of voice data including 22 hours of parallel normal voices. This paper describes how we designed the corpus and summarizes the specifications. The corpus is available at our project page[^1].'
address: |
Graduate School of Information Science and Technology, The University of Tokyo,\
7-3-1 Hongo Bunkyo-ku, Tokyo 133–8656, Japan.
bibliography:
- 'tts.bib'
title: 'JVS corpus: free Japanese multi-speaker voice corpus'
---
**Index Terms**: voice corpus, Japanese, speech synthesis
Introduction
============
Thanks to developments in deep learning techniques, studies on speech have been targeted actively [@hinton12dnnasr; @oord16wavenet; @takamichi17moment; @saito18advss]. Nowadays, speech synthesis, e.g., text-to-speech, singing voice synthesis, voice conversion, and speech coding, is becoming a machine learning task. Easily accessible voice corpora help to not only accelerate speech-related research but also improve the reproductivity of a study. In 2017, we released a large-scaled Japanese speech corpus, named the JSUT corpus [@sonobe17jsut], for end-to-end text-to-speech synthesis. The corpus included 10 hours of reading-style speech data uttered by a single native Japanese speaker and all pronunciations of daily-use characters and individual readings in Japanese [@joyokanji]. Since Oct. 2017, the project page [@jsut_corpus] was accessed more than 6,000 times (75% from Japan and 25% from foreign countries) from more than 60 countries. We believe that the JSUT corpus has become one of the most used Japanese corpora for modern speech synthesis research [@ueno19multispeakerend2endtts; @luo19waveletf0feature].
Towards more general purposes of speech-related research, this paper introduces a new Japanese voice corpus, named the *JVS* (Japanese versatile speech) corpus. The corpus is designed to have many benefits for many types of users as follows.
- **High-quality format**: The audio files are sampled at 24 kHz, encoded at 16 bit, and formatted in RIFF WAV.
- **High-quality recording**: The recordings were controlled by a professional sound director and done in a recording studio.
- **Many speakers**: The corpus includes 100 native Japanese speakers, and all of the speakers are professional, e.g., voice actor/actress.
- **Many styles**: Each speaker utters not only normal speech but also whisper and falsetto voices.
- **Large in scale**: In total, the corpus contains 30 hours of voice data.
- **Parallel/non-parallel utterances**: Each speaker utters parallel, i.e., common among speakers, and non-parallel, i.e., completely different among speakers, utterances.
- **Many tags**: The corpus includes not only voice data but also transcriptions, gender information, $F_0$ ranges, speaker similarity, and phoneme alignments.
- **Free for research**: The corpus is free to use for research in academic institutions and commercial companies.
- **Easily accessible**: The corpus is freely downloadable online.
The next section describes how we designed the corpus.
Corpus design
=============
The corpus consists of the following four sub-corpora. Their names are formatted as *\[NAME\]\[NUM\_UTT\]*. *\[NUM\_UTT\]* indicates the number of utterances per speaker.
- **parallel100**: 100 parallel normal (reading-style) utterances
- **nonpara30**: 30 non-parallel normal utterances
- **whisper10**: 10 whisper utterances
- **Falsetto10**: 10 falsetto utterances
The directory structures of the corpus are listed below. The speaker name is formatted as *jvs\[SPKR\_ID\]*. *\[SPKR\_ID\]* indicates the speaker ID with the range of 1 through 100.
Sub-corpora
-----------
This section describes how we designed the four sub-corpora.
### parallel100
Parallel voices, i.e., utterances that are common among speakers, are used for voice conversion [@toda07_MLVC; @stylianou88], speaker factorization [@lu13factor], multi-speaker modeling [@ueno19multispeakerend2endtts], and so on. We used 100 phonetically-balanced sentences of the sub-corpus “voiceactress100”[^2] of the JSUT corpus [@sonobe17jsut], and we let speakers utter the sentences. This corpus contains not only the audio files but also the transcriptions (stored in “parallel100/transcript\_utf8.txt”) and phoneme alignment (stored in “parallel100/lab”).
### nonpara30
The use of non-parallel voices, i.e., utterances that are completely different among speakers, is a challenging but more realistic situation than that of parallel voices. Sentences to be uttered are randomly selected from the JSUT corpus excluding its sub-corpus “voiceactress100.” Each speaker uttered 30 utterances that are different among speakers. This sub-corpus also includes transcriptions and phoneme alignments. Note that, the sentences are not phonetically balanced unlike the sub-corpora “parallel100.”
### whisper10
Whispering is used to quietly communicate, i.e., convey secret information without being overheard. Analysis [@ito05whisperanalysis], synthesis [@petrushin10whispertts], recognition [@jou05whisperrecognition] and conversion [@toda12bodyconductedvc] of whispered voices have the potential to augment our silent-speech communication. The first five sentences of this sub-corpus are the same as those of the sub-corpus “parallel100,” and they are parallel among speakers. The remaining five sentences are the same to those of the sub-corpus “nonpara30,” and they are non-parallel among speakers. Namely, ten utterances per speaker are parallel between whispered voices and normal voices.
### Falsetto10
Falsetto is a vocal register occupying the $F_0$ range that is higher than normal voices. The physiology of falsetto is different from that of normal voices [@childers91vocalqualityfactor], and the analysis and synthesis of falsetto are remaining tasks for signal processing-based vocoders. The first five sentences of this sub-corpus are the same as those of the sub-corpus “parallel100.” The remaining five sentences are the same as those of the sub-corpus “nonpara30” but different to those of the sub-corpus “whisper10.” Namely, five utterances are parallel among speakers, ten are parallel between normal voice and falsetto, and five are parallel between whisper and falsetto.
Tags
----
This section describes some of the annotation results.
- **$F_0$ range (gender\_f0range.txt)**: Typical pitch extractors, e.g., [@kawahara99; @morise16world; @reaper], have a range for $F_0$ search, and the setting is critical for the results ultimately obtained for the voices. This corpus contains manually annotated $F_0$ ranges per speaker for his/her normal voices.
- **Speaker similarity (speaker\_similarity\_.csv)**: Perceptual similarity between speakers is useful for selecting speakers (or models) [@lanchantin14mavm] and modeling speaker space [@saito19perceptual]. This corpus contains perceptual similarity scores between all pairs of speakers of each gender.
- **Duration (duration.txt)**: Duration, i.e., data size, and speech rate are also included. Phoneme-level duration is calculated from the results of phoneme alignments.
Results of data collection
==========================
Mininum \[min.\] Average \[min.\] Maximum \[min.\] Total (100 speaker) \[hour\]
------------------------------ ------------------ ------------------ ------------------ ------------------------------ --
parallel100 (100 utterances) 10.11 (jvs020) 13.11 18.24 (jvs084) 22
nonpara30 (30 utterances) 2.12 (jvs099) 2.62 3.86 (jvs036) 4.4
whisper10 (10 utterances) 0.95 (jvs045) 1.24 1.69 (jvs018) 2.0
falsetto10 (10 utterances) 0.90 (jvs045) 1.18 1.61 (jvs035) 2.0
30.4
Corpus specs
------------
We hired 100 native Japanese professional speakers, which included 49 male and 51 female speakers. Their voices were recorded in a recording studio. Recording for each speaker was done within one day. The recordings were controlled by a professional sound director. The voices were originally sampled at 48 kHz and downsampled to 24 kHz by SPTK [@sptk]. The 16-bit/sample RIFF WAV format was used. Sentences (transcriptions) were encoded in UTF-8. The full context and monophone labels were automatically generated by Open JTalk [@ojtalk]. The phoneme alignments were automatically generated by Julius [@lee01julius]. $F_0$ ranges were manually annotated in accordance with hands-on voice conversion [@toda19vchandson]. The WORLD vocoder [@morise16world; @morise16d4c] extracted $F_0$. Commas were added between breath groups. For annotating perceptual similarity scores, we followed Saito et al.’s study [@saito19perceptual] and used a crowdsourcing service, Lancers [@lancers], which is a famous crowdsourcing service in Japan. Each listener scored the perceptual similarity for each pair of speakers from $-3$ (completely different) and $+3$ (very similar). A final score for each speaker pair was obtained by averaging listeners’ scores. Ten different listeners scored each speaker pair, and 1,000 listeners participated in total.
Analysis
--------
### Duration
[Table \[tb:duration\_stats\]]{} lists the statistics for speaker-wise duration. This corpus contains 26 hours of normal voices and 4 hours of other-style voices. Each speaker uttered approximately 15.7 minutes of normal voices, 1.24 minutes of whispered voices, and 1.18 minutes of falsetto. In the sub-corpus “parallel100,” the transcription was common among speakers, but the duration was very different; speaker “jvs084” uttered 1.8 times slower than speaker “jvs020.”
### Perceptual speaker similarity
[Fig. \[fig:eps/simmat\_jvs\_females.pdf\]]{} shows matrices of perceptual similarity scores. For example, the most similar pair was “jvs019” and “jvs096.” Also, a speaker that was most dissimilar from the other speakers was “jvs010.”
Conclusion
==========
In this paper, we constructed a corpus named the JVS corpus. The corpus was designed for speech-related research using multi-speaker and multi-style voices. Text data of the corpus is licensed as shown in the LICENCE file in the JSUT corpus [@jsut_corpus]. The tags are licensed with CC BY-SA 4.0. The audio data may be used for
- Research by academic institutions
- Non-commercial research, including research conducted within commercial organizations
- Personal use, including blog posts.
Our project page at <https://sites.google.com/site/shinnosuketakamichi/research-topics/jvs_corpus> describes the terms for commercial use.
Acknowledgements
================
Part of this work was supported by the GAP foundation program of the University of Tokyo and the MIC/SCOPE \#182103104.
{width="1.0\linewidth"}
\[fig:eps/simmat\_jvs\_females.pdf\]
[^1]: <https://sites.google.com/site/shinnosuketakamichi/research-topics/jvs_corpus>
[^2]: The original sentences are included in the Voice Actress Corpus [@voiceactresscorpus], and the one included in the JSUT corpus had commas added at the phrase break positions.
|
---
abstract: 'The distillability of bipartite entangled state as seen by moving observers has been investigated. It is found that the same initial entanglement for a state parameter $\alpha$ and its “normalized partner” $\sqrt{1-\alpha^2}$ will be degraded as seen by moving observer. It is shown that in the ultra relativistic limit, the state does not have distillable entanglement for any $\alpha$.'
author:
- 'Shahpoor Moradi$^1$, [^1]'
title: Degradation of entanglement in moving frames
---
Relationship between special relativity and quantum information theory is discussed by many authors [@p1]. Peres *et al* [@p2] investigated the relativistic properties of spin entropy for a single, free particle of spin$-1/2$. They show that the usual definition of quantum entropy has no invariant meaning in special relativity. The reason is that, under a Lorentz boost, the spin undergoes a Wigner rotation whose direction and magnitude depend on the momentum of the particle. Even if the initial state is a direct product of a function of momentum and a function of spin, the transformed state is not a direct product. Lamata *et al*[@pa] define weak and strong criteria for relativistic isoentangled and isodistillable states to characterize relative and invariant behavior of entanglement and distillability. In this letter, we choose a generic state as the initial entangled state and we will try to show that the entanglement is degraded as seen by the relativistically observer. This help us to understand the relationship between special relativity and quantum information theory. The initial nonmaximal entangled state is |=\^[(a)]{}\_1([\_a]{})\^[(b)]{}\_1([\_b]{}) +\^[(a)]{}\_2([\_a]{})\^[(b)]{}\_2([\_b]{}),where $\alpha$ is some number that satisfies $|\alpha|\in(0,1)$. Here ${\mathbf{p}_a}$ and ${\mathbf{p}_b}$ are the corresponding momentums vectors of particles $A$ and $B$ and \^[(a)]{}\_1([\_a]{})=g([\_a]{})|0=|0,[\_a]{}=(
[c]{} g([\_a]{})\
0\
), \^[(a)]{}\_2([\_a]{})=g([\_a]{})|1=|0,[\_b]{}=(
[c]{} 0\
g([\_a]{})\
). For simplicity assume that they are sufficiently well localized around momenta ${\mathbf{p}}$, Under the Lorentz transformation the states (2) and (3) transformed as [@pa] =(
[c]{} b\_1()\
b\_2()\
)=(
[c]{} (\_/2)\
(\_/2)\
)g()=\_(\_/2)|0,+ (\_/2)|1,, =(
[c]{} -b\_2()\
b\_1()\
)=(
[c]{} - (\_/2)\
(\_/2)\
)g()=-(\_/2)|0,+ (\_/2)|1,, where $\theta_{{\mathbf{p}}}$ is Wigner angle satisfies the relation \_= ,here $\cosh\xi=(1-\beta^2)^{-1/2}$ where $\beta$ is boost speed and $\cosh\delta=p_0/m$. Now under Lorentz transformation the state transformed to (after tracing over momentum eigenket $|{\mathbf{p}}\re$ ), $$|\Phi\re_{\Lambda}=\left(\alpha
\cos^2(\theta_{{\mathbf{p}}}/2)+\sqrt{1-\alpha^2}\sin^2(\theta_{{\mathbf{p}}}/2)\right)|00\re+$$$$\sin(\theta_{{\mathbf{p}}}/2)\cos(\theta_{{\mathbf{p}}}/2)(\alpha-\sqrt{1-\alpha^2})(|01\re+|10\re)+$$(\^2(\_/2)+\^2(\_/2))|11. A very popular measure for the quantification of bipartite quantum correlations is the concurrence [@H]. This quantity can be defined C()={0,\_1-\_2-\_3-\_4}. with $\lambda_i$ being the square roots of the eigenvalues of $\rho_{AB}(\sigma_y\otimes\sigma_y)\rho^*_{AB}(\sigma_y\otimes\sigma_y)$ where the asterisk denotes complex conjugation and and $\sigma_y=\left(%
\begin{array}{cc}
0 & -i \\
i & 0 \\
\end{array}%
\right).$ Now the concurrence for this state is C(|\_)=2 .Which is the Lorentz invariant concurrence. To be more precise one should take wave packets in momentum space, with Gaussian momentum distributions $g({\mathbf{p}})=\pi^{-3/4}w^{-3/2}\exp\left(-|{\mathbf{p}}|^2/2w^2\right)$. If we trace the momentum degrees of freedom we obtain the usual entangled state $ |\phi\re=\alpha|00\re +\sqrt{1-\alpha^2}|11\re$. The general density matrix for two particle systems with momentums ${\mathbf{p}}_a$ and ${\mathbf{p}}_b$ is given by \_=\_[ijkl=1,2]{}C\_[ijkl]{}\_i(\_a)\_j(\_b) \[\_l(’\_a)\_m(’\_b)\]\^. For state $(1)$ the coefficients $C_{ijkl}$ are $$C_{1111}=\alpha^2,\;\;\;\;C_{2222}=1-\alpha^2,$$C\_[1122]{}=C\_[2211]{}= For obtaining the Lorentz transformation of (10), we need the relativistic properties of spin entropy for a single, free particle of spin$-1/2$. The quantum state of a spin-$\frac{1}{2}$ particle can be written in the momentum representation as follows ()=(
[c]{} a\_1()\
a\_2()\
), where (|a\_1()|\^2+|a\_2()|\^2)d=1.The density matrix corresponding to state $(6)$ is (’,”)=(
[cc]{} a\_1(’)a\_1(”)\^\* & a\_1(’)a\_2(”)\^\*\
a\_1(’)a\_2(”)\^\* & a\_2(’)a\_2(”)\^\*\
).By setting ${\mathbf{p}}'={\mathbf{p}}''={\mathbf{p}}$ and integrating over ${\mathbf{p}}$ we obtain the reduced density matrix for spin =(
[cc]{} 1+n\_z & n\_x-in\_y\
n\_x+in\_y & 1-n\_z\
), where the Bloch vector $\mathbf{n}$ is given by n\_z=(|a\_1()|\^2-|a\_2()|\^2)d=1, n\_x-in\_y=a\_1() a\_2()\^\*d.Now under Lorentz boost density matrix (10) transformed into $$\Lambda \rho_{\Phi} \Lambda
^{\dag}=\sum_{ijkl=1,2}C_{ijklmn}\Lambda
(p_a)\Psi_i({\mathbf{p}}_a)\otimes \Lambda
(p_b)\Psi_j({\mathbf{p}}_b)$$\^.
The reduced density matrix for spin is obtained by setting ${\mathbf{p}}_a={\mathbf{p}}'_a$, ${\mathbf{p}}_b={\mathbf{p}}'_b$ and tracing over momentum $$\tau=Tr_{{\mathbf{p}}_a{\mathbf{p}}_b}[\Lambda \rho_{\Phi}
\Lambda ^{\dag}]=\sum_{ijkl=1,2}C_{ijkl}Tr_{{\mathbf{p}}_a}
\left\{\Lambda ({{p}}_a)\Psi_i({\mathbf{p}}_a)[\Lambda
({{p}}_a)\Psi_k({\mathbf{p}}_a)]^{\dag}\right\}$$Tr\_[\_b]{} {([[p]{}]{}\_b)\_j(\_b)\[([[p]{}]{}\_b)\_l(\_b)\]\^}. To leading order $w/m\ll 1$ we have n\_z=n1-()\^2,n\_x=n\_y0,It can be appreciated in Eq. $(19)$ that the expression is decomposable in the sum of the tensor products of $2\times 2$ spin blocks, each corresponding to each particle. We compute now the different blocks, corresponding to the four possible tensor products of the states $(3)$ and $(4)$: Tr\_ {(p)\_1()\[(p)\_l()\]\^}=(
[cc]{} 1+n & 0\
0 & 1-n\
), Tr\_ {(p)\_2()\[(p)\_2()\]\^}=(
[cc]{} 1-n & 0\
0 & 1+n\
) ,Tr\_ {(p)\_1()\[(p)\_2()\]\^}=(
[cc]{} 0 & 1+n\
-(1-n) & 0\
), Tr\_ {(p)\_2()\[(p)\_1()\]\^}=(
[cc]{} 0 & -(1-n)\
1+n& 0\
). With the help of Eqs . $(21)$-$(24)$, it is possible to compute the effects of the Lorentz transformation, associated with a boost in the $x$ direction, on any density matrix of two spin-$1/2$ particles with factorized Gaussian momentum distributions. In particular density matrix (19) is reduced to =(
[cccc]{} 4\^2n+(1-n)\^2 & 0 & 0 & 2(1+n\^2)\
0 & 1-n\^2 & -2(1-n\^2) & 0\
0 & -2(1-n\^2) & 1-n\^2 & 0\
2(1+n\^2) & 0 & 0 & -4\^2n+(1+n)\^2\
).We can apply now the positive partial transpose criterion [@p3] to know whether this state is entangled and distillable. The partial transpose criterion provides a sufficient condition for the existence of entanglement in this case: if at least one eigenvalue of the partial transpose is negative, the density matrix is entangled; but a state with positive partial transpose can still be entangled. It is the well-known bound or nondistillable entanglement [@wid]. Partial transpose of density matrix (25) yields $$\tau^{T}=\frac{1}{4}\left(%
\begin{array}{cccc}
4\alpha^2n+(1-n)^2 & 0 & 0 &-2\alpha\sqrt{1-\alpha^2}(1-n^2) \\
0 & 1-n^2 & 2\alpha\sqrt{1-\alpha^2}(1+n^2) & 0 \\
0 & 2\alpha\sqrt{1-\alpha^2}(1+n^2) & 1-n^2 & 0 \\
-2\alpha\sqrt{1-\alpha^2}(1-n^2) & 0 & 0 & -4\alpha^2n+(1+n)^2\\
\end{array}%
\right).$$ It is possible diagonalize $\tau^{T}$ and get it’s eigenvalues \_1 =(1-n\^2)+(1+n\^2),\_2 =(1-n\^2)-(1+n\^2),\_3 =(1+n\^2)+,\_4 =(1+n\^2)-.For $0<n,\alpha<1$ eigenvalues $\lambda_1$, $\lambda_3$ and $\lambda_4$ are always positive. The eigenvalue $\lambda_2$ is negative for $\alpha\sqrt{1-\alpha^2}>R$ where R=.In this range the logarithmic negativity takes the form N=\_2{ (1+n\^2)(1+2)}. In ultra relativistic limit $n\rightarrow 0$: $
N\rightarrow\log_2\left\{
\frac{1}{2}+\alpha\sqrt{1-\alpha^2}\right\}$, then the state does not have distillable entanglement for any $\alpha$. For the rest frame $n=1$: $N=\log_2\left\{ 1+2\alpha\sqrt{1-\alpha^2}\right\}$. In the range $0<\alpha<1/\sqrt{2}$ the larger $\alpha$, the stronger the initial entanglement; but in the range $1/\sqrt{2}<\alpha<1$ the larger $\alpha$, the weaker the initial entanglement. For finite velocity, the monotonic decrease of $N$ with increasing boost speed for different $\alpha$ means that the entanglement of the initial state is lost due to Wigner rotation. From Fig.1 it is found that the entanglement in moving frame, for $\alpha$ and it’s normalized partner $\sqrt{1-\alpha^2}$, will be degraded as boost speed increases. Here we calculate the concurrence which is defined as follows C=2=2 .For density matrix (25) we have \_A=\_B=(
[cc]{} \^2n+(1-n)/2 & 0\
0 & -\^2n+(1+n)/2\
) ,then C= .In non relativistic limit $(n=1)$ we have: $C=2\alpha\sqrt{1-\alpha^2}$ and in ultrarelativistic limit $(n=0)$: $C=1/2$. It is interesting that for maximally entangled state as $\alpha=1/\sqrt{2}$ for all values of $n$ concurrence is $1$.
[a]{}
A. Peres and D. R. Terno, Rev. Mod. Phys. [**76**]{}, 93 (2004)(References therein) A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. [**88**]{}, 230402 (2002). L. Lamata, M. A. Martin-Delgado, and E. Solano, Phys. Rev.Lett. [**97**]{}, 250502 (2006). . Hill, W. K. Wootters, Phys. Rev. Lett. [**78**]{}, 5022 (1997). A. Peres, Phys. Rev. Lett. [ **77**]{}, 1413 (1996). G. Vidal and R. F. Werner, Phys. Rev. A [**65**]{}, 032314 (2002)
![Plot of negativity versus $n$ and $\alpha$[]{data-label=""}](fig){width="4in"}
[^1]: e-mail: shahpoor.moradi@gmail.com
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.